中文摘要：

接触问题是岩石等非连续介质研究中的关键力学问题。基于三维接触问题的实际物理意义,分别在法向和切向建立等价的互补模型。针对互补模型呈现出的强非线性性质,提出一个新的光滑逼近函数,当该函数中的参数趋于0^＋时,它等价于原来的互补模型。由于该逼近函数具有C^1连续,相应的Jacobian矩阵在任何条件下非奇异,这使得常规的Newton法及Newton族算法可以顺利地求解。同时,通过方向向量的引入,将已有研究在二维摩擦接触问题中所提出的约束函数法推广到三维,解决了三维接触问题中由于方向角的周期性带来的求解稳定性问题。在此基础上,建立三维点面接触有限元模型,并用经典算例验证该方法的有效性和适应性。

英文摘要：

The contact problem is one of the key mechanical problems in the discontinuous medium such as rock. Based on the physical meaning of 3D contact problem, the equivalent complementary models were established in normal and tangential directions, respectively. A new approximating smooth function was proposed for the nonlinear property of the complementary model. The approximating function is equivalent to the complementary model when the parameter tends to 0＋. Since the approximation function is C1 continuous, the corresponding Jacobian matrix is nonsingular under any condition which enables the successful solution for the conventional Newton algorithm. By introducing the direction vector, the constraint function method proposed in 2D frictional contact problems was extended to 3D ones. Hence the stability problem caused by the periodicity of the direction angle in 3D contact problems was resolved. Then, the 3D point-surface contact for finite element model was established. At last, the validity of the proposed method was verified with several classical cases.