p-Approximation axisymmetric shell elements for heat conduction in laminated composites

Abstract An axisymmetric shell finite element formulation for heat conduction in laminated composites is presented, where the temperature approximation in the direction of the shell thickness can be of an arbitrary polynomial order p . This is accomplished by introducing additional nodal variables in the element temperature approximation corresponding to the Lagrange interpolating polynomials in the shell thickness direction. The resulting temperature approximation consists of nodal approximation functions and nodal temperatures as well as derivatives of the nodal temperatures in the shell thickness direction. This temperature approximation has an important hierarchical property, i.e. the approximation functions and the nodal variables corresponding to an approximation order p are a subset of those corresponding to order p + 1. The element formulation ensures C 0 continuity or smoothness of temperature across inter-element boundaries. Weak formulation of the Fourier heat conduction equation for globally orthotropic material is constructed. The element properties are derived using the weak formulation (or the associated quadratic functional) 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 and internal heat generation) are all hierarchical. The formulation permits any desired order temperature distribution in the shell thickness direction without remodeling. 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. The geometry of the laminated shell is described by the element nodes at the middle surface of the element and the lamina thicknesses. Numerical examples are presented to demonstrate the accuracy, efficiency and overall superiority of the present formulation. The results are also compared with the corresponding h -approximation models using axisymmetric solid elements.