Stationary points of nonlinear plate theories. (English) Zbl 1368.74014
The author deals with nonlinear plate models expressing for instance thin growing tissues in biology or elastic sheets obtained by controlled polymerization. The reference configuration of the sheet is modelled by a bounded domain \(S\subset \mathbb{R}^2\). The Willmore functional given by
\[
W(u)=\frac 14 \int_S H^2\,d\mu_g+\int_{\partial S}\kappa_g\,d\mu_{g\partial}
\]
plays an important role in his investigations. The relevant case for thin film elasticity is the restriction of the Willmore functional to isometric immersions
\[
W_g^{2,2}(S)=\{u\in W^{2,2}(S,\mathbb{R}^3):(\nabla u)^\top (\nabla u)=g\;\text{a.e. in}\;S\}.
\]
Reviewer: Igor Bock (Bratislava)
MSC:
| 74B20 | Nonlinear elasticity |
| 74G55 | Qualitative behavior of solutions of equilibrium problems in solid mechanics |
References:
| [1] | Bartels, Sören; Hornung, Peter, Bending paper and the Möbius strip, J. Elasticity, 119, 1-2, 113-136 (2015) · Zbl 1464.74025 |
| [2] | Bauer, M.; Kuwert, E., Existence of minimizing Willmore surfaces of prescribed genus, Int. Math. Res. Not. IMRN, 10, 553-576 (2003) · Zbl 1029.53073 |
| [3] | Bohle, C.; Peters, G. P.; Pinkall, U., Constrained Willmore surfaces, Calc. Var. Partial Differential Equations, 32, 2, 263-277 (2008) · Zbl 1153.53037 |
| [4] | Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext (2011), Springer: Springer New York · Zbl 1220.46002 |
| [5] | Ciarlet, P. G., Mathematical elasticity. Vol. III. Theory of shells, Studies in Mathematics and its Applications, vol. 29 (2000), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam · Zbl 0953.74004 |
| [6] | Cohn-Vossen, S. E., Bendability of surfaces in the large, Uspekhi Mat. Nauk, 1, 33-76 (1936) · Zbl 0016.22501 |
| [7] | Efrati, E.; Sharon, E.; Kupferman, R., Elastic theory of unconstrained non-Euclidean plates, J. Mech. Phys. Solids, 57, 4, 762-775 (2009) · Zbl 1306.74031 |
| [8] | Efrati, E.; Sharon, E.; Kupferman, R., Hyperbolic non-Euclidean elastic strips and almost minimal surfaces, Phys. Rev. E, 83 (2013) |
| [9] | Efrati, Efi, Non-Euclidean ribbons: generalized Sadowsky functional for residually-stressed thin and narrow bodies, J. Elasticity, 119, 1-2, 251-261 (2015) · Zbl 1320.53008 |
| [10] | Efrati, Efi; Sharon, Eran; Kupferman, Raz, Buckling transition and boundary layer in non-Euclidean plates, Phys. Rev. E (3), 80, 1, Article 016602 pp. (2009) |
| [11] | Friesecke, G.; James, R. D.; Müller, S., A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55, 11, 1461-1506 (2002) · Zbl 1021.74024 |
| [12] | Friesecke, G.; James, R. D.; Müller, S., A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Arch. Ration. Mech. Anal., 180, 2, 183-236 (2006) · Zbl 1100.74039 |
| [13] | Friesecke, Gero; James, Richard D.; Mora, Maria Giovanna; Müller, Stefan, Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence, C. R. Math. Acad. Sci. Paris, 336, 8, 697-702 (2003) · Zbl 1140.74481 |
| [14] | Gemmer, J.; Venkataramani, S., Shape transitions in hyperbolic non-Euclidean plates, Soft Matter, 9, 34, 8151-8161 (2013) |
| [15] | Geymonat, G.; Sánchez-Palencia, E., On the rigidity of certain surfaces with folds and applications to shell theory, Arch. Ration. Mech. Anal., 129, 1, 11-45 (1995) · Zbl 0830.73040 |
| [16] | Han, Q.; Hong, J.-X., Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Mathematical Surveys and Monographs, vol. 130 (2006), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1113.53002 |
| [17] | Hartman, Philip; Nirenberg, Louis, On spherical image maps whose Jacobians do not change sign, Amer. J. Math., 81, 901-920 (1959) · Zbl 0094.16303 |
| [18] | Helfrich, W., Elastic properties of lipid bilayers: theory and possible experiments, Z. Naturforsch. A, C28, 636-703 (1973) |
| [19] | Hornung, P., Euler-Lagrange equation and regularity for flat minimizers of the Willmore functional, Comm. Pure Appl. Math., 64, 3, 367-441 (2011) · Zbl 1209.49061 |
| [20] | Hornung, P., Continuation of infinitesimal bendings on developable surfaces and equilibrium equations for nonlinear bending theory of plates, Comm. Partial Differential Equations, 38, 1368-1408 (2013) · Zbl 1456.74016 |
| [21] | Hornung, P., Another remark on constrained von Kármán theories (2014) · Zbl 1371.74193 |
| [22] | Hornung, P.; Velčić, I., Regularity of intrinsically convex \(W^{2, 2}\) surfaces and the derivation of homogenized bending shell models (2015) |
| [23] | Hornung, Peter, Fine level set structure of flat isometric immersions, Arch. Ration. Mech. Anal., 199, 3, 943-1014 (2011) · Zbl 1242.53061 |
| [24] | Isanov, T. G., The continuation of infinitesimal bendings, Dokl. Akad. Nauk SSSR, 234, 6, 1257-1260 (1977) · Zbl 0376.53002 |
| [25] | Ivanova-Karatopraklieva, I.; Sabitov, I. Kh., Deformation of surfaces, I, (Problems in Geometry. Problems in Geometry, Itogi Nauki i Tekhniki, vol. 23 (1991), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform.: Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. Moscow). (Problems in Geometry. Problems in Geometry, Itogi Nauki i Tekhniki, vol. 23 (1991), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform.: Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. Moscow), J. Math. Sci., 70, 2, 1685-1716 (1994), 187 (in Russian), translated in · Zbl 0835.53003 |
| [26] | Ivanova-Karatopraklieva, I.; Sabitov, I. Kh., Bending of surfaces, II, J. Math. Sci., 74, 3, 997-1043 (1995) · Zbl 0861.53002 |
| [27] | Klein, Y.; Efrati, E.; Sharon, E., Shaping of elastic sheets by prescription of non-Euclidean metrics, Science, 315, 5815, 1116-1120 (2007) · Zbl 1226.74013 |
| [28] | Klimentov, S. B., Extension of higher-order infinitesimal bendings of a simply connected surface of positive curvature, Mat. Zametki, 36, 3, 393-403 (1984) · Zbl 0581.53002 |
| [29] | Kolegaeva, E. M.; Fomenko, V. T., Continuation of infinitesimal bendings of surfaces to analytic bendings under external constraints, Mat. Zametki, 45, 2, 30-39 (1989), 141 · Zbl 0721.53006 |
| [30] | Korte, A. P.; Starostin, E. L.; van der Heijden, G. H.M., Triangular buckling patterns of twisted inextensible strips, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467, 2125, 285-303 (2011) · Zbl 1219.74017 |
| [31] | Kupferman, R.; Solomon, J. P., A Riemannian approach to reduced plate, shell, and rod theories, J. Funct. Anal., 266, 5, 2989-3039 (2014) · Zbl 1305.74022 |
| [32] | Kuwert, E.; Schätzle, R., Gradient flow for the Willmore functional, Comm. Anal. Geom., 10, 2, 307-339 (2002) · Zbl 1029.53082 |
| [33] | Kuwert, E.; Schätzle, R., Removability of point singularities of Willmore surfaces, Ann. of Math. (2), 160, 1, 315-357 (2004) · Zbl 1078.53007 |
| [34] | Kuwert, E.; Schätzle, R., Minimizers of the Willmore functional under fixed conformal class (2008) |
| [35] | Lewicka, M.; Mora, M. G.; Pakzad, M. R., The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells, Arch. Ration. Mech. Anal., 200, 3, 1023-1050 (2011) · Zbl 1291.74130 |
| [36] | Lewicka, M.; Ochoa, P.; Pakzad, M. R., Variational models for prestrained plates with Monge-Ampère constraint, Differential Integral Equations, 28, 9-10, 861-898 (2015) · Zbl 1363.74063 |
| [37] | Lods, V.; Mardare, C., The space of inextensional displacements for a partially clamped linearly elastic shell with an elliptic middle surface, J. Elasticity, 51, 2, 127-144 (1998) · Zbl 0921.73206 |
| [38] | Nirenberg, L., The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math., 6, 337-394 (1953) · Zbl 0051.12402 |
| [39] | Nirenberg, L., Rigidity of a class of closed convex surfaces, (Lang, R., Nonlinear Problems (1963), University of Wisconsin Press), 177-193 · Zbl 0111.34402 |
| [40] | Pakzad, M. R., On the Sobolev space of isometric immersions, J. Differential Geom., 66, 1, 47-69 (2004) · Zbl 1064.58009 |
| [41] | Pinkall, U.; Sterling, I., Willmore surfaces, Math. Intelligencer, 9, 2, 38-43 (1987) · Zbl 0616.53049 |
| [42] | Pogorelov, A. V., Bendings of Surfaces and Stability of Shells, Translations of Mathematical Monographs, vol. 72 (1988), American Mathematical Society: American Mathematical Society Providence, RI, translated from Russian by J.R. Schulenberger · Zbl 0656.73026 |
| [43] | Rivière, T., Analysis aspects of Willmore surfaces, Invent. Math., 174, 1, 1-45 (2008) · Zbl 1155.53031 |
| [45] | Schätzle, R., The Willmore boundary problem, Calc. Var. Partial Differential Equations, 37, 3-4, 275-302 (2010) · Zbl 1188.53006 |
| [46] | Schygulla, J., Willmore minimizers with prescribed isoperimetric ratio, Arch. Ration. Mech. Anal., 203, 3, 901-941 (2012) · Zbl 1288.74027 |
| [47] | Simon, L., Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom., 1, 2, 281-326 (1993) · Zbl 0848.58012 |
| [48] | Starostin, E. L.; van der Heijden, G. H.M., The shape of a Möbius strip, Nat. Mater., 6, 563-567 (2007) |
| [49] | Starostin, E. L.; van der Heijden, G. H.M., Force and moment balance equations for geometric variational problems on curves, Phys. Rev. E (3), 79, 6, Article 066602 pp. (2009) |
| [50] | Starostin, E. L.; van der Heijden, G. H.M., Equilibrium shapes with stress localisation for inextensible elastic Möbius and other strips, J. Elasticity, 119, 1-2, 67-112 (2015) · Zbl 1469.74081 |
| [51] | Vekua, I. N., Verallgemeinerte analytische Funktionen (1963), Akademie-Verlag: Akademie-Verlag Berlin, Herausgegeben von Wolfgang Schmidt · Zbl 0108.07703 |
| [52] | Weiner, J. L., On a problem of Chen, Willmore, et al., Indiana Univ. Math. J., 27, 1, 19-35 (1978) · Zbl 0343.53038 |
| [53] | Weyl, H., Über die Starrheit der Eiflächen und konvexen Polyeder, Berliner Sitzungsber., 250-266 (1917) · JFM 46.1115.02 |
| [54] | Willmore, T. J., Riemannian Geometry, Oxford Science Publications (1993), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York · Zbl 0797.53002 |
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