Matrix representations for 3D strain-gradient elasticity

Auffray, Nicolas; Quang, H. Le; He, Qi‐Chang

doi:10.1016/j.jmps.2013.01.003

Cited by 109 publications

(107 citation statements)

References 42 publications

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“…For A the question was solved in 2D by Auffray et al (2009a), the space of sixth-order tensors is more complex since it is divided in 8 classes. For the 3D case, the number of symmetry classes increases since C is now divided into 8 classes (Forte and Vianello, 1996), and A into 17 classes (Olive and Auffray, 2013;Auffray et al, 2013). At the present time, these questions remain open for the fifth-order tensor spaces M and M ♯ , both in 2D and 3D.…”

Section: Synthesismentioning

confidence: 92%

“…Until now, C and A, the vector spaces of C and A, have been investigated, both in a 2D and 3D euclidean spaces (Mehrabadi and Cowin, 1990;Forte and Vianello, 1996;Auffray et al, 2009aAuffray et al, , 2013. Also, the answers to the following three questions have been provided: (a) How many symmetry classes and which symmetry classes do C and A have?…”

Section: Synthesismentioning

confidence: 99%

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A complete description of bi-dimensional anisotropic strain-gradient elasticity

Auffray

Dirrenberger

Rosi

2015

International Journal of Solids and Structures

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102

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In the present paper spaces of fifth-order tensors involved in bidimensional strain gradient elasticity are studied. As a result complete sets of matrices representing these tensors in each one of their anisotropic system are provided. This paper completes and ends some previous studies on the subject providing a complete description of the anisotropic bidimensional strain gradient elasticity. It is proved that this behavior is divided into 14 non equivalent anisotropic classes, 8 of them being isotropic for classical elasticity. The classification and matrix representations of the acoustical gyrotropic tensor are also provided, these results may find interesting applications to the study of waves propagation in dispersive micro-structured-media.

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Section: Synthesismentioning

confidence: 92%

Section: Synthesismentioning

confidence: 99%

A complete description of bi-dimensional anisotropic strain-gradient elasticity

Auffray

Dirrenberger

Rosi

2015

International Journal of Solids and Structures

Self Cite

102

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“…The anisotropic case involving a large number of additional parameters is much more challenging from the identification perspective. Extended homogenization procedures can be used to identify the whole set of parameters from the consideration of an underlying periodic microstructure, as reviewed in [68][69][70].…”

Section: (B) Scalar Microstrainmentioning

confidence: 99%

“…The higher order term involves a sixth-rank tensor of elasticity moduli which is symmetric and assumed definite positive [70]. The stress-strain relations (4.6) become…”

Section: (A) Finite Microstrain Tensor Modelmentioning

confidence: 99%

Nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage

Forest¹

2016

Proc. R. Soc. A.

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International audienceThe construction of regularization operators presented in this work is based on the introduction of strain or damage micromorphic degrees of freedom in addition to the displacement vector and of their gradients into the Helmholtz free energy function of the constitutive material model. The combination of a new balance equation for generalized stresses and of the micromorphic constitutive equations generates the regularization operator. Within the small strain framework, the choice of a quadratic potential w.r.t. the gradient term provides the widely used Helmholtz operator whose regularization properties are well known: smoothing of discontinuities at interfaces and boundary layers in hardening materials, and finite width localization bands in softening materials. The objective is to review and propose nonlinear extensions of micromorphic and strain/damage gradient models along two lines: the first one introducing nonlinear relations between generalized stresses and strains; the second one envisaging several classes of finite deformation model formulations. The generic approach is applicable to a large class of elastoviscoplastic and damage models including anisothermal and multiphysics coupling. Two standard procedures of extension of classical constitutive laws to large strains are combined with the micromorphic approach: additive split of some Lagrangian strain measure or choice of a local objective rotating frame. Three distinct operators are finally derived using the multiplicative decomposition of the deformation gradient. A new feature is that a free energy function depending solely on variables defined in the intermediate isoclinic configuration leads to the existence of additional kinematic hardening induced by the gradient of a scalar micromorphic variable

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“…Therefore Ela M is divided into 28 symmetry classes 7 .This large number of symmetry classes has to be compared with the 8 symmetry classes of classical elasticity [6], and the 17 symmetry classes of second order elasticity [24,25]. Straightforward applications of our formula now give •Type I subgroups …”

Section: Mindlin Strain-gradient Elasticitymentioning

confidence: 99%

Analytical expressions for odd-order anisotropic tensor dimension

Auffray

2014

Comptes Rendus. Mécanique

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According to the symmetries of the matter the number of coefficients needed to define a tensorial relation varies. It is well-known that in linear elasticity the number of generic coefficients varies from 21 for a complete anisotropic material to 2 in case of isotropy. In a previous contribution, we provided analytical expressions that give the number of generic anisotropic coefficients in any anisotropic system for an even-order tensor. In the present note, we aim at extending the previous results to the case of odd-order tensors. As an illustration, the dimension of any anisotropic system for third-order piezoelectricity tensors and of the fifth-order coupling tensors of Mindlin's strain-gradient elasticity are determined. To cite this article: N. Auffray, C. R. Mécanique 341 (2013). RésuméExpressions analytiques donnant la dimension d'un tenseur anisotrope d'ordre impair. En fonction des symétries qu'un milieu possède, le nombre de coefficients génériques nécessaireà la définition d'une loi tensorielle varie. Dans le contexte de l'élasticité linéaire, si le milieu ne presente aucune symmétrie 21 coefficientś elastiques sont génériquement nécessaire, tandis que dans le cas de l'isotropie ce nombre se réduità 2. Dans une précédente note, nous avions dérivé des formules analytiques donnant, dans le cas d'un tenseur pair, le nombre de coefficients génériques nécessaire pour chaque type d'anisotropie. Le but de cette nouvelle contribution, est de compléter ces formules en lesétendant au cas des tenseurs impairs. En guise d'illustration, nous calculerons, pour l'ensemble des systèmes d'anisotropie possibles, la dimension des tenseurs piezoélectriques (ordre 3) ainsi que des tenseurs de couplage de la théorie de l'élasticitéà gradient (ordre 5). Pour citer cet article : N. Auffray, C. R. Mécanique 341 (2013).

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Matrix representations for 3D strain-gradient elasticity

Cited by 109 publications

References 42 publications

A complete description of bi-dimensional anisotropic strain-gradient elasticity

A complete description of bi-dimensional anisotropic strain-gradient elasticity

Nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage

Analytical expressions for odd-order anisotropic tensor dimension

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