We approximate the displacement field in a shell by a fifth-order Taylor–Young expansion in thickness. The model is derived from the truncation of the potential energy at fifth order. The equilibrium equations imply local constraints on the through-thickness derivatives of the zero-order displacement field. This leads to an analytical expression for the two-dimensional potential energy of a shell in terms of the zero-order displacement field and its derivatives that include non-standard transverse shearing and normal stress energy. Then we derive the equation of equilibrium and the boundary conditions.
BallJM. Some open problems in elasticity. In NewtonPHolmesPWeinsteinA (eds) Geometry, mechanics and dynamics. New York, NY: Springer, 2002, 3–59.
2.
FrieseckeGJamesRDMüllerS. Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity theory by Gamma-convergence. CR Acad Sci Paris Ser I2002; 336: 697–702.
3.
CiarletPG. Introduction to linear shell theory. Amsterdam: Elsevier, 1998.
4.
PruchnickiE. Nonlinearly elastic membrane model for heterogeneous shells: A formal asymptotic approach. J Elast2006; 84: 245–280.
5.
FrieseckeGJamesRDMüllerS. A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence. Arch Rat Mech Anal2006; 180: 183–236.
6.
CiarletPGRoquefortA. Justification of a two-dimensional nonlinear shell model of Koiter’s type. Chin Ann Math2001; 22B: 129–144.
7.
KoiterWT. On the nonlinear theory of thin elastic shells. Proc Kon Ned Akad Wetensch1966; B69: 1–54.
8.
CiarletPGLodsV. Asymptotic analysis of linearly elastic shells I: Justification of membrane shell equations. Arch Rat Mech Anal1996; 36: 119–161.
9.
CiarletPGLodsVMiaraB. Asymptotic analysis of linearly elastic shells II: Justification of flexural shell equations. Arch Rat Mech Anal1996; 136: 163–190.
10.
SteigmannDJ. Extension of Koiter’s linear shell theory to materials exhibiting arbitrary symmetry. Int J Eng Sci2012; 46: 654–676.
11.
PruchnickiE. Two-dimensional model of order h5 for the combined bending, stretching and transverse shearing of homogeneous plates derived from three-dimensional elasticity. Math Mech Solids2014; 19(5): 477–490.
12.
PruchnickiE. Two-dimensional model for the combined bending, stretching and shearing of shells: A general approach and application to laminated cylindrical shells derived from three-dimensional elasticity. Math Mech Solids2014; 19(5): 491–501.
13.
CiarletPGPaumierJC. A justification of the Marguerre-von Kármán equations. Comput Mech1986; 1: 177–202.
14.
SteigmannDJ. Two-dimensional models for the combined stretching of plates and shells on three-dimensional linear elasticity. Int J Eng Sci2008; 51: 216–232.
15.
FicheraG. Existence theorems in elasticity. In TruesdellC (ed.) Handbuch der physik, volume VIa/2. Berlin: Springer, 1972, 347–389.
16.
DikmenM. Theory of thin elastic shells. London: Pitman, 1982.
17.
PruchnickiE. Two-dimensional model for the combined bending, stretching and transverse shearing of laminated plates derived from three-dimensional elasticity. Math Mech Solids2010; 16(3): 304–316.
18.
LovelockDRundH. Tensors, differential forms, and variational principles. New York, NY: Wiley, 1975.
