In this paper, the fracture mechanics analysis in functionally graded materials and structures (FGMs) is presented. The elemental differential method, is extended to simulate the fracture behaviors of the functionally graded materials, in which the system of equations is established directly based on the equilibrium equations. The first and second order differentiations of the shape functions are utilized to interpolate the geometrical and physical variables within the isoparametric elements. A novel collocation strategy is introduced to construct the system of equations by the governing equations and the traction equilibrium equations according to the nodal distributions in the mesh grids of the structures. Furthermore, a mixed collocation element differential method is further proposed to handle the singular points in the computation domains such as the crack tips and structural corners. The weak-form formulations, such as the weighted residuals approach, are utilized to establish the system of equations for nodes within the domain of elements. Thus, the strong–weak form method can combine the superiorities of the standard finite element methods and the strong-form methods for the aspects of easily constructing shape functions and directly generating system of equations. Numerical examples about the stress intensity factors of the static and dynamic problems in functionally graded materials are presented to validate the proposed methods.

本文介绍了功能梯度材料和结构（FGM）的断​​裂力学分析。扩展了元素微分方法，以模拟功能梯度材料的断裂行为，其中直接根据平衡方程建立方程组。形状函数的一阶和二阶微分用于对等参元素内的几何和物理变量进行插值。引入了一种新颖的配置策略，根据结构网格网格中的节点分布，通过控制方程和牵引平衡方程构造方程组。此外，进一步提出了一种混合搭配元微分方法来处理计算域中的奇异点，如裂纹尖端和结构拐角。弱形式公式（例如加权残差法）用于建立元素域内节点的方程组。因此，在容易构造形状函数和直接生成方程组的方面，强弱形式方法可以将标准有限元方法和强形式方法的优点结合起来。给出了有关功能梯度材料中静态和动态问题的应力强度因子的数值示例，以验证所提出的方法。用于建立单元域内节点的方程组。因此，在容易构造形状函数和直接生成方程组方面，强弱形式方法可以将标准有限元方法和强形式方法的优点结合起来。给出了有关功能梯度材料中静态和动态问题的应力强度因子的数值示例，以验证所提出的方法。用于建立单元域内节点的方程组。因此，在容易构造形状函数和直接生成方程组的方面，强弱形式方法可以将标准有限元方法和强形式方法的优点结合起来。给出了有关功能梯度材料中静态和动态问题的应力强度因子的数值示例，以验证所提出的方法。