Simulation of the Crimping Process by Implicit and Explicit Finite Element Methods
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1 Simulation of the Crimping Process by Implicit and Explicit Finite Element Methods Stéphane Kugener AMP France ABSTRACT Calculation procedures for implicit and explicit codes applied to insulation crimping are presented and discussed. Some known equations are reviewed as a basis for discussion and to support engineering statements. Implicit finite element simulations were run under ANSYS 1, explicit simulations under OPTRIS 2. Critical aspects of the simulation are identified and discussed for each code. Strategies for the analysis are suggested and compared. For three cases of industrial importance correlation between theoretically predicted behavior and that of actual parts is established and discussed for two wire sizes and two different tool geometries. INTRODUCTION High-level computer-aided engineering (CAE) has become a relatively common activity. A large variety of codes for mechanical, thermal, fluid dynamics, or general process simulation are available at a reasonable cost, but their use also requires a reasonable level of expertise. Only a few years ago, many algorithms used in these codes were considered proprietary research tools. The latest trend in CAD/CAM systems technology, particularly the full integration of CAE tools, linear static analyses and plasticity analyses increased the level of computing aids available to the designer. CAE technology has past the step the analysis was based primarily on the geometry of the final product. The engineer can now include material behavior and process parameters in simulations of expected product or process performance. The move beyond linear approaches brought to an end the convenience of direct solutions and pushbutton analyses, and introduced the convergence uncertainties associated with nonlinear algorithms. Now, mathematical simulation of engineering problems can become a numerical adventure, with system crashes and restarts, a solution is obtained only through iterative steps and a rigorous calculation strategy. Clear identification of potential numerical difficulties and increased experience with the adapted codes will result in more efficient procedures and increased benefits. Application of these general viewpoints to a concrete problem of industrial importance is shown in this paper. Because of its complexity and technological importance the crimping process, here termed insulation crimping, is used as an example. It involves a metal forming process with path-dependent properties and requires highly nonlinear calculations. Therefore, it offers also an excellent opportunity to compare implicit and explicit finite element formulations. As a specific example, insulation crimping of a MIC I contact for two wire gauges of 1 mm 2 and 3 mm 2 was chosen. The contact is shown in Figure 1. To verify the validity of the mathematical simulations, tests were conducted at the manufacturing level on a T2 application tool, with an approximate crimping speed of 0.1 m/s. The setup comprised a rigid, nonsymmetric punch, a rigid anvil, an insulation crimp barrel of elastoplastic brass (CuZn 15), and an elastoplastic copper wire with polyethylene (PE) insulation. a Following common mathematical terminology in an explicit formulation the solutions, here the displacements u, are explicitly expressed as a function of other variables and parameters. In an implicit formulation the solution is contained within a function and its values have to be extracted, which is usually done through numerical algorithms. Copyright 2004 by Tyco Electronics Corporation. All rights reserved. 8 S. Kugener AMP Journal of Technology Vol. 4 June, 1995
2 ANSYS uses the following Newmark time integration method (see Ref. 1 for details): and are Newmark integration parameters and t is the time increment. This approach leads to the same type of inversion problem as before Figure 1. MIC I contact with insulation crimping as used in the study. The corresponding finite element model is 2-dimensional (2D), assuming plane stress. It is meshed with 680 2D, 4-node solid elements and 2400 contact elements with friction. Its parametric structure allows analysis of many types of crimps by setting a few, properly chosen geometrical parameters. the matrix (c 0 [M] + c l [c] + c 2 [K]) has to be inverted. Note that c... c 2 are constants. 0 Explicit formulation is based on the fundamental differential equation of dynamics OPTRIS uses central difference integration with THE MODEL Implicit and Explicit Formulations Chosen for the Model Implicit formulation allows a pseudo-static and a dynamic approach. In the pseudostatic case acceleration and velocity forces are neglected, the principle of virtual work leads to which yields with [K] = stiffness matrix, {u} = displacement pseudo vector, {F} = external force pseudo vector. The displacement is found directly by inversion of the rigidity matrix, which is a classical inversion problem. It uses Gauss elimination, with its usual matrix conditioning problems. In this case, computer time is proportional to the square of the mean wavefront size. For the implicit dynamic case acceleration and velocity forces are carried, it is with [M] = mass matrix, [c] = damping matrix. Because [M] is a diagonal mass matrix, the inversion is direct. The only difficulty is to ensure stable time integration by introducing the condition that the time increments be very small, that is L = some characteristic element dimension, = longitudinal velocity of sound, t = smallest transit time of dilatational waves to cross any element in the mesh. 3 The smallness of the time increment is an obvious disadvantage. If it is sufficiently small to be negligible for fast dynamic events, it is also highly penalizing for low speed phenomena such as most metal forming processes. However, the analysis can be accelerated by increasing AMP Journal of Technology Vol. 4 June, 1995 S. Kugener 9
3 L artificially the mass density of the material, b L the die velocity if inertia effects remain negligible. 4 For the crimping process considered here, the die velocity is about 0.1 m/s which is very slow for the default time increment. An increase of the speed to 1 m/s did not result in a noticeable inertia effect. Correction of the Stress Versus Strain Function A prime difficulty in applying finite element analysis to problems of the kind considered here is the lack of experimentally determined, true, nonlinear stress vs. strain curves. To be meaningful, finite element analysis needs as input ordered pairs of stress and strain at a given time t, i.e., the function t = true stress at time t, F = normal force, S t = true cross sectional area at time t, t = true strain at time t. Material manufacturers make available specific pairs of { ; }-values. Examples are 0.2% (the stress at 0.2% plastic strain as referred to the yield point), the tensile strength r, and the strain at rupture r. From these values one can construct a bilinear stress vs. strain relationship. With some experience and familiarity with the behavior of materials it is possible to smoothen that function and obtain a more realistic tensile test curve. From standardized tensile tests are available the uncorrected curves S 0 = initial cross sectional area, = (L - L 0 )/L 0, L 0 = initial length. This curve is reasonably accurate for small deformations, but is not acceptable for realistic, in-depth mechanical engineering analyses. However, such a curve can be corrected according to Francois et al. 5 by applying large deformation theory for incompressible materials and using the first-order approximations Then the corrected curves can be simulated by a multilinear model. This technique gave satisfactory results b An increase of two orders of magnitude provides a benefit of one order in cpu time. describing for brass contacts the behavior of the metal under consideration here. The case that included the behavior of the wires was more difficult to simulate. Additional considerations were required. First, the 19 strands of copper would lead to too large a model if internal sliding were taken into account. This phenomenon was modelled by introducing a copper core, thus artificially reducing the rigidity modulus of the copper. Furthermore, the behavior of the insulation became of critical importance, because one has to simulate simultaneously two effects. One of them is the large strain while the crimp voids are being filled, the other is the 20% reduction of the cross section under plane strain assumptions that would not allow expansion in the third dimension. Copper shows no noticeable compression under the conditions considered here. The behavior of the polyethylene insulation was simulated by assuming elastic deformation with a tensile modulus of 100 MPa, which is about one order of magnitude below that of most PE versions. Because of the incompressibility requirement, the plasticity codes could not be used. The problems associated with defining the behavior of the materials are the same for implicit and explicit analyses. Contact Element Management The crimping process is mainly a surface-on-surface sliding problem with a large number of elements sliding simultaneously on each other. The stability of the mechanism and to a certain extent also the final shape of the deformed parts is determined by the sliding friction taking place during crimping. This makes it important to include sliding friction in the models. The contact element management will be more difficult, because an initial equilibrium has to be found between the extrinsic force caused by the punch and the anvil reaction force, eventually through contact oscillation or rotation. ANSYS, in both the static and dynamic modes, simulates structure interactions with 3-node point to surface contact elements. Each node potentially in contact with an element has to be meshed, which penalizes the rigidity matrix size. Our model is composed of 2400 contact elements for only 680 structural elements. Contact interaction is solved in three steps: (1) (2) (3) A pinball algorithm fixes the close/open status of the gap. The next step determines whether penetration occurred or not. The third step applies the compatibility conditions if penetration occurs. A combination of penalization and Lagrange multiplier methods was used here, which in static code is the only one taking friction into account. The normal force, F n, is calculated by 10 S. Kugener AMP Journal of Technology Vol. 4 June, 1995
4 and i+l = Lagrange multiplier at iteration step (i + 1), K n = stiffness of the contact element, g = penetration depth, g = a user-defined penetration tolerance, p = an internally computed factor with p < 1. (14) [R] = orthogonal rotation matrix, [D R c ] = right stretch matrix. Then a strain tensor known as Hencky tensor d can be defined as [U] is the stretch matrix. Values of [ ] are numerically obtained through the spectral decomposition of [U] This algorithm is known for being more stable than the penalization method, and for avoiding conditioning problems 6,7 but still has convergence problems for the following reasons: The contact stiffness value is critical and can only be set iteratively. Different values have to be tuned along the punch to allow convergence. These values depend on the rigidity and plasticity of the local structure. Inadequately low stiffness values cause improper penetration and sometimes sticking. Inadequately high values affect substep size and convergence rate, and can generate major instabilities by introducing, in a single step, a resulting normal force exceeding the upper limit tolerated by the structure. In many cases this algorithm may cause an infinite number of numerical oscillations between several contacts. For explicit codes, the very small time increment permits verification of contact compatibility node by node and at each time step. This procedure is based on a contact search/tracking algorithm and a double loop master/slave algorithm which determines normal and tangential forces. 8 The formulation has the advantages of being very effective, almost transparent for the user, and will not affect the problem size. Large Strain/Deflection Formulation The common assumption of small displacements cannot be applied to crimping. Therefore, a Lagrangian formulation has to be used instead of the usual first-order tensor in the Eulerian formulation [ ij ] = first-order strain tensor, {u i } = displacement vector. In this case, a strain tensor is formed from the polar decomposition of the Lagrangian deformation gradient i are the eigenvalues and i are the eigenvectors of [U]. The Hencky formulation will not conserve orthogonality during deformation and may cause interaction problems with other algorithms using the assumption of orthotropic elasticity or kinetic stress hardening. For this reason, an isotropic stress hardening law for the brass was used. This kind of algorithm is powerful and relatively economical in computer time. However, it penalizes the analysis because it needs an eigenvalue extraction from the [U] matrix. Furthermore, it may simulate a realistic force/displacement curve but with a negative derivative, which may cause the Newton-Raphson e solution algorithm to crash. The latter instability could be avoided with a Riks arc length algorithm l0, which is not yet available to cover plasticity. In the dynamic analysis the instability will be dampened by inertia forces. Explicit codes do not need large strain/deformation formulation because the displacements are reactivated at each substep, very small time increments support small perturbation assumptions, the behavior under stress is described by true stress/ strain curves. Buckling and Bifurcation Instabilities Simulation of the crimping process is very delicate because of several physical instabilities. Some of them are caused by friction. If it is very high, the barrel may collapse on one side of the anvil. Transverse buckling of the barrel wall, towards the inside, seems to be numerically possible. The two barrel ends buttress together before one of them moves under the other. Flat ends emphasize this behavior. Because of the rolling mechanism of the barrel, the combined effects of friction and the mechanics itself can cause instability. When the barrel end begins to slide in the crimping radius, a large part of the vertical forces is supported by friction because of a long area of contact. But c The corresponding left turn expression would be [F] = [D L ] [R] with [D L ] being the left stretch matrix. d For details on the Hencky tensor see for instance reference 9. e For details see reference 1. AMP Journal of Technology Vol. 4 June, 1995 S. Kugener 11
5 when the barrel begins to roll, it decreases the contact area thus creating the condition for snap-through. In some cases, particularly when the wire is thin, fracture may occur at the bottom of the barrel. Only few explicit codes can handle fracture mechanics. As discussed above, large strain/deflection can partially simulate these phenomena with the Hencky tensor. With the implicit code, an initial buckling analysis would have made it possible to calculate buckling modes and critical loads. Post buckling analysis is not perfect for either implicit or explicit codes. Such formulations have to take into account defect rate and bifurcation equations which are not available in ANSYS and in OPTRIS. ll In static analysis, these phenomena can create major problems by generating infinite displacements or force/ displacement curves with a negative slope which usually crashes the Newton-Raphson algorithm of ANSYS. In all cases, dynamic analysis allows to pass through these instabilities because of the dampening effect of the inertia forces that also reduces noise in the results. COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS In a first example the deformation of an idealized, simple, geometric model was simulated with ANSYS. An implicit, dynamic approach was used. The result is shown in Figure 2. Because of the long time required for the simulation, the method was not extended to model realistic, industrially important crimps. Figure 2. Strain distribution in a simplified, non-industrial crimp simulated by ANSYS. Results of explicit code simulations are shown in Figures 3 to 5 together with photographs taken on actual samples in the corresponding stages of the crimping process. These figures permit a comparison of deformations as predicted by the simulation with those experienced in real life. While the photos taken on experimental samples show only deformations, the graphs produced by the simulation show also the distribution of the strain. This feature can be used for targeted modifications of existing designs, material selection, and process parameters. Figure 3 permits comparison of deformations obtained at three different crimping stages with those of a prototype, small punch, and a thin wire of 1 mm 2 cross section. Shown are the strain contours at the beginning of crimping, the beginning of overlap during the process, and at its end. In all cases, the agreement between experimental results and those predicted by the explicit code simulation is excellent. The general phenomena are well described. Some differences appear in the curvature of the barrel ends. This can be understood by considering that the tests were static, which emphasizes buttress and buckling phenomena because of the higher coefficient of static friction. Under true dynamic conditions, this effect is smoothed by inertia forces and the lower coefficient of dynamic friction. The deflection of the left-handed barrel side is less pronounced. This effect was taken care of by introducing a larger coefficient of friction. In Figure 4, results of experimental and simulated crimps are compared for the initial large punch and a thin wire of 1 mm 2 cross section. In the photographs on the left side, the deformation of the experimental sample is shown. The photos on the right side present the results of the simulation, which include deformation and strain. Here only the two stages beginning and end are covered. Figure 5 comprises examples of experimentally observed and predicted deformations and shows the strain distribution revealed by the simulation. As in Figure 4, the results are from a setup utilizing the initial large punch but now with a thick wire of 3 mm 2. Shown are only the conditions existing at the beginning of crimping. In all simulations the contact rotated a little to the right. This phenomenon is not confirmed by the test results shown in the figures. The reason is that in the tests the contact position was corrected initially to place the wire. However, very frequently such a rotation was observed in other experimental studies. The final configuration of the case described in Figure 4 was simulated poorly because fracture was not included in the algorithms. The high strain values computed in the simulation indicate that such had happened. Finally, it is interesting to note that, as shown by the simulation, friction has a major effect on the crimping process. STRATEGIES FOR OPTIMAL SIMULATION Based on the previous discussions, it appears that an explicit code will offer the best approach mainly because it allows a direct solution without considering in advance problems with contact element management. In this case, no special strategy has to be developed. If necessary, only speed increase options should be considered. 12 S. Kugener AMP Journal of Technology Vol. 4 June, 1995
6 Figure 3. Comparison of results of experimental and simulated crimps at three different stages for prototype small punch and thin wire. The simulation used explicit code. Experimental results in the left, simulated results in the right hand column. Shown are the strain contours at (a) the beginning of crimping, (b) the beginning of overlap, (c) the end of crimping. An important result of the study presented in this paper is that implicit static analysis was inadequate for simulating the crimping process. The Newton-Raphson algorithm has convergence problems around the physical instabilities of the system modelled. Various numerical precautionary measures were tried, but proved to be useless. Dynamic, implicit analysis will give a solution but only through a difficult iterative process. If this approach is chosen, a rigorous strategy is recommended considering the following points: Smoothed, true stress/strain curves must be entered. Most refined algorithms are required in form of versions including large deflection, large strain and friction. Contact stiffness values must be set iteratively and individually. AMP Journal of Technology Vol. 4 June, 1995 S. Kugener 13
7 Figure 4. Comparison of results of experimental and simulated crimps at two different stages for the initial large punch and thin wire. The simulation used explicit code. Experimental results in the left, simulated results in the right-hand column. Shown are the strain contours at (a) the beginning of crimping, (b) the end of crimping. Figure 5. Comparison of results of experimental and simulated crimps at two different stages for the initial large punch and thick wire. The simulation used explicit code. Experimental results on the left, simulated results on the right. Shown are the strain contours at the beginning of crimping. A contact stiffness value of K n = E/100 is a good initial value. The penetration tolerance g should be as large as possible, taking into consideration that the penalization force F p = K n. The tolerance for convergence of the solution should be set to the minimum for a first successful run and subsequently increased for more accurate results. Convergence tools such as predictors, bisections and linesearch algorithms should be avoided as they may 14 S. Kugener AMP Journal of Technology Vol. 4 June, 1995
8 create more problems than they really solve. Varying the time increment is generally more effective. In some cases, a larger time increment helps circumvent a local difficulty. SUMMARY This study shows that a complex metal-forming process can be simulated accurately and that finite element methods have reached a degree of maturity to make them useful as design tools and for process analysis. Explicit codes have to be considered for the future. They are expected to become of particular importance for die optimization and process simulation. Before choosing a specific approach for the simulation, a thorough problem definition including a discussion of anticipated difficulties should be prepared. The latter is particularly important because choosing the wrong simulation tool can lead to undesirable, lengthy calculation procedures or even to a dead end. For crimping simulations an explicit code is superior to an implicit one, especially because of the large number of contact elements used in the mathematical model. ACKNOWLEDGMENTS The author would like to thank Jean-Pierre Picaud for his contributions in the area of parametric geometry and Richard Hughes whose input greatly supported and enhanced the static analysis. He also thanks the members of AT&T DATAID for their collaboration especially, Nicolas Wolik, Projects Director, Scientific Department, who gave permission to run the ANSYS dynamic analysis, and Fabrice Villeval, Analysis Engineer in charge of fast dynamic simulation at the Scientific Department, who piloted the OPTRIS explicit analyses based on AMP s initial parametric deck. REFERENCES 1. ANSYS Theoretical Manual (Swanson Analysis Systems, Inc., Houston, PA, 1992). 2. OPTRIS, Theoretical Manual (Dynamic Software, Les Ulis, 1991). 3. D.P. Flanagan and T. Belytschko, Eigenvalues and stable time steps for the uniform strain Hexahedron and Quadrilateral, ASME J. of Applied Mechanics, 51, (1984). N. Rebelo, J.C. Nagtegaal and L.M. Taylor, Industrial application of implicit and explicit finite element method to forming process, WAM, D. Francois, A. Pineau, A. Zaoui, Elasticity et Plasticité (Ed. Hermes, Paris, 1993). Fortin/Glowinski, Méthodes de Lagrangiens Augmentés, Col. Méthodes Mathématiques pour l informatique (Dunod, Paris, 1982). D.A. Pierre, M.J. Lowe, Mathematical Programming via Augmented Lagrangians (Addison Wesley, Reading, MA, 1975). K. Sweizerhoff, Metal forming simulation with explicit time integration, International LS-DYNA 3D conference, Birmingham, U. K., April G.G. Weber and A.N. Zava Plangos, An objective time integration procedure for isotropic rate-independent elastic-plastic constitutive equations, Internatl. J. of Plasticity, 6, (1990). New Features of ANSYS Rev. 5.0 A (Swanson Analysis Systems, Inc., Houston, PA, 1993). C. Gontier, Stabilité des structures (Ed. Ecole Centrale, Paris, 1989). Stéphane Kugener is Advanced Projects Engineer in the Engineering Department of AMP de France. Stéphane Kugener received an Engineer Diploma in Structural Mechanics from Ecole Speciale des Travaux Publics of Paris in He specialized in Solid Mechanics and Scientific Calculation through a 3rd year at Ecole Centrale de Paris in From 1990 to 1991 he worked as civil engineer on seismic verification of bridges and dams in the engineering department of SOCOTEC. Since joining AMP in 1992, he supervised CAE activities and developed use of numerical simulation for injection molding, mechanical and thermal electric analyses. He is a member of AMP s European Moldflow Working Group. AMP Journal of Technology Vol. 4 June, 1995 S. Kugener 15
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