Non‐Gaussian positive‐definite matrix‐valued random fields with constrained eigenvalues: Application to random elasticity tensors with uncertain material symmetries

Johann Guilleminot · Christian Soize

31 citations0/0 files storedInternational Journal for Numerical Methods in Engineering

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Probabilistic and Robust Engineering Design Elasticity and Material Modeling Composite Material Mechanics

Abstract

Abstract This paper is devoted to the construction of a class of prior stochastic models for non‐Gaussian positive‐definite matrix‐valued random fields. The proposed class allows the variances of selected random eigenvalues to be specified and exhibits a larger number of parameters than the other classes previously derived within a nonparametric framework. Having recourse to a particular characterization of material symmetry classes, we then propose a mechanical interpretation of the constraints and subsequently show that the probabilistic model may allow prescribing higher statistical fluctuations in given directions. Such stochastic fields turn out to be especially suitable for experimental identification under material symmetry uncertainties, as well as for the development of computational multi‐scale approaches where the randomness induced by fine‐scale features may be taken into account. We further present a possible strategy for inverse identification, relying on the sequential solving of least‐square optimization problems. An application is finally provided. Copyright © 2011 John Wiley & Sons, Ltd.

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References

1991Stochastic Finite Elements: A Spectral Approach 2004Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure 2000A nonparametric model of random uncertainties for reduced matrix models in structural dynamics 2007Microstructural Randomness and Scaling in Mechanics of Materials 2005Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators 2005Maximum likelihood estimation of stochastic chaos representations from experimental data 2008Statistical volume element method for predicting microstructure–constitutive property relations 2009Generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions

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