Multi‐scale constitutive modelling of heterogeneous materials with a gradient‐enhanced computational homogenization scheme

2002/05/03 by V. Kouznetsova, V.G. Kouznetsova, M.G.D. Geers +3 · 810 citations
Engineering · Materials Science · Mathematics · #Boundary value problem #Cauchy stress tensor #Compatibility (geochemistry) #Composite Material Mechanics #Composite material #Constitutive equation #Continuum mechanics #Elasticity and Material Modeling #Finite element method #Geometry #Homogenization (climate) #Length scale #Materials science #Mathematical analysis #Mathematics #Mechanics #Microstructure #Nonlocal and gradient elasticity in micro/nano structures #Physics #Representative elementary volume #Tensor (intrinsic definition) #Thermodynamics

paper · doi:10.1002/nme.541

published in International Journal for Numerical Methods in Engineering 54(8), 1235-1260 (Wiley)

openalex publication_date 2002/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

Abstract A gradient‐enhanced computational homogenization procedure, that allows for the modelling of microstructural size effects, is proposed within a general non‐linear framework. In this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element (RVE). This enables us to incorporate the microstructural size and to account for non‐uniform macroscopic deformation fields within the microstructural cell. Every microstructural constituent is modelled as a classical continuum and the RVE problem is formulated in terms of standard equilibrium and boundary conditions. From the solution of the microstructural boundary value problem, the macroscopic stress tensor and the higher‐order stress tensor are derived based on an extension of the Hill–Mandel condition. This automatically delivers the microstructurally based constitutive response of the higher‐order macro continuum and deals with the microstructural size in a natural way. Several examples illustrate the approach, particularly the microstructural size effects. Copyright © 2002 John Wiley & Sons, Ltd.

Citations

• Elastic properties of reinforced solids: Some theoretical principles
• Elastic materials with couple-stresses

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