A multi‐director formulation for elastic—viscoelastic layered shells

A linear finite element formulation for the analysis of multi-layered shells comprised of linear elastic and viscoelastic layers is presented. The elastic and viscoelastic layers may occupy arbitrary locations and the formulation is appropriate for thick and thin shells. The concept of a multi-director field defined over a reference surface is employed for the description of the initial geometry and motion of the multi-layered shell. The kinematical theory incorporated in the three-dimensional variational formulation describes, within individual layers, the effects of transverse shear and transverse normal strain to arbitrary orders in the layer thickness co-ordinate. These kinematics have ‘local support’ over a layer and prove to be convenient and accurate in application. All stresses are computed through three-dimensional constitutive equations and the usual ‘zero normal stress’ shell hypothesis is not employed. Layer material properties are assumed to be isotropic, although this is not a restriction of the formulation. Sufficiently general constitutive equations for the viscoelastic layers are presented in rate form and an accurate and efficient product algorithm is introduced for their temporal integration. Finite element formulations using both resultant and continuum approaches are developed and compared. Observations and suggestions on the use of reduced/selective integration in the presence of high-order kinematics are made and a number of numerical examples are presented to illustrate the capability of the formulation.