Numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials

Abstract

The homogenization of nonlinear heterogeneous materials is much more difficult than the homogenization of linear ones. This is mainly due to the fact that the general form of the homogenized behavior of nonlinear heterogeneous materials is unknown. At the same time, the prevailing numerical methods, such as concurrent methods, require extensive computational efforts. A simple numerical approach is proposed to compute the effective behavior of nonlinearly elastic heterogeneous materials at small strains. The proposed numerical approach comprises three steps. At the first step, a representative volume element (RVE) for a given nonlinear heterogeneous material is defined, and a loading space consisting of all the boundary conditions to be imposed on the RVE is discretized into a sufficiently large number of points called nodes. At the second step, the boundary condition corresponding to each node is prescribed on the surface of the RVE, and the resulting nonlinear boundary value problem, is solved by the finite element method (FEM) so as to determine the effective response of the heterogeneous material to the loading associated to each node of the loading space. At the third step, the nodal effective responses are interpolated via appropriate interpolation functions, so that the effective strain-energy, stress–strain relation and tangent stiffness tensor of the nonlinear heterogeneous material are provided in a numerically explicit way. This leads to a non-concurrent nonlinear multiscale approach to the computation of structures made of nonlinearly heterogeneous materials. The first version of the proposed approach uses multidimensional cubic splines to interpolate effective nodal responses while the second version of the proposed approach takes advantage of an outer product decomposition of multidimensional data into rank-one tensors to interpolate effective nodal responses and avoid high-rank data. These two versions of the proposed approach are applied to a few examples where nonlinear composites whose phases are characterized by the power-law model are involved. The numerical results given by our approach are compared with available analytical estimates, exact results and full FEM or concurrent multilevel FEM solutions.