Uncertainty analysis of effective elasticity tensors using quaternion-based global optimization and Monte-Carlo method

Danek, Tomasz; Kochetov, Mikhail; Slawiński, Michael A.

doi:10.1093/qjmam/hbt004

Cited by 17 publications

(30 citation statements)

References 21 publications

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“…The elasticity parameters of the effective transversely isotropic tensor relative to the weighted Frobenius norm are obtained using the Gram matrix, in this case 5 × 5, whose entries are given in expression (9). The effective parameters-under an a priori assumption that the x 3 -axis coincides with the rotation-symmetry axis-arẽ …”

Section: Effective Transversely Isotropic Tensormentioning

confidence: 99%

See 1 more Smart Citation

Effective Elasticity Tensors in Context of Random Errors

Danek¹,

Kochetov

Slawiński

2015

J Elast

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We introduce the effective elasticity tensor of a chosen material-symmetry class to represent a measured generally anisotropic elasticity tensor by minimizing the weighted Frobenius distance from the given tensor to its symmetric counterpart, where the weights are determined by the experimental errors. The resulting effective tensor is the highestlikelihood estimate within the specified symmetry class. Given two material-symmetry classes, with one included in the other, the weighted Frobenius distance from the given tensor to the two effective tensors can be used to decide between the two models-one with higher and one with lower symmetry-by means of the likelihood ratio test.

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Section: Effective Transversely Isotropic Tensormentioning

confidence: 99%

“…(In this paper, we use the term "experimental errors" sensu lato, since c cannot be measured directly: a Hookean solid is itself a model of a physical material.) In that context, Danek et al [9] used the Monte-Carlo method to examine error perturbations of effective tensors determined by the unweighted Frobenius norm for all symmetry classes.…”

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confidence: 99%

Effective Elasticity Tensors in Context of Random Errors

Danek¹,

Kochetov

Slawiński

2015

J Elast

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“…In this case, the identification problem is reduced to determination of the symmetric part, which is the closest in some metric to a given tensor, as e.g., in [38][39][40][41][42][43][44][45]. Note that depending on the choice of the metric, the separated parts may have special properties.…”

Section: Introductionmentioning

confidence: 99%

“…There may exist multiple local and global minimums, so that the direct use of determined numerical methods is obstructed. However, for the majority of practical applications, there is no need for an exact approximation of the solutions to a global minimum, so that the optimization problems can be solved by using the heuristic algorithms, e.g., the particle swarm optimization method [71] employed in [39,41]. It seems plausible to use as a criterion of assigning a material by its properties to one or another class a small value of the residual caused by the related approximation in its constitutive equation.…”

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confidence: 99%

On Elastic Symmetry Identification for Polycrystalline Materials

Трусов¹,

Ostapovich²

2017

Symmetry

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Abstract:The products made by the forming of polycrystalline metals and alloys, which are in high demand in modern industries, have pronounced inhomogeneous distribution of grain orientations. The presence of specific orientation modes in such materials, i.e., crystallographic texture, is responsible for anisotropy of their physical and mechanical properties, e.g., elasticity. A type of anisotropy is usually unknown a priori, and possible ways of its determination is of considerable interest both from theoretical and practical viewpoints. In this work, emphasis is placed on the identification of elasticity classes of polycrystalline materials. By the newly introduced concept of "elasticity class" the union of congruent tensor subspaces of a special form is understood. In particular, it makes it possible to consider the so-called symmetry classification, which is widely spread in solid mechanics. The problem of identification of linear elasticity class for anisotropic material with elastic moduli given in an arbitrary orthonormal basis is formulated. To solve this problem, a general procedure based on constructing the hierarchy of approximations of elasticity tensor in different classes is formulated. This approach is then applied to analyze changes in the elastic symmetry of a representative volume element of polycrystalline copper during numerical experiments on severe plastic deformation. The microstructure evolution is described using a two-level crystal elasto-visco-plasticity model. The well-defined structures, which are indicative of the existence of essentially inhomogeneous distribution of crystallite orientations, were obtained in each experiment. However, the texture obtained in the quasi-axial upsetting experiment demonstrates the absence of significant macroscopic elastic anisotropy. Using the identification framework, it has been shown that the elasticity tensor corresponding to the resultant microstructure proves to be almost isotropic.

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“…Several researchers-among them, Gazis et al [13], Moakher and Norris [18], Kochetov and Slawinski [16,17], Danek et al [9]-examined relations between a generally anisotropic Hookean solid and its symmetric counterparts, which invoke the concept of distance within the space of Hookean solids. Voigt [23] and Norris [19] examined, in particular, relations between a generally anisotropic solid and its most symmetric counterpart: the closest isotropic solid.…”

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confidence: 99%

2-Norm Effective Isotropic Hookean Solids

Bos

Slawiński

2014

J Elast

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In the physical realm, an elasticity tensor that is computed based on measured numerical quantities with resulting numerical errors does not belong to any symmetry class for two reasons: (1) the presence of errors, and more intrinsically, (2) the fact that the symmetry classes in question are properties of Hookean solids, which are mathematical objects, not measured physical materials. To consider a good symmetric model for the mechanical properties of such a material, it is useful to compute the distance between the measured tensor and the symmetry class in question. One must then of course decide on what norm to use to measure this distance.The simplest case is that of the isotropic symmetry class. Typically, in this case, it has been common to use the Frobenius norm, as there is then an analytic expression for the closest element and it is unique. However, for other symmetry classes this is no longer the case: there are no analytic formulas and the closest element is not known to be unique.Also, the Frobenius norm treats an n × n matrix as an n 2 -vector and makes no use of the matrices, or tensors, as linear operators; hence, it loses potentially important geometric information. In this paper, we investigate the use of an operator norm of the tensor, which turns out to be the operator Euclidean norm of the 6 × 6 matrix representation of the tensor, in the expectation that it is more closely connected to the underlying geometry.We characterize the isotropic tensors that are closest to a given anisotropic tensor, and show that in certain circumstances they may not be unique. Although this may be a computational disadvantage in comparison to the use of the Frobenius norm-which has analytic expressions-we suggest that, since we work with only 6 × 6 matrices, there is no need to be extremely efficient and, hence, geometrical fidelity must trump computational considerations.

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Uncertainty analysis of effective elasticity tensors using quaternion-based global optimization and Monte-Carlo method

Cited by 17 publications

References 21 publications

Effective Elasticity Tensors in Context of Random Errors

Effective Elasticity Tensors in Context of Random Errors

On Elastic Symmetry Identification for Polycrystalline Materials

2-Norm Effective Isotropic Hookean Solids

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