A Minimal Integrity Basis for the Elasticity Tensor

Olive, Marc; Kolev, Boris; Auffray, Nicolas

doi:10.1007/s00205-017-1127-y

Cited by 33 publications

(49 citation statements)

References 84 publications

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“…However, the authors did not provide a final answer to the problem. A minimal set of 297 generators for the invariant algebra of the 3D elasticity tensor was finally obtained in 2017, by some of the present authors, in [29], which definitively solved this old problem (see also [31] for a tensorial expression of these generators, who were first expressed in [29] using transvectants of binary forms).…”

Section: Introductionmentioning

confidence: 86%

Generic separating sets for three-dimensional elasticity tensors

et al. 2019

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We define what is a generic separating set of invariant functions (a.k.a. a weak functional basis) for tensors. We produce then two generic separating sets of polynomial invariants for 3D elasticity tensors, one made of 19 polynomials and one made of 21 polynomials (but easier to compute) and a generic separating set of 18 rational invariants. As a byproduct, a new integrity basis for the fourth-order harmonic tensor is provided.Date: January 1, 2019. 2010 Mathematics Subject Classification. 74E10 (15A72 74B05).

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Section: Introductionmentioning

confidence: 86%

Generic separating sets for three-dimensional elasticity tensors

et al. 2019

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“…It was noted that the Olive-Auffray integrity basis is actually a minimal integrity basis [14]. In 2017, Olive, Kolev and Auffray [16] gave a minimal integrity basis of the elasticity tensors, with 297 invariants. Very recently, a number of new results appeared.…”

Section: Introductionmentioning

confidence: 99%

An irreducible polynomial functional basis of two-dimensional Eshelby tensors

Zhang

Ming

Chen

2019

Appl. Math. Mech.-Engl. Ed.

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Representation theorems for both isotropic and anisotropic functions are of prime importance in both theoretical and applied mechanics. In this article, we discuss about two-dimensional Eshelby tensors (denoted as M (2) ), which have wide applications in many fields of mechanics. Based upon the complex variable method, we obtain an integrity basis of ten isotropic invariants of M (2) . Since an integrity basis is always a polynomial functional basis, we further confirm that this integrity basis is also an irreducible polynomial functional basis of M (2) .(4.14)H, L, K, θ 1 , θ 2 and θ 3 are independent to each other. Moreover, we introduce six scalarvalued functions of H, L, K, θ 1 , θ 2 and θ 3 as below:(4.15) By some calculations, we have Thus, J 11 , . . . , J 16 are polynomials in J 1 , . . . , J 7 . In view of this, they are also polynomial invariants of M (2) and we only need to testify that any polynomial invariant of M (2) is polynomial in J 1 , . . . , J 16 .Recalling that each non-zero monomial

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“…In 1994, Boehler, Kirillov and Onat [1] studied the polynomial basis of anisotropic invariants of the elasticity tensor. In 2017, Olive, Kolev and Auffray [4] presented a minimal integrity basis of isotropic invariants of the elasticity tensor, with 297 invariants. It is well-known that the number of invariants with the same degree in a minimal integrity basis of some tensors is always fixed [7].…”

Section: Introductionmentioning

confidence: 99%

Two irreducible functional bases of isotropic invariants of a fourth-order three-dimensional symmetric and traceless tensor

Chen

et al. 2019

Mathematics and Mechanics of Solids

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The elasticity tensor is one of the most important fourth order tensors in mechanics. Fourth order three-dimensional symmetric and traceless tensors play a crucial role in the study of the elasticity tensors. In this paper, we present two isotropic irreducible functional bases of a fourth order three-dimensional symmetric and traceless tensor. One of them is the minimal integrity basis introduced by Smith and Bao in 1997. It has nine homogeneous polynomial invariants of degrees two, three, four, five, six, seven, eight, nine and ten, respectively. We prove that it is also an irreducible functional basis. The second irreducible functional basis also has nine homogeneous polynomial invariants. It has no quartic invariant but has two sextic invariants. The other seven invariants are the same as those of the Smith-Bao basis. Hence, the second irreducible functional basis is not contained in any minimal integrity basis. The Odd Degree Invariants in These Two BasesIn the Smith and Bao basis and the mixed functional basis, there are four odd degree invariants J 3 , J 5 , J 7 and J 9 , and six even degree invariants J 2 , J 4 , J 6 , K 6 , J 8 and J 10 . A notable property is that when D changes its sign, the odd degree invariants change their signs, while the even degree invariants are unchanged. Using this property, it is relatively easy to show that each of J 3 , J 5 , J 7 and J 9 is not a function of the other three odd degree invariants and the six even degree invariants J 2 , J 4 , J 6 , K 6 , J 8 and J 10 .Proposition 4.1. We have the following four conclusions.(a) If there is a fourth order three-dimensional symmetric and traceless tensor D such that J 5 = J 7 = J 9 = 0 but J 3 = 0, then J 3 is not a function of J 2 , J 4 , J 5 , J 6 , K 6 , J 7 , J 8 , J 9 and J 10 .(b) If there is a fourth order three-dimensional symmetric and traceless tensor D such that J 3 = J 7 = J 9 = 0 but J 5 = 0, then J 5 is not a function of J 2 , J 3 , J 4 , J 6 , K 6 , J 7 , J 8 , J 9 and J 10 .(c) If there is a fourth order three-dimensional symmetric and traceless tensor D such that J 3 = J 5 = J 9 = 0 but J 7 = 0, then J 7 is not a function of J 2 , J 3 , J 4 , J 5 , J 6 , K 6 , J 8 , J 9 and J 10 .(d) If there is a fourth order three-dimensional symmetric and traceless tensor D such that J 3 = J 5 = J 7 = 0 but J 9 = 0, then J 9 is not a function of J 2 , J 3 , J 4 , J 5 , J 6 , K 6 , J 7 , J 8 and J 10 .Proof. We now prove conclusion (a). If there is a fourth order three-dimensional symmetric and traceless tensor D such that J 5 = J 7 = J 9 = 0 but J 3 = 0, then we may consider −D.J 2 , J 4 , J 5 , J 6 , K 6 , J 7 , J 8 , J 9 and J 10 are unchanged, but J 3 changes its sign. This implies that J 3 is not a function of J 2 , J 4 , J 5 , J 6 , K 6 , J 7 , J 8 , J 9 and J 10 . The other three conclusions (b), (c) and (d) can be proved similarly.We now have the following theorem.Theorem 4.2. Each of J 3 , J 5 , J 7 and J 9 is not a function of the other three odd degree invariants and the six even degree invariants J 2 , J 4 , J 6 , ...

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A Minimal Integrity Basis for the Elasticity Tensor

Cited by 33 publications

References 84 publications

Generic separating sets for three-dimensional elasticity tensors

Generic separating sets for three-dimensional elasticity tensors

An irreducible polynomial functional basis of two-dimensional Eshelby tensors

Two irreducible functional bases of isotropic invariants of a fourth-order three-dimensional symmetric and traceless tensor

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