After the classical researches of Christoffel, Hugoniot, Hadamard, and Duhem on waves in elastic materials, it might seem that little remains to be learned. Such is not the case. As for most parts of mechanics, it has been necessary in the last decade to go over the matter again, not only so as to free the conceptual structure from lingering linearizing and to fix it more solidly in the common foundation of modern mechanics, but also so as to derive from it specific predictions satisfying modern needs for contact between theory and rationally conceived experiment. After reading the recent papers by Toupin & Bernstein [1961, 1] and by Hayes & Rivlin [1961, 2], I have seen that more can be learned than is there proved. In the present paper I follow Toupin & Bernstein’s approach to the general theory yet try to achieve the elegant and explicit directness of Ericksen’s earlier treatment of isotropic incompressible materials . At the same time, all the results of Hayes & Rivlin are obtained in shorter but more general form as immediate corollaries.
Elastic Material Wave Speed Isotropic Material Hyperelastic Material Singular Surface
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Baker, M., & J. L. Ericksen: Inequalities restricting the form of the stress- deformation relations for isotropic solids and Reiner-Rivlin fluids. J. Washington Acad. Sci. 44, 33–35. (1954)MathSciNetGoogle Scholar
Truesdell, C.: Das ungelöste Hauptproblem der endlichen Elastizitätstheorie. Z. angew. Math. Mech. 36, 97–103. (1954)MathSciNetGoogle Scholar
Noll, W.: A mathematical theory of the mechanical behavior of continuous media. Arch. Rational Mech. Anal. 2 (1958/59), 197–226.ADSzbMATHCrossRefGoogle Scholar
Rivlin, R. S.: The constitutive equations for certain classes of deformations. Arch. Rational Mech. Anal. 3, 304–311. (1959)MathSciNetzbMATHGoogle Scholar
Rivlin, R.S.: Constitutive equations for classes of deformations. Viscoelasticity: Phenomenological Aspects. New York: Academic Press 93–108. (1960)Google Scholar
Toupin, R. A., & B. Bernstein: Sound waves in deformed perfectly elastic materials; the acoustoelastic effect. J. Acoust. Soc. Amer. 33, 216–225. (1961)MathSciNetADSCrossRefGoogle Scholar
Hayes, M., & R. S. Rivlin: Propagation of a plane wave in an isotropic elastic material subject to pure homogeneous deformation. Arch. Rational Mech. Anal. 8, 15–22. (1961)MathSciNetADSzbMATHCrossRefGoogle Scholar
Toupin, R. A.: Some relations between waves, stability, uniqueness criteria, and restrictions on the form of the energy function in elasticity theory. Lecture delivered at a meeting in Newcastle-upon-Tyne. (1961)Google Scholar