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Symmetry and bifurcation in three-dimensional elasticity. (English) Zbl 0536.73011

[For parts I and II see ibid. 80, 295-331 (1982; Zbl 0509.73018), and the review above (Zbl 0536.73010).]
Without the physical assumption that the undeformed state of a body is stress free (i.e., the manifold may or may not have an embedding in \({\mathbb{R}}^ 3\) on which the stress function is zero), but with the assumption that the stress is the derivative of a potential, a function of the deformation gradient, the authors complete their description of the equilibrium solutions for a system of forces acting on this body, counting the number of such solutions and their stability. Besides this the authors extend their investigation to the existence and number of solutions to the bifurcation of the stress potential under material symmetry.
Examples are given of formal Signorini series, of loads and shapes of various kinds. The treatment is mainly theoretical and heavily mathematical. These three papers make a major contribution to the use of mathematics in continuum mechanics.
Reviewer: J.J.Cross

MSC:

74B20 Nonlinear elasticity
74G60 Bifurcation and buckling
35B32 Bifurcations in context of PDEs
35B35 Stability in context of PDEs
74G99 Equilibrium (steady-state) problems in solid mechanics
74H99 Dynamical problems in solid mechanics
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References:

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