Completely hierarchical p-version axisymmetric shell element for nonlinear heat conduction in laminated composites

Abstract

This paper presents a completely hierarchical p-version finite element formulation for an axisymmetric shell element for nonlinear conduction in laminated composites where the element temperature field can be of arbitrary polynomial orders p ξ and P η in the longitudinal (ξ) and the transverse (η) directions of the shell element. The approximation functions and the corresponding nodal variables for the axisymmetric shell element are derived by first, constructing the one-dimensional hierarchical approximation functions of orders p ξ and p η and the corresponding hierarchical nodal variable operators for each of the two directions ξ and η and then taking their product. This procedure gives the approximation functions and the nodal variables for the axisymmetric shell element that correspond to the polynomial orders p ξ and p η . The element approximation functions as well as the nodal variables are hierarchical and therefore the element matrices and the vectors corresponding to orders p ξ and p η are a subset of those corresponding to the polynomial orders ( p ξ + 1) and ( p η + 1). The formulation ensures C 0 continuity across the interelement boundaries. A weak formulation of the Fourier heat conduction equation with temperature dependent thermal conductivities, internal heat generation, film coefficients and radiation effects for globally orthotropic material is constructed. The element properties are derived using the weak formulation and the hierarchical element approximation. This formulation is extended for generally orthotropic material behavior where the material directions are not necessarily parallel to the global axes. Further extension of the formulation for laminated composites is accomplished by incorporating the material properties of each layer through numerical integration of the element matrix for each layer. The element matrices and the equivalent heat vectors (due to convection, distributed heat flux, radiation and internal heat generation) are all hierarchical. The formulation permits any desired order temperature distribution in the shell thickness direction without remodelling. There is no restriction on the number of layers and the lay up pattern of the layers. Each layer can be generally orthotropic. The material directions and the layer thicknesses may vary from point to point within each layer. Because of the temperature dependence of thermal conductivities, internal heat generation, film coefficients and radiation boundaries; the resulting discretized equations of equilibrium are nonlinear. These equations are solved using modified Newton-Raphson method. The tangent matrix is derived using the nonlinear equations of equilibrium. The modified tangent matrix is symmetric and provides excellent convergence during the equilibrium iteration process. Numerical examples are presented to demonstrate the accuracy, efficiency and overall superiority of the present formulation and the extremely good and fast convergence of the iteration process. Comparisons are made with available results in the literature.