巴西圆盘中心裂纹摩擦接触问题的逐点Lagrange乘子法

A POINT-BY-POINT LAGRANGE MULTIPLIER METHOD FOR MODELING BRAZILIAN DISK CENTRAL CRACK WITH CONTACT AND FRICTION

采用逐点Lagrange乘子法求解巴西圆盘中心裂纹在压剪荷载作用下裂纹面可能发生的摩擦接触问题。为了避免传统的Lagrange乘子法中总刚度阵求逆的困难,将Lagrange乘子逐点转到局部坐标系下,采用Gauss-Seidel迭代法求解法向和切向乘子,同时注意在求解的过程中对切向乘子约束修正,待所有点乘子求解完成后再变换到整体坐标系下迭代求解位移。与传统接触算法相比,该算法无需对总刚度阵求逆,降低了求解规模,提高了计算效率。通过该方法计算了巴西圆盘中心裂纹两种典型情况下的应力强度因子,计算结果与文献比较,吻合良好。考虑不同荷载角和裂纹长度对位移,应力强度因子和接触区的影响,并对不同摩擦系数下应力强度因子的影响进行了分析。结果表明:忽略裂纹接触摩擦作用,应力强度因子可能被高估。

A new algorithm-Point-by-Point Lagrange multiplier method（PLMM） is proposed in analysis of the contact and friction problem of the face of the center crack on cracked Brazilian disk specimens subjected to shear-compressive loading.To avoid the global stiffness inversion,the Lagrange multiplier is transformed to local coordinate system at first.Then the iterative solutions of normal and tangential multipliers are obtained by using Gauss-Seidel iteration method with Point-by-Point technique and more attention are paid to the constraint modification of the tangential multiplier.The point multipliers are transformed to global coordinate system to solve for displacements iteratively.The comparison between the proposed method and the ordinary contact algorithm shows that the PLMM,involving no search of the inverse global stiffness matrix,requires less computer storage and has high computational efficiency.The stress intensity factors（SIFs） of Brazilian disk central crack with contact and friction are calculated under the two typical cases using the proposed method,showing good agreement with the literatures.The effects of the load angle and the interface crack length on displacements,SIFs and contact region are investigated.Besides,the numerical results indicate that the friction has a significant influence on the SIFs and the SIFs may be overestimated without considering crack surface contact and friction.