Uncertainties in material models and their influence on structural behavior and reliability are important considerations in analysis and design of structures. In this article, a methodology based on the evidence theory is presented for uncertainty quantification of constitutive models. The proposed methodology is applied to Johnson–Cook plasticity model while considering various sources of uncertainty emanating from experimental stress–strain data as well as method of fitting the model constants and representation of the nondimensional temperature. All uncertain parameters are represented in interval form. Rules for agreement, conflict, and ignorance relationships in the data are discussed and subsequently used to construct a belief structure for each uncertain material parameter. The material model uncertainties are propagated through nonlinear crush simulation of an aluminium alloy 6061-T6 circular tube under axial impact load. Surrogate modeling and global optimization techniques are used for efficient calculation of the propagated belief structure of the tube response, whereas Yager’s aggregation rule of evidence is used for multi-model consideration. Evidence-based uncertainty in the structural response is measured and presented in terms of belief, plausibility, and plausibility-decision values.