In the Self-Contact Problem in Nonlinear Elasticity

Nifa, Anass; Bouali, Bouchta

doi:10.1007/s00205-021-01752-2

Published journal article · 2022

In the Self-Contact Problem in Nonlinear Elasticity

Anass Nifa · Bouchta Bouali

Archive for Rational Mechanics and Analysis 243, 1433–1448 (2022).

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Abstract

In this paper, we consider the minimization problem of $3$ -dimensional nonlinear hyperelastic bodies moving in $\mathbb{R}^{3}$ , which enables frictionless self-contact and forbids self-intersection. For this, we define a new class of admissible deformations based on a natural homotopy constraint. We study strictly orientation-preserving Sobolev maps in this new class and their global invertibility properties from a topological point of view. In this fashion, we prove that, under suitable hypotheses, such maps are actually homeomorphisms. Applying this result to the mixed displacement-traction problem, the existence of homeomorphic minimizers is shown for nonlinear stored energy functions with suitable properties.

MSC 2020

55M25: Degree, winding number
26B10: Implicit function theorems, Jacobians, transformations with several variables
74B20: Nonlinear elasticity
49S05: Variational principles of physics

Mathematics Subject Classification 2020

Publication details

DOI: 10.1007/s00205-021-01752-2
Published online: 31 January 2022
Journal: Archive for Rational Mechanics and Analysis
Volume / pages: 243 / 1433–1448
Subjects: Nonlinear elasticity; self-contact; Sobolev mappings; global invertibility
Language: English

Cite this paper

Anass Nifa and Bouchta Bouali. In the Self-Contact Problem in Nonlinear Elasticity. Archive for Rational Mechanics and Analysis 243, 1433–1448 (2022). doi:10.1007/s00205-021-01752-2

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References

As listed in the manuscript bibliography.

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[2]
J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Archive for Rational Mechanics and Analysis, 63 (1976), pp. 337–403.
[3]
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M. Giaquinta, G. Modica, and J. souček, Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity, Archive for Rational Mechanics and Analysis, 106 (1989), pp. 97–159.
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D. Henao and C. Mora-Corral, Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity, Arch. Ration. Mech. Anal., 197 (2010), pp. 619–655.
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———, Regularity of inverses of Sobolev deformations with finite surface energy, J. Funct. Anal., 268 (2015), pp. 2356–2378.
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S. Müller and S. J. Spector, An existence theory for nonlinear elasticity that allows for cavitation, Archive for Rational Mechanics and Analysis, 39 (1995), pp. 1 – 66.
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T. Qi, Almost-everywhere injectivity in nonlinear elasticity, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 109 (1988), p. 79–95.
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K. Rajala, Reshetnyak’s theorem and the inner distortion, Pure and Applied Mathematics Quarterly, 7 (2011), pp. 411–424.
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J. S. Raymond, Local inversion for differentiable functions and the darboux property, Mathematika, 49 (2002), p. 141–158.
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———, Space mappings with bounded distortion, Sib. Math. J., 8 (1968), pp. 466–487.
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V. Šverák, Regularity properties of deformations with finite energy, Archive for Rational Mechanics and Analysis, 100 (1988), pp. 105–127.
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J. Väisälä, Minimal mappings in euclidean spaces, Ann. Acad. Sci. Fenn. Ser. A I, 366 (1965), pp. 1–22.
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J. Ball, J. Currie, and P. Olver, Null lagrangians, weak continuity, and variational problems of arbitrary order, Journal of Functional Analysis, 41 (1981), pp. 135 – 174.

[reference-2] [2]
J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Archive for Rational Mechanics and Analysis, 63 (1976), pp. 337–403.

[reference-3] [3]
J. M. Ball, Global invertibility of sobolev functions and the interpenetration of matter, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 88 (1981), pp. 315–328.

[reference-4] [4]
P. Bauman and D. Phillips, Univalent minimizers of polyconvex functionals in two dimensions, Arch. Ration. Mech. Anal., 126 (1994), pp. 161–181.

[reference-5] [5]
P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity, Archive for Rational Mechanics and Analysis, 97 (1987), pp. 171–188.

[reference-6] [6]
P. G. Ciarlet and J. Nečas, Unilateral problems in nonlinear three-dimensional elasticity, Arch. Ration. Mech. Anal., 87 (1985), pp. 319–338.

[reference-7] [7]
S. Conti and C. De Lellis, Some remarks on the theory of elasticity for compressible Neohookean materials, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), 2 (2003), pp. 521–549.

[reference-8] [8]
I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications, Oxford lecture series in mathematics and its applications, Clarendon Press, 1995.

[reference-9] [9]
M. Giaquinta, G. Modica, and J. souček, Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity, Archive for Rational Mechanics and Analysis, 106 (1989), pp. 97–159.

[reference-10] [10]
D. Henao and C. Mora-Corral, Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity, Arch. Ration. Mech. Anal., 197 (2010), pp. 619–655.

[reference-11] [11]
———, Regularity of inverses of Sobolev deformations with finite surface energy, J. Funct. Anal., 268 (2015), pp. 2356–2378.

[reference-12] [12]
S. Hencl and P. Koskela, Lectures on Mappings of Finite Distortion, Lecture Notes in Mathematics, Springer International Publishing, 2014.

[reference-13] [13]
T. Iwaniec and J. Onninen, Deformations of finite conformal energy: existence and removability of singularities, Proc. Lond. Math. Soc. (3), 100 (2010), pp. 1–23.

[reference-14] [14]
O. Martio and W. P. Ziemer, Lusin’s condition (n) and mappings with nonnegative jacobians, Michigan Mathematical Journal, 39 (1992), pp. 495 – 508.

[reference-15] [15]
A. Molchanova and S. Vodopyanov, Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity, Calc. Var. Partial Differ. Equ., 59 (2020), p. 25. Id/No 17.

[reference-16] [16]
S. Müller, Higher integrability of determinants and weak convergence in L¹, J. Reine Angew. Math., 412 (1990), pp. 20–34.

[reference-17] [17]
S. Müller, T. Qi, and B. Yan, On a new class of elastic deformations not allowing for cavitation, Annales de l’I.H.P. Analyse non linéaire, 11 (1994), pp. 217–243.

[reference-18] [18]
S. Müller and S. J. Spector, An existence theory for nonlinear elasticity that allows for cavitation, Archive for Rational Mechanics and Analysis, 39 (1995), pp. 1 – 66.

[reference-19] [19]
T. Qi, Almost-everywhere injectivity in nonlinear elasticity, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 109 (1988), p. 79–95.

[reference-20] [20]
K. Rajala, Reshetnyak’s theorem and the inner distortion, Pure and Applied Mathematics Quarterly, 7 (2011), pp. 411–424.

[reference-21] [21]
J. S. Raymond, Local inversion for differentiable functions and the darboux property, Mathematika, 49 (2002), p. 141–158.

[reference-22] [22]
Y. Reshetnyak and H. McFaden, Space Mappings with Bounded Distortion, Advances in Soviet Mathematics, American Mathematical Society, 1989.

[reference-23] [23]
Y. G. Reshetnyak, On the stability of conformal mappings in multidimensional spaces, Sib. Math. J., 8 (1968), pp. 69–85.

[reference-24] [24]
———, Space mappings with bounded distortion, Sib. Math. J., 8 (1968), pp. 466–487.

[reference-25] [25]
V. Šverák, Regularity properties of deformations with finite energy, Archive for Rational Mechanics and Analysis, 100 (1988), pp. 105–127.

[reference-26] [26]
J. Väisälä, Minimal mappings in euclidean spaces, Ann. Acad. Sci. Fenn. Ser. A I, 366 (1965), pp. 1–22.

[reference-27] [27]
S. K. Vodop’yanov and V. M. Gol’dshtein, Quasiconformal mappings and spaces of functions with generalized first derivatives, Siberian Mathematical Journal, 17 (1976), pp. 399–411.