弱形式求积元法计算线弹性和弹塑性应力强度因子研究

Recently with the development of fracture mechanics, problems to be solved have been more and more complicate and diversified. How to establish an accurate and efficient computational fracture mechanics method to solve the problems has been a hot research topic in this field. A weak form Quadrature Element Method was proposed recently. Its application to various classical continuum mechanics problems shows its high precision and high efficiency. The method is to be used in the present study. Weak form descriptions of a linear elastic crack and an elasto-plastic crack are to be formulated by a sub-region mixed energy principle. Integrals involved in the formulation are to be evaluated by an efficient numerical integration scheme. Partial derivatives at the integration points are to be approximated by differential quadrature analogs. A system of algebraic equations is eventually derived from the variational principle and stress intensity factors are extracted from solving the equations. The computed results are to be compared to available theoretical or numerical solutions to discuss the features of the method. The computation of stress intensity factors using the method not only provides a new approach for fracture mechanics research, but also lays out a foundation for investigations of more complex fracture mechanics problems.

近年来，随着断裂力学研究的日益深入，需要求解的问题日趋复杂化和多样化。如何建立高精度、高效率的计算断裂力学方法来解决这些问题已成为该领域的一大研究热点。弱形式求积元法是最近提出的一种新型数值计算方法。它已成功应用于许多典型连续介质力学问题，体现出高精度、高效率等特点。本项目拟应用该方法，通过分区混合能量原理建立线弹性和弹塑性裂纹的弱形式描述，对描述中的积分进行数值积分近似，再对各积分点上的微分采用微分求积法进行近似，从而利用变分原理将问题转化为代数方程组来求解得到应力强度因子。该方法的计算特点将通过和已有解析解或数值解结果相比较来进行讨论。该方法对应力强度因子的计算将不仅为断裂力学研究提供一种新方法，而且为其研究更复杂断裂力学问题奠定基础。

精确、高效地计算任意裂纹问题的应力强度因子一直是断裂力学领域的重点和难点。本研究应用新近提出的弱形式求积元法，对线弹性和弹塑性应力强度因子进行了计算。首先从分区广义变分原理出发，建立了裂纹尖端附近区域内余能、余下区域内势能、以及两者交界面上混合功的表达式；其次借助裂纹尖端附近的应力渐近场解来直接引入应力强度因子；再次应用弱形式求积元法对所建立能量表达中的被积分式进行Lagrange插值离散，插值离散节点选用Lobatto非均匀分布节点（包含插值域的两个端点），便于边界条件施加，且积分可利用Gauss-Lobatto积分近似；然后利用微分求积法对各离散点上的微分进行近似；最终根据分区广义变分原理的驻值条件，得到一系列包含待求解余能区裂纹尖端应力渐近场系数和势能区位移的代数方程组，求解该方程组，即可直接得到应力强度因子。通过多个经典I、II、III型及复合型裂纹问题的求解，验证了该方法的可行性；通过计算结果和已有参考文献结果的对比，表明该方法具有极高的精度和效率；对该方法中影响计算结果精度的参数，例如余能区的尺寸大小、插值离散节点个数等，进行了参数敏感性分析，给出了相应的推荐取值。本研究成功实现了该方法对非连续介质问题的首次分析，奠定了其在断裂力学领域中的应用基础，为接下来研究更复杂断裂力学问题开辟了道路。