The objective of the current research is the investigation into possible non-linear influence of hydrostatic pressure on yielding of asymmetric (exhibiting the so-called "strength-differential effect") anisotropic sheet metals. To reach this aim, two yield functions are developed, called here "non-linear pressure sensitive criteria I and II," (NPC-1 and NPC-2). In addition, the non-associated flow rules are employed for these new criteria. The yield functions are defined as non-linearly dependent on hydrostatic pressure, while the plastic potential functions are introduced to be pressure insensitive. To calibrate these criteria, the yield functions need 10 directional experimental yield stresses and the plastic potential functions need eight Lankford coefficients data points. Four well-known anisotropic sheet metals with different structures, namely AA 2008-T4, a Face Centered Cubic material (FCC), AA 2090-T3, a Face Centered Cubic material (FCC), AZ31, a hexagonal closed packed material (HCP) and high-purity
The elastic properties of a spherical heterogeneous structure with isotropic periodic components is analyzed and a methodology is developed using the two-scale asymptotic homogenization method (AHM) and spherical assemblage model (SAM). The effective coefficients are obtained via AHM for two different composites: (a) composite with perfect contact between two layers distributed periodically along the radial axis; and (b) considering a thin elastic interphase between the layers (intermediate layer) distributed periodically along the radial axis under perfect contact. As a result, the derived overall properties via AHM for homogeneous spherical structure have transversely isotropic behavior. Consequently, the homogenized problem is solved. Using SAM, the analytical exact solutions for appropriate boundary value problems are provided for different number of layers for the cases (a) and (b) in the spherical composite. The numerical results for the displacements, radial and circumferential stresses for both methods are compared considering a spherical composite material loaded by an inside pressure with the two cases of contact conditions between the layers (a) and (b).
A class of constitutive relations for elastic bodies has been proposed recently, where the linearized strain tensor is expressed as a nonlinear function of the stress tensor. Considering this new type of constitutive equation, the initial boundary value problem for such elastic bodies has been expressed only in terms of the stress tensor. In this communication, this new type of nonlinear wave equation is studied for the case of a one-dimensional straight bar. Conditions for the existence of the travelling wave solutions are given and some self-similar solutions are obtained.
The presence of an interface can influence the thermo-mechanical response of a body. This influence is especially pronounced at small scales where the interface area to bulk volume ratio significantly increases. Since the thermo-mechanical properties of an interface can differ from those of the bulk, within interface continuum theory an interface is endowed with its own thermo-mechanical energetic structure. To date, the effects of interface in-plane damage on the thermo-mechanical response of a highly conductive interface have not been accounted for. Therefore in this contribution the computational aspects of thermo-mechanical solids containing highly conductive interfaces subject to in-plane degradation are studied.
To this end, the equations governing a fully non-linear transient problem are given. They are solved using the finite element method. The results are illustrated through a series of three-dimensional numerical examples for various interfacial parameters. In particular a comparison is made between the results of the intact and the degraded highly conductive interface formulation.
A variational model is used to study the behavior of a flexible but inextensible loop spanned by a liquid film, with the objective of explaining the stability and buckling of flat circular configurations. Loops made from filaments with intrinsic curvature and/or intrinsic twist density are considered, but attention is restricted to filaments with circular cross sections and uniform mechanical properties. Loops made with intrinsic curvature but no intrinsic twist density exhibit in-plane and out-of-plane buckling modes corresponding to stable solution branches that bifurcate from the branch of flat circular solutions and out-of-plane buckling occurs at a lower value of the dimensionless surface tension of the liquid film than does in-plane buckling. Additionally, however, the destabilizing influence of the intrinsic curvature can be countered by increasing the torsional rigidity relative to the flexural rigidity. For a loop with both intrinsic curvature and intrinsic twist density, only one branch of stable solutions bifurcates from the flat circular solution branch, the in-plane and out-of-plane buckling modes are intertwined, and bifurcation occurs at a value of the dimensionless surface tension less than that governing the behavior of loops made from filaments that are intrinsically rectilinear. Moreover, increasing the torsional rigidity relative to the flexural rigidity has no or little stabilizing effect if the loop is either too short or too long and, in contrast to what occurs for loops with only intrinsic curvature, if the intrinsic twist density is sufficiently large then the destabilizing influence of the intrinsic curvature cannot be countered by increasing the torsional rigidity relative to the flexural rigidity, regardless of the length of the loop.
For homogeneous higher-gradient elasticity models we discuss frame-indifference and isotropy requirements. To this end, we introduce the notions of local versus global SO(3)-invariance and identify frame-indifference (traditionally) with global left SO(3)-invariance and isotropy with global right SO(3)-invariance. For specific restricted representations, the energy may also be local left SO(3)-invariant as well as local right SO(3)-invariant. Then we turn to linear models and consider a consequence of frame-indifference together with isotropy in nonlinear elasticity and apply this joint invariance condition to some specific linear models. The interesting point is the appearance of finite rotations in transformations of a geometrically linear model. It is shown that when starting with a linear model defined already in the infinitesimal symmetric strain
Fibre-reinforced plates and shells are finding an increasing interest in engineering applications; in most cases dynamic phenomena need to be taken into account. Consequently, effective and robust computational tools are sought in order to provide reliable results for the analysis of such structural models. In this paper the mixed assumed-strain laminated plate element, previously used for static analyses, has been extended to the dynamic realm. This model is derived within the framework of the so-called First-order Shear Deformation Theory (FSDT). What is peculiar in this assumed-strain finite element is that in-plane strain components are modeled directly; the corresponding stress components are deduced via constitutive law. By enforcing the equilibrium equations for each lamina, and taking continuity requirements into account, the out-of-plane shear stresses are computed and, finally, constitutive law provides the corresponding strains. The resulting global strain field depends only on a fixed number of parameters, regardless of the total number of layers. Since the proposed element is not locking-prone, even in the thin plate limit, and provides an accurate description of inter-laminar stresses, an extension to the dynamic range seems to be particularly attractive. The same kinematic assumptions will lead to the formulation of a consistent mass matrix. The element, developed in this way, has been extensively tested for several symmetric lamination sequences; comparison with available analytical solutions and with numerical results obtained by refined 3-D models are also presented.
We propose deducing from three-dimensional elasticity a one dimensional model of a beam when the lateral boundary is not free of traction. Thus the simplification induced by the order of magnitude of transverse shearing and transverse normal stress must be removed. For the sake of simplicity we consider a beam with rectangular cross section. The displacement field in rods can be approximated by using a Taylor–Young expansion in transverse dimension of the rod and we truncate the potential energy at the fourth order. By considering exact equilibrium equations, the highest-order displacement field can be removed and the Euler–Lagrange equations are simplified.
We study the distension-induced gradient capillarity in membrane bud formation. The budding process is assumed to be primarily driven by diffusion of transmembrane proteins and acting line tensions on the protein-concentrated interface. The proposed model, based on the Helfrich-type potential, is designed to accommodate inhomogeneous elastic responses of the membrane, non-uniform protein distributions over the membrane surface and, more importantly, the thickness distensions induced by bud formations in the membrane. The latter are employed via the augmented energy potential of bulk incompressibility in a weakened manner. By computing the variations of the proposed membrane energy potential, we obtained the corresponding equilibrium equation (membrane shape equation) describing the morphological transitions of the lipid membrane undergoing bud formation and the associated thickness distensions. The effects of lipid distension on the shape equation and the necessary adjustments to the accompanying boundary conditions are also derived in detail. The resulting shape equation is solved numerically for the parametric representation of the surface which has one-to-one-correspondence with the membrane surface under consideration. The proposed model successfully predicts the bud formation phenomenon on a flat lipid membrane and the associated thickness distensions of the membrane and demonstrates a smooth transition from one phase to the other (including necking domains). It is also found that the final deformed configuration is energetically favorable and therefore is stable. Finally, we show that the inhomogeneous thickness deformation on the membrane in response to transmembrane protein diffusion makes a significant contribution to the budding and necking processes of the membrane.
A continuum model, based on a theory of electromagnetic media with microstructure, is exploited to deal with rigid conductors endowed with polarization and magnetization. Charge carriers are considered as a continuum superimposed to the microstructured conductor where the density of bound charges depends on the internal degrees of freedom of the continuum particle. The non-linear dynamical model is formulated, deriving the mechanical balance laws that are coupled with the electromagnetic field equations. A reduction to the micropolar linear case is performed in order to analyze admissible solutions in the form of one-dimensional plane waves. Dispersion equations are derived for modes pertaining to longitudinal and transverse fields and the effects of conductivity and polarization are evidentiated. Polariton modes, arising from the dynamics of microdeformation, are also discussed.
The solution to the elasticity problem in three-dimensional polyhedral domains in the vicinity of an edge around which the material properties depend on the angular angle is addressed. This asymptotic solution involves a family of eigenpairs and their shadows which are being computed by means of p-finite element methods. In particular the examples we give explicitly provide the asymptotic solution for cracks and V-notch edges and explore the eigenvalues as a function of the change in material properties in the angular direction. We demonstrate that the singular exponents may change considerably by changing the material properties variation in the angular direction. These eigenpairs are necessary to allow the extraction of the edge stress intensity functions.
A report on the workshop Computational mechanics of generalized continua and applications to materials with microstructure (Catania 29–31 October 2015) is provided. The constructive atmosphere that was present at the workshop in 2012 and the Euromech Colloquium in 2014, both in Cisterna di Latina, was repeated for this workshop in Catania. The objective of this meeting was to bring together experts within the CNRS International Associate Laboratory (LIA) Francois Cosserat–Tullio Levi Civita Coss & Vita in order to discuss topics of common interest. Particularly, the workshop was dedicated to the following projects of LIA: (i) computational mechanics of generalized continua; and (ii) nonlinearity and stability in continuous media. Also subjects related to the application of generalized continua to multiscale and smart materials were discussed. Approximately 25 Coss & Vita LIA members and other experts (mostly from France and Italy) gathered together, including PhD students and those students from the Scuola Superiore di Catania.
An analytical solution method is presented for the transient axial response of one-dimensional structures subjected to impact loading. The transient structural response is expressed as a series of the impact loading function and its progressive shifting values depending on both material position and time scale. The governing differential equation on the impact force is derived considering the general solution and constitutive equation of the contact interface. The analytical solution of differential equation yields a recursive function describing the impact force and displacement function of the total geometry. Depending on the contact surface roughness and the material properties, an impact function is introduced as a base function for impact response. The procedure is implemented to determine the shock waves generated at the collision of the elastic rod on the rigid surface and two elastic rods. The analytical solution also derives the steady-state response of the structure after the impact loading.
This paper is devoted to a rigorous analysis of an equilibrium problem for a two-dimensional homogeneous anisotropic elastic body containing a thin isotropic elastic inclusion. The thin inclusion is modelled within the framework of Euler–Bernoulli beam theory. Partial delamination of the inclusion from the elastic body results in the appearance of an interfacial crack. We deal with nonlinear conditions that do not allow the opposing crack faces to penetrate each other. We derive a formula for the first derivative of the energy functional with respect to the regular crack perturbation along the interface, which is related to energy release rates. It is proved that the energy release rates associated with crack translation and self-similar expansion are represented as path-independent integrals along smooth curves surrounding one or both crack tips. The path-independent integrals consist of regular and singular terms and are analogues of the well-known Eshelby–Cherepanov–Rice J-integral and Knowles–Sternberg M-integral.
Experimental testing on dry woven fabrics exhibits a complex set of evidences that are difficult to completely describe using classical continuum models. The aim of this paper is to show how the introduction of energy terms related to the micro-deformation mechanisms of the fabric, in particular to the bending stiffness of the yarns, helps in the modelling of the mechanical behaviour of this kind of materials. To this aim, a second gradient, hyperelastic, initially orthotropic continuum theory is proposed to model fibrous composite interlocks at finite strains. In particular, the present work explores the relationship between the onset of wrinkling during the simulation of the deep drawing of a woven fabric and the use of a second gradient model. It is shown that the introduction of second gradient terms accounting for the description of in-plane and out-of-plane bending rigidities decreases the onset of wrinkles during the simulation of deep drawing.
In this work, a quadratic energy, roughly proportional to the square of the curvature of the fibres, is presented and implemented in the simulations. This simple constitutive assumption allows the effects of the second gradient energy on both the wrinkling description and the numerical stability of the model to be clearly shown. The results obtained in second gradient simulations are descriptive of the experimental evidence of deep drawing whose description is targeted in this work. The present paper provides additional evidence of the fact that first gradient continuum theories alone cannot be considered fully descriptive of the behaviour of dry woven composite reinforcements. On the other hand, the proposed second gradient model for fibrous composite reinforcements opens the way both to the more accurate simulation of complex forming processes and to the possibility of controlling the onset of wrinkles.
The behavior of a bistable strut for variable geometry structures was investigated in this paper. A three-hinged arch subjected to a central concentrated load was used to study the effect of symmetric imperfections on the behavior of the bistable strut. Based on a nonlinear strain–displacement relationship, the virtual work principle was adopted to establish both the pre-buckling and buckling nonlinear equilibrium equations for the symmetric snap-through buckling mode. Then the critical load for symmetric snap-through buckling was obtained. The results show that the axial force is in compression before the arch is buckled, but it becomes in tension after buckling. Thus, the previous formulas cannot be used for the analysis of post-buckling behavior of three-hinged shallow arches. Then, the principle of virtual work was also used to establish the post-buckling equilibrium equations of the arch in the horizontal and vertical directions as well as the static boundary conditions, which are very important for bistable struts.
We examine the contribution of crack bridging and surface elasticity to the elastic interaction between a mode III finite crack and a screw dislocation. The surface effect on the crack faces is incorporated by using the continuum-based surface/interface model of Gurtin and Murdoch. The crack faces are subjected to a bridging force which is assumed to be proportional to the crack opening displacement, whereas the bridging stiffness is allowed to vary arbitrarily along the crack. By considering a continuous distribution of both screw dislocations and line forces on the crack, the boundary value problem is reduced to two decoupled first-order Cauchy singular integro-differential equations. After the expansion of the unknown line dislocation and line force densities and the known variable bridging stiffness into Chebyshev polynomials, these singular integro-differential equations are solved numerically using the collocation method. Owing to the incorporation of surface elasticity, the stresses at the crack tips only exhibit the weak logarithmic singularity when the dislocation is located on the real axis where the crack is located, whereas in the case when the dislocation is not on the real axis, the stresses at the crack tips exhibit both the weak logarithmic and the strong square-root singularities. The two densities, the crack opening displacement across the crack faces and the image force acting on the screw dislocation are specifically calculated. We note that crack bridging only exerts an effect on the line dislocation density but has no influence on the line force density. In addition, we demonstrate that both surface elasticity and crack bridging can reduce the strengths of the logarithmic stress singularity at the crack tips and the magnitude of the crack opening displacement across the crack faces. Our results also clearly show that both crack bridging and surface elasticity exert a significant influence on the magnitude and direction of the image force acting on the screw dislocation.
The present work is concerned with the thermoelasticity theory of Green and Naghdi of type I, II and III. By considering a mixed initial-boundary value problem for an isotropic medium in the context of all three models of type I, II and III in a unified way, we derive an identity in terms of the temperature and potential. On the basis of this identity, we establish the domain of influence theorem for the Green–Naghdi-II model. This theorem implies that for a given bounded support of thermomechanical loading, the thermoelastic disturbance generated by the pair of temperature and potential of the system vanishes outside a well-defined bounded domain. This domain is shown to depend on the support of the load, that is, on the initial and boundary data. It is also shown that under Green–Naghdi-II model, the thermoelastic disturbance propagates with a finite speed that is dependent on the thermoelastic parameters.
A mathematical model describing the quasistatic process of frictional contact between a nonlinearly elastic body and an elastic-rigid foundation is considered. The contact is modeled by normal compliance with unilateral constraint, and the friction by a slip-dependent version of Coulomb’s law. A weak formulation of the problem is derived and, under a smallness assumption on contact and friction functions, an existence result is proved by using incremental techniques, Kakutani’s fixed point theorem, and compactness, monotonicity, and lower semicontinuity arguments.
Edge waves in a three-dimensional plate are considered as the singularity cases of the obliquely incident and reflection fields of Lamb modes and horizontally polarized shear (SH) modes, solved by mode matching based on a real orthogonal relation or the boundary collocation technique. With that, the dispersion curves of the edge waves in a duralumin plate are evaluated. The leaky edge waves are predicted and the one corresponding to quasi-resonance at free-edge in a plane strain plate are verified together with the pure edge waves. The high-frequency features of the dispersion curves are discussed.
A discrete system constituted of particles interacting by means of a centroid-based law is numerically investigated. The elements of the system move in the plane, and the range of the interaction can be varied from a more local form (first-neighbours interaction) up to a generalized nth order interaction. The aim of the model is to reproduce the behaviour of deformable bodies with standard (Cauchy model) or generalized (second gradient) deformation energy density. The numerical results suggest that the considered discrete system can effectively reproduce the behaviour of first and second gradient continua. Moreover, a fracture algorithm is introduced and some comparison between first- and second-neighbour simulations are provided.
There has been a differential-geometric interpretation which associates each elastic thin rod with a curve on the three-dimensional unit sphere equipped with a Riemannian metric related to the bending and twisting stiffnesses of the rod. In this paper, we exploit this interpretation to study the stability of anisotropic, naturally straight, helical equilibrium rods with clamped ends. Such a rod is called geodesic here if its associated curve is a geodesic, or equivalently, its twisting stiffness equals one of the bending stiffnesses. We establish criteria for geodesic equilibria to be stable and possibly unstable separately, and develop a scheme predicting unstable non-geodesic equilibria. We also present an example to emphasize the necessity of examining whether a helical equilibrium rod is geodesic or not when one is concerned with its stability.
We study the formation of membrane budding in model lipid bilayers with the budding assumed to be driven by means of diffusion of trans-membrane proteins over a composite membrane surface. The theoretical model for the lipid membrane incorporates a modified Helfrich-type formulation as a special case. In addition, a spontaneous curvature is introduced into the model in order to accommodate the effect of the non-uniformly distributed proteins in the bending response of the membrane. Furthermore, we discuss the effects of line tension on the budding of the membrane, and the necessary adjustments to the boundary conditions. The resulting shape equation is solved numerically for the parametric representation of the surface, which has one to one correspondence to the membrane surface in consideration. Our numerical results successfully predict the vesicle formation phenomenon on a flat lipid membrane surface, since the present analysis is not restricted to the conventional Monge representation often adopted to the problems of this kind for the obvious computational simplicity, despite its limited capability to describe the deformed configuration of membranes. In addition, we show that line tension at the interface of the protein-concentrated domain makes a significant contribution to the bud formation of membranes.
The complex process of bone fracture healing is driven by a set of mechanobiological and biochemical factors. In the present paper, a mathematical model of the angiogenesis effect on bioresorbable bone graft healing is proposed. The synthesis of bone tissue and resorption of bone and bone substitute material are stimulated by adjacent strain energy, and in the meantime regulated by a set of geometry and biochemical factors. The most important new elements included in formulation of this model are the effect of sufficient and insufficient nutrients supply, dependence of actor cell number on pore surface, and dependence of sensor cell number on bone mass.
The proposed mathematical formulation was implemented in FEM software COMSOL. A simple example was selected to perform numerical simulations in order to check the effect of gap size and nutrients diffusion rate on healing process. Values of selected parameters introduced in the proposed model were estimated on the basis of experimental results reported in the literature. Agreement between the results of numerical simulations and experimental studies was observed.
This paper presents a Generalised Beam Theory formulation to study the partial interaction behaviour of two-layered prismatic steel–concrete composite beams. The novelty of the proposed approach is in its capacity to handle the deformability of the shear connections at the interface between the slab and steel beam in both the longitudinal and transverse directions in the evaluation of the deformation modes. This method falls within a category of cross-sectional analyses available in the literature for which a suitable set of deformation modes, including conventional, extension and shear, is determined from dynamic analyses of discrete planar frame models representing the cross-section. In this context, the shear connections are modelled using shear deformable spring elements. As a result, the in-plane partial shear interaction behaviour is accounted for in the planar dynamic analysis during the evaluation of the conventional and extension modes, while the longitudinal partial interaction behaviour associated with the shear modes is included in the out-of-plane dynamic analyses. In the case of the conventional modes, the longitudinal slip is accounted for in the post-processing stage where the warping displacements are determined. A numerical example of a composite box girder beam is presented and its structural response investigated for different levels of shear connection stiffness in both the longitudinal and transverse directions. The accuracy of the numerical results is validated against those obtained with a shell finite element model implemented in ABAQUS/Standard software.
We study the class of q-Fourier multiplier operators
In the framework of the Generalized Beam Theory (GBT) a new cross-section analysis is proposed, specifically suited for nonlinear elastic thin-walled beams (TWB). The approach is developed according to the nonlinear Galerkin method (NGM), which calls for the evaluation of nonlinear (passive) trial functions, to be used in conjunction with linear (active) trial functions, in describing the displacement field. The set of (quadratic) trial functions is determined here by requiring that the classic Vlasov’s kinematic hypotheses of the linear theory (i.e. (a) transverse inextensibility and (b) unshearability) are satisfied also in the nonlinear sense. The linear field is described by the so-called conventional displacements, by neglecting non-conventional displacements, which violate Vlasov’s hypotheses. The nonlinear trial functions thus generated are innovative deformation fields, which describe extensional and shear displacements in a different way from that of the non-conventional fields of the linear theory. In particular, they consist of non-constant tangential and out-of-plane displacements of the cross-section profile, able to ensure inextensibility and unshearability of all the plate elements, by balancing the second-order strains induced by the conventional displacements. Since nonlinear trial functions do not increase the number of the unknowns, the GBT spirit, as a reduction method, is preserved. A very promising example is discussed, which shows that equilibrium paths can be determined by using few linear trial functions in conjunction with the corresponding nonlinear trial functions, supplying good results when compared with burdensome finite-element solutions.
The aim of the present paper is the analysis of a two-dimensional continuum with two families of inextensible fibers that are orthogonal in the initial configuration. In the first part of the work, a new formulation is presented, in which the problem is reduced to a standard nonlinear constrained minimization, while in the second part of the work several numerical investigations are presented considering different boundary conditions with respect to standard symmetric bias extensional tests. The conceptual framework can be recognized in the researches by Pipkin and Rivlin on inextensible nets. Furthermore, an implicit version of the Rivlin representation of the generic placement for a two-dimensional sheet with two families of inextensible fibers is provided by considering the angles of the fiber directors as degrees of the freedom of the formulation. In this way the first gradient formulation is given in terms of two angle fields only.
In this paper we numerically simulate the phenomenon of bone growth in bone defects as driven by external mechanical excitation. Bone growth is accounted for through a continuum model that allows simulation of the filling of a defect. The influence of the model boundary conditions is also discussed. Two and three dimensional simulations are presented, explicitly showing the bone regeneration process inside the cavity on a weekly basis. Numerical results are qualitatively compared with literature experimental data from a rat calvarial defect exposed to low-intensity pulsed ultrasound. The obtained results show trend correlations with the targeted phenomenological observations and allow us to perform a first evaluation of the proposed model parameters to be optimized for clinically relevant situations, even if a systematic experimental campaign is still needed to precisely identify the bio-mechanical parameters involved.
Sets of classical invariants are used to characterize the mechanical behaviour of elastic solids with preferred directions. In this paper, we prove, for an n-preferred direction anisotropy, that only
In this paper, we show how very relevant mathematical works of P.I. Plotnikov, publicized by L.C. Evans and M. Portilheiro, can be used to model the effects of cyclic hysteresis phenomena for flows in unsaturated porous media. We draw particular attention to the example of a model for water–air flows, simplified for purposes of illustration. First, an unstable "spinodal" interval is artificially introduced. Then S.L. Sobolev’s method of dynamic regularization allows associating with the continuity equations additional information in the form of entropy type inequalities. The asymptotic limits of viscous approximate solutions generate effects of irreversibility and the expected clockwise hysteresis loop.
Vortex-induced vibrations at lock-in conditions are modeled through generalized van der Pol-Duffing oscillators endowed with frequency-dependent coefficients, taking inspiration from fluid-elastic models. Accordingly, it is found that the limit-cycle amplitude and the non-linear frequency are mutually dependent (feedback effect), differently from the classic oscillator behavior. Consequently, the mechanical non-linearities, which are often believed to be unimportant, do affect the amplitude of motion. Examples concerning an ideal one degree-of-freedom van der Pol-Duffing oscillator and a two degree-of-freedom model, coarsely representative of a tower building, confirm the importance of this approach also from a technical point of view. Thus, non-linear geometric terms and modal interaction (even in non-resonant cases) can lead to non-negligible modifications of purely aeroelastic problems.
On the basis of the introduction of the stiffness matrix of tensegrity structures, the eigenvalue analysis is carried out to study the influence of the prestress level on the stiffness of tensegrity structures. The triangular prismatic tensegrity structure, the star-shaped tensegrity structure and the star-shaped tensegrity structure with a central strut are selected as the numerical examples. The analytical results show that some eigenvalues increase linearly with the prestress level, whereas other eigenvalues firstly increase and then decrease or firstly decrease and then increase with the increase of the prestress level. This is because the stiffness matrix of the tensegrity structures is mainly composed of the material stiffness matrix and geometric stiffness matrix. As the contribution of these two parts of stiffness to eigenvalue models is different, the trends of eigenvalue variations are different with the increase of the prestressed level.
A new dynamic two-dimensional friction model is developed that is based on the bristle theory. Actually, it is the Reset Integrator Model converted into a two-dimensional space. Usually, two-dimensional friction models are in fact one-dimensional models that are rotated into a slip velocity direction. However, this common approach cannot be applied to the bristle model. That is why the idea of a two-dimensional bristle is presented. The bristle’s deformation is described using polar coordinates. The carried-out numerical simulation of a planar oscillator has proved that the new model correctly captures the mechanism of smoothing dry friction by dither applied in both a perpendicular and co-linear way regarding body velocity. Furthermore, the introduced mathematical model captures two-dimensional stick-slip behaviour. Cartesian slip velocity components are the only inputs to the model. In addition, our proposed model allows one to describe friction anisotropy using bristle parameters. The paper contains the results of an experimental verification of the new friction model, conducted with a special laboratory rig employed to investigate a two-dimensional motion in the presence of dither as well as to validate our numerical results.
This paper is concerned with the kinematics of unilateral constraints in multibody dynamics. These constraints are related to the contact between parts and the principle of impenetrability of matter and have the property that they may be active, in which case they give rise to constraint forces, or passive, in which case they do not give rise to constraint forces. In order to check whether the constraint is active or passive a distance function between parts of the multibody is required. The paper gives a rigorous definition of the distance function and derives certain of its properties. The unilateral constraint may then be expressed in terms of this distance function. The paper analyses the transitions from passive constraints to active and vice versa. Sufficient regularity of the transplacements of the parts and their boundary surfaces will lead to specific properties of the time derivative of the distance function. When the unilateral constraint is active then the parts are geometrically in contact and there is a certain contact surface that, in specific cases, may degenerate into a point. If the parts are in mechanical contact over the contact surface then there will be an interaction between the parts given by contact forces, such as normal and friction forces. Parts in contact may be at rest relative to one another, over the contact surface, or they may be in relative sliding motion. The transition from non-sliding contact to sliding and from sliding to non-sliding is discussed and necessary conditions on the relative velocity and the traction vector are derived. Appropriate complementary conditions are then formulated. These are instrumental when the technique of linear complementarity is used in order to find solutions to the equations of motion.
To investigate complex physical phenomena, bi-dimensional models are often an interesting option. It allows spatial couplings to be produced while keeping them as simple as possible. For linear physical laws, constitutive equations involve the use of tensor spaces. As a consequence the different types of anisotropy that can be described are encoded in tensor spaces involved in the model. In the present paper, we solve the general problem of computing symmetry classes of constitutive tensors in
A one-dimensional problem for a thermoelastic half-space is considered within the context of the theory of generalized thermoelastic diffusion with one relaxation time. The bounding surface is traction free and subjected to a time dependent thermal shock. Two types of boundary conditions are considered, deterministic and stochastic. A permeating substance is considered in contact with the bounding surface. Laplace transform technique is used to obtain the solution in the transformed domain by using a direct approach. Analysis of wave propagation in the medium is presented. The solution in the physical domain for temperature, displacement, stress, concentration and chemical potential are obtained in an approximate manner. Numerical results are carried out and represented graphically.
Two-dimensional anti-plane time-harmonic dynamic Green’s functions for a coated circular inhomogeneity in an infinitely extended matrix with spring- or membrane-type imperfect interfaces are presented. The inhomogeneity, coating and matrix are all assumed to be piezoelectric and transversely isotropic. By using the Bessel function expansions, explicit solutions for the electromechanical fields induced by a time-harmonic anti-plane line force and line charge located in the unbounded matrix, the annular coating and the circular inhomogeneity are derived. The present solutions can recover the anti-plane Green’s functions for some special cases, such as the dynamic or quasi-static Green’s functions of piezoelectricity with perfect interfaces, as well as the dynamic or quasi-static Green’s functions for a two-phase composite with perfect or imperfect interfaces. By means of detailed discussions, selected calculated results are graphically shown to demonstrate the dependence of the electromechanical fields on the circular frequency and the interface properties as well as the coating and size of the inclusion.
The purpose of this paper is to prove the relation
Harmonic holes are designed to leave undisturbed the mean stress in an uncut body subjected to a system of prescribed remote loadings. The role of residual surface tension in the design of harmonic holes is an important consideration, which is usually neglected at the macroscale but remains a significant factor in the design of such holes at the nanoscale. We consider the identification of the geometry of a single harmonic hole in an elastic plane subjected to uniform remote loading when residual surface tension is incorporated into the model of deformation. The geometry of the hole is defined by a conformal mapping with certain unknown coefficients determined from a system of non-linear equations. We illustrate our results with several examples. In particular, we show that for a given remote loading and surface tension, the shapes obtained exhibit strong size-dependency. Moreover, we find that the incorporation of the effect of surface tension greatly extends the range of admissible uniform remote loadings that guarantee the existence of harmonic holes.
In this paper, a numerical method based on finite elements is used to study the phenomena of resorption and growth of bone tissue and resorption of the biomaterial in the neighborhood of a dental implant fixture of the type IntraMobil Zylinder. The mechanical stimulus that drives these processes is a linear combination of strain energy and viscous dissipation. To simulate the implant, an axisymmetric model has been used from the point of view of the geometry; the material behavior is described in the poro-visco-elastic frame. The external action is represented by a load variable with sinusoidal law characterized by different frequencies. Investigated aspects are the influence of the load frequency and of the lazy zone on the remodeling process.
Using a classical non-linear theory, we analytically investigate possible ways for transforming the shape of a curved elastic membrane while keeping it tensioned and moderately strained. This is a critical issue because, as a rule, membranes must be considerably stretched in order to avoid wrinkling and slackening. If the final configuration is fixed, the membrane can be cut and formed according to the final shape, but this cannot be done if more configurations, considerably distant from one another, have to be achieved. Nevertheless, we propose large transformation movements that can be obtained starting from flat membranes while maintaining their strain as limited. We discuss in detail the paradigmatic example of the hyperbolic-paraboloid-shaped membrane. These opportunities are suitable for applications of transformable architecture because they do not require excessive tensioning, compatible with the strength of materials used for this kind of structures.
Peridynamics is a nonlocal continuum mechanics theory where its governing equation has an integro-differential form. This paper specifically uses bond-based peridynamics. Typically, peridynamic problems are solved via numerical means, and analytical solutions are not as common. This paper analytically evaluates peristatics, the static version of peridynamics, for a finite one-dimensional rod as well as a special case for two dimensions. A numerical method is also implemented to confirm the analytical results.
Axial free vibration of a nanobar carrying a nanoparticle is studied based on the nonlocal elasticity theory and Love’s assumption. By considering inertia of radial motion during longitudinal vibration, a governing equation for a nanobar–mass oscillation system is derived via Hamilton’s principle. An exact frequency equation is obtained and an approximate simple expression for the fundamental-mode resonance frequency is given. The size effect of the resonance frequencies is elucidated. The classical Love bar theory and the nonlocal bar theory can be recovered from two special cases by setting the nonlocal parameter and Poisson’s ratio to zero, respectively. Numerical examples are given to show the influence of the nonlocal scaling parameter and attached mass on the resonance frequencies and frequency shifts. Identification formulas for estimating the mass of an attached nanoparticle and predicting the nonlocal parameter are established through the frequency change.
We examine the neutrality of an elliptical inhomogeneity embedded in a particular class of compressible hyperelastic materials of harmonic type when a uniform Piola stress field is prescribed in the surrounding matrix. The present method is based on a spring-type imperfect interface model and on the proper choice of imperfect interface function, which realizes the same degree of imperfection in both normal and tangential directions. The analysis indicates that, in general, the neutral shape and the single imperfect interface function are both dependent on the magnitude of the prescribed stress field.
Temperature-rate-dependent thermoelasticity is a theory of thermoelasticity in which the stress, entropy and heat flux are permitted to depend on the rate of change of temperature and the temperature gradient, as well as the usual variables of temperature and deformation gradient. This has the effect of introducing two relaxation times into the equations of thermoelasticity. Another important effect is that heat now travels at a finite speed rather than the infinite speed implied by the diffusion equation. In an isotropic temperature-rate-dependent thermoelastic material, it is found that four plane harmonic waves may propagate: two purely elastic transverse waves and two longitudinal waves that are dispersive and attenuated. All four waves are stable in the sense that their amplitude remains bounded. An alternative theory that forces heat to travel at finite speed is that of generalized thermoelasticity, in which the rate of change of heat flux also appears in the heat conduction equation, thereby introducing a relaxation time. Two different methods of combining the effects of temperature-rate-dependent thermoelasticity and generalized thermoelasticity are discussed and it is found that the transverse waves are unaltered but that one or both of the longitudinal waves become unstable.
Peridynamics is a non-local continuum mechanics that replaces the differential operator embodied by the stress term
If the characteristic length
where
where
This paper proposes an application of surface elasticity theory in the analysis of the contact problem at nanoscale. The Fourier integral transform method is adopted to derive the fundamental solutions for the contact problem. As a special case, the deformation induced by a tangential triangle distribution force is discussed in detail. The results indicate some interesting characteristics in nano-contact mechanics, which are distinctly different from those in the macro-contact problem. This study is helpful to characterize and measure the mechanical properties of soft materials through nano-indentation.
Within the framework of micromorphic elasticity theory, a finite element approach capable of capturing the microstructure effect is developed to describe the bending behavior of microplates. To this end, the micromorphic theory is generally formulated first. The matrix representation of this formulation is then given from which a prism micromorphic element, including the effects of micro-deformation degrees of freedom of material particles, is proposed. The element is applied to the bending problem of micromorphic rectangular and circular plates subject to different boundary conditions. Selected numerical results are presented to show the microstructure influence on the bending of plates with various geometrical parameters. It is revealed that the element is capable of predicting the mechanical behavior of micromorphic continua in an efficient way.
An asymptotic solution is suggested for a thin isotropic spherical shell subject to uniform external pressure and concentrated load. The pressure is the main load and a concentrated lateral load is considered as a perturbation that decreases buckling pressure. First, the post-buckling solution of the shell under uniform pressure is constructed. A known asymptotic result for large deflections is used for this purpose. In addition, an asymptotic approximation for small post-buckling deflections is obtained and matched with the solution for large deflections. The proposed solution is in good agreement with numerical results. An asymptotic formula is then derived, with the load-deflection diagrams analyzed for the case of combined load. Buckling load combinations are calculated as limiting points in the load-deflection diagrams. The sensitivity of the spherical shell to local perturbations under external pressure is analyzed. The suggested asymptotic result is validated by a finite element method using the ANSYS simulation software package.
A rigorous formulation is presented for the frictionless axisymmetric interaction of a rigid disk with a two-layered inhomogeneous medium. The materials are considered to be linearly elastic transversely isotropic materials and the exponential variation of properties along the depth of each layer is assumed in order to model the effect of inhomogeneity. The disk is considered to be at the top of the coating layer or embedded at the interface of the two media. By satisfying the boundary conditions, the problem leads to a dual integral equation and is reducible to a Fredholm integral equation of the second kind which is solved numerically. The numerical solutions survey the effect of material inhomogeneity through two special cases: a functionally graded coating on a homogeneous transversely isotropic half-space and a homogeneous transversely isotropic coating on an inhomogeneous half-space. The accuracy of the numerical solutions is verified by comparison with the corresponding solution for homogeneous material in the literature.
A new approach for the determination of the global minimum time for the case of the brachistochronic motion of the Chaplygin sleigh is presented. The new approach is based on the use of the shooting method in solving the corresponding two-point boundary-value problem and defining either the crossing points of surfaces or the crossing points space of curves in a three-dimensional space of two costate variables and the time of the brachistochronic motion of the sleigh. A number of examples for multiple extremals of the Chaplygin sleigh brachistochrone problem are provided. In these examples, the global minimum is the solution to which the minimum time of motion corresponds.
The basic feature of the peridynamic model considered here is a continuum description of a material’s behavior as the integrated nonlocal force interactions between infinitesimal particles. In contrast to classical local and nonlocal theories, the peridynamic equation of motion introduced by Silling (J Mech Phys Solids 2000; 48: 175–209) is free of any spatial derivatives of displacement. A theory of thermoelastic composite materials (CMs) with nonlocal thermoperistatic properties of multiphase constituents of arbitrary geometry is analyzed for statistically homogeneous CMs subjected to homogeneous loading. A generalization of the Hill’s equality to peristatic composites is proved. The classical representations of effective elastic moduli through the mechanical influence functions for elastic CMs are generalized to the case of peristatics, and the energetic definition of effective elastic moduli is proposed. The general results establishing the links between the effective properties (effective elastic moduli, effective thermal expansion) and the corresponding mechanical and transformation influence functions are obtained by the use of the decomposition of local fields into load and residual fields. Effective properties of thermoperistatic CM are expressed through the introduced local stress polarization tensor averaged over the extended inclusion phase. A detected similarity of results for both the peristatic and locally elastic composites is explained fundamentally, as the methods used for obtaining the results widely exploit the Hill’s condition and the self-adjointness of the stress operator. However, the representation of effective properties for composites with both the local thermoelastic and nonlocal thermoperistatic properties do not always coincide. Therefore, the representation of effective eigenfields through mechanical influence functions generalizing Levin’s representation does not in general hold for thermoperistatic CMs; this is demonstrated for a one-dimensional numerical example.
A major drawback of the study of cracks within the context of the linearized theory of elasticity is the inconsistency that one obtains with regard to the strain at a crack tip, namely it becoming infinite. In this paper we consider the problem within the context of an elastic body that exhibits limiting small strain wherein we are not faced with such an inconsistency. We introduce the concept of a non-smooth viscosity solution which is described by generalized variational inequalities and coincides with the weak solution in the smooth case. The well-posedness is proved by the construction of an approximation problem using elliptic regularization and penalization techniques.
We propose a modification of the Hamiltonian formalism which can be used for dissipative systems. This work continues a previous work on Hamiltonian inclusions and advances by the introduction of a symplectic version of the Brezis–Ekeland–Nayroles principle. As an application we show how standard plasticity can be treated in our formalism.
In this paper we investigate a thermoelastic theory obtained from the Taylor approximation for the heat flux vector proposed by Choudhuri. This new thermoelastic theory gives rise to interesting mathematical questions. We here prove a uniqueness theorem and instability of solutions under the relaxed assumption that the elasticity tensor can be negative. Later we consider the one-dimensional and homogeneous case and we prove the existence of solutions. We finish the paper by proving the slow decay of the solutions. That means that the solutions do not decay in a uniform exponential way. This last result is relevant if it is compared with other thermoelastic theories where the decay of solutions for the one-dimensional case is of exponential way.
In this paper we venture a new look at the linear isotropic indeterminate couple-stress model in the general framework of second-gradient elasticity and we propose a new alternative formulation which obeys Cauchy–Boltzmann’s axiom of the symmetry of the force-stress tensor. For this model we prove the existence of solutions for the equilibrium problem. Relations with other gradient elastic theories and the possibility of switching from a fourth-order (gradient elastic) problem to a second-order micromorphic model are also discussed with the view of obtaining symmetric force-stress tensors. It is shown that the indeterminate couple-stress model can be written entirely with symmetric force-stress and symmetric couple-stress. The difference of the alternative models rests in specifying traction boundary conditions of either rotational type or strain type. If rotational-type boundary conditions are used in the integration by parts, the classical anti-symmetric nonlocal force-stress tensor formulation is obtained. Otherwise, the difference in both formulations is only a divergence-free second-order stress field such that the field equations are the same, but the traction boundary conditions are different. For these results we employ an integrability condition, connecting the infinitesimal continuum rotation and the infinitesimal continuum strain. Moreover, we provide the orthogonal boundary conditions for both models.
This paper addresses dynamics modelling and control of mechanical systems subjected to constraints due to the tasks they are to perform. The researched problem refers to the control of a manipulator end-effector motion subjected to a constraint on an acceleration change, that is, jerk. In the presented approach, the acceleration and jerk constraints are incorporated into a system constrained dynamics and then passed to its control dynamics using a unified approach, which is based on an advanced dynamics modelling method, that is, on the generalized programmed motion equations method. This enables one to derive motion equations for systems subjected to high-order constraints. The novelty of this approach consists of the formulation of a unified representation of constraints by differential equations, merging them into system constrained dynamics and then using nonlinear control theory techniques to design controllers for tracking motions along the pre-specified constraints.
The dynamics of a system consisting of a rotating rigid hub and a flexible composite thin-walled beam is discussed. The nonclassical effects like material anisotropy, rotary inertia and transverse shear are considered in the mathematical model of the structure. Moreover, the hub mass moment of inertia is taken into account. The differential equations of motion featuring beam bending–twist elastic coupling are derived using the Hamilton principle, and the Galerkin method is applied in order to reduce the partial differential governing equations to the ordinary differential equations. Parametric studies are conducted to evaluate beam stiffness coefficients depending on the fiber lamination angle. Next, numerical results are obtained to investigate the impact of hub to beam relative inertia on the natural frequencies of the structure. Cases of forced vibrations of the system are examined where the driving torque is considered as the sum of a constant (mean value) and a periodic component. Simulations show the importance of the hub inertia on the complete system dynamics. A shift of the resonance zones and a vibration absorption are observed.
The effects of nonlinear hysteretic damping on the post-critical behaviour of the visco-elastic Beck’s beam are discussed in this paper. The model consists of an inextensible and shear-undeformable cantilever beam, internally and externally damped, loaded at the free end by a follower force. Equations are derived in finite kinematics by taking as the configuration variable the rotation
A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite wedge-shaped inclusion, which meets the interface at a right angle and is loaded with normal forces is considered. The normal contact stresses along the contact line are determined and the behavior of the contact stresses in the neighborhood of singular points is established. By using methods of the theory of analytic functions, the problem is reduced to a singular integro-differential equation in a finite interval. Using an integral transformation a Riemann problem is obtained, the solution of which is presented in explicit form.
Quasistatic rate-independent damage combined with linearized plasticity with hardening at small strains is investigated. Fractional-step time discretization is devised with the purpose of obtaining a numerically efficient scheme, possibly converging to a physically relevant stress-driven solution, which however is to be verified a posteriori using a suitable integrated variant of the maximum-dissipation principle. Gradient theories both for damage and for plasticity are considered to make the scheme numerically stable with guaranteed convergence within the class of weak solutions. After finite-element approximation, this scheme is computationally implemented and illustrative 2-dimensional simulations are performed.
Based on the two-dimensional discrete Fast Fourier Transformation (FFT) method, a semi-analytical solution is developed for calculating the elastic fields of dislocation loops within isotropic bimaterials, where the imperfect interface can be described as two types of models: (a) dislocation-like and (b) force-like. Calculation examples of dislocation loops within Al–Cu bimaterials are performed to verify the reliability of the semi-analytical approach. Effects of constant matrix for the dislocation-like and force-like models on the interface elastic fields are studied, and it is shown that the interface elastic field is remarkably influenced by the interface conditions. Comparisons between perfect-bonding, dislocation-like and force-like imperfect interface models are performed to study the effects of interface conditions on the in-plane and out-of-plane elastic fields across the bimaterial interface plane.
For two unequal collinear cracks under compression, which crack would propagate first, the longer one or the shorter one, and how they affected each other was studied. By using complex stress function theory and considering crack surface friction, the analytical formula of stress intensity factors (SIFs) for an infinite plane containing two unequal collinear cracks was obtained and the analytical results have been validated through numerical simulation by employing ABAQUS code, photoelastic experiments, and compressive tests. The results show that the numerical, photoelastic, and compressive test results agree well with the analytical result. Finally, the effects of crack length, crack surface friction, and crack interval distance between two crack tips on SIFs were analyzed, and the results show that the SIF values at the longer crack tips are always higher than those at the shorter crack tips; for each crack, the SIF value at the internal tip is always larger than that at the external tip.
The aim of this paper is to model the macroscopic response of light-activated shape memory polymers (LASMPs) subject to mechanical loadings and exposure to light at certain wavelengths and frequencies. When exposed to external stimuli of mechanical, thermal, photochemical and other origins, polymers undergo microstructural changes, e.g., scission, cross-linking, crystallization, etc. These microstructural changes affect the macroscopic performance of the polymers. In this study, in order to incorporate the effect of microstructural changes on the macroscopic response of light-activated shape memory polymers, we formulate constitutive models based on the notion that the natural configuration of the body under consideration evolves during its response. The theoretical framework appeals to a multinetwork approach consisting of two microstructural networks, which are the original network and the new network formed owing to a light activation. An important distinction between the approach considered here and the usual multinetwork approaches is that there is no conversion of one network to another; instead, what we have is the formation of a second network owing to the linking of photosensitive particles that get linked due to light irradiation. Furthermore, two different constitutive models are considered. The first model assumes the two networks are isotropic. The second model takes into account the directional preference of the second network that is formed. Both these models build on the work of Sodhi and Rao, which is based on the framework developed by Rajagopal and Srinivasa. Several classical boundary value problems involving homogeneous and inhomogeneous deformations are studied. We also investigate two nonlinear constitutive relations and different loading modes. The results highlight the differences in the responses when isotropic and anisotropic models are considered.
The present work is concerned with a very recently proposed heat conduction model—an exact heat conduction model with a delay term for an anisotropic and inhomogeneous material—and some important theorems within this theory. A generalized thermoelasticity theory was proposed based on the heat conduction law with three phase-lag effects for the purpose of considering the delayed responses in time due to the micro-structural interactions in the heat transport mechanism. However, the model defines an ill-posed problem in the Hadamard sense. Subsequently, a proposal was made to reformulate this constitutive equation of heat conduction theory with a single delay term and the spatial behavior of the solutions for this theory have been investigated. A Phragmen–Lindelof type alternative was obtained and it has been shown that the solutions either decay in an exponential way or blow-up at infinity in an exponential way. The obtained results are extended to a thermoelasticity theory by considering the Taylor series approximation of the equation of heat conduction to the delay term and a Phragmen–Lindelof type alternative was obtained for the forward and backward in time equations. In the present work, we consider the basic equations concerning this new theory of thermoelasticity for an anisotropic and inhomogeneous material and make an attempt to establish some important theorems in this context. A uniqueness theorem has been established for an anisotropic body. An alternative characterization of the mixed initial-boundary value problem is formulated and a variational principle as well as a reciprocity principle is established.
Since the first studies dedicated to the mechanics of deformable bodies (by Euler, D’Alembert, Lagrange) the principle of virtual work (or virtual velocities) has been used to provide firm guidance to the formulation of novel theories. Gabrio Piola dedicated his scientific life to formulating a continuum theory in order to encompass a large class of deformation phenomena and was the first author to consider continua with non-local internal interactions and, as a particular case, higher-gradient continua. More recent followers of Piola (Mindlin, Sedov and then Richard Toupin) recognized the principle of virtual work (and its particular case, the principle of least action) as the (only!) firm foundation of continuum mechanics. Mindlin and Toupin managed to formulate a conceptual frame for continuum mechanics which is able to effectively model the complex behaviour of so-called architectured, advanced, multiscale or microstructured (meta)materials. Other postulation schemes, in contrast, do not seem able to be equally efficient. The present work aims to provide a historical and theoretical overview of the subject. Some research perspectives concerning this theoretical approach are outlined in the final section.
The 64 m diameter Sardinia Radio Telescope (SRT), located near Cagliari (Italy), is the world’s second largest fully steerable radio telescope with an active surface. Among its peculiarities is the capability of modifying the configuration of the primary mirror surface by means of electromechanical actuators. This capability enables, within a fixed range, balancing of the deformation caused by external loads. In this way, the difference between the ideal shape of the mirror (which maximizes its performance) and the actual surface can be reduced. The control loop of the radio telescope needs a procedure that is able to predict SRT deformation, with the required accuracy, in order to reduce deviation from the ideal shape. To achieve this aim, a finite element model that can accurately predict the displacements of the structure is required. Unfortunately, the finite element model of the SRT, although very refined, does not give completely satisfactory results, since it does not take into account essential pieces of information, for instance, thermal strains and assembly defects. This paper explores a possible update of the finite element model using only the benchmark data available, i.e. the photogrammetric survey developed during the setup of the reflecting surface. This updating leads to a significant reduction in the differences between photogrammetric data and results of the numerical model. The effectiveness of this tuning procedure is then assessed.
Classical invariants, despite most of them having unclear physical interpretation and not having experimental advantages, have been extensively used in modeling nonlinear magneto-elastic materials. In this paper, a new set of spectral invariants, which have some advantages over classical invariants, is proposed to model the behavior of transversely isotropic nonlinear magneto-elastic bodies. The novel spectral invariant formulation, which is shown to be more general, is used to analytically solve some simple magneto-mechanical boundary value problems. With the aid of the proposed spectral invariants it is possible to study, in a much simpler manner, the effect of different types of deformations on the response of the magneto-elastic material.
A field theory of deformation and fracture is presented. Applying the principle of local symmetry to the law of elasticity, this theory is capable of describing elastic deformation, plastic deformation, and fracture of solids based on the same theoretical basis. Using the Lagrangian formalism, the theory derives field equations analogous to the Maxwell equations of electrodynamics. The field equations yield wave solutions that represent the spatiotemporal behaviors of the velocity and rotation fields of solids under deformation. The dynamics of elastic deformation and plastic deformation are differentiated by the form of the longitudinal force acting on a unit volume. In the field equations, this longitudinal effect acts as the source term. In the elastic dynamics, the source term represents a restoring (energy-conservative) force proportional to the displacement from the equilibrium, and in the plastic dynamics it represents an energy-dissipative force proportional to the local velocity. Both effects are interpreted as the solid’s reaction to the external load. Fracture is characterized by the final stage of deformation, where the solid loses both energy-conservative and energy-dissipative reaction mechanisms. These behaviors are observed as different forms in the wave characteristics of the dynamics. Elastic deformation is characterized by longitudinal compression waves, while plastic deformation is characterized by transverse decaying waves. In the transitional stage from the elastic to the plastic regime, a solitary wave is generated if a certain condition is satisfied. Experimental observations of solids that exhibit these wave characteristics of the deformation field are presented.
This study presents an analysis of the stress-partitioning mechanism for fluid saturated poroelastic media in the transition from drained (e.g. slow deformations) to undrained (e.g. fast deformation) flow conditions. The goal of this analysis is to derive fundamental solutions for the general consolidation problem and to show how Terzaghi’s law is recovered as the limit undrained flow condition is approached. The approach is based on a variational macroscopic theory of porous media (VMTPM). First, the linearized form of VMTPM is expressed in a u–p dimensionless form. Subsequently, the behavior of the poroelastic system is investigated as a function of governing dimensionless numbers for the case of a displacement controlled compression test. The analysis carried out in this study produced two crucial results. First, in the limit of undrained flow, it confirmed that the solutions of the consolidation problem recover Terzaghi’s law. Second, it was found that a dimensionless parameter (
The Sardinia Radio Telescope (SRT), located near Cagliari (Italy), is the world’s second largest fully steerable radio telescope endowed with an active-surface system. Its primary mirror has a quasi-parabolic shape with a diameter of 64 m. The configuration of the primary mirror surface can be modified by means of electro-mechanical actuators. This capability ensures, within a fixed range, the balancing of the deformation caused, for example, by loads such as self-weight, thermal effects and wind pressure. In this way, the difference between the ideal shape of the mirror (which maximizes its performances) and the actual surface can be reduced. In this paper the authors describe the characteristics of the SRT, the close-range photogrammetry (CRP) survey developed in order to set up the actuator displacements, and a finite element model capable of accurately estimating the structural deformations. Numerical results are compared with CRP measurements in order to test the accuracy of the model.
Lipid membranes are versatile biological structures that undergo significant structural remodelling, often triggered by instabilities. Since they invariably possess heterogeneous properties, owing to the presence of multiple lipid species and their interactions with proteins, heterogeneity can have a significant impact on their equilibrium state and stability. In this work, we use curvature elasticity to derive the generalized stability criterion for heterogeneous lipid membranes. Our formulation entertains strain energies that go beyond the Helfrich energy and exhibit higher-order dependence on curvature invariants or spatially varying properties.
Several algorithms consisting in ‘non-standard’ versions of the Multiple Scale Method are illustrated for ‘difficult’ bifurcation problems. Preliminary, the ‘easy’ case of bifurcation from a cluster of distinct eigenvalues is addressed, which requires using integer power expansions, and it leads to bifurcation equations all of the same order. Then, more complex problems are studied. The first class concerns bifurcation from a defective eigenvalue, which calls for using fractional power expansions and fractional time-scales, as well as Jordan or Keldysh chains. The second class regards the interaction between defective and non-defective eigenvalues. This problem also requires fractional powers, but it leads to differential equations which are of a different order for the involved amplitudes. Both autonomous and parametrically excited non-autonomous systems are studied. Moreover, the transition from a codimension-3 to a codimension-2 bifurcation is explained. As a third class of problems, singular systems possessing an evanescent mass, as Nonlinear Energy Sinks, are considered, and both autonomous systems undergoing Hopf bifurcation and non-autonomous systems under external resonant excitation, are studied. The algorithm calls for a suitable combination of the Multiple Scale Method and the Harmonic Balance Method, the latter is applied exclusively to the singular equations. Several applications are shown, to test the effectiveness of the proposed methods. They include discrete and continuous systems, autonomous, parametrically and externally excited systems.
We present an asymptotic two-dimensional plate model for linear magneto-electro-thermo-elastic sensors and actuators, under the hypotheses of anisotropy and homogeneity. Four different boundary conditions pertaining to electromagnetic quantities are considered, leading to four different models: the sensor–actuator model, the actuator–sensor model, the actuator model and the sensor model. We validate the obtained two-dimensional models by proving weak convergence results. Each of the four plate problems turns out to be decoupled into a flexural problem, involving the transversal displacement of the plate, and a certain partially or totally coupled membrane problem.
This semi-inverse method is similar to that used in the so-called Saint-Venant problem for cylindrical three-dimensional first-gradient linear homogeneous and isotropic materials. This semi-inverse method is similar to that used by Saint-Venant to solve the omonimus problem for cylindrical three-dimensional first-gradient linear homogeneous and isotropic materials. Two examples are also presented. It is found that wedge forces are necessary to maintain the body in equilibrium and that these are not an artefact of the double application of the divergence theorem in the second-gradient material derivations.
We consider a constitutive model for the behavior of elastic flexoelectric materials including strain gradient fields and polarization gradient fields. This model is based on a stored elastic energy density function which depends on four independent variables: the polarization field and the polarization field gradient as well as the strain field and the strain field gradient. A generalized Toupin variational approach is utilized to find the governing equations (constitutive relations, equilibrium equations and boundary conditions) of the material. The present model is then applied to the problem of a thick walled cylindrical tube of elastic isotropic flexoelectric material, subjected to axisymmetric loading. The resulting radial displacement field noticeably differs from the elastic and strain gradient elastic cases.
The thermodynamics of open systems exchanging mass, heat, energy, and entropy with their environment is examined as a convenient unifying framework to describe the evolution of growing solid bodies in the context of volumetric growth. Following the theory of non-equilibrium thermodynamics (NET) introduced by De Donder and followers from the Brussels School of Thermodynamics, the formulation of the NET of irreversible processes for multicomponent solid bodies is shortly reviewed. In the second part, extending the framework of NET to open thermodynamic systems, the balance laws for continuum solid bodies undergoing growth phenomena incorporating mass sources and mass fluxes are expressed, leading to a formulation of the second principle highlighting the duality between irreversible fluxes and conjugated driving forces. A connection between NET and the open system thermodynamic formulation for growing continuum solid bodies is obtained by interpreting the balance laws with source terms as contributions from an external reservoir of nutrients.
In this study, we consider the kinematics of the hyperbolic plane and define the notions of inflection curve, circling-point curve, cubic of twice stationary curvature curve, and cubic of thrice stationary curve. We also obtain Cartesian and parametric equations of these curves and illustrate some special cases. Finally, we investigate the hyperbolic Ball point, the ordinary and the sixth-order Burmester points in the case of a finite instant pole.
We discuss the so-called two-temperature model in linear thermoelasticity and provide a Hilbert space framework for proving well-posedness of the equations under consideration. With the abstract perspective of evolutionary equations, the two-temperature model turns out to be a coupled system of the elastic equations and an abstract ordinary differential equation (ODE). Following this line of reasoning, we propose another model which is entirely an abstract ODE. We also highlight an alternative method for a two-temperature model, which might be of independent interest.
Since internal architecture greatly influences crucial factors for tissue regeneration, such as nutrient diffusion, cell adhesion and matrix deposition, scaffolds have to be carefully designed, keeping in mind case-specific mechanical, mass transport and biological requirements. However, customizing scaffold architecture to better suit conflicting requirements, such as biological and mechanical ones, remains a challenging issue. Recent advances in printing technologies, together with the synthesis of novel composite biomaterials, have enabled the fabrication of various scaffolds with defined shape and controlled in vitro behavior. Thus, the influence of different geometries of the assemblage of the matrix and scaffold on the remodeling processes of living bone and artificial material should be investigated. To this end, two implant shapes are considered in this paper, namely a circular inclusion and a rectangular groove of different aspect ratios. A model of a mixture of bone tissue and bioresorbable material with voids was used to numerically analyze the physiological balance between the processes of bone growth and resorption and artificial material resorption in a plate-like sample. The adopted model was derived from a theory for the behavior of porous solids in which the matrix material is elastic and the interstices are void of material.
The well-known developments in elastostatics concerning the equi-stressness criterion of optimality for two-dimensional multi-connected unbounded solids under the bulk-dominating load are generalized toward the transient three-dimensional case with rotational symmetry. This paper advances our previous work by focusing specifically on explicitly identifying the optimal equi-stress surfaces through a simple regular integral equation which involves the single-layer potential kernel associated with the axially symmetric Laplacian. Its two-dimensional analogue is also obtained as a competitive counterpart to the commonly used complex-variable formalism. In both cases, the equations are reformulated as a minimization problem, solved numerically with a standard genetic algorithm over a wide variety of governing parameters thus permitting comparison of the shape optimization results in spatial and plane elasticity for multi-connected domains.
In this paper, the analytical solution of the multi-step homogenization problem for multi-rank composites with generalized periodicity made of elastic materials is presented. The proposed homogenization scheme may be combined with computational homogenization for solving more complex microstructures. Three numerical examples are presented, concerning locally periodic stratified materials, matrices with wavy layers and wavy fiber-reinforced composites.
This paper studies two collinear impermeable interface cracks in bonded piezoelectric finite wedges. The radial edges are subjected to anti-plane concentrated forces and in-plane electric surface charges. The circular boundary is considered as traction free and electrically insulated. The finite complex transform developed in the literature for elastic wedges is extended to electro-elasticity for piezoelectric wedges. Based on the finite Mellin transforms and series expansions, the crack surface conditions are expressed analytically in the form of a set of simultaneous standard singular integral equations, which can be solved numerically by the Lobatto–Chebyshev method. The Normalized Intensity Factors (NIFs) around crack tips are obtained for the elastic and electric field in explicit forms. Dependence of the NIFs on the crack length and crack interval, as well as the wedge angle, is discussed in detail. Crack interaction is graphically demonstrated and analyzed. Results show that extension of one crack length generally enhances the NIFs for both tips of the other crack. Crack interval significantly affects the NIFs of the inner crack tips of the collinear cracks. The formulation and results in this paper include, as special cases, some wedge problems presented in the literature. These analyses are expected to provide some guidance for the design of piezoelectric composite wedges.
The lack of suitable computational methods has significantly restricted the creativity of engineers in designing the materials to be used in technological applications. When one wants exact analytical solutions for a given physical system, then usually drastic and restrictive simplifying assumptions are needed. In particular, homogeneity of physical and geometrical properties at lower length-scale is the standard assumption in continuum mechanics. On the other hand, it is well-known since the pioneering work of Gabrio Piola, and then re-established in the works by Mindlin, Toupin, Green, Adkins and Germain, that it is possible to synthetically describe microscopic inhomogeneity by means of field theories incorporating additional kinematical fields. The characteristic length-scale affecting macro-behavior can even be of the order of nanometers, in which case the intuition due to Richard Feynman about the importance of quantum effects at macro-scale could open the path to technological advancements. In the present paper we review some of the literature in the field and try to indicate some research perspectives that seem to us potentially ground-breaking. In particular, following the suggestion of Professor dell’Isola, we briefly describe his concept of pantographic lattices and sheets whose importance in nano-technology could be relevant.
The Mittag-Leffler relaxation function,
Within the framework of nonlinear elasticity we analyze instability of a uniformly compressed circular two-layered plate with an initially compressed or stretched layer. For a constitutive relation of the material we use the incompressible neo-Hookean model. We assume that the lower layer is subjected to radial tension or compression. As a result in this layer there are initial strains and stresses. The two-layered plate is subjected to a uniform lateral compression. We study the stability of the plate with the use of the static Euler method. Within the method we determine loading parameters for which the linearized boundary-value problem has non-trivial solutions. We derive the three-dimensional linearized equilibrium equations for each layer. The solutions of the latter equations are obtained with the help of the Fourier method. The equation for critical strains is derived. We present an analysis of dependence of critical stress resultants on the initial strains and stiffness parameters.
The objective of this paper is to study the bulk acoustic wave (BAW) propagation velocities in transversely isotropic piezoelectric materials, aluminum nitride, zinc oxide, cadmium sulfide and cadmium selenide. The bulk acoustic wave velocities are computed for each direction by solving the Christoffel’s equation based on the theory of acoustic waves in anisotropic solids exhibiting piezoelectricity. These values are calculated numerically and implemented on a computer by Bisection Method Iterations Technique (BMIT). The modification of the bulk acoustic wave velocities caused by the piezoelectric effect are graphically compared with the velocities in the corresponding non-piezoelectric materials. The results obtained in this study can be applied to signal processing, sound systems and wireless communication in addition to the improvement of surface acoustic wave (SAW) devices and military defense equipment.
In the present paper, we investigate the propagation of an harmonic plane wave propagating with assigned frequency by implementing the thermoelasticity theory based on a fractional order heat conduction law where the fractional order parameter
The aim of the present contribution is concerned with the interactions of thermoelastic displacements, temperatures and stresses for the three-phase-lag and Green–Naghdi heat equations in a functionally graded transversely isotropic plate subjected to a spatially varying heat source. The upper surface of the plate is stress free with prescribed surface temperature while the lower surface of the plate rests on a rigid foundation and is thermally insulated. The governing equations for displacement and temperature fields are obtained in the Laplace–Fourier transform domain by applying Laplace and Fourier transform techniques. The inversion of double transform has been done numerically. The numerical inversion of Laplace transform is done by using a method based on the Fourier Series expansion technique. The solution to the analogous problem for isotropic material is obtained by taking suitable nonhomogeneity parameters. A comparative study between isotropic material and transversely isotropic material is also shown. In absence of nonhomogeneity, the results corresponding to Green–Naghdi II, Green–Naghdi III and 3P lag models agree with the results of the existing literature.
This paper considers instantaneous impulses in multibody dynamics. Instantaneous impulses may act on the multibody from its exterior or they may appear in its interior as a consequence of two of its parts interacting by an impact imposed by a unilateral constraint. The theory is based on the Euler laws of instantaneous impulses, which may be seen as a complement to the Euler laws for regular motions. Based on these laws, and specific continuum properties of the quantities involved, local balance laws for momentum and moment of momentum, involving instantaneous impulses and introducing the Cauchy impulse tensor, are derived. Thermodynamical restrictions on the impulse tensor are formulated based on the dissipation inequality. By stating a principle of virtual work for instantaneous impulses, and demonstrating its equivalence to Euler’s laws, Lagrange’s equations are derived. Lagrange’s equations are convenient to use in the case of multibody dynamics containing rigid as well as flexible parts. A central theme of this paper is the discussion of the interaction between parts of the multibody and their relation to geometrical and kinematical constraints. This interaction is severely affected by the presence of friction, which is notoriously difficult to handle. In a preparation for this discussion we first consider the one-point impact between two rigid bodies. The importance of the so-called impact tensor for this problem is demonstrated. In order to be able to handle the impact laws of Poisson and Stonge, an impact process, governed by a system of ordinary differential equations, is defined. Within this model phenomena, such as slip stop, slip start and slip direction reversal, may be handled. For a multibody with an arbitrary number of parts and multiple impacts, the situation is much more complicated and certain simplifications have to be introduced. Equations of motion for a multibody, consisting of rigid parts and in the presence of ideal bilateral constraints and unilateral constraints involving friction, are formulated. Unique solutions are obtained, granted that the mass matrix of the multibody system is non-singular, the constraint matrices satisfy specific full rank conditions and that the friction is not too high.
In this paper we extend the formulation of the equilibrium problem for the nonlinear bending-torsion theory for curved rods on rods with rectifiable middle curve. We also prove the existence of solutions for the associated boundary value problem when the middle curve is piecewisely smooth.
For a self-similarly subsonically dynamically expanding Eshelby inclusion, we show by an analytic argument (based on the analyticity of the coefficients of the ensuing elliptic system and the Cauchy–Kowalevska theorem) that the particle velocity vanishes in the whole interior domain of the expanding inclusion. Since the acceleration term is thus zero in the interior domain in the Navier equations of elastodynamics, this reduces to an Eshelby problem. The classical Hill jump conditions across the interface of a region with transformation strain are expanded here to dynamics when the interface is moving with inertia satisfying the Hadamard jump conditions. The validity of the Eshelby property and the determination of the constrained strain from the dynamic Eshelby tensor in the interior domain allow one to fully determine from the Hill jump conditions the stress across the moving phase boundary of a self-similarly expanding ellipsoidal Eshelby inhomogeneous inclusion. The driving force can then be obtained. Self-similar motion grasps the early response of the system.
The complex variable method is used to obtain a solution for stress distribution around cutout of oval shape (and its variance) in an infinite plate having un-symmetric material properties with respect to mid plane. The mapping function used is for an oval shape but, using a different shape and size, a constant oval shape of different size as well as shapes such as a circle, ellipse, square, rectangle and eye are obtained. The stress functions are explicitly solved by incorporating the condition of single-valuedness of the out-of-plane displacement and the Schwarz formula along the hole boundary. The effects of the geometry, stacking sequence, material properties and loading angle on stresses and moments around a hole are studied. Some of the results are compared with existing literature and found to be in close agreement.
We study the variational significance of the "order-of-differentiation" symmetry condition of strain gradient elasticity. This symmetry condition stems from the fact that in strain gradient elasticity, one can interchange the order of differentiation in the components of the second displacement gradient tensor. We demonstrate that this symmetry condition is essential for the validity of free variational formulations commonly employed for deriving the field equations of strain gradient elasticity. We show that relying on this additional symmetry condition, one can restrict consideration to strain gradient constitutive equations with a considerably reduced number of independent material coefficients. We explicitly derive a symmetry unified theory of isotropic strain gradient elasticity with only two independent strain gradient material coefficients. The presented theory has simple stability criteria and its factorized displacement form equations of equilibrium allow for expedient identification of the fundamental solutions operative in specific theoretical and application studies.
A computational algorithm for solving anelastic problems in finite deformations is introduced. The presented procedure, termed the Generalised Plasticity Algorithm (GPA) hereafter, takes inspiration from the Return Mapping Algorithm (RMA), which is typically employed to solve the Karush–Kuhn–Tucker (KKT) system arising in finite elastoplasticity, but aims to modify and extend the RMA to the case of more general flow rules and strain energy density functions as well as to non-classical formulations of elastoplasticity, in which the plastic variables are not treated as internal variables. To assess its reliability, the GPA is tested in two different contexts. First, it is used for solving two classical problems (a shear-compression test and the necking of a circular bar). In both cases, the GPA is compared with the RMA in terms of structural set-up, computational effort and flexibility, and its convergence is evaluated by solving several benchmarks. Some restrictions of the classical form of the RMA are pointed out, and it is shown how these can be overcome by adopting the proposed algorithm. Second, the GPA is applied to characterise the mechanical response of a biological tissue that undergoes large deformations and remodelling of its internal structure.
In this paper we formulate a geometric theory of nonlinear thermoelasticity that can be used to calculate the time evolution of temperature and thermal stress fields in a nonlinear elastic body. In particular, this formulation can be used to calculate residual thermal stresses. In this theory the material manifold (natural stress-free configuration of the body) is a Riemannian manifold with a temperature-dependent metric. Evolution of the geometry of the material manifold is governed by a generalized heat equation. As examples, we consider an infinitely long circular cylindrical bar with a cylindrically symmetric temperature distribution and a spherical ball with a spherically-symmetric temperature distribution. In both cases we assume that the body is made of an arbitrary incompressible isotropic solid. We numerically solve for the evolution of thermal stress fields induced by thermal inclusions in both a cylindrical bar and a spherical ball, and compare the linear and nonlinear solutions for a generalized neo-Hookean material.
The coupling interaction of a piezoelectric screw dislocation with a bimaterial containing a circular inclusion is investigated by the complex potential method and conformal mapping technique. Explicit series solutions are obtained and then cast into new expressions with the coupling interaction effects separated. The new expressions converge much more rapidly and their one-order approximation formulae have satisfactory accuracy in many cases. According to the generalized Peach–Koehler formula, the image force acting on the screw dislocation is explicitly obtained and numerically studied to reveal the coupling interaction arising from multiple material properties as well as the geometry of inhomogeneous phases. In all regions, the coupling interaction has a significant influence on the number, location and stability of dislocation equilibrium points. In particular, the inclusion can reverse the image force within a region in the material on the other side of the interface.
In this paper, we examine the influence of swelling on the bulging bifurcation of inflated thin-walled cylinders under axial loading. We provide the bifurcation criteria for a membrane cylinder subjected to combined axial loading, internal pressure and swelling. We focus here on orthotropic materials with two preferred directions which are mechanically equivalent and are symmetrically disposed. Arterial wall tissue is modeled with this class of constitutive equation and the onset of bulging is considered to give aneurysm formation. It is shown that swelling may lead to compressive hoop stresses near the inner radius of the tube, which could have a potential benefit for preventing aneurysm formation. The effects of the axial stretch, the strength of the fiber reinforcement and the fiber winding angle on the onset of bifurcation are investigated. Finally, a boundary value problem is studied to show the robustness of the results.
The paper deals with nets formed by two families of fibers (cords) which can grow shorter but not longer, in a deformation. The nets are treated as two-dimensional continua in the three-dimensional space. The inextensibility condition places unilateral constraint on the partial derivatives y ,1 and y ,2 of the deformation
We discuss several issues regarding material homogeneity and strain compatibility for materially uniform thin elastic shells from the viewpoint of a three-dimensional theory, with small thickness, as well as a two-dimensional Cosserat theory. A relationship between inhomogeneity and incompatibility measures under the two descriptions is developed. More specifically, we obtain explicit forms of intrinsic dislocation density tensors characterizing the inhomogeneity of a dislocated Cosserat shell. We also formulate a system of governing equations for the residual stress field emerging out of strain incompatibilities which in turn are related to inhomogeneities. The equations are simplified for several cases under the Kirchhoff–Love assumption.
Various issues raised by the mechanics of an octopus’s arm are presented from the point of view of continuum mechanics. First, we study the conditions for a mixture of a number of fiber bundles to span the space of stress tensors. The geometry of a fiber bundle, as reflected in the direction and density of the fibers, is described by a vector field, or, more precisely, by a differential two-form. For the stress analysis, we propose a three-dimensional continuum model of the arm that is based on constitutive data available for muscle fibers. An analysis of small deformations superimposed on an activated configuration of the muscle bundles reveals the stiffening mechanism which enables the arm to support various external loadings, even if the geometry and the number of the fiber groups does not allow the tensions in them to span the space of stress tensors.
For many purposes in continuum mechanics it has been found useful to decompose the deformation gradient into two factors, which result from some elastic process through which destressing is achieved at a material point. There is an essential rotational non-uniqueness in these factors; however, some subfactors are uniquely defined. In particular, a certain unique right stretch tensor is identified, which serves as a convenient independent variable for describing the anisotropic elastic response of a solid from its evolving stress-free intermediate configurations.
This paper demonstrates a fractal system of thin plates connected with hinges. The system can be studied using the methods of the mechanics of solids with internal degrees of freedom. The structure is deployable, and initially, it is similar to a small-diameter one-dimensional manifold, which occupies significant volume after deployment. The geometry of solids is studied using the method of the moving hedron. The relationships enabling the definition of the geometry of the introduced manifolds are derived based on the Cartan structure equations. The proof substantially makes use of the fact that the fractal consists of thin plates that are not long compared to the sizes of the system. The mechanics are described for solids with rigid plastic hinges between the plates, and the hinges are made of shape memory material. Based on the ultimate load theorems, estimates are performed to specify the internal pressure that is required to deploy the package into a three-dimensional (3D) structure and the heat input needed to return the system into its initial state. Some possible applications of the smooth 3D manifolds are demonstrated.
Many large deformation constitutive models for the mechanical behavior of solid materials make use of the multiplicative decomposition
In this paper we derive explicit expressions for the Green’s functions in the case of an anisotropic elastic half-space and bimaterial subjected to a line force and a line dislocation. In contrast to previous studies in this area, our analysis includes the contributions of both anisotropic surface elasticity and surface van der Waals interaction forces. By means of the Stroh sextic formalism, analytical continuation and the state-space approach, the corresponding boundary value problem is reduced to a system of six (for a half-space) or 12 (for a bimaterial) coupled first-order differential equations. By employing the orthogonality relations among the corresponding eigenvectors, the coupled system of differential equations is further decoupled to six (for a half-space) or 12 (for a bimaterial) independent first-order differential equations. The latter is solved analytically using exponential integrals. In addition, we identify four and seven non-zero intrinsic material lengths for a half-space and a bimaterial, respectively, due entirely to the incorporation of the surface elasticity and surface van der Waal forces. We prove that these material lengths can be only either real and positive or complex conjugates with positive real parts.
This work considers the inflation and extension of an elastomeric tubular membrane when its material exhibits a time-dependent response. Three different models for time-dependent response are considered: finite linear viscoelasticity, Pipkin–Rogers non-linear viscoelasticity, and thermally induced chemorheological degradation. The first two are based on different assumptions about stress relaxation effects while the third accounts for time-dependent microstructural changes due to simultaneous scission and re-cross-linking of macromolecular network junctions. Each of these models describes a material response that softens with time. It is shown that the constitutive equations for all three models are included in a general non-linear single-integral constitutive equation.
In previous work, for elastic membranes, the material is fixed and a localized bulge may form as the load increases. In this work, the load is specified, and a localized bulge may form as the membrane material undergoes a time-dependent response. It is assumed that the extension and inflation histories are initially uniform, but there may be a time when a localized bulge-like deformation starts to form. This is treated as branching from the uniform extension and inflation history. For times beyond this ‘branching time’, the governing equations are satisfied by both the continuation of the initial uniform deformation history and the branched deformation history for the bulge. A unified condition for determining this branching time, applicable to all three models, is derived in terms of the general non-linear single-integral constitutive equation. Post-branching response is not considered here.
The present work considers the mechanical behavior of a non-classical elastic membrane with two independent bending rigidities. Our interest focuses on the case when the ratio of the Gaussian bending rigidity to the common flexural rigidity falls within the non-classical range, which cannot be covered by a classical elastic plate with an admissible positive Poisson ratio. Our results for a rectangular elastic membrane with two opposite free edges show that its deflection under a uniform transverse pressure could be considerably (even more than twice) larger than a classical elastic plate under otherwise identical conditions, while its lowest fundamental frequency and critical buckling force could be considerably (even more than 50%) lower than a classical elastic plate under otherwise identical conditions. These unexpected results suggest that, unlike classical elastic plates whose actual mechanical behavior are often insensitive to the value of the admissible positive Poisson ratio, actual mechanical behavior of a non-classical elastic membrane with two independent bending rigidities could be very sensitive to the exact values of the two independent bending rigidities. Therefore, the exact value of the Gaussian bending rigidity could be crucial for such non-classical elastic membranes. Only knowing the values of the flexural rigidity and Poisson ratio is insufficient for accurate prediction of mechanical modeling of such non-classical elastic membranes with two independent bending rigidities (such as some biomembranes and atom-thick monolayer membranes).
In this paper the traction contact problems for Stokes equation are discussed and the Stokes equation is considered in a mixed formulation. We prove the existence and uniqueness of the weak solution for a mixed formulation of Stokes equation with traction contact. The traction contact is described by subdifferential boundary conditions. For this problem we present a variational formulation in a form of a hemivariational inequality for the velocity field.
This paper gives a concise summary of the general theoretical framework suitable to describe shells with solid-like and liquid-like behavior. Thin-shell kinematics are considered and used to derive the equilibrium equations from linear- and angular-momentum balance. Based on the mechanical power balance and the mechanical dissipation inequality, the constitutive equations for the hyperelastic material behavior of constrained shells are derived and their material stability is examined. Various constitutive examples are considered and assessed for their stability. The governing weak form of the formulation is derived and decomposed into in-plane and out-of-plane components. The presented work provides a very general framework for a unified description of solid and liquid shells and illustrates what leads to their loss of material stability. This framework serves as a basis for developing computational shell formulations based on rotation-free shell discretizations. Therefore the full linearization of the formulation is also presented here.
This paper concerns an equilibrium problem for a two-dimensional elastic body with a thin Timoshenko elastic inclusion and a thin rigid inclusion. It is assumed that the inclusions have a joint point and we analyze a junction problem for these inclusions. The existence of solutions is proved and the different equivalent formulations of the problem are discussed. In particular, the junction conditions at the joint point are found. A delamination of the elastic inclusion is also assumed. In this case, the inequality-type boundary conditions are imposed at the crack faces to prevent a mutual penetration between the crack faces. We investigate the convergence to infinity and zero of a rigidity parameter of the elastic inclusion. It is proved that in the limit, we obtain a rigid inclusion and a zero rigidity inclusion (a crack).
The goal of this work is to shed some light on the problem of stress concentrations near nanoscale pores (nanovoids) in metallic thin-plates subjected to mechanical load. The limitations of classical elasticity at the nanoscale can be mitigated by the incorporation of a coherent surface model. The disturbance of the elastic field due to a nanovoid in an elastic thin-plate can be determined using a three-dimensional displacement formulation. Numerical results suggest that the surface energy and corresponding surface stress of the nanovoid significantly alter the local stress distribution and the relevant stress concentrations. The magnitude of this effect depends on parameters like the void size, film thickness, applied load, and material properties of the thin-plate and the void surface. The results of the study suggest that nanoporous thin-plates could be optimized for lower stress concentrations and might be less vulnerable to fracture, at least when subjected to uniaxial tensile loads.
We consider a mathematical model that describes the equilibrium of an elastic body in frictional contact with a moving foundation. The contact is modeled with a multivalued normal compliance condition with unilateral constraint, associated with a sliding version of Coulomb’s law of dry friction. We present a description of the model, list the assumptions on the data and derive its primal variational formulation, in terms of displacement. Then we prove an existence and uniqueness result, Theorem 3.1. We proceed with a penalization method in the study of the contact problem for which we present a convergence result, Theorem 4.1. Finally, under additional hypotheses, we consider a variational formulation of the problem in terms of the stress, the so-called dual variational formulation, and prove an equivalence result, Theorem 5.3. The proofs of the theorems are based on arguments of monotonicity, compactness, convexity and lower semicontinuity.
This paper investigates the linear steady state problem of several moving cracks in a functionally graded magneto-electro-elastic strip subjected to anti-plane mechanical and in-plane electric and magnetic loading. For simplicity, it is assumed that the properties of the strip vary continuously according to exponential functions along the thickness of the functionally graded piezoelectric piezomagnetic (FGPP) layer. By combining the dislocation method and integral transform technique, an exact solution in closed form to this problem is obtained. Electro-magneto-mechanical loads are applied on the crack surfaces, which are assumed to be magneto-electrically impermeable. Numerical examples are presented to show the interesting mechanical and electromagnetic coupling phenomena induced by multi-crack interactions. Finally, the effects of crack velocity, material constants, and geometric parameters upon the field intensity factors are studied. The results are useful for the design of the magneto-electro-elastic structures.
Axisymmetric deformation of a circular corrugated diaphragm is defined by the equations formulated in terms of projections of displacements on the axes of the principal plane equidistant from the tops of corrugations. The optimal design problem is considered basing on the homogenization of these equations. As a result, a diaphragm consisting of a flat central part and a corrugated part with variable amplitude is obtained.
The second-order laminated plate theory is utilized in this work to analyze orthotropic composite plates with asymmetric delamination. First, a displacement field satisfying the system of exact kinematic conditions is presented by developing a double-plate system in the uncracked plate portion. The basic equations of linear elasticity and Hamilton’s principle are utilized to derive the system of equilibrium and governing equations. As an example, a delaminated simply supported plate is analyzed using Lévy plate formulation and the state-space model by varying the position of the delamination along the plate thickness. The displacements, strains, stresses and the J-integral are calculated by the plate theory solution and compared with those by linear finite-element calculations. The comparison of the numerical and analytical results shows that the second-order plate theory captures very well the mechanical fields. However, if the delamination is separated by only a relatively thin layer from the plate boundary surface, then the second-order plate theory approximates badly the stress resultants and so the mode-II and mode-III J-integrals and thus leads to erroneous results.
This paper is devoted to a review of the linear isotropic theory of micropolar elasticity and its development with a focus on the notation used to represent the micropolar elastic moduli and the experimental efforts taken to measure them. Notation, not only the selected symbols but also the approaches used for denoting the material elastic constants involved in the model, can play an important role in the micropolar elasticity theory especially in the context of investigating its relationship with the couple-stress and classical elasticity theories. Two categories of notation, one with coupled classical and micropolar elastic moduli and one with decoupled classical and micropolar elastic moduli, are examined and the consequences of each are addressed. The misleading nature of the former category is also discussed. Experimental investigations on the micropolar elasticity and material constants are also reviewed where one can note the questionable nature and limitations of the experimental results reported on the micropolar elasticity theory.
Temperature and thermal stress distributions in a two-dimensional infinite thin plate subjected to a moving heat source with variable power and velocity are obtained by solving quasi-static thermoelasticity equations analytically with the aid of a thermoelastic displacement potential. The results show good agreement with experimental data for a stationary source with constant power and with a steady-state analytical solution in the open literature. It is shown that the quasi-static solution can predict changes of the thermal stress field during the movement of the heat source, and can give the effect of changes of power and velocity of the heat source on the thermal stress field during its movement.
The present work is concerned with an in-depth analysis of plane harmonic waves in a thermoelastic medium under two-temperature thermoelasticity with two relaxation parameters. After the mathematical formulation of the present problem, we obtain the dispersion relation solution of harmonic plane waves propagating in the medium. The transverse wave is observed to be not affected by the thermal field and the longitudinal wave is coupled with the thermal field. Hence, special attention is paid on two different modes of longitudinal plane wave. One is predominantly elastic and the other is predominantly thermal in nature. The asymptotic expressions for the phase velocity, specific loss and many other important wave characteristics are derived in the cases of very high- and low-frequency regions for both the elastic and thermal mode longitudinal waves. Numerical results of these wave components are obtained for the intermediate values of frequency and the results are illustrated graphically in order to verify the analytical results. On the basis of analytical and numerical results a thorough analysis of the effects of the thermal relaxation parameters on various wave characteristics is presented. A detailed comparison of the results in the context of the present model with the corresponding results of three other models is also provided and several findings regarding the prediction of the present model as compared to other models are highlighted. The present investigation brought out the effects of the second relaxation parameter and the two-temperature parameter on the propagation of a plane harmonic wave through the medium.
In this study, we aim to compare the optimal homotopy analysis solution with the exact solution of the thermoelastic interactions problem in an isotropic hollow cylinder. The thermoelastic interactions in a hollow cylinder in the context of the theory of generalized thermoelasticity with one relaxation time (Lord and Shulman’s theory) are considered. An application of a hollow cylinder is investigated where the inner surface is traction-free and subjected to a decaying-with-time thermal field, while the outer surface is traction-free and thermally isolated. The mathematical model is solved by analytical method and optimal homotopy analysis method (OHAM). In addition, the convergence of the obtained homotopy analysis method solution is discussed explicitly. Numerical results for the temperature distribution, displacement and radial stress are represented graphically. The accuracy of the optimal homotopy analysis method is validated by comparing the analytical and exact solutions for the field quantities.
In the present paper, a rational report on Euromech 563, Generalized continua and their applications to the design of composites and metamaterials (Cisterna di Latina 17–21 March 2014), is provided.
The frank and constructive spirit which animated the workshop by Dell’Isola et al. (ZAMM 2014; 94(5): 367–372) also characterized Euromech Colloquium 563. All presentations were video-recorded and are freely available online at the address http://www.memocsevents.eu/euromech563/?page_id=1013. The topics treated were selected by the organizers in order to allow a comparison of the available experimental evidence with the predictive capability of current theoretical models. The numerical investigations selected and presented aimed to make more effective the aforementioned comparison. The interested reader will find more details about the colloquium at the dedicated webpage
The well-known Kolosov–Muskhelishvili (KM) representation of the Airy function for 2D stress analysis in complex variable terms is enhanced by combining it with Walsh wavelets decomposition. It allows us to perform general analytical derivations up to the maximum extent possible which, in turn, provides a basis for developing a new stress computation algorithm readily incorporated into the routine single scale KM scheme. The mathematical treatment of the wavelet application is supported by a number of examples where non-trivial closed-form solutions are known and serve as a benchmark for numerical simulations. The comparison shows that the proposed framework has better performance than the conventional Fourier transform, especially when it comes to non-smooth stress distributions.
The displacement field in rods can be approximated by using a Taylor–Young expansion in transverse dimension of the rod. These involve that the highest-order term of shear is of second order in the transverse dimension of the rod. Then we show that transverse shearing energy is removed by the fourth-order truncation of the potential energy and so we revisit the model presented by Pruchnicki. Then we consider the sixth-order truncation of the potential which includes transverse shearing and transverse normal stress energies. For these two models we show that the potential energies satisfy the stability condition of Legendre–Hadamard which is necessary for the existence of a minimizer and then we give the Euler–Lagrange equations and the natural boundary conditions associated with these potential energies. For the sake of simplicity we consider that the cross-section of the rod has double symmetry axes.
The mechanical behavior of monolithic cementitious materials is known to be significantly affected by their granular nature. This paper describes an approach to incorporate the effects of granularity for cementitious material based upon the granular micromechanics paradigm and thermomechanics. As a result, macro-scale constitutive equations that utilize grain-scale force–displacement relationships have been derived. In this derivation, the connections between grain-scale free energy and dissipation and the macro-scale behavior are established based upon the thermomechanics framework. Expression for Cauchy stress tensor is obtained in terms of the free energy of grain-pair interactions. In addition, the free energy and dissipation potential of grain-pair interactions are used to find a grain-scale Clausius–Duhem type inequality. Thus, inter-granular constitutive equations are obtained in the usual manner with the exception that these are based upon simple physically motivated free energy and dissipation functions formulated in terms of grain-pair interactions. The derived model is used to simulate a wide variety of experimental tests that have been reported in the literature to evaluate material response under multi-axial conditions. The results are analyzed with the aim to reveal the connections between grain-scale mechanisms and the macro-scale behavior.
In the present analytical study, we consider the problem of a nanocrack with surface elasticity interacting with a screw dislocation. The surface elasticity is incorporated by using the continuum-based surface/interface model of Gurtin and Murdoch. By considering both distributed screw dislocations and line forces on the crack, we reduce the interaction problem to two decoupled first-order Cauchy singular integro-differential equations which can be numerically solved by the collocation method. The analysis indicates that if the dislocation is on the real axis where the crack is located, the stresses at the crack tips only exhibit the weak logarithmic singularity; if the dislocation is not on the real axis, however, the stresses exhibit both the weak logarithmic and the strong square-root singularities. Our result suggests that the surface effects of the crack will make the fracture more ductile. The criterion for the spontaneous generation of dislocations at the crack tip is proposed.
This paper studies wave propagation in a poroelastic solid bar with polygonal cross-section under plane-strain conditions. The boundary conditions on the surface of the cylinder whose base curve is polygon are satisfied by means of the Fourier expansion collocation method. The frequency equations are discussed for both symmetric and antisymmetric modes in the framework of Biot’s theory of poroelastic solids. For illustration purposes, sandstone saturated materials and bony material are considered. The numerical results were computed as the basis of relevant material data . Phase velocity is computed against the wavenumber for various cross-sections and results are presented graphically.
The stress–strain relation of a no-tension material, used to model masonry structures, is determined by the nonlinear projection of the strain tensor onto the image of the convex cone of negative-semidefinite stresses under the fourth-order tensor of elastic compliances. We prove that the stress–strain relation is indefinitely differentiable on an open dense subset
We studied the time evolution problem driven by growth for a non-Euclidean ball which at its initial state is equipped with a non-compatible distortion field. The problem is set within the framework of non-linear elasticity with large growing distortions. No bulk accretive forces are considered, and growth is only driven by the stress state. We showed that, when stress-driven growth is considered, distortions can evolve along different trajectories which share a common attracting manifold; moreover, they eventually attain a steady and compatible form, to which there corresponds a stress-free state of the ball.
A new second-order formulation is obtained for elastic wave propagation in 2D media bounded by a perfectly matched layer (PML). The formulation uses a complex coordinate stretching approach with a two-parameter stretch function. The final system, consisting of just two second-order displacement equations along with four auxiliary equations, is smaller than existing formulations, thereby simplifying the problem and reducing the computational cost. With the help of a plane-wave analysis, the stability of the continuous formulation is examined. It is shown that by increasing the scaling parameter in the stretch function, any existing instability is moved to higher spatial frequencies. Since discrete models cannot resolve frequencies beyond a certain limit, this can lead to significant computational stability improvements. Numerical results are shown to validate our formulation and to illustrate the improved stability that can be achieved with certain anisotropic media that have known issues.
We study the contribution of surface piezoelectricity to the anti-plane deformations of a hexagonal piezoelectric material weakened by a crack. The surface piezoelectricity is incorporated by using an extended version of the continuum-based surface/interface model of Gurtin and Murdoch. The original boundary value problem is finally reduced to a system of two coupled first-order Cauchy singular integro-differential equations by considering a distribution of line dislocations and electric-potential-dislocations on the crack. Through a diagonalization strategy, the coupled system can be transformed into two independent singular integro-differential equations, each of which contains only one single unknown function and can be numerically solved by the collocation method. Our solution demonstrates that the stresses, strains, electric displacements and electric displacements exhibit the logarithmic singularity at the crack tips. The obtained solution is further used to predict the size-dependent effective electroelastic properties of a piezoelectric solid containing multiple nanocracks with surface piezoelectricity within the framework of non-interaction approximation.
The piston problem for a hyperelastic hyperbolic conservative model where the stored energy is given in separable form is studied. The eigenfields corresponding to the hyperbolic system are of three types: linearly degenerate fields (corresponding to the contact characteristics), the fields which are genuinely nonlinear in the sense of Lax (corresponding to longitudinal waves), and, finally, nonlinear fields which are not genuinely nonlinear (corresponding to transverse waves). Taking the initial state free of stresses, we presented possible auto-similar solutions to the piston problem. In particular, we have shown that the equations admit transverse shock waves having a remarkable property: the solid density is decreasing through such a shock, it is thus a ‘rarefaction’ shock.
This paper focuses on the determination of the complete set of ideal orientations of FCC materials in the equibiaxial tension mode of deformation. The simulations are based on the numerical procedure developed by the authors in which, a rate-sensitive crystal plasticity model with Secant hardening law was employed. The resulting nonlinear system of equations is solved by the modified Newton–Raphson method. An Euler space scanning method is used to obtain the ideal orientations of a deformation mode. In this method some initial orientations which are evenly spaced in the Euler space are selected and their evolutions into the ideal orientations are tracked. To verify the accuracy of the presented Euler space scanning method for FCC crystal structures, the ideal orientations of plane strain compression loading are calculated and compared with the existing experimental results. It is observed that this method can predict all of the reported ideal orientations of this deformation mode accurately. Afterward, by applying this method to the equibiaxial tension mode of deformation, eight lines of ideal orientations E1–E8 are resulted. Finally, the major characteristics of the obtained ideal orientations along with the crystal evolution patterns are thoroughly discussed.
This paper constructs multiple elastic inclusions with prescribed uniform internal strain fields embedded in an infinite matrix under given uniform remote anti-plane shear. The method used is based on the sufficient and necessary conditions imposed on the boundary values of a holomorphic function, which guarantee the existence of the holomorphic function in a multiply connected region. The unknown shape of each of the multiple inclusions is characterized by a polynomial conformal mapping with a finite number of unknown coefficients. With the aid of Cauchy’s integral formula and Faber series, these unknown coefficients are determined by a system of nonlinear equations. Detailed numerical examples are shown for multiple inclusions with various prescribed uniform internal strain fields, for symmetrical inclusions and for inclusions whose shapes are independent of the remote loading, respectively. It is found that the admissible range of uniform internal strain fields for multiple inclusions is moderately larger than the admissible range of the uniform internal strain field for a single elliptical inclusion under the same remote loading. In particular, specific conditions on the prescribed uniform internal strain fields and elastic constants of the multiple inclusions are derived for the existence of symmetric inclusions and rotationally symmetrical inclusions. Moreover, for any two inclusions among multiple inclusions of shapes independent of the remote loading, it is shown that the ratio between the uniform internal strain fields inside the two inclusions equals a specific ratio determined by the shear moduli of the two inclusions and the matrix.
The geometry of parallelizable manifolds – that is, teleparallelism – is summarized in the language of local frame fields. Some problems in continuum mechanics that relate to the couple stresses that are produced in the bending and twisting of prismatic beams and wires are then discussed. It is then shown that by going to a higher dimensional analogue of the geometry that one uses for one-dimensional deformable objects, one is basically using the methods of teleparallelism in the context of the Cosserat approach to deformable bodies.
The interaction energy is a fundamental ingredient in the modeling of the macroscopic behavior of shape memory alloys and several attempts have been carried out to derive explicit formulas for it by micromechanical methods. The available models vary in sophistication according to the level of detail in the description of the underlying microstructure. There is, however, a common issue with most micromechanical estimates: they tend to overestimate the values of the interaction energy. While various solutions to this problem can be envisaged in the framework of multi-variant models by enriching the description of the microstructure, it seems that similar remedies are not yet available in the two-phases setting. In this work the quantitative relevance of this effect is evaluated, in a sample case, showing that it may lead to violations of the second law of thermodynamics. A new class of micromechanical estimates based on a two-phases microstructure is then proposed and used to model, in an overall way, the secondary accommodation phenomena that take place around product phase regions. The proposed expressions for the interaction energy turn out to yield physically plausible values, consistent with the thermodynamical bounds.
The present paper reports on the application of the VFC (Variational Feedback Control) to the mitigation of the vibrational motion of continuous structures. The control method has been very recently proposed in the context of mechanical engineering to obtain suspensions characterized by better comfort performances. In this paper the same methodology is indeed applied to a simply nonlinear supported beam that is forced by a seismic excitation. The earthquake shaking, modelled as a random displacement of the beam hinges, is moderated by the presence of a controllable damper interposed between the beam and the oscillating ground, and its damping is subjected to the VFC controller. The inputs processed by the controller are the accelerations measured by two sensors, one on the ground, and the second on the beam. Numerical results are examined and some preliminary considerations are presented in which the considered method is applied to piezoelectromechanical beams.
In the present work, we shall consider some common models in linear thermo-elasticity within a common structural framework. Due to the flexibility of the structural perspective we will obtain well-posedness results for a large class of generalized models allowing for more general material properties such as anisotropies, inhomogeneities, etc.
In this paper, the problem of the circular orthotropic bars with multiple cracks is investigated based on the Saint-Venant torsion theory. The solution to the problem of an orthotropic bar weakened by a Volterra-type screw dislocation is first obtained by means of the finite Fourier sine transform. In this research, the bar is assumed to be subjected to an axial net torsion when finding the dislocation solution. The closed form solution is then derived for displacement and stress fields in the bar. At the next step, the dislocation solution is employed to derive a set of Cauchy singular integral equations for analysis of the circular bar with curved cracks. The solution to these equations is used to determine the torsional rigidity of the bar and the stress intensity factors of the tips of the cracks. The paper is furnished with several examples of a single crack and multiple cracks.
The aim of the present work is the detection of corrosion damage along the inaccessible part of the boundary of a body under investigation. The data of the problem, besides all the information relative to the domain such as the geometry and the conductivity of the body, are the prescribed current fluxes and voltage measurements on the accessible part of the boundary. This constitutes, in general, a nonlinear inverse problem whose ill-posed feature requires a suitable solution procedure. The strategy proposed here is based on a linearization of the Robin boundary condition on the inaccessible part of the boundary and on the identification of a resistivity parameter related to the corroded surface. Besides giving a strategy to evaluate the corrosion damage parameter, this paper tries to sketch a sensitivity analysis of the computed solution with respect to all factors affecting the available information relative to the accessible boundary, such as the quantity and quality of data and the unavoidable errors corrupting the compatibility of the measured data.
We consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is assumed to be quasistatic and the material behaviour is described by a viscoelastic constitutive law with damage. The friction and contact are modeled with subdifferential boundary conditions. We derive the variational formulation of the problem which is a coupled system of a hemivariational inequality for the velocity and a parabolic variational inequality for the damage field. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of time-dependent stationary inclusions and a fixed point theorem.
The bifurcation and chaotic motion of a fully simply-supported thin rectangular plate considering nonlinear deflection subjected to axial subsonic airflow and transverse harmonic excitation is analyzed. Based on von Karman’s large deformation theory, the partial differential equation of motion of the structural system is formulated using Hamilton’s principle, and it is transformed into a set of ordinary differential equations (ODEs) through Galerkin’s method. The three-dimensional (3D) aerodynamic pressure induced by the transverse motion of the plate is derived from the linear potential flow theory, and the validity of the aerodynamic model is verified. For the structural system, Melnikov’s method is adopted to give an analytical expression of the necessary condition for the chaotic motion of the plate. The influences of the flow velocity and the external harmonic excitation on the chaotic motion of the plate are investigated. Numerical simulations are carried out to obtain the bifurcation diagrams, displacement time histories, phase portraits, and Poincaré maps of the nonlinear system to verify the validity of the analytical results. The results show that when the flow velocity increases, the plate will be unstable, and chaotic motion of the plate will occur.
Chiral effects cannot be described by means of the classical theories of continua. In the context of the strain gradient theory of porous elastic solids we study the deformation of a chiral cylinder subjected to torsion, extension and bending by terminal couples. This work is motivated by recent interest in using the chiral continuum as a model for some auxetic materials, bones and carbon nanotubes. The problem is reduced to the study of some two-dimensional problems. We show that the torsion of a chiral cylinder is accompanied by extension, bending, and a variation of the volume fraction field. The solution is used to investigate the deformation of a circular cylinder.
Polyconvexity is an important mathematical condition imposed on a strain energy function. In particular, it is sufficient for the ellipticity of the constitutive equation and for the material stability and becomes especially crucial in the context of nonlinear elasticity. In combination with another condition referred to as coercivity, polyconvexity ensures existence of the global minimizer of the total elastic energy which implies a solution of a boundary value problem. While a great variety of polyconvex energies are known for isotropic and have recently been proposed for anisotropic elastic solids, there are, to the best of our knowledge, no results on polyconvexity for electro- and magneto-elastic materials. In the present paper, we extend the notion of polyconvexity to the coupled electro- and magneto-elastic response and formulate polyconvex free energy functions for electro- and magneto-sensitive elastomers. In analogy to the purely elastic response, these free energy functions will ensure the positive features of the constitutive equations mentioned above, although a strict mathematical proof of this fact should be supplied later. The proposed model is able to describe the electro- and magnetostriction and demonstrates good agreement with the corresponding experimental data.
This paper deals with the three-dimensional analysis of the near-resonant regimes of a point load, moving steadily along the surface of a coated elastic half-space. The approach developed relies on a specialized hyperbolic–elliptic formulation for the wave field, established earlier by the authors. Straightforward integral solutions of the two-dimensional perturbed wave equation describing wave propagation along the surface are derived along with their far-field asymptotic expansions obtained using the uniform stationary phase method. Both sub-Rayleigh and super-Rayleigh cases are studied. It is shown that the singularities arising at the contour of the Mach cones typical of the super-Rayleigh case, are smoothed due to the dispersive effect of the coating.
The purpose of this paper is to establish a method of obtaining closed-form solutions in isotropic hyperelasticity using the complementary energy, the Legendre transform of the strain energy function. Using the complementary energy, the stress becomes the independent variable and the strain the dependent variable. Some of the implications of this formulation of the equations are explored and illustrative examples of solutions for spherical and cylindrical inflation for several forms of the complementary energy are presented.
According to the widespread opinion of historians, modern mechanics is the result of a revolutionary phenomenon that occurred during the Renaissance, during the so-called scientific revolution, because of the concourse of many factors including the discovery of fundamental mathematical and scientific Archimedean, or more generally Hellenistic, texts, the spread of a Platonic vision of reality in which mathematics played a key role and more in general the affirmation of a mechanistic natural philosophy. I have a different opinion; according to it modern mechanics was primarily born as a natural development of a mechanics carried out by more or less remote mathematicians, of whom Archimedes was an illustrious exponent only. Besides mechanics, these mathematicians developed other sciences known as mixed mathematics: optics, astronomy, music, which are at the basis of modern mathematical physics. This article intends to support this latter opinion making reference to original documents.
In the present paper, the stability of a nonlinear elastic cylindrical tube made of micropolar material is analyzed. It is assumed that the elastic properties of the tube vary through the wall thickness. The problem is studied for the case of axial compression of the tube under internal and external hydrostatic pressure. Applying linearization the neutral equilibrium equations have been derived, which describe the perturbed state of the tube. By solving these equations numerically the critical curves and corresponding buckling modes have been found, and the stability regions have been constructed in the planes of loading parameters (relative axial compression, relative internal or external pressure). Using these results, the influence of elastic properties, as well as the size of the tube, on the loss of stability is studied. Special attention has been paid to the analysis of how the pattern of change in elastic parameters affects the stability of a cylindrical tube made of micropolar material.
In this paper we study the mathematical model which describes quasistatic frictional contact problems between a deformable body and a rigid foundation. In this model the contact is bilateral and the behaviour of the material is described by a viscoelastic constitutive law with time delay. The variational formulation of the mathematical model is given as a hemivariational inequality of elliptic type. Based on some recent results for the elliptic hemivariational inequality and Banach’s fixed point theorem we prove existence and uniqueness of the solution for the obtained hemivariational inequality.
The paper is concerned with coordinate representations for rigid parts in multibody dynamics. The discussion is based on the general theory of the dynamics of multibody systems under constraints. General configuration coordinates are introduced and the requirement of their regularity is discussed. The use of Euler angles, quaternions and linear coordinates, where quaternions and linear coordinates require constraint conditions, is analysed in detail. These coordinate systems are all shown to be regular. Equations of motion are formulated, using Lagrange’s as well as Euler’s equations, and they are supplemented by the appropriate constraint conditions in the cases of quaternions and linear coordinates. Mass matrices are derived, and in terms of the Euler angles the mass matrix components are products of trigonometric functions whereas in terms of quaternions the matrix components are quadratic polynomials. Using linear coordinates gives rise to a constant mass matrix. Thus, there is a decreasing degree of complexity, regarding mass matrix components, when going from Euler angles to linear coordinates. This is obtained at the expense of an increasing gross number of degrees of freedom and the necessary introduction of constraint conditions. The different equations of motion obtained are compared with respect to their structural complexity. In all representations the components of the angular velocity are explicitly calculated. This is not always the case in previous investigations of this subject. The paper also gives a new proof of the well-known relation between angular velocity and unit quaternions and their time derivative.
In this work, we consider the two-dimensional problem of a thermoelastic half-space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The bounding surface of the half-space is taken to be traction free and subjected to a time-dependent thermal shock. Laplace and Fourier transform techniques are used. The solution in the transformed domain is obtained by a direct approach. Numerical inversion techniques are used to obtain the inverse double transforms. Numerical results are represented graphically and discussed.
In this paper we provide an expression for the border of the safe domain associated with projectile trajectories in a medium with a quadratic drag force. The curve defining the safe domain is given in parametric coordinates involving some integrals. These integrals have to be evaluated numerically after solving an integral equation. We show that not all the trajectories are necessary to obtain the safe domain.
Morphoelastic rods are thin bodies which can grow and can change their intrinsic curvature and torsion. We deduce a system of equations that rule accretion and remodeling in a morphoelastic rod by combining balance laws involving non-standard forces with constitutive prescriptions filtered by a dissipation principle that takes into account both standard and non-standard working. We find that, as in the theory of three-dimensional bulk growth proposed [DiCarlo, A and Quiligotti, S. Mech Res Commun 2002; 29: 449–456], it is possible to identify a universal coupling mechanism between stress and growth, conveyed by an Eshelbian driving force.
Ideal orientations are one of the material characteristics of the applied mode of deformation. The transfer of material texture to orientations near specific ideal orientations can improve the mechanical properties of the material. In this paper, we focus on the determination of ideal orientations of BCC crystals under the equibiaxial tension mode of deformation. To do this, an Euler space scanning method based on a crystal plasticity approach is presented. In this method some initial orientations which are evenly spaced in the Euler space are selected and their evolutions into the ideal orientations are tracked. The loading is applied incrementally until all of the lattice spin components become permanently zero. The rate sensitive crystal plasticity model with power law hardening is employed and the resulting nonlinear system of equations is solved by the modified Newton–Raphson method. In order to verify the simulation results, the ideal orientations of rolling textures are calculated. A comparison of the obtained results with the existing experimental data demonstrates that all of the reported ideal orientations are satisfactorily predicted. Afterward, preferred orientations for equibiaxial tension mode of deformation which have not been reported previously in the literature are calculated. This analysis resulted in eight fibers EF1–EF8 together with a plane of ideal orientations for equibiaxial tension loading. The effects of symmetry of the crystal structure and loading on the obtained ideal orientations are finally discussed.
This work is concerned with a circular inhomogeneity bonded to an infinite matrix though N concentric circular interphase layers within the context of Kirchhoff theory for isotropic laminated plates. An elegant and effective procedure is proposed to obtain the stress resultant fields within the internal inhomogeneity and the surrounding matrix under thermomechanical loadings. The boundary value problem is finally reduced to two coupled linear algebraic equations and four coupled linear algebraic equations that determine the six real coefficients of the stress resultant field in the inhomogeneity. In particular, the average stress resultants within the inhomogeneity can be directly determined from these six real coefficients. The other six unknown real coefficients, which control the stress resultant field in the surrounding matrix, can then be simply obtained. The effect of the N interphase layers on the stress resultant field is demonstrated by their influence on these 12 real coefficients. The obtained solution is further applied to the design of neutral and harmonic circular elastic inhomogeneities with a single interphase layer.
Based on complex function theory, an analytical solution for the dynamic stress concentration due to an arbitrary cylindrical cavity in an infinite inhomogeneous medium is investigated. Two different conformal mappings are introduced to solve scattering by an arbitrary cavity for the Helmholtz equation with variable coefficient through the transformed standard Helmholtz equation with a circular cavity. By assuming that the medium density continuously varies in the horizontal direction with an exponential law and the elastic modulus is constant, the complex-value displacements and stresses of the inhomogeneous medium are explicitly obtained. The distribution of the dynamic stress for the case of an elliptical cavity are obtained and discussed in detail via a numerical example. The results show that the wavenumber, inhomogeneous parameters, and different values of aspect ratio have a significant influence on the dynamic stress concentration factors around the elliptical cavity.
We consider a one-dimensional system of Lennard-Jones nearest- and next-to-nearest-neighbour interactions. It is known that if a monotone parameterization is assumed then the limit of such a system can be interpreted as a Griffith fracture energy with an increasing condition on the jumps. In view of possible applications to a higher-dimensional setting, where an analogous parameterization does not always seem reasonable, we remove the monotonicity assumption and describe the limit as a Griffith fracture energy where the increasing condition on the jumps is removed and is substituted by an energy that accounts for changes in orientation (‘creases’). In addition, fracture may be generated by ‘macroscopic’ or ‘microscopic’ cracks.
We discuss a completely forgotten work of the geologist GF Becker on the ideal isotropic nonlinear stress–strain function (Am J Sci 1893; 46: 337–356). Due to the fact that the mathematical modelling of elastic deformations has evolved greatly since the original publication we give a modern reinterpretation of Becker’s work, combining his approach with the current framework of the theory of nonlinear elasticity.
Interestingly, Becker introduces a multiaxial constitutive law incorporating the logarithmic strain tensor, more than 35 years before the quadratic Hencky strain energy was introduced by Heinrich Hencky in 1929. Becker’s deduction is purely axiomatic in nature. He considers the finite strain response to applied shear stresses and spherical stresses, formulated in terms of the principal strains and stresses, and postulates a principle of superposition for principal forces which leads, in a straightforward way, to a unique invertible constitutive relation, which in today’s notation can be written as
Here, G is the shear modulus and K is the bulk modulus. For Poisson’s number = 0 the formulation is hyperelastic and the corresponding strain energy
has the form of the maximum entropy function.
We approximate the displacement field in a shell by a fifth-order Taylor–Young expansion in thickness. The model is derived from the truncation of the potential energy at fifth order. The equilibrium equations imply local constraints on the through-thickness derivatives of the zero-order displacement field. This leads to an analytical expression for the two-dimensional potential energy of a shell in terms of the zero-order displacement field and its derivatives that include non-standard transverse shearing and normal stress energy. Then we derive the equation of equilibrium and the boundary conditions.
We investigate the behaviour of an elastic body which is in frictional contact with a foundation on a part of the boundary, and which can come into contact with a rigid obstacle on another part of the boundary. We associate this physical setting with two mechanical models. Every model is mathematically described by a boundary value problem which consists of a system of partial differential equations associated with a displacement condition, a traction condition, a frictional contact condition and a frictionless unilateral contact condition. In both models the unilateral contact is described by Signorini’s condition with non-zero gap. The difference between the models is given by the frictional condition we use. In the first model we use a condition with prescribed normal stress. In the second one, we use a frictional bilateral contact condition. The weak solvability of the boundary value problems we propose herein relies on an abstract generalized saddle point problem. Abstract existence, uniqueness and boundedness results as well as abstract convergence results of a regularization are established. Then, we discuss the existence, the uniqueness, the boundedness and the approximation of the weak solutions based on the abstract results.
In this paper, we propose two mechanical models for the single walled carbon nanotubes: (a) a discrete mechanical model at the nanoscale level based on replacing the atomic structure by a discrete lattice of elastic bars and (b) a homogenized continuous mechanical model based on atomistic experimental results on the stretching and angular variations of carbon atomic bonds. Two different interatomic mechanical models (linear and exponential) for the stretching–compression behavior of the carbon–carbon atomic bond have been used. The numerical results show that the homogenized asymptotic model is a good approximation of the full discrete model. The former has the advantage of requiring very little computational resources whereas the latter requires a huge computational cost hence limiting its application to structures with a small number of atoms and atomic bonds. Moreover, the obtained results show that the axial Young’s modulus is almost insensitive to the radius and the chirality but it is very sensitive to the choice of the interatomic mechanical models and their corresponding elastic constants. The linear interatomic model seems to be invalid when used for large deformation of the nanotubes, i.e. it is valid only for small displacements. On the other hand, and most importantly, the stress–strain curves obtained from the homogenized model using the exponential interatomic potential show that the nanotube type determines completely its mechanical characteristics. For instance, the tensile strength and elongation at break are the largest in the case of the armchair nanotubes and the smallest in the case of the zigzag nanotubes.
We consider a mathematical model which describes the equilibrium of an elastic body in frictional contact with an obstacle. The contact is modelled with normal compliance and unilateral constraint, associated with a slip-dependent version of Coulomb’s law of dry friction. We present a weak formulation of the problem, then we state and prove an existence and uniqueness result of the solution. The proof is based on arguments of elliptic quasivariational inequalities. We also study the finite element approximations of the problem and derive error estimates. Finally, we provide numerical simulations which illustrate both the behaviour of the solution related to the frictional contact conditions and the convergence order of the error estimates.
A transversely isotropic visco-hyperelastic constitutive model is provided for soft tissues, which accounts for large deformations, high strain rates, and short-term memory effects. In the first part, a constitutive model for quasi-static deformations of soft tissues is presented, in which a soft tissue is simulated as a transversely isotropic hyperelastic material composed of a matrix and reinforcing fibers. The strain energy density function for the soft tissue is additively decomposed into two terms: a neo-Hookean function for the base matrix, and a polyconvex polynomial function of four invariants for the fibers. A comparison with existing experimental data for porcine brain tissues and bovine pericardium shows that this new model can well represent the quasi-static mechanical behavior of soft tissues. In the second part, a viscous potential is proposed to describe the rate-dependent short-term memory effects, resulting in a visco-hyperelastic constitutive model. This model is tested for a range of strain rates from 0.1 /s to 90 /s and for multiple loading scenarios based on available experimental data for porcine and human brain tissues. The model can be applied to other soft tissues by using different values of material and fitting parameters.
A thin rectangular plate is modelled as an (initially flat) shell. Following Koiter, the two fundamental forms of the deformed middle surface are then used to define the strain measures of the body. On the middle surface of the plate two local coordinates are introduced: we will call them longitudinal and transversal, respectively. It is assumed that the components of the displacement field which characterize the middle surface kinematics can be expressed as a product of two functions: one defined along the longitudinal coordinate and one defined along the transversal coordinate. Given an explicit expression of the latter functions, the 2D plate fields are reduced to 1D ones. The functions of the transversal coordinate are chosen so that the stretch along it together with the membrane shear vanish. It is worth noting that the linearization of these constraints leads to the well-known Vlasov’s assumptions. It is shown that by modelling each side of a thin walled beam as a 1D continuum, the entire assembly can be reduced to a 1D model as well. This procedure gives rise to an hyperelastic 1D beam model in which at least the warping effect is taken into account. The main features of the model are shown by means of some simple applications.
The present work is concerned with the thermoelastic interactions in a linear, homogeneous, unbounded medium with a cylindrical cavity in the context of the theory of two-temperature thermoelasticity with two relaxation parameters. This theory takes into account that heat conduction in deformable bodies depends on two different temperatures: the conductive temperature and the thermodynamic temperature. The surface of the cavity is assumed to be stress free and subjected to a thermal shock. The problem is formulated on the basis of a thermoelastic model with two relaxation parameters proposed by Green and Lindsay and the two-temperature model suggested by Youssef in a unified way in order to make a detailed comparison between the two models by deriving the short-time approximated solutions for fields analytically. The results are also compared with the earlier findings. Significant dissimilarities among different models of thermoelasticity are pointed out. This study highlights a very significant feature of a two-temperature thermoelastic model.
A new representation for viscoelastic functions, the tensor relaxation-creep duality representation, is introduced. The derivation of a tensor time-differential constitutive equation for anisotropic viscoelastic materials using this new presentation is presented. The relaxation-creep duality characteristic ingrained in the new representation enables the interconversion of viscoelastic functions, which is not possible with the conventional Prony series representation of viscoelastic functions. The new representation therefore offers a better representation of the physics of viscoelasticity leading to a reduced number of viscoelastic parameters required to describe a viscoelastic function. The new representation has been demonstrated on two anisotropic viscoelastic crystallographic systems: (i) the symmetric systems with material- and time-independent eigenvectors and (ii) the symmetric systems with material-dependent but time-independent eigenvectors.
The purpose of the present work is to give a continuum model that can capture bending effects for free-standing graphene monolayers taking material’s symmetry properly into account. Starting from the discrete picture of graphene modelled as a hexagonal 2-lattice, we give the arithmetic symmetries. Confined to weak transformation neighbourhoods one is able to work with the geometric symmetry group. Use of the Cauchy–Born rule allows the transition from the discrete case to the continuum case. At the continuum level we use a surface energy that depends on an in-plane strain measure, the curvature tensor and the shift vector. Dependence of the energy on the curvature tensor allows for incorporating bending effects into the model. Dependence on the shift vector is motivated by the fact that discretely graphene is a 2-lattice. We lay down the complete and irreducible set of invariants for this surface energy amenable to available representation theory. This way we obtain the expression for the surface stress as well as the surface couple stress tensor, the first being responsible for the in-plane deformations and the second for the out-of-plane motions. Forms for the elasticities of the material are given accompanied by the field equations. The model, in its simplest form, predicts 13 independent scalar variables in the constitutive relations to be observed in experiments. The framework presented is valid for both materially and geometrically nonlinear theories. We also present the case where symmetry changes at the continuum level, without taking into account how energy behaves at the transition regime.
By using the specially constructed Kolosov–Muskhelishvili potentials a concise formulation of 2D elastostatic problems for general regular structures with multiphase nested inclusions is obtained in complex-variable terms. Analytical averaging of the stress/strain fields over the representative cell of the structure gives its effective moduli in the perturbation-like form that was known thus far only for less complicated phase arrangements. These derivations are further extended to prove the existence of the equi-stress nested inclusions under the square symmetry of the structure. In sharp contrast to the one-phase (homogeneous) inclusion, they no longer saturate the attainable Hashin–Shtrikman bounds on the effective bulk modulus, but continue to be a subject by themselves.
A novel approach to analyze kink banding failure in fiber reinforced composites is presented in this paper. By considering a single fiber–matrix unit cell and utilizing a fiber deformation representation that can simultaneously account for instability due to increasing fiber amplitude and fiber rotation, a sequence of events that lead to the formation of a kink band is elucidated. It is shown that peak compressive strength does not necessarily correspond to the onset of yielding (non-linearity) in the matrix and is dictated by the interaction between matrix non-linearity and initial fiber misalignment. Predictions of compressive strength and kink band formation, utilizing the new model, are found to be in good agreement with reported experimental data and associated higher-order finite element analysis.
A thin-walled structure is homogeneously embeddable if it can be obtained by carving it out of a homogeneous material block or, in other words, if it is materially defect-free. Explicit analytic and geometric conditions for the embedded homogeneity of plane linearly elastic beams are derived and discussed. In the geometric setting, a prominent role is played by the properties of the hodograph of uniaxial material tensor fields defined on the beam axis.
We consider an interface crack whose crack faces are coated with a thin reinforcing film of separate elastic material. We obtain asymptotic solutions, which, in the case of plane elasticity, demonstrate that the addition of the thin film effectively eliminates the well-known oscillatory behavior of the displacement and stress fields in the vicinity of the crack tip leading to a strong square-root stress singularity. In the case of anti-plane elasticity, the effect of the reinforcement is to reduce the order of the stress singularity at the crack tip. In addition, we demonstrate that the reinforcement induces a displacement field which is smooth locally and bounded at the crack tip.
Cosserat (micropolar, asymmetric) elasticity can better predict the mechanical behavior of the materials with a characteristic length scale than the classical theory of elasticity. However, the area of fracture modelling in a Cosserat medium is not widely presented in the literature. The simulation of cracks in a Cosserat medium using the eXtended Finite Element Method (XFEM) is presented in this paper. The proper crack tip enrichment of the translational and microrotational fields is important for the robustness and efficiency of the XFEM/Cosserat model. Using the example of an edge crack of the Mode I in this paper, we have shown that the values of the J-integral for the Cosserat solid are higher than those for the equivalent classical elastic solid. The difference becomes significant for the cases of materials with strong micropolar properties. The dependence of the fracture behavior of the crack of different sizes on the Cosserat elastic constants was found to be significant. Therefore, it is recommended that the special tip enrichment is considered for the microrotational field, and a careful analysis of the material parameters is performed in order to model fracture in a Cosserat medium.
The orthogonal polynomial approach has been used to solve the guided wave propagation in structures for about 20 years, but was limited until now to one-dimensional structures, that is, structures with a finite dimension in only one direction, such as horizontally infinite flat plates and axially infinite hollow cylinders. This paper extends the orthogonal polynomial approach to two-dimensional structures, that is, structures with finite dimensions in two directions, with illustration on the multilayered rectangular rod. Through numerical comparison with the available results, the validity of the extended polynomial approach is illustrated. The dispersion curves and displacement distributions of various layered rectangular rods are obtained to show their guided wave characteristics.
This report looks at the published literary sources on methods and approaches, which are based on fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) to solve the continuous and discrete mechanics problems. The problems solutions and comparative analysis results of fractional and classical models are presented. The report’s goal is to show an efficiency of using fractional calculus to describe the mechanical processes.
In this paper, an analytical solution for the displacement components and electric potential function is proposed for a piezothermoelastic long solid cylinder under general axisymmetric loadings. Governing equations, including thermal effects, for transversely isotropic piezoelectric material are presented in a cylindrical coordinate system. The most general form of the complementary and particular solutions for the displacement components and electric potential function are obtained in terms of a cosine and sinuous Fourier integral, as well as the temperature distribution function. Then, the general form of stress distributions and electric field are obtained using the constitutive equations. Finally, under some specific type of thermal boundary conditions, i.e. when the cylinder surface is under the heat convection of a flowing fluid or prescribed temperatures, as well as external pressure and electric charge, related stress and electric fields are obtained.
It is the aim of the paper to present a new point of view on rotational elasticity in a nonlinear setting using orthogonal matrices. The proposed model, in the linear approximation, can be compared to the well-known equilibrium equations of static linear elasticity. An appropriate kinetic energy is identified, and we present a dynamical model of rotational elasticity. The propagation of elastic waves in such a medium is studied and we find two classes of waves, transversal rotational waves and longitudinal rotational waves, both of which are solutions of the nonlinear partial differential equations. For certain parameter choices, the transversal wave velocity can be greater than the longitudinal wave velocity. Moreover, parameter ranges are identified where the model describes an auxetic material. However, in all cases the potential energy functional is positive definite. Finally, we couple the rotational waves to linear elastic waves to study the behaviour of the coupled system. We find wave-like solutions to the coupled equations and can visualise our results with the help of suitable figures.
A curved beam element based on the Timoshenko model and non-uniform rational B-splines (NURBS) interpolation both for geometry and displacements is presented. Such an element can be used to suitably analyse plane-curved beams and arches. Some numerical results will explore the effectiveness and accuracy of this novel method by comparing its performance with those of some accurate finite elements proposed in the technical literature, and also with analytical solutions: for the cases where such closed-form solutions were not available in the literature, they have been computed by exact integration of the governing differential equations. It is shown that the presented element is almost insensitive to both membrane- and shear-locking, and that such phenomena can be easily controlled by properly choosing the number of elements or the NURBS degree.
This paper addresses the problem of plate bending for a square weakened with a full-strength hole including the origin of coordinates. The vertices of the square lie on the coordinate axes and its neighbourhoods are cut by the same smooth full-strength arcs which are symmetric with respect to the coordinate axes. Rigid bars are attached to each component of the broken line of the outer boundary of the plate. The plate bends under the action of concentrated moments applied to the middle points of the bars. The unknown part of the boundary is free from external forces. Using the methods of complex analysis, the unknown part of the boundary is found under the condition that the tangential–normal moment takes a constant value. Numerical analysis is performed and the corresponding graphs are constructed.
In this paper, the propagation behavior of the surface Bleustein–Gulyaev (B-G) waves in a piezoelectric layered half-space is investigated. The governing equations of the coupled waves are obtained. The boundary conditions are assumed in such a way that the displacements, shear stress, electric potential, and electric displacements are continuous across the interface between the layer and the substrate together with the traction-free boundary at the surface of the layer. The electrically open and short conditions at the surface are adopted to solve the problem. The phase velocity is numerically calculated for the electric open and short cases for different thicknesses of the layer and wave number. The phase velocity equation of the B-G wave is obtained by an analytical technique when a layered half-space with an identical piezoelectric layer and substrate but with opposite polarization is utilized. The electromechanical coupling factor in the layered piezoelectric structures is discussed. The results obtained in this paper will be very useful for the engineering application of B-G waves.
In this paper the linearized equations of motion in multibody dynamics are derived. Explicit expressions for the coefficient matrices are presented and given their physical interpretations. The equations of motion are presented in terms of the mechanical stiffness, its adjoint and the associated differential operators. It is demonstrated how the adjoint matrix may be used to find solutions to the associated algebraic eigenvalue problem. The case of multiple roots of the characteristic equation will result in a generalized eigenvalue problem involving the notion of a Jordan chain. Qualitative properties of the spectrum are derived without explicitly solving the characteristic equation. Finally, the mechanical admittance and its spectral representations are discussed.
In this paper we prove the existence and uniqueness of the weak solution for a dynamic thermoviscoelastic problem describing contact problem between the body and foundation. The process is dynamic, the material behaviour is described by nonlinear viscoelastic law, strongly coupled with the thermal effects. The contact is modelled by nonmonotone subdifferential boundary conditions. The mechanical damage of the material is described by a parabolic equation. We use recent results from the theory of hemivariational inequailities and fixed point theorems.
This paper illustrates Gabrio Piola’s view on continuum models, especially how contact actions are defined. Piola presented his mechanical theory before the 1850s, in an attempt to generalize Lagrange’s analytical mechanics. He conceived, among the rest, an ideal state for physical bodies (which nowadays we would call a natural state), a very general set of what we would now call state variables, and obtained balance equations via the superposition of a rigid infinitesimal motion on the present configuration. These views look quite modern even today and seem to historically precede among other things the introduction of structured continua.
We propose a model for thermo-elastic beams, consistent with the theory of linear three-dimensional thermo-elasticity and deduced by a suitable version of the principle of virtual powers. Dimensional reduction is achieved by postulating convenient a priori representations for mechanical and thermal displacements, the latter playing the role of an additional kinetic variable. Such representations are regarded as internal constraints, some involving the first, others, the second, gradient of deformation and thermal displacements; these constraints are maintained by reactive stresses and hyper-stresses of the type occurring in non-simple elastic materials of grade two, and by reactive entropy influxes and hyper-influxes.
In this work the problem of a thermoelastic thick plate that is of infinite extent and finite thickness with a permeating substance in contact with one of the bounding planes is considered in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The upper surface is taken to be traction free, subjected to a time-dependent thermal shock and the chemical potential, also assumed to be a known function of time. The lower surface of the plate is laid on a rigid foundation, which is thermally insulating. The Laplace and Hankel transform techniques are used. The analytical solution in the transform domain is obtained by using a direct approach. The inverse of the double transform is obtained by using a numerical method based on Fourier expansion techniques. The temperature, displacement, stress and concentration, as well as the chemical potential, are obtained. Numerical computations are carried out and represented graphically.
The influence of periodical imperfections on the buckling load value of shallow elastic closed conical shells undergoing external pressure was investigated by numerical analysis. Results are compared with experimental data. It is established that initial imperfections significantly influence the buckling pressure value only for large imperfection amplitude (an order of magnitude more than the shell thickness). The existence is established of the effect of a ‘static resonance’.
We study the stress field of a three-phase composite in which an internal elliptical inclusion is bonded to the surrounding matrix through an interphase layer. The linearly elastic materials occupying both the inclusion and the matrix are generally anisotropic, whereas the interphase layer is made of an isotropic elastic material. The two interfaces of the three-phase composite are confocal ellipses. Two conditions are found that ensure that the internal in-plane and anti-plane stress field is uniform. When these conditions are met, the mean stress within the isotropic interphase layer is also uniform. A real form expression of the internal uniform stress field inside the inclusion is derived. Several examples are presented to demonstrate and validate the obtained results.
A series of molecular-dynamics simulations of the classic Taylor impact test is performed by using a flat-ended monocrystalline nanoscale projectile made of the Lennard-Jones two-dimensional solid. The nanoprojectile striking velocities range from 0.75 to 7 km/s. These atomistic simulations offer insight into nature of fragment distributions and evolution of state parameters. According to the simulation results, the cumulative distribution of fragment sizes in the course of this non-homogeneous fragmentation process for hypervelocity impacts appears to be well represented by the bimodal-exponential distribution commonly observed during high-energy uniform fragmentation events. For more moderate impact velocities, the cumulative distribution of fragment sizes, in addition to the bimodal-exponential part, exhibits a large-fragment tail. Temporal evolutions on instantaneous kinetic temperature, stress and strain invariants are presented and discussed. Scaling relations between temperature/temperature rate and kinematic rates of deformation are suggested.
We present a three-dimensional constitutive model for NiTi polycrystalline shape memory alloys exhibiting transformations between three solid phases (austenite, R-phase, martensite). The ‘full modelling sequence’ comprised of formulation of modelling assumptions, construction of the model, mathematical analysis and numerical implementation and validation is presented. Namely, by formulating micromechanics-inspired modelling assumptions we concentrate on describing the dissipation mechanism: a refined form of this description makes our model especially useful for complex loading paths. We then embed the model into the so-called energetic framework (extended to our case) while taking advantage of describing the dissipation mechanism through the so-called dissipation distance. We prove the existence of energetic solutions to our model by a backward Euler scheme. This is then implemented into finite element software, and numerical simulations compared with experiments are also presented.
Theories of second-gradient elastic materials have been constructed either through the notion of an interstitial energy flux, an additional term to be included in the balance of energy, or through an appropriate extension of the power of internal stresses via a hyperstress applied to the second-order gradient of velocity. This article provides a critical comparison of such apparently alternative points of view and shows how, in some cases, they can be reconciled with each other through an appropriate choice for the expression of the working of the internal stresses. This is accomplished by allowing the hyperstress to be non-symmetric.
This work is aimed at emphasising the relationship between metric and deformation, under the light of a novel formalism for the Polar Decomposition Theorem. All results are first presented in the classical formalism of Cauchy’s celebrated theorem, and then in the proposed alternative formalism. Although the latter requires a little more work to be established, it allows for directly defining all strain tensors as "covariant", i.e. with both feet being covectors. Emphasis is also placed on how, in the absence of the metric structure, the available mathematical tools are restricted to the deformation gradient alone. Along with these main results, and in the didactical intention that permeates this work, several hints are given, which could be useful in teaching Continuum Mechanics, e.g. the rigorous definition of the determinant of the deformation gradient in Riemannian manifolds, and a caveat on the definition of the spatial Hencky logarithmic strain. The setting is that of modern Continuum Mechanics, based on the description given by Differential Geometry in terms of differentiable manifolds. However, passing to the simpler case of affine spaces takes almost no effort, paying attention to keeping the distinction between vectors and covectors, and therefore allowing the matrices representing the metric tensors to differ from the unit matrix.
This paper presents a general model description for the contact of surface tension driven systems. The example system of a liquid droplet in contact with a deformable solid substrate is considered. This can be easily modified to consider two liquids or two solids in contact. The surface kinematics, essential to the modeling of surface tension, are described here in curvilinear coordinates. In particular modeling focus are the contact conditions at the contact boundary, where a wetting ridge may develop. It is shown that in the case of quasi-statics and hyperelasticity the governing equations can be derived from a global potential that accounts for contact as well as the energy storage within the bulk and surface domains. Altogether, 21 Euler–Lagrange equations are derived in this manner. Apart from these strong form equations, the governing weak form as well as its complete linearization, which are required for computational methods, are also discussed. It is shown that the governing equations can be further simplified into a reduced set of equations that are then suitable for an efficient computational implementation of the system. Computational solution methods are not discussed here, as the present focus is on the theory and its implications. A few remarks on analytical solutions, as well as a simple computational example, are given nonetheless. An auxiliary benefit of this work is a summary of the variation and linearization of the kinematical and constitutive equations of the system.
This paper considers effective strain tensors within the context of linear elastic equilibrium theory. The elastic properties of structured materials are often averaged over subvolumes of various scales inside the material. For subvolumes smaller than a representative volume element, simple volume-averaging of the stress and strain may not preserve the elastic energy. We introduce an averaging process which preserves the energy for all boundary conditions. This averaging process emphasizes the parts of the material which carry the most stress. Here the effective strain is weighted by the local stress, and can be interpreted as an average strain over all paths taken by loads and forces through the volume. This alternative effective strain may be especially appropriate for materials with voids, such as foams and granular matter, as the averaging only involves the material itself. For uniform boundary conditions the weighted strain matches the volume-averaged strain.
This paper investigates the properties of this weighted strain tensor. First, for each path taken by loads and forces through the volume we can measure a net length as well as a net extension due to the linear deformation. The weighted effective strain equals the ratio of average length to average extension, where the averaging is over all possible force paths. Thus this method provides a connection to load path analysis.
Secondly, even when the average rotation within the subvolume is zero, there may be local fluctuations in the rotation field. These rotations can act like a mechanism, transferring elastic energy between boundaries or degrees of freedom. The effective strain defined here highlights this mechanism effect.
A general plate model based on the peridynamic theory of solid mechanics is presented. The model is derived as a two-dimensional approximation of the three-dimensional bond-based theory of peridynamics via an asymptotic analysis. The resulting plate theory is demonstrated using a specially designed peridynamics code to simulate the fracture of a brittle plate with a central crack under tensile loading.
Gabrio Piola’s scientific papers have been underestimated in mathematical physics literature. Indeed, a careful reading of them proves that they are original, deep and far-reaching. Actually, even if his contribution to the mechanical sciences is not completely ignored, one can undoubtedly say that the greatest part of his novel contributions to mechanics, although having provided a great impetus to and substantial influence on the work of many preeminent mechanicians, is in fact generally ignored. It has to be remarked that authors Capecchi and Ruta dedicated many efforts to the aim of unveiling the true value of Gabrio Piola as a scientist; however, some deep parts of his scientific results remain not yet sufficiently illustrated. Our aim is to prove that non-local and higher-gradient continuum mechanics were conceived already in Piola’s works and to try to explain the reasons for the unfortunate circumstance which caused the erasure of the memory of this aspect of Piola’s contribution. Some relevant differential relationships obtained in Piola (Memoria intorno alle equazioni fondamentali del movimento di corpi qualsivogliono considerati secondo la naturale loro forma e costituzione, 1846) are carefully discussed, as they are still too often ignored in the continuum mechanics literature and can be considered the starting point of Levi-Civita’s theory of connection for Riemannian manifolds.
The most general situation of the reinforcement of a plate with multiple holes by several patches is considered. There is no restriction on the number and the location of the patches. Two types of patch attachment are considered: only along the boundary of the patch or both along the boundary of the patch and the boundaries of the holes which this patch covers. The unattached boundaries of the holes may be loaded with given in-plane stresses. The mechanical problem is reduced to a system of singular integral equations which can be further reduced to a system of Fredholm equations. A new numerical procedure for the solution of the system of singular integral equations is proposed in this paper. It is demonstrated on numerical examples that this procedure has advantages in the case of multiple patches and holes and allows achievement of better numerical convergence with less computational effort.
This paper considers the damping induced on a single degree-of-freedom system when it is coupled at one end of a wave-guide in which waves are radiated producing an energy loss in the oscillator motion that appears as a damping effect. In general, the whole system is described by the equation of the motion of the harmonic oscillator coupled with the wave equation of the propagation field. Hiding the variable that describes the wave motion and expressing it in terms of the oscillator vibration, a new equation for the oscillator is determined. In general, a nonconventional damping effect is born from the coupling terms. This paper examines cases in which the induced damping effect is of fractional-derivative type. This point of view produces physical examples of the way that simple mechanical structural systems, familiar to engineers, can exhibit fractional damping, a concept that does not always have a clear physical interpretation.
The governing equations for a micromorphic theory of electromagneto-elastic dielectrics are derived by a variational formulation. Balance equations and boundary conditions are obtained assuming the internal energy as dependent on macro and micro-strain variables. A micropolar linear model is derived and the evolution equations for dipole and quadrupole are exploited to arrive at an expression for the polarization density. The present model is applied to the simple shear static problem for an isotropic dielectric layer subject to an external field. The resulting shear displacement and electric potential noticeably differ from the classical elastic case.
The objective of this paper is to use some historical instances to explain in detail the meaning of mathematical physics theories in selected historical periods. The paper presents, in order, the first instances of applications of mathematics to physics, the first appearance of something resembling modern mathematical physics and a particular kind of mathematical physics theory, called rational mechanics. At the end of the paper, epistemological considerations clarify the difference between physical, mathematical, and mathematical physics theories.
We first give a complete analysis of the dispersion relation for travelling waves propagating in a pre-stressed hyperelastic membrane tube containing a uniform flow. We present an exact formula for the so-called pulse wave velocity, and demonstrate that as any pre-stress parameter is increased gradually, localized bulging would always occur before a superimposed small-amplitude travelling wave starts to grow exponentially. We then study the stability of weakly and fully nonlinear localized bulging solutions that may exist in such a fluid-filled hyperelastic membrane tube. Previous studies have shown that such localized standing waves are unstable under pressure control in the absence of a mean-flow, whether the fluid inertia is taken into account or not. Stability of such localized aneurysm-type solutions is desired when aneurysm formation in human arteries is modelled as a bifurcation phenomenon. It is shown that in the near-critical regime axisymmetric perturbations are governed by the Korteweg–de Vries equation, and so the associated (weakly nonlinear) aneurysm solutions are (orbitally) stable with respect to axisymmetric perturbations. Stability of the fully nonlinear aneurysm solutions are studied numerically using the Evans function method. It is found that for each wall–fluid density ratio there exists a critical mean-flow speed above which no axisymmetric unstable modes can be found, which implies that a fully nonlinear aneurysm solution may be completely stabilized by a mean flow.
We study well-posedness for the relaxed linear elastic micromorphic continuum model with symmetric Cauchy force-stresses and curvature contribution depending only on the micro-dislocation tensor. In contrast to classical micromorphic models our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. Another interesting feature concerns the prescription of boundary values for the micro-distortion field: only tangential traces may be determined which are weaker than the usual strong anchoring boundary condition. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch, and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes.
We present a mathematical model of structural reorganisation in a fibre-reinforced composite material in which the fibres are oriented statistically, i.e. obey a probability distribution of orientation. Such a composite material exemplifies a biological tissue (e.g. articular cartilage or a blood vessel) whose soft matrix is reinforced by collagen fibres. The structural reorganisation of the composite takes place as fibres reorient, in response to mechanical stimuli, in order to optimise the stress distribution in the tissue. Our mathematical model is based on the Principle of Virtual Powers and the study of dissipation. Besides incompressibility, our main hypothesis is that the composite is characterised by a probability density distribution that measures the probability of finding a family of fibres aligned along a given direction at a fixed material point. Under this assumption, we describe the reorientation of fibres as the evolution of the most probable direction along which the fibres are aligned. To test our theory, we compare our simulations of a benchmark problem with selected results taken from the literature.
We derive a one-dimensional model for the displacement of a longitudinally stressed elastic rod starting from a cylindrical three-dimensional linearized prestressed elastic body with a small diameter. The prestress is due to the prior elastic deformation of an isotropic, homogeneous, elastic body. We deduce the scaling of forces by a formal asymptotic expansion. Then we prove that the family of solutions of three-dimensional problems converges to the solution of the prestressed rod model. The energy of the limit model is the sum of the classical bending energy of the elastic rod and the energy of the prestressed string.
This contribution is the third part in a series devoted to the fundamental link between discrete particle systems and continuum descriptions. The basis for such a link is the postulation of the primary continuum fields such as density and kinetic energy in terms of atomistic quantities using space and probability averaging.
In this part, solutions to the flux quantities (stress, couple stress, and heat flux), which arise in the balance laws of linear and angular momentum, and energy are discussed based on the Noll’s lemma. We show especially that the expression for the stress is not unique. Integrals of all the fluxes over space are derived. It is shown that the integral of both the microscopic Noll–Murdoch and Hardy couple stresses (more precisely their potential part) equates to zero. Space integrals of the Hardy and the Noll–Murdoch Cauchy stress are equal and symmetric even though the local Noll–Murdoch Cauchy stress is not symmetric. Integral expression for the linear momentum flux and the explicit heat flux are compared to the virial pressure and the Green–Kubo expression for the heat flux, respectively.
It is proven that in the case when the Dirac delta distribution is used as kernel for spatial averaging, the Hardy and the Noll–Murdoch solution for all fluxes coincide.
The heat fluxes resulting from both the so-called explicit and implicit approaches are obtained and compared for the localized case. We demonstrate that the spatial averaging of the localized heat flux obtained from the implicit approach does not equate to the expression obtained using a general averaging kernel. In contrast this happens to be true for the linear momentum flux, i.e. the Cauchy stress.
A homogenization framework is developed that accounts for the effect of size at the micro- or nanoscale. This is achieved by endowing the interfaces of the micro- or nanoscopic features with their own independent structure, using the theory of surface elasticity. Following a standard small-strain approach for the microscopic deformation in terms of the macroscopic strain tensor, a Hill-type averaging condition is used to link the two scales. A procedure for determining overall effective properties in the case of composites with elastic components and elastic material interfaces is presented. A special example of multilayered composites demonstrates the correlation between a material interface and a very thin interphase layer.
The classical three-dimensional Cerruti problem of an isotropic half-space subjected to a concentrated tangential load on its surface is revisited here in the context of dipolar gradient elasticity. This generalized continuum theory encompasses the analytical possibility of size effects, which are absent in the classical theory, and has proven to be very successful in modelling materials with complex microstructure. The dipolar gradient elasticity theory assumes a strain-energy density function, which besides its dependence upon the standard strain terms, depends also on strain gradients. In this way, this theory can be viewed as a first-step extension of classical elasticity. The solution method is based on integral transforms and is exact. Of special importance is the behaviour of the new solution near to the point of application of the load where pathological singularities exist in the classical solution (based on the standard theory). The present results show departure from the ones predicted by the classical elasticity theory. Indeed, continuous and bounded displacements are found at the point of application of the load. Such a behaviour of the displacement field is, of course, more natural than the singular behaviour present in the classical solution.
In the setting of an n-dimensional Euclidean space, the duality between velocity fields on the class of admissible bodies and Cauchy fluxes is studied using tools from geometric measure theory. A generalized Cauchy flux theory is obtained for sets whose measure theoretic boundaries may be as irregular as flat (n – 1)-chains. Initially, bodies are modeled as normal n-currents induced by sets of finite perimeter. A configuration space comprising Lipschitz embeddings induces virtual velocities given by locally Lipschitz mappings. A Cauchy flux is defined as a real valued function on the Cartesian product of (n – 1)-currents and locally Lipschitz mappings. A version of Cauchy’s postulates implies that a Cauchy flux may be uniquely extended to an n-tuple of flat (n – 1)-cochains. Thus, the class of admissible bodies is extended to include flat n-chains and a generalized form of the principle of virtual power is presented. Wolfe’s representation theorem for flat cochains enables the identification of stress as an n-tuple of flat (n – 1)-forms representing the flat (n – 1)-cochains associated with the Cauchy flux.
This short note addresses a misconception that seems to be prevalent with regard to the use of compatibility conditions for the strains in both the linearized and non-linear theories of elasticity, when one uses a "stress-based" approach.
We consider a residually stressed plate-like body having the shape of a cylinder of cross-section and thickness h, subjected to a system of loads proportional to a positive multiplier . We look for the smallest value of the multiplier such that the plate buckles, the so-called critical multiplier. The critical multiplier is computed by minimizing a functional whose domain of definition is a collection of vector fields defined in the three-dimensional region = x (–h/2,+h/2). We let -> 0 and we show that if the residual stress and the incremental stress induced by the applied loads scale with in a suitable manner, then the critical multiplier converges to a limit that can be computed by minimizing a functional whose domain is a collection of scalar fields defined on the two-dimensional region . In the special case of null residual stress, the Euler–Lagrange equations associated to this functional coincide with the classical equations governing plate buckling.
An adhesive unilateral contact between visco-elastic bodies at small strains and in a Kelvin–Voigt rheology is scrutinized, neglecting inertia. The flow-rule for debonding the adhesive is considered rate independent, unidirectional, and non-associative due to dependence on the mixity of modes of delamination, namely Mode I (opening) needs (= dissipates) less energy than Mode II (shearing). Such mode-mixity dependence of delamination is a very pronounced (and experimentally confirmed) phenomenon typically considered in engineering models. An efficient semi-implicit-in-time FEM discretization leading to recursive quadratic mathematical programs is devised. Its convergence and thus the existence of weak solutions is proved. Computational experiments implemented by BEM illustrate the modeling aspects and the numerical efficiency of the discretization.
The aim of this paper is to study the existence of solutions and some approximations for a class of implicit evolution variational inequalities that represents a generalization of several quasistatic contact problems in elasticity. Using appropriate estimates for the incremental solutions, the existence of a continuous solution and convergence results are proved for some corresponding internal approximation and backward difference scheme. To solve the fully discrete problems, general additive subspace correction algorithms are considered, for which global convergence is proved and some error estimates are established.
A Griffith crack centrally bridged by a single grain in piezoelectric ceramics is modeled and investigated theoretically. Applying the Fourier transform method and the semi-permeable crack-face boundary conditions, various fracture parameters involving the stress intensity factors, the jumps of electrical potential and elastic displacement across the crack, and the energy release rate are given in closed forms. The total force and the electric potential acting on the bridging grain are obtained explicitly. Numerical results are calculated to show the effects of applied electrical fields and the permittivity of crack interior on the fracture parameters of concern. Under the assumption that the crack growth depends on the pull-out of the bridging grain, the obtained results reveal that applied positive electrical loadings enhance the crack growth and applied negative ones retard the crack growth. When one measures the crack length, applied positive electrical loadings will decrease the crack length and applied negative ones may increase the crack length. The above phenomena show that the crack-face bridging grains have contributions to various effects of applied electrical fields on the crack growth of piezoelectric solids.
We provide a full analytical solution for the problem of a generalized plane strain circular hollow cylinder subjected to axisymmetric loading conditions. The matrix of the cylinder obeys a micromorphic plasticity theory as proposed by Gologanu, Leblond, Perrin and Devaux. The solution gives explicit expressions for the displacement, the strain and its gradient, as well as the ordinary and generalized stress fields. The newly derived solution satisfies the equilibrium equations and is shown to be an extension of the solution of the same model problem using (von Mises) classical plasticity theory.
Through the analysis of the wave propagation in infinite two-dimensional periodic frame materials, this paper illustrates the complexity of their dynamic behavior. Assuming the frame size is small compared to the wavelength, the homogenization method of periodic discrete media coupled with normalization is used to identify the macroscopic behavior at the leading order. The method is applied on a frame material with the vertical elements stiffer than the horizontal elements. Such a material is highly anisotropic and presents a large contrast between the rigidities of the possible mechanisms. Thus the waves associated with different kinematics appear in different frequency ranges. Moreover, the stiffer elements can deform in bending at the macroscopic scale. The equivalent continuum is a second-grade medium at the leading order and shear waves can be dispersive. A criterion is proposed to easily determine when this bending effect has to be taken into account.
Simplified non-linear dynamical equations of circular cylindrical shells are obtained on the basis of asymptotic analysis. The original partial differential equations (PDEs), which have eighth order in the longitudinal coordinate, are replaced by two sets of equations that have fourth order on this coordinate. The first set contains non-linear and dynamic PDEs. They describe the oscillations in the inner region of the shell. For small non-linearities, they coincide with the well-known linear semi-membrane dynamical equations. Ordinary differential equations of the edge effect allow one to satisfy all the boundary conditions. They are linear and quasistatic. An important question about the boundary conditions for the simplified equations is considered.We also discuss the solution of the obtained boundary value problems.
The monotonicity of the linear viscoelastic functions, namely, the shear creep compliance, the Young’s relaxation modulus, the stretch creep compliance, the P-wave relaxation modulus, the Lamé’s first function, and the time-dependent Poisson’s ratio, were examined analytically and numerically. It was shown that both the Lamé’s first function and time-dependent Poisson’s ratio can be non-monotonic. Furthermore, in contrast to the reports by other researchers, the values of the time-dependent Poisson’s ratio were found to be bounded by the limits between – 1 and 0.5 after the physical constraints of the bulk and shear relaxation moduli are taken into account.
We present a model of uterine contractions that consists of tissue mechanics, electrical activity, and intrauterine pressure. Uterine contractions were simulated to measure the impact that ±3% changes in mechanical and electrical properties have on peak intrauterine pressure and the duration above 90% peak pressure. Peak pressure was overwhelmingly affected by changes in one mechanical property (maximum stress generated from cellular contraction), whereas duration above 90% peak pressure was overwhelmingly affected by changes in two electrical properties (the recovery rate and activation threshold of the action potential). In contrast, changes in other properties (e.g. pacemaker location and diffusion coefficient) had an impact on intrauterine pressure one order of magnitude smaller. Additionally, intrauterine pressure was strengthened by decreasing the fraction of fasciculi aligned in the longitudinal rather than the circumferential direction.
In this paper the algebraic structure of the isotropic nth-order gradient elasticity is investigated. In the classical isotropic elasticity it is well known that the constitutive relation can be broken down into two uncoupled relations between the elementary part of the strain and the stress tensors (deviatoric and spherical). In this paper we demonstrate that this result can not be generalized because in 2nd-order isotropic elasticity there exist couplings between elementary parts of higher-order strain and stress tensors. Therefore, and in certain way, nth-order isotropic elasticity have the same kind of algebraic structure as anisotropic classical elasticity. This structure is investigated in the case of 2nd-order isotropic elasticity, and moduli characterizing the behavior are provided.
In this paper several damage equations are analysed with respect to their properties at damage initialization. This is particularly important for soft biological tissues since two different loading regimes have to be clearly distinguished: the physiological domain where no damage evolution should be considered and the supra-physiological domain where damage evolves. At the transition between these two domains the behaviour of different damage models may influence the convergence of the Newton iteration when solving, for example, nonlinear finite element problems. It is shown that the model proposed by Balzani et al. (Comput Meth Appl Mech Eng 2012; 213–216: 139–151) a priori ensures smooth tangent moduli. In addition to that, a new damage function is proposed able to describe a slow damage evolution at damage initialization also providing smooth tangent moduli. Using this new damage function the approach given by Balzani et al. (Acta Biomater 2006; 2(6): 609–618) can also be modified such that smooth tangent moduli are guaranteed. Numerical analyses of a circumferentially overstretched artery are performed and show that no convergence problems are observed at the transition from the undamaged to the damaged domain, even when a model is used that has non-smooth tangent moduli.
In conventional presentations of continuum mechanics, the reference placement of a solid body is taken to be independent of the observer. Furthermore, the Euclidean spaces associated with different frames of reference are taken to be identical. In this paper these assumption are abandoned and different framings, as well as reference placements, will correspond to different Euclidean spaces. This requires a modified formulation of the principle of frame indifference for constitutive response functions. Consequences of this approach are investigated in terms of transformation formulas for kinematical and dynamical quantities and the reduction of constitutive equations for simple materials. A relation between material symmetry groups, relative to different framings, is derived.
We apply the fractional order theory of thermoelasticity to a one-dimensional problem for a spherical cavity subjected to a thermal shock. The predictions of the theory are discussed and compared with those for the generalized theory of thermoelasticity.
The paper concerns the analysis of equilibrium problems for 2D elastic bodies with thin inclusions modeled in the framework of Timoshenko beams. The first focus is on the well-posedness of the model problem in a variational setting. Then delaminations of the inclusions are considered, forming a crack between the elastic body and the inclusion. Nonlinear boundary conditions at the crack faces are considered to prevent a mutual penetration between the faces. The corresponding variational formulations together with weak and strong solutions are discussed. The model contains various physical parameters characterizing the mechanical properties of the inclusion, such as flexural and shear stiffness. The paper provides an asymptotic analysis of such parameters. It is proved that in the limit cases corresponding to infinite and zero rigidity, we obtain rigid inclusions and cracks with the non-penetration conditions, respectively. Finally, exemplary networks of Timoshenko beams are considered as inclusions as well.
We study the possibility of toroidal twist-like bifurcations for an isotropic Levinson–Burgess compressible elastic tube subject to a pure circular shear. We intend to model forms of instability for solids that are analogous to the classical Taylor–Couette patterns observed in the flow of viscous fluids. We first establish that the axisymmetric circular shear deformation is a fundamental solution of the equilibrium problem, and then investigate the possibility that this primary deformation may bifurcate into an axially periodic toroidal twist-like mode by analyzing the related incremental boundary-value problem. The analysis of the bifurcation problem and the evaluation of the critical load are carried out by following a novel effective procedure, based on the Magnus expansion.
The purpose of this paper is to describe the behavior of the elasticities of quartz across the α–β phase transformation at room pressure. In spite of the known dynamic behavior of quartz, we adhere to the prejudice that an equilibrium theory is sufficient for this task, and propose a Landau model based on a three-phase, two-order-parameter polynomial free energy. One of the order parameters, describing a suitable displacement of the oxygen atoms, produces an isostructural transition just above the temperature of the α–β phase transformation. Subsequently, the latter takes place with the joint contribution of the second structural order parameter, associated with another displacement of the oxygens. The free energy consists of the minimal number of monomials necessary to produce the two mentioned transitions and only those; in spite of this minimality, the results from theory appear to reasonably agree with the experimental data. A posteriori, we reconstruct the phase diagram in the plane of the coefficients of the quadratic monomials.
The aim of this contribution is to compute the effective in-plane tension and shear behaviour of textile-like elastic materials, that is, plates or shells with a periodic micro-structure composed of long woven or knitted fibres. The knitting or weaving results in multiple periodic contact between fibres. Mathematically the problem can be formulated as an in-plane elasticity problem defined in a heterogeneous domain with -periodic micro-structure, including multiple microcontact between the structural components, which is described by the Signorini and Tresca-friction contact conditions. The asymptotic analysis and homogenized limit for such problems was recently obtained by Damlamian et al. via periodic unfolding strategy. These results are briefly recalled in the paper. The obtained two-scale algorithm is implemented for some specific textile materials, which are macroscopic shells with a 3-D microstructure including contact. For each macroscopic deformation state, a contact problem in the periodicity cell or representative volume element is solved and the corresponding non-linear macroscopic stress–strain relation is obtained. The results are illustrated by the simulation of woven and knitted textiles.
A novel and rational approach based on Lie analysis is proposed to investigate the mechanical behaviour of materials exhibiting experimental master curves. Our approach relies on the idea that the mechanical response of materials is associated with hidden symmetries; the general objective of this contribution is to reveal those symmetries from measurements and to construct constitutive laws from them. This approach provides a priori two ways of formulating constitutive laws from data as well as the possibility of predicting new master curves and material charts. The first part of the paper is devoted to the presentation of the general methodology. Afterwards, the strategy is applied to the uniaxial creep and rupture behaviour of a Chrome-Molybdene alloy (9Cr1Mo) at different temperatures and stress levels. Constitutive equations for creep and rupture master responses are identified for this alloy and validated on experimental data.
The equations of motion for a beam are derived from the three-dimensional continuum mechanical point of view. The kinematics involve bending, torsion and shearing as well as cross-sectional displacements. The transplacement of the beam from an arbitrary curved reference placement is considered. For slender beams so called tubular coordinates may be used as global coordinates in the reference placement. An explicit geometrical characterization of slender beams is given. The equations of motion for the slender beam are derived and some consequences of the power theorem are presented. Using the principle of virtual power the classical local beam equations are obtained along with equations representing the local balance of bi-momentum. Constitutive assumptions are introduced where the dependence of stress on the natural deformation measures for the beam is obtained by assuming that the beam consists of a St Venant-Kirchhoff elastic material. Simplified stress-strain relations may be obtained using so called torsion free coordinates.
In this paper a stationary action principle is proved to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments. We remark that these fluids are sometimes also called Korteweg–de Vries or Cahn–Allen fluids. In general, continua whose deformation energy depends on the second gradient of placement are called second gradient (or Piola–Toupin, Mindlin, Green–Rivlin, Germain or second grade) continua. In the present paper, a material description for second gradient continua is formulated. A Lagrangian action is introduced in both the material and spatial descriptions and the corresponding Euler–Lagrange equations and boundary conditions are found. These conditions are formulated in terms of an objective deformation energy volume density in two cases: when this energy is assumed to depend on either C and C or on C–1 and C–1, where C is the Cauchy–Green deformation tensor. When particularized to energies which characterize fluid materials, the capillary fluid evolution conditions are recovered. A version of Bernoulli’s law valid for capillary fluids is found and useful kinematic formulas for the present variational formulation are proposed. Historical comments about Gabrio Piola’s contribution to analytical continuum mechanics are also presented.
We study the basic properties of tensor random fields (TRFs) of the wide-sense homogeneous and isotropic kind with generally anisotropic realizations. Working within the constraints of small strains, attention is given to antiplane elasticity, thermal conductivity, classical elasticity and micropolar elasticity, all in quasi-static settings albeit without making any specific statements about the Fourier and Hooke laws. The field equations (such as linear and angular momentum balances and strain–displacement relations) lead to consequences for the respective dependent fields involved. In effect, these consequences are restrictions on the admissible forms of the correlation functions describing the TRFs.
Motivated by the localized nature of elastic instabilities in radially stretched thin annular plates, we investigate the resistance to buckling of such configurations in the case when their mechanical properties are piecewise constant. By considering a plate consisting of two sub-annular regions perfectly bonded together and with different linear elastic properties, the neutral stability envelope corresponding to the case when radial constant displacement fields are applied on the inner and outer edges of the plate is investigated numerically in considerable detail. These results are complemented by an asymptotic reduction strategy that provides a greatly simplified eigenproblem capable of describing the original buckling problem in the limit of very thin plates.
By applying the eigenvalue approach methodology, a general solution scheme has been developed for a one dimensional problem on a fractional order generalized thermoelasticity in a half space containing an instantaneous heat source. The solution has been achieved in closed form in the Laplace transform domain for the perturbed temperature field and other field variables. The analysis on the numerical results and graphs has been presented for the space-time domain.
In this work, a thermomechanically coupled rolling problem with nonlocal contact, Coulomb’s friction and heat exchange boundary conditions, for incompressible rigid-plastic, temperature-, equivalent strain- and strain-rate-dependent materials, is considered. A coupled variational formulation, consisting of a nonlinear variational inequality for the velocity, a nonlinear variational equation for the temperature and an evolution equation for the equivalent strain, is derived. A variable stiffness parameters method is proposed, its convergence is proved and existence and uniqueness results are obtained.
A simple and efficient ‘complete dynamic approach’ is proposed, and named GBT-D, to evaluate a suitable basis of modes for the elastic analysis of thin-walled members in the framework of Generalized Beam Theory (GBT). The basis includes conventional and non-conventional modes, the latter accounting for transverse extension and membrane shear strain of the plate elements forming the cross-section, which identically vanish in the former set. The method relies on the solution of two distinct eigenvalue problems, governing the in-plane and the out-of-plane free oscillations of a segment of a thin-walled beam. Both the eigenvalue problems, differential in origin, and defined on a one-dimensional spatial domain, are transformed into an algebraic problem by means of a discretization carried out at the cross-section middle line. Numerical examples are then presented to outline the ease of use of the proposed method considering a single plate, an open cross-section and a partially closed one. Member analyses are also performed for the simplest boundary conditions, to validate the accuracy of the proposed GBT-D approach against finite element method results and analytical solutions, highlighting the importance in including the non-conventional modes.
This paper is the second part of a series dedicated to reviewing the fundamental link between discrete and continuum formulations, which is established by space averaging followed by probability density averaging. On obtaining the continuum balance laws of mass and linear momentum in the part I, here the balance laws of angular momentum and energy are re-derived from a discrete (atomistic/molecular) description. Different approaches (explicit and implicit) for the consideration of the potential energy are reviewed. Thereby for the explicit approach ambiguous possibilities for the localization of the potential energy are briefly discussed. Thereby we conclude that the explicit approach is preferable from the practical application point of view, however it becomes cumbersome when applied to multi-body interactions systems, whereas the implicit approach has no ambiguity in the localization of the potential energy to each particle and is easily applicable to any multi-body potential. Possible solutions for continuum fluxes (couple stresses, heat flux) are postponed until part III of the series.
The presence of nanoparticles in an elastic solid introduces disturbances that can vary significantly from the ones predicted by classical elasticity. In this study, we were able to determine that nanoscale effects introduced via a coherent interface model can indeed result in complex, non-local displacement and stress fields near the free surface of a substrate. The specific geometry is defined by a spherical nanoparticle near a straight boundary. The system is loaded either through a far-field uniaxial tension or a transformation strain (eigenstrain) in the particle itself. The elastic field can be fully determined using a three-dimensional displacement formulation that incorporates a well-established interface model.
The equilibrium equations of no-tension (masonry-like) bodies are analyzed. Unlike the existing proofs of the existence of the solution by Anzellotti or Giaquinta and Giusti, the present proof does not employ the uniform safe load condition. It is based on the assumption of the absence of a suitably defined collapse mechanisms. The collapse mechanism belongs to a generalized space BD(cl) of displacements of bounded deformation on the closure cl of the body . This generalized displacement can have a jump discontinuity on the boundary of the body and the generalized strain is a measure on the closure of the body (instead of the standard interpretation as a measure supported by the interior). The equilibrium solution, however, belongs to the classical space of displacements of bounded deformation BD().
We study the kinetics of a step propagating along a twin boundary in a cubic lattice undergoing an antiplane shear deformation. To model twinning, we consider a piecewise quadratic double-well interaction potential with respect to one component of the shear strain and harmonic interaction with respect to another. We construct semi-analytical traveling wave solutions that correspond to a steady step propagation and obtain the kinetic relation between the applied stress and the velocity of the step. We show that this relation strongly depends on the width of the spinodal region where the double-well potential is non-convex and on the material anisotropy parameter. In the limiting case when the spinodal region degenerates to a point, we construct new solutions that extend the kinetic relation obtained in the earlier work of Celli, Flytzanis and Ishioka into the low-velocity regime. Numerical simulations suggest stability of some of the obtained solutions, including low-velocity step motion when the spinodal region is sufficiently wide. When the applied stress is above a certain threshold, nucleation and steady propagation of multiple steps are observed.
Fiber alignment in biological tissues is created and maintained by the cells, which respond to mechanical stimuli arising from properties of the surrounding material. This coupling between mechanical anisotropy and tissue remodeling can be modeled in nonlinear elasticity by a fiber-reinforced hyperelastic material where remodeling is represented as the change in fiber orientation. Here, we study analytically a simple model of fiber reorientation in a rectangular elastic tissue reinforced by two symmetrically arranged families of fibers subject to constant external loads. In this model, the fiber direction tends to align with the maximum principal stretch. We characterize the global behaviour of the system for all material parameters and applied loads, and show that provided the fibers are tensile initially, the system converges to a stable equilibrium, which corresponds to either complete or intermediate fiber alignment.
The dislocation density tensor at the macroscale may be obtained by using two seemingly disparate definitions given by Nye and Arsenlis and Parks. Nye’s definition depends on counting the dislocations crossing a Burgers circuit of infinitesimal area at the macroscale, where as Arsenlis and Parks’s definition is defined as an integrated property of dislocations within an infinitesimal volume. In this paper, it is shown that Arsenlis and Parks’ and Nye’s definitions for the dislocation density tensor are equivalent when conditions on the length scales of the spacing and curvature of the dislocation lines are obeyed. It is also shown that the definition by Arsenlis and Parks, which can be easily employed in microscopic dislocation dynamics simulations, follows the fundamental extensive property of the Burgers vector, namely, the total Burgers vector of a Burgers circuit is the sum of Burgers vectors of individual dislocation lines intersecting the circuit.
This paper studies the internal stress field of a three-phase elliptical inclusion that is bonded to an infinite matrix through an interphase layer when the matrix is subjected to a linearly distributed in-plane stress field at infinity. Two conditions are found that ensure that the internal non-uniform stress field is simply a linear function of the two coordinates. For given material and geometric parameters of the composite, these conditions can be considered as two restrictions on the applied non-uniform loadings. When these two conditions are met, elementary-form expressions of the stresses in all the three phases are derived. In particular, it is found that the mean stress within the interphase layer is also a linear function of the coordinates. If the interphase layer and the matrix have the same elastic constants, the satisfaction of the two conditions will result in a harmonic inclusion under a prescribed non-constant field.
A plane problem for a poled transversely isotropic piezoelectric plane cut along two equal collinear straight cracks is considered. It is assumed that the electrical yielding occurs at the continuations of the cracks due to the applied mechanical and electrical loadings. We model these crack continuations as the zones with constant cohesive saturation limit electrical displacement. The Stroh formalism and a complex variable technique are adopted to obtain the analytic solution of the problem. Closed-form expressions are derived for the developed saturation zone length, the crack opening displacement, the crack opening potential drop, the stress intensity factors, and the energy release rate. A qualitative numerical case study is presented for ceramics PZT-4, PZT-5H, and BaTiO3 to study the effects of various parameters as follows: developed saturation zone length and prescribed load, stress intensity factor, energy release rate, and crack opening displacement on crack growth resistance. The energy release rate and the stress intensity factor variations are investigated with respect to the inter-crack distance. The results obtained are presented graphically and discussed.
At any point in space the material properties of the myocardium are characterized as orthotropic, that is, there are three mutually orthogonal axes along which both electrical and mechanical parameters differ. To investigate the role of spatial structural heterogeneity in an orthotropic material, electro-mechanically coupled models of the left ventricle (LV) were used. The implemented models differed in their arrangement of fibers and sheets in the myocardium, but were identical otherwise: (i) a generic homogeneous model, where a rule-based method was applied to assign fiber and sheet orientations, and (ii) a heterogeneous model, where the assignment of the orthotropic tissue structure was based on experimentally obtained fiber/sheet orientations. While both models resulted in pressure–volume loops and metrics of global mechanical function that were qualitatively and quantitatively similar and matched well with experimental data, the predicted deformations were strikingly different between these models, particularly with regard to torsion. Thus, the simulation results strongly suggest that heterogeneous structure properties play an important nonnegligible role in LV mechanics and, consequently, should be accounted for in computational models.
Quasi-static motions are motions for which inertial effects can be neglected, to the first order of approximation. It is crucial to be able to identify the quasi-static regime in order to efficiently formulate constitutive models from standard material characterization test data. A simple non-dimensionalization of the equations of motion for continuous bodies yields non-dimensional parameters which indicate the balance between inertial and material effects. It will be shown that these parameters depend on whether the characterization test is strain- or stress-controlled and on the constitutive model assumed. A rigorous definition of quasi-static behaviour for both strain- and stress-controlled experiments is obtained for elastic solids and a simple form of a viscoelastic solid. Adding a rate dependence to a constitutive model introduces internal time-scales and this complicates the identification of the quasi-static regime. This is especially relevant for biological soft tissue as this tissue is typically modelled as being a non-linearly viscoelastic solid. The results obtained here are applied to some problems in cardiac mechanics and to data obtained from simple shear experiments on porcine brain tissue at high strain rates.
The simple analytical expressions for the effective moduli and related quantities of interest in a planar doubly periodic matrix-inclusion structure previously obtained for only one inclusion in a cell are generalized on a finite number of non-intersecting inclusions each having its own local properties. As before, all phases are treated as linear and isotropic with perfectly bonding along smooth material interfaces. The derivations are performed by the complex variable technique applied to the quasi-periodic Weierstrassian zeta-function. Special attention is given to the equi-stress inclusion shapes (ESSs) where the analytical development can be fully completed. In particular, they are proved to saturate the multi-phase Hashin–Shtrikman bounds on the effective bulk modulus. The necessary condition of the ESSs existence is also found, although the question of whether they really exist is left aside. The results obtained form a basis for further numerical analysis of the attendant direct and optimization problems which are briefly discussed.
The motion of the material particles due to viscoelastic properties of the half-space is considered when the propagating surface wave has an assigned frequency. The detailed examination shows that for Rayleigh waves in classical elasticity as well as for the viscoelastic Rayleigh-type surface waves the sense of the particle path along the ellipse is retrograde on the surface of the half-space and changes into direct only once at depth about one seventh to one quarter of the wavelength. When there is a viscoelastic surface wave that does not satisfy the adopted four criteria for behaviour at infinity (the case of viscoelastic Rayleigh-type surface wave) but only two of them, this additional wave is direct on the surface of the half-space and does not change or may change many times the sense of the particle path along the ellipse with the distance from the stress-free surface of the half-space. In contrast to the elastic case, the ellipse axes in the viscoelastic case are not parallel and orthogonal to the surface of the half-space, respectively. Their orientation depends on the magnitude of the viscous part of the Lamé moduli and tends to a constant at great depth. The numerical computations in the paper refer to some typical values of the complex Lamé moduli and to some real materials.
We develop an elastic–isotropic rod model for superhelical DNA structures where the helical angle is varying as a function of the arc-length. Our motivation for a variable helical angle comes from some experiments and simulations on DNA braids where complex superhelical structures have been observed. The helical solutions are minimizers of a free energy consisting of elastic, entropic and electrostatic terms. These minimizers are obtained within a variational framework where the end-points of the helices are allowed to be variable so that the length of the superhelix is computed as part of the solution. Considering variable curvature solutions brings up the possibility of finding more complex DNA structures because for two (or more) interwound helices there is a geometrical lock-up helical angle which puts a limit on the length of a superhelix. We perform calculations with different ionic concentrations and study the effects of lock up for braided structures. We also extend the variable curvature model to study the formation of plectonemes in the presence of multivalent salts where the supercoiling radius can be regarded as a constant prescribed by the balance of attractive and repulsive forces in DNA–DNA interactions, and provide analytical solutions in terms of elliptic functions for the supercoil parameters.
Many living structures are coated by thin films, which have distinct mechanical properties from the bulk. In particular, these thin layers may grow faster or slower than the inner core. Differential growth creates a balanced interplay between tension and compression and plays a critical role in enhancing structural rigidity. Typical examples with a compressive outer surface and a tensile inner core are the petioles of celery, caladium, or rhubarb. While plant physiologists have studied the impact of tissue tension on plant rigidity for more than a century, the fundamental theory of growing surfaces remains poorly understood. Here, we establish a theoretical and computational framework for continua with growing surfaces and demonstrate its application to classical phenomena in plant growth. To allow the surface to grow independently of the bulk, we equip it with its own potential energy and its own surface stress. We derive the governing equations for growing surfaces of zero thickness and obtain their spatial discretization using the finite-element method. To illustrate the features of our new surface growth model we simulate the effects of growth-induced longitudinal tissue tension in a stalk of rhubarb. Our results demonstrate that different growth rates create a mechanical environment of axial tissue tension and residual stress, which can be released by peeling off the outer layer. Our novel framework for continua with growing surfaces has immediate biomedical applications beyond these classical model problems in botany: it can be easily extended to model and predict surface growth in asthma, gastritis, obstructive sleep apnoea, brain development, and tumor invasion. Beyond biology and medicine, surface growth models are valuable tools for material scientists when designing functionalized surfaces with distinct user-defined properties.
We model cardiac muscle contractions in the framework of finite elasticity with large distortions and couple a mechanical model with reaction–diffusion equations representing electrophysiological activity. Both models are implemented using anisotropic constitutive relations: we use stress–strain relations for fiber-reinforced materials, and anisotropic diffusion tensors for both the membrane potential and calcium ions. The effects of these choices on the electromechanical behavior are presented and discussed.
The application of nonlinear elasticity concepts to the mechanical modeling of soft biomaterials is currently the subject of intense investigation. For fibrous soft biomaterials, some specific strain-energy density models for anisotropic hyperelastic materials have been proposed in the literature that are particularly useful as they reflect the typical J-shaped stress–stretch stiffening response due to collagen fibers that is observed experimentally. These models have the feature of incorporating the increased stiffness of the collagen fibers with deformation and involve a maximum fiber stretch (or locking stretch). Here we apply such models to the analysis of the fracture or tearing of fibrous soft biomaterials. Attention is focused on a particular fracture test namely the trousers test where two legs of a cut specimen are pulled horizontally apart out of the plane of the test piece. It is shown that, in general, the location of the cut in the specimen plays a key role in the fracture analysis, and that the effect of the cut position depends crucially on the degree of strain–stiffening. This dependence is characterized explicitly for the specific strain–stiffening constitutive models considered. The effects of anisotropy and strain–stiffening on the fracture toughness (resistance to tearing) are also examined.
The post-buckling problem of a large deformed beam is analyzed using the canonical dual finite element method (CD-FEM). The feature of this method is to choose correctly the canonical dual stress so that the original non-convex potential energy functional is reformulated in a mixed complementary energy form with both displacement and stress fields, and a pure complementary energy is explicitly formulated in finite dimensional space. Based on the canonical duality theory and the associated triality theorem, a primal–dual algorithm is proposed, which can be used to find all possible solutions of this non-convex post-buckling problem. Numerical results show that the global maximum of the pure-complementary energy leads to a stable buckled configuration of the beam, while the local extrema of the pure-complementary energy present unstable deformation states. We discovered that the unstable buckled state is very sensitive to the number of total elements and the external loads. Theoretical results are verified through numerical examples and some interesting phenomena in post-bifurcation of this large deformed beam are observed.
The force on a screw dislocation in a multiphase laminated elastic structure is determined, both as an infinite integral and as a sum of multiple image terms. When the dislocation is near an interface, it is found that dislocation mobility is largely dependent on the ratio of the rigidities of the adjacent materials, and the perpendicular distance of the dislocation from the interface becomes the important length scale. When the dislocation is far from the interface layers, it is found that a combination of an odd number of alternate strata of two different materials has a stabilizing influence on the dislocation. Further, if two dissimilar semi-infinite planes are separated by any number of dissimilar thick layers, then the force on a screw dislocation at large distances is independent of the physical properties of the interface layers.
A dynamic crystal plasticity model for a low-symmetric β-cyclotetramethylene-tetranitramine single crystal with only limited operative slip systems has been developed, accounting for nonlinear elasticity, volumetric coupling with deviatoric behavior and thermo-dynamical consistence. Based on the decomposition of the stress tensor, a modified equation of state for anisotropic materials is employed. Simulations of the planar impact on the β-cyclotetramethylene-tetranitramine single crystal show good agreement with existing particle velocity data in the case of (110) and (011). Pressure snapshots, the dislocation density, the shear stress and the strain localization for β-cyclotetramethylene-tetranitramine single crystal under shocked loading are discussed. The present model provides more insights into a range of complex, orientation-dependent elastic and inelastic behaviors in shocked explosive crystals than isotropic elastic–plastic constitutive descriptions. The proposed formulation and algorithms can also be applied to study other low-symmetric crystals under high-pressure shocked loading that deform mainly by crystallographic slip.
The analytical treatment of an axisymmetric rigid punch indentation of an isotropic half-space reinforced by a buried extensible thin film is addressed. With the aid of appropriate displacement potential functions, Hankel transforms, and some mathematical techniques, the mixed boundary value problem under consideration is reduced to a Fredholm integral equation of the second kind. The most interesting results of the problem, including the equivalent normal stiffness of the system and the contact stress distribution beneath the rigid punch, are expressed in terms of the solution of the obtained Fredholm integral equation. Some limiting cases corresponding to inextensible and extremely extensible thin films, a surface-stiffened half-space, axisymmetric surface loading, and infinite embedment are studied and the available results in the literature are used for verification purposes. Some dimensionless plots are provided to show the effects of extensibility of the membrane on the system stiffness. It is observed that neglecting the elastic behavior of thin films and utilizing the simplified inextensible assumption for their modeling can lead to unrealistic predictions. It is shown that the axisymmetric problems concerning the elastic half-spaces including surface effects can be equivalently modeled by a surface-reinforced elastic medium. Taking the surface effects into account, the problem of an isotropic half-space under the action of a surface patch load is treated by employing the present approach.
In this paper we discuss the consequences of the distributional approach to dislocations in terms of the mathematical properties of the auxiliary model fields such as displacement and displacement gradient which are obtained directly from the main model field here considered as the linear strain. We show that these fields cannot be introduced rigourously without the introduction of gauge fields or, equivalently, without cuts in the Riemann foliation associated with the dislocated crystal. In a second step we show that the space of bounded deformations follows from the distributional approach in a natural way and discuss the reasons why it is adequate to model dislocations. The case of dislocation clusters is also addressed, as it represents an important issue in industrial crystal growth while from a mathematical point of view, peculiar phenomena might appear at the set of accumulation points. The elastic–plastic decomposition of the strain within this approach is also given a precise meaning.
This paper is concerned with the determination of the thermoelastic stress, strain and conductive temperature in a piezoelastic half-space body in which the boundary is stress free and subjected to thermal loading in the context of the fractional order two-temperature generalized thermoelasticity theory (2TT). The two-temperature three-phase-lag (2T3P) model, two-temperature Green–Naghdi model III (2TGNIII) and two-temperature Lord–Shulman (2TLS) model of thermoelasticity are combined into a unified formulation introducing unified parameters. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain that is then solved by the state-space approach. The numerical inversion of the transform is carried out by a method based on Fourier series expansion techniques. The numerical estimates of the quantities of physical interest are obtained and depicted graphically. The effect of the fractional order parameter, two-temperature and electric field on the solutions has been studied and comparisons among different thermoelastic models are made.
Let n ≥ 2. We prove a condition on $$f\in {C}^{2}\left({\hbox{ R }}_{+},\hbox{ R }\right)$$ for the convexity of f det on PSym(n), namely that f det is convex on PSym(n) if and only if
\[ {f}^{\prime\prime }\left(s\right)+\frac{n-1}{ns}\cdot {f}^{\prime }\left(s\right)\ge 0\phantom{\rule{0.25em}{0ex}}\hbox{ and }\phantom{\rule{0.25em}{0ex}}{f}^{\prime }\left(s\right)\le 0\phantom{\rule{0.25em}{0ex}}\forall \phantom{\rule{0.25em}{0ex}}s\in {\hbox{ R }}_{+}. \]
This generalizes the observation that C ↦ –ln det C is convex as a function of C.
The model of an edge dislocation inside a elastic thin layered semi-infinite matrix is utilized to study the interaction between an edge dislocation and the elastic thin layered semi-infinite matrix. The analytic solutions of complex functions of an edge dislocation in the matrix and in the thin layer are obtained by applying a complex potential approach in conjunction with the techniques of the image method. With the aid of the obtained stress fields and the Peach–Koehler formula, the explicit expressions for the glide and climb forces acting on the edge dislocation are obtained. The results indicate that, the material elastic dissimilarity, the thickness of thin layer and the position of the dislocation have a significant influence on the glide and climb forces acting on the dislocation. The glide and climb forces will increase with the decrease in thickness of thin layer when an edge dislocation is in the matrix or in the layer. In addition, they will increase with a decrease in the shear modulus ratio of the layer to the matrix, when the matrix is softer than the layer and an edge dislocation is in the matrix or in the layer.
An active elastodynamic cloak destructively interferes with an incident time harmonic in-plane (coupled compressional/shear) elastic wave to produce zero total elastic field over a finite spatial region. A method is described which explicitly predicts the source amplitudes of the active field. For a given number of sources and their positions in two dimensions it is shown that the multipole amplitudes can be expressed as infinite sums of the coefficients of the incident wave decomposed into regular Bessel functions. Importantly, the active field generated by the sources vanishes in the far-field. In practice the infinite summations are clearly required to be truncated and the accuracy of cloaking is studied when the truncation parameter is modified.
In this paper the use of control volumes in continuum mechanics is discussed. The global balance equations for momentum and moment of momentum for general control volumes are presented and their equivalence to the balance equations for material bodies is demonstrated. We also consider an approach based on the formulation of a principle of virtual power for control volumes. Based on this principle Lagrange’s equations for a control volume are derived. A specialization to so-called rigid control volumes is presented together with a didactic example concerning rocket propulsion. Finally, the balance of energy for a control volume is briefly discussed.
In this paper reflection and transmission of compression and shear waves at structured interfaces between second-gradient continua is investigated. Two semi-infinite spaces filled with the same second-gradient material are connected through an interface which is assumed to have its own material properties (mass density, elasticity and inertia). Using a variational principle, general balance equations are deduced for the bulk system, as well as jump duality conditions for the considered structured interfaces. The obtained equations include the effect of surface inertial and elastic properties on the motion of the overall system. In the first part of the paper general 3D equations accounting for all surface deformation modes (including bending) are introduced. The application to wave propagation presented in the second part of the paper, on the other hand, is based on a simplified 1D version of these equations, which we call "axial symmetric" case.
Constitutive modeling within peridynamic theory considers the collective deformation at each time of all the material within a -neighborhood of any point of a peridynamic body. The assignment of the parameter , called the horizon, is treated as a material property. The difference displacement quotient field in this neighborhood, rather than the extension scalar field, is used to generate a three-dimensional state-based linearly elastic peridynamic theory. This yields an enhanced interpretation of the kinematics between bonds that includes both length and relative angle changes. A free energy function for a linearly elastic isotropic peridynamic material that contains four material constants is proposed as a model, and it is used to obtain the force vector state and the associated modulus state for this material. These states are analogous to, respectively, the stress field and the fourth-order elasticity tensor in classical linear theory. In the limit of small horizon, we find that only three of the four peridynamic material constants are related to the classical elastic coefficients of an isotropic linear elastic material, with one of the three constants being arbitrary. The fourth peridynamic material constant, which accounts for the coupling effect of both bond length and relative angle change, has no effect on the limit, but remains a part of the peridynamic model. The determination of the two undetermined constants is the subject of future investigation. Peridynamic models proposed elsewhere in the literature depend on the deformation state through its dilatational and deviatoric parts and contain only two peridynamic material constants, in analogy to the classical linear elasticity theory. Observe from above that our model depends on both length and relative angle changes, as in classical linear theory, but, otherwise, is not limited to having only two material constants. In addition, our model corresponds to a nonordinary material, which represents a substantial break with classical models.
This paper is concerned with the geometrically non-linear theory of 6-parametric elastic shells with drilling degrees of freedom. This theory establishes a general model for shells, which is characterized by two independent kinematic fields: the translation vector and the rotation tensor. Thus, the kinematical structure of 6-parameter shells is identical to that of Cosserat shells. We show the existence of global minimizers for the geometrically non-linear 2D equations of elastic shells. The proof of the existence theorem is based on the direct methods of the calculus of variations essentially using the convexity of the energy in the strain and curvature measures. Since our result is valid for general anisotropic shells, we analyze the particular cases of isotropic shells, orthotropic shells and composite shells separately.
For laminated shells, the displacement field can be approximated in each layer by a third-order Taylor–Young expansion in thickness. Then we are motivated to consider a simplified theory based on the thickness-wise expansion of the potential energy truncated at third-order in thickness. The equilibrium equations imply local constraints on the through-thickness derivatives of the zero-order displacement field in each layer. These lead to an analytical expression for the two-dimensional potential energy of cylindrical shells in terms of the zero-order displacement field, and its derivatives, that includes non-standard transverse shearing and normal stress energy. As a consequence this potential energy satisfies the stability condition of Legendre–Hadamard which is necessary for the existence of a minimizer.
This study concerns the mathematical modeling of anisotropic and transversely inhomogeneous slender piezoelectric bars. Such rod-like structures are employed as passive sensors aimed at measuring the displacement field on the boundary of an underlying elastic medium excited by an external source. Based on the coupled three-dimensional dynamical equations of piezoelectricity in the quasi-electrostatic approximation, a set of limit problems is derived using formal asymptotic expansions of the electric potential and elastic displacement fields. The nature of these problems depends strongly on the choice of boundary conditions, therefore, an appropriate set of constrains is introduced in order to derive one-dimensional models that are relevant to the measurement of a displacement field imposed at one end of the bar. The structure of the first-order electric and displacement fields as well as the associated coupled limit equations are determined. Moreover, the properties of the homogenized material parameters entering these equations are investigated in various configurations. The obtained one-dimensional models of piezoelectric sensors are analyzed, and it is finally shown how they enable the identification of the boundary displacement associated with the probed elastic medium.
For homogeneous plates, the highest order term of transverse shear and normal stresses is of second order in thickness. To take this effect into account, we show that the thickness-wise expansion of the potential energy must be truncated at least from fifth order in thickness. The equilibrium equations imply local constraints on the through-thickness derivatives of the zeroth-order displacement field. These lead to an analytical expression for two-dimensional potential energy in terms of the zeroth-order displacement field and its derivatives, which include non-standard shearing and transverse normal energies and coupled stretching–shearing, bending–shearing and stretching–transverse normal energies. As a consequence, this potential energy satisfies the stability condition of Legendre–Hadamard, which is necessary for the existence of a minimizer.
Universal relations hold in all members that belong to a certain class of bodies, and they are therefore useful in designing experiments in which all members belonging to the particular class of bodies can be tested. It has been shown recently that the class of elastic bodies is much larger than the classical Cauchy elastic bodies. It has also been shown that such elastic bodies have firm thermodynamic underpinnings. In this short paper, we discuss universal relations that hold for a large sub-class of bodies which belong to this new class of elastic bodies. To be more precise, we consider a class of compressible isotropic elastic solids that are a sub-class of the new class of elastic bodies. We show that practically all the universal solutions which are possible in classical Cauchy elastic bodies are also possible within the context of the sub-class of elastic bodies that we consider.
The link between atomistic quantities and continuum fields has been the subject of research for at least half a century. Nevertheless, there are still many open questions and misleading discussions in the literature. Therefore, based on the fundamental principles of classical mechanics and statistical physics we construct the basic framework for the link between the atomistic and continuum worlds. In doing so, considerable attention is paid to the central force decomposition and multi-body potentials, balance of angular momentum for the system of particles and its relationship to the extended third Newton axiom and the difference between the theorem of change of kinetic energy and the energy balance law. A number of general theorems related to the convolution properties of statistically averaged quantities, as well as their rates are also proven. These theorems make the derivation of balance equations far simpler when compared to the approaches used by others. Such theorems also make the link between space–time averaging and space–probability averaging more transparent.
In this contribution the balance laws of mass and linear momentum are derived. The remaining balance laws of angular momentum and energy as well as the particular forms of fluxes, such as the stress, are discussed in the follow-up contributions of this series.
This work develops a rigorous variational upper bound for the difference between thermal fields generated in uniform media and thermal fields generated in heterogeneous media, for the same external loading. The bound can be calculated in a simple manner, with knowledge only of the heterogeneous material properties and a relatively easy-to-compute thermal field associated with a uniform medium. In order to evaluate the bound, the difficult-to-compute thermal field, associated with the heterogeneous material, does not need to be calculated. Three-dimensional numerical examples are provided to illustrate the results.
The problem of the brachistochronic motion of a mechanical system composed of rigid bodies and variable-mass particles is solved. The laws of the time-rate of mass variation of the particles as well as relative velocities of the expelled (or gained) masses are assumed to be known. The system moves in an arbitrary field of known potential and nonpotential forces. Applying Pontryagin’s minimum principle along with singular optimal control theory, a corresponding two-point boundary value problem is obtained. The appropriate numerical procedure based on the shooting method to solve the obtained two-point boundary value problem is presented. The considerations in the paper are illustrated by an example of determining the brachistochronic motion of a system composed of a rigid rod and two variable-mass particles attached to the rod.
The main goal of the present paper is to introduce certain duals and transpositions of a high-order tensor playing an important role in continuum mechanics. Emphasis is also placed on the comparison of the duals and the transpositions. In contrast to the duals, the transpositions depend on a metric of an underlying metric space. A high-order tensor is itself a representation of a multilinear function on a tensor space, obtained by means of the multilinear extension. The duals and the transpositions of a high-order tensor are identified as multilinear maps defined by the generalized scalar and the inner product, respectively. Consequently, the duals and the transpositions distinguish and define symmetries and symmetry-preserving transformation rules of a co-, contra- and mixed-variant tensor, respectively. As an application in continuum mechanics, the duals and the transpositions of a usual fourth-order tensor are defined and are employed to the determination of symmetries involved.
The problem of finite elastic deformation of a long rectangular rubber block which is deformed in a perturbed cylindrical configuration is examined here, and is motivated from the problem of determining surface rippling that is observed in bent multi-walled carbon nanotubes. The problem of finite elastic bending of a tube is considerably more complicated than the geometrically simpler problem of finite elastic bending of a rectangular block. Accordingly, we examine here the simpler block problem which is assumed to be sufficiently long so that the out of plane end effects may be ignored. The general equations governing plane strain deformations of an isotropic incompressible perfectly elastic Mooney material, which models rubber-like materials, are used to determine small superimposed deformations upon the well-known controllable family for the deformation of rectangular blocks into a sector of a solid bounded by two circular arcs. Traction free boundary conditions are assumed in an average sense along the bounding circular arcs. Physically realistic rippling is found to occur and typical numerical values are used to illustrate the solution graphically.
In this paper, torsional analysis of hollow bars with specially shaped non-circular cross-sections is presented. The well-known formulation of the torsion problem in complex variable theory is used to obtain an expression for shear stresses, angle of twist and cross-sectional warping. The most important part of the solution of the torsion problem here is to obtain an appropriate closed-form conformal mapping for specially shaped cross-sections. The obtained mapping is used to achieve a solution for torsion problem of bars with specially shaped hollow cross-section. Finally, the calculated results are compared with the finite-element method solution. The presented solution is capable of accurately predicting the angle of twist, warping, and shear stress in the studied hollow bars.
In this paper we consider three particular systems of the isothermal viscoelasticity with voids. Our main aim is to see how the coupling terms in the dissipative parts can be used to obtain qualitative properties which are not satisfied when the coupling terms are not present.
In the theory of dislocations, the Burgers vector is usually defined by referring to a crystal structure. Using the notion of affine development of curves on a differential manifold with a connection, we give a differential geometric definition of the Burgers vector directly in the continuum setting, without making use of an underlying crystal structure. As opposed to some other approaches to the continuum definition of the Burgers vector, our definition is completely geometric, in the sense that it involves no ambiguous operations such as the integration of a vector field: when we integrate a vector field, it is a vector field living in the tangent space at a given point in the manifold. For a body with distributed dislocations, the material manifold, which describes the geometry of the stress-free state of the body, is commonly taken to be a Weitzenböck manifold, i.e. a manifold with a metric-compatible, flat connection with torsion. We show that for such a manifold, the density of the Burgers vector calculated according to our definition reproduces the commonly stated relation between the density of dislocations and the torsion tensor.
The theory of singular dislocations is placed within the framework of the theory of continuous dislocations using de Rham currents. For a general n-dimensional manifold, an (n – 1)-current describes a local layering structure and its boundary in the sense of currents represents the structure of the dislocations. Frank’s rules for dislocations follow naturally from the nilpotency of the boundary operator.
To extend classical micro and nanomechanics of inclusions and inhomogeneities from bulk phase only to interface-featured multi-phase, we formulated a solution procedure for evaluating the significance of interface stress on embedded nanoparticles. The methodology allows, for instance, analytical determination of the influential effects of interface stress on elastic fields of both nanoparticles and matrices within the general framework of continuum theory of bulk and interface elasticity. A thorough curvilinear analysis of a general Euclidean interface is performed with the aid of field theory and applied to facilitate the integration of displacement continuity and traction jump boundary conditions of interface elasticity into the classical formulation of displacement potentials. For illustration purposes, the solution methodology is applied to a spherical nanoparticle embedded in an infinite substrate subjected to most general far-field tension and eigenstrain loads. The developed model extends the solution limit of interface-induced length-scale effects on embedded ellipsoidal nanoparticles to the solution availability of their corresponding classical counterparts, provided that interface material properties are given.
We consider an -parametrized collection of cylinders of cross section , where R2, and of fixed length . By Korn’s inequality, there exists a positive constant K such that |symu|2d3x ≥ K |u|2d3x provided that u H1(;R3) satisfies a condition that rules out infinitesimal rotations. We show that K/2 converges to a strictly positive limit, and we characterize this limit in terms of certain parameters that depend on the geometry of and on .
The axisymmetric bending of a functionally graded circular plate, which is subjected to a concentrated transverse force at the center, is investigated based on a generalization of England’s method. The complex variable function method adopted in this paper involves four analytic functions of the complex variable, in which the unknown constants can be determined from cylindrical boundary conditions, similar to that in classical plate theory. The axisymmetric bending problem of a functionally graded circular plate concentrically loaded at the center can be converted into a bending problem of a functionally graded annular plate subject to shear forces uniformly distributed over the inner boundary. The elasticity solutions of a transversely isotropic and functionally graded circular plate subject to a concentrated force at the center are then derived, from which elasticity solutions for a transversely isotropic (or isotropic) and homogeneous circular plate can be obtained from the analysis as a special case. Finally, numerical examples are presented to compare the proposed analytical solutions with those of a finite-element method. The results show that the proposed elasticity solutions are very simple and usable.
In this paper, we use the weight function for an elliptical crack embedded in an infinite elastic media in conjunction with the alternating method to derive the exact analytical solution for the stress intensity factor for a semi-elliptical surface crack subjected to an arbitrary mode I loading.
The theory of material evolution is specialized to accommodate density-preserving remodeling of isotropic materials. It is assumed that in the pure mechanical case, the rate of evolution depends on the stress, deformation and the evolution. The dissipation inequality and the conservation of density indicate that the driving force for the evolution is the symmetric deviatoric part of the Mandel stress. The isotropic tensor-valued function representation theorem is used to show that there are 18 different admissible evolution modes. Assuming that the dissipation inequality takes a quadratic form, each evolution mode is driven by an associated configurational force. In the proposed evolution model each mode is governed by a single material constant corresponding to viscosity. Moreover, consistent evolution criteria are developed such that evolution arises only if a certain threshold is reached.
The aim of this work is to present a general homogenization framework with application to magnetorheological elastomers under large deformation processes. The macroscale and microscale magnetomechanical responses of the composite in the material and spatial description are presented and the conditions for a well-established homogenization problem in Lagrangian description are identified. The connection between the macroscopic magnetomechanical field variables and the volume averaging of the corresponding microscopic variables in the Eulerian description is examined for several types of boundary conditions. It is shown that the use of kinematic and magnetic field potentials instead of kinetic field and magnetic induction potentials provides a more appropriate homogenization process.
This paper presents some new applications of a model of ductile rupture of porous ductile materials proposed by Gologanu, Leblond, Perrin and Devaux (GLPD). The first application is concerned with the relationship between this model and the class of generalized standard materials. We show that the GLPD model fits in this class, provided that the porosity is not allowed to change and the hypothesis of linearized theory is adopted. The advantage of this property is that it automatically warrants the unicity of the solution for the ‘projection’ problem of the (supposedly) elastic stress predictor onto the GLPD’s yield locus. The second application leads to some exact analytical solutions of the GLPD model constitutive equations for the problems of an elastic hollow sphere in the framework of linearized theory, and viscous in large deformations. Comparisons between the numerical predictions of the GLPD model and the analytical solutions confirm the robustness of the numerical scheme used to implement this model into SYSTUS© finite element (FE) code. Thus, these exact analytical solutions can be used to validate the implementation of the GLPD model in another finite element code.
In the third application, comparisons between experimental and numerical load vs. displacement curves for an axisymmetric pre-cracked specimen made of a typical stainless steel are found to yield satisfactory results.
We consider an abstract mixed variational problem which consists of a system of an evolutionary variational equation in a Hilbert space X and an evolutionary inequality in a subset of a second Hilbert space Y, associated with an initial condition. The existence and the uniqueness of the solution is proved based on a fixed point technique. The continuous dependence on the data was also investigated. The abstract results we obtain can be applied to the mathematical treatment of a class of frictional contact problems for viscoelastic materials with short memory. In this paper we consider an antiplane model for which we deliver a mixed variational formulation with friction bound dependent set of Lagrange multipliers. After proving the existence and the uniqueness of the weak solution, we study the continuous dependence on the initial data, on the densities of the volume forces and surface tractions. Moreover, we prove the continuous dependence of the solution on the friction bound.
The anti-plane deformation of an uncracked transversely isotropic sector is studied for two different types of boundary conditions. The solution of the problem is accomplished by means of two methods for each boundary condition, namely using the finite Fourier cosine transform and the method of separation of variables. The closed-form solutions are obtained for displacement and stress fields in the whole domain for each kind of boundary conditions. In the special cases, i.e. finite wedges, the results for the stress field show an exact agreement with those in the literature. In the following, the stress analysis of a transversely isotropic sectors weakened by a circular crack as well as radial crack is accomplished. The resultant singular integral equation of the Cauchy type are solved numerically and, finally, the stress intensity factors of the crack tips versus the ratios of the material properties are plotted and discussed.
The thermoelasticity problem in a thick-walled sphere is solved analytically using finite Hankel transform. Time-dependent boundary conditions are prescribed on the inner surface of the sphere, where for the mechanical boundary conditions the tractions are prescribed on both the inner and the outer surfaces of the hollow sphere. Obtaining the distribution of the temperature throughout the sphere, the quasi-static and the dynamical structural problem is solved and closed-form relations are derived for stress components. On solving the dynamical thermoelasticity problem, a thermal shock was observed after plotting the results. Using the obtained plots, the velocity of the spherical dilatation wave is computed and compared with the one from the classical formula and the results are compared with those in the literature, where possible.
Numerical solutions of a one-dimensional model of screw dislocation walls (twist boundaries) are explored. The model is an exact reduction of the three-dimensional system of partial differential equations of Field Dislocation Mechanics. It shares features of both Ginzburg–Landau (GL)-type gradient flow equations and hyperbolic conservation laws, but is qualitatively different from both. We demonstrate such similarities and differences in an effort to understand the equation through simulation. A primary result is the existence of spatially non-periodic, extremely slowly evolving (quasi-equilibrium) cell-wall dislocation microstructures practically indistinguishable from equilibria, which however cannot be solutions to the equilibrium equations of the model, a feature shared with certain types of GL equations. However, we show that the class of quasi-equilibria comprising a spatially non-periodic microstructure consisting of fronts is larger than that of the GL equations associated with the energy of the model. In addition, under applied strain-controlled loading, a single dislocation wall is shown to be capable of moving as a localized entity, as expected in a physical model of dislocation dynamics, in contrast to the associated GL equations. The collective evolution of the quasi-equilibrium cell-wall microstructure exhibits a yielding-type behavior as bulk plasticity ensues, and the effective stress–strain response under loading is found to be rate-dependent. The numerical scheme employed is non-conventional, since wave-type behavior has to be accounted for, and interesting features of two different schemes are discussed. Interestingly, a stable scheme conjectured by us to produce a non-physical result in the present context nevertheless suggests a modified continuum model that appears to incorporate apparent intermittency.
A Lagrangian formulation with nonlocality is investigated in this paper. The nonlocality of the Lagrangian is introduced by a new nonlocal argument that is defined as a nonlocal residual satisfying the zero mean condition. The nonlocal Euler–Lagrangian equation is derived from the Hamilton’s principle. The Noether’s theorem is extended to this Lagrangian formulation with nonlocality. With the help of the extended Noether’s theorem, the conservation laws relevant to energy, linear momentum, angular momentum and the Eshelby tensor are determined in the nonlocal elasticity associated with the mechanically based constitutive model. The results show that the conservation laws exist only in the form of the integral over the whole domain occupied by the body. The localization of the conservation laws is discussed in detail. We demonstrate that not every conservation law corresponds to a local equilibrium equation. Only when the nonlocal residual of conservation current exists, can a conservation law be transformed into a local equilibrium equation by localization.
Using the q2-Laplace transform and its inverse transform introduced early on by Hahn and deeply studied by Abdi we have prove that the q-analogues of the heat and wave equations are linked as in the classical case of Bragg and Dettman. As an application, we proved first, through the q-wave polynomials, that the q-Hermite and the q-little Jacobi polynomials are related. Second, we have given a q-analogue of the Poisson kernel studied by Fitouhi and Annabi.
In this paper we derive a theory for a linearly elastic residually stressed rod through an asymptotic analysis based on -convergence.
This research explores elastodynamics and wave propagation in fractal micropolar solid media. Such media incorporate a fractal geometry while being modelled constitutively by the Cosserat elasticity. The formulation of the balance laws which govern the mechanics of fractal micropolar solid media is presented. Four eigenvalue-type elastodynamic problems admitting closed-form analytical solutions are introduced and discussed. A numerical procedure to solve general initial boundary value wave propagation problems in three-dimensional micropolar bodies exhibiting geometric fractality is then applied. Verification of the numerical procedure is discussed using the analytical solutions.
According to the theory of materially uniform but inhomogeneous bodies two geometric structures can be defined on the body manifold when a uniform reference is known. The first is given by the material connection and the second by the intrinsic Riemannian metric. Two important classes of curves related with these structures are the geodesic and the autoparallel curves. The goal of the present contribution is to define and characterize mechanically these kind of curves for materially uniform but inhomogeneous bodies. We propose the use of these curves for constructing the stress-free non-Euclidean material manifold that plays the role of the reference configuration for the dislocated problem. A generic scheme for the construction of this manifold based on geodesics/autoparallel curves is given as well as a discussion related with the field of internal stresses. Attention is then focused on a continuous distribution of edge dislocations. We solve numerically the geodesic equation that corresponds to a solid body with a continuous symmetry group. By using the L2 norm for the dislocation density tensor we conclude that the higher the dislocation density the greater the deviation from the straight line is. For the same distribution of dislocations, but for a solid body with a discrete symmetry group, we solve analytically the autoparallel curves.
In this paper, the problem of two bonded anisotropic finite wedges with an interface crack subjected to anti-plane shear loading is investigated. The boundary conditions of radial edges are treated as traction–displacement conditions. The traction-free condition is applied on the circular segment of the wedge. Finite complex transforms, which have complex analogies to the standard finite Mellin transforms, have been used in order to solve this problem. The traction-free condition of the crack faces is expressed in an analytical form of a singular integral equation. The resultant singular integral equations are then solved numerically by use of the Chebyshev polynomial. The stress intensity factors at the crack tips are exhibited in various conditions. It is seen that, in general, the stress intensity factor is a function of the material property and apex angle of the wedge.
In this paper the interaction between two rigid parts of a multibody, connected by an ideal spherical joint equipped with a visco-elastic torsion bushing element, is derived. The model allows for arbitrary relative rotations of the parts and involves a non-linear torsion stiffness of the bushing. An expression for the interaction between the parts is derived and a specialization to the isotropic bushing element is presented.
The present work is concerned with the elasto-thermo-diffusion interactions inside a spherical shell whose inner and outer boundaries are traction free and are subjected to a step input in temperature under generalized thermoelastic diffusion. The chemical potential is also assumed to be a function of time on the boundaries of the shell. The problem is studied by employing three different theories of thermoelastic diffusion in a unified way. The integral transform technique is used to obtain the general solution. Using a numerical method and computer programming, the final solution in the physical domain is obtained. The results corresponding to thermoelastic medium, that is, in the absence of diffusion, are also found in a particular case. The influence of diffusion on variation of physical fields due to the cross effects arising from the coupling of temperature, diffusion and strain fields inside the shell is analyzed with the help of graphically plotted results of different field variables in the thermoelastic medium and thermoelastic diffusive medium. The variation of this influence with time under different theories is also discussed. The significant points are highlighted.
We consider two mathematical models, which describe the frictional contact between a deformable body and a foundation. In both models the process is assumed to be quasistatic, the material is viscoelastic, and the friction is given by a subdifferential boundary condition. In the first model the contact is described with a univalued condition between the normal stress and the normal displacement and in the second model it is described with a subdifferential condition, which links the normal stress and the normal velocity. For each model we derive a variational formulation, which is in the form of a history-dependent hemivariational inequality for the velocity field. Then we prove the existence of a weak solution and, under additional assumptions, its uniqueness. The proof is based on a recent result on history-dependent hemivariational inequalities.
The paper provides a rigorous analysis of the dispersion spectrum of shear horizontal elastic waves in periodically stratified solids. The problem consists of an ordinary differential wave equation with periodic coefficients, which involves two free parameters (the frequency) and k (the wavenumber in the direction orthogonal to the axis of periodicity). Solutions of this equation satisfy a quasi-periodic boundary condition which yields the Floquet parameter K. The resulting dispersion surface (K, k) may be characterized through its cuts at constant values of K, k and that define the passband (real K) and stopband areas, the Floquet branches and the isofrequency curves, respectively. The paper combines complementary approaches based on eigenvalue problems and on the monodromy matrix M. The pivotal object is the Lyapunov function (2, k2) 1/2traceM = cos K which is generalized as a function of two variables. Its analytical properties, asymptotics and bounds are examined and an explicit form of its derivatives obtained. Attention is given to the special case of a zero-width stopband. These ingredients are used to analyse the cuts of the surface (K, k). The derivatives of the functions (k) at fixed K and (K) at fixed k and of the function K(k) at fixed are described in detail. The curves (k) at fixed K are shown to be monotonic for real K, while they may be looped for complex K (i.e. in the stopband areas). The convexity of the closed (first) real isofrequency curve K(k) is proved thus ruling out low-frequency caustics of group velocity. The results are relevant to the broad area of applicability of ordinary differential equation for scalar waves in 1D phononic (solid or fluid) and photonic crystals.
The interconversions between relaxation moduli and creep compliances, including stretch, shear, bulk parts, and the time-dependent Poisson’s ratio, are derived by using the relaxation-creep duality representation. The relaxation-creep duality representation for the viscoelastic functions introduced in this paper is composed of an exponential function that characterizes the relaxation behavior and a complementary one that characterizes the creep behavior. All viscoelastic functions can be represented as the same form. The new sets of coefficients, called the modulating constants, between viscoelastic functions obey the elastic-like interconversions, and do not involve the characteristic times. The relationships of characteristic times between those functions are also derived. These interconversion formulas can then be calculated easily. Three literatures are referenced to calculate the consistency of the viscoelastic functions via the new interconversions introduced in this work. The Young’s relaxation modulus in one literature is not consistent to the shear one in another literature. By assuming a constant bulk modulus, the modified Young’s relaxation modulus and time-dependent Poisson’s ratio that was derived by the new interconversions can meet the measured curves and can be consistent to the shear creep compliance in the literatures. The fitted data from experiments can then be checked via the new mathematical interconversions for the consistency.
According to the theory of materially uniform but inhomogeneous bodies, when the symmetry group of a material is continuous it induces a non-uniqueness to the uniform reference. Therefore, it is possible, that by manipulating the symmetry group the inhomogeneity of the material – namely, the dislocations – may cancel out. A solid mathematical framework is constructed in order to describe this situation using the language of exterior calculus. We set down a system of exterior differential equations which when solved render the totality of the uniform references that may be healed by a given symmetry group. From a mathematical point of view these equations have the form of Cartan’s equations of structure. We present the generic set of solutions of these equations and then specialize to the particular case of an isotropic solid body.
Two variational principles for nonlinear magnetoelastostatics are studied, considering a magnetosensitive body completely surrounded by free space extending to infinity. The functionals depend on the deformation function as one of the independent variables, and on either the scalar magnetic potential or the magnetic vector potential as the independent magnetic variable. Alternative representations for the energy densities are given for free space, from which simple expressions for the first and second variations of the functionals are obtained.