RIKEN Review No. 30 (September, 2000): Focused on High Performance Computing in RIKEN 1999 A node-to-point contact element strategy and its applications Hui-Lin Xing ∗1 and Akitake Makinouchi ∗2 Materials Fabrication Laboratory, RIKEN Based on the characters of the explicit time integration algorithm, a reliable and efficient contact element strategy, named as node-to-point contact element strategy, is proposed and applied to handle the static or quasi-static multi- deformation-body contact with friction. The Coulomb friction model governs the friction behavior with an additional limit on the allowable shear stress, which is treated as a flow plasticity rule. The penalty method is adopted to impose the normal and the sticking contact. Finally, numerical examples of contact between finite deformation bodies are presented to show the efficiency and stability of this algorithm. 1. Introduction There is currently much interest in deformation analysis of multiple elasto-plastic bodies in contact. For details, see (Wriggers 1995). In the current FEM analysis, both the dynamic-explicit FEM and the static-implicit FEM are avail- able corresponding to the different problems. However, con- vergence is still a problem in implicit analysis, especially when three-dimensional large deformation contact problems with sliding friction are encountered. This is partly due to the iteration solution method and its corresponding serious requirement, such as no drastic change of the contact state and the deformation state, more smooth contact surface def- inition (Nagtegaal and Taylor 1991, Ling et al . 1997). Al- though many efforts have been made as above, there still exist problems to be overcome (Parisch 1997, Zhong 1993). Thus dynamic-explicit FEM seems to be used increasely, even for problems, which are characterized as static or quasi-static ones, but it is also well known that it is difficult for dynamic- explicit FEM to predict the stress distribution with a high accuracy. Therefore, a static-explicit method is employed in this paper to simulate static or quasi-static contact problems. Contact problems are characterized by contact constraints that are imposed on contacting boundaries. We propose an arbitrarily shaped isoparametric contact element with fric- tion to be applied in our code and introduce an explicit time integration algorithm to overcome the existing problem. Nu- merical examples of contact between multiple elastoplastic bodies in finite deformation are presented to show the effi- ciency and stability of this algorithm. 2. General consideration and notation 2.1 Kinematics Consider two bodies B 1 and B 2 with surfaces S 1 and S 2 , respectively, to contact on an interface Sc, and Sc = S 1 ∩ S 2 , S α c = Sc ∩ s α , where superscript α =1, 2 refers to body B α (as shown in Fig. 1). Let the union of the two bodies be de- noted by B : B = B 1 ∪B 2 , be the unit normal vector of the contact surface, be the unit tangential vector along the rel- *1 e-mail address: xing@postman.riken.go.jp *2 e-mail address: akitake@postman.riken.go.jp Fig. 1. Bodies in contact with each other. ative sliding direction on the contact surface, and = × . Thus and form a tangent plane to the contact surface. When contact occurs, the following conditions should be sat- isfied for unilateral contact gn =0, ˙ gn =˙ 1 · 1 +˙ 2 · 2 =0 on Sc, (1) α · α ≤ 0, 1 + 2 =0 on Sc, (2) where gn is the gap normal to the contact surface and α is the contact stress on S α c . The so-called slave-master concept is widely used for the im- plementation of contact analysis. Assume that one of the bodies, B 1 , is the slave and the material points on its con- tact surface are called slave nodes; and the other body B 2 is the master and the material points on its contact surface are called master nodes. Contact (master) segments that span master nodes cover the contact surface of the master body. Therefore, the above problem can be regarded as a contact between a slave node and a point of a master segment. And a slave node makes contact with only one segment, but one seg- ment can make contact with one or more slave nodes at the same time. This is the basic assumption of the node-to-point contact element strategy (Xing and Makinouchi, 1998). 2.2 Constitutive equation of contact 2.2.1 Normal contact stress We choose the penalty method to treat the normal con- 35