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Quasiconvex relaxation of isotropic functions in incompressible planar hyperelasticity

Part of: Existence theories Elastic materials Functions of several variables Real functions

Published online by Cambridge University Press:  11 June 2019

Robert J. Martin ,
Jendrik Voss ,
Patrizio Neff  and
Ionel-Dumitrel Ghiba
Robert J. Martin
Affiliation:
Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg–Essen, Campus Essen, Thea-Leymann-Straße 9, 45127Essen, Germany (robert.martin@uni-due.de; max.voss@uni-due.de) (patrizio.neff@uni-due.de)
Jendrik Voss
Affiliation:
Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg–Essen, Campus Essen, Thea-Leymann-Straße 9, 45127Essen, Germany (robert.martin@uni-due.de; max.voss@uni-due.de) (patrizio.neff@uni-due.de)
Patrizio Neff
Affiliation:
Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg–Essen, Campus Essen, Thea-Leymann-Straße 9, 45127Essen, Germany (robert.martin@uni-due.de; max.voss@uni-due.de) (patrizio.neff@uni-due.de)
Ionel-Dumitrel Ghiba
Affiliation:
Department of Mathematics, Alexandru Ioan Cuza University of Iaşi, Blvd. Carol I, no. 11, 700506, Iaşi, Romania and Octav Mayer Institute of Mathematics of the Romanian Academy, Iaşi Branch, 700505Iaşi, (dumitrel.ghiba@uaic.ro)
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Abstract

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In this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued function W; SL(2) → ℝ with W(RF) = W(FR) = W(F) for all F ∈ SL(2) and all R ∈ SO(2), where SL(2) and SO(2) denote the special linear group and the special orthogonal group, respectively. In order to obtain our result, we combine earlier work by Dacorogna and Koshigoe on the relaxation of certain conformal planar energy functions with a recent result on the equivalence between polyconvexity and rank-one convexity for objective and isotropic energies in planar incompressible nonlinear elasticity.

Keywords

quasiconvexityrank-one convexitypolyconvexityquasiconvex envelopesnonlinear elasticityincompressibilityhyperelasticityrelaxationmicrostructure

MSC classification

Primary: 26B25: Convexity, generalizations
Secondary: 26A51: Convexity, generalizations 49J45: Methods involving semicontinuity and convergence; relaxation 74B20: Nonlinear elasticity

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2620 - 2631
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Royal Society of Edinburgh 2019

References

1Abeyaratne, R.. Discontinuous deformation gradients in plane finite elastostatics of incompressible materials. J. Elast. 10, (1980), 255–293.CrossRefGoogle Scholar
2Alibert, J.-J. and Dacorogna, B.. An example of a quasiconvex function that is not polyconvex in two dimensions. Arch. Ration. Mech. Anal. 117 (1992), 155–166.CrossRefGoogle Scholar
3Aubert, G.. Necessary and sufficient conditions for isotropic rank-one convex functions in dimension 2. J. Elast. 39 (1995), 31–46.CrossRefGoogle Scholar
4Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1976), 337–403.CrossRefGoogle Scholar
5Ball, J. M.. Constitutive inequalities and existence theorems in nonlinear elastostatics. In Nonlinear analysis and mechanics: Heriot-Watt symposium (ed.Knops, R.), vol. 1, pp. 187–241 (Boston: Pitman Publishing Ltd., 1977).Google Scholar
6Buttazzo, G., Dacorogna, B. and Gangbo, W.. On the envelopes of functions depending on singular values of matrices. Bollettino dell'Unione Matematica Italiana, VII. Ser., B 8 (1994), 17–35.Google Scholar
7Conti, S.. Quasiconvex functions incorporating volumetric constraints are rank-one convex. Journal de Mathématiques Pures et Appliquées 90 (2008), 15–30.Google Scholar
8Conti, S. and Hackl, K.. Analysis and computation of microstructure in finite plasticity, vol. 78 (Switzerland: Springer, 2015).Google Scholar
9Dacorogna, B.. Direct methods in the calculus of variations, 2nd edn, Applied Mathematical Sciences, vol. 78 (Berlin: Springer, 2008).Google Scholar
10Dacorogna, B. and Haeberly, J.-P.. Some numerical methods for the study of the convexity notions arising in the calculus of variations. ESAIM: Mathematical Modelling and Numerical Analysis 32 (1998), 153–175.CrossRefGoogle Scholar
11Dacorogna, B. and Koshigoe, H.. On the different notions of convexity for rotationally invariant functions. Annales de la faculté des sciences de Toulouse: Mathématiques 2 (1993), 163–184.Google Scholar
12Dacorogna, B. and Marcellini, P.. A counterexample in the vectorial calculus of variations. In Material instabilities in continuum mechanics (ed.Ball, J. M.), pp. 77–83 (USA: Oxford Science Publications, 1988).Google Scholar
13Dunn, J. E., Fosdick, R. and Zhang, Y.. Rank 1 convexity for a class of incompressible elastic materials. In Rational continua, classical and new: a collection of papers dedicated to Gianfranco Capriz on the occasion of his 75th birthday (eds Podio-Guidugli, P. and Brocato, M.), pp. 89–96 (Milano: Springer Milan, 2003).CrossRefGoogle Scholar
14Fosdick, R. and MacSithigh, G.. Minimization in incompressible nonlinear elasticity theory. J. Elast. 16 (1986), 267–301.Google Scholar
15Ghiba, I.-D., Neff, P. and Šilhavỳ, M.. The exponentiated Hencky-logarithmic strain energy. Improvement of planar polyconvexity. Int. J. Non. Linear. Mech. 71 (2015), 48–51, doi: 10.1016/j.ijnonlinmec.2015.01.009Google Scholar
16Ghiba, I.-D., Martin, R. J. and Neff, P.. Rank-one convexity implies polyconvexity in isotropic planar incompressible elasticity. Journal de Mathématiques Pures et Appliquées 116 (2018), 88–104.Google Scholar
17Hencky, H.. Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen. Zeitschrift für technische Physik 9 (1928), 215–220, available at www.uni-due.de/imperia/md/content/mathematik/ag_neff/hencky1928.pdfGoogle Scholar
18Hencky, H.. Welche Umstände bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen Körpern?. Zeitschrift für Physik 55 (1929), 145–155, available at www.uni-due.de/imperia/md/content/mathematik/ag_neff/hencky1929.pdf.CrossRefGoogle Scholar
19Kinderlehrer, D., James, R., Luskin, M. and Ericksen, J. L.. Microstructure and phase transition, vol. 54 (Berlin, Germany: Springer Science & Business Media, 2012).Google Scholar
20Knowles, J. K. and Sternberg, E.. On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch. Ration. Mech. Anal. 63 (1976), 321–336.CrossRefGoogle Scholar
21Knowles, J. K. and Sternberg, E.. On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics. J. Elast. 8 (1978), 329–379.CrossRefGoogle Scholar
22Le Dret, H. and Raoult, A.. The quasiconvex envelope of the Saint Venant–Kirchhoff stored energy function. Proceedings of the Royal Society of Edinburgh Section A: Mathematics 125 (1995), 1179–1192.CrossRefGoogle Scholar
23Martin, R. J., Ghiba, I.-D. and Neff, P.. Rank-one convexity implies polyconvexity for isotropic, objective and isochoric elastic energies in the two-dimensional case. Proceedings of the Royal Society Edinburgh A 147A (2017), 571–597, available at arXiv:1507.00266.CrossRefGoogle Scholar
24Martin, R. J., Voss, J., Ghiba, I.-D., Sander, O. and Neff, P.. The quasiconvex envelope of conformally invariant planar energy functions in isotropic hyperelasticity. submitted (2018), available at arXiv:1901.00058.Google Scholar
25Mielke, A.. Necessary and sufficient conditions for polyconvexity of isotropic functions. J. Convex. Anal. 12 (2005), 291.Google Scholar
26Neff, P., Lankeit, J., Ghiba, I.-D., Martin, R. J. and Steigmann, D. J.. The exponentiated Henckylogarithmic strain energy. Part II: coercivity, planar polyconvexity and existence of minimizers. Zeitschrift für angewandte Mathematik und Physik 66 (2015), 1671–1693, doi: 10.1007/s00033-015-0495-0CrossRefGoogle Scholar
27Neff, P., Eidel, B. and Martin, R. J.. Geometry of logarithmic strain measures in solid mechanics. Arch. Ration. Mech. Anal. 222 (2016), 507–572, available at arXiv:1505.02203. doi: 10.1007/s00205-016-1007-xGoogle Scholar
28Rosakis, P. and Simpson, H. C.. On the relation between polyconvexity and rank-one convexity in nonlinear elasticity. J. Elast. 37 (1994), 113–137.Google Scholar
29Šilhavỳ, M.. The mechanics and thermodynamics of continuous media. Texts and Monographs in Physics (Berlin, Germany: Springer, 1997).Google Scholar
30Šilhavỳ, M.. Rank 1 convex hulls of isotropic functions in dimension 2 by 2. Mathematica Bohemica 126 (2001), 521–529.CrossRefGoogle Scholar