Constitutive laws and existence questions in incompressible nonlinear elasticity. (English) Zbl 0648.73013

The paper is a real contribution to the study of incompressible nonlinear elastic materials. The first part of the paper is devoted to the specification of the general concept of incompressible constitutive laws and its mathematical justification, the concept of incompressible elastic bodies, and some results regarding the existence and uniqueness of hydrostatic pressure fields. The considered constitutive law prescribes only the deviatoric part of the Cauchy stress, while its trace appears as a kind of Lagrange multiplier associated with a restricted principle of virtual work. The second part of the paper contains local existence results and regularity properties for the pure displacement problem, with dead or live loads, of an incompressible elastic body around a natural state.
It is proved that if the body force is sufficiently small, in a suitable chosen space, then there exists a unique deformation in \(W^{m+2,q}(\Omega)^ 3\) and a unique pressure field in \(W^{m+1,q}(\Omega)^ 3/{\mathbb{R}}\) strongly satisfying the equilibrium equations. Here m is an integer and \(q\in (1,\infty)\), with \((m+1)q>3\) in the case of dead loads, and \(mq>3\) for live loads. The inverse and implicit function theorems and variational methods are used to obtain the above existence and regularity results. We mention that these results are of different nature from those obtained by J. M. Ball [Arch. Ration. Mech. Anal. 63, 337-403 (1977; Zbl 0368.73040)].
The paper is of high mathematical level and is throwing light upon some subtle aspects regarding the behaviour of incompressible materials. The author is carefully pointing out the origins of his results and their connections with the already known results in the field.
Reviewer: Gh.Gr.Ciobanu


74B20 Nonlinear elasticity
74B05 Classical linear elasticity
74A20 Theory of constitutive functions in solid mechanics
74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)


Zbl 0368.73040
Full Text: DOI


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