Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications

高等学校計算数学学報 = Numerical mathematics

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.

http://www.mate.tue.nl/mate/pdfs/2091.pdf

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

https://hal.archives-ouvertes.fr/hal-01389078/document

A computational approach to handle complex microstructure geometries

Multi-scale second-order computational homogenization of multi-phase materials : a nested finite element solution strategy

Effective properties of composite materials with periodic microstructure : a computational approach

A numerical method making use of fast Fourier transforms has been proposed in Moulinec and Suquet (1994, 1998) to investigate the effective properties of linear and non-linear composites. This method is based on an iterative scheme the rate of convergence of which is proportional to the contrast between the phases. Composites with high contrast (typically above 104) or infinite contrast (those containing voids or rigid inclusions or highly non-linear materials) cannot be handled by the method. This paper presents two modified schemes. The first one is an accelerated scheme for composites with high contrast which extends to elasticity a scheme initially proposed in Eyre and Milton (1999). Its rate of convergence varies as the square root of the contrast. The second scheme, adequate for composites with infinite contrast, is based on an augmented Lagrangian method. The resulting saddle-point problem involves three steps. The first step consists of solving a linear elastic problem, using the fast Fourier transform method. In the second step, a non-linear problem is solved at each individual point in the volume element. The third step consists of updating the Lagrange multiplier. Applications of this scheme to rigidly reinforced and to voided composites are shown. Copyright © 2001 John Wiley & Sons, Ltd.

A computational scheme for linear and non-linear composites with arbitrary phase contrast

https://hal.archives-ouvertes.fr/hal-00088331/document

Jean-Claude Michel

Papers

Effective properties of composite materials with periodic microstructure : a computational approach

A computational scheme for linear and non-linear composites with arbitrary phase contrast

Nonuniform transformation field analysis

A Computational Method Based on Augmented Lagrangians and Fast Fourier Transforms for Composites with High Contrast

Computational analysis of nonlinear composite structures using the Nonuniform Transformation Field Analysis