On a certain class of elastic materials with nonelliptic energy densities
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).
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J. M. Ball, Convexity conditions and existence theorem in non-linear elasticity, Arch. Rat. Mech. Anal. 63, 337–403 (1977)
- 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 431861, DOI https://doi.org/10.1007/BF00279991
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B. Dacorogna, A generic result for non-convex problems in the calculus of variations, J. Func. Analysis, in press
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V. Tvergaard, A. Needleman and K. K. Lo, Flow localization in the plane strain tensile test, J. Mech. Phys. Solids, in press
N. Triantafyllidis, A. Needleman and V. Tvergaard, On the development of shear bands in pure bending, Int. J. Solids Struct., in press.
- Ivar Ekeland and Roger Temam, Analyse convexe et problèmes variationnels, Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). Collection Études Mathématiques. MR 0463993
J. N. Hutchinson and K. W. Neale, Sheet necking, II : time-independent behavior, in Mechanics of sheet metal forming, Plenum Press, NY, 1978
C. B. Morrey, Multiple integrals in the calculus of variations, Springer, Berlin, 1966
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S. S. Antman, Ordinary differential equations of non-linear elasticity, Parts I, II, Arch. Rat. Mech. Anal. 61 307–393 (1976)
J. M. Ball, Convexity conditions and existence theorem in non-linear elasticity, Arch. Rat. Mech. Anal. 63, 337–403 (1977)
J. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rat. Mech. Anal. 63, 321 (1977)
J. L. Ericksen, Equilibrium of bars, J. Elasticity 5, 191–201 (1975)
B. Dacorogna, A generic result for non-convex problems in the calculus of variations, J. Func. Analysis, in press
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)
V. Tvergaard, A. Needleman and K. K. Lo, Flow localization in the plane strain tensile test, J. Mech. Phys. Solids, in press
N. Triantafyllidis, A. Needleman and V. Tvergaard, On the development of shear bands in pure bending, Int. J. Solids Struct., in press.
I. Ekeland and R. Temam, Analyse convexe et problemes variationnels, Dunod, Gauthier-Villars, Paris, 1974
J. N. Hutchinson and K. W. Neale, Sheet necking, II : time-independent behavior, in Mechanics of sheet metal forming, Plenum Press, NY, 1978
C. B. Morrey, Multiple integrals in the calculus of variations, Springer, Berlin, 1966
L. C. Young, Lectures on the calculus of variations and optimal control theory, W. B. Saunders, 1969
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© Copyright 1982
American Mathematical Society
