Numerical homogenization ofN-component composites including stochastic interface defects

Abstract

The purpose of this paper is to present a mathematical formulation and numerical analysis for a homogenization problem of random elastic composites with stochastic interface defects. The homogenization of composites so de"ned is carried out in two steps: (i) probabilistic averaging of stochastic discontinuities in the interphase region, (ii) probabilistic homogenization by extending the e!ective modules method to media random in the micro-scale. To obtain such an approach the classical mathematical homogenization method is formulated for n-component composite with random elastic components and implemented in the FEMbased computer program. The article contains also numerous computational experiments illustrating stochastic sensitivity of the model to interface defects parameters and verifying statistical convergence of probabilistic simulation procedure. saturated media [20}22]. Nowadays, in deterministic homogenization, new numerical methods have been developed such as the self-consistent Mori}Tanaka model [23] or FEM-like bounds on e!ective properties [24]; numerous new applications of known methods have been discussed [25,26]. On the other hand, problems of composites strength in the context of their e!ective properties [27,28], homogenization of multiphase media [29], inverse homogenization problems [30] and many others have also been studied. It should be underlined that the problem discussed is very often related to optimization techniques [28,31,32] being an important aspect considering numerous engineering applications of the latter. Such problems are solved by using variational upper and lower bounds of the e!ective composite characteristics [28,33,34].Stochastic concepts in the area of homogenization appeared "rst in the late 1960s [1]. They gradually evolved in the 1970s [35] and 1980s [36}38], but these approaches were either simple variational upper and lower bounds or did not lead to any numerical realizations. Advanced mathematical theories [39, 40] and computational methodologies have been emerging only recently. The most important appears to be: the Delaunay networks approach [41], the Voronoi Cell Finite Element Method [42}43] and the Monte-Carlo method [40, 44].All the methods developed so far, no matter whether deterministic or probabilistic, have one general disadvantage*they as a rule neglect defects appearing on the boundary between the composite components (some deterministic models were presented in [5, 45}46]). It has been proved by laboratory tests that these defects are the dominating factors in the evaluation of non-linear behavior and the strength of composite structures [47,48]. There are many recent papers dealing with deterministically modeled interface defects considering physical mechanisms [47,49] and computational aspects [50}52]. On the other hand, the micro-mechanical formulations for the contact problems [53}56] neglect their real random character e!ectively considering the expected values of these defects only.A "rst formulation for a random composite with stochastic interf...

Multiscale homogenization of n-component composites with semi-elliptical random interface defects

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