Abstract：A new extended Hellinger-Reissner variational principle for non-homogeneous material has been developed which provides a convenient procedure for deriving the stiffness matrix of a non-homogeneous element that can be subdivided into regions of different material properties. In such case some of stress components along the interface and the displacement across the interface may become discontinuous. This formulation can also be used for thick laminated plates in which transverse shear stress of each layer is independent. One kind of new assumed stress hybrid multilayer element with a traction-free cylindrical surface is derived by the use of the extended principle. The stresses of each layer are interpolated by the natural coordinates and are obtained by the use of internal displacement as weight function to impose the equilibrium conditions in a variational sense. The stresses also satisfy the traction-free conditions over the cylindrical surface exactly. The continuous conditions of the displacement between the layers and the elements are relaxed by using Lagrange multipliers respectively. Numerical results show that the present elements are much more efficiency than the ordinary assumed stress hybrid elements and the conventional assumed displacement elements for analyzing stress distribution around difference types of cylindrical notches in thin to thick laminated composites when very coarse meshes are used.
2015, Vol. 36 
