We consider a second-order thin curved film whose behavior is governed by an energy made up of a first-order nonlinear part depending on the gradient of the deformation augmented by a quadratic second-order part depending on the tensor of second derivatives of the deformation. We carry out a 3D–2D analysis through an asymptotic expansion in powers of the thickness of the film as it tends to zero.
BoulakiaMGuerreroS.On the interaction problem between a compressible fluid and a Saint-Venant Kirchhoff elastic structure. Adv Diff Equ2017; 22(1/2): 1–48.
IgnatRZorgatiH.Dimension reduction and optimality of the uniform state in a phase-field-crystal model involving a higher-order functional. J Nonlinear Sci2020; 30(1): 261–282.
12.
ChermisiMDal MasoGFonsecaI, et al. Singular perturbation models in phase transitions for second-order materials. Indiana Univ Math J2011; 60: 367–409.
13.
CicaleseMSpadaroE-NZeppieriC-I.Asymptotic analysis of a second-order singular perturbation model for phase transitions. Calc Var Partial Differ Equ2011; 41: 127–150.
14.
ŠilhavýM.Equilibrium of phases with interfacial energy: a variational approach. J Elast2011; 105: 271–303.
15.
JaviliAMcBrideASteinmannP.Thermomechanics of solids with lower-dimensional energetics: on the importance of surface, interface, and curve structures at the nanoscale. A unifying review. Appl Mech Rev2013; 65: 010802.
16.
LevitasV-IWarrenJ-A.Phase field approach with anisotropic interface energy and interface stresses: large strain formulation. J Mech Phys Solids2016; 91: 94–125.
17.
GrandiDKružíkMMaininiE, et al. Equilibrium for multiphase solids with eulerian interfaces. J Elast2020; 142: 409–431.
18.
BernadouM.Finite element methods for thin shell problems. Hoboken, NJ: John Wiley & Sons, 1996.
19.
ChapelleDBatheKJ.The finite element analysis of shells (computational fluid and solid mechanics). 2nd ed.Berlin: Springer, 2011.
20.
CiarletPGDestuynderP.A justification of the two-dimensional linear plate model. J Mécanique1979; 18(2): 315–344.
21.
CiarletPG.A justification of the Von Kármán equation. Arch Rational Mech Anal1980; 73: 349–189.
22.
FriedrichsKODresslerRF.A boundary-layer theory for elastic plates. Commun Pure Appl Math1961; 14: 1–33.
23.
GoldenveizerAL.Derivation of an approximate theory of shells by means of asymptotic integration of the equations of the theory of elasticity. J Appl Math Mech1963; 27(4): 593–608.
24.
Sanchez-PalenciaE.Statique et dynamique des coques minces. I. Cas de flexion pure non inhibée. C R Acad Sci Paris Sér I1989; 309: 411–417.
25.
Sanchez-PalenciaE.Statique et dynamique des coques minces. II. Cas de flexion pure inhibée. C R Acad Sci Paris Sér I1989; 309: 531–537.
26.
Sanchez-PalenciaE.Passage à la limite de l‘élastiocité tridimensionnelle à la théorie asymptotique des coques minces. C R Acad Sci Paris Sér II1990; 311: 909–916.
27.
CiarletPGLodsV.Analyse asymptotique des coques linéairement élastiques. I. Coques membranaires. C R Acad Sci Paris Sér I1994; 319(1): 863–868.
28.
CiarletPGLodsVMiaraB.Analyse asymptotique des coques linéairement élastiques. I. Coques en flexion. C R Acad Sci Paris Sér I1994; 319(1): 95–100.
29.
MiaraB.Analyse asymptotique des coques membranaires non linéairement élastiques. C R Acad Sci Paris Série I. 1994; 318: 689–694.
30.
CastineiraGRodriguez-ArósA.Linear viscoelastic shells: an asymptotic approach. Asympt Anal2018; 107(3–4): 169–201.
31.
CastineiraGRodriguez-ArósA.On the justification of viscoelastic elliptic membrane shell equations. J Elasticity2018; 130(1): 85–113.
32.
ChenXDaiHPruchnickiE.On a consistent rod theory for a linearized anisotropic elastic material. I: asymptotic reduction method. Math Mech Solids2021; 26(2): 217–229.
33.
RaoultA.Doctoral dissertation, Analyse mathématique de quelques modéles de plaques et de poutres élastiques ou elasto-plastiques, Paris, 1988.
34.
CiarletPG.Plates and junctions in elastic multi-structures: an asymptotic analysis. Berlin: Springer-Verlag, 1990.
35.
CiarletPG.Mathematical elasticity: three dimensional elasticity. Vol. I. Amsterdam: North Holland, 1988.
36.
CiarletPG. Mathematical elasticity: theory of plates. Vol. II. Amsterdam: North Holland, 1997.
37.
CiarletPG.Mathematical elasticity: theory of shells. Vol. III. Amsterdam: North Holland, 2000.
De GiorgiE. Sulla convergenza di alcune successioni di integrali del tipo dell area. Rend Mat1975(8): 277–294.
40.
De GiorgiEFranzoniT. Su un tipo di convergenza variazionale. Atti Accad Naz Lincei1975; 58: 842–850.
41.
BraidesA.Γ-convergence for beginners (Oxford lecture series in mathematics and its applications). Oxford: Oxford University Press, 2002.
42.
Dal MasoG. An introduction to Γ-convergence. Boston, MA: Birkhauser, 1993.
43.
AcerbiEButtazzoBPercivaleD.A variational definition for the strain energy of an elastic string. J Elast1991; 25: 137–148.
44.
BabadjianJFZappaleEZorgatiH, Dimensional reduction for energies with linear growth involving the bending moment. J Math Pures Appl2008; 90(6): 520–549.
45.
BraidesAFonsecaIFrancfortG.3D-2D asymptotic analysis for inhomogeneous thin films. Indiana Univ Math J2000; 49(4): 1367–1404.
46.
FonsecaIFrancfortG.3D-2D asymptotic analysis of an optimal design problem for thin films. J für Die Reine und Angew Math1998; 1998(505): 173–202.
47.
Le DretHRaoultA. The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J Math Pures Appl1995; 75: 551–580.
48.
Le DretHRaoultA. The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J Nonlinear Sci1996; 6: 59–84.
49.
ZorgatiH.A Γ-convergence result for thin curved films bonded to a fixed substrate with a noninterpenetration constraint. Chin Ann Math Ser B2006; 27(6): 615–636.
ZorgatiH.Films courbés minces ferromagnétiques. C R Math Acad Sci Paris2005; 340(1): 81–86.
52.
PantzO.Quelques problèmes de modélisation en élasticité non linéaire. Doctoral dissertation, Université de Pierre et Marie Curie, Paris, 2001.
53.
TrabelsiK.Nonlinearly elastic thin plate models for a class of Ogden materials. I: the membrane model. Anal Appl2005; 3(2): 195–221.
54.
MeunierN.Recursive derivation of one-dimensional models from three-dimensional nonlinear elasticity. Math Mech Solids2008; 13(2): 172–194.
55.
PantzO.Derivation of the nonlinear membrane plate models from the three-dimensional elasticity. C R Acad Sci Paris Ser I Math2000; 331(2): 171–174.
56.
Anza-HafsaOMandallenaJ-P.The nonlinear membrane energy under the constraint “”. Bull Sci Math2008; 132(4): 272–291.
57.
El JarroudiM. Homogenization of a nonlinear elastic fibre-reinforced composite: a second gradient nonlinear elastic material. J Math Anal Appl2013; 403: 487–505.
