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Injectivity and self-contact in nonlinear elasticity. (English) Zbl 0628.73043
Self-contact may occur in any traction problem within the theory of large deformations of (hyperelastic) bodies; in fact, self-contact may lead to non-trivial equilibrium placements even when the body is totally unloaded and under no condition of place.
The authors were the first to adress the problem of self-contact in hyperelasticity; their results, announced in C. R. Acad. Sci., Paris, Sér. I 301, 621-624 (1985; Zbl 0585.73018) are proved here in full. Variational methods are used and the solution of the mixed problem under dead body load and zero surface traction is taken to be the global minimizer of the total energy (thus alternative solutions corresponding to local minima are not considered). The stored-energy function w is supposed to satisfy Ball’s conditions of polyconvexity, coerciveness and singularity $$(W\to +\infty$$ as det $$F\to 0+$$; $$F=\nabla \phi$$ being the displacement gradient). Interpenetration is denied by the condition $$\int_{\Omega}\det F dx\leq vol \phi (\Omega)$$ imposed upon all allowable functions (here $$\Omega$$ is the set occupied by the body in the reference placement); in fact, by a classical formula, $$\int_{\phi (\Omega)}card \phi^{-1}(x')dx'=\int_{\Omega}\det Fdx$$. Actually, as the negation of interpenetration introduces anyway a sort of unilateral constraint, the authors impose on the body also a condition of confinement without frictin ($$\phi$$ ($${\bar \Omega}$$) must belong to a closed set with smooth boundary).
The main theorem assures the existence of at least one minimizer; other theorems describe properties of the minimizer under the presumption that it be sufficiently smooth (for instance card $$\phi$$ $${}^{-1}(x')=1$$ except when x’$$\in \phi (\partial \Omega)\cap int B$$, and then card $$\phi$$ $${}^{-1}(x')=1$$ or 2; in the latter case there are two opposite outer normal vectors to $$\phi$$ ($$\partial \Omega)$$ at x’).
Reviewer: G.Capriz

##### MSC:
 74B20 Nonlinear elasticity 74S30 Other numerical methods in solid mechanics (MSC2010)
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##### References:
 [1] Antman, S. S. [1976]: Ordinary differential equations of nonlinear elasticity II: Existence and regularity theory for conservative boundary value problems, Arch. Rational Mech. Anal. 61, 353–393. · Zbl 0354.73047 [2] Antman, S. S., & H. Brezis [1978]: The existence of orientation-preserving deformations in nonlinear elasticity, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. II (R. J. Knops, Editor), pp. 1–29, Pitman, London. [3] Ball, J. M. [1977]: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63, 337–403. · Zbl 0368.73040 [4] Ball, J. M. [1981a]: Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Royal Soc. Edinburgh 88 A, 315–328. · Zbl 0478.46032 [5] Ball, J. M. [1981b]: Remarques sur l’existence et la régularité des solutions d’élastostatique nonlinéaire, in Recent Contributions to Nonlinear Partial Differential Equations, pp. 50–62, Pitman, Boston. [6] Ball, J. M., J. C. Currie & P. J. Olver [1981]: Null Lagrangians, weak continuity and variational problems of arbitrary order, J. Functional Analysis 41, 135–174. · Zbl 0459.35020 [7] Ciarlet, P. G. [1985]: Elasticité Tridimensionnelle, Masson, Paris. · Zbl 0572.73027 [8] Ciarlet, P. G. [1987]: Mathematical Elasticity, Vol. 1, North-Holland, Amsterdam. · Zbl 0612.73060 [9] Ciarlet, P. G., & P. Destuynder [1979]: A justification of a nonlinear model in plate theory, Comput. Meth. Applied Mech. Engrg. 17/18, 227–258. · Zbl 0405.73050 [10] Ciarlet, P. G., & G. Geymonat [1982]: Sur les lois de comportement en élasticité non-linéaire compressible, C. R. Acad. Sci. Paris Sér. II 295, 423–426. · Zbl 0497.73017 [11] Ciarlet, P. G., & J. Nečas [1985a]: Unilateral problems in nonlinear, three-dimensional elasticity, Arch. Rational Mech. Anal. 87, 319–338. · Zbl 0557.73009 [12] Ciarlet, P. G., & J. Nečas [1985b]: Injectivité presque partout, auto-contact, et noninterpénétrabilité en élasticité non linéaire tridimensionnelle, C. R. Acad. Sci. Paris Sér. I 301, 621–624. [13] Germain, P. [1972]: Mécanique des Milieux Continus, Tome 1, Masson, Paris. · Zbl 0242.73005 [14] Gurtin, M. E. [1981]: Introduction to Continuum Mechanics, Academic Press, New York. · Zbl 0559.73001 [15] Hanyga, A. [1985]: Mathematical Theory of Non-Linear Elasticity, Polish Scientific Publishers, Warszawa, and Ellis Horwood, Chichester. · Zbl 0561.73011 [16] Marcus, M., & V. J. Mizel [1973]: Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems, Bull. Amer. Math. Soc. 79, 790–795. · Zbl 0275.49041 [17] Marsden, J. E., & T. J. R. Hughes [1983]: Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs. · Zbl 0545.73031 [18] Meisters, G. H., & C. Olech [1973]: Locally one-to-one mappings and a classical theorem on Schlicht functions, Duke Math. J. 30, 63–80. · Zbl 0112.37702 [19] Nečas, J. [1967]: Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris. [20] Noll, W. [1978]: A general framework for problems in the statics of finite elasticity, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, pp. 363–387, North-Holland Mathematics Studies, Amsterdam. · Zbl 0415.73046 [21] 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 [22] Rado, T., & P. V. Reichelderfer [1955]: Continuous Transformations in Analysis, Springer-Verlag, Berlin. · Zbl 0067.03506 [23] Schwartz, L. [1967]: Cours d’Analyse, Hermann, Paris. [24] Smith, K. T. [1983]: Primer of Modern Analysis, Springer-Verlag, New York, Second Edition. · Zbl 0517.26001 [25] Truesdell, C., & W. Noll [1965]: The non-linear field theories of mechanics, Handbuch der Physik, Vol. III/3, 1–602. · Zbl 1068.74002 [26] Valent, T. [1979]: Teoremi di esistenza e unicità in elastostatica finita, Rend. Sem. Mat. Univ. Padova 60, 165–181. · Zbl 0425.73011 [27] Valent, T. [1982]: Local theorems of existence and uniqueness in finite elastostatics, in Finite Elasticity (D. E. Carlson & R. T. Shield, Editors), pp. 401-421. · Zbl 0512.73038 [28] Vodopyanov, S. K., V. M. Goĺdshtein, & Yu. G. Reshetnyak [1979]: On geometric properties of functions with generalized first derivatives, Russian Math. Surveys 34, 19–74. · Zbl 0429.30017 [29] Wang, C.-C, & C. Truesdell [1973]: Introduction to Rational Elasticity. Noordholf, Groningen.
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