Hostname: page-component-669899f699-rg895 Total loading time: 0 Render date: 2025-04-27T01:22:04.717Z Has data issue: false hasContentIssue false
  • English
  • Français

Sobolev regularity of the inverse for minimizers of the neo-Hookean energy satisfying condition INV

Part of: Manifolds Existence theories Elastic materials Equilibrium (steady-state) problems

Published online by Cambridge University Press:  11 April 2025

Panas Kalayanamit Open the ORCID record for Panas Kalayanamit [Opens in a new window]
Panas Kalayanamit*
Affiliation:
Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins, Avenida Libertador Bernardo O’Higgins, Rancagua, Chile (panas.kalayanamit@uoh.cl)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the existence and regularity of minimizers of the neo-Hookean energy in the closure of classes of deformations without cavitation. The exclusion of cavitation is imposed in the form of the divergence identities, which is equivalent to the well-known condition (INV) with $\operatorname{Det} = \operatorname{det}$. We show that the neo-Hookean energy admits minimizers in classes of maps that are one-to-one a.e. with positive Jacobians, provided that these maps are the weak limits of sequences of maps that satisfy the divergence identities. In particular, these classes include the weak closure of diffeomorphisms and the weak closure of homeomorphisms satisfying Lusin’s condition N. Moreover, if the minimizers satisfy condition (INV), then their inverses have Sobolev regularity. This extends a recent result by Doležalová, Hencl, and Molchanova by showing that the minimizers they obtained enjoy extra regularity properties and that the existence of minimizers can still be obtained even when their coercivity assumption is relaxed.

Keywords

Cavitationcondition INVdivergence identitieselastic deformationneo-Hookeanregular minimizersobolev regularity

MSC classification

Primary: 74B20: Nonlinear elasticity
Secondary: 49J45: Methods involving semicontinuity and convergence; relaxation 49Q15: Geometric measure and integration theory, integral and normal currents 74G65: Energy minimization
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 21
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1977), 337–403.CrossRefGoogle Scholar
Ball, J. M.. Global invertibility of Sobolev functions and the interpenetration of matter. Proc. R. Soc. Edinb. Sect. A, Math. 88 (1981), 315–328.CrossRefGoogle Scholar
Ball, J. M.. Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. R. Soc. Lond. Ser. A. 306 (1982), 557–611.Google Scholar
Ball, J. M., Currie, J. C. and Olver, P. J.. Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41 (1981), 135–174.CrossRefGoogle Scholar
Ball, J. M. and Murat, F.. $W^{1,p}$-Quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984), 225–253.CrossRefGoogle Scholar
Barchiesi, M., Henao, D. and Mora-Corral, C.. Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity. Arch. Ration. Mech. Anal. 224 (2017), 743–816.CrossRefGoogle Scholar
Barchiesi, M., Henao, D., Mora-Corral, C. and Rodiac, R.. Harmonic dipoles and the relaxation of the neo-Hookean energy in 3D elasticity. Arch. Ration. Mech. Anal. 247 (2023), .CrossRefGoogle Scholar
Barchiesi, M., Henao, D., Mora-Corral, C. and Rodiac, R.. A Relaxation Approach to the Minimization of the Neo-Hookean Energy in 3D. SIAM Journal on Mathematical Analysis. 56 (2024), 7830–7845.CrossRefGoogle Scholar
Blatz, P. J.. On the Thermostatic Behavior of elastomers, , (Springer US, Boston, MA, 1971).CrossRefGoogle Scholar
Cañulef Aguilar, V. and Henao, D.. A lower bound for the void coalescence load in nonlinearly elastic solids. Interfaces Free Bound. 21 (2019), 409–440.CrossRefGoogle Scholar
Ciarlet, P. G.. Mathematical elasticity. Volume I: Three-dimensional elasticity. Of Stud. Math. Appl., Vol. 20 (North-Holland, Amsterdam, etc., 1988).Google Scholar
Conti, S. and De Lellis, C.. Some remarks on the theory of elasticity for compressible Neohookean materials. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5). 2 (2003), 521–549.Google Scholar
De Lellis, C. and Ghiraldin, F.. An extension of the identity Det = det. C. R. Math. Acad. Sci. Paris. 348 (2010), 973–976.CrossRefGoogle Scholar
Doležalová, A., Hencl, S. and Malý, J.. Weak Limit of Homeomorphisms in $W^{1,n-1}$ and (INV) Condition. Arch. Ration. Mech. Anal. 247 (2023), .CrossRefGoogle Scholar
Doležalová, A., Hencl, S. and Molchanova, A.. Weak limit of homeomorphisms in $W^{1,n-1}$: Invertibility and lower semicontinuity of energy. ESAIM: COCV. 30 (2024), .Google Scholar
Evans, L. C. and Gariepy, R. F.. Measure Theory and Fine Properties of functions (CRC Press, Boca Raton, 1992).Google Scholar
Federer, H.. Geometric measure theory (Springer-Verlag, Berlin, 1969) (Class. Math. Reprint of the 1969 edition).Google Scholar
Gent, A. N.. Cavitation in rubber: a cautionary tale. Rubber Chem. Technol. 63 (1990), 49–53.CrossRefGoogle Scholar
Giaquinta, M., Modica, G. and Souček, J.. Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 106 (1989), 97–159.CrossRefGoogle Scholar
Hardt, R. and Lin, F. -H.. A remark on $H^ 1$ mappings. Manuscr. Math. 56 (1986), 1–10.CrossRefGoogle Scholar
Henao, D.. Cavitation, invertibility, and convergence of regularized minimizers in nonlinear elasticity. J. Elasticity. 94 (2009), 55–68.CrossRefGoogle Scholar
Henao, D. and Kalayanamit, P.. The positivity of the Jacobian in the weak limit of generalised axisymmetric maps. ArXiv:2409.19473. (2024).Google Scholar
Henao, D. and Mora-Corral, C.. Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity. Arch. Ration. Mech. Anal. 197 (2010), 619–655.CrossRefGoogle Scholar
Henao, D. and Mora-Corral, C.. Fracture surfaces and the regularity of inverses for BV deformations. Arch. Ration. Mech. Anal. 201 (2011), 575–629.CrossRefGoogle Scholar
Henao, D. and Mora-Corral, C.. Lusin’s condition and the distributional determinant for deformations with finite energy. Adv. Calc. Var. 5 (2012), 355–409.CrossRefGoogle Scholar
Henao, D. and Mora-Corral, C.. Regularity of inverses of Sobolev deformations with finite surface energy. J. Funct. Anal. 268 (2015), 2356–2378.CrossRefGoogle Scholar
Henao, D., Mora-Corral, C. and Oliva, M.. Global invertibility of Sobolev maps. Adv. Calc. Var. 14 (2021), 207–230.CrossRefGoogle Scholar
Henao, D., Mora-Corral, C. and Xu, X.. A numerical study of void coalescence and fracture in nonlinear elasticity. Comput. Methods Appl. Mech. Eng. 303 (2016), 163–184.CrossRefGoogle Scholar
Henao, D. and Serfaty, S.. Energy estimates and cavity interaction for a critical-exponent cavitation model. Commun. Pure Appl. Math. 66 (2013), 1028–1101.CrossRefGoogle Scholar
Hencl, S. and Malý, J.. Jacobians of Sobolev homeomorphisms. Calc. Var. Partial Differ. Equ. 38 (2010), 233–242.CrossRefGoogle Scholar
Hencl, S. and Onninen, J.. Jacobian of weak limits of Sobolev homeomorphisms. Adv. Calc. Var. 11 (2018), 65–73.CrossRefGoogle Scholar
Horgan, C. O. and Polignone, D. A.. Cavitation in nonlinearly elastic solids: a review. Appl. Mech. Rev. 48 (1995), 471–485.CrossRefGoogle Scholar
Krömer, S.. Global invertibility for orientation-preserving Sobolev maps via invertibility on or near the boundary. Arch. Ration. Mech. Anal. 238 (2020), 1113–1155.CrossRefGoogle Scholar
Kristensen, J.. A necessary and sufficient condition for lower semicontinuity. Nonlinear Anal. Theory Methods Appl. Ser. A, Theory Methods. 120 (2015), 43–56.CrossRefGoogle Scholar
Kumar, A. and Lopez-Pamies, O.. The poker-chip experiments of Gent and Lindley (1959) explained. J. Mech. Phys. Solids. 150 (2021), .CrossRefGoogle Scholar
Llavona, J. G.. Approximation of continuously differentiable functions. In North-Holland Math. Stud, Vol. 130 (Elsevier, Amsterdam, 1986).Google Scholar
Müller, S.. Weak continuity of determinants and nonlinear elasticity (Continuité faible des déterminants et applications à l’élasticité non linéaire). C. R. Acad. Sci. Paris, Sér. I. 307 (1988), 501–506.Google Scholar
Müller, S., Qi, T. and Yan, B. S.. On a new class of elastic deformations not allowing for cavitation. Ann. Inst. Henri Poincaré, Anal. Non LinéAire. 11 (1994), 217–243.CrossRefGoogle Scholar
Müller, S. and Spector, S. J.. An existence theory for nonlinear elasticity that allows for cavitation. Arch. Rational Mech. Anal. 131 (1995), 1–66.CrossRefGoogle Scholar
Malý, J.. Weak lower semicontinuity of polyconvex integrals. Proc. R. Soc. Edinb. Sect. A, Math. 123 (1993), 681–691.CrossRefGoogle Scholar
Mora-Corral, C.. Quasistatic evolution of cavities in nonlinear elasticity. SIAM J. Math. Anal. 46 (2014), 532–571.CrossRefGoogle Scholar
Negrón–Marrero, P. V. and Sivaloganathan, J.. Cavitation of a spherical body under mechanical and self-gravitational forces. Proc. R. Soc. Edinb. A: Math. 154 (2024), 1–31.CrossRefGoogle Scholar
Sivaloganathan, J. and Spector, S. J.. On the existence of minimizers with prescribed singular points in nonlinear elasticity. J. Elasticity. 59 (2000), 83–113.CrossRefGoogle Scholar
Sivaloganathan, J. and Spector, S. J.. On the optimal location of singularities arising in variational problems of nonlinear elasticity. J. Elasticity. 58 (2000), 191–224.CrossRefGoogle Scholar
Šverák, V.. Regularity properties of deformations with finite energy. Arch. Ration. Mech. Anal. 100 (1988), 105–127.CrossRefGoogle Scholar