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**Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity.**
*(English)*
Zbl 0589.73017

This paper concerns the establishment of the uniqueness of solutions in simple displacement boundary value problems in the nonlinear theory of homogeneous hyperelasticity for smooth equilibrium configurations satisfying certain boundary conditions, for which the stored energy vector W is of rank-one-convex and it is strictly quasi-convex at an (n\(\times n)\)-matrix F, such that \(x\to Fx+b\), where b is an n-constant vector. In such cases it was proved that the homogeneous deformation of the body is the only smooth equilibrium solution.

Furthermore, for the interesting case of radial deformations of an isotropic sphere it may, then, be shown that all smooth radial solutions are of the form \(x\to \lambda x\) \((\lambda >0)\), provided that the stored energy W is either strictly rank-one-convex and W is strictly quasi- convex, or W is strictly rank-one-convex and W is quasi convex. At the end of the paper the previous results are adpated to boundary value problems for incompressible materials.

Furthermore, for the interesting case of radial deformations of an isotropic sphere it may, then, be shown that all smooth radial solutions are of the form \(x\to \lambda x\) \((\lambda >0)\), provided that the stored energy W is either strictly rank-one-convex and W is strictly quasi- convex, or W is strictly rank-one-convex and W is quasi convex. At the end of the paper the previous results are adpated to boundary value problems for incompressible materials.

Reviewer: P.S.Theocaris

### MSC:

74B20 | Nonlinear elasticity |

74G30 | Uniqueness of solutions of equilibrium problems in solid mechanics |

74H25 | Uniqueness of solutions of dynamical problems in solid mechanics |

### Keywords:

equilibrium solutions; quasiconvexity; stored energy density; displacement boundary value problems; homogeneous hyperelasticity; smooth equilibrium configurations
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\textit{R. J. Knops} and \textit{C. A. Stuart}, Arch. Ration. Mech. Anal. 86, 233--249 (1984; Zbl 0589.73017)

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### References:

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