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The quasiconvex envelope of conformally invariant planar energy functions in isotropic hyperelasticity. (English) Zbl 1475.74012

The main result of the paper concerns an explicit formula for the quasiconvex enveloppe of conformally invariant elastic energies \(W\) in the two-dimensional setting. If \(\operatorname{GL}^+(2)\) denotes the group of \(2\times 2\) matrices with positive determinant, a function \(W:\operatorname{GL}^+(2)\rightarrow \mathbb R\) is called conformally invariant iff \(W(AFB)=W(F)\) for all \(A,B\in\{aR\in \operatorname{GL}^+(2)\vert a\in\mathbb{R}_+^\star\), \(R\in \operatorname{SO}(2)\}\) where \(\operatorname{SO}(2)\) denotes the special orthogonal group. Conformally invariance requirement is equivalent to isotropy (i.e., \(W(FR)=W(F)\) for all \(F\in \operatorname{GL}^+(2)\) and \(R\in \operatorname{SO}(2)\)) and material objectivity (i.e., \(W(QF)=W(F)\) for all \(F\in \operatorname{GL}^+(2)\) and \(Q\in \operatorname{SO}(2)\)) augmented by the requirement of invariance to isochoric deformations (i.e., \(W(aF)=W(F)\) for all \(F\in \operatorname{GL}^+(2)\) and \(a\in\mathbb{R}_+^\star\)). The objectivity and isotropy provide the representation of the energy in terms of the singular values of \(F\), i.e., \(W(F)=g(\lambda_1,\lambda_2)\) for some function \(g:(0,\infty)\times(0,\infty)\rightarrow \mathbb{R}\); the additional invariance with respect to isochoric deformations provides the representation \(W(F)=h(\lambda_1/\lambda_2)\) for some function \(h:(0,\infty)\rightarrow\mathbb{R}.\) The paper also discusses more general properties related to the \(W^{1,p}\)-quasiconvex envelope on \(\operatorname{GL}^+(n)\) which ensure that a version of Dacorogna formula can be applied to conformally invariant elastic energies in the two-dimensional setting.

MSC:

74B20 Nonlinear elasticity
74G65 Energy minimization in equilibrium problems in solid mechanics
30C62 Quasiconformal mappings in the complex plane
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