On the stability of cavitating equilibria
Author:
J. Sivaloganathan
Journal:
Quart. Appl. Math. 53 (1995), 301-313
MSC:
Primary 73C50; Secondary 49J45, 73G05, 73H99, 73V25
DOI:
https://doi.org/10.1090/qam/1330654
MathSciNet review:
MR1330654
Full-text PDF Free Access
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S. S. Antman and P.V. Negron-Marrero, The remarkable nature of radially symmetric equilibrium states of aelotropic nonlinearly elastic bodies, J. Elasticity 18, 131–164 (1987)
J. M. Ball, Constitutive inequalities and existence theorems in nonlinear elastostatics, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. 1 (R. J. Knops, ed.), Pitman, London, 1977, pp. 187–241
J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A 306, 557–611 (1982)
J. M. Ball, J. C. Currie, and P. J. Olver, Null Lagrangians, weak continuity and variational problems of arbitrary order, J. Funct. Anal. 41, 135–174 (1981)
J. M. Ball and F. Murat, $W^{1, P}$-Quasiconvexity and variational problems for multiple integrals, J. Funct. Anal. 58, 225–253 (1984)
J. M. Ball and F. Murat, Remarks on rank-one convexity and quasiconvexity, Ordinary and Partial Differential Equations, Vol. III, Dundee, 1990; Pitman Research Notes Ser. 254, 25–37 (1991)
P. G. Ciarlet, Mathematical Elasticity, Vol. 1: Three-Dimensional Elasticity, North Holland, Amsterdam, 1988
C. O. Horgan, Void nucleation and growth for compressible nonlinearly elastic materials: an example, Internat. J. Solids Structures 29, 279–291 (1992)
C. O. Horgan and R. Abeyaratne, A bifurcation problem for a compressible nonlinearly elastic medium: growth of a microvoid, J. Elasticity 16, 189–200 (1986)
C. O. Horgan and T. J. Pence, Cavity formation at the center of a composite incompressible nonlinearly elastic sphere, J. Appl. Mech. 56, 302–308 (1989)
C. O. Horgan and T. J. Pence, Void nucleation in tensile loading of a composite incompressible nonlinearly elastic sphere, J. Elasticity 21, 61–82 (1989)
R. D. James and S. J. Spector, The formation of filamentary voids in solids, J. Mech. Phys. Sol. 39, 783–813 (1991)
R. D. James and S. J. Spector, Remarks on $W^{1, p}$-quasiconvexity, interpenetration of matter, and function spaces for elasticity, Ann. Inst. H. Poincaré-Anal. nonlinéaire 9, 263–280 (1992)
C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 1965
F. Meynard, Cavitation dans un milieu hyperelastique, Thése de Doctorat, École Polytechnique Federale de Lausanne, 1990
F. Meynard, Existence and nonexistence results on the radially symmetric cavitation problem, Quart. Appl. Math. 50, 201–226 (1992)
K. A. Pericak-Spector and S. J. Spector, Nonuniqueness for a hyperbolic system: cavitation in nonlinear elastostatics, Arch. Rational Mech. Anal. 101, 293–317 (1988)
J. Sivaloganathan, Cavitation, the incompressible limit, and material inhomogeneity, Quart. Appl. Math. 49, 521–541 (1991)
J. Sivaloganathan, The generalised Hamilton-Jacobi inequality and the stability of equilibria in nonlinear elasticity, Arch. Rational Mech. Anal. 107, 347–369 (1989)
J. Sivaloganathan, Implications of rank-one convexity, Ann. Inst. H. Poincaré Anal. Non Linéaire 5, 99–118 (1988)
J. Sivaloganathan, Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity, Arch. Rational Mech. Anal. 96, 96–136 (1986)
J. Sivaloganathan, A field theory approach to stability of equilibria in radial elasticity, Math. Proc. Cambridge Philos. Soc. 99, 589–604 (1986)
C. A. Stuart, Radially symmetric cavitation for hyperelastic materials, Ann. Inst. H. Poincaré Anal. Non Linéaire 2, 33–66 (1985)
C. Truesdell and W. Noll, The non-linear field theories of mechanics, Handbuch der Physik III/3, Springer-Verlag, Berlin, 1965
S. S. Antman and P.V. Negron-Marrero, The remarkable nature of radially symmetric equilibrium states of aelotropic nonlinearly elastic bodies, J. Elasticity 18, 131–164 (1987)
J. M. Ball, Constitutive inequalities and existence theorems in nonlinear elastostatics, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. 1 (R. J. Knops, ed.), Pitman, London, 1977, pp. 187–241
J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A 306, 557–611 (1982)
J. M. Ball, J. C. Currie, and P. J. Olver, Null Lagrangians, weak continuity and variational problems of arbitrary order, J. Funct. Anal. 41, 135–174 (1981)
J. M. Ball and F. Murat, $W^{1, P}$-Quasiconvexity and variational problems for multiple integrals, J. Funct. Anal. 58, 225–253 (1984)
J. M. Ball and F. Murat, Remarks on rank-one convexity and quasiconvexity, Ordinary and Partial Differential Equations, Vol. III, Dundee, 1990; Pitman Research Notes Ser. 254, 25–37 (1991)
P. G. Ciarlet, Mathematical Elasticity, Vol. 1: Three-Dimensional Elasticity, North Holland, Amsterdam, 1988
C. O. Horgan, Void nucleation and growth for compressible nonlinearly elastic materials: an example, Internat. J. Solids Structures 29, 279–291 (1992)
C. O. Horgan and R. Abeyaratne, A bifurcation problem for a compressible nonlinearly elastic medium: growth of a microvoid, J. Elasticity 16, 189–200 (1986)
C. O. Horgan and T. J. Pence, Cavity formation at the center of a composite incompressible nonlinearly elastic sphere, J. Appl. Mech. 56, 302–308 (1989)
C. O. Horgan and T. J. Pence, Void nucleation in tensile loading of a composite incompressible nonlinearly elastic sphere, J. Elasticity 21, 61–82 (1989)
R. D. James and S. J. Spector, The formation of filamentary voids in solids, J. Mech. Phys. Sol. 39, 783–813 (1991)
R. D. James and S. J. Spector, Remarks on $W^{1, p}$-quasiconvexity, interpenetration of matter, and function spaces for elasticity, Ann. Inst. H. Poincaré-Anal. nonlinéaire 9, 263–280 (1992)
C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 1965
F. Meynard, Cavitation dans un milieu hyperelastique, Thése de Doctorat, École Polytechnique Federale de Lausanne, 1990
F. Meynard, Existence and nonexistence results on the radially symmetric cavitation problem, Quart. Appl. Math. 50, 201–226 (1992)
K. A. Pericak-Spector and S. J. Spector, Nonuniqueness for a hyperbolic system: cavitation in nonlinear elastostatics, Arch. Rational Mech. Anal. 101, 293–317 (1988)
J. Sivaloganathan, Cavitation, the incompressible limit, and material inhomogeneity, Quart. Appl. Math. 49, 521–541 (1991)
J. Sivaloganathan, The generalised Hamilton-Jacobi inequality and the stability of equilibria in nonlinear elasticity, Arch. Rational Mech. Anal. 107, 347–369 (1989)
J. Sivaloganathan, Implications of rank-one convexity, Ann. Inst. H. Poincaré Anal. Non Linéaire 5, 99–118 (1988)
J. Sivaloganathan, Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity, Arch. Rational Mech. Anal. 96, 96–136 (1986)
J. Sivaloganathan, A field theory approach to stability of equilibria in radial elasticity, Math. Proc. Cambridge Philos. Soc. 99, 589–604 (1986)
C. A. Stuart, Radially symmetric cavitation for hyperelastic materials, Ann. Inst. H. Poincaré Anal. Non Linéaire 2, 33–66 (1985)
C. Truesdell and W. Noll, The non-linear field theories of mechanics, Handbuch der Physik III/3, Springer-Verlag, Berlin, 1965
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Article copyright:
© Copyright 1995
American Mathematical Society
