Numerical solution of the integral equations of elasto-plasticity for a homogeneous medium with several heterogeneous inclusions

The work is devoted to the calculation of stress and strain fields in a homogeneous elasto-plastic medium with a finite number of heterogeneous inclusions. The medium is subjected to an arbitrary external stress field. Elasto-plastic behavior of the medium is described by the equations of the incremental theory of plasticity with isotropic hardening. For the numerical solution, the external stress field applied to the medium is divided on a consequence of small steps, and the problem is linearized at every step. The linearized problem is reduced to the solution of the volume integral equations for the stress field increment inside the inclusions and in the regions involved in the plastic deformations. Then, these equations are discretized using Gaussian approximating functions. For such functions, the elements of the matrix of the discretized problems are calculated in explicit analytical forms. If the approximating nodes compose a regular grid, the matrix of the discretized problem obtains the Toeplitz properties, and the product of such a matrix and a vector can be calculated by the Fast Fourier Transform technique. The latter accelerates essentially the process of iterative solution of the discretized problems. The proposed method is mesh free, and the coordinates of the approximating nodes and elasto-plastic properties of the material at the nodes are the only information required for carrying out the method. Distributions of stresses and plastic strains in the media with isolated inclusions are compared with the finite element calculations. The influence of the number of approximating nodes and the rate of hardening of the material on the convergence of the numerical solutions is analyzed.