Model reduction by mean-field homogenization in viscoelastic composites. I. Primal theory

A homogenization scheme for viscoelastic composites proposed by Lahellec & Suquet (2007 Int. J. Solids Struct. 44, 507-529 (doi:10.1016/j.ijsolstr.2006.04.038)) is revisited. The scheme relies upon an incremental variational formulation providing the inelastic strain field at a given time step in terms of the inelastic strain field from the previous time step, along with a judicious use of Legendre transforms to approximate the relevant functional by an alternative functional depending on the inelastic strain fields only through their first and second moments over each constituent phase. As a result, the approximation generates a reduced description of the microscopic state of the composite in terms of a finite set of internal variables that incorporates information on the intraphase fluctuations of the inelastic strain and that can be evaluated by mean-field homogenization techniques. In this work we provide an alternative derivation of the scheme, relying on the Cauchy-Schwarz inequality rather than the Legendre transform, and in so doing we expose the mathematical structure of the resulting approximation and generalize the exposition to fully anisotropic material systems.