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Ellipticity and deformations with discontinuous gradients in finite elastostatics. (English) Zbl 0731.73023
This is an interesting (and, for the explicitness of its results, I would say endearing) contribution to the problem in the title. Circumstances are already known where loss of convexity of the elastic potential is synonymous with the existence of equilibrium shocks. The main goal of the author here is to deduce, in a constructive way, conditions on the constitutive hyperelastic law which are necessary and sufficient for the material to sustain equilibrium shocks; but there are also a number of side results, regarding the invertibility of a traction response mapping, which offer a hint for a mechanical interpretation of the ellipticity condition. A section is devoted to the derivation of conditions for the existence of equilibrium deformations with piecewise constant gradients; an explicit representation is achieved of all pairs of associated deformation gradients, hence the question of existence of shocks is answered in a constructive manner.
Reviewer: G.Capriz (Pisa)

MSC:
74B20 Nonlinear elasticity
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