In this paper, a consistent finite-strain shell theory for incompressible hyperelastic materials is formulated. First, for a shell structure made of an incompressible material, the three-dimensional (3D) governing system is derived through the variational approach, which is composed of the mechanical field equation and the constraint equation. Then, series expansions of the independent variables are conducted about the bottom surface and along the thickness direction of the shell. The recursive relations of the coefficient functions in the series expansions can be derived from the original 3D governing system. Further from the top surface boundary condition, a 2D vector shell equation is obtained, which represents the local force-balance of a shell element. The associated edge boundary conditions are also proposed. It is verified that shell equation system is consistent with the 3D variational formulation. The weak formulation of the shell equation is established for future numerical calculations. To show the validity of the shell theory, the axisymmetric deformations of a spherical and a circular cylindrical shell made of incompressible neo-Hookean materials are studied. By comparing with the exact solutions, it is shown that the asymptotic solutions obtained from the shell theory attain the accuracy of O(h2).
LoveAEH.On the small free vibrations and deformation of thin elastic shells. Philos Trans R Soc Lond Ser A1888; 179: 491–546.
2.
TimoshenkoSWoinowsky-KriegerS.Theory of Plates and Shells, Vol. 2. New York: McGraw-Hill, 1959.
3.
KoiterWTSimmondsJG.Foundations of Shell Theory. New York: Springer, 1973.
4.
CiarletPG.Mathematical Elasticity, Vol. III: Theory of Shells. Amsterdam: North-Holland, 2000.
5.
CiarletPG.An introduction to differential geometry with applications to elasticity. J Elast2005; 78: 1–215.
6.
PietraszkiewiczWGorskiJ.Shell Structures: Theory and Applications, Vol. 3. Boca Raton, FL: CRC Press, 2013.
7.
KirchhoffGR.ber das Gleichgewicht und Bewegungen einer elastischen Scheibe. J reine angew Math1850; 40: 51–88.
8.
ReissnerE.On the Theory of Thin Elastic Shells. Ann Arbor, MI: JW Edwards, 1949, pp. 231–247.
9.
NaghdiPM.On the theory of thin elastic shells. Q Appl Math1957; 14: 369–380.
10.
KoiterWT. A consistent first approximation in the general theory of thin elastic shells. In: Theory of Thin Elastic Shells. Amsterdam: North-Holland, 1960, pp. 12–33.
11.
Gol’denveizerAL.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: 903–924.
12.
KaplunovJDKossovitchLYNoldeEV.Dynamics of thin walled elastic bodies. New York: Academic Press, 1998.
13.
AghalovyanL.Asymptotic theory of anisotropic plates and shells. Singapore: World Scientific, 2015.
14.
BazarenkoNAIsrailevichVI.Asymptotic solution of the elasticity problem for a hollow, finite length, thin cylinder. J Appl Math Mech1965; 29: 1219–1238.
15.
VilenskaiaTVIsrailevichVI.Asymptotic behavior of the solution of the problem of elasticity for a thin spherical shell. J Appl Math Mech1967; 30: 342–361.
16.
HamdouniAMilletO.Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. Part I: the shallow shells. Arch Mech2003; 55: 135–176.
17.
HamdouniAMilletO.Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. Part II: the strongly curved shells. Arch Mech2003; 55: 177–220.
KienzlerRSchneiderP.Consistent theories of isotropic and anisotropic plates. J Theoret Appl Mech2012; 50: 755–768.
20.
CiarletPGDestuynderP.A justification of a nonlinear model in plate theory. Comput Meth Appl Mech Eng1979; 17: 227–258.
21.
DestuynderP.Sur une Justification des Modeles de Plaques et de Coques par les Methodes Asymptotiques. PhD thesis, Universite Pierre et Marie Curie, Paris, 1980.
22.
Le DretHRaoultA. The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity. J Math Pures Appl1995; 74: 549–578.
23.
Le DretHRaoultA. The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J Nonlin Sci1996; 6: 59–84.
24.
FrieseckeGJamesRDMoraMG. Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Γ -convergence. C R Math. 2003; 336: 697–702.
25.
SimoJCRifaiMSFoxDD.On a stress resultant geometrically exact shell model. Part IV: Variable thickness shells with through-the thickness stretching. Comput Meth Appl Mech Eng1990; 81: 91–126.
26.
BüchterNRammERoehlD.Three-dimensional extension of nonlinear shell formulation based on the enhanced assumed strain concept. Int J Numer Meth Eng1994; 37: 2551–2568.
27.
ReddyJArciniegaR.Shear deformation plate and shell theories: from stavsky to present. Mech Adv Mater Struct2004; 11: 535–582.
28.
SteigmannDJ.Extension of Koiter’s linear shell theory to materials exhibiting arbitrary symmetry. Int J Eng Sci2012; 51: 216–232.
29.
SteigmannDJ.Koiter’s shell theory from the perspective of three-dimensional nonlinear elasticity. J Elast2013; 111: 91–107.
30.
MakowskiJStumpfH.Finite strains and rotations in shells. In: Finite Rotations in Structural Mechanics. Berlin: Springer, 1986, pp. 175–194.
31.
ItskovM.A generalized orthotropic hyperelastic material model with application to incompressible shells. Int J Numer Meth Eng2001; 50: 1777–1799.
LiHChermisiM.The von Kármán theory for incompressible elastic shells. Calculus Variations Partial Differ Eqns2013; 48: 185–209.
34.
NaghdiPM. The theory of plates and shells. In: Handbuch der Physik, Vol. VIa/2. Heidelberg: Springer, 1972, pp. 425–640.
35.
ZhilinPA.Mechanics of deformable directed surfaces. Int J Solids Struct1976; 12: 635–648.
36.
RubinMB.Cosserat theories: shells, rods and points. Dordrecht: Springer Science and Business Media, 2000.
37.
AltenbachJAltenbachHEremeyevVA.On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch Appl Mech2010; 80: 73–92.
38.
AltenbachHEremeyevV.Thin-walled structural elements: classification, classical and advanced theories, new applications. In: Shell-like Structures: Advanced Theories and Applications. Cham: Springer, 2017, pp. 1–62.
39.
LibaiASimmondsJG.The Nonlinear Theory of Elastic Shells. Cambridge: Cambridge University Press, 1998.
40.
DaiHHSongZL.On a consistent finite-strain plate theory based on a 3-D energy principle. Proc R Soc Lond Ser A2014; 470: 2171.
41.
SongZLDaiHH.On a consistent finite-strain shell theory based on 3-D nonlinear elasticity. Int J Solids Struct2016; 97: 137–149.
42.
WangJSongZLDaiHH.On a consistent finite-strain plate theory for incompressible hyperelastic materials. Int J Solids Struct2016; 78: 101–109.
43.
OgdenR.Non-linear Elastic Deformations. New York: Ellis Horwood, 1984.
44.
ChadwickP.Continuum Mechanics: Concise Theory and Problems (2nd edn). New York: Dover Publications, 1999.
