基于比例边界有限元法的磁电弹性材料的静态及动态断裂分析

Magnetoelectroelastic material is a novel smart material and possesses the capability of converting energy among mechanical,electric and magnetic fields. Due to this attractive feature, the magnetoelectroelastic materials have found increasingly applications in construction of sensors, transducers, actuators, memory and recording devices, etc. However, the magnetoelectroelastic materials are brittle inherently and they have a tendency to develop cracks even in manufacturing process.Therefore, it is of great importance to study the fracture behaviour of the magnetoelectroelastic materials.The finite element method (FEM) is one of the most frequently used numerical methods for crack problems. However, it is of low efficiency because high mesh resolution is required around the crack tip even when singular elements are used. The scaled boundary finite element method is a recently developed semi-analytical numerical approach combining advantages of the finite element method and the boundary element method. Based on taking full advantages of the scaled boundary finite element method, this project aims at the establishment of an accurate and efficient numerical method for the analysis of cracked magnetoelectroelastic materials subjected to mechanical,electric and magnetic loading. The corresponding computational program of this method is also developed in this project.One of the most important advantages of this method is that it allows accurate intensity factors of various physical quantities can be determined directly from their definition,and hence no special crack-tip treatments, such as refining the crack-tip mesh or using singular elements,are necessary. The completeness of this project has important theoretical significance and wide application prospect.

磁电弹性材料是一种新型的智能材料。这类材料能够实现机械能、电能和磁能之间的相互转换。这种特性使其越来越广泛地应用于制作传感器、换能器、制动器、记忆与记录装置等电子产品。然而，由于固有的脆性，磁电弹性材料甚至在制作的过程中就存在裂纹。因此，对磁电弹性材料的断裂行为进行研究尤为重要。目前多采用有限元法对其进行研究，然而这种方法在裂尖区即使用奇异单元也需要很大的网格密度，求解效率低。比例边界有限元法是一种新型的半解析数值方法，集合了有限元法和边界元法的优点。本项目拟在充分发挥比例边界有限元法优点的基础上，针对力、电、磁耦合荷载作用下的磁电弹性材料断裂问题，构建一种精确、高效的数值方法，并编制相应的计算程序。这种方法具有明显的优势，即裂纹尖端各物理量的强度因子可以直接根据其定义推导出来，不需要使用有限元法和边界元法必须的网格加密或特殊的奇异单元。该项目的完成具有重要的理论意义和广泛的应用前景。

磁电弹材料是一种具有磁场、电场与弹性场耦合性质的新型复合材料，可用于制作传感器和驱动器等智能元件。然而，由于其固有的脆性，磁电弹性材料在制作和使用过程中不可避免会产生裂纹或孔洞，从而导致缺陷附近的物理场奇异，影响结构的完整性，并可能引发结构功能失效。因此，研究磁电弹性材料的断裂问题，具有重要的理论和实际意义，是智能结构设计和评估的重要基础和前提。为了获得更为精确高效的磁电弹性材料裂纹分析的方法，基于改进的插值型移动最小二乘法，提出了磁电弹性材料静态和动态断裂分析的插值型无单元伽辽金方法，这种方法可以直接根据定义求得应力强度因子、电位移强度因子和磁感应强度因子。这种方法只需要在求解域的边界上采用无单元伽辽金法进行数值离散，减少了一个空间维数，并且不需要边界元法所需要的基本解。在没有离散的径向采用解析的方法进行求解，从而具有较高的计算精度。在改进的插值型移动最小二乘法中，不仅形函数满足Kronecker delta函数性质，而且权函数是非奇异的。此外，改进的插值型移动最小二乘法计算形函数时待定系数比传统的移动最小二乘法少一个。项目培养了多名青年教师，研究生等人才，发表了多篇学术论文，为国家和社会发展创造了良好的社会经济效益。

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