On micro-to-macro transitions for multi-scale analysis of non-linear heterogeneous materials: unified variational basis and finite element implementation

This work describes a homogenization‐based multi‐scale procedure required for the computation of the material response of non‐linear microstructures undergoing small strains. Such procedures are important for computer modelling of heterogeneous materials when the length‐scale of heterogeneities is s...

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Bibliographic Details
Published in:International journal for numerical methods in engineering Vol. 87; no. 1-5; pp. 149 - 170
Main Authors: Perić, D., de Souza Neto, E. A., Feijóo, R. A., Partovi, M., Molina, A. J. Carneiro
Format: Journal Article
Language:English
Published: Chichester, UK John Wiley & Sons, Ltd 08.07.2011
Wiley
Subjects:
Boundaries
Computation
Computer simulation
Exact sciences and technology
finite element
Finite element method
Fundamental areas of phenomenology (including applications)
homogenization
Inelasticity (thermoplasticity, viscoplasticity...)
Mathematical analysis
Mathematical models
micro-to-macro transitions
Microcavities
multi-scale analysis
Nonlinearity
Physics
Solid mechanics
Structural and continuum mechanics
Representative volume element
Averaging method
Non linear material
Small dimension
Constraint
finite element
multi-scale analysis
Homogenization
Boundary condition
Modeling
Axiomatic
micro-to-macro transitions
Global local method
Finite element method
Heterogeneous material
Inelasticity
Elastoplasticity
Multiscale method
Plasticity
Variational calculus
Non linear effect
Microstructure
ISSN:0029-5981, 1097-0207, 1097-0207
Online Access:Get full text
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Description
Summary:This work describes a homogenization‐based multi‐scale procedure required for the computation of the material response of non‐linear microstructures undergoing small strains. Such procedures are important for computer modelling of heterogeneous materials when the length‐scale of heterogeneities is small compared to the dimensions of the body. The described multi‐scale procedure relies on a unified variational basis which, apart from the continuum‐based variational formulation at both micro‐ and macroscales of the problem, also includes the variational formulation governing micro‐to‐macro transitions. This unified variational basis leads naturally to a generic finite element‐based framework for homogenization‐based multi‐scale analysis of heterogenous solids. In addition, the unified variational formulation provides clear axiomatic basis and hierarchy related to the choice of boundary conditions at the microscale. Classical kinematical constraints are considered over the representative volume element: (i) Taylor, (ii) linear boundary displacements, (iii) periodic boundary displacement fluctuations and (iv) minimal constraint, also known as uniform boundary tractions. In this context the Hill‐Mandel averaging requirement, which links microscopic and macroscopic stress power, plays a fundamental role in defining the microscopic forces compatible with the assumed kinematics. Numerical examples of both microscale and two‐scale finite element simulations of elasto‐plastic material with microcavities are presented to illustrate the main features and scope of the described computational strategy. Copyright © 2010 John Wiley & Sons, Ltd.
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ISSN:0029-5981
1097-0207
1097-0207
DOI:10.1002/nme.3014