Large deformation problem of hyperbolic shallow shells with bimodular effect: A perturbation-variation method

As an important analytical method, the perturbation-variation method combines the advantages of perturbation method and variational method, and is widely used for the solution of nonlinear plate and shell problems. In this study, the perturbation-variation method is used to solve the large deformation problem of a flexible hyperbolic thin shallow shells with different moduli in tension and compression (bimodular effect). First, we establish the governing equations expressed in terms of three displacement components, in which the bimodular effect of materials is considered in physical equations and the large deformation of shell is considered in geometrical equations. Next, we use the perturbation-variation method to solve the governing equations established, obtaining the perturbation-variation solution up to third-order. During the application of the method, the displacements are expressed into perturbation formulas of all levels first, and the unknown constants and functions in the perturbation formulas are determined by the extreme value conditions of the functional established (variational principle), hence the perturbation-variation method gets its name from this practice. The numerical results from FEM basically agree with the perturbation-variation solution obtained. The method proposed may be extended into the solution of similar partial differential equations established in applied fields.