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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。
The contact problem is a key mechanical problem in the research of rock and other non-continuous media. Based on the actual physical meaning of the three-dimensional contact problem, an equivalent complementary model is established in the normal and tangential directions respectively. Aiming at the strong nonlinearity of the complementary model, a new smooth approximation function is proposed. When the parameters in the function converge to 0 ~ +, it is equivalent to the original complementary model. Since the approximation function has C ~ 1 continuity, the corresponding Jacobian matrix is non-singular under any conditions, which makes the conventional Newton’s method and the Newton’s algorithm successfully solved. At the same time, by introducing the direction vector, the constraint function method proposed by the existing research in the two-dimensional frictional contact problem is generalized to three-dimensional, which solves the problem of the stability of solution in the three-dimensional contact problem due to the periodicity of the directional angle. On this basis, a three-dimensional finite element model of point contact is established, and the validity and adaptability of the method are verified by classical numerical examples.