On Bhattacharyya relative entropy in a homogenization of composite materials

Abstract

The main idea of this work is an application of relative entropy in the numerical analysis of probabilistic divergence between original material tensors of the composite constituents and its effective tensor in the presence of material uncertainties. The homogenization method is based upon the deformation energy of the representative volume elements for the fiber‐reinforced and particulate composites and uncertainty propagation begins with elastic moduli of the fibers, particles, and composite matrices. Relative entropy follows a mathematical model originating from Bhattacharyya probabilistic divergence and has been applied here for Gaussian distributions. The semi‐analytical probabilistic method based on analytical integration of polynomial bases obtained via the least squares method fittings enables for determination of the basic probabilistic characteristics of the effective tensor and the relative entropies. The methodology invented in this work may be extended toward other probability distributions and relative entropies, for homogenization of nonlinear composites and also accounting for some structural interface defects.

Uncertainty propagation, entropy, and relative entropy in the homogenization of some particulate composites

“…It has been documented that qualitative results obtained with Shannon and relative entropies agree very well with those obtained before using probabilistic moments analyses, 15,25 so the proposed methodology looks promising for engineering problems in composites. It should be mentioned that the research presented in this paper is an extension of the initial application of Bhattacharyya relative entropy in the homogenization method based on the interface stress approach presented recently by the author as a communication 54 …”

“…It should be mentioned that the research presented in this paper is an extension of the initial application of Bhattacharyya relative entropy in the homogenization method based on the interface stress approach presented recently by the author as a communication. 54 The paper consists of four sections beyond this Introduction. The first one presents a mathematical formulation of the homogenization problem, its entropy, and its relative entropy determination.…”

Probabilistic relative entropy in elasto‐plasticity using the iterative generalized stochastic stress‐based Finite Element Method

“…This is completed not only to validate such a probabilistic approach in addition to the Monte‐Carlo simulation or versus some semi‐analytical technique. Another research goal here is to determine the reliability index 12 using the traditional First Order Reliability Method (FORM) 13 and to contrast it with the relative probabilistic entropy apparatus 14 . Preliminary results document with no doubt an efficient determination of the first four probabilistic characteristics of the resulting stresses.…”

On probabilistic entropies application in uniaxial deformation of hyper‐elastic materials

“…A deformation of a hyper‐elastic specimen is treated as Gaussian random variable having a priori given expectation varying within the given experimental bounds and some specific standard deviation relevant to the experimental error level. Probabilistic computational analysis is delivered here with the use of the stochastic perturbation technique [3] and this analysis includes numerical determination of strain‐dependent fluctuations of the first two probabilistic characteristics, that is expectations and coefficients of variations of the resulting tensile stress and has been focused on Shannon entropy calculation [4]. Finally, a relative entropy proposed by Bhattacharyya [5] is introduced to quantify a distance in‐between increasing tensile stress value and its admissible counterpart.…”