On the equivalence between spatial and material volume averaging of stress in large strain multi-scale solid constitutive models

This paper discusses some equivalence relationships for large strain multi-scale solid constitutive models based on the volume averaging of the microscopic stress and deformation gradient fields over a representative volume element (RVE). The analysis is carried out within a purely kinematically-based variational framework. Sufficient conditions are established under which the volume average of the microscopic first Piola–Kirchhoff stress over the reference (or material) configuration of the RVE is mechanically equivalent to the average of the microscopic Cauchy stress field over the deformed (or spatial) configuration. The Taylor (or homogeneous deformation) assumption, the affine boundary condition (or linear RVE boundary displacements) as well as the periodic boundary displacement fluctuations constraint – all widely adopted in multi-scale constitutive modelling – satisfy the sufficient conditions. For these classes of models, the choice of the material or spatial configuration for stress volume averaging is immaterial. Some averaging equivalence relations involving the homogenised Kirchhoff stress are also briefly discussed.