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Dynamic cavitation with shocks in nonlinear elasticity

Published online by Cambridge University Press:  14 November 2011

K. A. Pericak-Spector  and
Scott J. Spector
K. A. Pericak-Spector
Affiliation:
Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901-4408, U.S.A
Scott J. Spector
Affiliation:
Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901-4408, U.S.A
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Abstract

The hyperbolic system of conservation laws that govern the motion of a homogeneous isotropic, nonlinearly elastic body is shown to have a discontinuous solution for a class of stored-energy functions of slow growth. This solution is admissible by the usual entropy criterion and is in fact preferred by the entropy-rate criterion over the smooth equilibrium solution to the same problem. The existence of such a dissipative solution shows that the equilibrium solution is dynamically unstable. This instability cannot be ascertained by linearisation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 4 , 1997 , pp. 837 - 857
Copyright
Copyright © Royal Society of Edinburgh 1997

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References

1Ball, J. M.. Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. Roy. Soc. London Ser. A 306 (1982), 557–611.Google Scholar
2Choski, R.. The singular limit of a hyperbolic system and the incompressible limit of solutions with shocks and singularities in nonlinear elasticity. Q. Appl. Math, (to appear).Google Scholar
3Chou-Wang, M. S. and Horgan, C. O.. Cavitation in nonlinear elastodynamics for neo-Hookean materials. Internat. J. Engrg. Sci. 27 (1989), 967–73.CrossRefGoogle Scholar
4Dafermos, C. M.. The entropy rate admissibility criterion for solutions of hyperbolic conservation laws. J. Differential Equations 14 (1973), 202–12.Google Scholar
5Dafermos, C. M.. Hyperbolic systems of conservation laws. In Systems of Nonlinear Partial Differential Equations, ed. Ball, J. M., 25–70 (Amsterdam: D. Reidel, 1983).Google Scholar
6Gent, A. N.. Cavitation in rubber: a cautionary tale. Rubber Chem. Tech. 63 (1991), G49–G53.CrossRefGoogle Scholar
7Gent, A. N. and Lindley, P. B.. Internal rupture of bonded rubber cylinders in tension. Proc. Roy. Soc. London Ser. A 249 (1958), 195–205.Google Scholar
8Gent, A. N. and Tompkins, D. A.. Surface energy effects for small holes or particles in elastomers. J. Polymer Sci. Part A–2 7 (1969), 1483–8.CrossRefGoogle Scholar
9Horgan, C. O. and Polignone, D. A.. Cavitation in nonlinearity elastic solids: A review. Appl. Mech. ROT. 48(1995), 471–85.CrossRefGoogle Scholar
10James, R. D. and Spector, S. J.. The formation of filamentary voids in solids. J. Mech. Phys. Solids 39 (1991), 783–813.CrossRefGoogle Scholar
11Lax, P. D.. Hyperbolic systems of conservation laws. Comm. Pure Appl. Math. 10 (1957), 537–66.Google Scholar
12Lax, P. D.. Shock waves and entropy. In Contributions to Functional Analysis, ed. Zaratonelo, E. A., 603–34 (New York: Academic Press, 1976).Google Scholar
13Müller, S. and Spector, S. J.. An existence theory for nonlinear elasticity that allows for cavitation. Arch. Rational Mech. Anal. 131 (1995), 1–66.Google Scholar
14Ogden, R. W.. Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids. Proc. Roy. Soc. London Ser. A 328 (1972), 567–83.Google Scholar
15Pericak-Spector, K. A. and Spector, S. J.. Nonuniqueness for a hyperbolic system: Cavitation in nonlinear elastodynamics. Arch. Rational Mech. Anal. 101 (1988), 293–317.CrossRefGoogle Scholar
16Shearer, M.. Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type. Arch. Rational Mech. Anal. 93 (1986), 45–59.CrossRefGoogle Scholar
17Sivaloganathan, J.. Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. Arch. Rational Mech. Anal. 96 (1986), 97–136.CrossRefGoogle Scholar
18Slemrod, M.. Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Rational Mech. Anal. 18 (1983), 301–15.Google Scholar
19Spector, S. J.. Linear deformations as global minimizers in nonlinear elasticity, Q. Appl. Math. 52 (1994), 59–64.CrossRefGoogle Scholar
20Stuart, C. A.. Radially symmetric cavitation for hyperelastic materials. Anal. Nonlineaire 2 (1985), 33–66.Google Scholar
21Stuart, C. A.. Estimating the critical radius for radially symmetric cavitation. Q. Appl. Math. 51 (1993), 251–63.Google Scholar
22Yan, B.. Cavitation solutions to homogeneous Van der Waals type fluids involving phase transitions. Q. Appl. Math. 53 (1995), 721–30.CrossRefGoogle Scholar