On a certain class of elastic materials with nonelliptic energy densities

Kikuchi, N.; Triantafyllidis, N.

doi:10.1090/qam/678196

Authors: N. Kikuchi and N. Triantafyllidis
Journal: Quart. Appl. Math. 40 (1982), 241-248
MSC: Primary 73G10
DOI: https://doi.org/10.1090/qam/678196
MathSciNet review: 678196
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Abstract: In this paper we are concerned with the behavior of a certain class of hyperelastic materials with non-elliptic strain energy density functions that satisfy very weak growth conditions at infinity. It is shown, by means of an example, that solutions of boundary-value problems involving these materials do not exist within the usual admissible spaces in view of the severe discontinuities involved (displacement discontinuities).

[1] S. S. Antman, Ordinary differential equations of non-linear elasticity, Parts I, II, Arch. Rat. Mech. Anal. 61 307-393 (1976)
[2] J. M. Ball, Convexity conditions and existence theorem in non-linear elasticity, Arch. Rat. Mech. Anal. 63, 337-403 (1977)
[3] J. K. Knowles and Eli Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rational Mech. Anal. 63 (1976), no. 4, 321–336 (1977). MR 0431861, https://doi.org/10.1007/BF00279991
[4] J. L. Ericksen, Equilibrium of bars, J. Elasticity 5 (1975), no. 3–4, 191–201 (English, with French summary). Special issue dedicated to A. E. Green. MR 0471528, https://doi.org/10.1007/BF00126984
[5] B. Dacorogna, A generic result for non-convex problems in the calculus of variations, J. Func. Analysis, in press
[6] P. J. Blatz and W. L. Ko, Application of finite elasticity theory to the deformation of rubbery materials, Trans. Soc. Rheol. 6, 223-251 (1962)
[7] V. Tvergaard, A. Needleman and K. K. Lo, Flow localization in the plane strain tensile test, J. Mech. Phys. Solids, in press
[8] N. Triantafyllidis, A. Needleman and V. Tvergaard, On the development of shear bands in pure bending, Int. J. Solids Struct., in press.
[9] Ivar Ekeland and Roger Temam, Analyse convexe et problèmes variationnels, Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). Collection Études Mathématiques. MR 0463993
Ivar Ekeland and Roger Temam, Convex analysis and variational problems, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. Translated from the French; Studies in Mathematics and its Applications, Vol. 1. MR 0463994
[10] J. N. Hutchinson and K. W. Neale, Sheet necking, II : time-independent behavior, in Mechanics of sheet metal forming, Plenum Press, NY, 1978
[11] C. B. Morrey, Multiple integrals in the calculus of variations, Springer, Berlin, 1966
[12] L. C. Young, Lectures on the calculus of variations and optimal control theory, Foreword by Wendell H. Fleming, W. B. Saunders Co., Philadelphia-London-Toronto, Ont., 1969. MR 0259704