A hyperelastic extended Kirchhoff-Love shell model with out-of-plane normal stress: I. Out-of-plane deformation

This is the first part of a two-part article on a hyperelastic extended Kirchhoff-Love shell model with out-of-plane normal stress. We present the derivation of the new model, with focus on the mechanics of the out-of-plane deformation. Accounting for the out-of-plane normal stress distribution in the out-of-plane direction affects the accuracy in calculating the deformed-configuration out-of-plane position, and consequently the nonlinear response of the shell. The improvement is beyond what we get from accounting for the out-of-plane deformation mapping. By accounting for the out-of-plane normal stress, the traction acting on the shell can be specified on the upper and lower surfaces separately. With that, the new model is free from the midsurface location in terms of specifying the traction. We also present derivations related to the variation of the kinetic energy and the form of specifying the traction and moment acting on the upper and lower surfaces and along the edges. We present test computations for unidirectional plate bending, plate saddle deformation, and pressurized cylindrical and spherical shells. We use the neo-Hookean and Fung's material models, for the compressible- and incompressible-material cases, and with the out-of-plane normal stress and without, which is the plane-stress case.

Automated summary

This article presents a hyperelastic extended Kirchhoff-Love shell model that takes into account the out-of-plane normal stress. The derivation of the new model is provided, with a focus on the mechanics of out-of-plane deformation. By considering the distribution of out-of-plane normal stress, the accuracy of calculating the deformed-configuration out-of-plane position and the nonlinear response of the shell are improved. The new model allows for specifying the traction acting on the shell on the upper and lower surfaces separately, without relying on the midsurface location. Test computations for different shell deformations and material models are also presented.