An efficient homogenization method for composite materials with elasto-plastic components

The homogenization problem for elasto-plastic media with arrays of isolated inclusions (matrix composite) is considered. A combination of self-consistent and numerical methods is used for calculation of the overall response of such composites under quasi-static loading. Elasto-plastic properties of the medium and the inclusions are described by the equations of the incremental theory of plasticity with isotropic hardening. For the construction of the average stress–strain relations of the composites, the process of external loading is divided into a sequence of small steps, and the problem is linearized at every step. The self-consistent effective field method allows reducing the homogenization problem at every step to the calculation of stresses and elasto-plastic deformations in a composite cell that contains a finite number of inclusions. The linearized problems are formulated in terms of volume integral equations for the stress or elastic strain field increments in the cell. For the numerical solution, these equations are discretized by Gaussian approximating functions concentrated in a set of nodes that cover the composite cell. For such functions, elements of the matrix of the discretized problems are calculated in explicit analytical forms. If the approximating nodes form a regular grid, the matrix of the discretized problem has Toeplitz’s properties, and the matrix–vector products of such matrices can be calculated by the fast Fourier transform technique. The latter accelerates substantially the process of iterative solution of the discretized problems. The dependencies of the overall stress–strain curves on the number of inclusions inside the cell are studied in the 2D and 3D cases. The inclusions that are stiffer or softer then the matrix are considered. The predictions of the method are compared with the finite element calculations available in the literature.