Fritz J. and Dolores H. Russ College of Engineering and Technology

Transcription

1 A FINITE ELEMENT METHOD FOR RING ROLLING PROCESSES A Dissertation Presented to The Faculty of the Fritz J. and Dolores H. Russ College of Engineering and Technology Ohio University In Partial Fulfillment of the Requirement for the Degree Doctor of Philosophy Lohitha Dewasurendra June, 1998

2 CONTENTS 1. INTRODUCTION., BACKGROUND THEORETICAL FORMULATION CONTINUOUS REMESHING ARBITRARY LAGRANGIAN EULERIAN (ALE) FORhluLATION LAGRANGIAN COMPUTATION WITH ARBITRARY LAGRANGFAN EULERIAN (ALE) MESH UPDATE Lagrangian Formulation with Rigid-viscoplastic Material Constitutive Relation Rigid- Viscoplastic Constitutive Relationship IMPLEMENTATION..., LAGRANGIAN COMPUTATION I Mathematical Formulation Contact Boundhry Condition Time Integration TRANSFEROFFIELDV~LESTOTHE~R~ALMESH REGENERATION OF THE ANALYSIS MESH VIRTUAL ROTATION UPON STEADY ST^ HEAT TRANSFER DURING VIRTUAL ROTATION PERFORMANCE OF THE TWO MESH SYSTEM APPROACH COMPARISON OF THE TWO MESH SYSTEM APPROACH AGAINST THE CONVENTIONAL METHOD Analysis with the conventional FEM... 76

3 FIGURE 5-5 : THE STARTING AMS AM, THE MMS USED EN THE DEL ANALYSIS FIGURE 5-6 : S m DISTR~BU~ON PREDICTED BY THE DEL ANALYSIS FIGURE 5-71 STRAIN RATE DISTRIBUTION PREDICTED BY DEL ANALYSIS FIGURE 5-8: SCHEMATIC REPRESENTATION OF TRANSFER OF DEFORMATION HISTORY BETWEEN MMS AND AMS FIGURE 6-1 : LMTIAL MESH SETUP FOR RECTANGULAR RING ROLLING PROCESS FIGURE 6-2: TOTAL RAJN I ' S 20% REDUCTION FIGURE 6-3: STRAIN RATE AT 20% REDUCTION FIGURE 6-4: TOTAL STRAIN AT 40% REVOLUTIONS FIGURE 6-5: STRAIN RATE AT 40% REDUCTION FIGURE 64: R ~ GROWTH G RGURE 6-7: PROGRESSION OF SECTION REDUCTION FIGURE 6-8: EXPERIMENTAL VALIDATION - SIDE SPREAD [ FIGURE 6-9: CONTACT ZONE FIGURE 6-10: INITIAL MESH SETZTP FOR THE RING BILLET RGURE 6-11 : ROLL PROFILE AM, BILLET PLACEMENT FIGURE 6-12 : TOTAL STRAIN AT 26 REVOLUTIONS ( 13.4% REDUCTION) FIGURE 6-1 3: LATERAL SPREAD AFTER 26 REVOLUTIONS (1 3.4% REDUCTION) FIGURE 6-14: TOTAL STRAIN AFTER 76 REVOLUTIONS (38.4% REDUCTION) RGURE 6-15: LATERAL SPREAD -R 76 REVOLUTIONS (38.4% REDUCTION) FIGURE 6-16: TOTAL STRAZN 113 REVOLUTIONS (57% REDUCTION) FIGURE 6-17: LATERAL SPREAD AFTER 113 REVOLUTIONS (57% REDUCTION) FIGURE 6-18: ROLL PROFILE F u FIGURE 6-19: RING GROWTH FIGURE 6-20: COMPARISON WITH'EXPERIMENTAL RESULTS FIGURE 7-1 : ANALYSIS MESH AND ITS DEL MESH MOTION FOR STRIP ROLLING ( S ~ HALF) Y FIGURE A-2: NODAL CONNECTIVITY OF A HEXAHEDRAL ELEMENT

4 I. Introduction Until recently the metal forming process design was considered more an art than a science. Limited availability of mathematical techniques that can predict the complex material flow under non-linear conditions can be identified as the primary reason for this. The process designer who is only equipped with years of experience typically solves the metal forming problem through costly trial and error based method. In many cases the result will not be the most cost-effective optimum design that can deliver the required product properties. The material flow pattern and the strain distribution in the workpiece are predicted using past experience which is often challenged by new part geometries and new alloys. Computer simulation of metal forming processes can effectively replace the costly shop floor trials by fast and effective realistic computer trials. Among the currently available computer simulation techniques, the Finite Element Method (FEM) can be identified as the most comprehensive method, which is rapidly gaining popularity in the forming industry. However, until recently, mathematical techniques and the required computer power were major barriers for any effective scientific forming process design methodology. Today, modem high performance workstations equipped with effective FEA tools and modern visualization techniques can serve as a virtual forgmg plant or a rolling mill that can not only simulate the actual forming operation but also give comprehensive details such as material flow within the workpiece, die wear, temperature distribution in the workpiece and the die, required soak time, and the dwell

5 2 time etc. These details can certainly assist the design engineer to alter the entire forming process including the die geometries, forming sequence, and heating sequence, etc., with significantly reduced lead-time. However, penetration of FEA technology in the metal forming industry has been relatively siow compared to other areas such as structural engineering. This slow acceptance may be attributed to the difficulties ranging from the large computational burden to large user interaction time. A typical industrial forging problem usually requires regeneration of the finite element mesh several times due to excessive mesh distortion resulting from large plastic deformation. Until recently, automatic remeshing was available only in two-dimensional problems due to restrictions on available elements and mesh generation techniques. Therefore, remeshing in three-dimensional applications was primarily done manually, thus making the application of FEA quite questionable in many three-dimensional situations. The finite element formulations for bulk forming problems were limited to the usage of only hexahedral elements. On the other hand, the automatic mesh generation techniques were limited to tetrahedral elements in threedimensional geometries. Therefore, automation of the analysis with auto-remeshing feature was not feasible for three-dimensional applications. The recent advances in element technology have however successfully bridged this gap by making tetrahedral elements available for bulk forming analysis. Chenot and co-workers[l] and Dewasurendra and co-workers[2] have already demonstrated the power of these new element technologies when integrated with auto-remeshing techniques. The

6 3 computational requirement has now become the main focus as the forming process design requires multiple analyses with different die configurations and process parameters. It is the standard practice to reduce computation time by employing fine elements in regons that undergo deformation and coarse elements elsewhere thereby reducing the overall size of the numerical problem. This technique is readily applicable in some forming operations such as cogging and heading operations where the deformation is confined to a relatively small region in the workpiece. In addition, the deformation zone is relatively stationary with respect to the workpiece in these forming processes. In forming operations such as rolling, extrusion, ring rolling, and disk rolling, the deformation zone is not stationary with respect to the material. This class of processes can be identified as continuous deformation processes where the deformation zone travels along the workpiece. The conventional FEA with above technique of using a fine element region in the deformation zone and coarse elements elsewhere is not readily applicable in this class of problems. As a result, applicability of FEA for such processes for meaningful results within practical time limits is quite questionable. This research is aimed at developing a novel technique to harvest the capabilities of FEA within acceptable time limits without sacrificing the accuracy. Using ring rolling as the case study, a comprehensive technique was developed and successfully implemented. Ring rolling is the most compute intensive continuous forming process which requires prohibitively large amount computer resources if conventional FEM is utilized. As shown

7 4 in the subsequent chapters, the method implemented in this research is extremely effective in industrial three dimensional ring rolling problems and it can be easily extended to rest of the continuous forming problems. With this method, compute speed for relatively simpler processes such as profiied strip rolling and extrusion processes may be potentially elevated up to near interactive levels.

8 2. Background Axi-symmetrical components - especially rings - are essential components in most of the engneered products. Since their cross-section remains constant along the circumference, they are ideal candidates for rolling. The early demand for ring products came with the growth of the locomotive industry where the rail wheel was well suited for the ring rolling process. Today, the manufacturing of seamless rings is a growing industry. Most of the critical components with axi-symmetric geometry and stringent requirements on structural integrity around the circumference are good candidates for ring rolling processes. Casing of jet turbine engines[3], propulsion cones of stealth submarines[4], outer racers of large bearings[5-61, locomotive wheels, nuclear reactor parts, flanges of various geometries, and Ring gears[7-81 are typical products that are made by ring rolling. The cross sectional geometry may range from a simple rectangular shape to highly complex profiles. The ring rolling process can deliver high near net or net shaped rings with no seems or flashes with no draft (or very minimal draft). Another very important advantage of the ring rolling process is the favorable grain flow that it yields[9]. As the ring deforms, the grain boundaries conform to the ring contour providing a surface highly resistant to surface cracks since the grain boundaries lie parallel to the surface. Circumferencially spread grains in ring rolling also deliver much superior properties compared to forged parts.

9 6 Rolling of profiled rings typically involves a ring billet being rolled in two radial rolls - one or both driven externally. Additional idling rolls known as guide rolls may be present on the sides of the ring to maintain the position of the ring at the center. Some cases may involve additional axial rolls to control the axial growth of the ring. Based on the roll setup and the geometry of the ring being rolled, ring rolling processes can be divided into following major categories. a) Radial Ring Rolling: - The ring billet is rolled with only two ralal rolls (with axes parallel to the ring axis) and guide rolls. Figure 2-1 (a) shows a schematic of a simple radial ring rolling process. b) Radial-Axial Ring Rolling: - The ring billet is rolled with two radial rolls and two axial rolls. The axial rolls may impose the profile in the axial direction or they may control the axial growth of the ring. These processes are sometimes called two-pass ring rolling. The axial rolls are typically placed diametrically opposite to the radial rolls. Figure 2-1 (b) shows a schematic of a two pass radial-axial ring rolling process. c) Wheel Rolling and other Special Ring Rolling: - The ring billet is rolled between radial and axial rolls as required by the geometry. Some cases may involve an enclosure die and axial rolls to form the billet into the enclosure die. This type of ring rolling is sometimes called closed die ring

10 7 rolling. Products include locomotive wheels, gears, aircraft engine components, and bearing rings. Figure 2-1 (c) shows a schematic of a simple wheel rolling process.

11 (a) Single Pass Radial Ring Rolling (b) Two Pass Radial-Axial Ring Rolling (c) Special Ring Rolling (Wheel Rolling) Figure 2-1: Various Ring Rolling Processes[9]

12 9 Ring rolling inherits many advantages in manufacturing seamless rings with hgh structurai integrity. At the same time, the design process is faced with unique problems that are not common to any other forming process. A simplified ring rolling process involving a ring billet, main roil, pressure roll, and guide rolls is schematically shown in Figure 2-2. As seen in the figure, the deformation zone is confined to an area within the proximity of the roll contact that is very small compared to the overall geometry. 1 rigid zone Figure 2-2: Schematic Representation of a Ring Rolling Process In a ring rolling mill, the main roll is externally driven while the pressure roll is advanced towards the main roll reducing the gap between the two. Consequently, the

13 10 ring billet undergoes deformation while rotating through the continuously decreasing roll gap. Ring billet then increases its diameter to compensate for the cross sectional reduction while forming any contours that may be imposed upon it by the rolls. Unlike in straight strip or profiled rolling processes, higher reductions cannot be achleved in ring rolling processes in a single pass due to the slippage that can take place during rolling. This significantly limits the feed rate leading to a very high number of revolutions of the ring needed to achieve the required reduction. In straight profiled rolling, the primary concern is to fill the roll gap. But in ring rolling, the final diameter of the ring also becomes a design requirement in addition to the roll profile fill. An ideal design should achieve both these targets simultaneously. In some case, if profile fill occurs before it reaches the final required diameter, defiling may occur as the billet is rolled further. Ring rolling is a complex process. The process controls must be effective and consistent for good product quality. Today's state-of-the-art ring rolling mills are equipped with electronic sensing mechanisms and feed back controls to ensure the product quality. For example, the force on the guide rolls must be carefully adjusted for maintaining the ring position. Too high guide roll forces might distort the ring into an oval shape. Due to the complexity of the process and its parameters, the process design is extremely difficult and mostly dependent on trial and error techniques. Computer aided analysis can play a major roli in the design of the ring rolling process with accurate predictions including ring profile formation and the diametric

14 11 growth. Over the years, many attempted to predict the final outcome of ring rolling processes using analytical and physical modeling methods. Simulation using model materials is reported to deliver close-to-life predictions. Thorough study with model materials could lead to development of dimensionless parameters which permit the use of scaled down ring rolling mills and ring blanks providing more economy. Maekawa and co-workers[ll-121 used plasticine as the modeling material for the study of the deformation pattern in ring rolling. The plasticine blanks were prepared with black and white checkered cross sections to analyze the material flow within the workpiece. Mamalis and co-workers[l3-171 conducted experimental work with terullium lead as the modeling material to study the spread and defects in ring rolling. A Wax material was used to model ring rolling processes by Boucly and co-workers[18]. Among the analytical methods, early analytical methods were confined to approximate methods including; Empirical formulae Volume mapping Slab method Slip line fields method

15 Upper bound method. The empirical formulae are generally limited to each class of ring rolling operations with specific materials and processing parameters therefore results cannot be generalized for other ring rolling operations. Typically (material and process dependant) empirical formulae can only be developed for ring growth and roll torque predictions in simple ring geometries. The volume mapping is capable of predicting the diametric growth in closed pass ring rolling operations. Application in operations with open roll passes involve approximations in lateral spread usually computed based on equivalent rectangular cross sections where empirical formulae are readily available. This approach cannot predict roll forces or torque. The slab method uses a very coarse discretization of the ring into slabs where the slabs are assumed to be planar during the deformation. However, the shape of the slabs and the deformed ring has to be predetermined. Thus, the slab method can be used to predict the roll force and the torque in pass ring rolling operations. Yang and Ryoo [ applied this method to predict the roll force and the torque in profile ring rolling of an L-section. Hawkyard and co-workers[21] used slip line fields method to predict roll separating force in ring rolling processes with rectangular cross-sections.

16 13 Among the various flavors of upper bound solution techniques, the methods developed by Doege and Aboutour[22] seems to be very effective in analysis of ring rolling operations. This method involves finding a kinematically admissible velocity field, which leads to a minimum forming energy. The velocity field is first expressed in terms of parameters, which will be computed by minimization of the forming energy. However, it is difficult to express an admissible velocity field for problems involving complex shapes and general material flow especially in transient conditions. To overcome this difficulty, the deformation zone is discretized into radal elements. Therefore, thls technique is also called the Upper Bound Element Techque (UBET). The forming energy is expressed on each radial element using the following assumptions. The deformation is limited to the deformation zone only. Radial sections remain plane during the deformation process. The strain rate is distributed linearly within each element. Once the velocity field is computed, strain, strain-rate and the roll tractions can be readily calculated. Application of the UBET was further demonstrated by Hahn[23] where the ring cross sections made of circular arcs (grooves, fillets) and straight lines were modeled

17 with reasonable accuracy. Basically, this research enhanced the UBET element library by introducing higher order (sides with circular arcs) elements. 14 Among the various analytical techniques, the finite element method (FEM) is the most comprehensive tool that can accurately analyze the ring rolling process (or any other large deformation problem) under very general conditions. However, mathematical overhead is significant compared to the approximate models. Early applications of FEA in ring rolling problems were limited to two-dimensional problems. Yang and Kim [24,26] applied the rigid-plastic finite element analysis for plane strain ring rolling from which predictions of strain-rate, strain, roll torque and roll separating force were made. Due to the very small reductions present in the ring rolling process, the contact area between the dies (rolls) and the workpiece is relatively small which demands very fine discretization for modeling the contact condition between the rolls and the ring billet. Metal forming problems that undergo large deformations are non-linear in nature because of the non-linear geometric conditions and the non-linear material flow behavior. Mechanics of these problems are expressed according to the Lagrangian description where the frame of reference is attached to the material. During the Lagrangian analysis, total deformation process is divided into a number of finite time steps and governing equations are solved for the velocity field. The finite element mesh geometry is updated with the current incremental displacement field after each time step. It should be noted that during the calculation of the velocity field in finite element analysis, time step

18 should be small enough so that the nodes do not skip the contact region during this geometric updating process. -- \-. *-. '\ Coarse mesh Coarse mesh /.,- Pressure Pressure, roll roll.b<, '-. -,, Fine Fine mesh. mesh,._. J' -z , --. Main roll 2---:b Main roll ' -1 Figure 2-3: Motion of the fine mesh region during ring rolling As explained above, the element size and the size of the time increment are limited to very small values in finite element analysis of ring rolling processes. Consequently, a large number of elements and time increments are required to simulate a single revolution of the ring rolling operation. Situation is further complicated by the fact that ring rolling usually involves a large number of revolutions (>SO). The conventional Lagrangian type of formulation would require a prohibitively large computation time making the applications of FEA impractical in industrial ring rolling problems. Total number of finite elements has to be kept at a minimum level in order to lower the simulation time. Unfortunately, size of the time increment cannot be increased without compromising the accuracy of the model. A common practice to reduce the size

19 16 of the model is to employ fine elements in the deforming area to accurately capture the deformation and coarse elements elsewhere to ensure the continuity of the workpiece. Almost all commercially available large deformation based FEA tools are based on Lagrangian formulation where the finite element mesh is etched in the material. As a result, the finite element mesh is updated with the material flow. If such a Lagrangian approach is used to solve ring rolling problems with conventional mesh updating schemes, the fine mesh region will move away from the deformation zone as shown in Figure 2-3 since the deformation zone is not stationary with respect to the material coordinates. In other words, the ring can be seen as rotating through a deformation zone that is stationary with respect to the spatial coordinates. With this observation it deemed logical to investigate the Arbitrary Lagrangian Eulerian (ALE) description where the finite element mesh can be given an arbitrary motion irrespective of the material flow. The primary objective of using ALE mesh motion is to reduce the total number of elements by employrng fine and coarse elements in different regions and maintaining their respective location with respect to the material deformation and the contact conditions through a customized mesh motion. Thls enables significantly smaller models produce results of equal quality without compromising the accuracy of the prediction thereby making FEA applicable to processes such as ringrolling, spin-forming, and other similar forming processes.

20 Theoretical Formulation formulations. Large defbrrnation problems can be analyzed using the following finite element (a) Penalty formulation with rigid-viscoplastic material constitutive behavior. (b) Mixed formulation with rigid-viscoplastic materiai constitutive behavior. Each technique has its own merits toward specific applications. For example, problems associated with residual stresses require the use of EIasto-plastic formulation. Almost all industrial hot forming operations can be modeled using the rigid-viscoplastic material constitutive model that is significantly simpler than the elasto-plastic model. Penalty formulation with the rigid-viscoplastic material model is the simplest among above methods and almost all the commerciaiiy available software tools were originally based on this formulation. However, mathematical difficulties can arise with certain classes of elements due to the incompressibility constraint inherent to rigid-viscopiastic model. Consequently, only a limited selection of elements was available for rigidviscoplastic based penalty formulation. On the other hand, the elasto-viscoplastic model involves rigorous mathematics and usually requires a considerably large computation

21 18 time compared to the rigid-viscoplastic analysis. Even though a wide selection of elements can be theoretically handled, most of the useful elements such as linear tetrahedral elements do not offer acceptable accuracy with the elasto-plastic formulation. The mixed formulation introduces pressure at each pressure node that requires solution of a larger matrix equation in each non-linear iteration. However, this formulation offers superior performance and allows the use of a broad spectrum of elements whereas only quadrilateral and hexahedra1 elements are used with the penalty based rigid-viscoplastic method and the elasto-plastic formulation. The latter formulations do not deliver accurate results with other useful elements such as tetrahedral elements. The geometric representation of the tetrahedral element does not provide enough modes of deformation under the isochoric conditions. Therefore tetrahedral element is not suitable for incompressible flow problems with rigid-viscoplastic and elasto-plastic formulations. Analysis of metal forming processes often requires regeneration of the finite element mesh of the deforming body as the existing Lagrangian mesh becomes severely distorted as the deformation progresses. Automation of the regeneration of the mesh for the deforming body is an essential requirement for any realistic use of the finite element method in solving large deformation problems such as metal forming processes. Current mesh generation techniques can deliver triangle as well as quadrilateral elements in twodimensional geometries. Therefore the penalty formulation with quadrilateral elements is well suited for two-dimensional plain strain and axi-symmetric problems. However, available mesh generation techniques can only deliver tetrahedral elements for three-

22 19 dimensional geometries. Therefore, the practicality of the rigid-viscoplastic penalty method is very questionable in three-dimensional applications where the remeshing has to be done manually. The most recent advances in the element technology however, addressed this issue by introducing the most useful elements such as linear tetrahedral element with the mixed formulation (pressure-velocity formulation) [ where automatic remeshing can be easily incorporated. However, it is advisable to use hexahedral elements in three-dimensional cases and quadrilateral elements in two-dimensional cases of forming problems such as rolling, ring-rolling and extrusion where the geometry is somewhat simpler in the longitudinal direction. Release of finite element nodes from the dies as the material flows out of the contact zone usually adds additional non-linear iterations in the simulation process. In the above mentioned forming problems, the transition of finite element nodes through the contact zone is much more frequent than in forging problems. The use of hexahedral and quadrilateral elements makes the node release from the dies at regular intervals and a group of nodes are released in each occasion. These elements reduce the need of additional non-linear iterations necessary for release of die-contact nodes. The penalty formulation with above elements is utilized in this research with the rigid-viscoplastic material constitutive model that is well suited for large deformation processes. As discussed in earlier sections, the efficiency of the Finite Element Method can be further enhanced for ring rolling by customizing the mesh motion. The same

23 20 technique can be applied to other similar flow dominated forming problems such as strip rolling, extrusion, thread rolling, and spin forming etc. Following is a detailed explanation on different techniques that can be used for customizing the finite element mesh motion irrespective of the material displacement field. It should be noted that these different techniques could utilize all the above different formulations in the analysis.

24 3.7 Continuous remeshing The most rudimentary method of customizing the finite element mesh motion is to use the traditional Lagrangian method and repeatedly regenerate the mesh after each time step of the analysis process whle maintaining the required mesh pattern. The field variables can be transferred to the new mesh using the weighted average method or least squares method that is more accurate but slower than the former. Ths technique has been tried in the past to eliminate the need for remeshing in three-dimensional problems. Backward extrusion, and shear extrusion are prime candidates for this technique. In these problems, remeshing was done by simply relocating the internal nodes using mesh smoothing techniques such as Laplacian smoothing. Theoretically, ALE mesh motion can be achieved by continuous remeshing by maintaining pre-determined nodal locations while keeping the same mesh topology. For ring rolling, thread rolling and similar processes, the new mesh can be constructed by maintaining the angular locations of the nodes so that the fine mesh region and the coarse mesh region are maintained in their respective locations. As outlined before, the problem size has to be significantly cut down for finite element analysis to be practical in industrial ring rolling problems. The mesh needs to be customized so that a fine mesh is employed in the deformation zone and coarse elements elsewhere in the ring billet. Customization of the mesh motion for ring rolling problems is done merely by freezing the angular motion of the mesh and letting it follow the material in other directions.

25 However, this requires relocation of the boundary nodes that can pose serious problems in repeated remeshing because the boundary is already discretized. Figure 3-1 shows the effect of repeated remeshing on the discretized geometry. The geometric discretization errors can be overcome by using geometric smoothing methods such as fitting quadratic parametric surfaces over surrounding facets as shown in Figure 3-2. However, this process needs a well-structured mesh on the free surface of the deforming body. Fortunately, circular geometries such as ring billets do have a well-structured mesh on the surface where each boundary node is shared by four element faces. Actual Original discretized geometry no. 2 Figure 3-1: Effect of Remeshing on the Discretized Geometry

26 Fitted Figure 3-2: Smooth Surface Approximation over Discretized Surface The geometric smoothing can be further enhanced by using the nature of the mesh in the ring billet. A well structured mesh in the ring billet is composed of hexahedral elements that can be generated by revolving the mesh in the cross-section of the billet (quadrilaterals). Therefore, the nodes in the billet can be arranged in individual continuous rings of nodes. These individual nodal groups can be used in the geometric smoothing process more effectively and accurately than using the surface mesh itself Curve fitting can be done using three or more nodes for relocation of nodes during the continuous remeshing. Figure 3-3 shows the use of three nodes (nearest node and two surrounding nodes) to relocate a boundary node. This method is very attractive for circular geometnes such as the ring except for the deformation zone where the circularity of the geometry is temporarily lost due to roll indentation. In the proximity of the entry

27 24 and exit sections of the deformation zone, the curve fitting adds approximation errors. The errors accumulate in these locations to unacceptable levels during the repeated remeshing and there is no robust technique to alleviate the problem. New node Old nodes Discretized \\ Predetermined, angular position Figure 3-3: Using an Arc through Three Points to Rebuild the Surface. The method of continuous remeshing can be very useful in some problems where the need for remeshing can be completely eliminated or delayed. Unfortunately, ring rolling problems demand relocation of the boundary nodes as well as the interior nodes. Any cumulative error introduced on the geometric position of the boundary nodes can grow to unacceptable levels in ring rolling due to large number of ring revolutions involved in the process. The repeated relocation of the boundary nodes therefore prevents using this technique of continuous remeshing in a meaningful manner in ring rolling problems.

28 3.2 Arbitrary Lagrangian Eulerian (ALE) Formulation Lagrangian description has been widely used to define the mechanics of deformation in problems with large material displacements. It uses the material coordinate system as the frame of reference. When the Lagrangian description is used with the finite element method, the mesh needs to be updated with the material displacements. Metal forming problems are usually handled with the so-called updated Lugrungiun Description where the finite element mesh is continuously updated after each non-linear iteration. Eulerian description, on the other hand, is traditionally used in fluid flow problems where the frame of reference is fixed in space. This method is very useful in steady state flow problems. The following can be highlighted as the advantages of updated Lagrangian description in metal forming applications over the Eulerian description. a) Lagrangian description introduces less complex governing equations due to the absence of material convection through element boundaries. b) Updating history dependent parameters is straightforward because material elements and finite elements coincide with each other.

29 26 c) Definition of the evolving boundary of the continuum does not need extra constraints because the boundary is defined with finite element nodes. However, there are also significant practical disadvantages associated with the Lagrangan description. The finite element mesh may become computationally irregular (severely distorted) when the workpiece undergoes large deformations. Severe distortion of the mesh is a result of lack of control over the mesh movement allowed in the formulation. Since the finite element mesh is tied to the material, the mesh has to follow the material flow leading to an entangled mesh in most bulk forming problems. In such cases, the deformed object has to be remeshed with improved mesh quality and the deformation history must be carefully transferred to the new mesh integration points. On the other hand, the Eulerian description uses a finite element mesh that is fixed in space. Eulerian formulation is usually used to solve problems in fluid mechanics where the region of interest is fixed in space. A major advantage of the Eulerian formulation is that the finite element mesh does not distort with the material flow. Therefore the Eulerian schemes are not threatened by remeshing problems as in the Lagrangan case. Several researchers[ have used a Lagrangian and Eulerian combined approach to extract advantages of both schemes. This combined approach is named as Arbitrary Lagrangian Eulerian (ALE) description where the finite element mesh can be given an arbitrary motion irrespective of the material flow. Lagrangian and Eulerian descriptions can be

30 identified as extreme cases of the spectrum of ALE description. ALE formulation inherits following advantages over conventional Lagrangian fbrrnulation. a) The finite element mesh can be controlled so that severe mesh distortion can be either eliminated or delayed. Finite element mesh does not have to follow the material flow. In the case of ring rolling (or any other similar process), finite element mesh can be given a customized motion irrespective of the material flow so that the regions with different mesh densities remain where they are supposed to be. b) Boundary conditions can be accurately modeled because the motion of the boundary elements can be controlled to represent a more accurate boundary area. It is this feature that attracted many researchers to use the ALE techniques for the analysis of fluid - structure interaction problems.

31 The basic governing equations in ALE description can be expressed as[34] Continuity: Momentum: Where, J = Jacobian, J = It1 x = Material coordinate X = Spatial coordinate p = Material density vi = i - component of material velocity wi = i - component of grid velocity au = ij - component of the Cauchy stress tensor The convection term in the momentum equation can be easily handled in metal forming problems. But, the convection term in the continuity equation poses sigruficant problems in satisfling continuity in rigid visco-plastic penaity formulation. In the

32 29 penalty formulation, treating the material as nearly incompressible satisfies the continuity. This condition is satisfied by penalizing volumetric strain in the material. Convection terms can be taken into account in the computation of volumetric strain. However, it should be noted that due to the penalty approach, any slightest error in computation of the convection term could lead to gross errors in the stress field. Though it was desirable to release the finite element mesh from the material for efficient usage of FEA in problems such as rolling and ring-rolling, ALE description introduces more problems while maintaining the mesh within the continuum of the workpiece. The finite element mesh must never leave the boundary of the object. This constraint can be introduced as an essential boundary condition on the finite element mesh by imposing zero material convection through the free sudace. At the same time finite element grid is free to slide on the boundary without leaving it. This ALE mesh constraint on the free boundary can be identified as one of the major disadvantages in ALE technique in metal forming applications. The essential boundary condition for the ALE mesh motion can be expressed as: (v-w)-n (3-3) where, v is the material velocity, w is the grid velocity, and n is the normal to the boundary.

33 30 The normal to the boundary surface at gnd points is not explicitly defined due to the discretization of the geometry. A simple averaging techmque may work only if a smaller relative velocity is present at the grid point. Still this may introduce errors at sharp corners. However, ths pauses a major dificulty in ring rolling problems due the relatively higher relative velocity (v - w) present at the boundary grid points. Accumulation of any errors in imposing this boundary condition over hundreds of time steps can lead to artificial distortions of the boundary. Several researchers have applied ALE formulation to solve metal forming problems in order to avoid reconstruction of the finite element mesh due to severe distortion. Ghosh and Kikuchi[34] used ALE formulation to solve numerous metal forming problems including back extrusion and side extrusion. Their primary intention was to maintain the finite element mesh computationally sound using adaptive gnd motion schemes. They were able to adjust the mesh so that discretization errors are minimized at the same time. They did not have the dificulty in maintaining boundary nodes because they were treated as Lagrangian while some selected interior nodes were treated as ALE. Another difficulty introduced by the ALE formulation is the updating of material history at element integration points that do not correspond to the same material points during the non-linear iterations. The fully implicit formulation that allows the use of large time steps with unconditional stability requires the finite element mesh updated

34 during the non-linear iteration process. The conventional Lagrangian formulation poses no difficulties as the mesh integration points correspond to the same material points during the entire forming process (provided that no remeshes are performed). But, the relative motion of the element integration points with respect to the material introduced by the ALE formulation requires additional computation to update the material history at the integration points. The time dependent variables at material points and the mesh integration points are related through the expression[34]: where, P t X - the time dependent variable - time - material coordinate X = spatial coordinate wk = k-component of the mesh velocity vk = k-component of the material velocity Xk - - k-component of the nodal coordinate Evaluation of the terms in the above equation can introduce a sigmficant amount of computational efrort in three-dimensional problems where the brick elements contain eight integration points. Therefore, only the so called direct iteration technique where the mesh is not updated during the non-linear iterations is suited for the compute intensive problems such as ring rolling processes when analyzed using the ALE

35 32 formulation. The direct iteration however is only conltionally stable. The small time steps in ring rolling problems which is restricted by other constraints such as the size of the contact area however is unlikely to pose any problems with respect to the stability of the direct iteration technique. Only one history update is required upon the acceptance of the solution in direct iteration technique.

36 3.3 Lagrangian Computation With Arbitrary Lagrangian Eulerian (ALE) Mesh update Another promising method to customize the finite element mesh motion in a large deformation problem is to use the Lagrangian computation and continuous remeshing with the help of another geometric entity. The second geometric entity can be in any form of geometric representation of the actual geometry of the deforming workpiece. Thls does not have to participate in the finite element analysis, but it must help accurate maintenance of the geometric deformation and related field variables. Such an alternative concept would eliminate the difficulties encountered in both ALE formulation and continuous remeshing as discussed in previous sections and deliver superior results without sacrificing accuracy. A logical choice for such accurate but simplistic representation of the geometry would be another mesh system which consists of very fine elements throughout the billet domain. The secondary mesh system can be updated and treated as a regular Lagrangian mesh that is etched in the material. Since the secondary mesh system is defined in the material coordinate system, it can be termed as the Material Mesh System (MMS). The primary mesh system can be termed as the Analysis Mesh System (AMS) which is used in the Lagrangian analysis to compute the velocity fieid. It can be automatically generated using the MMS imposing any customized mesh patterns. The customized mesh for the ring-rolling problems should consist of a fine mesh region in the deformation zone and a coarse mesh

37 34 elsewhere. A suitable choice of AMS and MMS for a radial ring rolling problem consisting of a main roll and a pressure roll without any axial rolls are shown in Figure 3-4. The analysis mesh is primarily used for the finite element analysis using the Lagrangian computation and the velocity field is then transported to the material mesh. The finely discretized material mesh is then updated with the material velocity field and the history dependent field variables such as strain. This concept of two mesh system can deliver the advantages of ALE formulation by being able to use a significantly reduced problem for the analysis without the difficulties encountered in the ALE implementation. It is therefore selected as the prime candidate for the ring rolling analysis. This concept also has great potential for application in other problems such as shape rolling, extrusion, and spin forming, thread rolling, orbital forging, etc.

38 Figure 3-4: A Typical Analysis Mesh and a Material Mesh Lagrangian Formulation with Rigid-viscoplastic Material Constitutive Relation The Lagrangan formulation is now well established for the large deformation problems. Among the various material models, the rigid visco-plastic constitutive relation has been found adequate and effective for bulk forming problems with large deformation. Kobayashi and co-workersr were among the first researchers who

39 36 developed and perfected the rigid-viscoplastic formulation for large deformation problems. With their success in application of the method in industrial forming problems, almost all the available commercial tools picked up this method as their primary solution for forming problems. Following is a brief description of the rigid-viscoplastic formulation. Deformable body Surface with prescribed velocity Figure 3-5: A Deformable Body Subjected to Boundary Conditions Consider a workpiece subjected to boundary condition as shown in Figure 3-5. When inertia forces are negligble, the Cauchy's equilibrium equation can be written as[30]:

40 where, qi = ji - component of the Cauchy stress tensor f; - I - component of the specific body force. The force boundary condition on the surface S,can be expressed as: where, ti - i - component of the prescribed surface traction n~ = j - component of the surface normal vector Using the Galerkin's weighted residual approach, the weak form of the equilibrium equation can be written as: where, & = i - component of the virtual incremental velocity, - 0 on S,, Integrating the first term in the first integral by parts: Using equations (3-7) and (3-8) and using the Gauss theorem:

41 where, di L, = -, ij - component of the spatial gradient of velocity oxj dsv, 6L, = -, ij- component of the virtual increment of the velocity gradient. Decomposing velocity gradient into the symmetric and skew symmetric parts: where, D. rl = i(* + 2) y - 2 zj ( I ii.i, ij - component of the rate-of-deformation tensor, ij - component of the spin tensor The virtual increments of these quantities can be related as: Substituting equation (3-11) in (3-9) and using q, mij = 0:

42 Equation (3-12) is known as the weak form of the Cauchy's equilibrium equation. Th~s equation and the material constitutive relationshp are used to derive the finite element governing equations. This equation is suitable for rate type material constitutive 3 9 relations where the Cauchy stress is related to the rate-of-deformation. The above equation contains instantaneous parameters such as the rate-of-deformation and the Cauchy stress. Therefore, unlike in elasto-plastic formulations, rigorous time integration schemes and computation of co-rotational quantities are not required for the analysis of large deformation problems. Evaluation of co-rotational quantities can be very compute intensive in large scale problems. The rigid-viscoplastic analysis is known to be about four times faster than the elasto-plastic analysis Rigid-Viscoplastic Constitutive Relationship The rigid viscoplastic relationshp has been provenc sufficiently accurate and effective for application in metal forming problems where the workpiece is subjected to very large deformations at high temperatures. The constitutive relationship relates the total Cauchy stress to the deviatoric component of the current rate-of-deformation (or the natural strain rate). The elastic component of the strain is neglected. The basic relationship for rigid viscopiastic material is given by[44]:

43 where, DL. = ij - component of the deviatoric rate-of-deformation tensor Y F - Viscosity constant f = -- 1, Huber - ~ s etype s yield function k k - strain hardening parameter - 0; - ij - component of the deviatoric Cauchy stress tensor Therefore, if I; 5 0 the material is treated rigid. Otherwise, the rate-of- deformation is proportional to the gradient off with respect to o. Taking the derivative off from equation (3-14) and substituting in equation (3-1 3): The effective strain-rate is defined as: Substituting from equation (3-15):

44 where, a = d; aha;, the effective stress From equations (3-16) and (3-17), deviatoric component of the Cauchy stress tensor can be related to the deviatoric component of the rate-of-deformation tensor as: where, R - 2 a = _, the material constant. a!. = RD;... rl (3-18) The above equation provides a simple relation between the deviatoric components of the rate-of-deformation tensor and the total Cauchy stress that is very convenient in the numerical implementation. However, the mean stress remains indeterminate due to the material incompressibility. This is resolved by treating the material slightly compressible. The traditional implementations of the rigid-viscoplastic formulation used a term analogous to bulk viscosity to determine the mean stress. Thls term is also known as the penalty constant in rigid-viscoplastic formulation. The mean stress was computed by multiplying the volumetric strain-rate with a very large (typically, of the order lows) constant. A drawback of this method is the accumulation of the volumetric loss during the simulation. Typical forming problems experienced 1-5%

45 42 volume loss during time steps. Using a larger value for the penalty constant can reduce the volume loss. However, larger values for the penalty constant tend to make the global stiffness matrix ill-conditioned, making the round-off errors large and causing difficulties in the convergence of the non-linear iteration process. ALPID[4 I] is a prime example of this type of implementation. This difficulty was overcome by using a term analogous to the bulk modulus instead of the bulk viscosity where the mean stress is evaluated using the total volumetric strain. This process can be seen as penalizing the total volumetric strain instead of the volumetric strain-rate. There is no accumulation of the volume loss under this approach. Therefore, a lower penalty constant could be used to deliver better results with faster numerical convergence in the non-linear iteration compared to that in the earlier approach. With the new approach, the volume loss in a typical forming problem remains around 0.1%-0.2% irrespective of the number of time steps. The mean stress is computed as: where, K - Penalty constant ETH = Thermal strain

46 The volumetric strain is computed as: where, v - Current volume ov = Original volume The numerical value for the penalty parameter K has to be sufficiently large for accurate simulation of the (nearly) incompressible material deformation problems. The numerical trials revealed that a value of IOOR is appropriate for K. The terms in the governing equilibrium equation (eqn 3-12) can be rearranged as: where, 0, - Hydrostatic stress - 4, - Kroneker delta Substituting from equations (3-18) and (3-19) and rearranging terms, equation (3-2 I ) can be rewritten as:

47 44 An important property of the above equation is that all the terms in the left hand side of the equation are symmetric. The symmetric terms would yield a symmetric stiffness matrix that will greatly simplifjr the subsequent numerical procedures. This equation is now ready for linearization for the finite element implementation. The linearization for Newton-Raphson non-linear iterations is performed by evaluating the lef3 hand side of the equation in the unknown configuration at time t + At.

48 4. Implementation After a comprehensive study of all the techniques available for the customization of the mesh motion, the two-mesh system method was chosen as the best candidate for the finite element analysis of ring rolling processes. Continuous remeshing technique using a single mesh system and the rzle formulation were in fact impiemented prior to the selection of the two-mesh system concept. The implementations of different techniques revealed unforeseen problems inherent to these techniques. The magnitude of the computation effort and the repeated nature of approximations involved in the implementation of these techniques in the analysis of ring rolling processes could only be assessed through rigorous implementations. As outlined in the previous chapter, this comprehensive study revealed that the accumulation of geometric errors in both these schemes can lead to unacceptable results in ring rolling as well as other continuous forming problems. It should be noted that a successful mesh customization alone does not solve the practical limitations of analysis techniques for ring rolling problems. A typical ring rolling process requires a very large number of revolutions (>50) of rolling. On the other hand, relatively small die contact area limits the size of the time step in numerical methods to a very small value so that the potential contact nodes do not skip the die contact boundary condition during the time step. If large time steps are used, nodes that

49 46 undergo die contact may skip the contact region during the geometric update though the time increment leahng to gross errors in the prediction of deformation. An exaggeration of this effect is schematically shown in Figure 4-1. Figure 4-1 : A graphical representation of the geometric errors in the presence of large time steps Usually, time step for non-linear problems are increased by using fully implicit equations where the finite element stiffness matrix is formed on the unknown end

50 47 configuration. Implicit schemes make the non-linear problem unconditionally stable with respect to the size of the time step. The size of the time step allowable in non-linear solution by explicit methods is very limited due to the conditional stability of the solution method[ Typically explicit methods have been proven superior to implicit methods in problems with extremely large strain rates such as in impact problems, and high speed rolling problems[46]. Problems like ring rolling perform well with the implicit methods and the explicit methods would require a very large number of time steps malung the application uneconomical. Convergence however, slows with the increasing time step size. Yet, it is always cheaper to use larger time steps against larger number of small steps to simulate the same forming process. As already explained, the only barrier to using larger time steps is the geometric limitations inherent to the ring rolling operations. A typical ring rolling process with a deformation zone confined to a segment of about five degrees should not use time steps that allow rotations in excess of one degree. This type of small time steps would demand more than time increments to complete 100 revolutions of the ring. With implicit methods, a typical industrial problem can be usually solved in less than 100 time steps. Stiffer compute requirements in ring rolling problems therefore demand further improvements in the overall approach if the computation times are to be kept within reasonable limits. The ring rolling process can be identified as a continuous deformation process with little variation in the rate of deformation with time. This quasi-steady nature of the deformation taking place in the ring rolling operation can be exploited to further improve

51 48 the computational efficiency of the analysis. The overall deformation during the entire ring rolling problem can be approximated to a decomposed set of steady-state deformation processes each lasting only one ring revolution. The errors introduced in this approximation are expected to be negligible in rolling processes with small rates of reduction. Almost all the ring rolling processes are expected to perform well with this approximation without hampering the accuracy of the results. In reality, the progressive deformation imparts a linear variation of strains from roll exit to the roll gap entry section. The strain jump across the roll gap is equivalent to the reduction applied in a single ring rotation. This behavior is clearly represented in the strain plot in the ring billet shown in Figure 4-2. Yet, this variation becomes very small as a fraction of the average strain towards the end of the rolling operation and eventually gets further smoothed out with the final finishing rolling passes. In typical ring rolling problems, prediction of this variation of deformation around the circumference is not very important. The prediction of total deformation (such as total strain in the cross section) and the final dimensions are treated as critical in the design perspective. In this respect, the approximation of deformation decomposition into a set of steady state deformation processes deems appropriate for the compute intensive ring rolling process analysis.

52 Color Index Figure 4-2: Linear variation of total strain imparted by a typical ring rolling o~erations

53 4.1 Lagrangian Computation Lagrangian computation is carried out on the analysis mesh with appropriate boundary conditions for the ring rolling problem. Therefore, the difficulties encountered in ALE computation, such as the constraint of zero material convection through the boundary of the billet continuum, are avoided. Continuous interpolation of material constitutive data at the mesh integration points during non-linear iterations was also avoided because the mesh integration points follow the material during the Lagrangian time step. Incompressibility of the material can also be easily satisfied with Lagrangian description since there is no material convection through the element boundaries. Free boundary is accurately represented with existing finite elements as opposed to the ALE mesh motion where special care needs to be taken if boundary nodes are treated as ALE nodes. ALE boundary nodes need to remain on the free boundary of the billet continuum thereby allowing them to only slide along the boundary. Finite element discretization leaves no robust mathematical technique for accurate computation of the boundary tangential directions. This task of maintaining ALE tangential motion of free boundary nodes becomes nearly impossible for problems such as ring rolling processes where the ALE mesh motion needs to be repeatedly applied on the same set nodes. A comprehensive study revealed the practical limitations of the ALE technique for ring rolling and other problems alike and lead to choosing the two mesh system assisted Lagrangian computation.

54 4.1.1 Mathematical Formulation The mathematical formulation for large deformation problem was implemented with rigid viscoplastic material constitutive relation. The rigid-viscoplastic formulation is simple and adequate for the ring rolling problems. As derived in the previous chapter, the governing equilibrium equation can be written with the iteration counters as: In the filly updated Lagrangan analysis, the terms in the above equation are evaluated in the unknown configuration at t - 4. After rigorous linearization as presented in Appendix A, the final form of matm equation for deformation analysis is obtained as: where, &)'I) = The tangent stiffness matrix corresponding to the deviatoric component of the deformation. This matrix is evaluated on the geometry at if" iteration.

55 52 The gradient stiffness matrix corresponding to the dialational component of the deformation. The matrix is computed on the geometry at? iteration. The external load vector for (i+ iteration. The internal load vector due to the deviatoric component of the stress at iih iteration. The internal load vector due to the mean stress (dialational component of the deformation). This vector is evaluated on the geometry at?' iteration. dvil-l~ = The incremental nodal velocity vector (solution) for the next iteration. The iterations are continued until convergence is reached. The right hand-side of the equation (4-2) is the out-of-balance load after i-iterations. The numerical convergence of the non-linear iterations can be defined either by the magnitude of the norm of the incremental velocity vector or the out-of-balance load vector. The magnitude of the incremental velocity of each node can be used as a more stringent requirement on the acceptance of the convergence. However, the relative norm

56 of the incremental velocity vector is much suited for large deformation problems. Therefore, the] 53 convergence is defined by: where, /I. 1 = Euclidean norm ECO, - Convergence tolerance Contact Boundary Condition Die contact boundary condition plays a major role in metal forming problems. The accuracy of the results of the metal forming problem and the convergence of the numerical method greatly depends on the accuracy and the effectiveness of the treatment of the die contact boundary condition. The interaction between the die and the workpiece takes a very complex form during the deformation process. In the fully updated Lagrangian scheme, the contact area has to be re-computed after each non-linear iteration. The complexity of the treatment of the die-contact is further increased by the fact that the nodes of the die and the billet almost never coincide during the deformation.

57 54 The basic idea of the contact algorithm is to eliminate any penetration of the billet nodes into the die or visa-versa. Usually, the billet nodes are considered as the master nodes and the die nodes are treated as the slave nodes. The computation of the penetration of the billet nodes into the die surface elements can be considered as the core of the contact algorithm. hexahedrons on surface elements wedge on the edge # overlap thickness Figure 4-3: Imaginary thickness of the die surjace The elimination of the overlap between the billet nodes and the die surface is the next step of the contact algorithm. Two basic methods are available for the treatment of the contact boundary condition. The simplest method of overlap elimination is to

58 explicitly apply the Dirichlet velocity boundary condition on the billet contact nodes in the normal direction to the die surface. This method is known as the direct method. The stiffness matrix and the load vector need to be expressed in a local coordinate frame at each contact node where one axis is aligned with overlap vector (die normal). Thus, application of this method needs further additional computation. Applicability is limited when a node is in simultaneous contact with two distinct bodies (dies) with normal directions that are not mutually not perpendicular. The definition of the local coordinate system will be ambiguous in these situations. When the two normal vectors are mutually perpendicular, the local coordinate system can be defined in such a way that two axes are aligned with the two die normals. Frictional traction is applied in the direction opposite to the sliding velocity that is tangential to the die surface. A major drawback of this method is that it is not suitable for contact between deformable bodies where both surfaces deform under contact forces. This method is also susceptible to numerical difficulties due to the discretization of the die surface. The numerical oscillations in the contact computation can lead to non-convergence in the overall non-linear problem. However, thls method has been already implemented[47] and proven adequate for large deformation problems involving die contact where die surface can be considered rigid for contact computation. A more involved but efficient method is the penalty approach where the overlap is eliminated using application of appropriate nodal forces in the direction normal to the die surface. The main advantages of the penalty method are:

59 It is readily applicable to contact between two deformable bodies. The contact forces are applied to both bodies. 56 This method does not have any difficulty in situations where the billet node is in contact with two dies irrespective their die normals. Contact forces are appiied along each die normal. It does not require the stiffness matrix and the Ioad vector expressed in IocaI coordinate systems. NurnericaIIy superior with non-iinear iterations in fuily updated Lagrangian scheme. Less prone to nodal oscillation between adjacent die elements due to geometric discretization. A variation of this method known as the Augmented Lagrangian approach was used in the present research. It can also be viewed as a penalty method in an incremental form applied during the non-linear iterations. The incremental form further improves the convergence of the non-linear iterations since the contact forces are systematically increased or decreased as needed by the internal Ioad vector and the instantaneous overlap. In certain cases, the conditionally stable direct method of application of Dirichlet boundary condition can make the system of non-linear equations unstabie in the

60 presence of large time steps. The performance of the unconchtionally stable augmented Lagrangian method is far superior to the direct method when large time steps are used Time Integration One of the most important aspects in the implementation of DEL technique in the analysis of ring rolling problems is the selection of an appropriate time integration parameter. The time integration parameter usually has a profound effect on the results in large deformation problems when large rotations are involved. In the implementation of the fully updated Lagrangian scheme, the nodal coordinates are updated after each nonlinear iteration. where, 'Y - Nodal coordinate, At = time increment, Av = increment in the velocity a - time integration parameter. The preceding superscript indicates the time and the succeedmg superscript indicates the iteration counter. The value of a depends on the type of the integration scheme chosen. The nodal coordinates for the first iteration in a new time increment is computed as:

61 5 8 I ""x:" = 'X, + (At) v,...( 4-5) The terms in the right hand side of equation (4-5) are the nodal coordinate and the velocity from the previous time step. a= I: The fully implicit formula usually used in the large deformation problems. This value is known to improve the convergence of the non-linear problem. The contact computation is upto-date with this selection as the contact forces are computed and applied in the current configuration. In the fully implicit scheme, mechanical equilibrium is satisfied at the end of the time step that is more desirable with large time steps. One drawback of this method is that it has potential of making elements collapsed (severely distorted) during the iterations. a=/2: This is known as the Trapezoidal or the Galerkin formula. Sometimes used in large deformation analysis. Practical experience shows some deterioration in the convergence of non-linear iterations. a = 0: This is known as the Euler formula. Sometimes termed as direct iteration scheme where the stiffness matrix does not need re-evaluation if the material is not rate dependent. The geometric stiffness remains constant during non-linear time steps. The equilibrium is satisfied at the beginning of the time step. The convergence slower compared to above schemes.

62 59 This may be useful when the mesh (at least one element) is nearly collapsed. The element shapes do not change during iterations and the mesh is updated only after convergence. The large rotations that occur in ring rolling problems however require the selection of a very effective time integration scheme. If elasto-plastic formulation were used for the analysis the effect of large rotations would significantly add extra computation since elasto-plastic formulation requires updating tensorial components through the time increment into a co-rotational coordinate frame. The approximate formulae such as the method used by Hughes and co-workers[48] may not suitable for this application. The aforementioned formula uses the spin tensor that can be easily derived from the velocity gradient tensor with the Cayley-Hamilton formula that does not need any matrix inversions or computing eigenvalues. This formula has become very popular in recent years replacing the classical method that involves rigorous calculations including matrix inversions and computing eigenvalues. However, due to the extent of large rotations involved in ring rolling problems the only method is the classical R-U decomposition that can be mathematical quite expensive. R-U decomposition is done by decomposing the deformation gradient tensor into a product of the rotation tensor R that is an orthogonal tensor and the stretch tensor U. Fortunately, the rigd-viscoplastic formulation does not require transformation of any tensorial quantities between the different material coordinate frames as in the elasto-

63 60 plastic formulation. The Cauchy stress is computed using the rate-of-deformation tensor that is an instantaneous quantity derived using the velocity field in the current configuration. Therefore, it is not necessary perform R-U decomposition or perform any rotations of tensorial components. As mentioned earlier, the time-discretization offers many combinations to compute the current configuration using the velocity fields of the previous time step (time t) and the current time step (previous iteration). The most commonly used time-integration constant (a = 1) was used in the implementation for updating the finite element mesh. However, computation of gradients such as the spatial velocity gradient requires further attention due to the sensitivity of the penalty formulation to the accuracy of the volumetric component. The penalty constant tends to significantly magnifjr any errors in the volumetric component of the deformation gradient quantities that can cause gross errors in the entire solution. As mentioned earlier, the penalty formulation for rigid-viscoplastic analysis was slightly modified from the conventional method by using the volumetric strain instead of the volumetric strain-rate. Computation of the volumetric strain is more accurate with the selection of a = 1. The rate terms are still involved in the volumetric term in the stiffness matrix. The following examples demonstrate the importance of the reference configuration within the time step for the computation of the volumetric strain rate. The linear element shown in Figure 4-4 undergoes volumetric compression and then expansion during the rigid body motion. The variation of the instantaneous volumetric strain rate is show in Figure 4-5. as seen in this example, at mid point configuration, the

64 61 volumetric strain is computed as zero whereas the end configuration possesses non-zero volumetric strain during a pure rigid body rotation. Therefore, the mid-point configuration was used for computation of all rate (gradient) quantities and the end point configuration was used for the geometry related quantities. Figure 4-4: Pure Rotatlon of a Linear Element (Exaggerated)

65 Figure 4-5: Volumetric strain within the time step

66 4.2 Transfer of field variables to the material mesh Upon completion of the Lagrangan time increment on the analysis mesh, all the field variables are transferred to the material mesh where deformation history is stored undisturbed. In order to prevent cumulative errors during data transfer and interpolation from the analysis mesh to the material mesh, history variables were transferred as increments that are accurate during the current time increment. For example, increments of temperature and effective strain are used during the forward transfer to the material mesh from the anaiysis mesh. But the total values of those variables are used during the reverse transfer to the analysis mesh from material mesh. As a result there would not be any cumulative errors associated with the transfer process in the material mesh. This is one of the most important advantages in using two mesh systems as opposed to the one mesh system for both analysis and storage of history variable in DEL approach. Transfer of field variables was made more efficient by utilizing a customized mesh for ring rolling analysis. Mesh configuration of the cross section is kept constant in the ring that can be viewed as a rotation of the cross sectional mesh about the axis of the ring malung three dimensional solid hexahedrons. Material mesh is also generated to have identical mesh configuration in the cross section. Nodes in the analysis mesh are sequenced to form a number of arrays in such a way that each array contains numbers of nodes belong to one closed line in the ring with same location in the cross section. This can be seen as grouping nodes in circumferential direction. The Directional Eulerian Lagrangian (DEL) mesh motion is executed in such a way that the nodal motion is frozen along these lines.

67 64 In other wards motion is treated as Eulerian along the lines and Lagrangan in the plane normal the lines at each node. At the same time nodes in the material mesh can seen as traveling along these lines. These lines can be viewed as streak lines in the material flow. Figure 4-6 shows one streak line formed in the ring billet. Figure 4-6: A streakline in the finite element mesh in the ring billet

68 65 Upon formation of the streak lines in the ring billet, transfer of field variables can be carried with minimal computation expense. Field variables that are available at element integration points are first transferred to the nodes in the analysis mesh using a weighted average method. where P,, = parameter value at the ith node Pe2 = parameter value at the jfh element center det(j) = determinant of the Jacobian at the element center (element volume) e = number of elements sharing node i Each mater~al node IS located in the parent streak line in the analysis mesh and the streak line is scanned to find the surround pair of analysis node. Then field variables are interpolated with respect to the angular position of the material node in analysis mesh. This is schematically shown in Figure 4-7. This simple interpolation scheme associated with the streak lines of the material and analysis mesh help reduce the computation time during transfer of field variables significantly. Each and every element in the analysis mesh have to be scanned to locate each and every material node in the corresponding analysis mesh in order to interpolate field variables in the absence of

69 66 streak lines. Such an approach without streak lines becomes more computationally expensive in the case of boundary nodes that cannot be located in any of the analysis mesh elements due to the discretization, but have to be located outside the discretized object. In that case, the nearest element has to be considered when transferring history variables. All these dieculties were overcome by employing the above streak line approach.

70 4.3 Regeneration of the Analysis Mesh After transferring field variables to the material mesh, analysis mesh has to be regenerated with updated material mesh. DEL mesh motion is obtained by freezing the angular position of the analysis mesh. Location of the analysis mesh nodes are obtained by interpolating cylindrical coordinated of the surrounding material nodes with respect to the angular location as: r = r l ~ (P - (r2 - rl) (a2 - al) - - zl+ (B- al) (r2--i) (a2- al) where r I, r2, z l,d, a 1, and a2 are shown in Figure 4-7. j3 = constant (obtained from the initial user supplied analysis mesh)

71 Material node : Analysis node Figure 4-7: Location of an analysis node in the rnateriul mesh

72 4.4 Virtual Rotation Upon Steady State Ring rolling processes conducted at elevated temperatures usually complete the process within a relatively smaller number of ring revolutions (40) with higher reduction rates. However, some ring rolling processes including those performed at lower temperatures need relatively large number of revolutions (>75) owing to the low feed rates permissible with the prevailing processing conditions. Analysis of those processes still need prohibitively large amounts of computation time due to the low time step size and high number time increments needed by the simulation even with the DEL mesh update scheme. Due to the inherent low feed rates (<I% in reduction), deformation in ring rolling processes can be assumed to be in a quasi-steady state. This behavior can be exploited to further improve the overall computational effort. Under this assumption, the ring rolling process can be decomposed into a set of stead state processes each lasting one revolution in the rolling process. Figure 4-8 schematically shows one such steady-state process. The continuous mandrel motion is replaced by step wise motion applying the reduction in each revolution in a single step. With these assumptions, ring rolling analysis can be carried out until the process reaches a steady state. The steady state deformation parameters can be used to extrapolate and complete the current ring revolution. It was noticed that if such a stepwise motion is applied to the pressure roll, the deformation

73 reaches steady state within less than degrees of the ring rotation thereby avoiding the majority of the computation effort. 70 The extrapolation of the steady state deformation in each ring rotation done by first extracting the geometry and the deformation parameters in the ring cross section in the steady state portion. A slice representing steady state deformation is first selected in the steady state region. The cross sectional geometry is used to extrapolate the deformation. This is done by reconstructing a completely new ring using the r and z coordinates (in cylindrical coordinate frame) of the streak-lines in the cross section and 8 coordinates of the nodes in the original mesh. The r-coordinate is adjusted to represent the diametric growth of the ring. The diametric growth of the ring is computed using the steady state cross-sectional area and the original volume of the ring thereby maintaining the original volume. The new mesh for the material mesh and the analysis mesh will have the same mesh topology except the nodal locations are changed for the new geometry. The redistribution of the deformation history parameters are basically done by assigning element and nodal parameters extracted from the stead state portion. This process becomes very straight forward and computationally efficient because of the already formed streaklines in the analysis mesh. The repeated mesh reconstruction after each revolution also regularizes the mesh avoiding severe mesh distortion in the circumferential direction. However, the severe mesh distortion in the cross section can be only dealt with remeshing the ring. The volume loss inherent to rigid-viscoplastic formulation also avoided with the repeated ring reconstruction. Otherwise, obtaining a

74 71 near isochoric deformation will be difficult with the large number of time steps involved in the ring rolling analysis. Errors in the process of transfer of deformation parameters between the mesh systems are also reduced by this simplification. Had virtual rotation not been performed, the errors in the data transfer between the mesh systems would continue to accumulate at already formed sections. The axis of the newly constructed ring is kept parallel to the z-axis for rings with symmetric cross sections. 'The new ring axis is simply located by shifting the current axis by the amount of the diametric growth in the current ring revolution. However, ring with non-symmetric sections can form a conical shape if the rolling process is not appropriately designed. The imbalance in the sectional reductions present in rings with non-symmetric sections lead to this phenomenon of cone formation. If proper care is not taken the conical shape can be lost during regeneration of ring. However, the phenomenon of conical formation of the ring was also treated in the current implementation of the virtual rotation. This option is automaticaily turned off when symmetry boundary conditions are encountered. Conical formation is treated by locating the ring axis inclined to the z-axis. The inclination and the placement of the ring axis is done using a least square fit of a series of axes of elemental rings made by each element in the cross section. However, this method may not suitable for large conical angles where accurate prediction not critical. Such cases need immehate attention for process redesign. Rings with small conical formation defects are routinely accepted after treated

75 them with sizing operations and the quantitative prediction of the conical formation is important in such cases. 72 / Main Roll 1 Figure 4-8: Schematic of quasi-steadv state deformation in ring rolling

76 4.5 Heat transkr during virtual rotation Actual process time lost during virtual rotation of the ring upon reachng steady state has to be considered in order to model heat transfer accurately. Steady state portion which is already subjected to the heat transfer to the dies through the contact area has to undergo heat transfer to the surrounding environment until it comes back to the roll gap in the next ring revolution. Therefore, only exposed boundary condition is modeled during this phase of heat transfer analysis. The pseudo code of the overall strategy with heat transfer during virtual rotation is shown in Figure 4-9.

77 /for I I I I to nrev while (transient) do Compute velocity field for current step. Transfer incremental deformations to material mesh. Update the material mesh and history. Derive the analysis mesh from material mesh. Transfer history data to the analysis mesh. I 1 end while I I 1 Scan the ring for steady-state information. 1 I I I I Reconstruct the ring with steady-state infomation and terminate current ring revolution, Perform heat transfer analysis for the remaining time actually needed to complete the ring rdation. Update the Temperature distribution in the ring. / end for I Figure 4-9: Pseudo Code for Non-isothermal Ring-Rolling Analysis with intermittent Ring Reconstruction

78 5. Performance Of The Two Mesh System Approach The two mesh system approach implemented under ths research showed a remarkable speedup in extremely time consuming metal forming analyses such as the ring rolling process. In contrast to the other methods such as the 2% dimensional FEA[49] (based on finite element method and the slab analysis), the two mesh approach does not seem to deteriorate the accuracy of results. This chapter is dedicated for a critical analysis of the two mesh system approach against the conventional FEA techniques.

79 5. f Comparison of the two mesh system approach against the conventional method The comparison of the two methods was done through a simulation of a simple ring rolling process. For simplicity, a ring with a rectangular cross-section was used in the study. The process parameters of the ring rolling are shown in Table 5-1. Table 5- I: Paramelers of the Ring Rolling Analysis Used in the Performance Analysis Ring billet dimensions Reduction rate Main roll diameter Pressure roll diameter Speed of the hven roll 1 "xlwx 5.0" OD 0.05" per rev 9" 2.75" 5V.m Analysis with the conventional FEM Finite element mesh for the ring for the use of conventional Lagrangan based FEM was constructed with uniform mesh density in the circurnferencial direction of the ring. The deciding factor of the level of discretization in the circumference was the size of the smallest contact area between the workpiece and the dies. Due to the discretization of the surface, at least three to four layers of surface elements are required

80 77 to model the surface traction at the contact interface. Based on the dimensions of the rolls and the ring billet, the contact area between in main dnven roll and the ring billet was seen as the smaller of the two contact interfaces seen by the workpiece. A rough but conservative estimate for the size of the contact area between the main driven roll and the ring billet can be obtained by a simple computation assuming that the main roll makes half the indentation on the ring cross-section. However, it has been observed that in real life ring rolling operations, most of the thickness reduction takes place at the outer radius by the main dnven roll. For the overall indentation of 0.05" applied in the above ring rolling process, it can be shown that the main roll makes an arc of contact of 8.1 degrees measured at the center of the ring billet. Therefore, the ring billet was discretized with 360 layers of elements along the circumference, each element making one degree arc. This level of discretization is deemed sufficient for modeling the contact with about 8 layers of elements to represent the contact interface. The initial mesh for the ring billet is shown in Figure 5-1. Both the main roll and the pressure roll were modeled as rigid surfaces. Only a small portion of the rolls was used in the model thereby reducing the computation burden on the contact search algorithm. However, these small segments of the rolls need to be rotated back periodically to maintain the contact conditions. This need was eliminated using virtual rotations of the rolls which does not update the angular position of the rolls but feeds the roll surface velocity to the contact computation. The surface segments of the main roll and the pressure roll used in the model are shown in Figure 5-2.

81 The determination of the time step for rolling problems needs several considerations. The need for faster analysis of the problem leads to the usage of larger time steps without any deterioration of the convergence. Fortunately, the fully updated Lagrangian formulation is capable of delivering faster convergence rates in the presence of larger time steps. However, the smaller contact region and the faster material motion across the contact region limits the size of the time step. Too large time steps may cause some surface nodes to skip the contact region without experiencing the contact surface tractions during the geometry update process. It is always advisable to limit the time step in problems with material motion through the contact region so that the potential contact nodes experience the contact condition through two or three time steps. The ring rotation f-br this ring rolling problem is approximately about 5 rpm (30 degrees per second). A time step of 0.03 seconds was used in the analysis so that the contact nodes would advance through about 0.9 degrees of rotation during each time step. This time step leads the contact nodes to experience the contact boundary conditions during 9 time steps in each ring rotation. The analysis parameters used in the simulation are summarized in Table 5-2. The flow stress of the material (lead) was expressed as o = 0.24~*.~~ MPa I. 78

82 Table 5-2: Analysis Parameters used in the Conventional Lagrangian Analysis Number of elements 3600 Number of nodes Time step 0.03 sec Billet material Lead Total number of time steps 15

83 Figure 5-1: Starting Mesh of the Ring Billet for the Conventional Lagrangian Analysis