A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures

Abstract

In this paper, we address the construction of a prior stochastic model for nonGaussian deterministically-bounded positive-definite matrix-valued random fields in the context of mesoscale modeling of heterogeneous elastic microstructures. We first introduce the micromechanical framework and recall, in particular, Huet's Partition Theorem. Based on the latter, we discuss the nature of hierarchical bounds and define, under some given assumptions, deterministic bounds for the apparent elasticity tensor. Having recourse to the Maximum Entropy Principle under the constraints defined by the available information, we then introduce two random matrix models. It is shown that an alternative formulation of the boundedness constraints further allows constructing a probabilistic model for deterministically-bounded positive-definite matrix-valued random fields. Such a construction is presented and relies on a class of random fields previously defined. We finally exemplify the overall methodology considering an experimental database obtained from EBSD measurements and provide a simple numerical application.Key words: Micromechanics; Heterogeneous materials; Apparent elasticity tensor; Mesoscale modeling; Random field; Non-Gaussian. $ J. Guilleminot, A. Noshadravan, R. Ghanem and C. Soize, A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures,

A stochastic computational multiscale approach; Application to MEMS resonators

“…(33) and (34), can be expressed in terms of their corresponding Young's modulus E and Poisson ratio ν as C C C iso (E, ν) following Eq. (32). After solving the optimization problem, the bounds are found to be C C C iso L (E = 130.0 GPa, ν = 0.278) and C C C iso U (E = 187.9 GPa, ν = 0.181).…”

“…In [32] and [33], it was achieved with the help of a minimization procedure and Huet's partition theorem [40]. It can also be estimated directly from the stiffness matrix of the FE model following the developments in [16,17].…”

“…In order to ensure the existence of the expectation of the norm of the generated tensor inverses [29,30], we introduce a lower bound, as proposed in [31], when generating the meso-scale material tensors. Bounds were also introduced in the random field generator in the other works [32,33]. The generation of the spatially correlated random matrices is achieved by generating a multivariate random field.…”

Design optimization under uncertainties of a mesoscale implant in biological tissues using a probabilistic learning algorithm

“…This model has been used for modeling and identifying cortical bone in [61,63]. Note that extensions of this model can be found in [30,31,86] for some positive-definite lower and upper bounds introduced as constraints and also for random fields with any material symmetry property. Let [K] = E{[K(x)]} be the mean matrix in M + n , which corresponds to the positive-definite fourth-order tensor…”

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].…”