On local uniqueness in nonlinear elastodynamics
Author:
R. J. Knops
Journal:
Quart. Appl. Math. 64 (2006), 321-333
MSC (2000):
Primary 74B20; Secondary 74H25
DOI:
https://doi.org/10.1090/S0033-569X-06-01023-7
Published electronically:
May 3, 2006
MathSciNet review:
2243866
Full-text PDF Free Access
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Abstract: A conservation law, derived from properties of the energy-momentum tensor, is used to establish uniqueness of suitably constrained solutions to the initial boundary value problem of nonlinear elastodynamics. It is assumed that the region is star-shaped, that the data are affine, and that the strain-energy function is strictly rank-one convex and quasi-convex. It is shown how these assumptions may be successively relaxed provided that the class of considered solutions is correspondingly further constrained.
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b1 J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mechs. Anal. 63 (1976/77), 337-403.
b2 J. M. Ball, Strict convexity, strong ellipticity and regularity in the calculus of variations. Math. Proc. Camb. Phil. Soc. 87 (1980), 501-513.
b3 J. M. Ball, Some open problems in elasticity. In: Geometry, mechanics, and dynamics: volume in honor of the 60th birthday of J.E.Marsden (ed. by P. Newton, Philip Holmes and Alan Weinstein), Springer-Verlag, New York, 2002, pp 3-59.
br1 L. Brun, Sur l’unicité en thermoélasticité dynamique et diverses expressions analogues à la formule de Clapeyron. C. R. Acad. Sci. Paris 261 (1965), 2584-2587.
br2 L. Brun, Méthodes énergétiques dans les systémes évolutifs linéaires. Premiére Partie: Séparation des énergies. Deuxiéme Partie: Théorémes d’unicité. J. de Mech. 8 (1969), 125-166, 167-192.
c P. Chadwick, Applications of an energy-momentum tensor in non-linear elasticity. J. Elasticity 5 (1975), 249-258.
d C. M. Dafermos, The second law of thermodynamics and stability. Arch. Rational Mech. Anal. 70 (1979), 167-179.
d05 C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Second Edition. Springer-Verlag, Berlin, 2005.
jle J. L. Ericksen, Special topics in elastostatics. In: Yih, C-S (ed.) Advances in Applied Mechanics, Academic Press, New York, 1977, pp. 189-244.
e J. D. Eshelby, The elastic energy-momentum tensor. J. Elasticity 5 (1975), 321-335.
f D. C. Fletcher, Conservation laws in linear elastodynamics. Arch. Rational Mech. Anal. 60 (1976), 329-353.
gf I. M. Gelfand and S. V. Fomin, Calculus of Variations. Prentice-Hall, Englewood Cliffs, NJ, 1963.
g A. E. Green, On some general formulae in finite elastostatics. Arch. Rational Mech. Anal. 50 (1973), 73-80.
hi R. Hill, Energy-momentum tensors in elastostatics: some reflections on the general theory. J. Mech. Phys. Solids 34 (1986), 305-317.
hkm T. J. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyberbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rational Mech. Anal. 63 (1976), 273-294.
j F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Comm. Pure Appl. Mathematics 13 (1960), 551-585.
k R. J. Knops, On uniqueness in nonlinear homogeneous elasticity. In: Rational Continua Classical and New (ed. by P. Podio-Guidugli and M. Brocato) Springer, Milano, 2003, pp. 119-137.
kp R. J. Knops and L. E. Payne, Uniqueness in classical elasticity. Arch. Rational Mech. Anal. 27 (1967), 349-355.
kp71 R. J. Knops and L. E. Payne, Uniqueness Theorems in Linear Elasticity. Springer Tracts in Natural Philosophy, vol. 19, Springer-Verlag, Berlin, 1971.
klp R. J. Knops, H. A. Levine and L. E. Payne, Non-existence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics. Arch. Rational Mech. Anal. 55 (1974), 52-72.
ks R. J. Knops and C. A. Stuart, Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. Arch. Rational Mech. Anal. 86 (1984), 233-249.
kws J. K. Knowles and E. Sternberg, On a class of conservation laws in linearized and finite elastostatics. Arch. Rational Mech. Anal. 44 (1972), 187-211.
ma G. Maugin, Material Inhomogeneities in Elasticity. Chapman and Hall, London, 1993.
m S. Müller, Variational models for microstructure and phase transitions. In: Proc. C.I.M.E. Summer School “Calculus of Variations and Geometric Evolution Problems” (ed. by S. Hildebrandt and M. Struwe), Cetraro. Lecture Notes in Mathematics, vol. 1713, Springer-Verlag, Berlin, 1996, pp. 85-210.
s M. Sofer, A note on the growth of trajectories in nonlinear elastodynamics. J. Elasticity 28 (1992), 185-192.
w L. Wheeler, A uniqueness theorem for the displacement problem in finite elastodynamics. Arch. Rational Mech. Anal. 63 (1977), 183-189.
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Additional Information
R. J. Knops
Affiliation:
School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, U.K.
Keywords:
Nonlinear elastodynamics,
uniqueness,
constrained solutions
Received by editor(s):
July 25, 2005
Published electronically:
May 3, 2006
Article copyright:
© Copyright 2006
Brown University