Abstract
We construct two-dimensional, two-phase random heterogeneous microstructures by stochastic simulation using the planar Boolean model, which is a random collection of overlapping grains. The structures obtained are discretized using finite elements. A heterogeneous Neo-Hooke law is assumed for the phases of the microstructure, and tension tests are simulated for ensembles of microstructure samples. We determine effective material parameters, i.e., the effective Lamé moduli lambda ^* and mu ^*, on the macroscale by fitting a macroscopic material model to the microscopic stress data, using stress averaging over many microstructure samples. The effective parameters lambda ^* and mu ^* are considered as functions of the microscale material parameters and the geometric parameters of the Boolean model including the grain shape. We also consider the size of the Representative Volume Element (RVE) given a precision and an ensemble size. We use structured and unstructured meshes and also provide a comparison with the FE^2 method.
Highlights
The problem of relating effective constitutive parameters of composite material laws to their microstructure is common in many areas, e.g., in material science as well as in geoscience
In our computational homogenization approach, we use a hyperelastic, homogeneous comparison material and identify its parameters by minimizing the squares of the residuals in the equilibrium equation. This parameter identification method is well known in mechanics and often denoted equilibrium gap method (EGM) [9]; see Sect. 4.2
The notion of a representative volume element (RVE) was introduced by Hill [21]: the RVE should be as small as possible and large enough to be representative of the material; see [14] for an overview on the topic
Summary
The problem of relating effective constitutive parameters of composite material laws to their microstructure is common in many areas, e.g., in material science as well as in geoscience. Computational Mechanics inclusions in a matrix material These grains may have different shapes and sizes and their density is a model parameter. By applying numerical homogenization, using many microstructure samples, we obtain effective material parameters on the macroscale for various parameter sets of the Boolean model. The numerical homogenization approach is based on applying parameter identification to the microscopic problem, assuming a given material model on the macroscale. A sufficient fit could be achieved assuming a Neo-Hooke material model on the macroscale This has been observed earlier by Löhnert et al, who have fitted a macroscopic Neo-Hooke law to microscopic data [58,59,60]. We will use identified macro material parameters to perform a comparison with simulations using the FE2 computational homogenization method. For a comprehensive view on homogenization in solid mechanics, we refer to [43,64,73,76] and the references therein
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