zbMATH — the first resource for mathematics

Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. (English) Zbl 0589.73017
This paper concerns the establishment of the uniqueness of solutions in simple displacement boundary value problems in the nonlinear theory of homogeneous hyperelasticity for smooth equilibrium configurations satisfying certain boundary conditions, for which the stored energy vector W is of rank-one-convex and it is strictly quasi-convex at an (n\(\times n)\)-matrix F, such that \(x\to Fx+b\), where b is an n-constant vector. In such cases it was proved that the homogeneous deformation of the body is the only smooth equilibrium solution.
Furthermore, for the interesting case of radial deformations of an isotropic sphere it may, then, be shown that all smooth radial solutions are of the form \(x\to \lambda x\) \((\lambda >0)\), provided that the stored energy W is either strictly rank-one-convex and W is strictly quasi- convex, or W is strictly rank-one-convex and W is quasi convex. At the end of the paper the previous results are adpated to boundary value problems for incompressible materials.
Reviewer: P.S.Theocaris

74B20 Nonlinear elasticity
74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
Full Text: DOI
[1] Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63, 337-403, 1978. · Zbl 0368.73040 · doi:10.1007/BF00279992
[2] Ball, J. M., Constitutive inequalities and existence theorems in nonlinear elastostatics. Nonlinear Analysis and Mechanics: Heriot-Watt Symposium Vol. I. pp. 187-238. Pitman. London 1977.
[3] Ball, J. M., Strict convexity, strong ellipticity and regularity in the calculus of variations. Math. Proc. Camb. Phil. Soc. 87, 501-503, 1980. · Zbl 0451.35028 · doi:10.1017/S0305004100056930
[4] Ball, J. M., Existence of solutions in finite elasticity. Proc. IUTAM Symposium on Finite Elasticity. Lehigh University. August 1980 (Edited by D. E. Carlson and R. T. Shield), pp. 1-12.
[5] Ball, J. M., Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. Roy. Soc. Lond. A 306, 557-611, 1982. · Zbl 0513.73020 · doi:10.1098/rsta.1982.0095
[6] Chillingworth, D. R. J., J. E. Marsden & Y. H. Wan, Symmetry and bifurcation in three dimensional elasticity. Part I. Arch. Rational Mech. Anal. 80, 295-331, 1982. · Zbl 0509.73018 · doi:10.1007/BF00253119
[7] Chillingworth, D. R. J., J. E. Marsden & Y. H. Wan, Symmetry and bifurcation in three dimensional elasticity. Part II. Arch. Rational Mech. Anal. 83, 362-395, 1983. · Zbl 0536.73010 · doi:10.1007/BF00963840
[8] Ericksen, J. L., & R. A. Toupin, Implications of Hadamard’s conditions for elastic stability with respect to uniqueness theorems. Canad. J. Math. 8, 432-436, 1956. · Zbl 0071.39801 · doi:10.4153/CJM-1956-051-2
[9] Green, A. E., On some general formulae in finite elastostatics. Arch. Rational Mech. Anal. 50, 73-80, 1973. · Zbl 0286.73038 · doi:10.1007/BF00251294
[10] Gurtin, M. E., & S. J. Spector, On stability and uniqueness in finite elasticity. Arch. Rational Mech. Anal. 70, 152-165, 1970. · Zbl 0426.73038
[11] Hadamard, J., Leçons sur la propagation des ondes. Paris, Hermann 1903 (Reprinted by Dover, 1952). · JFM 34.0793.06
[12] Hill, R., On uniqueness and stability in the theory of finite elastic strain. J. Mech. Phys. Sols. 5, 229-241, 1957. · Zbl 0080.18004 · doi:10.1016/0022-5096(57)90016-9
[13] Hill, R., Bifurcation and uniqueness in non-linear mechanics of continua. Problems of Continuum Mechanics. Society for Industrial and Applied Mathematics. Philadelphia. 155-164, 1961.
[14] John, F., Uniqueness of non-linear elastic equilibrium for prescribed boundary displacements and sufficiently small strains. Comm. Pure Appl. Math. 25, 617-634, 1972. · Zbl 0287.73009 · doi:10.1002/cpa.3160250505
[15] Marsden, J. E., & Y. H. Wan, Linearization stability and Signorini series for the traction problem in elastostatics. Proc. Roy. Soc. Edin. A 95, 171-180, 1983. · Zbl 0533.73022
[16] Morrey, Jr., C. B., Quasi-convexity and the lower semi-continuity of multiple integrals, Pacific J. Math. 2, 25-53, 1952. · Zbl 0046.10803
[17] Spector, S., On uniqueness in finite elasticity with general loading. J. Elast. 10, 149-161, 1980. · Zbl 0426.73037 · doi:10.1007/BF00044500
[18] Spector, S. J., On uniqueness for the traction problem in finite elasticity. J. Elast. 12, 367-383, 1982. · Zbl 0506.73043 · doi:10.1007/BF00042210
[19] Truesdell, C., & W. Noll, The non-linear field theories of mechanics. Handbuch der Physik. Vol. III/3. Springer. Berlin. 1965. · Zbl 0779.73004
[20] Truesdell, C., & R. Toupin, Static grounds for inequalities in finite elastic strain. Arch. Rational Mech. Anal. 12, 1-33, 1963. · Zbl 0119.19201 · doi:10.1007/BF00281217
[21] Wan, Y. H., & J. E. Marsden, Symmetry and bifurcation in three-dimensional elasticity. Part III: Stressed reference configuration. Arch. Rational Mech. Anal. 84, 203-233, 1984. · Zbl 0536.73011 · doi:10.1007/BF00281519
[22] Wang, C.-C., & C. Truesdell, Introduction to Rational Elasticity. Noordhoff. Leyden. 1973.
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.