Invariant Properties for Finding Distance in Space of Elasticity Tensors

Bucataru, Ioan; Slawiński, Michael A.

doi:10.1007/s10659-008-9186-9

Cited by 19 publications

(11 citation statements)

References 25 publications

(65 reference statements)

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“…Several authors (see for instance Zener and Siegel (1949), Spoor et al (1995), Bucataru and Slawinski (2008), Moussaddy et al (2013), Ghossein and Lévesque (2014)) have proposed methods which can be used to estimate the deviation from mechanical isotropy. They differ from each other by the measure of the amplitude of the stiffness tensor (represented in matrix form adopting Voigt notation), as well as the number of the coefficients used in that measure.…”

Section: Definitions For Isotropymentioning

confidence: 99%

Numerically-aided 3D printed random isotropic porous materials approaching the Hashin-Shtrikman bounds

Zerhouni

Tarantino

Danas

2019

Composites Part B: Engineering

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The present study introduces a methodology that allows to combine 3D printing, experimental testing, numerical and analytical modeling to create random closed-cell porous materials with statistically controlled and isotropic overall elastic properties that are extremely close to the relevant Hashin-Shtrikman bounds.In this first study, we focus our experimental and 3D printing efforts to isotropic random microstructures consisting of single-sized (i.e. monodisperse) spherical voids embedded in a homogeneous solid matrix.The 3D printed specimens are realized by use of the random sequential adsorption method. A detailed FE numerical study allows to define a cubic representative volume element (RVE) by combined periodic and kinematically uniform (i.e. average strain or affine) boundary conditions. The resulting cubic RVE is subsequently assembled to form a standard dog-bone uniaxial tension specimen, which is 3D printed by use of a photopolymeric resin material. The specimens are then tested at relatively small strains by a proper multi-step relaxation procedure to obtain the effective elastic properties of the porous specimens.

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Section: Definitions For Isotropymentioning

confidence: 99%

Numerically-aided 3D printed random isotropic porous materials approaching the Hashin-Shtrikman bounds

Zerhouni

Tarantino

Danas

2019

Composites Part B: Engineering

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“…It is important to note that {Ã : A ∈ SO(3)} is a strict subgroup of SO (6); in other words, C is not required to be invariant under all orthogonal transformations in R 6 . If C is invariant under all A ∈ SO(3), then it is isotropic; its form is…”

Section: Notationmentioning

confidence: 99%

“…for all orthogonal matrices U, V ∈ R n×n , and, hence, is invariant under the embedding of SO (3) in SO (6), which is the operation stated by Eq. (3).…”

Section: Normsmentioning

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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“…Note that the definition and characterization of distances and associated projections have been extensively studied within the deterministic framework of theoretical elasticity: see [2] and [28] for discussions about distances in Ela; [29] and [4] for closed-form expressions of the projections expressed in matrix and vector forms respectively and [5] [21] [22] [23] for discussions regarding the definition of projections and closest approximations taking into account the symmetry reference frame. Physical and mathematical interpretations for the case of random elasticity matrices have been discussed quite recently in [12] [13] [14].…”

Section: Application Of the Generalized Approach To The Class Of Tranmentioning

confidence: 99%

Generalized stochastic approach for constitutive equation in linear elasticity: a random matrix model

Guilleminot

Soize

2011

Numerical Meth Engineering

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This work is concerned with the construction of stochastic models for random elasticity matrices, allowing either for the generation of elasticity tensors exhibiting some material symmetry properties almost surely (integrating the statistical dependence between the random stiffness components), or for the modeling of random media that requires the mean of a stochastic anisotropy measure to be controlled apart from the level of statistical fluctuations. To this aim, we first introduce a decomposition of the stochastic elasticity tensor on a deterministic tensor basis and consider the probabilistic modeling of the random components, having recourse to the MaxEnt principle. Strategies for random generation and estimation are further reviewed and the approach is exemplified in the case of a material that is transversely isotropic almost surely. In a second stage, we make use of such derivations to propose a generalized model for random elasticity matrices that takes into account, almost separately, constraints on both the level of stochastic anisotropy and the level of statistical fluctuations. An example is finally provided and shows the efficiency of the approach.

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Invariant Properties for Finding Distance in Space of Elasticity Tensors

Cited by 19 publications

References 25 publications

Numerically-aided 3D printed random isotropic porous materials approaching the Hashin-Shtrikman bounds

Numerically-aided 3D printed random isotropic porous materials approaching the Hashin-Shtrikman bounds

2-Norm Effective Isotropic Hookean Solids

Generalized stochastic approach for constitutive equation in linear elasticity: a random matrix model

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