Quasiconvex relaxation of isotropic functions in incompressible planar hyperelasticity

Martín, Randy; Voss, Jendrik; Neff, Patrizio; Ghiba, Ionel–Dumitrel

doi:10.1017/prm.2019.35

Cited by 6 publications

(9 citation statements)

References 27 publications

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“…W is rank-one convex if and only if φ is convex and nondecreasing. (1.8) These results also allow for an explicit calculation of the quasiconvex relaxation for conformally invariant and incompressible isotropic planar hyperelasticity [37,38].…”

Section: Introductionmentioning

confidence: 83%

“…with φ : [0, ∞) → R, and W SL is rank-one convex on SL(2) if and only if φ is monotone and convex [37]. For W SL = W | SL(2) , we find…”

Section: Condition D) the Inequality Can Be Obtained By Considering The Restrictionmentioning

confidence: 94%

“…Then rank-one convexity of W implies rank-one convexity [37] of W SL with respect to SL (2). Recall that any function W SL : SL(2) → R can be expressed in the form (1.8), i.e.,…”

Section: Condition D) the Inequality Can Be Obtained By Considering The Restrictionmentioning

confidence: 99%

See 2 more Smart Citations

Sharp Rank-One Convexity Conditions in Planar Isotropic Elasticity for the Additive Volumetric-Isochoric Split

Voss

Ghiba

Martín

et al. 2021

J Elast

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We consider the volumetric-isochoric split in planar isotropic hyperelasticity and give a precise analysis of rank-one convexity criteria for this case, showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce the conditions for rank-one convexity to a family of one-dimensional coupled differential inequalities. In particular, this allows us to derive a simple rank-one convexity classification for generalized Hadamard energies of the type $W(F)=\frac{\mu }{2} \hspace{0.07em} \frac{\lVert F \rVert ^{2}}{\det F}+f(\det F)$ W ( F ) = μ 2 ∥ F ∥ 2 det F + f ( det F ) ; such an energy is rank-one convex if and only if the function $f$ f is convex.

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Section: Introductionmentioning

confidence: 83%

“…with φ : [0, ∞) → R, and W SL is rank-one convex on SL(2) if and only if φ is monotone and convex [37]. For W SL = W | SL(2) , we find…”

Section: Condition D) the Inequality Can Be Obtained By Considering The Restrictionmentioning

confidence: 94%

See 1 more Smart Citation

Sharp Rank-One Convexity Conditions in Planar Isotropic Elasticity for the Additive Volumetric-Isochoric Split

Voss

Ghiba

Martín

et al. 2021

J Elast

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“…For n = 2, on the other hand, it is still an open question whether rank-one convexity implies quasiconvexity [6,8,13,18,21,31,[33][34][35]39]. In fact, many classes of functions on R 2×2 have been identified for which rank-one convexity even implies polyconvexity [7,19,25,26,32] and thus quasiconvexity.…”

Section: W ((1 − T)f + T(f + H))mentioning

confidence: 99%

A rank-one convex, nonpolyconvex isotropic function on with compact connected sublevel sets

Voss¹,

Ghiba

Martín

et al. 2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\textrm {GL}^{\!+}(2)$ of invertible $2\times 2$ - - matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander Mielke, it has been conjectured that the equivalence of rank-one convexity and polyconvexity holds for isotropic functions on $\textrm {GL}^{\!+}(2)$ as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function $W\colon \textrm {GL}^{\!+}(2)\to \mathbb {R}$ which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.

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“…For n = 2, on the other hand, it is still an open question whether rank-one convexity implies quasiconvexity [28,8,6,13,30,31,32]. In fact, many classes of functions on R 2×2 have been identified for which rank-one convexity even implies polyconvexity [7,29,22,18,23] and thus quasiconvexity.…”

Section: A Conjecture On Rank-one Convexity and Polyconvexitymentioning

confidence: 99%

A rank-one convex, non-polyconvex isotropic function on $\operatorname{GL}^+(2)$ with compact connected sublevel sets

Voss,

Ghiba,

Martin

et al. 2020

Preprint

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According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group GL + (2) of invertible 2 × 2 -matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander Mielke, it has been conjectured that the equivalence of rank-one convexity and polyconvexity holds for isotropic functions on GL + (2) as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function W : GL + (2) → R which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.

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Quasiconvex relaxation of isotropic functions in incompressible planar hyperelasticity

Cited by 6 publications

References 27 publications

Sharp Rank-One Convexity Conditions in Planar Isotropic Elasticity for the Additive Volumetric-Isochoric Split

Sharp Rank-One Convexity Conditions in Planar Isotropic Elasticity for the Additive Volumetric-Isochoric Split

A rank-one convex, nonpolyconvex isotropic function on with compact connected sublevel sets

A rank-one convex, non-polyconvex isotropic function on $\operatorname{GL}^+(2)$ with compact connected sublevel sets

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