We present two versions of a mathematical model for the evolution of wear in a thermoelastic beam resulting from frictional contact with a rigid moving surface. One version is quasistatic and the other dynamic. We show that the quasistatic problem allows for the decoupling of the mechanical and thermal aspects of the process. The problem reduces to that of the heat equation with nonlinear and history-dependent boundary conditions. Then the displacements, shear stresses, and wear can be obtained by quadrature. We establish the existence of a local weak solution for the problem and a partial uniqueness result and obtain conditions for the solution's further regularity. The dynamic problem consists of the heat equation coupled with the equation of motion and the Archard wear equation. We prove the existence and uniqueness of the local weak solution of this problem too.

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This paper presents a generalized two-step endochronic approach for estimating notch stresses and strains based on elastic stress solutions. In the first stress-controlled step, notch root strains are calculated from elastic stresses using a conventional uniaxial method, such as Glinka’s energy density method and Neuber’s rule. In the second strain-controlled step notch root stresses corresponding to the estimated local strains are calculated from the given material properties. Both stress-controlled and strain-controlled algorithms based on endochronic plasticity theory are presented herein. The proposed method is used to calculate multiaxial strains under monotonie and nonproportional loads. Various geometric constraints (plane stress, plane strain, and intermediate level) are also examined. The results are compared with experimental measurements by other researchers and with predictions from other models.