ИССЛЕДОВАНИЕ ИНВАРИАНТНОСТИ J-ИНТЕГРАЛА ПРИ СЛОЖНОМ СДВИГЕ

Abstract

The problem of determining the stress-strain state of an elastic-plastic beam with an edge crack under conditions of a complex shear is considered. A quasi-static external loading is given, which leads to complex loading processes at each point of the beam. Deformations are considered small. A numerical algorithm for solving the problem based on the application of the finite element method and the differential-nonlinear version of the plasticity theory taking into account microdeformation (Novozhilov, Kadashevich, Chernyakov) is proposed. The no-uniformity of plastic deformation is approximately taken into account by representing the plastic deformation tensor as a sum of elementary plastic deformations, each of which has its own yield surface and a system of internal micro-elastic forces. Mathematically, the boundary value problem is formulated as a nonlinear boundary value problem and the Cauchy problem with respect to the loading parameter. To solve the Cauchy problem, a step-by-step method that reduces the solution of the original problem to a sequence of solutions on time layers is used. At each step in time, using the iterative method, the nonlinear boundary value problem is reduced to a sequence of quasilinear problems that are solved by the finite element method. The quadrangular isoparametric finite element is chosen as the basis for the finite element method. To calculate the stiffness matrices of finite elements, an algorithm is used that makes it possible to reduce the number of multiple integrals to be calculated in the defining relations of the theory of plasticity taking into account microdeformations, up to the dimensionality of the loading trajectory. To calculate the J-integral, a direct method is used, which is based on its direct calculation on the basis of finite element method. When carrying out the calculations, St45 steel is chosen as the material of the beam, for which, when describing elastic-plastic deformation within the framework of the theory of plasticity taking into account microdeformations, it is permissible to use constants as universal material functions. Under various loading schemes, zones of plasticity in the section of the beam are constructed. The influence of the loading history on form and size of plastic zones and on the J-integral is investigated. It is shown that under the accepted loading schemes, within the admissible accuracy, the J-integral is invariant under both proportional and complex loading, and also that its magnitude depends on the loading history.