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,
In this work, we address the constitutive modeling, in a probabilistic framework, of the hyperelastic response of soft biological tissues. The aim is on the one hand to mimic the mean behavior and variability that are typically encountered in the experimental characterization of such materials, and on the other hand to derive mathematical models that are almost surely consistent with the theory of nonlinear elasticity. Towards this goal, we invoke information theory and discuss a stochastic model relying on a low-dimensional parametrization. We subsequently propose a two-step methodology allowing for the calibration of the model using standard data, such as mean and standard deviation values along a given loading path. The framework is finally applied and benchmarked on three experimental databases proposed elsewhere in the literature. It is shown that the stochastic model allows experiments to be accurately reproduced, regardless of the tissue under consideration.
Abstract. This paper is concerned with the construction of a new class of generalized nonparametric probabilistic models for matrix-valued non-Gaussian random fields. More specifically, we consider the case where the random field may take its values in some subset of the set of real symmetric positive-definite matrices presenting sparsity and invariance with respect to given orthogonal transformations. Within the context of linear elasticity, this situation is typically faced in the multiscale analysis of heterogeneous microstructures, where the constitutive elasticity matrices may exhibit some material symmetry properties and may then belong to a given subset M sym n (R) of the set of symmetric positive-definite real matrices. First of all, we present an overall methodology relying on the framework of information theory and define a particular algebraic form for the random field. The representation involves two independent sources of uncertainties, namely one preserving almost surely the topological structure in M sym n (R) and the other one acting as a fully anisotropic stochastic germ. Such a parametrization does offer some flexibility for forward simulations and inverse identification by uncoupling the level of statistical fluctuations of the random field and the level of fluctuations associated with a stochastic measure of anisotropy. A novel numerical strategy for random generation is subsequently proposed and consists in solving a family of Itô stochastic differential equations. The algorithm turns out to be very efficient when the stochastic dimension increases and allows for the preservation of the statistical dependence between the components of the simulated random variables. A Störmer-Verlet algorithm is used for the discretization of the stochastic differential equation. The approach is finally exemplified by considering the class of almost isotropic random tensors.
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
This paper is devoted to the modeling of compressible hyperelastic materials whose response functions exhibit uncertainties at some scale of interest. The construction of parametric probabilistic representations for the Ogden class of stored energy functions is specifically considered and formulated within the framework of Information Theory. The overall methodology relies on the principle of maximum entropy, which is invoked under constraints arising from existence theorems and consistency with linearized elasticity. As for the incompressible case discussed elsewhere, the derivation essentially involves the conditioning of some variables on the stochastic bulk and shear moduli, which are shown to be statistically dependent random variables in the present case. The explicit construction of the probability measures is first addressed in the most general setting. Subsequently, particular results for classical Neo‐Hookean and Mooney‐Rivlin materials are provided. Salient features of the probabilistic representations are finally highlighted through forward Monte‐Carlo simulations. In particular, it is seen that the models allow for the reproduction of typical experimental trends, such as a variance increase at large stretches. A stochastic multiscale analysis, where uncertainties on the constitutive law of the matrix phase are taken into account through the proposed approach, is also presented.
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