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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Vydáno v:	International journal for numerical methods in engineering Ročník 87; číslo 1-5; s. 149 - 170
Hlavní autoři:	Perić, D., de Souza Neto, E. A., Feijóo, R. A., Partovi, M., Molina, A. J. Carneiro
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Chichester, UK John Wiley & Sons, Ltd 08.07.2011 Wiley
Témata:	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
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Abstract	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.
AbstractList	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. 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.
Author	de Souza Neto, E. A. Perić, D. Partovi, M. Molina, A. J. Carneiro Feijóo, R. A.
Author_xml	– sequence: 1 givenname: D. surname: Perić fullname: Perić, D. email: d.peric@swansea.ac.uk organization: Civil and Computational Engineering Centre, School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, U.K – sequence: 2 givenname: E. A. surname: de Souza Neto fullname: de Souza Neto, E. A. organization: Civil and Computational Engineering Centre, School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, U.K – sequence: 3 givenname: R. A. surname: Feijóo fullname: Feijóo, R. A. organization: Laboratório Nacional de Computação Científica (LNCC/MCT), Av. Getúlio Vargas, 333, Quitandinha, Petrópolis-Rio de Janeiro, CEP 25651-075, Brazil – sequence: 4 givenname: M. surname: Partovi fullname: Partovi, M. organization: Civil and Computational Engineering Centre, School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, U.K – sequence: 5 givenname: A. J. Carneiro surname: Molina fullname: Molina, A. J. Carneiro organization: Civil and Computational Engineering Centre, School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, U.K
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Keywords	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
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Snippet	This work describes a homogenization‐based multi‐scale procedure required for the computation of the material response of non‐linear microstructures undergoing... This work describes a homogenization-based multi-scale procedure required for the computation of the material response of non-linear microstructures undergoing...
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SubjectTerms	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
Title	On micro-to-macro transitions for multi-scale analysis of non-linear heterogeneous materials: unified variational basis and finite element implementation
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