Study Of Semi-analytic Method Of Solving Functionally Graded Laminated Plated And Stress Singularities Of Elastoplastic V-notches

At present, there exist the difficulties in the mechanical analysis of functionally graded material (FGM) structures. In this thesis, a semi-analytic method for the bending and vibration analysis of orthotropic FGM rectangular plates with arbitrary thickness and material parameters being variable in the thickness direction was established. On the other hand, the plastic stress singularities of anti-plane V-notches in hardening materials and singularities of three-dimensional (3D) electromechanical fields of V-notched structures in piezoelectric materials were evaluated by the use of the interpolating matrix method. The main work and contribution of the thesis are given as follows:1 Based on 3D linear elastic theory and double trigonometric series expansion, a semi-analytical method for the bending analysis of the orthotropic FGM rectangular plates subjected to the simple supports along the four edges was established by the combination of the interpolating matrix method and the state space equations. Hence, the semi-analytic solutions of 3D displacement and stress components of FGM laminated plates with arbitrary thickness and material parameters being variable in the thickness direction were obtained. The present results are more accurate than ones obtained by the well-known state space method in the case of taking the same trigonometric series terms. Naturally, the semi-analytic method was successfully used to solve the bending problem of the isotropic plate, but the state space method is still difficult although it is a simpler problem. Here, the results of the finite element analysis for the FGM laminated structures have the considerable errors since there exist the stress singularities at the interface ends.2 With the displacement components as the basic variables, the governing equations for free vibration analysis of FGM rectangular thick plates with simple supports along the four edges were established, which are two-order ordinary differential equations with variable coefficients. Then the interpolating matrix method was adopted to solve the equations so that the semi-analytic solutions of the natural frequencies and associated eigenfunctions of the FGM laminated plates were obtained. The semi-analytic method is suitable for free vibration analysis of FGM rectangular plates with arbitrary thickness and material parameters being variable in the thickness direction. The results show that the method has the advantages of high accuracy, less computation, easy to use, etc.3 Based on the asymptotic extension of the displacement field near the notched tip, the governing equations for evaluating the stress singularities of the V-notches and cracks in the orthotropic Reissner’s plates and laminated structures were established. Then the interpolating matrix method is used to solve the derivative equations. Consequently, the leading stress singularity orders and the associated eigenfunctions of the V-notches and cracks were simultaneously obtained. The numerical results of the examples showed that the present method is more accurate and efficient than the published methods for analyzing the notches and cracks. In fact, it has not yet been found that the stress singularities of the V-notches in the orthotropic laminated structures are studied.4 Based on the asymptotic extension of the displacement field near the notched tip, the nonlinear governing equations for evaluation of plastic stress singularities of anti-plane V-notches in hardening materials were established. Then the interpolating matrix method was used to solve the nonlinear eigenvalue problems by an iteration process. Hence, the leading plastic stress singularity orders and the associated eigenfunctions of anti-plane V-notches and cracks had been obtained. The numerical examples demonstrated that the present method has high computed precision. It has sufficiently eliminated the difficulty in the evaluation of plastic stress singularities of anti-plane V-notches in hardening material’s.5 A new way was proposed to evaluate the singularity at the vertex of 3D column-shaped V-notch encountered in piezoelectric structures. Based on the asymptotic assumption for the physical field near the notch tip, the singular functions of 3D electromechanical field near the V-notches are expressed by the eigenfunction expansion approach. The characteristic differential equations with the singular parameters of the notches were built from 3D linear elastic theory and Maxell equations. By applying the interpolating matrix method to solve the differential equations, the leading singular orders near the notch tip were achieved. The associated eigenvectors of the displacement and stress fields in the notch tip region have been determined with the same degree of accuracy, which is advantageous in analyzing the electro-mechanical coupling singular field by using the first-order derivatives of displacements and electric potential functions.