Journal of Elasticity

, Volume 84, Issue 2, pp 99–129 | Cite as

Local and Global Injective Solution of the Rotationally Symmetric Sphere Problem



There are problems in the classical linear theory of elasticity whose closed form solutions, while satisfying the governing equations of equilibrium together with well-posed boundary conditions, predict the existence of regions, often quite small, inside the body where material overlaps. Of course, material overlapping is not physically realistic, and one possible way to prevent it combines linear theory with the requirement that the deformation field be injective. A formulation of minimization problems in classical linear elasticity proposed by Fosdick and Royer [3] imposes this requirement through a Lagrange multiplier technique. An existence theorem for minimizers of plane problems is also presented. In general, however, it is not certain that such minimizers exist. Here, the Euler–Lagrange equations corresponding to a family of three-dimensional problems is investigated. In classical linear elasticity, these problems do not have bounded solutions inside a body of anisotropic material for a range of material parameters. For another range of parameters, bounded solutions do exist but yield stresses that are infinite at a point inside the body. In addition, these solutions are not injective in a region surrounding this point, yielding unrealistic behavior such as overlapping of material. Applying the formulation of Fosdick and Royer on this family of problems, it is shown that both the displacements and the constitutive part of the stresses are bounded for all values of the material parameters and that the injectivity constraint is preserved. In addition, a penalty functional formulation of the constrained elastic problems is proposed, which allows to devise a numerical approach to compute the solutions of these problems. The approach consists of finding the displacement field that minimizes an augmented potential energy functional. This augmented functional is composed of the potential energy of linear elasticity theory and of a penalty functional divided by a penalty parameter. A sequence of solutions is then constructed, parameterized by the penalty parameter, that converges to a function that satisfies the first variation conditions for a minimizer of the constrained minimization problem when this parameter tends to infinity. This approach has the advantages of being mathematically appealling and computationally simple to implement.

Mathematics Subject Classifications (2000)

74B05 74E10 74G65 74G70 74S05 

Key words

anisotropy elasticity energy minimization Finite Element Method singularity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aguiar, A.R., Fosdick, R.L.: A singular problem in incompressible nonlinear elastostatics. Math. Models Methods Appl. Sci. 10, 1181–1207 (2000)MathSciNetGoogle Scholar
  2. 2.
    Aguiar, A.R., Fosdick, R.L.: Self-intersection in elasticity. Int. J. Solids Struct. 38, 4797–4823 (2001)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Fosdick, R.L., Royer-Carfagni, G.: The constraint of local injectivity in linear elasticity theory. Proc. R. Soc. Lond., A 457, 2167–2187 (2001)MATHADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Fosdick, R.L., Royer-Carfagni, G.: A penalty interpretation for the lagrange multiplier fields in incompressible multipolar elasticity theory. Math. Mech. Solids 10, 389–413 (2005)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Golub, G.H., van Loan, C.F.: Matrix Computations. 3rd ed., John Hopkins University, Baltimore (1996)MATHGoogle Scholar
  6. 6.
    John, F.: Plane strain problems for a perfectly elastic material of harmonic type. Commun. Pure Appl. Math. XIII, 239–296 (1960)CrossRefGoogle Scholar
  7. 7.
    Kim, S.J., Kim, J.H.: Finite element analysis of laminated composites with contact constraint by extended interior penalty methods. Int. J. Numer. Methods Eng. 36, 3421–3439 (1993)MATHCrossRefGoogle Scholar
  8. 8.
    Knowles, J.K., Sternberg, E.: On the singularity induced by certain mixed boundary conditions in linearized and nonlinear elastostatics. Int. J. Solids Struct. 11, 1173–1201 (1975)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Lekhnitskii, S.G.: Anisotropic Plates. Gordon and Breach Science, New York (1968)Google Scholar
  10. 10.
    Luenberger, D.G.: Linear and Nonlinear Programming. 2nd ed., Addison-Wesley, Reading (1984)MATHGoogle Scholar
  11. 11.
    Malkus, D.S., Hughes, T.J.R.: Mixed finite element methods – Reduced and selective integration technique: A unification of concepts. Comput. Methods Appl. Mech. Eng. 15, 63–81 (1978)CrossRefMATHGoogle Scholar
  12. 12.
    Obeidat, K., Stolarski, H., Fosdick, R., Royer-Carfagni, G.: Numerical analysis of elastic problems with injectivity constraints. In: European Conference on Computational Mechanics (ECCM-2001) (2001)Google Scholar
  13. 13.
    Oden, J.T., Kikuchi, N., Song, Y.J.: Reduced integration and exterior penalty methods for finite element approximations of contact problems in incompressible elasticity. TICOM Report – University of Texas at Austin, Austin, TX 80 (1980)Google Scholar
  14. 14.
    Oden, J.T., Kim, S.J.: Interior penalty methods for finite element approximations of the signorini problem in elastostatics. Comput. Math. Appl. 8, 35–56 (1982)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Simo, J.C., Taylor, R.L.: Penalty function formulations for incompressible nonlinear elastostatics. Comput. Methods Appl. Mech. Eng. 35, 107–118 (1982)MATHCrossRefGoogle Scholar
  16. 16.
    Sokolnikoff, I.S.: Mathematical Theory of Elasticity. 2nd ed., McGraw-Hill, New York (1956)MATHGoogle Scholar
  17. 17.
    Ting, T.C.T.: The remarkable nature of radially symmetric deformation of spherically uniform linear anisotropic elastic solids. J. Elast. 53, 47–64 (1999)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Zienkiewicz, O.C., Taylor, R.L., Too, J.B.: Reduced integration technique in general analysis of plates and shells. Int. J. Numer. Eng. 3, 275–290 (1971)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of Structural EngineeringUniversity of São PauloSão CarlosBrazil

Personalised recommendations