A numerical two-scale homogenization scheme: the FE2-method
A wide class of micro-heterogeneous materials is designed to satisfy the advanced challenges of modern materials occurring in a variety of technical applications. The effective macroscopic properties of such materials are governed by the complex interacti
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J¨org Schr¨ oder Institute of mechanics, Dept. Civil Engineering, Fac. of Engineering, University of Duisburg-Essen Corresponding author: [email protected] Abstract A wide class of micro-heterogeneous materials is designed to satisfy the advanced challenges of modern materials occurring in a variety of technical applications. The effective macroscopic properties of such materials are governed by the complex interaction of the individual constituents of the associated microstructure. A sufficient macroscopic phenomenological description of these materials up to a certain order of accuracy can be very complicated or even impossible. On the contrary, a whole resolution of the fine scale for the macroscopic boundary value problem by means of a classical discretization technique seems to be too elaborate. Instead of developing a macroscopic phenomenological constitutive law, it is possible to attach a representative volume element (RVE ) of the microstructure at each point of the macrostructure; this results in a two-scale modeling scheme. A discrete version of this scheme performing finite element (FE) discretizations of the boundary value problems on both scales, the macro- and the micro-scale, is denoted as the FE2 -method or as the multilevel finite element method. The main advantage of this procedure is based on the fact that we do not have to define a macroscopic phenomenological constitutive law; this is replaced by suitable averages of stress measures and deformation tensors over the microstructure. Details concerning the definition of the macroscopic quantities in terms of their microscopic counterparts, the definition/construction of boundary conditions on the RVE as well as the consistent linearization of the macroscopic constitutive equations are discussed in this contribution. Furthermore, remarks concerning stability problems on both scales as well as their interactions are given and representative numerical examples for elasto-plastic microstructures are discussed.
J. Schröder, K. Hackl (Eds.), Plasticity and Beyond, CISM International Centre for Mechanical Sciences, DOI 10.1007/978-3-7091-1625-8_1, © CISM, Udine 2014
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Introduction
For the analysis of micro-heterogeneous materials, we define two different scales, the macroscopic scale (coarse scale) and the microscopic scale (fine scale). The fine scale is assumed to be the scale of the heterogeneities of characteristic length l, whereas the characteristic length of the coarse scale is denoted by L. If we assume that the domain size at the fine scale is sufficient for homogenization requirements, then the separation of scales expressed by lL (1) has to hold. A homogenized – that means effective macroscopic – description of the micro-heterogeneous material requires the definition of a representative volume element (RVE) or a statistically homogeneous volume element, which is here assumed to be possible. In classical works, effective quantities of micro-heterogeneous media, such as stiffness or compliance tensors, have been discussed by Voigt (1910
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