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

Abstract 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.

[1]  Pierre Suquet,et al.  Computational analysis of nonlinear composite structures using the Nonuniform Transformation Field Analysis , 2004 .

[2]  Sia Nemat-Nasser,et al.  On two micromechanics theories for determining micro–macro relations in heterogeneous solids , 1999 .

[3]  Kenjiro Terada,et al.  Two-scale kinematics and linearization for simultaneous two-scale analysis of periodic heterogeneous solids at finite strain , 2003 .

[4]  David L. McDowell,et al.  A multiscale multiplicative decomposition for elastoplasticity of polycrystals , 2003 .

[5]  Christian Miehe,et al.  Strain‐driven homogenization of inelastic microstructures and composites based on an incremental variational formulation , 2002 .

[6]  Damijan Markovic,et al.  Strong coupling methods in multi-phase and multi-scale modeling of inelastic behavior of heterogeneous structures , 2003 .

[7]  V. Kouznetsova,et al.  Multi-scale second-order computational homogenization of multi-phase materials : a nested finite element solution strategy , 2004 .

[8]  J. Schröder,et al.  Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains , 1999 .

[9]  R. Hill.  A self-consistent mechanics of composite materials , 1965 .

[10]  J. Michel,et al.  Effect of a nonuniform distribution of voids on the plastic response of voided materials: a computational and statistical analysis , 2005 .

[11]  M. Gurtin,et al.  An introduction to continuum mechanics , 1981 .

[12]  E Weinan,et al.  Multiscale modeling of the dynamics of solids at finite temperature , 2005 .

[13]  Sergiy Melnyk,et al.  Shape optimization of two-phase inelastic material with , 2005 .

[14]  W. Becker,et al.  Effective stiffness and microscopic deformation of an orthotropic plate containing arbitrary holes , 2004 .

[15]  Sia Nemat-Nasser,et al.  Averaging theorems in finite deformation plasticity , 1999 .

[16]  Kohei Yuge,et al.  Two-scale finite element analysis of heterogeneous solids with periodic microstructures , 2004 .

[17]  Stefanie Reese,et al.  Meso-macro modelling of fibre-reinforced rubber-like composites exhibiting large elastoplastic deformation , 2003 .

[18]  M. Lambrecht,et al.  Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals , 2002 .

[19]  Serdar Göktepe,et al.  A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage , 2005 .

[20]  C. Miehe,et al.  Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy , 2003 .

[21]  J. Michel,et al.  Effective properties of composite materials with periodic microstructure : a computational approach , 1999 .