功能梯度磁电弹材料切口强度边界元法研究

The notch propagation is the source of the fracture in functionally graded magneto-electro-elastic structures. Investigating the mechanical and physical reasons for the notch propagation is an important subject for improving the mechanical property for the notched structures. A new method, in which the singularity characteristic analysis and the boundary element method are combined to analyze the strength of the notched structures, is to be proposed in this proposal. The singularity characteristic analysis can describe the singularity physical fields at the notch tip properly. At the same time, the computing amount of the boundary element method is small. Firstly, the mathematical reason for the nearly singularity integrals in the magneto-electro-elastic boundary integral equations will be explored. The research on adopting the subtraction method to realize the semi-analytical calculation for the nearly singular integrals will be executed. The boundary element method based on the state space method will be built for further decreasing the computational work. Then, the study will focus on the numerical solution for the differential equations with changing coefficients, which is established from the characteristic analysis for the functionally graded magneto-electro-elastic notch. The rule of the heterogeneous of the materials on the notch singularity will be considered. At last, the study will be concentrated on the combination technique for the asymptotic expansions at the notch tip district and the boundary integral equations established on the far notch tip region. So that, the complete physical fields at the notch tip sector, which including the singular components, the constant components and the non-singular components, could be efficiently calculated. The mechanism for decreasing the singularity and increasing the fracture mechanical property will be discussed. The results of the proposed project would provide the guide for improving the strength of the functionally graded magneto-electro-elastic notched structures.

切口开裂是功能梯度磁电弹材料构件断裂的起源，查清切口开裂的物理力学机制，提升构件抗裂能力是一重要的研究课题。项目提出联合奇性特征分析法和边界元法共同分析含切口构件的强度，充分发挥前者能有效描述奇异物理场和后者工程计算量小的优势。首先，找寻磁电弹材料边界积分方程中几乎奇异积分产生的数学原因，研究采用扣除法来实现对其半解析计算，发展基于状态空间法的边界元法减小分析功能梯度材料结构的计算量。其次，研制功能梯度磁电弹材料切口变系数奇性特征微分方程组的数值解法，分析材料非均匀性对切口奇性的影响规律。最后，研究描述切口尖端奇异场量的渐近展开式和模拟远场非奇异区域的边界积分方程的联合方案，实现对切口尖端包括奇异量、常量和高阶非奇异量的完整物理场的有效计算。探索各种奇异场量和非奇异场量对切口强度的影响规律，研究相应的降奇和增韧机制，为提高功能梯度磁电弹材料切口强度提供理论指导。

功能梯度磁电弹材料切口尖端会产生奇异的力、电、磁场，严重的奇异物理场会使得结构断裂或被电荷击穿。项目围绕打造功能梯度磁电弹材料的数值分析工具、分析切口尖端的力电磁奇异场、尖端奇异场对切口断裂强度的影响等科学问题展开研究，完成了申请书的计划内容，研究成果如下：. 基于混合变分原理，将叠层功能梯度磁电弹复合材料平面方向的物理量采用传统的边界元法离散插值，而竖向的物理量采用精细积分法求解，利用界面连续条件将边界积分方程和状态空间方程耦合起来，建立了状态空间边界元法。. 针对薄层结构中边界积分方程将遇到的几乎奇异积分问题，分析了其产生的数学原因，将积分核剖分为奇异项和非奇异项，对非奇异部分直接运用高斯数值积分，对奇异部分反复运用分部积分运算导出了解析公式，从而实现了几乎奇异积分的半解析计算，使得状态空间边界元法可以有效分析薄层功能梯度磁电弹材料的各类物理场。.从应力平衡方程和电磁麦克斯韦方程组出发，基于Williams渐近展开，利用特征分析技巧，推导了压电材料、压电压磁材料、角度非均质材料、径向非均质材料切口的奇性特征微分方程组，再采用变量代换法，将非线性特征问题转化为线性特征问题，研制了求解变系数特征微分方程的插值矩阵解法，可以有效计算各类磁电弹材料切口的奇性指数和特征角函数。基于数值计算结果，研究了降低磁电弹材料切口奇异性的材料配比方案，为提高功能梯度磁电弹材料切口强度提供理论指导。. 将磁电弹材料切口尖端的奇性特征展开和状态空间边界元法联合，求解出了切口尖端物理场Williams渐近展开的展开系数，从而获得了磁电弹材料切口的完整奇异物理场，可以根据需要计算切口尖端的奇异物理场和高阶非奇异物理场。采用断裂力学准则，预测了含切口功能梯度磁电弹材料的结构强度。. 本项目研究结果，为提升功能梯度磁电弹材料构件抗裂能力提供借鉴。