zbMATH — the first resource for mathematics

Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. (English) Zbl 0628.73018
In this very detailed work, the author examines the problem of uniqueness of certain three-dimensional regular and singular radial solutions to the equilibrium equations of nonlinear elasticity for an isotropic material. Two cases are considered - when the homogeneous elastic body in its reference state is a ball B and when it is a spherical shell \(B^{\epsilon}\). The uniqueness of regular and cavitating equilibria are established and this is the core result of the paper. It is then used to investigate the asymptotic behaviour of solutions to the mixed problem for \(B^{\epsilon}\) in the limit \(\epsilon\to 0\). The sense in which equilibrium solutions for \(B^{\alpha}\) approximate those for B when \(\epsilon\) is small, is also determined.
This is a well written paper whose results should be of interest for theoreticians working in the field of nonlinear elasticity. It should be mentioned, as done by the author, that exactly analogous results hold for corresponding two-dimensional problems.
Reviewer: H.Ramkissoon

74B20 Nonlinear elasticity
74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
74K25 Shells
Full Text: DOI
[1] R. A. Adams (1975), Sobolev Spaces, New York, Academic Press. · Zbl 0314.46030
[2] J. M. Ball (1982), Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. Roy. Soc. A 306, 557–611. · Zbl 0513.73020 · doi:10.1098/rsta.1982.0095
[3] J. M. Ball (1984), Differentiability properties of symmetric and isotropic functions, Duke Math. J. 50, 699–727. · Zbl 0566.73001 · doi:10.1215/S0012-7094-84-05134-2
[4] J. M. Ball, J. C. Currie & P. J. Olver (1981), Null Lagrangians, weak continuity and variational problems of arbitrary order, J. Funct. Anal. 41, 135–174. · Zbl 0459.35020 · doi:10.1016/0022-1236(81)90085-9
[5] L. Cesari (1983), Optimization-theory and applications, Springer-Verlag, Berlin-Heidelberg-New York. · Zbl 0506.49001
[6] T. B. Cox & J. R. Low (1974), An investigation of the plastic fracture of AISI 4340 and 18 Nickel-200 Grade maraging steels, Mat. Trans. 5, 1457–1470. · doi:10.1007/BF02646633
[7] A. E. Green (1973), On some general formulae in finite elastostatics, Arch. Rational Mech. Anal. 50, 73–80. · Zbl 0286.73038 · doi:10.1007/BF00251294
[8] J. W. Hancock & M. J. Cowling (1977), The initiation of cleavage by ductile tearing, Fracture 1977, 2, ICF 4, Waterloo, Canada.
[9] P. Hartman (1973), Ordinary differential equations, Wiley. · Zbl 0281.34001
[10] C. O. Horgan & R. Abeyaratne (1985), A bifurcation problem for a compressible nonlinearly elastic medium: growth of a microvoid, (to appear in J. of Elasticity).
[11] R. J. Knops & C. A. Stuart (1984), Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity, Arch. Rational Mech. Anal. 86, 233–249. · Zbl 0589.73017 · doi:10.1007/BF00281557
[12] C. A. Stuart (1985), Radially symmetric cavitation for hyperelastic materials, Ann. Inst. H. Poincaré, Anal. Non. Linéaire 2, 33–66. · Zbl 0588.73021 · doi:10.1016/S0294-1449(16)30411-5
[13] C. Truesdell & W. Noll (1965), The non-linear field theories of mechanics, Handbuch der Physik III/3, Berlin, Heidelberg, New York, Springer.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.