On the variationally consistent computational homogenization of elasticity in the incompressible limit

Abstract

Background: Computational homogenization is a well-established approach in material modeling with the purpose to account for strong micro-heterogeneity in an approximate fashion without excessive computational cost. However, the case of macroscopically incompressible response is still unresolved. Methods: The computational framework for Variationally Consistent Homogenization (VCH) of (near) incompressible solids is discussed. A canonical formulation of the subscale problem, pertinent to a Representative Volume Element (RVE), is established, whereby complete macroscale incompressibility is obtained as the limit situation when all constituents are incompressible. Results: Numerical results for single RVEs demonstrate the seamless character of the computational algorithm at the fully incompressible limit. Conclusions: The suggested framework can seamlessly handle the transition from (macroscopically) compressible to incompressible response. The framework allows for the classical boundary conditions on the RVE as well as the generalized situation of weakly periodic boundary conditions.