Applied and Engineering Versions of the Theory of Elastoplastic Processes of Active Complex Loading. Part 1: Conditions of Mathematical Well-Posedness and Methods for Solving Boundary Value Problems

摘要

In the proposed theory of plasticity, the deviator constitutive relation has a trinomial form (the vectors of stresses, stress rates, and strain rates, which are formed form the deviators, are coplanar) and contains two material functions; one of these functions depends on the modulus of the stress vector, and the other, on the angle between the stress vector and the strain rate, the length of the deformation trajectory arc, and the moduli of the stress and strain vectors. The spherical parts of the stress and strain tensors satisfy the relations of elastic variation in the volume. We obtain conditions on the material functions of the model which ensure the mathematical well-posedness of the statement of the initial-boundary value problem (i.e., the existence and uniqueness of the generalized solution, and its continuous dependence on the external loads). We also describe the scheme for solving the initial-boundary value problem step by step using the model and present the expression for the Jacobian of the boundary value problem at the time step. These results are formalized as a subprogram for prescribing the mechanical properties of the user material in the finite-element complex ABAQUS, which allows one to calculate the structure deformations on the basis of the proposed theory.