- Email: [email protected]
Simulation of the orthogonal metal cutting process using an arbitrary Lagrangian–Eulerian finite-element method
Journal of Materials Processing Technology 103 (2000) 267±275
Simulation of the orthogonal metal cutting process using an arbitrary Lagrangian±Eulerian ®nite-element method M. Movahhedy, M.S. Gadala*, Y. Altintas Department of Mechanical Engineering, University of British Columbia, 2324 Main Mall, Vancouver, Canada V6T 1N5 Received 3 March 1999
Abstract Two different ®nite-element formulations, the Lagrangian and the Eulerian, have been used extensively in the modeling of the orthogonal metal cutting process. Each of these formulations has some disadvantages that make it inef®cient for modeling the cutting process. In this paper, it is shown that a more general formulation, the arbitrary Lagrangian±Eulerian (ALE) method may be used to combine the advantages and avoid the shortcomings of both of the previous methods. It is also shown that due to the characteristics of the cutting process, this formulation offers the most ef®cient modeling approach. Some preliminary results of this approach are presented to demonstrate its capabilities and potential in simulating the cutting process. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Orthogonal metal cutting process; ALE; Finite-element method
1. Introduction Metal cutting is one of the most common manufacturing operations, and a great deal of research has been devoted in understanding the mechanics of this process, with the objective of obtaining more effective cutting tools and more ef®cient manufacturing process plans. Traditionally, these objectives have been achieved by experimentation and prototyping. In recent years, however, with the surge in computational power and capacity, the focus has turned to numerical simulation of the process with the objective of replacing costly experimentation and prototyping with numerical simulation. Many such simulations have appeared in the literature. Despite the many simulation attempts reported in the literature, it is perhaps fair to say that cutting simulation has not become a common tool in industry, perhaps due to doubts about the reliability of such results and lack of suf®cient veri®cation. Much of this is related to the inherent complexity of the process, which probably makes it one of the most challenging processes from the numerical point of view. Although the actual machining process is simpli®ed to the most basic 2D plane-strain problem, orthogonal cutting, * Corresponding author. Tel.: 1-604-822-2777; fax: 1-604-822-2403. E-mail address: [email protected] (M.S. Gadala)
there are still many issues to deal with in such a numerical simulation. The cutting process usually involves large deformation of the material at very high strain rates (103±107 sÿ1) with the strains being much greater than unity. The chip produced is in contact with tool face in a highly pressured zone causing sticking friction, which will transform to sliding friction further up the tool face. Large plastic work and high friction generate an enormous amount of thermal energy in that locality, causing an increase in the temperature of the metal. The strain rate and temperature have opposite effects on the properties of the material. Considering that all of these phenomena occur in a very tiny region around the tip of the tool, the complexity of the process becomes evident. In a real machining process, there are tool wear and vibrations, also, which affect the process. The stress, strain, strain rate and temperature variables are all dependent on the cutting parameters such as the feed rate, and the cutting speed as well as on the geometrical features of the tool such as the rake angle and the nose radius. Further complicating the case is the fact that it is very dif®cult, if not impossible, to accurately obtain the ¯ow model of the material under these conditions. Apart from the complexities of the actual cutting process, there are aspects that make the numerical simulation of the cutting process particularly dif®cult. As with the analysis of
0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 4 8 0 - 5
268
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
other metalworking operations, large deformation of the material results in the distortion of the elements, causing deterioration of the results. On the other hand, unconstrained ¯ow of material occurs over free boundaries, which form a large area of the material boundaries. The boundary conditions change during the course of the analysis as the contact between the tool and the chip is being developed. However, more importantly, and unlike many forming processes, there is cutting action in which the bulk of the material is branched. This action is in a sense similar to what occurs in elastic±plastic crack-propagation problems. Simulating such cutting action is perhaps one of the most challenging aspects in this analysis. In this paper, the authors focus on the numerical aspects of the simulation of the cutting process and propose an alternative approach to deal with them. 2. Numerical aspects of cutting simulation The ®nite-element method has been largely used for the simulation of metalworking operations. From the viewpoint of numerical formulation, these analyses may be divided into two major approaches: the Lagrangian approach and the Eulerian approach. In the Lagrangian approach, a natural approach for solid-mechanics analysis, the FE mesh consists of material elements that cover the region of analysis exactly. These elements are attached to the material and are deformed with the deformation of the workpiece. This approach is particularly very convenient when unconstrained ¯ow of material is involved, since the FE mesh will accurately represent the material boundaries during the course of the analysis. The Eulerian approach, on the other hand, is more suitable for ¯uid-¯ow problems involving a control volume. In this method, the mesh consists of elements that are ®xed in space and cover the control volume, and the material properties are calculated at ®xed spatial locations as the material ¯ows through the mesh. This approach has also been used to model the large deformation of solids, mostly in metal forming analyses. Due to the ®xed nature of the approach, it is most suitable for cases where there are minimal free boundaries, i.e., where the boundaries of the material region are known a priori. Examples of these cases are closed-die forging and extrusion problems. Focusing on cutting process simulation, both approaches have been used extensively to model this process. In addition to relative ease in formulation and implementation, what makes the Lagrangian approach very attractive for this process is that the boundaries of the workpiece are mostly traction-free, and even the boundaries that are not free on the tool face are changing during the course of analysis. In this approach, the analysis can be started from indentation to the incipient stage to steady state. The chip is formed easily to its proper thickness, and no a priori assumption is needed about the shape of the chip. This approach, however, has some disadvantages too. With the deformation of the mate-
rial in front of the tool tip, the elements are greatly distorted such that sometimes the mesh has to be regenerated. More importantly, the simulation of cutting action at the tip of the tool has been traditionally achieved by separation of nodes in front of the tool tip, similar to what is usually done for the propagation of a self-similar crack in fracture-mechanics problems. Despite its simplicity, this node-separation method creates many problems that adversely affect the results. First and foremost is the choice of the criterion based on which a node in front of the tool tip is allowed to separate. Many criteria have been used in the literature, ranging from simple geometrical or strain criteria to more complex fracture-mechanics criteria [1±13]. Even if the type of criterion is chosen properly, there is no physical indication as to what criterion value should be adopted. All of this makes the separation criterion more of arbitrary nature, with great effects on the results. A critical look at the literature shows that whilst most researchers have used this approach, there is no consensus on the appropriate criterion. Usui and Shirakashi [1] use a geometrical criterion based on the separation of a node when it is suf®ciently close to the tool tip. The node is split into two nodes, one moving up along the rake face and the other remaining on the machined surface. Komvopoulos and Erpenbeck [2], Zhang and Bagchi [3], and Shih [4] use a similar distance criterion. Whilst these studies all agree that the value of the critical distance at which the node splits is crucial to the success of simulation, they use different values. In [2], a distance tolerance of 0.5L is used, where L is the side length of the element in front of the tool tip, whereas in [3], a value of (0.1±0.3)L is adopted and in [4], an arbitrary value based on trial runs is used. Carroll and Strenkowski [5] use a separation criterion based on the effective plastic strain at the node nearest to the cutting edge. The typical criterion values in their analysis are between 0.4 and 0.65 depending on the cutting conditions. Xie et al. [6] use such a criterion with a critical value of 0.5, whilst Hashemi et al. [7] adopt values that ranged from 0.6 to 1.5. Iwata et al. [8] employ a stress-based ductile fracture criterion as the chip-formation criterion. Ceretti et al. [9] use a damage criterion. Lin and Lin [10] note that the critical value for a criterion based on the effective plastic strain will be dependent on cutting parameters such as the depth of cut, thus such a criterion lacks generality. They propose a criterion based on the strain energy density, which is de®ned as a material constant obtained from the tensile test. Chen and Black [11] emphasize that the separation criterion should not be treated only as a numerical parameter, but as a material constant that may be measured experimentally. They point out, however, that the critical strain energy density determined from the uniaxial tensile test cannot be representative of the cutting operation, and that combinations of two or more separation criteria may also be used. Huang and Black [12] study various combinations and magnitudes of separation criteria and conclude that whilst
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
269
Fig. 1. Lagrangian cutting analysis with a 308 rake angle tool. Effective plastic strain contours are also shown.
the type of criterion may not be crucial, its magnitude has a great effect on the stress and strain distributions in the chip and the machined surface. They prefer a geometrical criterion for steady-state cutting simulation, because it is easier to determine. Imperative to using a node-splitting method is that there has to be a pre-de®ned parting line of the nodes, in front of the tool tip, that will be ``unzipped'' as the tool advances. In addition to this requiring on a priori assumption, it sometimes creates problems, as the parting line may be pushed out of position during the course of deformation and thus the target nodes will not be in front of the tool tip at the time of separation. The magnitude of the separation criterion has a profound effect on this, and may have to be relaxed to avoid it. On the other extreme, too relaxed a criterion creates an unrealistic material gap, which moves in front of the tool tip. This gap is commonly observed in the node-splitting approach, but not in experimental studies involving ductile materials. This crack also shifts the points of maximum stress from the tip of the tool to a point further up the tool, affecting the predicted shear angle. Finally, the use of parting line effectively restricts the analysis only to cutting with sharp-edge tools. There are also other side effects to node separation; it creates unbalanced forces in the analysis domain which should be treated. Also, a Lagrangian approach with node separation does not perform well for small or negative rake angles, including problems involving tools with chamfer or a nose radius. Although most of the above problems can be alleviated by using a denser mesh in the locality of the tool tip, that in itself increases computational intensity, and heightens the need for rezoning of the mesh. Therefore, the problem becomes highly mesh-dependent.
To demonstrate some of the problems pertinent to the Lagrangian approach with the node-splitting method, the authors performed a series of simulations using the updated Lagrangian approach and shear friction on the tool face. The material data are the same, as will be reported in Section 4 of this paper. The nodes are separated based on a critical distance criterion, the value of which is ®ne tuned in trial runs to ®nd the best compromise between having a smaller crack ahead of the tool and avoiding the shifting of the parting line. Fig. 1 shows a simulation with a 308 rake angle tool. It can be seen that a crack is moving ahead of the tip, and that due to shifting of the point of maximum stress further up on the tool, the shear plane does not emanate from the tool tip. The inadequacy of the node-splitting approach for smaller rake angles is shown in Fig. 2 which illustrates a simulation with a 108 rake angle tool. Although the critical distance is ®ne tuned to secure the best result, it is seen that the machined surface is unrealistically wavy due to shifting of the parting line during the process. A larger criterion value will alleviate this problem at the expense of having a larger crack ahead of the tool. Fig. 3 shows the normal and tangential forces on the tool in Lagrangian simulation. The oscillation in the forces is due to successive separation of nodes in front of the tool tip. Every time, a node is split, the forces on the tool are suddenly reduced, and then gradually increased until the next node is released. The dif®culties with the Lagrangian formulation with the node-splitting method have prompted some researchers to look for alternative methods for simulating the cutting action. For example, Ceretti et al. [9] used a scheme that deletes the elements that reach a critical damage value. Obviously, this approach violates the fundamental law of continuity due to loss of volume, and thus the mesh around
270
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
Fig. 2. Lagrangian cutting analysis with a 108 rake angle tool. Contours of effective plastic strain are also shown.
the tip should be very ®ne to minimize this volume loss. Furthermore, the method is too sensitive to the criterion value, as sometimes elements in places other than those neighboring to the tool tip may be targeted for deletion. The authors also report that after deletion of damaged elements, the domain boundaries have to be smoothed or else the subsequent remeshing attempt will fail. Overall, this approach seems to be too intrusive on the part of the analyst. Modi®cation to this approach may result in better simulation. As is done in crack-propagation problems, and instead of deleting the element completely, the stiffness characteristics of the element may be adjusted to re¯ect a zero or very small value in the separation direction. Such modi®cation, however, will still suffer from the other problems of the element-deletion method.
Fig. 3. Normal and tangential forces in Lagrangian cutting analysis.
A viable alternative for node splitting within a Lagrangian analysis is presented by simulating the cutting action as the continuous indentation and ¯ow of the material around the tool. Based on some experimental evidence, Madhavan et al. [14] suggested that the indentation might be the actual chipformation mechanism in ductile materials. In the indentation approach, cutting is simulated by forcing the tool into the material in small increments, causing the ¯ow of the material around the tool tip. As the tool advances, the FE mesh is distorted and has to be regenerated. In [14], the remeshing is triggered when the effective plastic strain exceeds a value of 0.25 at any point in the workpiece. Sekhon and Chenot [15] used a similar approach in which the FE mesh is checked at the end of each increment and the mesh is automatically regenerated if a distorted element is detected. Marusich and Ortiz [16] used continuous remeshing in combination with a mesh-smoothing algorithm in order to resolve mesh distortion problems. In their approach, elements are targeted for re®nement when their plastic power contents exceed a prescribed tolerance. Although the indentation approach overcomes many of the problems pertinent to the node-splitting approach, it comes at a high computational cost. First, the mesh has to be very ®ne around the tool tip, or else the material will overlap the tool in between the nodal points, and subsequently, the remeshing of the workpiece will amount to the loss of the overlapped material. Second, due to highly localized deformation and ®neness of elements, the mesh has to be regenerated frequently. In exemplary simulations performed using DEFORM-2D software and its automatic remeshing module, the entire region had to be remeshed at intervals of less than 10 incremental steps, and still considerable volume of material had to be cut out due to overlapping with the tool in every remeshing. Finally, once the FE mesh is regenerated,
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
material variables have to be interpolated from the old mesh to the new mesh. Since the two meshes may be considerably different from each other, this can introduce errors, which will eventually deteriorate the results. It should also be noted that even when the difference between the old and the regenerated mesh is small, the interpolation process is approximate and cannot account for the history of deformation. Also, there will be a speci®c need to use very small incremental steps and high computational power. As an example, Marusich and Ortiz [16] report that a typical simulation (including thermal analysis) involving 1100 six-node triangular elements required one million incremental steps which took 20 h of CPU time on a DEC 3000 workstation. It is obvious that the computational cost of this approach is sometimes prohibitive. In the Eulerian approach, on the other hand, none of the problems pertinent to the Lagrangian formulation exist in the modeling of the cutting action. The cutting action is modeled as the ¯ow of the material around the tool. In this respect, it is similar to the above-mentioned indentation. However, since the mesh is ®xed spatially, no mesh distortion occurs and consequently, no remeshing is required. The density of the mesh is only dictated by the expected gradients of stress and strain. Thus, the Eulerian approach is more ef®cient computationally and is best for modeling the region around the tool tip, at least for ductile materials. This approach, however, is not suitable for modeling the unconstrained ¯ow of material on free boundaries. The fact that the mesh is spatially ®xed means that an a priori assumption should be made about the shape of the chip [5,17±19] and the contact conditions. Because of the ®xed mesh, the contact length and the free boundaries of the chip cannot be developed as natural results of deformation, and adjustment of the boundaries are necessary outside of the solution domain. Commonly, an iterative procedure is employed in which the velocities at free boundary nodes are checked to ensure that their components normal to the boundary are zero and that the boundary is a streamline. This involves changing the boundary, adjustment of the mesh and interpolation of variables until convergence is achieved. It should be noted that the Eulerian formulation contains convective terms which make material time derivatives much more dif®cult to handle. History-dependent material properties have to be interpolated in an approximate way and material points have to be traced back to the ®xed mesh points. Many attempts have been made to do this. For example, Abo-Elkhier et al. [23] used a method in which a hypothetical mesh is generated using the displaced material points and an interpolation using element shape functions is performed between the hypothetical mesh and the ®xed Eulerian mesh. Such a process is computationally very intensive, introduces interpolation errors and creates convergence problems. The above-mentioned problems make the implementation of an Eulerian method more dif®cult than the implementation of the Lagrangian method. In the Eulerian cutting analyses appearing in the literature, the ¯ow approach,
271
introduced initially by Zienkiewicz and Godbole [20], is used in which the elasticity of the material is neglected, and thus the residual stresses in the machined surface cannot be predicted. 3. Arbitrary Lagrangian±Eulerian formulation It can be seen that both the Lagrangian and the Eulerian methods are inef®cient in modeling some of the characteristics of the cutting process. The Lagrangian performs well in modeling unconstrained formation of chip boundaries and contact length, whilst the Eulerian is the method of choice as far as the material ¯ow around the tip of the tool is concerned. Therefore, it would be ideal if these two methods could be combined in a single analysis and each of them used where they perform the best. A more general approach, the ALE method provides this opportunity. In an ALE analysis, the FE mesh is neither attached to the material nor ®xed in space. The mesh, in general, has a motion that is independent of the material. This general approach can be reduced to Eulerian or updated Lagrangian analyses as two special cases. In the following sections a scheme for the ALE method and its application in cutting analysis are reported. 3.1. ALE procedure Starting from the principle of virtual work, an ALE formulation is derived in terms of two sets of velocities, those of the material points v and those of the grid points v. These velocities are in general independent of each other, but there exists a one-to-one mapping between the material and the computational (grid) domains, provided that the Jacobian of the mapping function is non-zero. To satisfy this, the boundaries of the two domain should coincide, requiring that: v ÿ v n 0
(1)
where n is the normal vector at any point on the boundary. The physical interpretation of Eq. (1) is that no normal convective velocity occurs across the boundary if the surface particles remain on the surface. To establish the mapping between the two domains, the time derivatives of a function f (x) in the computational reference system is expressed in terms of its material time derivative in the material reference system (x): df (2) f 0 f_ v ÿ v dx where f 0 and f_ are the computational and material time derivatives, respectively. Eq. (2) is important in the incremental analysis since it gives the relationship between the material-associated quantities and the grid-point-associated quantities, and thus makes it possible to track the material deformation history when ALE is used.
272
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
Using the above equation in the expression for virtual work and after linearization of the equation, the following ®nal incremental ALE formulation will result (the details can be found in [21]): Z @dui t @dt vj t @dt vk t @dt sij t t t _ s v ÿ s s ÿ v dV ij ik t ij t k k t t v @ xj @ xk @ xk @ t xk Z @dt vk @dt fij t dV ÿ dui t fiB t t vk ÿ t vk t @ xk @ xk tv Z @dt vk @dt vj @dt sij t dS ÿ dui t fiS t ÿ t sik t nj t t vk ÿ t vk t nj t @ xk @ xk @ xk ts Z Z dui t fiB dt V dui t fiS dt S tv
ts
where the left superscript t refers to variables at time t (the start of the incremental step), f B and f S denote body and surface forces, respectively, sij are components of Cauchy stress tensor, nj represents boundary normals, and dui is virtual displacement. Discretization of the above equation yields an unsymmetrical stiffness matrix with 2N unknown variables whilst having only N equilibrium equations. Supplementary equations are needed, therefore, to solve this set of equations. Supplementary equations are provided by assigning the desired grid velocities in each incremental step at the start of that step. Any desired criterion can be used in deciding the mesh motion, re¯ecting the arbitrary nature of the ALE. For example, the mesh motion can be designed such that mesh distortion is minimized and elements retain their regular shapes throughout the analysis. Without loss of generality, the mesh motion can be expressed in terms of material velocity in the following form: vi a i b i vi
(4)
(no sum on i), where a(i) and b(i) are arbitrary coef®cients decided by the mesh-motion scheme. This form of expression for vi is very helpful in two respects. First, it helps to treat the supplementary constraint equations on the element level, thus reducing the size of the global stiffness matrix to N instead of 2N, and second, it simpli®es degeneration of the ALE equations to the Eulerian and updated Lagrangian approaches as special cases: if a i b i 0;
Eulerian formulation
if a i 0 and b i 1;
updated Lagrangian formulation
Note that in a general ALE motion, both of the coef®cients can be non-zero. As indicated above, the stiffness matrix resulting from Eq. (3) is non-symmetrical in general. It may be shown that when speci®c constitutive relations are introduced, the ALE formulation may be considered a logical extension to an updated Lagrangian formulation [24]. In other words, problems formulated with ALE may be solved using a regular symmetric Gauss elimination solver as the case in total or updated Lagrangian methods, by simply handling some
terms for the non-symmetric contributions to the stiffness matrix as loading terms and moving them to the right-hand side of the equilibrium equations. To implement the ALE approach, beside the usual procedures in incremental plasticity problems such as stress integration, two special procedures are needed at the end of each increment; one for the extrapolation of material-associated properties and the other for specifying the expected motion for each degree of freedom of the mesh in the next incremental step. After the solution has converged in an incremental step, the positions of nodal points are updated in such a way that the material and grid points coincide at the start of the next step. This requires that the properties of the material nodal points such as stresses, strains, and reaction forces are mapped to the new nodal positions. Instead of geometric extrapolation, as the common approach in previous Eulerian formulations, a rearranged form of Eq. (2) is used for transformation: df Df Df u ÿ u dx
(5)
where u and u stand for material and grid displacements, respectively. This equation ensures consistency of the ALE procedure. For the mesh-motion scheme, the objective has been set to retain the regularity of the mesh throughout the analysis. The trans®nite method for mesh generation has been modi®ed to account for the boundary motions of a patch of elements, and is applied at the end of each increment to ®nd the positions of nodal points in the next step. This method reproduces the boundary nodes exactly, and only the internal nodes of a patch are adjusted. Alternatively, an approach based on isoparametric mapping may be used that can handle patches with varying element density [22]. 4. Simulation example In this section, how the ¯exibility of the ALE approach can be used to create more ef®cient simulation for metal cutting processes is explained. The idea is to use features of the Eulerian approach in an area close to the tool tip, whilst simultaneously taking advantage of features of the Lagrangian approach in modeling unconstrained ¯ow of the material on the free boundaries, which de®nes the shape and size of the chip. To achieve this, a start is made by assuming an initial chip size, but it should be noted that this assumption is not an inherent requirement of the ALE; by designing a more sophisticated mesh-motion scheme, all stages of the process can be modeled. An initial chip geometry is assumed here as a matter of convenience, because it considerably simpli®es the mesh-motion scheme, and because the steady-state of the cutting process is of interest. During the analysis, part of the mesh is Eulerian (®xed in space), whilst the motion on free boundaries is of Lagrangian nature, i.e. it follows the material deformation. The rest of the domain possesses a
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
273
Fig. 4. Mesh-motion scheme in ALE cutting analysis.
general ALE motion, and moves within an incremental step at a speed determined by the mesh-motion scheme. To avoid entanglement of nodes on Lagrangian boundaries, the positions of these nodes are also adjusted tangential to the boundary, as this does not violate Eq. (1). In other words, the nodes on the boundaries are Lagrangian in the normal direction and ALE in the tangential direction. The capabilities of ALE formulation are demonstrated in the example that follows. It should be noted that the present purpose is only to demonstrate these capabilities and a thorough cutting analysis is not intended. An elasto-plastic analysis with linear strain-hardening is performed on a material with a Young's modulus of 200 GPa and an initial yield stress of 414 MPa. The strain-hardening modulus is chosen to be 1 GPa. The strain rate and temperature effects on material properties are neglected in this analysis. Fig. 4 shows the initial con®guration of the ®niteelement model as well as the con®guration when the cutting process has reached steady state. It can be seen that the nodal
positions in the central region of the model and in the neighborhood of the tool remain ®xed throughout the analysis (the Eulerian region). Displacement boundary conditions are applied on the left and bottom sides of the model, which advance the workpiece towards the stationary tool. The tool is assumed to be rigid, with a rake angle of 108, a nose radius of 0.025 mm, and a cutting a depth of 0.25 mm. The model consists of about 1000 linear triangular elements, and a total movement of 1.5 mm is simulated in 1000 incremental steps. The mesh-motion design in this simulation is shown in Fig. 4. It can be seen that as the material advances towards the tool, the area between the left boundary and the ®xed area becomes smaller, whilst the area between the ®xed area and the right boundary becomes larger. The motion of the nodal points in these regions is designed in such a way that the elements in these areas retain their regular shape. Also, the chip length increases steadily during the analysis, this being achieved by expanding the elements at the chip tail.
274
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
Fig. 5. Contours of effective stress in the primary and secondary deformation zone.
Again, the mesh-motion scheme retains the mesh regularity and the elements are distributed evenly in the shear deformation zone as well as in the chip tail. It should be noted that the nodes on free boundaries are free in the direction normal to the boundary at any point. Fig. 5 shows the formation of the chip after 0.05, 0.5, 1.0, and 1.5 mm of tool advance into the material. It can be seen that the chip is formed steadily. It is also noted that the thickness and the shape of the chip are developed in a natural unconstrained fashion, not affected by the assumed initial shape of the chip; the chip adjusts its thickness automatically, and without any need for the iterative adjustment performed in a similar Eulerian analysis. Fig. 5 also shows the contours of effective stress in the cutting zone. The chip-formation process, the primary and secondary deformation zones, the stress patterns around the tool tip and the chip curling, are all apparent in this ®gure. The area at the tail of the chip corresponds to the assumed initial chip. Finally, the ¯ow of the material within the workpiece and the chip is shown in Fig. 6, where the velocity vectors are plotted. The velocity vectors around the tool tip
Fig. 6. Velocity vectors in the ALE cutting analysis.
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
clearly show the plastic ¯ow of the material around the nose. Overall, the stress- and strain-distributions and the chip geometry agree qualitatively with experimental and numerical analyses reported in the literature. 5. Conclusions An ALE ®nite-element formulation is described and then employed to model the metal cutting process. It is shown that both the Lagrangian and Eulerian analyses suffer from shortcomings in modeling this process, and that the strength of both methods can be combined to achieve a more ef®cient cutting simulation. The features of an ALE analysis of cutting process are as follows: 1. No node-separation criterion is necessary, and no change in mesh topology is needed. 2. The density of the mesh around the tool tip need not be very high, and thus the process is computationally ef®cient. 3. The chip formation occurs by continuous plastic ¯ow of the material around the tool. 4. The shape and the thickness of the chip are developed automatically in the process and there is no need for iterative adjustment of the boundaries. 5. Frequent remeshing and data interpolation are avoided, as the mesh motion becomes part of the solution procedure. The computational intensity of the solution is not affected, however, as the supplementary constraint equations are treated on the element level [21]. An example has been presented to demonstrate the capabilities of the ALE approach. Further work is still under way to improve the performance of the approach in the simulation of metal cutting processes; the results of full scale and thorough analyses of the cutting process are to be presented in forthcoming publications. References [1] E. Usui, T. Shirakashi, Mechanics of machining-from descriptive to predictive theory, on the art of cutting metals Ð 75 years later, ASME-PED 7 (1982) 13±15. [2] K. Komvopoulos, S.A. Erpenbeck, Finite element modeling of orthogonal cutting, ASME J. Eng. Ind. 113 (1991) 253±267. [3] B. Zhang, A. Bagchi, Finite element simulation of chip formation and comparison with machining experiment, ASME J. Eng. Ind. 116 (1994) 289±297. [4] A.J. Shih, Finite element simulation of orthogonal metal cutting, ASME J. Eng. Ind. 117 (1995) 84±93.
275
[5] J.T. Carroll, J.S. Strenkowski, Finite element models of orthogonal cutting with application to single point diamond turning, Int. J. Mech. Sci. 30 (1988) 899±920. [6] J.Q. Xie, A.E. Bayoumi, H.M. Zbib, A study on shear banding in chip formation of orthogonal machining, Int. J. Mach. Tools Manuf. 36 (1996) 835±847. [7] J. Hashemi, A.A. Tseng, P.C. Chou, Finite element modeling of segmental chip formation in high speed orthogonal cutting, J. Mater. Eng. Perform. 3 (1994) 712±721. [8] K. Iwata, A. Osakada, Y. Terasaka, Process modeling of orthogonal cutting by rigid-plastic ®nite element method, J. Eng. Mater. Technol. 106 (1984) 132±138. [9] J.E. Ceretti, P. Fallbehmer, W.T. Wu, T.J. Altan, Application of 2D FEM on chip formation in orthogonal cutting, J. Mater. Process. Technol. 59 (1996) 169±181. [10] Z.C. Lin, S.Y. Lin, A coupled ®nite element model of thermo-elastic± plastic large deformation for orthogonal cutting, ASME J. Eng. Mater. Technol. 114 (1992) 218±226. [11] A.G. Chen, J.T. Black, FEM modeling in metal cutting, Manuf. Rev. 7 (1994) 120±133. [12] J.M. Huang, J.T. Black, An evaluation of chip separation criteria for the FEM simulation of machining, J. Manuf. Sci. Eng. 118 (1996) 545±554. [13] T. Obikawa, E. Usui, Computational machining of titanium alloy, ®nite element modeling and a few results, J. Manuf. Sci. Eng. 118 (1996) 208±215. [14] V. Madhavan, S. Chandrasekar, T.N. Farris, Mechanistic model of machining as an indentation process, in: D.A. Stephenson, R. Stevenson (Eds.), Materials Issues in Machining III and the Physics of Machining Process, The Mineral, Metals and Materials Society, Vol. 2, 1993, pp. 187±209. [15] G.S. Sekhon, J.L. Chenot, Numerical simulation of continuous chip formation during non-steady orthogonal cutting, Eng. Comput. 10 (1993) 31±48. [16] T.D. Marusich, M. Ortiz, Modeling and simulation of high-speed machining, Int. J. Numer. Methods Eng. 38 (1995) 3675±3694. [17] J.S. Strenkowski, K.J. Moon, Finite element prediction of chip geometry and tool±workpiece temperature distribution in orthogonal metal cutting, ASME J. Eng. Ind. 112 (1990) 313±318. [18] K.W. Kim, H.-C. Sin, Development of a thermo-viscoplastic cutting model using ®nite element method, Int. J. Mach. Tools Manuf. 36 (1996) 379±397. [19] J.-S. Wu, J.R. Dillon, W.-Y. Lu, Thermo-viscoplastic modeling of machining process using a mixed ®nite element method, ASME J. Manuf. Sci. Eng. 118 (1996) 470±482. [20] O.C. Zienkiewicz, P.N. Godbole, Flow of plastic and visco-plastic solids with special reference to extrusion and forming processes, Int. J. Numer. Methods Eng. 8 (1974) 3±16. [21] J. Wang, M.S. Gadala, Formulation and survey of ALE method in nonlinear solid mechanics, Finite Elements Anal. Des. 24 (1996) 253. [22] M.S. Gadala, M. Movahhedy, On the mesh motion for ALE modeling of metal forming processes, submitted to Finite Elements in Analysis and Design. [23] M. Abo-Elkhier, G.A.E. Oravas, M.A. Dokainish, A consistent Eulerian formulation for large deformation analysis with reference to metal-extrusion process, Int. J. Non-linear Mech. 52 (1988) 37±52. [24] M.S. Gadala, J. Wang, ALE formulation and its application in solid mechanics, Comput. Methods Appl. Mech. Eng. 167 (1998) 33±55.
Simulation of the orthogonal metal cutting process using an arbitrary Lagrangian±Eulerian ®nite-element method M. Movahhedy, M.S. Gadala*, Y. Altintas Department of Mechanical Engineering, University of British Columbia, 2324 Main Mall, Vancouver, Canada V6T 1N5 Received 3 March 1999
Abstract Two different ®nite-element formulations, the Lagrangian and the Eulerian, have been used extensively in the modeling of the orthogonal metal cutting process. Each of these formulations has some disadvantages that make it inef®cient for modeling the cutting process. In this paper, it is shown that a more general formulation, the arbitrary Lagrangian±Eulerian (ALE) method may be used to combine the advantages and avoid the shortcomings of both of the previous methods. It is also shown that due to the characteristics of the cutting process, this formulation offers the most ef®cient modeling approach. Some preliminary results of this approach are presented to demonstrate its capabilities and potential in simulating the cutting process. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Orthogonal metal cutting process; ALE; Finite-element method
1. Introduction Metal cutting is one of the most common manufacturing operations, and a great deal of research has been devoted in understanding the mechanics of this process, with the objective of obtaining more effective cutting tools and more ef®cient manufacturing process plans. Traditionally, these objectives have been achieved by experimentation and prototyping. In recent years, however, with the surge in computational power and capacity, the focus has turned to numerical simulation of the process with the objective of replacing costly experimentation and prototyping with numerical simulation. Many such simulations have appeared in the literature. Despite the many simulation attempts reported in the literature, it is perhaps fair to say that cutting simulation has not become a common tool in industry, perhaps due to doubts about the reliability of such results and lack of suf®cient veri®cation. Much of this is related to the inherent complexity of the process, which probably makes it one of the most challenging processes from the numerical point of view. Although the actual machining process is simpli®ed to the most basic 2D plane-strain problem, orthogonal cutting, * Corresponding author. Tel.: 1-604-822-2777; fax: 1-604-822-2403. E-mail address: [email protected] (M.S. Gadala)
there are still many issues to deal with in such a numerical simulation. The cutting process usually involves large deformation of the material at very high strain rates (103±107 sÿ1) with the strains being much greater than unity. The chip produced is in contact with tool face in a highly pressured zone causing sticking friction, which will transform to sliding friction further up the tool face. Large plastic work and high friction generate an enormous amount of thermal energy in that locality, causing an increase in the temperature of the metal. The strain rate and temperature have opposite effects on the properties of the material. Considering that all of these phenomena occur in a very tiny region around the tip of the tool, the complexity of the process becomes evident. In a real machining process, there are tool wear and vibrations, also, which affect the process. The stress, strain, strain rate and temperature variables are all dependent on the cutting parameters such as the feed rate, and the cutting speed as well as on the geometrical features of the tool such as the rake angle and the nose radius. Further complicating the case is the fact that it is very dif®cult, if not impossible, to accurately obtain the ¯ow model of the material under these conditions. Apart from the complexities of the actual cutting process, there are aspects that make the numerical simulation of the cutting process particularly dif®cult. As with the analysis of
0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 4 8 0 - 5
268
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
other metalworking operations, large deformation of the material results in the distortion of the elements, causing deterioration of the results. On the other hand, unconstrained ¯ow of material occurs over free boundaries, which form a large area of the material boundaries. The boundary conditions change during the course of the analysis as the contact between the tool and the chip is being developed. However, more importantly, and unlike many forming processes, there is cutting action in which the bulk of the material is branched. This action is in a sense similar to what occurs in elastic±plastic crack-propagation problems. Simulating such cutting action is perhaps one of the most challenging aspects in this analysis. In this paper, the authors focus on the numerical aspects of the simulation of the cutting process and propose an alternative approach to deal with them. 2. Numerical aspects of cutting simulation The ®nite-element method has been largely used for the simulation of metalworking operations. From the viewpoint of numerical formulation, these analyses may be divided into two major approaches: the Lagrangian approach and the Eulerian approach. In the Lagrangian approach, a natural approach for solid-mechanics analysis, the FE mesh consists of material elements that cover the region of analysis exactly. These elements are attached to the material and are deformed with the deformation of the workpiece. This approach is particularly very convenient when unconstrained ¯ow of material is involved, since the FE mesh will accurately represent the material boundaries during the course of the analysis. The Eulerian approach, on the other hand, is more suitable for ¯uid-¯ow problems involving a control volume. In this method, the mesh consists of elements that are ®xed in space and cover the control volume, and the material properties are calculated at ®xed spatial locations as the material ¯ows through the mesh. This approach has also been used to model the large deformation of solids, mostly in metal forming analyses. Due to the ®xed nature of the approach, it is most suitable for cases where there are minimal free boundaries, i.e., where the boundaries of the material region are known a priori. Examples of these cases are closed-die forging and extrusion problems. Focusing on cutting process simulation, both approaches have been used extensively to model this process. In addition to relative ease in formulation and implementation, what makes the Lagrangian approach very attractive for this process is that the boundaries of the workpiece are mostly traction-free, and even the boundaries that are not free on the tool face are changing during the course of analysis. In this approach, the analysis can be started from indentation to the incipient stage to steady state. The chip is formed easily to its proper thickness, and no a priori assumption is needed about the shape of the chip. This approach, however, has some disadvantages too. With the deformation of the mate-
rial in front of the tool tip, the elements are greatly distorted such that sometimes the mesh has to be regenerated. More importantly, the simulation of cutting action at the tip of the tool has been traditionally achieved by separation of nodes in front of the tool tip, similar to what is usually done for the propagation of a self-similar crack in fracture-mechanics problems. Despite its simplicity, this node-separation method creates many problems that adversely affect the results. First and foremost is the choice of the criterion based on which a node in front of the tool tip is allowed to separate. Many criteria have been used in the literature, ranging from simple geometrical or strain criteria to more complex fracture-mechanics criteria [1±13]. Even if the type of criterion is chosen properly, there is no physical indication as to what criterion value should be adopted. All of this makes the separation criterion more of arbitrary nature, with great effects on the results. A critical look at the literature shows that whilst most researchers have used this approach, there is no consensus on the appropriate criterion. Usui and Shirakashi [1] use a geometrical criterion based on the separation of a node when it is suf®ciently close to the tool tip. The node is split into two nodes, one moving up along the rake face and the other remaining on the machined surface. Komvopoulos and Erpenbeck [2], Zhang and Bagchi [3], and Shih [4] use a similar distance criterion. Whilst these studies all agree that the value of the critical distance at which the node splits is crucial to the success of simulation, they use different values. In [2], a distance tolerance of 0.5L is used, where L is the side length of the element in front of the tool tip, whereas in [3], a value of (0.1±0.3)L is adopted and in [4], an arbitrary value based on trial runs is used. Carroll and Strenkowski [5] use a separation criterion based on the effective plastic strain at the node nearest to the cutting edge. The typical criterion values in their analysis are between 0.4 and 0.65 depending on the cutting conditions. Xie et al. [6] use such a criterion with a critical value of 0.5, whilst Hashemi et al. [7] adopt values that ranged from 0.6 to 1.5. Iwata et al. [8] employ a stress-based ductile fracture criterion as the chip-formation criterion. Ceretti et al. [9] use a damage criterion. Lin and Lin [10] note that the critical value for a criterion based on the effective plastic strain will be dependent on cutting parameters such as the depth of cut, thus such a criterion lacks generality. They propose a criterion based on the strain energy density, which is de®ned as a material constant obtained from the tensile test. Chen and Black [11] emphasize that the separation criterion should not be treated only as a numerical parameter, but as a material constant that may be measured experimentally. They point out, however, that the critical strain energy density determined from the uniaxial tensile test cannot be representative of the cutting operation, and that combinations of two or more separation criteria may also be used. Huang and Black [12] study various combinations and magnitudes of separation criteria and conclude that whilst
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
269
Fig. 1. Lagrangian cutting analysis with a 308 rake angle tool. Effective plastic strain contours are also shown.
the type of criterion may not be crucial, its magnitude has a great effect on the stress and strain distributions in the chip and the machined surface. They prefer a geometrical criterion for steady-state cutting simulation, because it is easier to determine. Imperative to using a node-splitting method is that there has to be a pre-de®ned parting line of the nodes, in front of the tool tip, that will be ``unzipped'' as the tool advances. In addition to this requiring on a priori assumption, it sometimes creates problems, as the parting line may be pushed out of position during the course of deformation and thus the target nodes will not be in front of the tool tip at the time of separation. The magnitude of the separation criterion has a profound effect on this, and may have to be relaxed to avoid it. On the other extreme, too relaxed a criterion creates an unrealistic material gap, which moves in front of the tool tip. This gap is commonly observed in the node-splitting approach, but not in experimental studies involving ductile materials. This crack also shifts the points of maximum stress from the tip of the tool to a point further up the tool, affecting the predicted shear angle. Finally, the use of parting line effectively restricts the analysis only to cutting with sharp-edge tools. There are also other side effects to node separation; it creates unbalanced forces in the analysis domain which should be treated. Also, a Lagrangian approach with node separation does not perform well for small or negative rake angles, including problems involving tools with chamfer or a nose radius. Although most of the above problems can be alleviated by using a denser mesh in the locality of the tool tip, that in itself increases computational intensity, and heightens the need for rezoning of the mesh. Therefore, the problem becomes highly mesh-dependent.
To demonstrate some of the problems pertinent to the Lagrangian approach with the node-splitting method, the authors performed a series of simulations using the updated Lagrangian approach and shear friction on the tool face. The material data are the same, as will be reported in Section 4 of this paper. The nodes are separated based on a critical distance criterion, the value of which is ®ne tuned in trial runs to ®nd the best compromise between having a smaller crack ahead of the tool and avoiding the shifting of the parting line. Fig. 1 shows a simulation with a 308 rake angle tool. It can be seen that a crack is moving ahead of the tip, and that due to shifting of the point of maximum stress further up on the tool, the shear plane does not emanate from the tool tip. The inadequacy of the node-splitting approach for smaller rake angles is shown in Fig. 2 which illustrates a simulation with a 108 rake angle tool. Although the critical distance is ®ne tuned to secure the best result, it is seen that the machined surface is unrealistically wavy due to shifting of the parting line during the process. A larger criterion value will alleviate this problem at the expense of having a larger crack ahead of the tool. Fig. 3 shows the normal and tangential forces on the tool in Lagrangian simulation. The oscillation in the forces is due to successive separation of nodes in front of the tool tip. Every time, a node is split, the forces on the tool are suddenly reduced, and then gradually increased until the next node is released. The dif®culties with the Lagrangian formulation with the node-splitting method have prompted some researchers to look for alternative methods for simulating the cutting action. For example, Ceretti et al. [9] used a scheme that deletes the elements that reach a critical damage value. Obviously, this approach violates the fundamental law of continuity due to loss of volume, and thus the mesh around
270
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
Fig. 2. Lagrangian cutting analysis with a 108 rake angle tool. Contours of effective plastic strain are also shown.
the tip should be very ®ne to minimize this volume loss. Furthermore, the method is too sensitive to the criterion value, as sometimes elements in places other than those neighboring to the tool tip may be targeted for deletion. The authors also report that after deletion of damaged elements, the domain boundaries have to be smoothed or else the subsequent remeshing attempt will fail. Overall, this approach seems to be too intrusive on the part of the analyst. Modi®cation to this approach may result in better simulation. As is done in crack-propagation problems, and instead of deleting the element completely, the stiffness characteristics of the element may be adjusted to re¯ect a zero or very small value in the separation direction. Such modi®cation, however, will still suffer from the other problems of the element-deletion method.
Fig. 3. Normal and tangential forces in Lagrangian cutting analysis.
A viable alternative for node splitting within a Lagrangian analysis is presented by simulating the cutting action as the continuous indentation and ¯ow of the material around the tool. Based on some experimental evidence, Madhavan et al. [14] suggested that the indentation might be the actual chipformation mechanism in ductile materials. In the indentation approach, cutting is simulated by forcing the tool into the material in small increments, causing the ¯ow of the material around the tool tip. As the tool advances, the FE mesh is distorted and has to be regenerated. In [14], the remeshing is triggered when the effective plastic strain exceeds a value of 0.25 at any point in the workpiece. Sekhon and Chenot [15] used a similar approach in which the FE mesh is checked at the end of each increment and the mesh is automatically regenerated if a distorted element is detected. Marusich and Ortiz [16] used continuous remeshing in combination with a mesh-smoothing algorithm in order to resolve mesh distortion problems. In their approach, elements are targeted for re®nement when their plastic power contents exceed a prescribed tolerance. Although the indentation approach overcomes many of the problems pertinent to the node-splitting approach, it comes at a high computational cost. First, the mesh has to be very ®ne around the tool tip, or else the material will overlap the tool in between the nodal points, and subsequently, the remeshing of the workpiece will amount to the loss of the overlapped material. Second, due to highly localized deformation and ®neness of elements, the mesh has to be regenerated frequently. In exemplary simulations performed using DEFORM-2D software and its automatic remeshing module, the entire region had to be remeshed at intervals of less than 10 incremental steps, and still considerable volume of material had to be cut out due to overlapping with the tool in every remeshing. Finally, once the FE mesh is regenerated,
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
material variables have to be interpolated from the old mesh to the new mesh. Since the two meshes may be considerably different from each other, this can introduce errors, which will eventually deteriorate the results. It should also be noted that even when the difference between the old and the regenerated mesh is small, the interpolation process is approximate and cannot account for the history of deformation. Also, there will be a speci®c need to use very small incremental steps and high computational power. As an example, Marusich and Ortiz [16] report that a typical simulation (including thermal analysis) involving 1100 six-node triangular elements required one million incremental steps which took 20 h of CPU time on a DEC 3000 workstation. It is obvious that the computational cost of this approach is sometimes prohibitive. In the Eulerian approach, on the other hand, none of the problems pertinent to the Lagrangian formulation exist in the modeling of the cutting action. The cutting action is modeled as the ¯ow of the material around the tool. In this respect, it is similar to the above-mentioned indentation. However, since the mesh is ®xed spatially, no mesh distortion occurs and consequently, no remeshing is required. The density of the mesh is only dictated by the expected gradients of stress and strain. Thus, the Eulerian approach is more ef®cient computationally and is best for modeling the region around the tool tip, at least for ductile materials. This approach, however, is not suitable for modeling the unconstrained ¯ow of material on free boundaries. The fact that the mesh is spatially ®xed means that an a priori assumption should be made about the shape of the chip [5,17±19] and the contact conditions. Because of the ®xed mesh, the contact length and the free boundaries of the chip cannot be developed as natural results of deformation, and adjustment of the boundaries are necessary outside of the solution domain. Commonly, an iterative procedure is employed in which the velocities at free boundary nodes are checked to ensure that their components normal to the boundary are zero and that the boundary is a streamline. This involves changing the boundary, adjustment of the mesh and interpolation of variables until convergence is achieved. It should be noted that the Eulerian formulation contains convective terms which make material time derivatives much more dif®cult to handle. History-dependent material properties have to be interpolated in an approximate way and material points have to be traced back to the ®xed mesh points. Many attempts have been made to do this. For example, Abo-Elkhier et al. [23] used a method in which a hypothetical mesh is generated using the displaced material points and an interpolation using element shape functions is performed between the hypothetical mesh and the ®xed Eulerian mesh. Such a process is computationally very intensive, introduces interpolation errors and creates convergence problems. The above-mentioned problems make the implementation of an Eulerian method more dif®cult than the implementation of the Lagrangian method. In the Eulerian cutting analyses appearing in the literature, the ¯ow approach,
271
introduced initially by Zienkiewicz and Godbole [20], is used in which the elasticity of the material is neglected, and thus the residual stresses in the machined surface cannot be predicted. 3. Arbitrary Lagrangian±Eulerian formulation It can be seen that both the Lagrangian and the Eulerian methods are inef®cient in modeling some of the characteristics of the cutting process. The Lagrangian performs well in modeling unconstrained formation of chip boundaries and contact length, whilst the Eulerian is the method of choice as far as the material ¯ow around the tip of the tool is concerned. Therefore, it would be ideal if these two methods could be combined in a single analysis and each of them used where they perform the best. A more general approach, the ALE method provides this opportunity. In an ALE analysis, the FE mesh is neither attached to the material nor ®xed in space. The mesh, in general, has a motion that is independent of the material. This general approach can be reduced to Eulerian or updated Lagrangian analyses as two special cases. In the following sections a scheme for the ALE method and its application in cutting analysis are reported. 3.1. ALE procedure Starting from the principle of virtual work, an ALE formulation is derived in terms of two sets of velocities, those of the material points v and those of the grid points v. These velocities are in general independent of each other, but there exists a one-to-one mapping between the material and the computational (grid) domains, provided that the Jacobian of the mapping function is non-zero. To satisfy this, the boundaries of the two domain should coincide, requiring that: v ÿ v n 0
(1)
where n is the normal vector at any point on the boundary. The physical interpretation of Eq. (1) is that no normal convective velocity occurs across the boundary if the surface particles remain on the surface. To establish the mapping between the two domains, the time derivatives of a function f (x) in the computational reference system is expressed in terms of its material time derivative in the material reference system (x): df (2) f 0 f_ v ÿ v dx where f 0 and f_ are the computational and material time derivatives, respectively. Eq. (2) is important in the incremental analysis since it gives the relationship between the material-associated quantities and the grid-point-associated quantities, and thus makes it possible to track the material deformation history when ALE is used.
272
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
Using the above equation in the expression for virtual work and after linearization of the equation, the following ®nal incremental ALE formulation will result (the details can be found in [21]): Z @dui t @dt vj t @dt vk t @dt sij t t t _ s v ÿ s s ÿ v dV ij ik t ij t k k t t v @ xj @ xk @ xk @ t xk Z @dt vk @dt fij t dV ÿ dui t fiB t t vk ÿ t vk t @ xk @ xk tv Z @dt vk @dt vj @dt sij t dS ÿ dui t fiS t ÿ t sik t nj t t vk ÿ t vk t nj t @ xk @ xk @ xk ts Z Z dui t fiB dt V dui t fiS dt S tv
ts
where the left superscript t refers to variables at time t (the start of the incremental step), f B and f S denote body and surface forces, respectively, sij are components of Cauchy stress tensor, nj represents boundary normals, and dui is virtual displacement. Discretization of the above equation yields an unsymmetrical stiffness matrix with 2N unknown variables whilst having only N equilibrium equations. Supplementary equations are needed, therefore, to solve this set of equations. Supplementary equations are provided by assigning the desired grid velocities in each incremental step at the start of that step. Any desired criterion can be used in deciding the mesh motion, re¯ecting the arbitrary nature of the ALE. For example, the mesh motion can be designed such that mesh distortion is minimized and elements retain their regular shapes throughout the analysis. Without loss of generality, the mesh motion can be expressed in terms of material velocity in the following form: vi a i b i vi
(4)
(no sum on i), where a(i) and b(i) are arbitrary coef®cients decided by the mesh-motion scheme. This form of expression for vi is very helpful in two respects. First, it helps to treat the supplementary constraint equations on the element level, thus reducing the size of the global stiffness matrix to N instead of 2N, and second, it simpli®es degeneration of the ALE equations to the Eulerian and updated Lagrangian approaches as special cases: if a i b i 0;
Eulerian formulation
if a i 0 and b i 1;
updated Lagrangian formulation
Note that in a general ALE motion, both of the coef®cients can be non-zero. As indicated above, the stiffness matrix resulting from Eq. (3) is non-symmetrical in general. It may be shown that when speci®c constitutive relations are introduced, the ALE formulation may be considered a logical extension to an updated Lagrangian formulation [24]. In other words, problems formulated with ALE may be solved using a regular symmetric Gauss elimination solver as the case in total or updated Lagrangian methods, by simply handling some
terms for the non-symmetric contributions to the stiffness matrix as loading terms and moving them to the right-hand side of the equilibrium equations. To implement the ALE approach, beside the usual procedures in incremental plasticity problems such as stress integration, two special procedures are needed at the end of each increment; one for the extrapolation of material-associated properties and the other for specifying the expected motion for each degree of freedom of the mesh in the next incremental step. After the solution has converged in an incremental step, the positions of nodal points are updated in such a way that the material and grid points coincide at the start of the next step. This requires that the properties of the material nodal points such as stresses, strains, and reaction forces are mapped to the new nodal positions. Instead of geometric extrapolation, as the common approach in previous Eulerian formulations, a rearranged form of Eq. (2) is used for transformation: df Df Df u ÿ u dx
(5)
where u and u stand for material and grid displacements, respectively. This equation ensures consistency of the ALE procedure. For the mesh-motion scheme, the objective has been set to retain the regularity of the mesh throughout the analysis. The trans®nite method for mesh generation has been modi®ed to account for the boundary motions of a patch of elements, and is applied at the end of each increment to ®nd the positions of nodal points in the next step. This method reproduces the boundary nodes exactly, and only the internal nodes of a patch are adjusted. Alternatively, an approach based on isoparametric mapping may be used that can handle patches with varying element density [22]. 4. Simulation example In this section, how the ¯exibility of the ALE approach can be used to create more ef®cient simulation for metal cutting processes is explained. The idea is to use features of the Eulerian approach in an area close to the tool tip, whilst simultaneously taking advantage of features of the Lagrangian approach in modeling unconstrained ¯ow of the material on the free boundaries, which de®nes the shape and size of the chip. To achieve this, a start is made by assuming an initial chip size, but it should be noted that this assumption is not an inherent requirement of the ALE; by designing a more sophisticated mesh-motion scheme, all stages of the process can be modeled. An initial chip geometry is assumed here as a matter of convenience, because it considerably simpli®es the mesh-motion scheme, and because the steady-state of the cutting process is of interest. During the analysis, part of the mesh is Eulerian (®xed in space), whilst the motion on free boundaries is of Lagrangian nature, i.e. it follows the material deformation. The rest of the domain possesses a
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
273
Fig. 4. Mesh-motion scheme in ALE cutting analysis.
general ALE motion, and moves within an incremental step at a speed determined by the mesh-motion scheme. To avoid entanglement of nodes on Lagrangian boundaries, the positions of these nodes are also adjusted tangential to the boundary, as this does not violate Eq. (1). In other words, the nodes on the boundaries are Lagrangian in the normal direction and ALE in the tangential direction. The capabilities of ALE formulation are demonstrated in the example that follows. It should be noted that the present purpose is only to demonstrate these capabilities and a thorough cutting analysis is not intended. An elasto-plastic analysis with linear strain-hardening is performed on a material with a Young's modulus of 200 GPa and an initial yield stress of 414 MPa. The strain-hardening modulus is chosen to be 1 GPa. The strain rate and temperature effects on material properties are neglected in this analysis. Fig. 4 shows the initial con®guration of the ®niteelement model as well as the con®guration when the cutting process has reached steady state. It can be seen that the nodal
positions in the central region of the model and in the neighborhood of the tool remain ®xed throughout the analysis (the Eulerian region). Displacement boundary conditions are applied on the left and bottom sides of the model, which advance the workpiece towards the stationary tool. The tool is assumed to be rigid, with a rake angle of 108, a nose radius of 0.025 mm, and a cutting a depth of 0.25 mm. The model consists of about 1000 linear triangular elements, and a total movement of 1.5 mm is simulated in 1000 incremental steps. The mesh-motion design in this simulation is shown in Fig. 4. It can be seen that as the material advances towards the tool, the area between the left boundary and the ®xed area becomes smaller, whilst the area between the ®xed area and the right boundary becomes larger. The motion of the nodal points in these regions is designed in such a way that the elements in these areas retain their regular shape. Also, the chip length increases steadily during the analysis, this being achieved by expanding the elements at the chip tail.
274
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
Fig. 5. Contours of effective stress in the primary and secondary deformation zone.
Again, the mesh-motion scheme retains the mesh regularity and the elements are distributed evenly in the shear deformation zone as well as in the chip tail. It should be noted that the nodes on free boundaries are free in the direction normal to the boundary at any point. Fig. 5 shows the formation of the chip after 0.05, 0.5, 1.0, and 1.5 mm of tool advance into the material. It can be seen that the chip is formed steadily. It is also noted that the thickness and the shape of the chip are developed in a natural unconstrained fashion, not affected by the assumed initial shape of the chip; the chip adjusts its thickness automatically, and without any need for the iterative adjustment performed in a similar Eulerian analysis. Fig. 5 also shows the contours of effective stress in the cutting zone. The chip-formation process, the primary and secondary deformation zones, the stress patterns around the tool tip and the chip curling, are all apparent in this ®gure. The area at the tail of the chip corresponds to the assumed initial chip. Finally, the ¯ow of the material within the workpiece and the chip is shown in Fig. 6, where the velocity vectors are plotted. The velocity vectors around the tool tip
Fig. 6. Velocity vectors in the ALE cutting analysis.
M. Movahhedy et al. / Journal of Materials Processing Technology 103 (2000) 267±275
clearly show the plastic ¯ow of the material around the nose. Overall, the stress- and strain-distributions and the chip geometry agree qualitatively with experimental and numerical analyses reported in the literature. 5. Conclusions An ALE ®nite-element formulation is described and then employed to model the metal cutting process. It is shown that both the Lagrangian and Eulerian analyses suffer from shortcomings in modeling this process, and that the strength of both methods can be combined to achieve a more ef®cient cutting simulation. The features of an ALE analysis of cutting process are as follows: 1. No node-separation criterion is necessary, and no change in mesh topology is needed. 2. The density of the mesh around the tool tip need not be very high, and thus the process is computationally ef®cient. 3. The chip formation occurs by continuous plastic ¯ow of the material around the tool. 4. The shape and the thickness of the chip are developed automatically in the process and there is no need for iterative adjustment of the boundaries. 5. Frequent remeshing and data interpolation are avoided, as the mesh motion becomes part of the solution procedure. The computational intensity of the solution is not affected, however, as the supplementary constraint equations are treated on the element level [21]. An example has been presented to demonstrate the capabilities of the ALE approach. Further work is still under way to improve the performance of the approach in the simulation of metal cutting processes; the results of full scale and thorough analyses of the cutting process are to be presented in forthcoming publications. References [1] E. Usui, T. Shirakashi, Mechanics of machining-from descriptive to predictive theory, on the art of cutting metals Ð 75 years later, ASME-PED 7 (1982) 13±15. [2] K. Komvopoulos, S.A. Erpenbeck, Finite element modeling of orthogonal cutting, ASME J. Eng. Ind. 113 (1991) 253±267. [3] B. Zhang, A. Bagchi, Finite element simulation of chip formation and comparison with machining experiment, ASME J. Eng. Ind. 116 (1994) 289±297. [4] A.J. Shih, Finite element simulation of orthogonal metal cutting, ASME J. Eng. Ind. 117 (1995) 84±93.
275
[5] J.T. Carroll, J.S. Strenkowski, Finite element models of orthogonal cutting with application to single point diamond turning, Int. J. Mech. Sci. 30 (1988) 899±920. [6] J.Q. Xie, A.E. Bayoumi, H.M. Zbib, A study on shear banding in chip formation of orthogonal machining, Int. J. Mach. Tools Manuf. 36 (1996) 835±847. [7] J. Hashemi, A.A. Tseng, P.C. Chou, Finite element modeling of segmental chip formation in high speed orthogonal cutting, J. Mater. Eng. Perform. 3 (1994) 712±721. [8] K. Iwata, A. Osakada, Y. Terasaka, Process modeling of orthogonal cutting by rigid-plastic ®nite element method, J. Eng. Mater. Technol. 106 (1984) 132±138. [9] J.E. Ceretti, P. Fallbehmer, W.T. Wu, T.J. Altan, Application of 2D FEM on chip formation in orthogonal cutting, J. Mater. Process. Technol. 59 (1996) 169±181. [10] Z.C. Lin, S.Y. Lin, A coupled ®nite element model of thermo-elastic± plastic large deformation for orthogonal cutting, ASME J. Eng. Mater. Technol. 114 (1992) 218±226. [11] A.G. Chen, J.T. Black, FEM modeling in metal cutting, Manuf. Rev. 7 (1994) 120±133. [12] J.M. Huang, J.T. Black, An evaluation of chip separation criteria for the FEM simulation of machining, J. Manuf. Sci. Eng. 118 (1996) 545±554. [13] T. Obikawa, E. Usui, Computational machining of titanium alloy, ®nite element modeling and a few results, J. Manuf. Sci. Eng. 118 (1996) 208±215. [14] V. Madhavan, S. Chandrasekar, T.N. Farris, Mechanistic model of machining as an indentation process, in: D.A. Stephenson, R. Stevenson (Eds.), Materials Issues in Machining III and the Physics of Machining Process, The Mineral, Metals and Materials Society, Vol. 2, 1993, pp. 187±209. [15] G.S. Sekhon, J.L. Chenot, Numerical simulation of continuous chip formation during non-steady orthogonal cutting, Eng. Comput. 10 (1993) 31±48. [16] T.D. Marusich, M. Ortiz, Modeling and simulation of high-speed machining, Int. J. Numer. Methods Eng. 38 (1995) 3675±3694. [17] J.S. Strenkowski, K.J. Moon, Finite element prediction of chip geometry and tool±workpiece temperature distribution in orthogonal metal cutting, ASME J. Eng. Ind. 112 (1990) 313±318. [18] K.W. Kim, H.-C. Sin, Development of a thermo-viscoplastic cutting model using ®nite element method, Int. J. Mach. Tools Manuf. 36 (1996) 379±397. [19] J.-S. Wu, J.R. Dillon, W.-Y. Lu, Thermo-viscoplastic modeling of machining process using a mixed ®nite element method, ASME J. Manuf. Sci. Eng. 118 (1996) 470±482. [20] O.C. Zienkiewicz, P.N. Godbole, Flow of plastic and visco-plastic solids with special reference to extrusion and forming processes, Int. J. Numer. Methods Eng. 8 (1974) 3±16. [21] J. Wang, M.S. Gadala, Formulation and survey of ALE method in nonlinear solid mechanics, Finite Elements Anal. Des. 24 (1996) 253. [22] M.S. Gadala, M. Movahhedy, On the mesh motion for ALE modeling of metal forming processes, submitted to Finite Elements in Analysis and Design. [23] M. Abo-Elkhier, G.A.E. Oravas, M.A. Dokainish, A consistent Eulerian formulation for large deformation analysis with reference to metal-extrusion process, Int. J. Non-linear Mech. 52 (1988) 37±52. [24] M.S. Gadala, J. Wang, ALE formulation and its application in solid mechanics, Comput. Methods Appl. Mech. Eng. 167 (1998) 33±55.