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Linear stress-strain relations in nonlinear elasticity. (English) Zbl 0990.74010

Acta Mech. 146, No. 1-2, 109-113 (2001); comments and reply ibid. 171, No. 3-4, 241-245 (2004).
Summary: We obtain nonlinear strain measures which are compatible with the existence of elastic potential and lead to a linear stress-strain relation. An unusual property of these measures is their dependence on material parameters. Standard strain measures commonly used in nonlinear elasticity are shown to be consistent with linear stress-strain relations only for particular cases of Poisson’s ratio. We also present the corresponding potentials for these cases.
Comment (summary) by K.Y. Volokh: Recently this journal published a work stating that the idea of geometrical nonlinearity within Hooke’s law is “no more than a widely accepted illusion since the linear stress-strain laws hold only for very nontrivial measures representing the corresponding strain tensor which depend on material parameters”. Since the linear stress-strain relations with nonlinear strains are, indeed, widely used in research and design the arguments of the authors of this work should be considered. Below it is shown where a flaw in these arguments is and why Hooke’s law with nonlinear strains is correct.

MSC:

74B20 Nonlinear elasticity
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[1] Antman, S. S.: The theory of rods. In: Handbuch der Physik, vol. VI a/2. Berlin: Springer 1972.
[2] Antman, S. S.: Nonlinear problems of elasticity. Applied Mathematical Sciences107. New York: Springer 1994.
[3] Drozdov, A.: Finite elasticity and viscoelasticity. Singapore: World Scientific 1996. · Zbl 0839.73001
[4] Eliseev, V. V.: The nonlinear dynamics of elastic rods. Appl. Math. Mech.52, 493-498 (1988). · Zbl 0708.73039
[5] Lurie, A. I.: Nonlinear theory of elasticity. North-Holland Series in Applied Mathematics and Mechanics 36. Amsterdam New York: North Holland 1990.
[6] Truesdell, C.: A first course in rational continuum mechanics. Maryland: The Johns Hopkins University 1972.
[7] Truesdell, C., Noll, W.: The nonlinear field theories of mechanics. In: Handbuch der Physik, vol. III/3, pp. 1-602. Berlin: Springer 1965. · Zbl 1068.74002
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