Application of the variational EIV approach to linear viscoelastic phases governed by several internal variables-Examples with the generalized Maxwell law

New developments regarding the variational approach originally proposed by Lahellec, N., and Suquet, P., Int. J. Solids Struct. 44, 507-529, 2007, also called EIV approach, are presented in the context of linear viscoelasticity. In previous work (Tressou et al., Eur. J. Mech./A Solids. 68, 104-116, 2018), we have demonstrated the efficiency of this approach as associated to different linear schemes and, as a result, its possible application to various microstructures. In the present paper, our aim is to enlarge the application domain of the EIV model in terms of constituents' viscoelastic laws that may be considered. Initially applied to linear viscoelastic phases described by a single, traceless, internal variable, the original formulation of the EIV model is here generalized to phases described by several internal variables with both spherical and deviatoric parts. The resulting generalized EIV approach is evaluated by comparison of the estimated phase average and effective responses to reference solutions obtained by full-field finite element (FE) simulations for a fiber-reinforced microstructure. Different linear viscoelastic laws with increasing complexity are considered to this aim. At first, a classical Maxwell model is considered for the matrix as in Lahellec and Suquet's original work, but now the internal variable (viscous strain) has a spherical part. Bulk and shear moduli have identical or different characteristic times, the matrix only or both constituents are viscoelastic. Then, the matrix is described by a generalized Maxwell model with two and four internal variables. Different distributions of relaxation times are studied for bulk and shear moduli in order to test the robustness of the model. In every studied situation, the agreement with reference solutions is satisfactory.

Automated summary

This paper presents the new developments of the EIV approach in the context of linear viscoelasticity and demonstrates its efficiency and application in various microstructures. The generalized EIV approach is evaluated and compared with reference solutions obtained by finite element simulations.