Computation Of Coefficients Of Crack-Tip Stress Fields Using The Weak-form Quadrature Element Method

Engineering structures exhibit various cracks inevitably. How to effectively reduce fracture failure of the structure due to the unstable crack propagation has been a difficulty to engineers. One of the essential reasons for the unstable crack propagation is crack-tip stress singularity. Therefore, investigating the intensity and distribution of the crack-tip stress field plays an important role in analyzing the crack state. The crack-tip stress field can be described by the Williams eigenfunction expansion. The undetermined coefficients in the expansion are called the coefficients of the crack-tip stress field. In the past, the coefficient of the first singular term has been considered to be the most crucial parameter that governs the unstable crack propagation. However, recent studies show that the coefficients of the higher order terms can also have nonnegligible effects on the crack propagation. To this end, how to efficiently and accurately calculate the coefficients of the crack-tip stress field, especially the higher order coefficients, becomes the major objective of this thesis. Solving this problem well will provide the necessary foundation for further analysis of the unstable crack extension.In order to obtain the coefficients of the crack-tip stress field directly, the cracked region is divided into two subregions, namely the complementary energy region and the potential energy region, first. The complementary energy region is a circular region with its center at the crack-tip, and its field variables are stresses; the potential energy region is the rest, and its field variables are displacements. Then according to the subregion generalized variational principle, the mixed energy functional of the whole region can be established. The integration and differentiation in the energy functional can be discretized and approximated using the weak-form quadrature element method. Based on the stationary condition of the variational principle, a system of algebraic equations containing the coefficients of the crack-tip stress field can be derived. Solving the equations finally, the coefficient of any order can be obtained directly.To show the characteristics of the proposed method, the coefficients of the crack-tip stress field for various conditions, including mode I, mode II, mixed mode I/II and mode III straight cracks or kink cracks, has been calculated in the thesis. The sensitivity of the involved parameters, such as the radius of the complementary energy region, the truncated number of the expansion terms, and the number of the quadrature nodes in a quadrature element, has been investigated. Proper values for the parameters have been recommended. As can be seen from the results, the proposed method can not only calculate the coefficients of the crack-tip stress field conveniently and directly, but also improve the formulation simplicity and the computation accuracy in comparison with other existing methods.