General formulation of the first-order perturbation-based stochastic homogenization method using many random physical parameters for multi-phase composite materials

In the numerical prediction of the homogenized macroscopic properties of an arbitrary heterogeneous material with periodic microstructure, a general formulation of the first-order perturbation-based stochastic homogenization method is presented in a discretized form based on the finite element method in order to consider the variability or uncertainty of the mechanical properties of the material models. Many random parameters are defined for each material model and for each component of the stress–strain matrix of the constituent’s material model. The first-order terms of the characteristic displacement are thoroughly studied both theoretically and numerically, and are also used in the verification of the developed computer code. The comparison with the Monte Carlo simulation also supports the proposed formulation.