On J. Ball’s Fundamental Existence Theory and Regularity of Weak Equilibria in Nonlinear Radial Hyperelasticity



In 1982, J. Ball formulated a pioneering theory on the existence and uniqueness of weak radial equilibria to the pure displacement boundary value problem associated with isotropic, frame-invariant strain-energy functions in nonlinear hyperelasticity. In the theory [Bal82], he posed the following question: “Does strong ellipticity (‘of the stored energy’) imply that all solutions to the equilibrium equations which pass through the origin and have finite energy are trivial?” J. Ball’s work depended critically on the number of elasticity dimensions.

In this chapter, we will present models in n-dimensional elasticity that establish that the answer to J. Ball’s question is negative. This work extends to higher dimensional elasticity the approach and results we presented, for the first time, on this question in [Ha07]. These models also provide further insight into another central, (very) difficult problem of nonlinear elasticity, namely, that of regularity of weak equilibria, which would be hard to gain by other methods such as the common, but delicate, phase plane analysis.


Equilibrium Solution Singular Solution Nonlinear Elasticity Strong Ellipticity Phase Plane Analysis 
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  1. [Ada75]
    Adams, R.A.: Sobolev Spaces, Academic Press, New York (1975).MATHGoogle Scholar
  2. [AB78]
    Antman, S.S., Brézis, H.: The existence of orientation-preserving deformations in nonlinear elasticity, in Nonlinear Analysis and Mechanics, Vol. II (Knops, R.J., ed.), Pitman, London (1978).Google Scholar
  3. [Bal77b]
    Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63, 337–403 (1977).MATHCrossRefGoogle Scholar
  4. [Bal82]
    Ball, J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. Roy. Soc. Lond. A, 306, 557–611 (1982).MATHCrossRefGoogle Scholar
  5. [BM85]
    Ball, J.M., Mizel, J.V.: One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. Arch. Rational Mech. Anal., 90, 325–388 (1985).MATHCrossRefMathSciNetGoogle Scholar
  6. [CS94]
    Chen, W.F., Saleeb, A.F.: Constitutive Equations for Engineering Materials, Elsevier, New York (1994).Google Scholar
  7. [Er73]
    Ericksen, J.L.: Loading devices and stability of equilibrium, in Nonlinear Elasticity (Dickey, R.W., ed.), Academic Press, New York (1973).Google Scholar
  8. [Eri83]
    Ericksen, J.L.: Ill-posed problems in thermoelasticity theory, in Systems of Nonlinear Partial Differential Equations (Ball, J.M., ed.), Reidel, Dordrecht (1983).Google Scholar
  9. [Ha07]
    Haidar, S.M.: Convexity conditions in uniqueness and regularity of equilibrium in nonlinear elasticity, in Integral Methods in Science and Engineering: Techniques and Applications (Constanda, C., Potapenko, S., eds.), Birkhäuser, Boston, MA (2007), 109–118.Google Scholar
  10. [Hai00]
    Haidar, S.M.: Existence and regularity of weak solutions to the displacement boundary value problem of nonlinear elastostatics, in Integral Methods in Science and Engineering, CRC Press, Boca Raton (2000), 161–166.Google Scholar
  11. [Mor66]
    Morrey, C.B.: Multiple Integrals in the Calculus of Variations, Springer, Berlin (1966).MATHGoogle Scholar
  12. [RE55]
    Rivlin, R.S., Ericksen, J.L.: Stress-deformation relations for isotropic materials. J. Rational Mech. Anal., 4, 323–425 (1955).MathSciNetGoogle Scholar
  13. [TN65]
    Truesdell, C., Noll, W.: The non-linear field theories of mechanics, in Handbuch der Physik, Vol. III/3 (Flugge, S., ed.), Springer, Berlin (1965).Google Scholar

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© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Grand Valley State UniversityAllendaleUSA

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