基于哈密尔顿体系的裂纹尖端应力奇性分析及计量

对弹性平面扇形域问题，将径向坐标模拟成时间坐标，通过适当的变换，将扇形域问题导向哈密尔顿体系，利用分离变量法及本征函数向量展开等方法，推导出裂纹尖端的应力奇性解的计算公式，结合变分原理，提出一种解决应力奇性计算的断裂分析元，将此分析元与有限元法相结合，可以进行某些断裂力学或复合材料等应力奇性问题的计算及分析，数值计算结果表明，该方法具有精度高，使用十分方便，灵活等优点，是哈密尔顿体系和辛数学优越性的一次具体体现。

Analysis and Calculation of Stress Singularity at Crack Tipof Hamiltonian System

The radial coordinate is simulated as the time coordinate, the governing equations of plane elasticity in sec -torial domain are transformed into Hamiltonian form via variable substitutes and variational principles. The method of separation of variables and the method of eigenfunction expansion are used to derive the calculation equations of stress singularities at crack tip. A new analytical element is presented to solve the stress singularities problems, and this kind of analytical element can be installed into FEM program systems, then some crack mechanics problems and composite stress singularity problems can be calculated by this element. The numerical results obtained through computer program demonstrate the correctness of the method proposed. It demonstrates that the Hamiltonian system theory and symplectic mathematics are in an advantageous position.