This paper presents the application of the Galerkin Finite Block Method (GFBM) to address cracked solids associated with Functionally Graded Materials (FGMs), leveraging the foundational principles of the Galerkin method. The equilibrium equations pertinent to FGMs are articulated in their weak form. Employing Chebyshev polynomials as shape functions, the GFBM integrates mapping techniques to accommodate irregular finite or semi-infinite physical domains. Boundary and continuity conditions are enforced through the Lagrange Multiplier Method. The domain integrals are calculated through either analytical or numerical integration. The proposed method can easily solve the edge crack or diagonal crack problem by using the crack opening displacement method. The accuracy and convergence of the proposed method are illustrated through a selection of numerical examples. The obtained numerical solutions are verified with analytical solutions and the results from the Finite Element Method.

中文翻译：

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功能梯度材料中裂纹固体的伽辽金有限块法和拉格朗日乘子法

本文介绍了伽辽金有限块方法 (GFBM) 的应用，利用伽辽金方法的基本原理来解决与功能梯度材料 (FGM) 相关的裂纹固体问题。与 FGM 相关的平衡方程以其弱形式阐明。 GFBM 采用切比雪夫多项式作为形状函数，集成了映射技术以适应不规则的有限或半无限物理域。边界和连续性条件通过拉格朗日乘子法强制执行。域积分通过解析积分或数值积分来计算。该方法可以利用裂纹张开位移法轻松解决边缘裂纹或斜角裂纹问题。通过一些数值例子说明了所提出方法的准确性和收敛性。所获得的数值解通过解析解和有限元法的结果进行了验证。