弹塑性杆非线性断裂问题的新型增强有限元法

A NOVEL AUGMENTED FINITE ELEMENT METHOD FOR NONLINEAR FRACTURE IN ELASTOPLASTIC BARS

针对一维弹塑性材料杆件的非线性断裂,该文提出了一种新型弹塑性增强有限元。该单元采用von Mises屈服准则和线性等向硬化模型描述开裂前的材料弹塑性变形,而结合利用内聚力关系来描述随后的裂纹萌生和非线性断裂过程。引入内部节点来描述单元内由于裂纹引起的位移不连续,通过单元凝聚获得含裂纹单元的刚度矩阵,并对数值稳定性问题进行了分析。通过与基于材料力学方法推导得到的解析解的对比验证了该新型弹塑性增强有限单元法在列式上的正确性和在数值上的高效性与精确性。

This paper presents a 1D elastoplastic augmented finite element method （A-FEM） that can deal with the nonlinear fracture in elastoplastic bars with significant plastic deformation. The new element employed the von Mises yield criterion and the linear isotropic hardening model for the pre-cracking elastoplastic deformation, and a cohesive law to account for the ensuing crack initiation and growth. Internal nodes were introduced to accommodate the discontinuous displacement field due to cohesive fracture but their degrees of freedom （DoFs） were eliminated via an efficient condensation procedure in each element. A mathematically exact element stiffness matrix in the piece-wise linear sense was thus derived, without any additional DoFs. An analytical elastoplastic solution based on the strength-of-material method has also been developed and employed to check the numerical efficiency and accuracy of the 1D elastoplastic A-FEM. Several numerical examples were conducted to demonstrate the correctness, efficiency and accuracy of the proposed elastoplastic A-FEM.