金属-陶瓷梯度材料热机械断裂力学模型研究的积分方程方法

The project mainly focuses on the research on the thermo-elastic fracture behavior of metal/ceramic functionally graded materials by using the new boundary conditions and the new method. Firstly, it is assumed that interface crack is partially insulated and the temperature drop across the crack surfaces is the result of the thermal resistant due to the heat conduction through the crack region. The new models of thermo-elastic fracture problem are studied by introducing the new crack surface contidions subject to the thermal / mechanical loading, dynamic thermal / mechanical loading or dynamic thermal / mechanical/ electrical loading according to the study of mechanics and mathematical theory. Secondly, considering the functionally graded coating as functionally graded piezoelectric / piezomagnetic materials, the thermo-elastic fracture behavior of metal/ceramic functionally graded materials are investigated with the shear modulus for the coordinates of an arbitrary linear function by introducing the magnetoelectrical permeability coefficient. Finally, in accordance with established theoretical models of thermo-elastic fracture, the static and dynamic fracture problem of the periodic cracks are reduced to solving a system of singular integral equation with the Hilbert kernel, which shows the inherent nature of the periodic problem. By using wavelet and periodic wavelet, the singular integral equations with the Hilbert kernel can be solved, which further improve the solution accuracy. This project would give some technical support for the design of high-quality functionally graded materials. In addition, we try to perfect the mathematical proof of convergence of generalized integral with parameters encountered in the solution process of mixed boundary value problems .

本项目主要侧重于新的边界条件、新方法在金属-陶瓷梯度材料热弹性断裂问题方面的研究。首先考虑在热/机械载荷、动态热/机械载荷、动态热/力/电载荷作用下，引入新的裂纹面边界条件，裂纹区域存在热阻，裂纹面部分绝热，依据力学和数学理论研究梯度材料新模型的热弹性断裂问题；其次考虑涂层为功能梯度压电/压磁材料，引入磁电渗透系数，研究剪切模量为坐标的任意线性函数时含有裂纹的金属-陶瓷梯度材料的热弹性断裂问题；最后按照建立的热弹性断裂理论模型，采用新的数学方法，将模型中周期裂纹断裂的静、动态问题转化为求解一组带Hilbert核的奇异积分方程组，展现周期问题的固有性质，利用小波和周期小波的方法数值求解这些奇异积分方程组，进一步提高求解的精度，从而对研究设计高品质的功能梯度材料给予一定的技术支持。另外，试图完善求解混合边值问题的过程中遇到含参数广义积分收敛性问题的数学证明。

本项目主要侧重于新的边界条件、新方法在金属-陶瓷梯度材料热弹性断裂问题方面的研究。首先考虑在热/机械载荷作用下, 引入新的裂纹面边界条件, 运用积分方程方法，利用积分变换技术，研究了功能梯度材料的热弹性断裂问题。其次，研究了周期裂纹、双周期裂纹断裂模型。将模型中周期裂纹断裂的静、动态问题转化为求解一组带 Hilbert 核的奇异积分方程组，并给出了数值求解方法。同时，研究了涂层为功能梯度压电/压磁材料的断裂和接触问题，研究了准晶材料的断裂和接触问题，研究了利用小波和周期小波数值求解奇异积分方程（组）问题。通过数值求解这些奇异积分方程（组）, 进一步提高了求解的精度, 从而对研究设计高品质的功能梯度复合材料给予一定的技术支持。本项目的成功实施, 加深了人们对功能梯度复合材料断裂力学的认识，为功能梯度材料断裂力学的深入研究奠定了基础。

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