Effective hyperelastic material parameters from microstructures constructed using the planar Boolean model

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 $$\lambda ^*$$ λ ∗ and $$\mu ^*$$ μ ∗ , on the macroscale by fitting a macroscopic material model to the microscopic stress data, using stress averaging over many microstructure samples. The effective parameters $$\lambda ^*$$ λ ∗ and $$\mu ^*$$ μ ∗ 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$$^2$$ 2 method.

Probabilistic Analysis of Composite Materials with Hyper-Elastic Components

“…Some other methods include averaging and tolerance-averaging methods [224,225], hydrodynamic limit methods [226], the Mori-Zwanzig formalism [227], renormalization group methods [228], variational methods [229], and homogenization methods [230][231][232][233][234][235][236]. The multiscale approach is very frequently used for the determination of the macroscopic (or effective) properties of composites, also in the stochastic context [237,238]. Some introduction to their theory is available, for example, in [239,240].…”