Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. (English) Zbl 0628.73018

In this very detailed work, the author examines the problem of uniqueness of certain three-dimensional regular and singular radial solutions to the equilibrium equations of nonlinear elasticity for an isotropic material. Two cases are considered - when the homogeneous elastic body in its reference state is a ball B and when it is a spherical shell \(B^{\epsilon}\). The uniqueness of regular and cavitating equilibria are established and this is the core result of the paper. It is then used to investigate the asymptotic behaviour of solutions to the mixed problem for \(B^{\epsilon}\) in the limit \(\epsilon\to 0\). The sense in which equilibrium solutions for \(B^{\alpha}\) approximate those for B when \(\epsilon\) is small, is also determined.
This is a well written paper whose results should be of interest for theoreticians working in the field of nonlinear elasticity. It should be mentioned, as done by the author, that exactly analogous results hold for corresponding two-dimensional problems.
Reviewer: H.Ramkissoon


74B20 Nonlinear elasticity
74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
74K25 Shells
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