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为得到用于分析奇异热流密度场的高效的有限元列式,针对不同材料中界面裂纹尖端的扇形区域,推导出二维热传导特征解问题的基本方程和边界条件的弱形式.利用特征方程展开方法,可获得分析裂纹尖端处二维热传导特征解的一维有限元列式.该列式只需对扇形区域在角度方向上离散,最后得到一个二次特征根矩阵的总体方程.求解该方程可得到二维热传导问题的特征解.数值计算表明,该方法可高效准确地求解奇异热流密度场特征解.
In order to obtain an efficient finite element formulation for analyzing the singular heat flux density field, the basic equations of the two-dimensional heat conduction characteristic solution and the weak form of the boundary conditions are derived for the fan-shaped area of the interface crack tip in different materials. Method, a one-dimensional finite element formulation for the analysis of the two-dimensional heat conduction characteristic solution at the crack tip can be obtained. This formula only needs to be discretized in the angular direction of the fan-shaped area and finally a general equation of the quadratic eigenvalue matrix is obtained. The characteristic solution of two-dimensional heat conduction problem can be obtained.The numerical results show that this method can efficiently and accurately solve the singular heat flux density field characteristic solution.