热冲击载荷下功能梯度材料动态断裂分析的无网格边界元法研究

Functionally graded materials (FGMs) often produce some cracks, when they work in a high temperature environment. If the crack growth under heat shock loadings, the material will invalid or damage. Therefore, it is very significant for study the dynamic fracture mechanics of functionally graded materials under heat shock loading. A new meshless boundary element method is proposed in this project, which is used to analysis the dynamic fracture mechanics of functionally graded materials under heat shock loading. The present method has obvious advantages, such as it is only discrete elements on the boundary, or it can solve the stress concentration problems effectively. In order to achieve this project, firstly we develop the theoretical model of the thermal shock problem of functionally graded materials, and use the fundamental solution of the approximate problem to establish the boundary integral equation by the weighted residual method. Secondly, the domain integrals will be transformed into boundary integrals using the radial integration method, and a pure boundary integral equation is produced. Finally , the stress intensity factor of crack tip is calculated by the crack opening displacements, according to the criterion of crack propagation to determine the crack propagation condition, the expansion direction and step length. The purpose of this project is to develop a numerical simulation technique for dynamic fracture analysis of functionally graded materials under thermal shock loading, and provide a scientific evidence for the safely use of functionally graded materials and also for the optimization design of the structures.

功能梯度材料在高温环境服役过程中经常会产生各种各样的裂纹，在热冲击载荷作用下裂纹扩展会导致材料失效甚至破坏，因此研究功能梯度材料在热冲击载荷下的断裂行为尤为重要。本项目提出利用无内部网格的边界元法分析功能梯度材料在热冲击载荷下的动态断裂力学问题，充分发挥边界元法只需在边界上划分网格以及在处理应力集中问题时的优势。首先，建立功能梯度材料热冲击问题的理论模型，利用其近似问题的基本解，通过加权余量法推导边界积分方程；其次，对边界积分方程中含有的域积分项，采用径向积分法将其转化为边界积分，形成纯边界积分方程；最后，通过裂纹张开位移计算裂纹尖端应力强度因子，再根据裂纹扩展准则确定裂纹的扩展条件、扩展方向和步长。本项目利用边界元法在断裂力学问题中的显著优点，拟开发适合热冲击载荷下功能梯度材料动态断裂分析的无内部网格边界元法数值模拟技术，为功能梯度材料的安全使用以及结构的优化设计提供科学依据。

功能梯度材料能有效地缓解热应力和残余应力，从而在航空航天和众多民用科技领域有广泛的应用，对功能梯度材料进行热应力分析和断裂分析可以为工程结构的优化设计和安全使用提供理论依据。本项目采用无内外网格的边界元法分析了功能梯度材料在热冲击载荷下的动态断裂力学问题，充分发挥了边界元法只需在边界上划分网格以及在处理应力集中问题时的优势。本项目在执行期间，首先利用近似问题的基本解，通过加权余量法推导了功能梯度材料热冲击问题的边界积分方程，并采用径向积分法将域积分转化为边界积分，为项目的实施奠定了基础；其次在计算近奇异积分时采用不等间隔单元子分技术，在保证计算精度的同时极大减少了近奇异积分的计算量，保证了所编制程序计算刚度矩阵时的计算效率；最后在求解微分方程组时，由于边界元法离散后形成的刚度矩阵为满阵，当方程组规模较大时需要耗费大量计算时间，因此开展了POD模型降阶法求解微分方程组，通过模型降阶法将成千上万个自由度的微分方程组降为只含几十个甚至几个未知量的微分方程组，极大提高了计算效率。本项目研究成果为功能梯度材料的热冲击问题提供了一种新的研究思路，同时也拓展了边界元法的应用范围。

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