Transactions on Engineering Sciences vol 1, 1993 WIT Press, ISSN

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1 A contact algorithm for explicit dynamic FEM analysis E. Anderheggen", D. Ekchian", K. Heiduschke", P. Bartelt^ "Swiss Federal Institute of Technology, Honggerberg, CH-8093 Zurich, Switzerland &#2%2 AG, Corporal Aesearc^ FI-^P^ Schaan, Principality of Liechtenstein ABSTRACT A numerical contact algorithm which can be used with an explicit time integration scheme in a finite element code is introduced. Its effectiveness is demonstrated on low velocity impact and penetration problems containing many contact surfaces. The treatment so far is two-dimensional and without friction. INTRODUCTION The increasing interest in dynamic penetration and wave propagation phenomena has heightened the need for simulation tools capable of modelling multi-body impacts. Engineers who have analysed such problems, have traditionally relied on one-dimensional analytical solutions [1]. These are valid as long as the system is truly one-dimensional and remains elastic. For complex two- or three-dimensional systems in which plastic deformations and friction effects occur, the finite element method (FEM) is often used. In the finite element analysis of impact and wave propagation problems, an explicit time integration scheme is usually employed to solve the system of equations which governs the dynamic response of the system. The explicit algorithm is ideal for this kind of problem since the stable or critical time step of the algorithm is near the time step required to

2 272 Contact Mechanics accurately follow the propagation of the stress waves. Implicit methods which require a much greater computational effort for each time step would be unpractical for this application. (For a more detailed explanation of the difference between explicit and implicit methods the reader is referred to a standard finite element text, such as [2]. This will not be discussed here). The explicit integration scheme has become a popular and standard method available in most commercial finite element packages. However, the numerical treatment of the contact or impact phenomena is not as well understood. The purpose of this paper is to introduce a numerical contact algorithm which can be used in conjunction with an explicit integration scheme in a finite element code. The contact algorithm belongs to the class of socalled "nodal projections algorithms" or "simplified Lagrangian methods" [3]. At present the new algorithm is two-dimensional and without friction. The method is general, i.e. it handles node-to-side contact and is not restricted to node-to-node impact. It requires no mass distribution as in [5] and has been implemented without damping. The method is unsymmetric in that it requires the definition of a "master" and "slave" surface. (However, work is presently underway to symmetrize the algorithm.) The method is non iterative which makes it computationally stable. In the following section, the numerical algorithm will be presented. Afterwards, several example problems will be discussed. The algorithm will be applied to a problem in which a nail is driven by a flying piston into a steel substrate. This problem is especially demanding since the analysis contains eight contacting surfaces. Before discussing such a complex example, the method will be used in the dynamic stress analysis of a simple bar impact. In this problem, the influence that the algorithm has on the stable time step of the explicit time integration will be demonstrated. In addition, the energy (balance) in the system will be carefully observed and analysed. The paper will not treat topics associated with impact and wave propagation effects such as strain-rate dependent plastic material behaviour under finite deformations.

3 Contact Mechanics 273 THE NUMERICAL ALGORITHM The basic idea behind the algorithm (and all nodal projection algorithms) is to first let the two contacting finite element bodies penetrate into each other within a single time step. At the end of the time step, additional incremental nodal displacements (and velocities and accelerations) are introduced into the system such that the non-penetration conditions are strictly enforced. This can be schematically shown as follows: Slave Nodes Master Nodes Figure 1: Definition of variables. The master node k penetrates the slave surface between node i and j. In Figure 1, C% is the contact force at the k-th "master node". It acts perpendicular to the "slave-side" between nodes i and j. Of course, if the node does not penetrate, C^=0. At node i are the slave forces C^ and C^, which act in the x and y directions, respectively. During contact, the master and slave forces must be in equilibrium. Hence, (CJ =[E]{CJ (1) where {CJ is the vector of Q slave node forces with two entries C^ and Cyj per slave node and {C^} is the vector of C% master node forces in the direction perpendicular to the penetrating side. The matrix [E] with components Eft represents the i-th slave node force in the x- and y-directions due to = 1. The additional incremental displacements {U^} and {U^j within the time step are for the master and the slave nodes, respectively

4 274 Contact Mechanics Figure 2: Definition of the length factor a and angle <j). (3) Assemble the narrowly banded matrix AtM[MJ-i + [EF [MJ-i [E]) (4) Solve the system of linear equations (6) for all the master contact forces C%: {C J = (At* ([MJ-i + [ET] [MJ-i [E])-i {P} (5) Determine the slave forces {CJ from equation (1) and correct the nodal displacements and velocities of the master and slave nodes considering the acceleration changes due to the contact forces. There are several salient features to the algorithm which deserve a few remarks: (1) The construction and solution of the contact equations is straight forward if the contact surfaces are "smooth". The search algorithm, finding the penetration lengths {P} and location on the slave sides [E], is also simple. However, in the vicinity of corners or abrupt changes in a contact surface, it is often not clear how to determine these values. Because of space requirements, this search algorithm will not be discussed here. (2) The master contact surface forces C^ found by solving the system of equations should be all positive. A non-positive C% implies no contact, which contradicts the fact that the node was found to be penetrating a slave surface. We have found that negative C^ forces sometimes do occur. They result when one master node k penetrates deeply into a slave surface. The adjacent master nodes (k-1, k+1) can have a negative force associated with them. A negative force is usually a signal that the time step is too large.

5 Contact Mechanics 275 {UJ= Af[MJ-i{CJ (3) where At is the time step; [MJ lumped mass matrices. and [MJ are the coitesponding diagonal Let {P} be the vector whose component P% represents the penetration length of the k-th master node into the slave side between nodes i and j. The kinematic non-penetration condition within a time-step requires that {tu + [EF {UJ = {P} (4) where the same matrix [E] as in equation (1) appears. Substituting equations (2) and (3) into equation (4) yields At' [MJ-i (C J + [E]T At* [MJ-i {C,} = {P} (5) Further substitution of equation (1) into (5) gives At* ([MJ-i + [EF [MJ-i [E]) (Cm) = {P} (6) which is a positive-definite narrowly banded equation system with master-forces {C,J as unknowns and the penetration lengths {P} as the right hand side. The size of the equation is given by the number of master nodes which penetrate the slave surface. The algorithm can be formulated in a simple five step procedure: (1) For all master nodes of all possible contact zones determine the penetrated slave-side (if any) and the penetration lengths P^ of the vector {P} ignoring contact. For each group of (one or more) master nodes penetrating adjacent slave sides (or the same slave-side) do: (2) Determine the coefficients E^ which are only functions of the length factor a which defines the position of the penetrating node k on the side i-j and the angle <j> of side i-j, see Figure 2.

6 276 Contact Mechanics EXAMPLES In the first example problem a 100 mm cylindrical steel (E = 210'000 N/mm*) rod with a velocity of 10 m/s impacts a 200 mm long rod with the same diameter and material. Poisson's ratio was artificially set to zero in order to make the impact as one-dimensional as possible. The ends of the rods areflat;both rods are free to move. The meshes contained axis symmetric four node elements [6] and were uniformly discretized, i.e. the regions near the impact zone were not more finely modelled. To analyse the effectiveness of the algorithm the time step of the explicit time integration scheme varied. Since the smallest element length, 1, was 2 mm and the wave propagation speed, c, was 5(10^) mm/s, the critical time step of the time integration is approximately A T < 1/c = 2 mm / 5 (1.0e6) mm /s = 400 ns. Figure 3 displays the contact force at the impact surface for various time steps. The contact force is defined as the sum of all master forces acting on the surface. It can be clearly seen that at a time step of half the critical time step (200 ns) the contact force oscillates strongly. At about oneforth the critical time step (100 ns) the results improve dramatically. At a time of 50 ns, the results resemble the one-dimensional analytical solution (a square pulse) with the exception of oscillations at the point of initial contact. Interestingly, the oscillations disappear very quickly and do not introduce strong oscillations in the axial stress. Figure 4 shows the axial stress at a point near the contacting surface of the longer bar. The figure shows that a time step of 200 ns (one-half the critical time step) the stress oscillations are clearly unacceptable. The results improve at a time step of 100 ns (one-forth the critical time step). A useful method to judge the correctness of a transient dynamic analysis is study the energy in the system [4]. Theoretically, the energy change in the system should be zero. Moreover, e = E-I-K = 0 where e is the energy change, E is the external energy is the system, which includes the initial kinetic energy and the energy due to surface tractions and body forces. K is the kinetic energy in the system; I is the internal energy. The initial kinetic energy in the system was approxi-

7 Contact Mechanics 277 mately 12 J. For the case At = 200 ns, an energy increase of 0.5 J was observed. This change (4 %) is judged to be unacceptable. At time steps under 100 ns, energy decreases of less than 0.1 J were observed (< 1 %). In the next example, the numerical simulation of nail driving tool is demonstrated. The tool is invaluable on construction sites where quick temporary (and permanent) fastenings are required. In this example problem a nail is driven by a piston with an initial velocity of 90 m/s into a 20 mm thick steel substrate. The tool is often used to fasten thin metal roofing material - so-called profile sheets - to steel beams. Hence, in this example, before the nail penetrates the steel substrate, it penetrates two 1 mm thick profile sheets as well. A washer is placed on the nail in order to stabilise it during penetration and to clamp the profile sheets on to the steel substrate. This example problem is similar to earth penetration problems [7] or dynamic rigid punch problems analysed in [8]. The FEM analysis of this example, as shown in Figure 5, contains eight contact surfaces: (1) piston-nail, (2) nail-washer, (3) nail-first profile sheet, (4) nail-second profile sheet, (5) nail-substrate, (6) washer-first profile sheet, (7) first profile sheet - second profile sheet and (8) second profile sheet substrate. It is important to point out that such complex problems are impossible to study analytically or even experimentally. Typically, a FEM analysis is carried out for a variety of reasons such as: (1) To determine the forces acting on the nail. (2) To approximate the pull-out force of the nail. (3) To determine the wave propagation effects in the piston (4) To study the deformations in the substrate, profile sheets and washer. (5) To optimise the clamping forces between the various components in the system. (6) To optimise the energy usage in the system. (7) To optimise the nail, washer and piston geometries. Figure 5 shows the deformed mesh plots of the system at several different times during the penetration. Of interest is the time t = 100 ns when the washer impacts the profile sheets which are momentarily pressed

8 278 Contact Mechanics against the steel substrate. Afterwards, the washer moves backwards on the conical nail shaft and since there is no contact between the profile sheets and washer, the profile sheets "relax" and spring back. Obviously, a bad washer design. The finite element calculation was carried out using a time step of 5 ns or an eighth of the critical time step. Figure 6 contains several interesting numerical results. Figure 6a contains the contact forces acting against the nail and piston over time. There are approximately in equilibrium with each other, as expected, and can be viewed as the deceleration and history of the nail and piston. Figure 6b shows the radial contact force between the substrate and nail. Finally, Figure 6c shows the rigid-body velocities of all the components in the system. Of interest is the velocity of the washer. It's initial velocity is the same as the nail and piston (90 m/s). Afterwards, it impacts the top profile sheet and in the absence of friction changes its direction and moves backward on the nail shaft. CONCLUSIONS/SUMMARY A contact algorithm for two-dimensional, frictionless explicit dynamic finite element analysis has been introduced. The algorithm is general in that it is not restricted to node-to-node impact, requires no mass distribution or damping. The algorithm has performed satisfactorily. In order for the algorithm to work well, i.e. to produce no severe oscillations in stress after impact and to minimise the energy change in the system, a smaller time step than the critical time step of the explicit time integration must be used. Chatter on the contact surfaces was also observed. However, if the time step was small enough, the chatter disappears quickly. Although the algorithm requires the solution of a small, banded system of equations for each time step, the contact algorithm has used less than five percent of the total solution time of all the problems the authors have run to date. Future work will be performed to make the algorithm independent of the definition of master and slave surfaces and to introduce friction.

9 Contact Mechanics w 60.0 u Figure 3: <Vv Time (c) The contact force (kn) vs time ( is) for (a) At = 200 ns, (b) At = 100 ns and (c) At = 50 ns.

10 280 Contact Mechanics o.o Time Time (C) Figure 4: The axial stress (N/mnf) vs time (jus) for (a) At = 200 ns, (b) At =100 ns and (c) At = 50 ns.

11 Contact Mechanics 281 (a) (b) r j~j~t~h Figure 5: Deformed mesh plots for the nail penetration example (a) T=0s, (b) T=lOO^s, (c) T=300 (is.

12 282 Contact Mechanics o fa to 0) Pi Time Figure 6a: Contact forces (kn) vs time (jus) between the piston and nail and the nail and substrate Time Figure 6b: Radial contact force (kn) vs. time ( is) between the nail and substrate piston-- : Nail :. Wa.sh.0r-. jsheet jl Sheet! Figure Time 6c: Rigid body velocities (m/s) vs. time ( is) of several components in the system.

13 Contact Mechanics 283 REFERENCES [1] Kolsky, H., Stress Waves in Solids, Dover Publications, Inc., New York, [2] Bathe, K., Finite Element Procedures in Engineering Analysis, Prentice-Hall Inc., Englewood Cliffs, New Jersey, [3] Hughes, T.J. R. and Belytschko, T., Nonlinear Finite Element Analysis, Lecture Notes to a Short Course Taught by, Paris, [4] Key, S. and Isay, C, 'On the Use and Implementation of an Energy Balance Equation in Explicit Transient Dynamic Analysis' in Computational Aspects of Contact, Impact and Penetration, edited by R. Kulak and L. Schwer, Elmepress International, Lausanne, [5] Hallquist, J., Goodreau G., and Benson D., 'Sliding Interfaces with Contact-Impact in Large-Scale Lagrangian Computations', Computer Methods in Applied Mechanics and Engineering 51 (1985). [6] Heiduschke, K., Anderheggen, E., Bartelt, P., 'Axissymmetric Three-Node Triangular and Four-Node Quadrilateral Finite Elements for Finite Elasto-Plasticity', To appear. [7] Chen, E., Reedy, E., 'Penetration into Geological Targets: Numerical Studies on Sliding Friction' in Computational Aspects of Contact, Impact and Penetration, edited by R. Kulak and L. Schwer, Elmepress International, Lausanne, [8] Hughes, et.al. 'A Finite Element Method for a Class of Contact- Impact Problems', Computer Methods in Applied Mechanics and Engi- 8(1976).

Abstract. 1 Introduction

Abstract. 1 Introduction Contact analysis for the modelling of anchors in concrete structures H. Walter*, L. Baillet** & M. Brunet* *Laboratoire de Mecanique des Solides **Laboratoire de Mecanique des Contacts-CNRS UMR 5514 Institut