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Must elastic materials be hyperelastic? (English) Zbl 1257.74017

Summary: Rivlin’s observation that elastic stress response that does not derive from a strain energy (Cauchy elasticity) admits negative work in closed cycles of deformations is demonstrated for specific isotropic stress responses and for a simple closed cycle of deformation. It is concluded that elastic materials must be hyperelastic.

MSC:

74B20 Nonlinear elasticity
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References:

[1] M. Reiner, Second-order Effects in Elasticity, Plasticity and Fluid Dynamics (1964) · Zbl 0125.00105
[2] Truesdell, C., Das ungelöste Hauptproblem der endlichen Elastizitätstheorie”, Zeitschrift für Angewandte Mathematik Mechanik (ZAMM) 36, 97-103 (1956). English translation in Foundations of Elasticity Theory (1965) · Zbl 0070.41703
[3] Baker, M., Inequalities restricting the form of the stress-deformation relations for isotropic elastic solids and Reiner-Rivlin fluids, Journal of the Washington Academy of Science 44, 33-35 (1954). Reprinted in Foundations of Elasticity Theory (1965)
[4] Batra, R., International Journal of Non-Linear Mechanics 36 pp 421– (2001) · Zbl 1345.74008
[5] Truesdell, C., The Non-linear Field Theories of Mechanics, Handbuch der Physik III/3 (1965)
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