A seven-parameter spectral/hp finite element formulation for isotropic, laminated composite and functionally graded shell structures

Abstract

In this paper we apply high-polynomial order spectral/hp basis functions in the numerical implementation of a novel 7-parameter continuum shell finite element formulation using general quadrilateral finite elements. The implementation is applicable to the analysis of isotropic, functionally graded and laminated composite shells undergoing fully geometrically nonlinear mechanical response. The shell finite element formulation is constructed in a purely displacement-based setting such that no mixed variational procedures are employed and full numerical integration of all quantities appearing in the virtual work statement is performed using high-order Gauss–Legendre quadrature rules. An efficient procedure for numerically integrating the discrete weak formulation through the shell thickness is implemented; hence no thin- or shallow-shell type restrictions are imposed on the element. For the case of laminated composites, we introduce a discrete tangent vector field, defined on the approximate shell mid-surface, that permits the use of skewed and/or arbitrarily curved elements in the numerical simulation of complex shell structures. To reduce computer memory requirements in the numerical implementation we adopt element-level static condensation, wherein, the interior degrees of freedom of each element are implicitly eliminated prior to assembly of the global sparse finite element coefficient matrix; an approach that results in storage requirements that are on par with traditional low-order finite element implementations. The accuracy and overall robustness of the developed shell element are illustrated through the solution of several nontrivial benchmark problems taken from the literature. These numerical studies provide evidence that the proposed shell element may be used to obtain accurate locking-free results and that large load increments can be employed (in the numerical simulation of shell structures undergoing very large deformations).