A simple computational homogenization method for structures made of linear heterogeneous viscoelastic materials

In this paper, a numerical multiscale method is proposed for computing the response of structures made of linearly non-aging viscoelastic and highly heterogeneous materials. In contrast with most of the approaches reported in the literature, the present one operates directly in the time domain and avoids both defining macroscopic internal variables and concurrent computations at micro and macro scales. The macroscopic constitutive law takes the form of a convolution integral containing an effective relaxation tensor. To numerically identify this tensor, a representative volume element (RVE) for the microstructure is first chosen. Relaxation tests are then numerically performed on the RVE. Correspondingly, the components of the effective relaxation tensor are determined and stored for different snapshots in time. At the macroscopic scale, a continuous representation of the effective relaxation tensor is obtained in the time domain by interpolating the data with the help of spline functions. The convolution integral characterizing the time-dependent macroscopic stress–strain relation is evaluated numerically. Arbitrary local linear viscoelastic laws and microstructure morphologies can be dealt with. Implicit algorithms are provided to compute the time-dependent response of a structure at the macroscopic scale by the finite element method. Accuracy and efficiency of the proposed approach are demonstrated through 2D and 3D numerical examples and applied to estimate the creep of structures made of concrete.