×

zbMATH — the first resource for mathematics

Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity. (English) Zbl 0677.73014
The authors consider several new classes of generalized (or weak) diffeomorphisms of an open set \(\Omega\) in \({\mathbb{R}}^ n\). Their main goal, in the particular case of three-dimensional hyperelasticity, is to minimize an elastic energy functional of the form \[ \quad F(u;\Omega)=\int_{\Omega}W(x,Du)dx, \] where W is a polyconvex function satisfying appropriate coercivity conditions, in such a way that the minimizers thus found be mappings that are globally invertible in some sense, i.e. deformations that are admissible from the viewpoint of mechanics.
The generalized diffeomorphisms are introduced as special subspaces of the space of rectifiable currents in the sense of geometric measure theory [cf. H. Federer, Geometric measure theory (1969; Zbl 0176.008)]. Usual diffeomorphisms are identified with the current \(T_ u\) that is integration on the graph of u. This point of view allows u and its inverse \(u^{-1}\) to play a symmetrical role. The first part of the paper is devoted to the description and properties of various classes of such currents that generalize the usual Sobolev spaces \(W^{1/p}(\Omega;{\mathbb{R}}^ n)\) and still have diffeomorphism-like properties. Keeping in mind applications to the calculus of variations, the authors prove several weak sequential closedness results in these classes. It should be noted that some of these results of weak continuity of determinants are proved in a more elementary fashion by S. Müller [C.R. Acad. Sci. Paris, 307, Sér. I, 501-506 (1988)].
Application of these results to three-dimensional elasticity yields existence theorems with globally invertible minimizers under coercivity hypotheses such as \[ | M(F)| =^{def}\{1+| F|^ 2+| Adj F|^ 2+| \det F|^ 2\}^{1/2}, \]
\[ W(x,F)\geq | M(F)|^ p+(| M(F)|^ q/| \det F|^{q-1}). \] Furthermore, the equilibrium equations in the deformed configuration are shown to be satisfied as well as conservation of energy. Finally, cavitation in elasticity is studied, cf. (*) J. M. Ball, Philos. Trans. R. Soc., A 306, 557-611 (1982; Zbl 0513.73020). It is shown that it cannot happen in the present framewok, and the connection with (*) is discussed in detail.
[Note that the authors have recently written an Erratum-Addendum to the present paper.]
Reviewer: H.LeDret

MSC:
74B20 Nonlinear elasticity
74S30 Other numerical methods in solid mechanics (MSC2010)
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49Q99 Manifolds and measure-geometric topics
58A15 Exterior differential systems (Cartan theory)
49Q20 Variational problems in a geometric measure-theoretic setting
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] E. Acerbi & E. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125–145. · Zbl 0565.49010
[2] S. S. Antman, Ordinary differential equations of nonlinear elasticity. II. Existence and regularity theory for conservative boundary value problems. Arch. Rational Mech. Anal. 61 (1976) 353–393. · Zbl 0354.73047
[3] S. S. Antman, Geometrical and analytical questions in nonlinear elasticity. In Seminar on nonlinear partial differential equations, Mathematical Sciences Research Institute (ed. S. S. Chern) Springer-Verlag, New York, 1984. · Zbl 0551.73018
[4] S. S. Antman & H. Brezis, The existence of orientation-preserving deformations in nonlinear elasticity. In Nonlinear Analysis and Mechanics: Heriot-Watt Symposium (ed. R. J. Knops), Vol. II, Pitman, London 1978.
[5] G. Anzellotti & M. Giaquinta, Existence of the displacement field for an elastoplastic body subject to Hencky’s law and von Mises yield condition. Manuscripta Math. 32 (1980) 101–136. · Zbl 0465.73022
[6] G. Anzellotti & M. Giaquinta, On the existence of the fields of stresses and displacements for an elasto-perfectly plastic body in static equilibrium. J. Math. Pures et Appl. 61 (1982) 219–244. · Zbl 0467.73044
[7] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337–403. · Zbl 0368.73040
[8] J. M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter. Proc. Roy. Soc. Edinburgh 88 A (1981) 315–328. · Zbl 0478.46032
[9] J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. Roy. Soc. A 306 (1982) 557–611. · Zbl 0513.73020
[10] J. M. Ball, Differentiability properties of symmetric and isotropic functions. Duke Math. J. 50, 3, (1984) 699–727. · Zbl 1077.74507
[11] P. J. Ciarlet & J. Nečas, Injectivity and self-contact in nonlinear elasticity. Arch. Rational Mech. Anal. 97 (1987) 171–188. · Zbl 0628.73043
[12] G. Dal Maso, Integral representation on BV({\(\Omega\)}) of {\(\Gamma\)}-limits of variational integrals. Manuscripta Math. 30 (1980) 387–416. · Zbl 0435.49016
[13] E. de Giorgi, Teoremi di semicontinuità nel calcolo delle variazioni. Istituto Nazionale di Alta Matematica, 1968–69.
[14] I. Ekeland & R. Temam, Convex analysis and variational problems, North Holland Amsterdam, 1976. · Zbl 0322.90046
[15] H. Federer, Geometric measure theory. Springer-Verlag, New York 1969. · Zbl 0176.00801
[16] H. Federer & W. Fleming, Normal and integral currents. Ann. of Math. 72 (1960) 458–520. · Zbl 0187.31301
[17] M. Giaquinta, G. Modica & J. Souček, Functionals with linear growth in the calculus of variations. Comm. Math. Univ. Carolinae 20 (1979) 143–171. · Zbl 0409.49006
[18] E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston 1984. · Zbl 0545.49018
[19] R. J. Knops & C. A. Stuart, Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. Arch. Rational Mech. Anal. 86 (1984) 233–249. · Zbl 0589.73017
[20] P. Marcellini, Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscripta Math. 51 (1985) 1–28. · Zbl 0573.49010
[21] P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. H. Poincaré, Analyse non linéaire 3 (1986) 391–409. · Zbl 0609.49009
[22] P. Marcellini, The stored energy of some discontinuous deformations in nonlinear elasticity, preprint. · Zbl 0679.73006
[23] C. B. Morrey, Multiple integrals in the calculus of variations, Springer, Berlin, 1966. · Zbl 0142.38701
[24] C. B. Morrey, Quasi-convexity and the semicontinuity of multiple integrals. Pacific J. Math. 2 (1952) 23–53. · Zbl 0046.10803
[25] P. Podio-Guiougli, G. Vergara Caffarelli & E. G. Virga, Discontinuous energy minimizers in nonlinear elastostatics: an example of J. Ball revisited. J. Elasticity 15 (1986) 75–96. · Zbl 0575.73021
[26] Yu. G. Reshetnyak, General theorems on semicontinuity and on convergence with a functional. Sibirskii Mat. Zhurnal 8 (1967) 1051–1069.
[27] Yu. G. Reshetnyak, Weak convergence of completely additive vector functions on a set. Sibirskii Mat. Zhurnal 9 (1968) 1386–1394.
[28] L. Simon, Lectures on geometric measure theory. Centre for Math. Analysis, Australian National University, Canberra n. 3, 1983. · Zbl 0546.49019
[29] J. Sivaloganathan, Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. Arch. Rational Mech. Anal., to appear. · Zbl 0628.73018
[30] J. Sivaloganathan, A field theory approach to stability of radial equilibria in non-linear elasticity. Math. Proc. Cambridge Phil. Soc., to appear. · Zbl 0612.73013
[31] J. Souček, On the structure of the space of deformations in nonlinear elasticity. Aplikace Matematiky, to appear.
[32] C. A. Stuart, Radially symmetric cavitation for hyperelastic materials. Ann. Inst. H. Poincaré, Analyse non linéaire 2 (1985) 33–66. · Zbl 0588.73021
[33] V. Šverák, Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal. 98 (1987).
[34] C. Truesdell & W. Noll, The non-linear field theories of mechanics. Handbuch der Physik III/3, Springer, Berlin 1965. · Zbl 0779.73004
[35] B. White, A new proof of the compactness theorem for integral currents. Preprint 1987.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.