Coordinate-free Characterization of the Symmetry Classes of Elasticity Tensors

Bóna, Andrej; Bucataru, Ioan; Slawiński, Michael A.

doi:10.1007/s10659-007-9126-0

Cited by 29 publications

(49 citation statements)

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“…When the random elasticity tensor is modeled by using the random matrix ensemble SE + defined in [45] [46] (see also §2.3.1), it can be shown that the mean and variance of the anisotropy measure increases almost linearly with the level of fluctuations of the elasticity tensor [21]. This property follows from the fact that each material symmetry can be defined (as a necessary -but not sufficient-condition) by the algebraic multiplicities of the elasticity tensor eigenvalues [40] [6], which are all equal to one for random matrices belonging to SE + and to other classical ensembles (because of the repulsion phenomena [32]). Two types of methods have been proposed so far in the literature for addressing the case when the two levels of fluctuations have to be controlled apart from one another.…”

Section: ∀X ∈ ω [C(x)] = [Q][c(x)][q]mentioning

confidence: 99%

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

Guilleminot¹,

Soize²

2013

Multiscale Model. Simul.

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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.

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Section: ∀X ∈ ω [C(x)] = [Q][c(x)][q]mentioning

confidence: 99%

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

Guilleminot¹,

Soize²

2013

Multiscale Model. Simul.

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“…The following cubic tensor, which has this orthonormal basis as its natural basis, has been proposed by Moakher and Norris [18]: with multiplicities m 1 = 1, m 2 = 2 and m 3 = 3, as expected in view of the coordinate-free characterization formulated by Bóna et al [4]. As we will see later, eigenvalue (4.5) is an orthogonal invariant.…”

Section: Distance To Cubic Symmetrymentioning

confidence: 81%

“…We refer to matrix (2.7) as Kelvin's notation [26, p. 110]. This notation has been used by several researchers; notably, Fedorov [11], Helbig [15], Chapman [5] and Bóna et al [4]. For subsequent use, we invoke the scalar product given by…”

Section: An Orthogonal Transformation a ∈ O(3) Acts On The Elasticitymentioning

confidence: 99%

Invariant Properties for Finding Distance in Space of Elasticity Tensors

Bucataru

Slawiński

2008

J Elasticity

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“…Major topics of interest in which the concept has been used are: the use of six eigenstiffnesses and orthogonal eigenstates for a better understanding of material behaviour (Rychlewski 1984;Annin and Ostrosablin 2008); different aspects of a spectral decomposition of the stiffness tensor (Theocaris and Philippidis 1991;Theocaris 2000;Bolcu et al 2010); the investigation of material symmetries and preferred deformation modes of anisotropic media, e.g., composite materials (Mehrabadi and Cowin 1990;Bóna et al 2007) including the relationship to fabric tensors (Moesen et al 2012) and deformation-induced anisotropy (Cowin 2011); the transformation of the properties of one anisotropic medium to the closest effective medium from a differing symmetry group (Norris 2006;Diner et al 2011;Kochetov and Slawinski 2009;Moakher and Norris 2006); wave attenuation and elastic constant inversion from wave traveltime data (Carcione et al 1998;Dellinger et al 1998). The inversion of Hooke's law in the case of incompressible or slightly compressible materials was studied by Itskov and Aksel (2002), while the use of the spectral decomposition of the stiffness tensor in a constitutive formulation for finite hyperelasticity in a finite element context was described in Dłuzewski and Rodzik (1998).…”

Section: The Kelvin Mappingmentioning

confidence: 99%

On advantages of the Kelvin mapping in finite element implementations of deformation processes

et al. 2016

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Classical continuum mechanical theories operate on three-dimensional Euclidian space using scalar, vector, and tensor-valued quantities usually up to the order of four. For their numerical treatment, it is common practice to transform the relations into a matrix-vector format. This transformation is usually performed using the so-called Voigt mapping. This mapping does not preserve tensor character leaving significant room for error as stress and strain quantities follow from different mappings and thus have to be treated differently in certain mathematical operations. Despite its conceptual and notational difficulties having been pointed out, the Voigt mapping remains the foundation of most current finite element programmes. An alternative is the so-called Kelvin mapping which has recently gained recognition in studies of theoretical mechanics. This article is concerned with benefits of the Kelvin mapping in numerical modelling tools such as finite element software. The decisive difference to the Voigt mapping is that Kelvin's method preserves tensor character, and thus the numerical matrix notation directly corresponds to the original tensor notation. Further benefits in numerical implementations are that tensor norms are calculated identically without distinguishing stress-or strain-type quantities, and tensor equations can be directly transformed into matrix equations without additional considerations. The only implementational changes are related to a scalar factor in certain finite element matrices, and hence, harvesting the mentioned benefits comes at very little cost.

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Coordinate-free Characterization of the Symmetry Classes of Elasticity Tensors

Cited by 29 publications

References 0 publications

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

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

Invariant Properties for Finding Distance in Space of Elasticity Tensors

On advantages of the Kelvin mapping in finite element implementations of deformation processes

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