由于功能梯度材料应用的广泛性，对功能梯度材料中裂纹的产生及其动态裂纹扩展的研究便显得尤其重要。本项目主要通过拓展静态平面弹性力学Muskhelishvil的解析函数边值问题方法，研究功能梯度材料线性弹性小变形情况下的简单裂纹的快速传播和止裂的动力学。首先假定裂纹以拟静态传播(在瞬时近似满足静态平衡)，建立功能梯度材料直线段裂纹传播的动力学模型，其基本方程是具有非线性边值条件的线性波动方程。然后利用发展的复变方法，把波动方程转化为以裂纹为边界的解析函数边值问题，分析决定解析边值问题解的存在性和解的结构的指标，从而研究裂纹稳定传播时边值问题的解的性态，探讨快速扩展裂纹顶端附近的应力场和应变场，给出裂纹是否止裂、分叉的判据，最后通过对等价的奇异积分方程进行数值计算，对裂纹传播与止裂的动力学进行定量分析。研究结果为功能梯度材料与结构的设计、裂纹检测与材料修复等提供有价值的理论指导。

Because of the wide application of functionally graded materials, it is particularly important to study the generation of cracks and the. dynamic crack propagation in functionally graded materials. In this project, we mainly study the dynamics of the propagation and arrest of simple cracks in functionally graded materials under the condition of linear elastic small deformation of functionally graded materials by extending the analytic function boundary value method of Muskhelishvil in static plane elasticity. First, we assume that the crack propagates in a quasi-static way (at approximately instantaneous static equilibrium) and establish a dynamic model of crack propagation in the straight line section of functionally graded material, whose basic equation is a linear wave equation with nonlinear boundary conditions. Then we employ the extended complex variable method, put the wave equation into the crack boundary boundary value problem of analytic function, analyze the index for deciding the existence and structure of solutions in analytic boundary value problem, furthermore, study the state of solutions of boundary value problems in the case of stable crack propagation, explore the stress and strain field near the tip of crack with rapid expansion, present the crack arrest and bifurcations criterion, finally carry out numerical calculation to quantitative analysis of fracture dynamics through the equivalent singular integral equation. The results provide valuable theoretical guidance for the design of functionally graded materials and structures, crack detection and material repair.