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Unilateral problems in nonlinear three-dimensional elasticity. (English) Zbl 0557.73009

The authors discuss the title topic. A series of theorems by J. M. Ball [ibid. 63, 337-403 (1977; Zbl 0368.73040)] on the existence of solutions are discussed in detail. Ball’s results are then extended to include unilateral boundary conditions corresponding to contact without friction and to locking constraints on the deformation.
The paper is theoretical, written in the notation of modern, continuum theory. It should be of interest and use to those concerned with existence and uniqueness of solutions in nonlinear elasticity.
Reviewer: R.L.Huston

MSC:

74B20 Nonlinear elasticity
74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
74A55 Theories of friction (tribology)
74M15 Contact in solid mechanics
74B99 Elastic materials
74H99 Dynamical problems in solid mechanics

Citations:

Zbl 0368.73040
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References:

[1] Ball, J. M. [1977]: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63, 337–403. · Zbl 0368.73040 · doi:10.1007/BF00279992
[2] Ciarlet, P. G. [1983]: Lectures on Three-Dimensional Elasticity, Tata Institute Lectures on Mathematics, Springer-Verlag, Berlin.
[3] Ciarlet, P. G. [1985]: Topics in Mathematical Elasticity, Vol. 1, North-Holland, Amsterdam.
[4] Ciarlet, P. G. & Geymonat, G. [1982]: Sur les lois de comportement en élasticité non-linéaire compressible, C. R. Acad. Sci. Paris Sér. A 295, 423–426. · Zbl 0497.73017
[5] Ciarlet, P. G. & Nečas, J. [1984]: Problèmes unilatéraux en élasticité non linéaire tri-dimensionnelle, C. R. Acad. Sci. Paris, Sér. A 298, 189–192.
[6] Duvaut, G. & Lions, J. L. [1972]: Les Inéquations en Mécanique et en Physique, Dunod, Paris. · Zbl 0298.73001
[7] Fichera, G. [1972]: Boundary value problems of elasticity with unilateral constraints, Handbuch der Physik, Vol. VIa/2, 391–424, Springer-Verlag, Berlin, Heidelberg, New-York.
[8] Germain, P. [1972]: Mécanique des Milieux Continus, Tome 1, Masson, Paris. · Zbl 0242.73005
[9] Guo Zhong-Heng [1980]: The unified theory of variational principles in nonlinear elasticity, Arch. Mech. 32, 577–596. · Zbl 0443.73009
[10] Gurtin, M. E. [1981]: Introduction to Continuum Mechanics, Academic Press, New York. · Zbl 0559.73001
[11] Hlaváček, I.; Haslinger, J.; Nečas, J. & Lovíšek, J. [1982]: Solution of Variational Inequalities in Mechanics, Alfa, Bratislava.
[12] Marsden, J. E. & Hughes, T. J. R. [1983]: Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs. · Zbl 0545.73031
[13] Nečas, J. [1967]: Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris.
[14] Nečas, J. [1983]: Introduction to the Theory of Nonlinear Equations, Teubner Texte zur Mathematik, Band 52, Leipzig.
[15] Nečas, J. & Hlaváček, I. [1981]: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Elsevier, Amsterdam.
[16] Ogden, R. W. [1972]: Large deformation isotropic elasticity: On the correlation of theory and experiment for compressible rubberlike solids, Proc. Roy. Soc. London A 328, 567–583. · Zbl 0245.73032 · doi:10.1098/rspa.1972.0096
[17] Prager, W. [1957]: On ideal locking materials, Transaction of the Society of Rheology 1, 169–175. · Zbl 0098.37702 · doi:10.1122/1.548818
[18] Truesdell, C. & Noll, W. [1965]: The non-linear field theories of mechanics, Handbuch der Physik, Vol. III/3, 1–602. · Zbl 1068.74002
[19] Wang, C.-C. & Truesdell, C. [1973]: Introduction to Rational Elasticity, Noordhoff, Groningen. · Zbl 0308.73001
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