Generalized stochastic approach for constitutive equation in linear elasticity: a random matrix model

Abstract

This work is concerned with the construction of stochastic models for random elasticity matrices, allowing either for the generation of elasticity tensors exhibiting some material symmetry properties almost surely (integrating the statistical dependence between the random stiffness components), or for the modeling of random media that requires the mean of a stochastic anisotropy measure to be controlled apart from the level of statistical fluctuations. To this aim, we first introduce a decomposition of the stochastic elasticity tensor on a deterministic tensor basis and consider the probabilistic modeling of the random components, having recourse to the MaxEnt principle. Strategies for random generation and estimation are further reviewed and the approach is exemplified in the case of a material that is transversely isotropic almost surely. In a second stage, we make use of such derivations to propose a generalized model for random elasticity matrices that takes into account, almost separately, constraints on both the level of stochastic anisotropy and the level of statistical fluctuations. An example is finally provided and shows the efficiency of the approach.

Stochastic Model and Generator for Random Fields with Symmetry Properties: Application to the Mesoscopic Modeling of Elastic Random Media

“…Such a strategy can be readily applied to all symmetry classes and allows the mean of the anisotropy measure to be imposed within a given range which depends on the mean value and level of statistical fluctuations of the elasticity tensor. An alternative generalized approach for random matrices has been proposed in [53] for the isotropic class and has been generalized to all symmetry classes in [19]. The proposed model is obtained by introducing a particular algebraic representation for fact that the later involves (and actually depends on) an additional measure -denoted by m in [26], Eq.…”

“…In this work, we build on the results obtained in [19] (for the random matrix case) and address the construction of a class of prior generalized stochastic models for elasticity tensor random fields. The main contributions are the derivation of a stochastic model for non-Gaussian tensor-valued random fields exhibiting some symmetry properties and the construction of a new random generator able to perform in high probabilistic dimensions.…”

“…In considering a particular algebraic decomposition of the non-Gaussian random field {[C(x)], x ∈ Ω}, namely: . Such an algebraic representation does allow for some flexibility in terms of anisotropy modeling (see [19] [53] for numerical evidences).…”

Multiscale Identification of the Random Elasticity Field at Mesoscale of a Heterogeneous Microstructure Using Multiscale Experimental Observations

“…We are interested in using a stochastic representation of C meso with a minimum of hyperparameters (dimension of vector b), which is adapted to the elliptic boundary value problem corresponding to the linear elastostatic problem. Parametric stochastic models have been proposed for real-valued stochastic fields [24,6,5,21], and for non-Gaussian tensor-valued random fields in the framework of the heterogeneous anisotropic linear elasticity [50,51,56,54,14], with important enhancements to take into account the material symmetry and the existence of elasticity bounds [25,26,27,28]. Hereinafter, the stochastic model for the apparent elastic tensorvalued random field C meso is based on the model proposed in [50] for a heterogeneous anisotropic microstructure at the mesoscale.…”

Stochastic Representations and Statistical Inverse Identification for Uncertainty Quantification in Computational Mechanics

“…An important step in the methodology is the construction and the use of algebraic prior stochastic models (APSM) of such a nonGaussian random field for which some advanced generators have been developed. The APSM and the associated generators presented and used hereinafter are those that have been published in [44,45,48,49,50,51,52,53,54,55,40,56,57,58]. Three illustrations are presented: the stochastic modeling of track irregularities for high-speed trains and its experimental identification [59], the stochastic continuum modeling of random interphases from atomistic simulations for a polymer nanocomposite [60], and the multiscale identification of the random elasticity field at mesoscale of a heterogeneous microstructure using multiscale experimental observations [61,62,63].…”