Random Stiffness Tensor of Particulate Composites with Hyper-Elastic Matrix and Imperfect Interface.

Damian Sokołowski ,Marcin Kamiński

doi:10.3390/ma14216676

Abstract

The main aim of this study is determination of the basic probabilistic characteristics of the effective stiffness for inelastic particulate composites with spherical reinforcement and an uncertain Gaussian volume fraction of the interphase defects. This is determined using a homogenization method with a cubic single-particle representative volume element (RVE) of such a composite and the finite element method solution. A reinforcing particle is spherical, located centrally in the RVE, surrounded by the thin interphase of constant thickness, and remains in an elastic reversible regime opposite to the matrix, which is hyper-elastic. The interphase defects are represented as semi-spherical voids, which are placed on the outer surface of this particle. The interphase is modeled as hyper-elastic and isotropic, whose effective stiffness is calculated by the spatial averaging of hyper-elastic parameters of the matrix and of the defects. A constitutive relation of the matrix is recovered experimentally by its uniaxial stretch. The 3D homogenization problem solution is based upon a numerical determination of strain energy density in the given RVE under specific uniaxial and biaxial stretches as well as under shear deformations. The analytical relation of the effective composite stiffness to the input uncertain parameter is recovered via the response function method, using a polynomial basis and an optimized order. Probabilistic calculations are completed using three concurrent approaches, namely the iterative stochastic finite element method (SFEM), Monte Carlo simulation and by the semi-analytical method. Previous papers consider the composite fully elastic, which limits the applicability of the resulting effective stiffness tensor computed therein. The current study voids this assumption and defines the composite as fully hyper-elastic, thus extending applicability of this tensor to strains up to 0.25. The most important research finding is that (1) the effective stiffness tensor is sensitive to random interface defects in its hyper-elastic range, (2) its resulting randomness is not close to Gaussian, (3) the semi-analytical method is not perfectly suited to stochastic calculations in this region of strains, as opposed to the linear elastic region, and (4) that the increase in random dispersion of defects volume fraction has a much higher effect on the stochastic characteristics of this stiffness tensor than fluctuation of the strain.

Highlights

The characteristics of the uniaxial coefficient are exclusively shown in part (a), the biaxial ones in part (b) and shearing ones in part (c) of these graphs. Colors on these graphs distinguish the three independent probabilistic methods applied, i.e., the iterative stochastic finite element method (ISFEM, red color), the Monte Carlo simulation (MCS, green color) and the semi-analytical method (SAM, blue color) that were used in the computations
The ISFEM and SAM produce continuous results in the entire domain of α(w) and εij, while the MCS discrete points are evenly distributed through this domain, which is a trait of the MCS
Carlo simulation (MCS, green color) skewness and kurtosis is considered, either the returns theand results semi-analytical method (SAM, blue color) that were used in the computations

Summary

Introduction

Multiscale computational methods and simulations have attracted scientists and engineers for many years, which has been documented by the comprehensive review presented in [1] As is expected, such multiscale approaches are frequently connected with averaging and with the homogenization method [2,3], where these two last techniques serve for remarkable reduction of computational complexity and material heterogeneity on a micro- or nano-scale. As it is known, some specific model reduction techniques [4] have been concurrently resolved to minimize computer effort and preserve numerical accuracy. The homogenization method in the context of multiscale modelling is employed in fluid mechanics, to model transport phenomena in some human tissues [9], for thermal transport simulation in polymer nanocomposites [10], to model electromagnetic and elastic couplings in CFRP composites [11], and to simultaneously carry out molecular simulations and multiscale homogenization in seepage and diffusion problems [12]

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Results

Conclusion