Some outstanding open problems of nonlinear elasticity are described. The problems range from questions of existence, uniqueness, regularity and stability of solutions in statics and dynamics to issues such as the modelling of fracture and self-contact, the status of elasticity with respect to atomistic models, the understanding of microstructure induced by phase transformations, and the passage from three-dimensional elasticity to models of rods and shells. Refinements are presented of the author’s earlier work Ball [1984a] on showing that local minimizers of the elastic energy satisfy certain weak forms of the equilibrium equations.


Weak Solution Equilibrium Solution Lagrange Equation Nonlinear Elasticity Young Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Acerbi E., G. Buttazzo and D. Percivale [1991], A variational definition of the strain energy for an elastic string. J. Elasticity, 25:137–148.MathSciNetzbMATHGoogle Scholar
  2. Acerbi E., I. Fonseca and N. Fusco [1997], Regularity results for equilibria in a variational model of fracture. Proc. Royal Soc. Edinburgh, 127A:889–902.MathSciNetGoogle Scholar
  3. Acerbi E., and N. Fusco [1984], Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal., 86:125–145.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Acerbi E., and N. Fusco [1988], A regularity theorem for minimizers of quasiconvex integrals. Arch. Rational Mech. Anal., 99:261–281.MathSciNetGoogle Scholar
  5. Ambrosio, L. [1989], Variational problems in SBV. Acta Appl. Math., 17:1–40.zbMATHMathSciNetCrossRefGoogle Scholar
  6. Ambrosio, L. [1990], Existence theory for a new class of variational problems. Arch. Rational Mech. Anal., 111:291–322.zbMATHMathSciNetCrossRefGoogle Scholar
  7. Ambrosio, L. and A. Braides [1995], Energies in SBV and variational models in fracture. In Homogenization and applications to material sciences (Nice 1995), volume 9 of GAKUTO Internat. Ser. Math. Sci. Appl., pages 1–22, Tokyo. Gakkötosho.Google Scholar
  8. Ambrosio, L. N. Fusco and D. Pallara [1997], Partial regularity of free discontinuity sets II. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24:39–62.MathSciNetzbMATHGoogle Scholar
  9. Ambrosio, L. N. Fusco and D. Pallara [2000], Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press.Google Scholar
  10. Ambrosio, L. and D. Pallara [1997], Partial regularity of free discontinuity sets I. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24:1–38.MathSciNetzbMATHGoogle Scholar
  11. Andrews, G. [1980], On the existence of solutions to the equation u tt=u xxt+σ(u x)x. J. Differential Eqns, 35:200–231.zbMATHCrossRefGoogle Scholar
  12. Antman, S. S. [1976], Ordinary differential equations of nonlinear elasticity. II. Existence and regularity theory for conservative boundary-value problem. Arch. Rational Mech. Anal., 61:353–393.zbMATHMathSciNetGoogle Scholar
  13. Antman, S. S. [1983], The influence of elasticity on analysis: Modern developments. Bull. Amer. Math. Soc., 9:267–291.zbMATHMathSciNetCrossRefGoogle Scholar
  14. Antman, S. S. [1995], Nonlinear Problems of Elasticity, volume 107 of Applied Mathematical Sciences. Springer-Verlag, New York.Google Scholar
  15. Antman, S. S. and P. V. Negrón-Marrero [1987], The remarkable nature of radially symmetric equilibrium states of aeolotropic nonlinearly elastic bodies. J. Elasticity, 18:131–164.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Antman, S. S. and J. E. Osborn [1979], The principle of virtual work and integral laws of motion. Arch. Rational Mech. Anal., 69:231–262.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Antman, S. S. and T. Seidman [1996], Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity. J. Differential Eqns, 124:132–185.MathSciNetCrossRefzbMATHGoogle Scholar
  18. Ball, J. M. [1977], Constitutive inequalities and existence theorems in nonlinear elastostatics. In R.J. Knops, editor, Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, Vol. 1. Pitman.Google Scholar
  19. Ball, J. M. [1977a], Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63:337–403.zbMATHGoogle Scholar
  20. Ball, J. M. [1980], Strict convexity, strong ellipticity, and regularity in the calculus of variations. Proc. Camb. Phil. Soc., 87:501–513.zbMATHGoogle Scholar
  21. Ball, J. M. [1981], Global invertibility of Sobolev functions and the interpenetration of matter. Proc. Royal Soc. Edinburgh, 88A:315–328.Google Scholar
  22. Ball, J.M. [1981a], Remarquessur l’existence et la régularité des solutions d’élastostatique non linéaire. In H. Berestycki and H. Brezis, editors, Recent Contributions to Nonlinear Partial Differential Equations. Pitman.Google Scholar
  23. Ball, J. M. [1982], Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. Royal Soc. London A, 306:557–611.zbMATHGoogle Scholar
  24. Ball, J. M. [1984], Differentiability properties of symmetric and isotropic functions. Duke Math. J., 51:699–728.zbMATHMathSciNetCrossRefGoogle Scholar
  25. Ball, J. M. [1984a], Minimizers and the Euler-Lagrange equations. In Trends and applications of pure mathematics to mechanics (Palaiseau, 1983), pages 1–4. Springer, Berlin.Google Scholar
  26. Ball, J. M. [1986], Minimizing sequences in thermomechanics. In Proc. Meeting on “Finite Thermoelasticity”, pages 45–54, Roma. Accademia Nazionale dei Lincei.Google Scholar
  27. Ball, J. M. [1989], A version of the fundamental theorem for Young measures. In M. Rascle, D. Serreand M. Slemrod, editors, Proceedings of conference on “Partial differential equations and continuum models of phase transitions,” pages 3–16. Springer Lecture Notes in Physics. No. 359.Google Scholar
  28. Ball, J. M. [1992], Dynamic energy minimization and phase transformations in solids. In Proceedings of ICIAM 91. SIAM.Google Scholar
  29. Ball, J. M. [1996], Nonlinear elasticity and materials science; a survey of some recent developments. In P.J. Aston, editor, Nonlinear Mathematics and Its Applications, pages 93–119. Cambridge University Press.Google Scholar
  30. Ball, J. M. [1996a], Review of Nonlinear Problems of Elasticity, by Stuart S. Antman. Bull. Amer. Math. Soc., 33:269–276.Google Scholar
  31. Ball, J. M. [1998], The calculus of variations and materials science. Quart. Appl. Math., 56:719–740.zbMATHMathSciNetGoogle Scholar
  32. Ball, J. M. [2001], Singularities and computation of minimizers for variational problems. In R. DeVore, A. Iserles and E. Suli, editors, Foundations of Computational Mathematics. Cambridge University Press.Google Scholar
  33. Ball, J. M. and C. Carstensen [1999], Compatibility conditions for microstructures and the austenite-martensite transition. Materials Science & Engineering A, 273–275:231–236.CrossRefGoogle Scholar
  34. Ball, J. M., C. Chu and R. D. James [1995], Hysteresis during stress-induced variant rearrangement. J. de Physique IV, C8:245–251.Google Scholar
  35. Ball, J. M., C. Chu and R. D. James [2002], Metastability and martensite. In preparation.Google Scholar
  36. Ball, J. M., P. J. Holmes, R. D. James, R. L. Pego and P. J. Swart [1991], On the dynamics of fine structure. J. Nonlinear Sci., 1:17–90.MathSciNetCrossRefzbMATHGoogle Scholar
  37. Ball, J. M. and R. D. James [2003], From Microscales to Macroscales in Materials. Book, in preparation.Google Scholar
  38. Ball, J. M. and R. D. James [2002], Incompatible sets of gradients and metastability. In preparation.Google Scholar
  39. Ball, J. M. and R. D. James [1987], Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal., 100:13–52.MathSciNetCrossRefzbMATHGoogle Scholar
  40. Ball, J. M. and R. D. James [1991], A characterization of plane strain. Proc. Roy. Soc. London A, 432:93–99.MathSciNetzbMATHGoogle Scholar
  41. Ball, J. M. and R. D. James [1992], Proposed experimental tests of a theory of fine microstructure, and the two-well problem. Phil. Trans. Roy. Soc. London A, 338:389–450.zbMATHGoogle Scholar
  42. Ball, J. M. and J. E. Marsden [1984], Quasiconvexity at the boundary, positivity of the second variation, and elastic stability. Arch. Rational Mech. Anal., 86:251–277.MathSciNetCrossRefzbMATHGoogle Scholar
  43. Ball, J. M. and V. J. Mizel [1985], One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equations. Arch. Rational Mech. Anal., 90:325–388.MathSciNetzbMATHGoogle Scholar
  44. Ball, J. M. and F. Murat [1984], W 1,p-quasiconvexity and variational problems for multiple integrals. J. Functional Analysis, 58:225–253.MathSciNetCrossRefzbMATHGoogle Scholar
  45. Bauman, P., N. C. Owen and D. Phillips [1991], Maximal smoothness of solutions to certain Euler-Lagrange equations from nonlinear elasticity. Proc. Royal Soc. Edinburgh, 119A:241–263.MathSciNetGoogle Scholar
  46. Bauman, P., N. C. Owen and D. Phillips [1991a], Maximum principles and a priori estimates for a class of problems from nonlinear elasticity. Annales de ľInstitut Henri Poincaré-Analyse non linéaire, 8:119–157.MathSciNetzbMATHGoogle Scholar
  47. Bauman, P., N. C. Owen and D. Phillips [1992], Maximum principles and a priori estimates for an incompressible material in nonlinear elasticity. Comm. in Partial Diff. Eqns, 17:1185–1212.MathSciNetzbMATHGoogle Scholar
  48. Bauman, P. and D. Phillips [1994], Univalent minimizers of polyconvex functionals in 2 dimensions. Arch. Rational Mech. Anal., 126:161–181.MathSciNetCrossRefzbMATHGoogle Scholar
  49. Ben Belgacem, H. [1997], Une méthode de Γ-convergence pour un modèle de membrane non linéaire. C. R. Acad. Sci. Paris Sér. I Math., 324:845–849.zbMATHGoogle Scholar
  50. Bhattacharya, K. [2001], Microstructure of martensite. A continuum theory with applications to the shape-memory effect. Oxford University Press, (to appear).Google Scholar
  51. Bhattacharya, K. and R. D. James [1999], A theory of thin films of martensitic materials with applications to microactuators. J. Mech. Phys. Solids, 47:531–576.MathSciNetCrossRefzbMATHGoogle Scholar
  52. Bianchini, S. and A. Bressan [2001], A center manifold technique for tracing viscous waves. Preprint.Google Scholar
  53. Blanc, X., C. Le Bris and P.-L. Lions [2001], Convergence de modèles moléculaires vers des modèles de mécanique des milieux continus. C. R. Acad. Sci. Paris Sér. I Math., 332:949–956.zbMATHGoogle Scholar
  54. Bourdin, B., G. A. Francfort and J.-J. Marigo [2000], Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids, 48:797–826.MathSciNetCrossRefzbMATHGoogle Scholar
  55. Braides, A. [1994], Loss of polyconvexity by homogenization. Arch. Rational Mech. Anal., 127:183–190.zbMATHMathSciNetCrossRefGoogle Scholar
  56. Braides A. [1998], Approximation of Free-Discontinuity Problems, volume 1694 of Lecture Notes in Mathematics. Springer-Verlag, Berlin.Google Scholar
  57. Braides A. and A. Coscia [1993], A singular perturbation approach to variational problems in fracture mechanics. Math. Models Methods Appl. Sci., 3:303–340.MathSciNetCrossRefzbMATHGoogle Scholar
  58. Braides A. and A. Coscia [1994], The interaction between bulk energy and surface energy in multiple integrals. Proc. Royal Soc. Edinburgh, 124A:737–756.MathSciNetGoogle Scholar
  59. Braides A., I. Fonseca and G. Francfort [2000], 3D-2D asymptotic analysis for inhomogeneous thin films. Indiana Univ. Math. J., 49:1367–1404.MathSciNetCrossRefzbMATHGoogle Scholar
  60. Braides A. and M. S. Gelli [2001a], Limits of discrete systems with long-range interactions. Preprint.Google Scholar
  61. Braides A. and M. S. Gelli [2001b], Limits of discrete sytems without convexity hypotheses. Preprint.Google Scholar
  62. Braides A., G. Dal Maso and A. Garroni [1999], Variational formulation for softening phenomena in fracture mechanics: the one-dimensional case. Arch. Rational Mech. Anal., 146:23–58.CrossRefzbMATHGoogle Scholar
  63. Bressan, A. [1988], Contractive metrics for nonlinear hyperbolic systems. Indiana J. Math., 37:409–421.zbMATHMathSciNetCrossRefGoogle Scholar
  64. Bressan, A. [1995], The unique limit of the Glimm scheme. Arch. Rational Mech. Anal., 130:205–230.zbMATHMathSciNetCrossRefGoogle Scholar
  65. Bressan, A. [2000], Hyperbolic Systems of Conservation Laws. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press.Google Scholar
  66. Bressan, A. and R. M. Colombo [1995], The semigroup generated by 2 × 2 conservation laws. Arch. Rational Mech. Anal., 133:1–75.MathSciNetCrossRefzbMATHGoogle Scholar
  67. Bressan, A., G. Crasta and B. Piccoli [2000], Well-posedness of the Cauchy problem for n × n systems of conservation laws. Mem. Amer. Math. Soc, 146(694).Google Scholar
  68. Bressan, A. and P. G. Le Floch [1997], Uniqueness of weak solutions to hyperbolic systems of conservation laws. Arch. Rational Mech. Anal., 140:301–317.MathSciNetCrossRefzbMATHGoogle Scholar
  69. Bressan, A. and P. Goatin [1999], Oleinik type estimates and uniqueness for n × n conservation laws. J. Differential Eqns, 156:26–49.MathSciNetCrossRefzbMATHGoogle Scholar
  70. Bressan, A. and M. Lewicka [2000], A uniqueness condition for hyperbolic systems of conservation laws. Discrete Contin. Dynam. Systems, 6:673–682.MathSciNetzbMATHGoogle Scholar
  71. Bressan, A., T.-P. Liu and T. Yang [1999], L 1 stability estimates for n × n conservation laws. Arch. Rational Mech. Anal., 149:1–22.MathSciNetCrossRefzbMATHGoogle Scholar
  72. Buttazzo, G. [1995], Energies on BV and variational models in fracture mechanics. In Curvature flows and related topics (Levico, 1994), volume 5 of GAKUTO Internat. Ser. Math. Sci. Appl., pages 25–36, Tokyo. Gakkötosho.Google Scholar
  73. Buttazzo, G. and M. Belloni [1995], A survey on old and recent results about the gap phenomenon. In Recent Developments in Well-Posed Variational Problems, pages 1–27, edited by R. Lucchetti and J. Revalski, Kluwer Academic Publishers, Dordrecht.Google Scholar
  74. Catto, I., C. Le Bris and P.-L. Lions [1998], The Mathematical Theory of Thermodynamic Limits: Thomas-Fermi Type Models. Oxford University Press.Google Scholar
  75. Cherepanov, G. P., editor [1998], Fracture. Krieger, Malabar, Fl.zbMATHGoogle Scholar
  76. Chillingworth, D. R. J., J. E. Marsden and Y. H. Wan [1982], Symmetry and bifurcation in three-dimensional elasticity, I. Arch. Rational Mech. Anal., 80:295–331.MathSciNetCrossRefzbMATHGoogle Scholar
  77. Chillingworth, D. R. J., J. E. Marsden and Y. H. Wan [1983], Symmetry and bifurcation in three-dimensional elasticity, II. Arch. Rational Mech. Anal., 83:363–395.MathSciNetCrossRefzbMATHGoogle Scholar
  78. Chlebík, M. and B. Kirchheim [2001], Rigidity for the four gradient problem, (to appear).Google Scholar
  79. Chu, C. and R. D. James [1993], Biaxial loading experiments on Cu-Al-Ni single crystals. In Experiments in Smart Materials and Structures, pages 61–69. ASME. AMD-Vol. 181.Google Scholar
  80. Chu, C. and R. D. James [1995], Analysis of microstructures in Cu-14.0%Al-3.9%Ni by energy minimization. J. de Physique IV, C8:143–149.Google Scholar
  81. Ciarlet, P. G. [2000], Un modèle bi-dimensionnel non linéaire de coque analogue à celui de W. T. Koiter. C. R. Acad. Sci. Paris Sér. I Math., 331:405–410.zbMATHMathSciNetGoogle Scholar
  82. Ciarlet, P. G. [1988], Mathematical Elasticity, Vol.I: Three-Dimensional Elasticity. North-Holland Publishing Co., Amsterdam.Google Scholar
  83. Ciarlet, P. G. [1997], Mathematical Elasticity. Vol. II: Theory of Plates. North-Holland Publishing Co., Amsterdam.Google Scholar
  84. Ciarlet, P. G. [2000], Mathematical Elasticity. Vol. III: Theory of Shells. North-Holland Publishing Co., Amsterdam.Google Scholar
  85. Ciarlet, P. G. and J. Nečas [1985], Unilateral problems in nonlinear three-dimensional elasticity. Arch. Rational Mech. Anal., 87:319–338.MathSciNetCrossRefzbMATHGoogle Scholar
  86. Ciarlet, P. G. and A. Roquefort [2000], Justification d’un modèle bi-dimensionnel non linéaire de coque analogue à celui de W. T. Koiter. C. R. Acad. Sci. Paris Sér. I Math., 331(5):411–416.MathSciNetzbMATHGoogle Scholar
  87. Coleman, B. D. and E.H. Dill [1973], On thermodynamics and the stability of motion of materials with memory. Arch. Rational Mech. Anal., 51:1–53.MathSciNetzbMATHGoogle Scholar
  88. Coleman, B. D. and W. Noll [1963], The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rational Mech. Anal., 13:167–178.MathSciNetCrossRefzbMATHGoogle Scholar
  89. Dacorogna, B. [1982], Quasiconvexity and relaxation of non convex variational problems. J. Funct. Anal., 46:102–118.zbMATHMathSciNetCrossRefGoogle Scholar
  90. Dacorogna, B. and P. Marcellini [1999], Implicit Partial Di erential Equations. Birkhäuser Boston Inc., Boston, MA.Google Scholar
  91. Dafermos, C. M. [1969], The mixed initial boundary-value problem for the equations of nonlinear one-dimensional viscoelasticity. J. Differential Eqns, 6:71–86.zbMATHMathSciNetCrossRefGoogle Scholar
  92. Dafermos, C. M. [1972], Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl., 38:33–41.zbMATHMathSciNetCrossRefGoogle Scholar
  93. Dafermos, C. M. [1996], Entropy and the stability of classical solutions of hyperbolic systems of conservation laws. In Recent Mathematical Methods in Nonlinear Wave Propagation (Montecatini Terme, 1994), volume 1640 of Lecture Notes in Math., pages 48–69, Berlin. Springer.Google Scholar
  94. Dafermos, C. M. [2000], Hyperbolic Conservation Laws in Continuum Physics, volume 325 of Grundlehren der Mathematischen Wissenschaften. Springer.Google Scholar
  95. Dafermos, C. M. and W. J. Hrusa [1985], Energy methods for quasilinear hyperbolic initial boundary-value problems. Arch. Rational Mech. Anal., 87:267–292.MathSciNetCrossRefzbMATHGoogle Scholar
  96. Dal Maso, G. [1993], An Introduction to Γ-convergence. Birkhäuser Boston Inc., Boston, MA.Google Scholar
  97. DeGiorgi, E. and T. Franzoni [1979], On a type of variational convergence. In Proceedings of the Brescia Mathematical Seminar, Vol. 3 (Italian), pages 63–101, Milan. Univ. Cattolica Sacro Cuore.Google Scholar
  98. Deam, R. T. and S. F. Edwards [1976], The theory of rubber elasticity. Philos. Trans. Roy. Soc. London Ser. A, 280:317–353.MathSciNetGoogle Scholar
  99. Demoulini, S. [2000], Weak solutions for a class of nonlinear systems of viscoelasticity. Arch. Rational Mech. Anal., 155:299–334.zbMATHMathSciNetCrossRefGoogle Scholar
  100. Demoulini, S., D. M. A. Stuart and A.E. Tzavaras [2000], Construction of entropy solutions for one-dimensional elastodynamics via time discretisation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 17:711–731.MathSciNetCrossRefzbMATHGoogle Scholar
  101. Demoulini, S., D. M. A. Stuart and A.E. Tzavaras [2001], A variational approximation scheme for three-dimensional elastodynamics with polyconvex energy. Arch. Rational Mech. Anal., 157:325–344.MathSciNetCrossRefzbMATHGoogle Scholar
  102. DiPerna, R. J. [1983], Convergence of approximate solutions of conservation laws. Arch. Rational Mech. Anal., 82:27–70.zbMATHMathSciNetCrossRefGoogle Scholar
  103. DiPerna, R. J. [1985], Compensated compactness and general systems of conservation laws. Trans. A.M.S., 292:283–420.Google Scholar
  104. Duhem, P. [1911], Traité ďÉnergetique ou de Thermodynamique Générale. Gauthier-Villars, Paris.Google Scholar
  105. Ebin, D. G. [1993], Global solutions of the equations of elastodynamics of incompressible neo-Hookean materials. Proc. Nat. Acad. Sci. U.S.A., 90:3802–3805.zbMATHMathSciNetGoogle Scholar
  106. Ebin, D. G. [1996], Global solutions of the equations of elastodynamics for incompressible materials. Electron. Res. Announc. Amer. Math. Soc., 2:50–59 (electronic).zbMATHMathSciNetCrossRefGoogle Scholar
  107. Ebin, D. G. and R.A. Saxton [1986], The initial value problem for elastodynamics of incompressible bodies. Arch. Rational Mech. Anal., 94:15–38.MathSciNetCrossRefzbMATHGoogle Scholar
  108. Ebin, D. G. and S.R. Simanca [1990], Small deformations of incompressible bodies with free boundary. Comm. Partial Differential Equations, 15:1589–1616.MathSciNetzbMATHGoogle Scholar
  109. Ebin, D. G. and S.R. Simanca [1992], Deformations of incompressible bodies with free boundaries. Arch. Rational Mech. Anal., 120:61–97.MathSciNetCrossRefzbMATHGoogle Scholar
  110. Edwards, S. F. and T.A. Vilgis [1988], The tube model theory of rubber elasticity. Rep. Progr. Phys., 51:243–297.MathSciNetCrossRefGoogle Scholar
  111. Ericksen, J. L. [1966], Thermoelastic stability. In Proc 5 th National Cong. Appl. Mech., pages 187–193.Google Scholar
  112. Ericksen, J. L. [1977b], On the formulation of St.-Venant’s problem. In Nonlinear analysis and mechanics: Heriot-Watt Symposium (Edinburgh, 1976), Vol. I, pages 158–186. Res. Notes in Math., No. 17. Pitman, London.Google Scholar
  113. Ericksen, J. L. [1977b], Special topics in elastostatics. In C.-S. Yih, editor, Advances in Applied Mechanics, volume 17, pages 189–244. Academic Press.Google Scholar
  114. Ericksen, J. L. [1983], Ill-posed problems in thermoelasticity theory. In Proceedings of a NATO/London Mathematical Society advanced study institute held in Oxford, July 25–August 7, 1982, pages 71–93. D. Reidel Publishing Co., Dordrecht.Google Scholar
  115. Euler, L. [1744], Additamentum I de curvis elasticis, methodus inveniendi lineas curvas maximi minimivi proprietate gaudentes. Bousquent, Lausanne. In Opera Omnia I, Vol. 24, 231–297.Google Scholar
  116. Evans, L. C. [1986], Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal., 95:227–268.zbMATHMathSciNetCrossRefGoogle Scholar
  117. Evans, L. C. and R. F. Gariepy [1987], Some remarks concerning quasiconvexity and strong convergence. Proc. Roy. Soc. Edinburgh, 106A:53–61.MathSciNetGoogle Scholar
  118. Fefferman, C. [1985], The thermodynamic limit for a crystal. Comm. Math. Phys., 98(3):289–311.zbMATHMathSciNetCrossRefGoogle Scholar
  119. Foccardi, M. and M. S. Gelli [2001], A finite-differences approximation of fracture energies for non-linear elastic materials. Preprint.Google Scholar
  120. Fonseca, I. [1988], The lower quasiconvex envelope of the stored energy function of an elastic crystal. J. Math. Pures Appl., 67:175–195.zbMATHMathSciNetGoogle Scholar
  121. Fonseca, I. and W. Gangbo [1995], Local invertibility of Sobolev functions. SIAM J. Math. Anal., 26:280–304.MathSciNetCrossRefzbMATHGoogle Scholar
  122. Foss, M. [2001], On Lavrentiev’s Phenomenon. PhD thesis, Carnegie-Mellon University.Google Scholar
  123. Francfort, G. A. and J.-J. Marigo [1998], Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids, 46:1319–1342.MathSciNetCrossRefzbMATHGoogle Scholar
  124. Friesecke, G. [2000], personal communication.Google Scholar
  125. Friesecke, G. and G. Dolzmann [1997], Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy. SIAM J. Math. Anal., 28:363–380.MathSciNetCrossRefzbMATHGoogle Scholar
  126. Friesecke, G. and R. D. James [2000], A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods. J. Mech. Phys. Solids, 48:1519–1540.MathSciNetCrossRefzbMATHGoogle Scholar
  127. Friesecke, G., R. D. James and S. Müller [2001], Rigorous derivation of nonlinear plate theory and geometric rigidity. C. R. Acad. Sci. Paris Sér. I Math., (to appear).Google Scholar
  128. Friesecke, G. and J. B. McLeod [1996], Dynamics as a mechanism preventing the formation of finer and finer microstructure. Arch. Rational Mech. Anal., 133:199–247.MathSciNetCrossRefzbMATHGoogle Scholar
  129. Friesecke, G. and J. B. McLeod [1997], Dynamic stability of non-minimizing phase mixtures. Proc. Roy. Soc. London Ser. A, 453:2427–2436.MathSciNetzbMATHCrossRefGoogle Scholar
  130. Friesecke, G. and F. Theil [2001], Validity and failure of the Cauchy-Born hypothesis in a 2D mass-spring lattice. Preprint.Google Scholar
  131. Giaquinta, M., G. Modica and J. Souček [1989], Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 106:97–159. Addendum, ibid., 109:385–392, 1990.MathSciNetCrossRefzbMATHGoogle Scholar
  132. Giaquinta, M., G. Modicaand J. Souček [1994], A weak approacht of inite elasticity. Calc. Var. Partial Differential Equations, 2:65–100.MathSciNetCrossRefzbMATHGoogle Scholar
  133. Giaquinta, M., G. Modica and J. Souček [1998], Cartesian Currents in the Calculus of Variations. Volumes I, II. Springer-Verlag, Berlin. Cartesian currents.Google Scholar
  134. Glimm, J. [1965], Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math., 18:697–715.zbMATHMathSciNetGoogle Scholar
  135. Green, A.E. and J.E. Adkins [1970], Large Elastic Deformations. Oxford University Press, second edition.Google Scholar
  136. Green, A. E. and W. Zerna [1968], Theoretical Elasticity. Clarendon Press, Oxford, second edition.zbMATHGoogle Scholar
  137. Greenberg, J. M., R. C. MacCamy and V. J. Mizel [1967], On the existence, uniqueness, and stability of solutions of the equations σ′(u x)u xx + λu xtx=ρ 0 u tt. J. Math. Mech., 17:707–728, 1967/1968.MathSciNetGoogle Scholar
  138. Gromov, M. [1986], Partial Differential Relations. Springer-Verlag, Berlin.zbMATHGoogle Scholar
  139. Gurtin, M. E. [1981], Topics in Finite Elasticity. SIAM, 1981.Google Scholar
  140. Hane, K. [1997], Microstructures in Thermoelastic Martensites. PhD thesis, Department of Aerospace Engineering and Mechanics, University of Minnesota.Google Scholar
  141. Hao, W., S. Leonardi and J. Nečas [1996], An example of irregular solution to a nonlinear Euler-Lagrange elliptic system with real analytic coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23:57–67.MathSciNetzbMATHGoogle Scholar
  142. Healey, T. J. [2000], Global continuation in displacement problems of nonlinear elastostatics via the Leray-Schauder degree. Arch. Rational Mech. Anal., 152:273–28.zbMATHMathSciNetCrossRefGoogle Scholar
  143. Healey, T. J. and P. Rosakis [1997], Unbounded branches of classical injective solutions to the forced displacement problem in nonlinear elastostatics. J. Elasticity, 49:65–78.MathSciNetCrossRefzbMATHGoogle Scholar
  144. Healey, T. J. and H. Simpson [1998], Global continuation in nonlinear elasticity. Arch. Rational Mech. Anal., 143:1–28.MathSciNetCrossRefzbMATHGoogle Scholar
  145. Hrusa, W. J. and M. Renardy [1988], An existence theorem for the Dirichlet problem in the elastodynamics of incompressible materials. Arch. Rational Mech. Anal., 102:95–117. Corrections ibid 110:373–375, 1990.MathSciNetCrossRefzbMATHGoogle Scholar
  146. Hughes, T. J. R., T. Kato and J.E. Marsden [1977], Well-posed quasilinear hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rational Mech. Anal., 63:273–294.MathSciNetCrossRefzbMATHGoogle Scholar
  147. James, R. D. and S. J. Spector [1991], The formation of filamentary voids in solids. J. Mech. Phys. Solids, 39:783–813.MathSciNetCrossRefzbMATHGoogle Scholar
  148. Jiang, S. and R. Racke [2000], Evolution equations in thermoelasticity. Chapman & Hall/CRC, Boca Raton, FL.zbMATHGoogle Scholar
  149. John, F. [1961], Rotation and strain. Comm. Pure Appl. Math., 14:391–413.zbMATHMathSciNetGoogle Scholar
  150. John, F. [1965], Estimates for the derivatives of the stresses in a thin shell and interior shell equations. Comm. Pure Appl. Math., 18:235–267.MathSciNetGoogle Scholar
  151. John, F. [1971], Refined interior equations for thin elastic shells. Comm. Pure Appl. Math., 24:583–615.zbMATHMathSciNetGoogle Scholar
  152. John, F. [1972a], Bounds for deformations in terms of average strains. In Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin), pages 129–144. Academic Press, New York.Google Scholar
  153. John, F. [1972b], Uniqueness of non-linear elastic equilibrium for prescribed boundary displacements and sufficiently small strains. Comm. Pure Appl. Math., 25:617–634.zbMATHMathSciNetGoogle Scholar
  154. John, F. [1988], Almost global existence of elastic waves of finite amplitude arising from small initial disturbances. Comm. Pure Appl. Math., 41:615–666.zbMATHMathSciNetGoogle Scholar
  155. Kato, T. [1985], Abstract Differential Equations and Nonlinear Mixed Problems. Lezioni Fermi. Scuola Normale Superiore, Pisa; Accademia Nazionale dei Lincei, Rome.Google Scholar
  156. Kinderlehrer, D. and P. Pedregal [1991], Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal., 115:329–365.MathSciNetCrossRefzbMATHGoogle Scholar
  157. Kinderlehrer, D. and P. Pedregal [1994], Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal., 4:59–90.MathSciNetzbMATHGoogle Scholar
  158. Kirchheim, B. [2001], Deformations with finitely many gradients and stability of quasiconvex hulls. C. R. Acad. Sci. Paris Sér. I Math., 332:289–294.zbMATHMathSciNetGoogle Scholar
  159. Knops, R. J. and C.A. Stuart [1984], Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. Arch. Rational Mech. Anal., 86:233–249.MathSciNetCrossRefzbMATHGoogle Scholar
  160. Knops, R. J. and E. W. Wilkes [1973], Theory of elastic stability. In S. Flugge, editor, Encyclopedia of Physics, volume VIa/1-4. Springer-Verlag, Berlin.Google Scholar
  161. Kohn, R. V. [1982], New integral estimates for deformations in terms of their nonlinear strains. Arch. Rational Mech. Anal., 78:131–172.zbMATHMathSciNetCrossRefGoogle Scholar
  162. Koiter, W. T. [1976], A basic open problem in the theory of elastic stability. In Applications of Methods of Functional Analysis to Problems in Mechanics (Joint Sympos., IUTAM/IMU, Marseille, 1975), pages 366–373. Lecture Notes in Math., 503. Springer, Berlin.Google Scholar
  163. Kristensen, J. [1994], Lower Semicontinuity of Variational Integrals. PhD thesis, Technical University of Lyngby.Google Scholar
  164. Kristensen, J. [1999], On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré, Anal. Non Linéaire, 16:1–13.zbMATHMathSciNetCrossRefGoogle Scholar
  165. Kristensen, J. and A. Taheri [2001], Partial regularity of strong local minimisers. Preprint.Google Scholar
  166. Lazzeri, A. and C. B. Bucknall [1995], Applications of a dilatational yielding model to rubber-toughened polymers. Polymer, 36:2895–2902.CrossRefGoogle Scholar
  167. Le Dret, H. [1990], Sur les fonctions de matrices convexes et isotropes. C. R. Acad. Sci. Paris Sér. I Math., 310:617–620.zbMATHGoogle Scholar
  168. Le Dret, H. and A. Raoult [1995a], From three-dimensional elasticity to nonlinear membranes. In Asymptotic methods for elastic structures (Lisbon, 1993), pages 89–102. de Gruyter, Berlin.Google Scholar
  169. Le Dret, H. and A. Raoult [1995b], The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl., 74:549–578.MathSciNetzbMATHGoogle Scholar
  170. Le Dret, H. and A. Raoult [1996], The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Sci., 6:59–84.MathSciNetCrossRefzbMATHGoogle Scholar
  171. Le Dret, H. and A. Raoult [1998], From three-dimensional elasticity to the nonlinear membrane model. In Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. XIII (Paris, 1994/1996), pages 192–206. Longman, Harlow.Google Scholar
  172. Le Dret, H. and A. Raoult [2000], Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results. Arch. Ration. Mech. Anal., 154:101–134.MathSciNetCrossRefzbMATHGoogle Scholar
  173. Lieb, E. H. and B. Simon [1977], The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math., 23:22–116.MathSciNetCrossRefGoogle Scholar
  174. Lin, P. [1990], Maximization of the entropy for an elastic body free of surface traction. Arch. Rational Mech. Anal., 112:161–191.zbMATHMathSciNetCrossRefGoogle Scholar
  175. Liu, T.-P. [1977], Initial boundary-value problems in gas dynamics. Arch. Rational Mech. Anal., 64:137–168.zbMATHMathSciNetGoogle Scholar
  176. Liu, T.-P. [1981], Admissible solutions of hyperbolic conservation laws. Memoirs AMS, 30 (240).Google Scholar
  177. Liu, T.-P. and T. Yang [1999a], L 1 stability for 2×2 systems of hyperbolic conservation laws. J. Amer. Math. Soc., 12:729–774.MathSciNetCrossRefzbMATHGoogle Scholar
  178. Liu, T.-P. and T. Yang [1999b], L 1 stability of conservation laws with coinciding hugoniot and characteristic curves. Indiana Univ. Math. J, 48:237–247.MathSciNetCrossRefzbMATHGoogle Scholar
  179. Liu, T.-P. and T. Yang [1999c], Well-posedness theory for hyperbolic conservation laws. Comm. Pure Appl. Math, 52:1553–1586.MathSciNetCrossRefzbMATHGoogle Scholar
  180. Love, A. E. H. [1927], A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, fourth edition (revised and enlarged); Reprinted by Dover, New York, 1944.Google Scholar
  181. Luskin, M. [1996], On the computation of crystalline microstructure. Acta Numerica, 5:191–258.zbMATHMathSciNetCrossRefGoogle Scholar
  182. Marsden, J. E. and T.J.R. Hughes [1983], Mathematical Foundations of Elasticity. Prentice-Hall.Google Scholar
  183. Meisters, G. H. and C. Olech [1963], Locally one-to-one mappings and a classical theorem on Schlicht functions. Duke Math. J., 30:63–80.MathSciNetCrossRefzbMATHGoogle Scholar
  184. Mielke, A. [1988], Saint-Venant’s problem and semi-inverse solutions in nonlinear elasticity. Arch. Rational Mech. Anal., 102:205–229. Corrigendum ibid. 110:351–352, 1990.zbMATHMathSciNetGoogle Scholar
  185. Mielke, A. [1990], Normal hyperbolicity of center manifolds and Saint-Venant’s principle. Arch. Rational Mech. Anal., 110:353–372.zbMATHMathSciNetCrossRefGoogle Scholar
  186. Mizel, V. J., M. Foss and W. J. Hrusa [2002], The Lavrentiev gap phenomenon in nonlinear elasticity, (to appear).Google Scholar
  187. Monneau, R. [2001], Justification de la théeorie non linéeaire de Kirchho-Love, comme application d’une nouvelle méethode d’inversion singulière. C. R. Acad. Sci. Paris Séer. I Math., (to appear).Google Scholar
  188. Morrey, C. B. [1952], Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math., 2:25–53.zbMATHMathSciNetGoogle Scholar
  189. Müller, S. [1988], Weak continuity of determinants and nonlinear elasticity. C. R. Acad. Sci. Paris Sér. I Math., 307:501–506.zbMATHGoogle Scholar
  190. Müller, S. [1999], Variational methods for microstructure and phase transitions. In Calculus of variations and geometric evolution problems, volume 1713 of Lecture Notes in Math., pages 85–210. Springer, Berlin.Google Scholar
  191. Müller, S., T. Qi and B. S. Yan [1994], On a new class of elastic deformations not allowing for cavitation. Ann. Inst. Henri Poincarè, Analyse Nonlinèaire, 11:217–243.zbMATHGoogle Scholar
  192. Müller, S. and S. J. Spector [1995], An existence theory for nonlinear elasticity that allows for cavitation. Arch. Rational Mech. Anal., 131:1–66.MathSciNetCrossRefzbMATHGoogle Scholar
  193. Müller, S. and V. Šverák [1996] Attainment results for the two-well problem by convex integration. In J. Jost, editor, Geometric analysis and the calculus of variations, pages 239–251. International Press.Google Scholar
  194. Müuller, S. and V. Šverãak [2001], Convex integration for Lipschitz mappings and counterexamples to regularity. Annals of Math., (to appear).Google Scholar
  195. Müller, S. and M. A. Sychev [2001], Optimal existence theorems for nonhomogeneous differential inclusions. J. Funct. Anal., 181:447–475.MathSciNetCrossRefzbMATHGoogle Scholar
  196. Muncaster, R. G. [1979], Saint-Venant’s problem in nonlinear elasticity: a study of cross sections.In Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, pages 17–75. Pitman, Boston, Mass.Google Scholar
  197. Muncaster, R. G. [1983], Saint-Venant’s problem for slender prisms. Utilitas Math., 23:75–101, 1983.zbMATHMathSciNetGoogle Scholar
  198. Nečas, J. [1977], Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity. In Theory of Nonlinear Operators, pages 197–206, Berlin. Akademie-Verlag.Google Scholar
  199. Ogden, R. W. [1972a], Large deformation isotropic elasticity — on the correlation of theory and experiment for incompressible rubberlike solids. Proc. Roy. Soc. London A, 326:562–584.Google Scholar
  200. Ogden, R. W. [1972b], Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids. Proc. Roy. Soc. London A, 328:567–583.zbMATHGoogle Scholar
  201. Ogden, R. W. [1984], Nonlinear Elastic Deformations. Ellis Horwood.Google Scholar
  202. Pantz, O. [2000], Dérivation des modèles de plaques membranaires non linéaires à partir de l’élasticité tri-dimensionnelle. C. R. Acad. Sci. Paris Sér. I Math., 331:171–174.zbMATHMathSciNetGoogle Scholar
  203. Pantz, O. [2001a], Quelques Problèmes de Modélisation en Élasticité Nonlinéaire. PhD thesis, Université Paris 6.Google Scholar
  204. Pantz, O. [2001b], Une justification partielle du modèle de plaque en flexion par Γ-convergence. C. R. Acad. Sci. Paris Sér. I Math., 332:587–592.zbMATHMathSciNetGoogle Scholar
  205. Pedregal, P. [1991], Parametrized Measures and Variational Principles, volume 30 of Progress in nonlinear differential equations and their applications. Birkhäuser, Basel.Google Scholar
  206. Pedregal, P. [1994], Jensen’s inequality in the calculus of variations. Differential Integral Equations, 7:57–72.zbMATHMathSciNetGoogle Scholar
  207. Pedregal, P. [2000], Variational Methods in Nonlinear Elasticity. SIAM, Philadelphia.zbMATHGoogle Scholar
  208. Pego, R. L. [1987], Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability. Arch. Rational Mech. Anal., 97:353–394.zbMATHMathSciNetCrossRefGoogle Scholar
  209. Penrose, O. [2001], Statistical mechanics of nonlinear elasticity. Markov Processes and Related Fields, (to appear).Google Scholar
  210. Pericak-Spector, K. A. and S. J. Spector [1997], Dynamic cavitation with shocks in nonlinear elasticity. Proc. Roy. Soc. Edinburgh, 127A:837–857.MathSciNetGoogle Scholar
  211. Phillips, D. [2001]. On one-homogeneous to elliptic systems in two dimensions. C. R. Acad. Sci. Paris Sér, I Nath., (to appear).Google Scholar
  212. Phillips, R. [2001]. Crystals, defects and microstructures. Cambridge University Press.Google Scholar
  213. Polignone, D. A. and C. O. Horgan [1993a], Cavitation for incompressible anisotropic non-linearly elastic spheres. J. Elasticity, 33:27–65.MathSciNetCrossRefzbMATHGoogle Scholar
  214. Polignone, D. A. and C. O. Horgan [1993b], Effects of material anisotropy and inhomogeneity on cavitation for composite incompressible anisotropic nonlinearly elastic spheres. Internat. J. Solids Structures, 30:3381–3416.MathSciNetCrossRefzbMATHGoogle Scholar
  215. Post, K. D. E. and J. Sivaloganathan [1997], On homotopy conditions and the existence of multiple equilibria in finite elasticity. Proc. Royal Soc. Edinburgh, 127 A:595–614.MathSciNetGoogle Scholar
  216. Potier-Ferry, M. [1981], The linearization principle for the stability of solutions of quasilinear parabolic equations. I. Arch. Rational Mech. Anal., 77:301–320.zbMATHMathSciNetCrossRefGoogle Scholar
  217. Potier-Ferry, M. [1982], On the mathematical foundations of elastic stability theory. I. Arch. Rational Mech. Anal., 78:55–72.zbMATHMathSciNetCrossRefGoogle Scholar
  218. Qi, Tang [1988], Almost-everywhere injectivity in nonlinear elasticity. Proc. Royal Soc. Edinburgh, 109 A:79–95.Google Scholar
  219. Qin, T. [1998], Symmetrizing the nonlinear elastodynamic system. J. Elasticity, 50:245–252.zbMATHMathSciNetCrossRefGoogle Scholar
  220. Racke, R. and S. Zheng [1997], Global existence and asymptotic behavior in nonlinear thermoviscoelasticity. J. Differential Equations, 134:46–67.MathSciNetCrossRefzbMATHGoogle Scholar
  221. Radin, C. [1987], Low temperature and the origin of crystalline symmetry. Internat. J. Modern Phys. B, 1:1157–1191.MathSciNetCrossRefGoogle Scholar
  222. Rybka, P. [1992], Dynamical modelling of phase transitions by means of viscoelasticity in many dimensions. Proc. Royal Soc. Edinburgh, 121 A:101–138.MathSciNetGoogle Scholar
  223. Serre, D. [2000], Systèmes de Lois de Conservation, Vols I, II. Diderot, Paris, 1996. English translation: Systems of Conservation Laws, Vols I,II, Cambridge Univ. Press, Cambridge.Google Scholar
  224. Shu, Y.C. [2000], Heterogeneous thin films of martensitic materials. Arch. Ration. Mech. Anal., 153:39–90.zbMATHMathSciNetCrossRefGoogle Scholar
  225. Sivaloganathan, J. [1986], Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. Arch. Rational Mech. Anal., 96:97–136.zbMATHMathSciNetCrossRefGoogle Scholar
  226. Sivaloganathan, J. [1989], The generalised Hamilton-Jacobi inequality and the stability of equilibria innonlin earelasticity. Arch. Rational Mech. Anal., 107:347–369.zbMATHMathSciNetCrossRefGoogle Scholar
  227. Sivaloganathan, J. [1995], On the stability of cavitating equilibria. Quart. Appl. Math., 53:301–313.zbMATHMathSciNetGoogle Scholar
  228. Sivaloganathan, J. [1999], On cavitation and degenerate cavitation under internal hydrostatic pressure. Proc. R. Soc. Lond. Ser. A, 455:3645–3664.zbMATHMathSciNetGoogle Scholar
  229. Sivaloganathan, J. and S.J. Spector [2000a], On the optimal location of singularities arising in variational problems of nonlinear elasticity. J. Elasticity, 58:191–224.MathSciNetCrossRefzbMATHGoogle Scholar
  230. Sivaloganathan, J. and S. J. Spector [2000b], On the existence of minimizers with prescribed singular points in nonlinear elasticity. J. Elasticity, 59:83–113. In recognition of the sixtieth birthday of Roger L. Fosdick (Blacksburg, VA, 1999).MathSciNetCrossRefzbMATHGoogle Scholar
  231. Sivaloganathan, J. and S. J. Spector [2001], A construction of infinitely many singular weak solutions to the equations of nonlinear elasticity. Preprint.Google Scholar
  232. Sobolevskii, P. E. [1966], Equations of parabolic type in Banach space. Amer. Math. Soc. Transl., 49:1–62.zbMATHGoogle Scholar
  233. Stoppelli, F. [1954], Un teorema di esistenza e di unicita relativo alle equazioni delľelastostatica isoterma per deformazioni finite. Recherche Mat., 3:247–267.zbMATHMathSciNetGoogle Scholar
  234. Stoppelli, F. [1955], Sulla svilluppibilita in serie de potenze di un parametro delle soluzioni delle equazioni delľelastostatica isoterma. Recherche Mat., 4:58–73, 1955.zbMATHMathSciNetGoogle Scholar
  235. Stringfellow, R. and R. Abeyaratne [1989], Cavitation in an elastomer — comparison of theory with experiment. Materials Science and Engineering A — Structural Materials Properties, Microstructure and Processing, 112:127–131.Google Scholar
  236. Stuart, C. A. [1985], Radially symmetric cavitation for hyperelastic materials. Ann. Inst. H. Poincaré. Anal. Non. Linéaire, 2:33–66.zbMATHMathSciNetGoogle Scholar
  237. Stuart, C. A. [1993], Estimating the critical radius for radially symmetric cavitation. Quart. Appl. Math., 51:251–263.zbMATHMathSciNetGoogle Scholar
  238. Šverák, V. [1988], Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal., 100:105–127.MathSciNetzbMATHCrossRefGoogle Scholar
  239. Šverák, V. [1991], Quasiconvex functions with subquadratic growth. Proc. Roy. Soc. Lond. A, 433:723–732.zbMATHCrossRefGoogle Scholar
  240. Šverák, V. [1992], Rank-one convexity does not imply quasiconvexity. Proc. Royal Soc. Edinburgh, 120A:185–189.Google Scholar
  241. Šverák, V. [1995], Lower-semicontinuity of variational integrals and compensated compactness. In Proc. International Congress of Mathematicians, Zurich 1994, Basel. BirkhaÜser.Google Scholar
  242. Šverák, V. and X. Yan [2000], A singular minimizer of a smooth strongly convex functional in three dimensions. Calc. Var. Partial Differential Equations, 10:213–221.MathSciNetCrossRefzbMATHGoogle Scholar
  243. Sychev, M. A. [1999], A new approach to Young measure theory, relaxation and convergence in energy. Ann. Inst. H. Poincaré Anal. Non Linéaire, 16:773–812.zbMATHMathSciNetCrossRefGoogle Scholar
  244. Sychev, M. A. [2001], Few remarks on differential inclusions. Preprint.Google Scholar
  245. Sylvester, J. [1985], On the Differentiability of O(n) invariant functions of symmetric matrices. Duke Math. J., 52:475–483.zbMATHMathSciNetCrossRefGoogle Scholar
  246. Tadmor, E. B., M. Ortiz and R. Phillips [1996], Quasicontinuum analysis of defects in solids. Phil. Mag. A, 73:1529–1563Google Scholar
  247. Taheri, A. [2001a], On Artin’s braid group and polyconvexity in the calculus of variations. Preprint.Google Scholar
  248. Taheri, A. [2001b], Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations. Preprint.Google Scholar
  249. Tartar, L. [1979], Compensated compactness and applications to partial differential equations. In R.J. Knops, editor, Nonlinear Analysis and Mechanics; Heriot-Watt Symposium, Vol. IV, pages 136–192. Pitman Research Notes in Mathematics.Google Scholar
  250. Tartar, L. [1982], The compensated compactness method applied to systems of conservation laws. In Systems of Nonlinear Partial Differential Equations, J. M. Ball, editor, pages 263–285. NATO ASI Series, Vol. C111, Reidel.Google Scholar
  251. Tartar, L. [1993], Some remarks on separately convex functions. In Proceedings of conference on Microstructures and phase transitions, IMA, Minneapolis, 1990.Google Scholar
  252. Tonelli, L. [1921], Fondamenti di Calcolo delle Variazioni, Volumes I, II. Zanichelli, 1921–23.Google Scholar
  253. Truesdell, C. and W. Noll [1965], The non-linear field theories of mechanics. In S. Flügge, editor, Handbuch der Physik, Berlin. Springer. Vol. III/3.Google Scholar
  254. Valent, T. [1988], Boundary Value ProblemsofFinite Elasticity, volume 31 of Springer Tracts in Natural Philosophy. Springer-Verlag.Google Scholar
  255. Vodop’yanov, S. K., V. M. Goľdshtein and Yu. G. Reshetnyak [1979], The geometric properties of functions with generalized first derivatives. Russian Math. Surveys, 34:19–74.CrossRefzbMATHGoogle Scholar
  256. Šilhavý, M. [1997], The Mechanics and Thermodynamics of Continuous Media. Springer.Google Scholar
  257. Šilhavý, M. [2000], Differentiability properties of rotationally invariant functions. J. Elasticity, 58:225–232.MathSciNetCrossRefzbMATHGoogle Scholar
  258. Wan, Y. H. and J. E. Marsden [1983], Symmetry and bifurcation in three-dimensional elasticity, Part III: Stressed reference configurations. Arch. Rational Mech. Anal., 84:203–233.MathSciNetCrossRefzbMATHGoogle Scholar
  259. Weiner, J. H. [1983], Statistical Mechanics of Elasticity. Wiley, New York.zbMATHGoogle Scholar
  260. Weinstein, A. [1985], A global invertibility theorem for manifolds with boundary. Proc. Royal Soc. Edinburgh, 99:283–284.zbMATHMathSciNetGoogle Scholar
  261. Xu, C.-Y. [2000], Asymptotic Stability of Equilibria for Nonlinear Semiflows with Applications to Rotating Viscoelastic Rods. PhD thesis, Department of Mathematics, University of California, Berkeley, 2000.Google Scholar
  262. Xu, C.-Y. and J.E. Marsden [1996], Asymptotic stability for equilibria of nonlinear semiflows with applications to rotating viscoelastic rods. I. Topol. Methods Nonlinear Anal., 7:271–297.MathSciNetzbMATHGoogle Scholar
  263. Young, L. C. [1969], Lectures on the Calculus of Variations and Optimal Control Theory. Saunders, 1969. Reprinted by A.M.S. Chelsea.Google Scholar
  264. Zhang, K. [1991], Energy minimizers in nonlinear elastostatics and the implicit function theorem. Arch. Rational Mech. Anal., 114:95–117.zbMATHMathSciNetCrossRefGoogle Scholar
  265. Zhang, K. [2001], A two-well structure and intrinsic mountain pass points. Calc. Var. Partial Differential Equations, 13:231–264.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 2002

Authors and Affiliations

  • John M. Ball

There are no affiliations available

Personalised recommendations