Existence and nonexistence results on the radially symmetric cavitation problem
Author:
François Meynard
Journal:
Quart. Appl. Math. 50 (1992), 201-226
MSC:
Primary 73G05; Secondary 73C50
DOI:
https://doi.org/10.1090/qam/1162272
MathSciNet review:
MR1162272
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Abstract: We investigate the problem of radially symmetric cavitation for a hyperelastic ball in $\mathbb {R}{^n} , n \ge 2$. The radial equilibrium equation is analyzed by a shooting argument. The basic formulation of the problem is the same as C. A. Stuart’s formulation in [10], but an asymptotic study of the solutions of the radial equilibrium equation allows us to enlarge the discussion of cavitation to cases that are excluded from the context of [10]. Finally, criteria for nonexistence to the problem of cavitation are briefly discussed. They have a physical interpretation through relations between the total energy and the radial stress.
- J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A 306 (1982), no. 1496, 557–611. MR 703623, DOI https://doi.org/10.1098/rsta.1982.0095
J. M. Ball, Minimizing sequences in Thermomechanics, Meeting on finite Thermoelasticity, Academia Nazionale dei Lincei, Roma. 1986
A. N. Gent and P. B. Lindley, Internal rupture of bonded rubber cylinders in tension, Proc. Roy. Soc. London Ser. A 249, 195–205 (1958)
- Morton E. Gurtin, An introduction to continuum mechanics, Mathematics in Science and Engineering, vol. 158, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 636255
- C. O. Horgan and R. Abeyaratne, A bifurcation problem for a compressible nonlinearly elastic medium: growth of a microvoid, J. Elasticity 16 (1986), no. 2, 189–200. MR 849671, DOI https://doi.org/10.1007/BF00043585
- Richard D. James and Scott J. Spector, The formation of filamentary voids in solids, J. Mech. Phys. Solids 39 (1991), no. 6, 783–813. MR 1120242, DOI https://doi.org/10.1016/0022-5096%2891%2990025-J
F. Meynard, Cavitation radiale d’un milieu homogène isotrope et hyperelastique. Ph.D. Thesis, Dept. of Mathematics EPFL 1990 (to appear)
- Jeyabal Sivaloganathan, Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity, Arch. Rational Mech. Anal. 96 (1986), no. 2, 97–136. MR 853969, DOI https://doi.org/10.1007/BF00251407
- J. Sivaloganathan, A field theory approach to stability of radial equilibria in nonlinear elasticity, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 3, 589–604. MR 830370, DOI https://doi.org/10.1017/S0305004100064513
- C. A. Stuart, Radially symmetric cavitation for hyperelastic materials, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 1, 33–66 (English, with French summary). MR 781591
- C. A. Stuart, Special problems involving uniqueness and multiplicity in hyperelasticity, Nonlinear functional analysis and its applications (Maratea, 1985) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 173, Reidel, Dordrecht, 1986, pp. 131–145. MR 852573
J. M. Ball, Discontinuous equilibrium solutions and cavitation in non-linear elasticity, Philos. Trans. Roy. Soc. London Ser. A 306, 557–611 (1982)
J. M. Ball, Minimizing sequences in Thermomechanics, Meeting on finite Thermoelasticity, Academia Nazionale dei Lincei, Roma. 1986
A. N. Gent and P. B. Lindley, Internal rupture of bonded rubber cylinders in tension, Proc. Roy. Soc. London Ser. A 249, 195–205 (1958)
M. E. Gurtin, An introduction to continuum mechanics, Academic Press, New York, 1981
C. O. Horgan and R. Abeyaratnne, A bifurcation problem for a compressible non-linearly elastic medium, growth of a micro-void, J. Elasticity 16, 189–200 (1986)
R. D. James and S. J. Spector, The formation of filamentary voids in solids, Institute for Mathematics and its Applications, University of Minnesota, preprint series, 572, 1989
F. Meynard, Cavitation radiale d’un milieu homogène isotrope et hyperelastique. Ph.D. Thesis, Dept. of Mathematics EPFL 1990 (to appear)
J. Sivaloganathan, Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity, Arch. Rational Mech. Anal. 96, 97–136 (1986)
J. Sivaloganathan, A field theory approach to stability of radial equilibria in non-linear elasticity, Math. Proc. Cambridge Philos. Soc. 99, 589–604 (1986)
C. A. Stuart, Radially symmetric cavitation for hyperelastic materials, Ann. Inst. H. Poincaré 2, 33–66 (1985)
C. A. Stuart, Special problems involving uniqueness and multiplicity, Hyperelasticity, Non-Linear Functional Analysis and its Applications, (S. P. Singh, ed.), D. Reidel Publishing Company, 1986, pp. 131–145
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© Copyright 1992
American Mathematical Society