Plates with incompatible prestrain of high order

Lewicka, Marta; Raoult, Annie; Ricciotti, Diego

doi:10.1016/j.anihpc.2017.01.003

Lewicka, Marta ; Raoult, Annie ; Ricciotti, Diego

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1883-1912.

Résumé
Abstract

On s'intéresse au comportement de structures minces d'épaisseur h dont l'énergie interne $E^{h}$ est régie par une métrique riemannienne tridimensionnelle G imposée, constante dans l'épaisseur, n'admettant pas nécessairement d'immersion isométrique. On sait que lorsque la restriction de G à la surface moyenne ω possède une immersion isométrique suffisamment régulière, c'est-à-dire appartenant à $W^{2, 2} (ω, R^{3})$ , alors $h^{- 2} \inf E^{h}$ admet une limite finie c quand h tend vers 0. Le modèle limite correspondant généralise le modèle de flexion non linéaire, classique pour la métrique euclidienne. Nous nous plaçons ici dans le cas où c vaut 0, ce qui équivaut à la nullité de trois des six coeffiecients du tenseur de courbure associé à G. Nous montrons qu'alors $\inf E^{h} \leq C h^{4}$ . Nous identifions la Γ-limite de $h^{- 4} E^{h}$ et montrons qu'elle généralise l'énergie de von Kármán. Elle s'exprime en fonction des déplacements incrémentaux par rapport à la surface définie par le modèle de flexion et de déformations tangentielles généralisées. De plus, nous montrons que l'infimum de ce modèle limite à l'ordre 4 n'est nul que si G admet une immersion isométrique, auquel cas $\min E^{h} = 0$ pour tout h.

We study the elastic behaviour of incompatibly prestrained thin plates of thickness h whose internal energy $E^{h}$ is governed by an imposed three-dimensional smooth Riemann metric G only depending on the variable in the midsurface ω. It is already known that $h^{- 2} \inf E^{h}$ converges to a finite value c when the metric G restricted to the midsurface has a sufficiently regular immersion, namely $W^{2, 2} (ω, R^{3})$ . The obtained limit model generalizes the bending (Kirhchoff) model of Euclidean elasticity. In the present paper, we deal with the case when c equals 0. Then, equivalently, three independent entries of the three-dimensional Riemann curvature tensor associated with G are null. We prove that, in such regime, necessarily $\inf E^{h} \leq C h^{4}$ . We identify the Γ-limit of the scaled energies $h^{- 4} E^{h}$ and show that it consists of a von Kármán-like energy. The unknowns in this energy are the first order incremental displacements with respect to the deformation defined by the bending model and the second order tangential strains. In addition, we prove that when $\inf h^{- 4} E^{h} \to 0$ , then G is realizable and hence $\min E^{h} = 0$ for every h.

MR Zbl

DOI : 10.1016/j.anihpc.2017.01.003

Mots clés : Non-Euclidean plates, Prestrained energy, Nonlinear elasticity, Calculus of variations

@article{AIHPC_2017__34_7_1883_0,
     author = {Lewicka, Marta and Raoult, Annie and Ricciotti, Diego},
     title = {Plates with incompatible prestrain of high order},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1883--1912},
     publisher = {Elsevier},
     volume = {34},
     number = {7},
     year = {2017},
     doi = {10.1016/j.anihpc.2017.01.003},
     mrnumber = {3724760},
     zbl = {1457.74121},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2017.01.003/}
}

TY  - JOUR
AU  - Lewicka, Marta
AU  - Raoult, Annie
AU  - Ricciotti, Diego
TI  - Plates with incompatible prestrain of high order
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2017
SP  - 1883
EP  - 1912
VL  - 34
IS  - 7
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2017.01.003/
DO  - 10.1016/j.anihpc.2017.01.003
LA  - en
ID  - AIHPC_2017__34_7_1883_0
ER  -

%0 Journal Article
%A Lewicka, Marta
%A Raoult, Annie
%A Ricciotti, Diego
%T Plates with incompatible prestrain of high order
%J Annales de l'I.H.P. Analyse non linéaire
%D 2017
%P 1883-1912
%V 34
%N 7
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2017.01.003/
%R 10.1016/j.anihpc.2017.01.003
%G en
%F AIHPC_2017__34_7_1883_0

Lewicka, Marta; Raoult, Annie; Ricciotti, Diego. Plates with incompatible prestrain of high order. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1883-1912. doi : 10.1016/j.anihpc.2017.01.003. http://www.numdam.org/articles/10.1016/j.anihpc.2017.01.003/

Bibliographie
Cité par

[1] Acerbi, E.; Buttazzo, G.; Percivale, D. A variational definition for the strain energy of an elastic string, J. Elast., Volume 25 (1991), pp. 137–148 | DOI | MR | Zbl

[2] Audoly, B.; Boudaoud, A. Self-similar structures near boundaries in strained systems, Phys. Rev. Lett., Volume 91 (2004)

[3] Bella, P.; Kohn, R.V. Metric-induced wrinkling of a thin elastic sheet, J. Nonlinear Sci., Volume 24 (2014), pp. 1147–1176 | DOI | MR | Zbl

[4] Ben Amar, M.; Müller, M.M.; Trejo, M. Petal shapes of sympetalous flowers: the interplay between growth, geometry and elasticity, New J. Phys., Volume 14 (2012)

[5] Bhattacharya, K.; Lewicka, M.; Schaffner, M. Plates with incompatible prestrain, Arch. Ration. Mech. Anal., Volume 221 (2016), pp. 143–181 | DOI | MR | Zbl

[6] Caillerie, D.; Sanchez-Palencia, E. A new kind of singular stiff problems and application to thin elastic shells, Math. Models Methods Appl. Sci., Volume 5 (1995), pp. 47–66 | DOI | MR | Zbl

[7] Caillerie, D.; Sanchez-Palencia, E. Elastic thin shells: asymptotic theory in the anisotropic and heterogeneous cases, Math. Models Methods Appl. Sci., Volume 5 (1995), pp. 473–496 | DOI | MR | Zbl

[8] Ciarlet, P.G. A justification of the von Kármán equations, Arch. Ration. Mech. Anal., Volume 73 (1980), pp. 349–389 | DOI | MR | Zbl

[9] Ciarlet, P.G. Mathematical Elasticity, vol. II: Theory of Plates, North-Holland, Amsterdam, 1997 | MR

[10] Ciarlet, P.G. Mathematical Elasticity, vol. III: Theory of Shells, North-Holland, Amsterdam, 2000

[11] Ciarlet, P.G.; Destuynder, P. A justification of the two-dimensional linear plate model, J. Méc., Volume 18 (1979), pp. 315–344 | MR | Zbl

[12] Ciarlet, P.G.; Destuynder, P. A justification of a nonlinear model in plate theory, Comput. Methods Appl. Mech. Eng., Volume 17–18 (1979), pp. 227–258 | Zbl

[13] Ciarlet, P.G.; Lods, V. Asymptotic analysis of linearly elastic shells. I. Justification of membrane shells equations, Arch. Ration. Mech. Anal., Volume 136 (1996), pp. 119–161 | MR | Zbl

[14] Ciarlet, P.G.; Lods, V. Asymptotic analysis of linearly elastic shells: generalized membrane shells, J. Elast., Volume 43 (1996), pp. 147–188 | DOI | MR | Zbl

[15] Ciarlet, P.G.; Lods, V.; Miara, B. Asymptotic analysis of linearly elastic shells. II. Justification of flexural shells equations, Arch. Ration. Mech. Anal., Volume 136 (1996), pp. 163–190 | MR | Zbl

[16] Conti, S.; Dolzmann, G. Derivation of elastic theories for thin sheets and the constraint of incompressibility, Analysis, Modeling and Simulation of Multiscale Problems, Springer, Berlin, 2006, pp. 225–247 | DOI | MR | Zbl

[17] Conti, S.; Dolzmann, G. Derivation of a plate theory for incompressible materials, C. R. Math. Acad. Sci. Paris, Volume 344 (2007), pp. 541–544 | DOI | MR | Zbl

[18] Conti, S.; Maggi, F. Confining thin sheets and folding paper, Arch. Ration. Mech. Anal., Volume 187 (2008), pp. 1–48 | MR | Zbl

[19] Dervaux, J.; Ben Amar, M. Morphogenesis of growing soft tissues, Phys. Rev. Lett., Volume 101 (2008) | DOI

[20] Dervaux, J.; Ciarletta, P.; Ben Amar, M. Morphogenesis of thin hyperelastic plates: a constitutive theory of biological growth in the Foppl–von Karman limit, J. Mech. Phys. Solids, Volume 57 (2009), pp. 458–471 | DOI | MR | Zbl

[21] Destuynder, P. Sur une justification des modèles de plaques et de coques par les méthodes asymptotiques, 1980 (Doctorat d'état, Paris)

[22] Destuynder, P. Comparaison entre les modèles tri-dimensionnels et bi-dimensionnels de plaques en élasticité, RAIRO. Anal. Numér., Volume 15 (1981), pp. 331–369 | DOI | Numdam | MR | Zbl

[23] Destuynder, P. A classification of thin shell theories, Acta Appl. Math., Volume 4 (1985), pp. 15–63 | DOI | MR | Zbl

[24] De Lellis, C.; Inauen, D.; Székelyhidi, L. Jr. A Nash–Kuiper theorem for $C^{1, \frac{1}{5} - δ}$ immersions of surfaces in 3 dimensions | arXiv | Zbl

[25] Efrati, E.; Sharon, E.; Kupferman, R. Elastic theory of unconstrained non-Euclidean plates, J. Mech. Phys. Solids, Volume 57 (2009), pp. 762–775 | DOI | MR | Zbl

[26] Fox, D.D.; Raoult, A.; Simo, J.C. A justification of nonlinear properly invariant plate theories, Arch. Ration. Mech. Anal., Volume 124 (1993), pp. 157–199 | MR | Zbl

[27] Friesecke, G.; James, R.; Mora, M.G.; Müller, S. Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence, C. R. Math. Acad. Sci. Paris, Volume 336 (2003), pp. 697–702 | DOI | MR | Zbl

[28] Friesecke, G.; James, R.; Müller, S. A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Commun. Pure Appl. Math., Volume 55 (2002), pp. 1461–1506 | DOI | MR | Zbl

[29] Friesecke, G.; James, R.; Müller, S. A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence, Arch. Ration. Mech. Anal., Volume 180 (2006), pp. 183–236 | DOI | MR | Zbl

[30] Gemmer, J.; Venkataramani, S. Shape selection in non-Euclidean plates, Physica D, Volume 240 (2011) no. 19, pp. 1536–1552 | DOI | MR | Zbl

[31] Hornung, P.; Lewicka, M.; Pakzad, R. Infinitesimal isometries on developable surfaces and asymptotic theories for thin developable shells, J. Elast., Volume 111 (2013) | DOI | MR | Zbl

[32] Kim, J.; Hanna, J.A.; Byun, M.; Santangelo, C.D.; Hayward, R.C. Designing responsive buckled surfaces by halftone gel lithography, Science, Volume 335 (2012), pp. 1201–1205 | MR | Zbl

[33] Klein, Y.; Efrati, E.; Sharon, E. Shaping of elastic sheets by prescription of non-Euclidean metrics, Science, Volume 315 (2007), pp. 1116–1120 | DOI | MR | Zbl

[34] Klein, Y.; Venkataramani, S.C.; Sharon, E. Experimental study of shape transitions and energy scaling in thin non-Euclidean plates, Phys. Rev. Lett., Volume 106 (2011), pp. 118303 | DOI

[35] Le Dret, H.; Raoult, A. The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., Volume 73 (1995), pp. 549–578 | MR | Zbl

[36] Le Dret, H.; Raoult, A. The membrane shell model in nonlinear elasticity: a variational asymptotic derivation, J. Nonlinear Sci., Volume 6 (1996), pp. 59–84 | DOI | MR | Zbl

[37] Lewicka, M. A note on the convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry, ESAIM Control Optim. Calc. Var., Volume 17 (2011), pp. 493–505 | DOI | Numdam | MR | Zbl

[38] Lewicka, M.; Li, H. Convergence of equilibria for incompressible elastic plates in the von Karman regime, Commun. Pure Appl. Anal., Volume 14 (January 2015) no. 1 | DOI | MR | Zbl

[39] Lewicka, M.; Mahadevan, L.; Pakzad, R. The Foppl–von Karman equations for plates with incompatible strains, Proc. R. Soc. A, Volume 467 (2011), pp. 402–426 | DOI | MR | Zbl

[40] Lewicka, M.; Mahadevan, L.; Pakzad, R. Models for elastic shells with incompatible strains, Proc. R. Soc. A, Volume 47 (2014) no. 2165 (1471–2946) | DOI

[41] Lewicka, M.; Mahadevan, L.; Pakzad, R. The Monge–Ampère constraint: Matching of isometries, density and regularity, and elastic theories of shallow shells, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 34 (2017), pp. 45–67 | DOI | Numdam | MR | Zbl

[42] Lewicka, M.; Mora, M.G.; Pakzad, R. Shell theories arising as low energy gamma-limit of 3d nonlinear elasticity, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume IX (2010), pp. 1–43 | Numdam | MR | Zbl

[43] Lewicka, M.; Mora, M.G.; Pakzad, R. The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells, Arch. Ration. Mech. Anal. (3), Volume 200 (2011), pp. 1023–1050 | MR | Zbl

[44] Lewicka, M.; Ochoa, P.; Pakzad, R. Variational models for prestrained plates with Monge–Ampere constraint, Differ. Integral Equ., Volume 28 (2015) no. 9–10, pp. 861–898 | MR | Zbl

[45] Lewicka, M.; Pakzad, R. Scaling laws for non-Euclidean plates and the $W^{2, 2}$ isometric immersions of Riemannian metrics, ESAIM Control Optim. Calc. Var., Volume 17 (2011), pp. 1158–1173 | DOI | Numdam | MR | Zbl

[46] Lewicka, M.; Pakzad, M. The infinite hierarchy of elastic shell models; some recent results and a conjecture, Infinite Dimensional Dynamical Systems, Fields Inst. Commun., vol. 64, 2013, pp. 407–420 | DOI | MR | Zbl

[47] Li, H.; Chermisi, M. The von Karman theory for incompressible elastic shells, Calc. Var. Partial Differ. Equ., Volume 48 (2013) no. 1–2, pp. 185–209 | MR | Zbl

[48] Liang, H.; Mahadevan, L. The shape of a long leaf, Proc. Natl. Acad. Sci. (2009) | DOI | MR | Zbl

[49] Liu, F.C. A Lusin property of Sobolev functions, Indiana Univ. Math. J., Volume 26 (1977), pp. 645–651 | MR | Zbl

[50] Miara, B.; Sanchez-Palencia, E. Asymptotic analysis of linearly elastic shells, Asymptot. Anal., Volume 12 (1996), pp. 41–54 | MR | Zbl

[51] Monneau, R. Justification of nonlinear Kirchhoff–Love theory of plates as the application of a new singular inverse method, Arch. Ration. Mech. Anal., Volume 169 (2003), pp. 1–34 | DOI | MR | Zbl

[52] Mora, M.G.; Scardia, L. Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density, J. Differ. Equ., Volume 252 (2012), pp. 35–55 | MR | Zbl

[53] Müller, S.; Pakzad, M.R. Convergence of equilibria of thin elastic plates – the von Kármán case, Commun. Partial Differ. Equ., Volume 33 (2008), pp. 1018–1032 | DOI | MR | Zbl

[54] Ortiz, M.; Gioia, G. The morphology and folding patterns of buckling-driven thin-film blisters, J. Mech. Phys. Solids, Volume 42 (1994), pp. 531–559 | DOI | MR | Zbl

[55] Raoult, A. Construction d'un modèle d'évolution de plaques avec terme d'inertie de rotation, Ann. Mat. Pura Appl., Volume CXXXIX (1985), pp. 361–400 | MR | Zbl

[56] Raoult, A. Analyse mathématique de quelques modèles de plaques et de poutres élastiques ou élasto-plastiques, 1988 (Doctorat d'état, Paris)

[57] Rodriguez, E.K.; Hoger, A.; McCulloch, A. Stress-dependent finite growth in finite soft elastic tissues, J. Biomech., Volume 27 (1994), pp. 455–467 | DOI

[58] Sanchez-Palencia, E. Passage à la limite de l'élasticité tridimensionnelle à la théorie asymptotique des coques minces, C. R. Acad. Sci. Paris Sér. II, Volume 311 (1990), pp. 909–916 | MR | Zbl

[59] Sanchez-Palencia, E. Statique et dynamique des coques minces. I. Cas de flexion pure non inhibée, C. R. Math. Acad. Sci., Volume 309 (1989), pp. 411–417 | MR | Zbl

[60] Sanchez-Palencia, E. Statique et dynamique des coques minces. II. Cas de flexion pure inhibée. Approximation membranaire, C. R. Math. Acad. Sci., Volume 309 (1989), pp. 531–537 | MR | Zbl

[61] Schmidt, B. Plate theory for stressed heterogeneous multilayers of finite bending energy, J. Math. Pures Appl., Volume 88 (2007), pp. 107–122 | DOI | MR | Zbl

[62] Sharon, E.; Roman, B.; Marder, M.; Shin, G.S.; Swinney, H.L. Buckling cascade in free thin sheets, Nature, Volume 419 (2002), pp. 579 | DOI

[63] Sharon, E.; Roman, B.; Swinney, H.L. Geometrically driven wrinkling observed in free plastic sheets and leaves, Phys. Rev. E, Volume 75 (2007) | DOI

[64] Trabelsi, K. Modeling of a membrane for nonlinearly elastic incompressible materials via gamma-convergence, Anal. Appl., Volume 4 (2006), pp. 31–60 | DOI | MR | Zbl

[65] Ware, T.H.; McConney, M.E.; Wie, J.J.; Tondiglia, V.P.; White, T.J. Voxelated liquid crystal elastomers, Science, Volume 347 (2015) no. 6225, pp. 982–984 | DOI

Cité par Sources :