On the Statistical Dependence for the Components of Random Elasticity Tensors Exhibiting Material Symmetry Properties

Abstract

This work is concerned with the characterization of the statistical dependence between the components of random elasticity tensors that exhibit some given material symmetries. Such an issue has historically been addressed with no particular reliance on probabilistic reasoning, ending up in almost all cases with independent (or even some deterministic) variables. Therefore, we propose a contribution to the field by having recourse to the Information Theory. Specifically, we first introduce a probabilistic methodology that allows for such a dependence to be rigorously characterized and which relies on the Maximum Entropy (Max-Ent) principle. We then discuss the induced dependence for the highest levels of elastic symmetries, ranging from isotropy to orthotropy. It is shown for instance that for the isotropic class, the bulk and shear moduli turn out to be independent Gamma-distributed random variables, whereas the associated stochastic Young modulus and Poisson ratio are statistically dependent random variables. Keywords Elasticity tensor • Statistical dependence • Elastic moduli • Probabilistic model • MaxEnt approach Mathematics Subject Classification (2000) 15A52 • 60B20 • 74A40 • 74B05 Johann Guilleminot and Christian Soize, On the statistical dependence for the components of random elasticity tensors exhibiting material symmetry properties, accepted for publication

Stochastic isotropic hyperelastic materials: constitutive calibration and model selection

“…Next, following [23][24][25][26][27], for the random nonlinear shear modulus µ(a 0 ), defined by (3.7), we set the mathematical expectations: 12) where, by the constraint (3.11), the mean value µ(a 0 ) is fixed and greater than zero, and the logarithmic constraint (3.12) implies that both µ(a 0 ) and µ(a 0 ) −1 are second-order random variables (i.e. they have finite mean and finite variance).…”

“…Specifically, stochastichyperelastic models were identified from experimental data consisting of the mean values and standard deviations of elastic stresses under finite strain deformations. Prior to this, in [26,27], a similar strategy was applied to the stochastic representation of tensor-valued random variables and random fields in linear elasticity. These strategies rely on the maximum entropy principle for a discrete probability distribution introduced by [28][29][30].…”

Effective Elasticity Tensors in Context of Random Errors

“…Herein, expressions (19)-(23) are stated as an example; they are not as immediately useful as expressions (16) and (17), because, unlike the latter, they give only a conditional effective tensor in the sense of the weighted Frobenius norm: the closest transversely isotropic tensor among those whose rotation-symmetry axis coincides with the x 3 -axis. In general, finding the effective transversely isotropic tensor requires an optimization under all orientations of the symmetry axis.…”

“…Recently, there has been much work, notably by Guilleminot and Soize [16,17] and by Noshadravan et al [22] on developing probabilistic models for random Hookean solids that ensure that the elasticity tensor satisfies the theoretical constraints of index symmetries and positive definiteness. Generally speaking, the more complex is the model, the bigger sample of realizations of the random elasticity tensor is required to fit the parameters.…”

Hybrid Sampling/Spectral Method for Solving Stochastic Coupled Problems

“…Further, we acknowledge that our choice of stochastic model is rather arbitrary; in engineering applications, adequate stochastic models can be inferred from experimental data using mathematical statistics methods or constructed using information-theoretic methods [10].…”