195 Introduction to Contact Problems 15.1 INTRODUCTION: THE GAP In many practical problems, the information required to develop a finite-element model, for example, the geometry of a member and the properties of its constituent materials, can be determined with little uncertainty or ambiguity. However, often the loads experienced by the member are not so clear. This is especially true if loads are transmitted to the member along an interface with a second member. This class of problems is called contact problems, and they are arguably the most common boundary conditions encountered in practical problems. The finite-element commu- nity has devoted, and continues to devote, a great deal of effort to this complex problem, leading to gap and interface elements for contact. Here, we introduce gap elements. First, consider the three-spring configuration in Figure 15.1. All springs are of stiffness k . Springs A and C extend from the top plate, called the contactor , to the bottom plate, called the target . The bottom of spring B is initially remote from the target by a gap g . The exact stiffness of this configuration is (15.1) From the viewpoint of the finite-element method, Figure 15.1 poses the following difficulty. If a node is set at the lowest point on spring B and at the point directly FIGURE 15.1 Simple contact problem. 15 k kg kg c = < ≥      2 3 δ δ . contactor P ABC δ kk k g target 0749_Frame_C15 Page 195 Wednesday, February 19, 2003 5:20 PM © 2003 by CRC CRC Press LLC 196 Finite Element Analysis: Thermomechanics of Solids below it on the target, these nodes are not initially connected, but are later connected in the physical problem. Furthermore, it is necessary to satisfy the nonpenetration constraint whereby the middle spring does not move through the target. If the nodes are considered unconnected in the finite-element model, there is nothing to enforce the nonpenetration constraint. If, however, the nodes are considered connected, the stiffness is artificially high. This difficulty is overcome in an approximate sense by a bilinear contact element. In particular, we introduce a new spring, k g , as shown in Figure 15.2. The stiffness of the middle spring ( B in series with the contact spring) is now denoted as k m , and (15.2) It is desirable for the middle spring to be soft when the gap is open ( g > δ ) and to be stiff when the gap is closed ( g ≤ δ ): (15.3) Elementary algebra serves to demonstrate that (15.4) Consequently, the model with the contact is too stiff by 0.5% when the gap is open, and too soft by 0.33% when the gap is closed (contact). One conclusion that can be drawn from this example is that the stiffness of the gap element should be related to the stiffnesses of the contactor and target in the vicinity of the contact point. FIGURE 15.2 Spring representing contact element. contactor target P AB C δ kk k k g k kk m g =+         − 11 1 . k kg kg g = > ≤      /100 δ δ 100 . k kkg kkg c ≈ +> +≤      2001 2099 . δ δ 0749_Frame_C15 Page 196 Wednesday, February 19, 2003 5:20 PM © 2003 by CRC CRC Press LLC Introduction to Contact Problems 197 15.2 POINT-TO-POINT CONTACT Generally, it is not known what points will come into contact, and there is no guarantee that target nodes will come into contact with foundation nodes. The gap elements can be used to account for the unknown contact area, as follows. Figure 15.3 shows a contactor and a target, on which are indicated candidate contact areas, d S c and d S t , containing nodes c1, c2,… ,cn, t1, t2,… ,tn. The candidate contact areas must contain all points for which there is a possibility of establishing contact. The gap (i.e., the distance in the undeformed configuration) from the i th node of the contactor to the j th node of the target is denoted by g ij . (For the purpose of this discussion, the gap is constant, i.e., not updated.) In point-to-point contact, the i th node on the contactor is connected to each node of the target by a spring with a bilinear stiffness. (Clearly, this element may miss the edge of the contact zone when it does not occur at a node.) It follows that each node of the target is connected by a spring to each of the nodes on the contactor. The angle between the spring and the normal at the contactor node is α ij , while the angle between the spring and the normal to the target is α ji . Under load, the i th contactor node experiences displacement u ij in the direction of the j th target node, and the j th target node experiences displace- ment u ji . For example, the spring connecting the i th contactor node with the j th target node has stiffness k ij , given by (15.5) in which δ ij = u ij + u ji is the relative displacement. The force in the spring connecting the i th contactor node and j th target node is f ij = k ij ( g ij ) δ ij . The total normal force experienced by the i th contactor node is f i = ∑ j f ij cos( α ij ). FIGURE 15.3 Point-to-point contact. k kg kg ij ijlower ij ij ijupper ij ij = < ≥      δ δ , candidate contactor contact surface candidate target contact surface dS c dS t c1 t1 t2 t3 c2 c3 α 31 k(g 31 ) 0749_Frame_C15 Page 197 Wednesday, February 19, 2003 5:20 PM © 2003 by CRC CRC Press LLC 198 Finite Element Analysis: Thermomechanics of Solids As an example of how the spring stiffness might depend upon the gap, consider the function (15.6) where γ , α , and ε are positive parameters selected as follows. When g ij − δ ij − γ > 0, k ij attains the lower-shelf value, k 0 ε , and we assume that ε << 1. If g ij − δ ij − γ < 0, k ij approaches the upper-shelf value, k 0 (1 − ε ). We choose γ to be a small value to attain a narrow transition range from the lower- to the upper-shelf values. In the range 0 < g ij − δ ij < γ , there is a rapid but continuous transition from the lower-shelf (soft) value to the upper-shelf (stiff) value. If we now choose α such that αγ = 1, k ij becomes k 0 /2, when the gap closes ( g ij = δ ij ). The spring characteristic is illustrated in Figure 15.4. The total normal force on a contactor node is the sum of the individual contact- element forces, namely (15.7) Clearly, significant forces are exerted only by the contact elements that are “closed.” FIGURE 15.4 Illustration of a gap-stiffness function. kg kgg ij ij ij ij ij ij ij () ()tan()(), − =+− −−−−−                 − δ εε π α δγ δγ 0 12 12 2 2 fkg tj ij ij ij ij ij i N c =− ∑ ( ) cos( ). δδ α k ij k 0 /2 k 0 ε k 0 ε k 0 δ ij –g ij 0749_Frame_C15 Page 198 Wednesday, February 19, 2003 5:20 PM © 2003 by CRC CRC Press LLC Introduction to Contact Problems 199 15.3 POINT-TO-SURFACE CONTACT We now briefly consider point-to-surface contact, illustrated in Figure 15.5 using a triangular element. Here, target node t3 is connected via a triangular element to contactor nodes c1 and c2. The stiffness matrix of the element is written as k([g 1 − δ 1 ], [g 2 − δ 2 ]) , in which g 1 − δ 1 is the gap between nodes t1 and c1, and is the geometric part of the stiffness matrix of a triangular elastic element. The stiffness matrix of the element can be made a function of both gaps. Total force normal to the target node is the sum of the forces exerted by the contact elements on the candidate contactor nodes. In some finite-element codes, the foregoing scheme is used to approximate the tangential force in the case of friction. Namely, an “elastic-friction” force is assumed in which the tangential tractions are assumed proportional to the normal traction through a friction coefficient. This model does not appear to consider sliding and can be considered a bonded contact. Advanced models address sliding contact and incorporate friction laws not based on the Coulomb model. 15.4 EXERCISES 1. Consider a finite-element model for a set of springs, illustrated in the following figure. A load moves the plate on the left toward the fixed plate on the right. What is the load-deflection curve of the configuration? For a finite-element model, an additional bilinear spring is supplied, as shown. What is the load-deflection curve of the finite-element model? Identify a k g value for which the load-deflection behavior of the finite- element model is close to the actual configuration. Why is the new spring needed in the finite-element model? FIGURE 15.5 Element for point-to-surface contact. candidate target contact surface candidate contactor contact surface element connecting node t3 with nodes c1 and c2 dS t dS c t2 t1 c1 c2 c3 t3 ˆ K ˆ K 0749_Frame_C15 Page 199 Wednesday, February 19, 2003 5:20 PM © 2003 by CRC CRC Press LLC 200 Finite Element Analysis: Thermomechanics of Solids 2. Suppose a contact element is added in the previous problem, in which the stiffness (spring rate) satisfies Suppose αγ = 1, k L = k/100, and k u = 100k. Compute the stiffness k for the configuration as a function of the deflection δ . F H L k k k k k kg kgg ij ij ij ij ij ij () ()tan()(). − =+− −−−−−                 − δ εε π α δγ δγ 0 12 12 2 2 0749_Frame_C15 Page 200 Wednesday, February 19, 2003 5:20 PM © 2003 by CRC CRC Press LLC . the load-deflection behavior of the finite- element model is close to the actual configuration. Why is the new spring needed in the finite -element model? FIGURE 15. 5 Element for point-to-surface. the geometric part of the stiffness matrix of a triangular elastic element. The stiffness matrix of the element can be made a function of both gaps. Total force normal to the target node is the sum of the. right. What is the load-deflection curve of the configuration? For a finite -element model, an additional bilinear spring is supplied, as shown. What is the load-deflection curve of the finite -element model?