A unified approach to invariants of plane elasticity tensors

Forte, Sandra; Vianello, Maurizio

doi:10.1007/s11012-014-9916-y

Cited by 18 publications

(27 citation statements)

References 10 publications

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“…It has been demonstrated that in 2D, there is only 4 different possibilities for the symmetry class of an elasticity tensor ( Forte and Vianello, 2014;He and Zheng, 1996;Vianello, 1997 ):…”

Section: Symmetry Classes and Stratamentioning

confidence: 99%

“…6 This is achieved through a two-step process ( Forte and Vianello, 2014;Itin and Hehl, 2013;2014;Vianello, 1997 ):…”

Section: Irreducible Decompositionmentioning

confidence: 99%

“…Those invariants were discovered and rediscovered at different times, hence multiple names are attached to them. But as demonstrated in an important review paper ( Forte and Vianello, 2014 ) these different approaches are strictly equivalent.…”

Section: Introductionmentioning

confidence: 99%

“…These last years have seen a renew of interest toward the use of tensor invariants in continuum mechanics Desmorat and Vannucci, 2013;Forte and Vianello, 2014;Olive and Auffray, 2014;de Saxcé and Vallée, 2013 ). This renew is driven by the will to better understand links that exist between microstructure and overall elastic behavior.…”

Section: Introductionmentioning

confidence: 99%

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Invariant-based reconstruction of bidimensional elasticity tensors

Auffray

Ropars

2016

International Journal of Solids and Structures

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a b s t r a c tIn the present paper a method to reconstruct the 2D elasticity tensor from observations is proposed. The novelty of the method, compared to other ones, is that this approach is based on the identification of the invariants, rather than the components, of the elasticity tensor. The main advantage is that all the information concerning the material, such as its symmetry class and the orientation of the tested sample, are obtained during the reconstruction process. We believe that such an approach based on intrinsic quantities may find interesting applications for the identification of mechanical parameters based on fullfield measurements.

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“…It has been demonstrated that in 2D, there is only 4 different possibilities for the symmetry class of an elasticity tensor ( Forte and Vianello, 2014;He and Zheng, 1996;Vianello, 1997 ):…”

Section: Symmetry Classes and Stratamentioning

confidence: 99%

“…6 This is achieved through a two-step process ( Forte and Vianello, 2014;Itin and Hehl, 2013;2014;Vianello, 1997 ):…”

Section: Irreducible Decompositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Invariant-based reconstruction of bidimensional elasticity tensors

Auffray

Ropars

2016

International Journal of Solids and Structures

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“…The analog problem in plane elasticity for the elasticity tensor in bidimensional space under the action of the orthogonal group O(2) has already been solved by numerous authors [89,44,14,90,37,7]).…”

Section: Introductionmentioning

confidence: 99%

A Minimal Integrity Basis for the Elasticity Tensor

Olive

Kolev

Auffray

2017

Arch Rational Mech Anal

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We definitively solve the old problem of finding a minimal integrity basis of polynomial invariants of the fourth-order elasticity tensor C. Decomposing C into its SO(3)-irreducible components we reduce this problem to finding joint invariants of a triplet (a, b, D), where a and b are second-order harmonic tensors, and D is a fourth-order harmonic tensor. Combining theorems of classical invariant theory and formal computations, a minimal integrity basis of 297 polynomial invariants for the elasticity tensor is obtained for the first time.

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Quadratic Invariants of the Elasticity Tensor

Itin

2016

J Elast

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We study the quadratic invariants of the elasticity tensor in the framework of its unique irreducible decomposition. The key point is that this decomposition generates the direct sum reduction of the elasticity tensor space. The corresponding subspaces are completely independent and even orthogonal relative to the Euclidean (Frobenius) scalar product. We construct a basis set of seven quadratic invariants that emerge in a natural and systematic way. Moreover, the completeness of this basis and the independence of the basis tensors follow immediately from the direct sum representation of the elasticity tensor space. We define the Cauchy factor of an anisotropic material as a dimensionless measure of a closeness to a pure Cauchy material and a similar isotropic factor is as a measure for a closeness of an anisotropic material to its isotropic prototype. For cubic crystals, these factors are explicitly displayed and cubic crystal average of an arbitrary elastic material is derived.

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A unified approach to invariants of plane elasticity tensors

Cited by 18 publications

References 10 publications

Invariant-based reconstruction of bidimensional elasticity tensors

Invariant-based reconstruction of bidimensional elasticity tensors

A Minimal Integrity Basis for the Elasticity Tensor

Quadratic Invariants of the Elasticity Tensor

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