Uniqueness and stability result for Cauchy’s equation of motion for a certain class of hyperelastic materials

Abstract

We consider Cauchy’s equation of motion for hyperelastic materials. The solution of this nonlinear initial-boundary value problem is the vector field which discribes the displacement which a particle of this material perceives when exposed to stress and external forces. This equation is of greatest relevance when investigating the behavior of elastic, anisotropic composites and for the detection of defects in such materials from boundary measurements. This is why results on unique solvability and continuous dependence from the initial values are of large interest in materials’ research and structural health monitoring. In this article we present such a result, provided that reasonable smoothness assumptions for the displacement field and the boundary of the domain are satisfied for a certain class of hyperelastic materials where the first Piola–Kirchhoff tensor is written as a conic combination of finitely many, given tensors.

Keywords:

• Cauchy’s equation of motion
• hyperelastic materials
• uniqueness and stability
• Cordes condition
• Gronwall’s lemma

AMS Subject Classifications:

• 35A01
• 35A02
• 35L20
• 35L70
• 74B20

Acknowledgements

The authors thank Dr. Frank Schöpfer and Dr. Frank Binder for many helpful discussions. This work was supported by the German Science Foundation (Deutsche Forschungsgemeinschaft, DFG) under Schu 1978/4-1 and Schu 1978/4-2.