Analysis Of The Stress Singularity Of Plane V-notches In Bonded Dissimilar Materials

The cases of V-notches of bonded dissimilar materials are frequently encountered in engineering applications. According to the linear theory of elasticity, there are several different stress singularities at a V-notch tip. The singularity reflects a stress concentration near the sharp V-notches. In this paper, a new way is proposed to determine the stress singularities for plane V-notch problems. Firstly, the asymptotic displacement field in the V-notch tip region is expressed as a series expansion with respect to the radial coordinate from the tip. On the basis of the strain-displacement relations of linear elastic theory, we can obtain the strain components. Then, from Hooke's law of plane problems, the plane stresses are expressed by the asymptotic displacement functions. Hence, the governing equations of the elastic theory are transformed into nonlinear eigenanalysis of ordinary differential equations (ODEs) with respect to the circumferential coordinate 9 around the V-notch tip. An alternative approach is adopted to transform the above equations into a linear eigenvalue problem of ODEs by introducing two new field variables. Considering the constraint conditions and stress status on the edges and interfaces near the V-notch tip, we can gain the boundary conditions corresponding to the ODEs from the plane V-notches in bonded dissimilar materials. At last, the evaluation of stress singularities near a V-notch tip is transformed to solving a linear eigenvalue problem of the ODEs.In this paper, the interpolating matrix method, which is a numerical method that had early been proposed to solve two point boundary value problems (BVPs) of ODEs, is further developed to treat the derived eigenvalue problem of ODEs from the V-notch problems. Hence, the singularity orders of the V-notch problem are determined through solving the corresponding ODEs by means of the interpolating matrix method. Meanwhile, the associated eigenvectors of the displacement and stress fields near the V-notches are also obtained. These functions are essential in calculating the displacement field, stress field and the generalized stress intensity factors of V-notches. The present method is used to solve the plane V-notch problems in bonded orthotropic multi-materials. The numerical results show that the new method is an effective and accurate tool for dealing with singularity orders of plane V-notches.