Symplectic approach has emerged a popular tool in dealing with elasticity problems especially for those with stress singularities. However, anisotropic material problem under polar coordinate system is still a bottleneck. This paper presents a subfield method coupled with the symplectic approach to study the anisotropic material under antiplane shear deformation. Anisotropic material around wedge tip is considered to be consisted of many subfields with constant material properties which can be handled by the symplectic approach individually. In this way, approximate solutions of the stress and displacement can be obtained. Numerical examples show that the present method is very accurate and efficient for such wedge problems. Besides, this paper has extended the application of the symplectic approach and provides a new idea for wedge problems of anisotropic material.
Xiaofei Hu,1 Weian Yao,1, ) and Zhaoxiang Fang2 1)State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China 2)Navigation College, Wuhan University of Technology, Wuhan 430063, China
With the help of the coordinate transformation technique, the symplectic dual solv- ing system is developed for multi-material wedges under antiplane deformation. A virtue of present method is that the compatibility conditions at interfaces of a multi-material wedge are expressed directly by the dual variables, therefore the governing equation of eigenvalue can be derived easily even with the increase of the material number. Then, stress singularity on multi-material wedges under antiplane deformation is investigated, and some solutions can be presented to show the validity of the method. Simultaneously, an interesting phenomenon is found and proved strictly that one of the singularities of a special five-material wedge is independent of the crack direction.
In the symplectic space composed of the original variables,displacements,and their dual variables,stresses,the symplectic solution for the composite laminates based on the Pipes-Pagano model is established in this paper.In contrast to the traditional technique using only one kind of variables,the symplectic dual variables include displacement components as well as stress components.Therefore,the compatibility conditions of displacement and stress at interfaces can be formulated simultaneously.After being introduced into the symplectic dual system,the uniform schemes,such as the separation of variables and symplectic eigenfunction expansion method,can be implemented conveniently to analyze composite laminate problems.An analytical solution for the free edge effect of composite laminates is obtained,showing the effectiveness of the symplectic dual method in analyzing composite laminates.
