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International Journal of Machine Tools & Manufacture 38 (1998) 1067–1087

FEA modeling and simulation of shear localized chip formation in metal cutting J.Q. Xiea,*, A.E. Bayoumib, H.M. Zbibc b

a Valenite, Inc., 31750 Sherman Drive, Madison Heights, MI 48071, USA Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, USA c Department of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164, USA

Received 2 January 1997; in final form 7 August 1997

Abstract The finite element analysis (FEA) has been applied to model and simulate the chip formation and the shear localization phenomena in the metal cutting process. The updated Lagrangian formulation of plane strain condition is used in this study. A strain-hardening thermal-softening material model is used to simulate shear localized chip formation. Chip formation, shear banding, cutting forces, effects of tool rake angle on both shear angle and cutting forces, maximum shear stress and plastic strain fields, and distribution of effective stress on tool rake face are predicted by the finite element model. The initiation and extension of shear banding due to material’s shear instability are also simulated. FEA was also used to predict and compare materials behaviors and chip formations of different workpiece materials in metal cutting. The predictions of the finite element analysis agreed well with the experimental measurements. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Metal cutting; Finite element analysis (FEA); Chip formation; Shear localization; Cutting tool

1. Introduction Improvements in manufacturing technologies require better modeling and simulation of metal cutting processes. Theoretical and experimental investigations of metal machining have been extensively carried out using various techniques. On the other hand, the complicated mechanisms usually associated in metal cutting, such as interfacial friction, heat generated due to friction and * Corresponding author. 0890-6955/98/$19.00 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 7 ) 0 0 0 6 3 - 1

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severe strain in the cutting region, and high strain rates, have also limited the theoretical modeling of chip formation. So, many researchers have been focusing on computer modeling and simulation of machining process to solve many complicated problems arising in the development of new technologies. Metal cutting is one of the areas in which flow localization can be generated around the primary and secondary deformation zones. It is believed that most serrated chips are caused by flow (shear) localization during the chip deformation. Flow localization results in deformation bands of intense shear dividing the chip into segments during cutting process. The shear band is a very thin layer with extremely concentrated shear strain which may cause the chip to become easily separated and broken. Although in some cases the serrated chip may not be completely broken, a certain amount of damage within the chip material is present because the strains inside the shear bands are very large. It is believed that a shear localized chip can break easily, which is helpful for chip disposal in automatic machining. One of the state-of-the-art efforts in manufacturing engineering is computer simulation of the machining process to predict power requirements, cutting forces and chip formation using numerical models. These computational models would have great value in reducing or even eliminating the number of trial-and-error experiments which traditionally are used for tool design, process selection, machinability evaluation, and chip breakage investigation. The difficulty of reaching a better theoretical understanding of the metal cutting process impelled researchers in the field to apply the finite element analysis to model the cutting process. More attention to the finite element method have been paid in the past decade in respect to its capability of numerically modeling different types of metal cutting problems. The advantage of the finite element method is that the entire complicated process can be automatically simulated using a computer. The finite element analysis, as a tool to numerically simulate the metal machining processes, is advancing steadily [1,5,7,9–14,16,17,21,28]. In contrast, it is also clear that more work still needs to be done in order for the metal cutting simulation and prediction to become parts of the computer integrated manufacturing (CIM) system. In this investigation, a finite element model of orthogonal metal machining process is developed to predict and simulate the shear localization, plastic deformation and shear stress field of the chip and workpiece, the stress distribution on tool face, and cutting forces. The developed model is based on a general purpose, non-linear, implicit, 2-D finite element software, NIKE2D [6]. In this finite element model, the workpiece material is constrained so that no displacement occurs at its boundaries, and the tool tip may penetrate the workpiece into the chip and machined surface along the parting line as the tool advances. A material failure criterion is set to allow the separation between the chip and workpiece to occur when a prescribed critical effective strain value is reached in the elements at the tool tip. In this case, a layer from the workpiece can be separated by the tool tip to form a chip. The workpiece material is assumed to obey a strain-hardening thermal-softening model, which results into an effective stress–strain curve that exhibits strain hardening followed by strain (thermal) softening. The transition from the hardening region to the softening region occurs when the stress reaches a maximum value at which material instability occurs, leading to adiabatic shear band development.

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2. Governing equations and material models 2.1. Governing equations The finite element method has been successfully implemented to study large deformation problems in metal forming. Metal cutting is similar to metal forming where large plastic deformation occurs. However, several modifications must be made in order to use the finite element technique in metal cutting to overcome the above mentioned complications. The basic idea of using the finite element method is to seek a solution to the momentum equation

ij,j + bi = ˙ i (i,j = 1,2,3),

(1)

where ij is the Cauchy stress tensor, is the current mass density, bi is the body force, i is the particle velocity in Cartesian coordinates, ˙ i is the acceleration, and j indicates partial differentiation with respect to xj. For elastic–plastic deformations, the strain rate tensor ⑀˙ ij is usually decomposed into an elastic part ⑀˙ eij and a plastic part ⑀˙ pij such that

⑀˙ ij =

1 ( + j,i), ⑀˙ ij = ⑀˙ eij + ⑀˙ pij . 2 i,j

(2)

From Hook’s Law we have ⴰ

⑀˙ eij = Ceijkl⫺ 1 kl; Ceijkl = 2G ␦ik ␦jl +

(3) 2G ␦ ␦ , 1 ⫺ 2 ij kl

where Ceijkl is the elasticity tensor, G and are the shear modulus and Poison’s ratio respectively, ⴰ and ij is the corotational stress rate given by ⴰ

ij = ˙ ij ⫺ ikjk + ij ik

(4)

with ij being the material spin. Using the von Mises flow rule, ⑀˙ pij is obtained as [26] ·

⑀˙ pij =

␥¯ sij ,

2¯

(5) ·

in Eq. (5), sij is the deviatoric stress, ␥¯ = (2⑀˙ pij ⑀˙ pij )1/2 is the effective strain rate, and

¯ =

冉 冊 1 s s 2 ij ij

1/2

is the effective shear stress. ·

The effective strain rate ␥¯ is determined from the yield criterion for rate-independent materials. For von Mises materials, the yield criterion is given by

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f = √J2 ⫺ ¯ (␥¯ ) = 0,

(6)

where J2 is the second invariant of sij. Then, from the consistency condition ˙f = 0 one can obtain the relation ⴰ

sij sij ∂¯ , ␥¯ = , h= 2√J2h ∂␥¯ ·

(7)

where h is the strain hardening/softening modulus. Combining Eqs (2), (3), (5) and (7) we obtain ⴰ

ij = Cijkl ⑀˙ ij ,

(8)

where Cijkl is the elastic–plastic incremental tensor and is given by Cijkl = Ceijkl ⫺ Cpijkl,

Cpijkl =

G2 s s . ¯ 2(G + h) ij kl

(9)

·

For rate-dependent materials the flow stress ¯ = K(␥¯ ,␥¯ ) and Eqs (4) and (5) yield

ⴰ

冉

ij = Ceijkl ⑀˙ kl ⫺

·

␥¯ 2¯

冊

skl .

(10)

These equations are implemented in the finite element modeling program. In the program, the convergence of the equilibrium iterations in the finite element analysis is determined by examining both the displacement norm 兩兩{⌬u}i兩兩 ⱕ ed, umax

(11)

and the energy norm: ({⌬u}i)T{Q}in + 1 ⱕ ee. ({⌬u}0)T{Q}0n + 1

(12)

In Eqs (11) and (12), {⌬u}0 and {⌬u}i are the increments in displacement in which the superscript denotes iteration number, umax is the maximum displacement norm over all of the n steps including the current iteration, {Q}n + 1 is the residual, ed and ee are tolerances that are typically 10⫺2 to 10⫺3 or smaller and usually are adjustable for different problems. If convergence is not attained and the solution is not divergent, the displacement is updated using a line search scheme and iterations are then continued. If the solutions are determined to be divergent, or convergence fails to occur within an assigned number of iteration, the stiffness matrix [K] is reformed using the current estimation of geometry before continuing equilibrium iteration [6].

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2.2. Material model for the flow stress In metal cutting, large plastic deformations take place accompanied by excessive heat generation. Both high temperature and high strain rate effects (among other factors), on the material behavior need to be considered when simulating the chip formation, especially for modeling the flow localization. Correct determination of the material constitutive model is the key to successfully simulate the formation of shear bands in metal cutting. The material behavior during the formation of the shear localized chip is very complicated and there has been no confirmed model to describe the behavior of workpiece material during the machining process to date. The majority of the energy dissipated in cutting is converted into heat in a very small zone, which makes the temperature distribution in the deformation zone complex. The appearance of the shear localization in the chip involves several mechanical, physical and thermophysical properties. Numerous experiments have shown that temperature plays an important role in the flow instability in chips. The high plastic deformation rate and severe tool–workpiece friction can quickly increase the rate of heat generation. The adiabatic or quasi-adiabatic condition may be reached resulting from the high net accumulation of heat. The temperature can become very high locally to result in further thermal softening in the workpiece material. In this case, the material strain-hardening capacity is reduced so that the instability may take place in a narrow band of the chip. It is believed that shear instability is directly caused by the material flow in the shear bands, and the workpiece material behavior of plastic deformation in cutting process is quite different from the material behavior in ordinary plastic deformation. Although various studies have been made to investigate the material behavior in the machining process, there is still no satisfied materials constitutive model as a function of strain, strain-rate, temperature, etc. to properly describe the material behavior during cutting process. In this investigation, the cutting tool is assumed to be rigidly elastic material with no plastic deformation, and the workpiece material is assumed to be elastoplastic as given by Eq. (9). An isotropic, non-linear hardening law is specified by defining the relationship between yield stress and effective plastic strain as a piecewise curve. This gives some flexibilities in specifying the material model for the flow stress to describe the effects of the thermal softening as a result of quasi-adiabatic heating from large and rapid plastic deformation, which occurs during chip formation. Thus, a mechanical model for the flow stress that exhibits strain hardening, which is followed by strain softening caused by thermal softening, is assumed. Figure 1 shows a schematic of the strain-hardening and thermal-softening model. It is known that plastic deformation takes place when the shear stress on the workpiece material reaches the yield point A in Fig. 1. Then, strain hardening begins as the plastic strain further increases from point A to point B. In this case, the stress at the yield point A is 802.69 MPa and the stress at the point B is 931.47 MPa [8]. Because of large plastic deformations and severe friction in metal cutting, a large amount of heat is generated in the deformation or shear zone of the workpiece material during the process of chip formation. Both the temperature and plastic deformation work in the shear zone do not have enough time to escape due to very high strain rates. Consequently, the temperature in the shear zone increases rapidly to elevated temperatures. This high temperature causes thermal softening in the shear zone which, in return, results in an increase of shear strain in the shear zone with a decreasing stress. This may explain the occurrence of shear localization in metal cutting under certain conditions.

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Fig. 1. Schematic of strain-hardening and thermal-softening material model.

Therefore, the strain-hardening and thermal-softening material model can properly simulate the material behavior in the metal cutting process. In this investigation, the workpiece material property up to point B is taken from a materials handbook. The thermal-softening part of the material property is arbitrarily composed so that the stress will decrease 50% from point B to point C; the strain at point C is twice that of strain at point B. 3. FEA simulation of orthogonal cutting process 3.1. Machining model and boundary conditions The linear quadrilateral elements are used for both the workpiece and the tool materials. Fig. 2 illustrates a schematic of the finite element modeling of an orthogonal cutting process. The top edge of the cutting tool is fixed in the vertical direction and the tool can move horizontally. The bottom edge of the workpiece is constrained in both directions and its left edge is fixed in the horizontal direction. The mesh density of the elements is determined with a convergence error analysis by running some pre-testing models. In this study, the workpiece is divided into 1764 elements and the cutting tool into 30 elements. The cutting process is eventually produced by applying a horizontal displacement boundary condition along the right edge (far side of the cutting edge) of the cutting tool (Fig. 2). The cutter starts from the outside of the workpiece and moves into it at a given cutting speed. The cutting speed can be changed by assigning different displacement boundary conditions. As the cutting tool advances into the workpiece and the chip separates from it, there exist frictions between the tool rake face and chip bottom surface, and between the tool flank face and the machined surface on the workpiece. Some corresponding frictional coefficients can be assigned

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Fig. 2. Finite element model of 2-D machining process.

to these contacting surfaces based on cutting conditions, especially cutting speed and the tool and workpiece materials. These frictions are quite complicated because both sliding and sticking frictions may be involved, and there is still no accurate model to describe the friction in metal cutting. The widely used friction model is the chip–tool friction which is a combination of sticking and sliding regions [2,4,18]. Sticking friction usually takes place in the region near the tool tip, while sliding friction takes place in the region near the point where the chip loses contact with the tool face. In this investigation, the coefficient of friction along the tool–chip interface is assigned based upon the assumption that sticking friction takes place over a distance of 0.01 mm from the tool tip and sliding friction takes place over the remaining part along the tool–chip interface. The friction between the tool flank face and the machined surface is assumed to be sliding friction. 3.2. Simulation of the chip formation To simulate the formation of a chip in the metal cutting process using the finite element method, a tie-break type slideline consisting of two sets of nodes on both surfaces simultaneously between the chip to be formed and the machined workpiece surface to be generated is specified. These two sets of nodes are initially tied together and are allowed to separate (break) when a specified criterion is reached. Although this slideline is the anticipated parting line between the chip and the machined workpiece surface, it does reflect the reality of the actual formation process of the

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chip and the machined surface. A criterion needs to be specified to the slideline so that the chip may separate from the workpiece along the slideline when the criterion is reached. The separation criterion used is the effective plastic strain in the two elements adjacent to the tool tip region. The effective plastic strain ⑀¯ (p) based on the von Mises criterion is calculated for every node at each time step. When the value of the effective plastic strain at a nodal point reaches a prescribed critical value ⑀¯ (p) 0 , that is

⑀¯ (p) ⱖ ⑀¯ (p) 0 ,

(13)

the ‘tie’ at this nodal point is broken and material separation occurs. In this case, the material which is originally tied together as one part is separated into the chip and the machined workpiece surface. As the tool advances into the workpiece, the nodes ahead of the tool tip continue to separate when the effective plastic strains at those nodes reach the separation criterion, and the chip formation continues. Figure 3 shows the chip formation process as the tool advances into the workpiece. Generally, the value of the separation criterion may be between 0.3 and 0.7. However, the separation criterion varies slightly for different workpiece materials and different cutting conditions. Different values of the separation criterion may not significantly affect chip geometry and stress distributions in the chip, but may affect the stress distributions in the machined surface and the effective plastic strain distributions both in the chip and in the machined surface. Therefore,

Fig. 3. Chip formation with localized shear band modeled by FEA.

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some experience may be needed in order to assign a reasonable value for the critical effective plastic strain. In this study, ⑀¯ (p) 0 = 0.5 was used. 4. Results and discussions 4.1. Experimental tests Several orthogonal cutting tests were carried out to verify the results from the finite element model. The orthogonal machining process was attained by cutting a cylindrical workpiece tube on one end on a CNC lathe [23]. This was to ensure that a plane strain condition could prevail during the process of chip formation. The cutting tool was the cemented carbide cutting tip which was installed on a tool holder. After installation, the cutting tool had a rake angle, ␣n, of 8° and flank clearance angle, n, of 7°. In this investigation, the sharp tool without flank wear was assumed and used for all the cutting tests. The workpiece material used in the cutting tests was Ti-6Al-4V titanium alloy. Some preliminary cutting tests were conducted to determine the range of cutting conditions such that the machining process was smooth to ensure the absence of chatter or built-up edge. The cutting speed was set at 5.0 m s⫺1 and the feed rate was set at 0.3 mm rev⫺1. All these are in accordance with the finite element machining model. The cutting forces were recorded using a Kistler piezoelectric dynamometer through a computer controlled data acquisition system. 4.2. Simulation of chip formation and shear banding The chips from cutting tests were collected to study the chip formation and flow localization. The chip specimens were mounted, polished metallographically, and etched. The specimens were then examined and photographed on a microscope. Figure 4 is a photograph of a shear localized chip with shear bands. Figure 5(a) shows a chip formation process with localized shear banding modeled by the finite element analysis. From Figs 4 and 5(a), it is apparent that the chip formation and shear banding modeled by FEA are very similar to those of chips experimentally formed. In metal cutting, the shear band forms along an angle usually called the shear angle. It can be seen from either Fig. 4 or Fig. 5(a) that localized plastic deformation takes place in a very narrow zone rather than in a plane as is often assumed when analysing the orthogonal machining process. The width of the shear band predicted by the finite element analysis is found to be about 0.03 mm which falls into within the experimentally measured values of 0.008 mm to 0.031 mm. It can be noted that the predicted width of the shear banding is not significantly affected by the element size. However, in order to correctly model the shear band width, different constitutive models that include size effects might be considered as suggested by Zbib and Aifantis (1988)[26] and by Zbib (1992)[27]. Chip formation, in practice, is an extremely quick process, especially at high cutting speeds. In metal machining, the cutting tool advances into the workpiece material and tool tip cuts the material off, accompanied with some large plastic deformation, as shown in Fig. 3. Certain strainhardening may take place in the workpiece material in this stage. As the cutting process continues, the tool tip and rake face further cut and press the workpiece material in the area ahead of the

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Fig. 4. Shear localized chip with shear bands (Ti-6Al-4V).

tool tip and cause more plastic deformation. The tremendous pressure as well as frictions, plastic deformation and material fracture near the tool tip, may all contribute to a very high temperature at the tool tip and its nearby area. Because of the very high speed of the cutter movement, the strain rate may reach as high as 105 or even higher, so that the workpiece material in front of the tool tip will be squeezed severely before it can flow along the tool rake face, as shown in Fig. 5(b). The tool–workpiece friction becomes more severe which will then generate more heat. The adiabatic or quasi-adiabatic condition may be reached due to the very high net accumulation of heat and high strain rate. In this case, the material near the tool tip continues to deform and the temperature can be very high locally in some areas of the workpiece, resulting in further thermal softening. This thermal softening reduces the material’s strain-hardening capacity so the instability takes place in a narrow band of the chip and finally forms an intense shear band. The very high pressure near the tool tip also causes sticking and sliding frictions between the tool rake face and the chip. The sticking of the workpiece material on the tool rake face is one of the reasons that form the built-up edge (BUE) on cutting tools. 4.3. Strain and stress The finite element analysis provides an effective method to numerically predict the stress and strain in the deformation area and cutting forces. Figure 6(a) and (b) show the contours of effective plastic strain for different tool positions. At the initial stage of cutting, plastic strains are relatively small and concentrate around the tool tip in both the chip and the machined surface, as shown in Fig. 6(a). As the cutting process continues until the shear band is fully formed, the plastic

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Fig. 5. The chip formation process as tool advances into the workpiece. (a) Tool is advancing into the workpiece. (b) Further deformed chip formation.

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Fig. 6. Contours of effective plastic strain. (a) One stage during cutting process. (b) When shear band is fully formed.

strains increase and spread out into a narrow zone where shear banding takes place as can be seen in Fig. 6(b). The maximum effective plastic strain reaches 2.3. These results explain some of the material instability, in particular the formation of shear banding in the primary deformation zone. Figure 6(a) and (b) indicate that high plastic strains at different time steps of cutting process

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occur near the tool tip; and the chip material above the shear zone does not have significant plastic deformation. This also illustrates the catastrophic plastic deformation that occurs in a limited shear zone. The contours of maximum shear stress at two different cutting stages are plotted in Fig. 7(a) and (b). By comparing the results in Fig. 7(a) and (b) with the results in Fig. 6(a) and (b), it can be seen that the field of the maximum shear stress is around the plastic deformation zone and approximately matches with the areas of the primary and secondary deformation zones. Unlike the effective plastic strain, the maximum shear stress is not confined to a narrow zone, although the shear instability does occur in a narrow area. As the cutting tool further advances into the workpiece material, the total area of the maximum shear stress expands as can be seen in Fig. 7(b), though the maximum shear stress keeps the same magnitude of approximately 480 N mm⫺2 for different cutting stages. This means that the maximum shear stress of the workpiece material is independent of the cutting stages. Shear instability will occur as long as the maximum shear stress is reached. This agrees with the result of ordinary materials test. Fig. 7(b) clearly shows the initiation of the shear banding in front of the tool tip. It can also be noted from Fig. 7(b) that the shear stress in the shear band just ahead of the tool tip is lower than that in the nearby area. This is the indication of shear band initiation due to material’s shear instability, because the low shear stress near the tool tip is apparently caused by the thermal softening of the workpiece material, as mentioned above. This also shows that in metal cutting, the chip does start to shear from the position of tool tip. Figure 8 plots the distribution of effective stress on the tool rake face predicted by the finite element analysis. Figure 9 is the relationship between the effective stress on tool face and the distance from tool tip. It can be seen that the effective stress has the maximum value at the tool tip and gradually diminishes to a negligible value at the position where the chip is no longer in contact with the tool rake face. This agrees qualitatively with the results by Komvopoulos et al. (1990) [11] and Usui et al. (1984) [22]. In addition, Figs 6 and 7 also show the existence of shear stress and plastic strain beneath the machined surface. It can be seen that the maximum stress induced at some distance beneath the machined surface, instead of on top of the machined surface. This is in good agreement with the earlier findings by the authors in characterizing the machined surface integrity [3,24,25]. This indicates that the finite element analysis may effectively be used as a tool to simulate and evaluate the stress, strain and integrity of the machined surface. 4.4. Cutting forces Figure 10 shows the predicted cutting force per unit width of cut. The Cutting force Fc is determined by integrating the stresses acting on both the tool rake face/chip and the flank face/machined interfaces. The actual cutting force can then be calculated by multiplying the cutting forces per unit width of cut by the actual width of cut. It can be seen from Fig. 10 that the cutting force increases from zero linearly to a peak value, then decreases and finally gradually approaches steady value. The steady value is the average cutting force usually measured in cutting tests. It is apparent that the finite element analysis can show more dynamic details of the predicted instantaneous cutting forces, while the ordinary cutting tests measure the average cutting forces. In this study, the width of cut was 5 mm. The steady value of the predicted cutting force per

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Fig. 7. Contours of effective shear stress (105 N mm ⫺ 2). (a) At an early stage of cutting process. (b) Indication of the shear banding initiation.

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Fig. 8. Distribution of effective stress on the tool rake face (105 N mm ⫺ 2).

Fig. 9.

Effective stress on the tool face versus distance from tool tip.

unit width of cut is used to calculate the predicted main cutting force, Fc. The measured average Fc is 921 N while the predicted Fc is 865 N. The predicted value of Fc is in good agreement with the experimentally measured value with an error of 6% in this case. It should be pointed out that there might be a difference between the actual machining process

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Fig. 10. Cutting forces predicted by finite element analysis.

and the finite element modeling of machining process. In actual machining, there is a ‘thrusting’ force which is induced by the feed rate. The thrusting force heavily depends on the magnitude of the feed rate and workpiece material, and is usually the dominant component in the measured thrust cutting force. Without a proper feed rate to induce a thrusting movement on cutting tool, the real machining process will not take place. So the experimentally measured cutting and thrust forces always include the thrusting force component. On the other hand, in the finite element modeling, such a thrusting force is not needed to enforce a feed rate in order to accomplish the cutting process. Therefore, such a thrusting force is not exerted when the boundary conditions were assigned. The lack of this thrusting force would largely reduce the predicted value of thrust force and slightly reduce the predicted cutting force. 4.5. Effects of tool rake angle A series of finite element simulated machining tests with different tool rake angles ranging from ⫺16° to 20° was carried out to study the effects of rake angle on shear band angle and cutting forces. Figure 11 shows a machining process modeled with a negative rake angle ␣n = ⫺8°. Fig. 12 shows the effect of tool rake angle on the shear angle as predicted by the FEA. In the practical range of tool rake angle from ⫺16° to 20° the shear band angle increases from about 36° to 55°. It can be seen that as the rake angle increases, the increase of shear angle slows down and tends to approach to a constant value; as the magnitude of rake angle decreases negatively, the decrease of shear angle also slows down and tends to approach to a constant value. These are in agreement with previously observed experimental results [4,14,20]. Figure 13 shows the effect of the rake angle on the cutting force. As the rake angle decreases from 20° to ⫺16°, the main cutting force, Fc, increases. This is apparent because decreasing the rake angle raises the force component in the direction of Fc. The trend of the effects of rake

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Fig. 11. Machining process modeled with a negative rake angle.

Fig. 12. Relationship between tool rake angle and shear banding angle.

angle on the main cutting force Fc agrees well with experimentally obtained results [12,15,20]. This, therefore, gives this FEA model an obvious advantage in machine tool, cutting tool and tool holder/fixture designs, because the tool geometry and other factors can be simulated and optimized in the computer without running a cutting test. In the real world, though, almost all machinings are three dimensional. It, therefore, may require

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Fig. 13. Effects of tool rake angle on cutting forces.

3-D modeling of the machining process in order to predict the results more accurately when FEA is applied in cutting tool design. 4.6. Material behavior modeling In this study, the material behavior in metal cutting was modeled and evaluated using a commercial FEA code, ABAQUS. Two workpiece materials with different material properties and constitutive relations were defined in the software’s user-defined subroutine. The intention here was to qualitatively illustrate the possibility of numerically simulating varied chip formations for different workpiece materials. So less emphasis was put on the accuracy of matching material’s constitutive models with real materials. One material was assumed as the Ti-6Al-4V titanium alloy with ductile failures which is defined by the modified Gurson constitutive model [19], in which the local failure between individual voids combined with the original Gurson constitutive relations could provide a means of tracking deformation and failure of a continuum element. Another material was assumed as an ordinary elastic–plastic low-carbon steel, AISI 1020, with the strain-hardening thermal-softening law, as described earlier in this paper. These two materials were modeled as both have the same cutting conditions. The cutting tool was modeled as a rigid body with the chip breaker. Figures 14 and 15 show the FEA modeled chip formations of titanium alloy and the low-carbon steel, respectively. It can be seen that the discontinuous chips were formed for the titanium alloy which is a typical material that shear localization usually takes place during machining. However, a continuous chip was formed for the low-carbon steel, as expected. Both results are in accordance with experimental observations. This shows that FEA can be used as an efficient numerical method to test the materials behaviors and/or chip formations for different workpiece materials without running cutting tests. This is especially convenient and cost-saving when only relative comparisons between

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Fig. 14.

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Discontinuous chips modeled by FEA.

Fig. 15. Continuous chip modeled by FEA.

different materials are needed. Chips could be produced, analysed and compared on the computer instead of on the real machine tools. This could be very helpful in cutting tool design optimization and in determining optimal machining process. Meanwhile, this also suggests that more accurate and complete material constitutive models are needed in order to correctly simulate the machining processes using finite element analysis. 5. Conclusions

1. A quasi-static finite element modeling of chip formation and shear banding in orthogonal metal cutting has been presented. The updated Lagrangian formulation for plane strain condition is used in this investigation. The ‘tiebreak’ slideline was used to separate the newly formed chip from the workpiece surface. The effective plastic strain is used as the material failure criterion.

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In order to model the shear banding during chip formation, a strain-hardening thermal-softening model for the flow stress is used. 2. The FEA simulations of chip formation and shear banding show that the finite element analysis can successfully model the metal machining process. In addition, cutting forces, distribution of effective stress on tool rake face, maximum shear stress and plastic strain fields in the workpiece material can also be predicted. 3. The simulated results show that the effective plastic strain is within a narrow area along the shear zone angle, and the field of the maximum shear stress matches with the area of the primary and secondary deformation zones. This finite element model predicted the detailed deformation in front of the tool tip and the initiation of the shear band. The comparison between predicted and experimentally measured cutting forces indicates that cutting forces, among others, can be reasonably simulated with the finite element analysis. However, the lack of thrusting force in FEA modeling may bring in some errors to the predicted cutting forces, especially to the thrust cutting force. The effect of tool rake angle on the shear band angle and cutting forces is also predicted by the FEA modeling. This implies that FEA could be an important part of a computer integrated manufacturing (CIM) system since several machining parameters can be simulated and optimized without running experiments. 4. The material behavior in metal cutting was successfully modeled by FEA. Both discontinuous and continuous chips were simulated and the computer modeled results are in accordance with experimental observations. Therefore, FEA can be used as an efficient numerical method to test the materials behaviors and/or chip formations for different workpiece materials without running cutting tests. Chips could be simulated, analysed and compared on the computer instead of on the real machine tools. This study also suggests that more accurate and complete material constitutive models are needed in order to correctly simulate the machining processes using finite element analysis.

References [1] M.M. Ahmad, W.A. Draper, R.T. Derricott, An application of the finite element method to the prediction of cutting tool performance, Int. J. Mach. Tools Manufact. 29 (2) (1989) 197. [2] J.A. Baily, Friction in metal machining—Mechanical aspects, International Journal of WEAR 31 (1975) 243. [3] A.E. Bayoumi, Q. Xie, M.N. Hamdan, Effects of cutting conditions on dynamic property of work material and machined surface integrity, International Journal of WEAR 146 (2) (1991) 301. [4] G. Boothroyd, Fundamentals of Metal Machining and Machine Tools, Scripta Book Company, 1975. [5] J.T. Carroll III, J.S. Strenkowski, Finite element models of orthogonal cutting with application to single point diamond turning, International Journal of Mechanical Sciences 30 (12) (1988) 899. [6] B. Engelmann, J.O. Hallquist, NIKE2D: A nonlinear, implicit, two-dimensional finite code for solid mechanics— User manual, Lawrence Livermore National Laboratory, UCRL-MA-105413, April 1991. [7] J.M. Huang, J.T. Black, An evaluation of chip separation criteria for the FEM simulation of machining, Trans. ASME Journal of Manufacturing Science and Engineering 118 (1996) 545. [8] H.J. Hucek (Ed.), Aerospace Structural Metals Handbook, 1987 Publication, Metals and Ceramics Information Center, Battelle Columbus Division, Columbus, Ohio, 1987. [9] K. Iwata, K. Osakada, Y. Terasaka, Process modeling of orthogonal cutting by the rigid-plastic finite element method, Trans. ASME Journal of Engineering Materials and Technology 106 (1984) 132.

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[10] B.E. Klamecki, Incipient chip formation in metal cutting—A three dimension finite element analysis, Ph.D. dissertation, University of Illinois at Urbana–Champaign, 1973. [11] K. Komvopoulos, S.A. Erpenbeck, Finite element modeling of orthogonal metal cutting, in: Computer Modeling and Simulation of Manufacturing Processes, ASME Winter Annual Meeting, MD-vol. 20, PED-vol. 48, November 1990, p. 1. [12] D. Lee, W.W. Wilkening, Material modeling and high speed machining process, in: Annual Technical Report to Advanced Machining Research Program (AMRP), General Electric Company, August 1982. [13] T.D. Marusich, M. Ortiz, Modeling and simulation of high-speed machining, International Journal for Numerical Methods in Engineering 38 (1995) 3675. [14] R.T. Sedgwick, H.E. Read, D.E. Wilkins, Numerical modeling of high-speed machining processes, in: Annual Technical Report to Advanced Machining Research Program (AMRP). General Electric Company, August 1982. [15] M.C. Shaw, Metal Cutting Principles, Clarendon Press, Oxford, 1984. [16] A.J.M. Shih, S. Chandrasekar, H.T.Y. Yang, Finite element simulation of metal cutting process with strain-rate and temperature effects, in: Fundamental Issues in Machining, ASME-WAM, PED-vol. 43, Dallas, Texas, November 1990, p. 11. [17] J.S. Strenkowski, J.T. Carroll III, A finite element model of orthogonal metal cutting, Trans. ASME J. of Eng. Ind. 107 (1985) 349. [18] E.M. Trent, Metal Cutting, 3rd ed., Butterworth–Heinemann Ltd., London, 1991. [19] V. Tvergaard, A. Needleman, Analysis of the cup–cone fracture in a round tensile bar, Acta Metall. 32 (1984) 157. [20] E. Usui, A. Hirota, Analytical prediction of three dimensional cutting process, Part 2, ASME J. of Eng. Ind., May 100 (1978) 229. [21] E. Usui, T. Shirakashi, Mechanics of machining—From ‘Descriptive’ to ‘Predictive’ theory, in: On the Art of Cutting Metals—75 Years Later, ASME Publication PED, vol. 7, 1982, p. 13. [22] E. Usui, T. Shirakashi, T. Kitagawa, Analytical prediction of cutting tool wear, Wear 100 (1984) 129. [23] Q. Xie, An analytical and finite element analysis of chip formation and shear banding in metal cutting process, Ph.D. dissertation, Washington State University, 1993. [24] Q. Xie, A.E. Bayoumi, L.A. Kendall, G.L. Sheldon, Study on residual stresses and tool wear induced by machining processes, in: Trans. NAMRC XVII, 1989, p. 170. [25] Q. Xie, A.E. Bayoumi, L.A. Kendall, Tool wear and its effects on the machined surface integrity, J. of Materials Shaping Tech. 8 (4) (1990) 255. [26] H.M. Zbib, E.C. Aifantis, On the localization and postlocalization behavior of plastic deformation, I. On the initiation of shear bands, Res Mechanica 23 (1988) 261. [27] H.M. Zbib, The strain gradient phenomenon in viscoplasticity: Theory and application to shear instability, in: S.I. Anderson et al. (Eds.), Modeling of Plastic Deformation and Its Engineering Application, Reso National Lab, Denmark, 1992, p. 525. [28] B. Zhang, A. Bagchi, A study of chip separation and its approximation in finite element simulation of continuous chip formation, in: D.A. Stephenson, R. Stephenson (Eds.), Materials Issues in Machining—II and the Physics of Machining Processes—II, Winter Annual Meeting of the ASME, Chicago, 1994, p. 157.

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FEA modeling and simulation of shear localized chip formation in metal cutting J.Q. Xiea,*, A.E. Bayoumib, H.M. Zbibc b

a Valenite, Inc., 31750 Sherman Drive, Madison Heights, MI 48071, USA Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, USA c Department of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164, USA

Received 2 January 1997; in final form 7 August 1997

Abstract The finite element analysis (FEA) has been applied to model and simulate the chip formation and the shear localization phenomena in the metal cutting process. The updated Lagrangian formulation of plane strain condition is used in this study. A strain-hardening thermal-softening material model is used to simulate shear localized chip formation. Chip formation, shear banding, cutting forces, effects of tool rake angle on both shear angle and cutting forces, maximum shear stress and plastic strain fields, and distribution of effective stress on tool rake face are predicted by the finite element model. The initiation and extension of shear banding due to material’s shear instability are also simulated. FEA was also used to predict and compare materials behaviors and chip formations of different workpiece materials in metal cutting. The predictions of the finite element analysis agreed well with the experimental measurements. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Metal cutting; Finite element analysis (FEA); Chip formation; Shear localization; Cutting tool

1. Introduction Improvements in manufacturing technologies require better modeling and simulation of metal cutting processes. Theoretical and experimental investigations of metal machining have been extensively carried out using various techniques. On the other hand, the complicated mechanisms usually associated in metal cutting, such as interfacial friction, heat generated due to friction and * Corresponding author. 0890-6955/98/$19.00 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 7 ) 0 0 0 6 3 - 1

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severe strain in the cutting region, and high strain rates, have also limited the theoretical modeling of chip formation. So, many researchers have been focusing on computer modeling and simulation of machining process to solve many complicated problems arising in the development of new technologies. Metal cutting is one of the areas in which flow localization can be generated around the primary and secondary deformation zones. It is believed that most serrated chips are caused by flow (shear) localization during the chip deformation. Flow localization results in deformation bands of intense shear dividing the chip into segments during cutting process. The shear band is a very thin layer with extremely concentrated shear strain which may cause the chip to become easily separated and broken. Although in some cases the serrated chip may not be completely broken, a certain amount of damage within the chip material is present because the strains inside the shear bands are very large. It is believed that a shear localized chip can break easily, which is helpful for chip disposal in automatic machining. One of the state-of-the-art efforts in manufacturing engineering is computer simulation of the machining process to predict power requirements, cutting forces and chip formation using numerical models. These computational models would have great value in reducing or even eliminating the number of trial-and-error experiments which traditionally are used for tool design, process selection, machinability evaluation, and chip breakage investigation. The difficulty of reaching a better theoretical understanding of the metal cutting process impelled researchers in the field to apply the finite element analysis to model the cutting process. More attention to the finite element method have been paid in the past decade in respect to its capability of numerically modeling different types of metal cutting problems. The advantage of the finite element method is that the entire complicated process can be automatically simulated using a computer. The finite element analysis, as a tool to numerically simulate the metal machining processes, is advancing steadily [1,5,7,9–14,16,17,21,28]. In contrast, it is also clear that more work still needs to be done in order for the metal cutting simulation and prediction to become parts of the computer integrated manufacturing (CIM) system. In this investigation, a finite element model of orthogonal metal machining process is developed to predict and simulate the shear localization, plastic deformation and shear stress field of the chip and workpiece, the stress distribution on tool face, and cutting forces. The developed model is based on a general purpose, non-linear, implicit, 2-D finite element software, NIKE2D [6]. In this finite element model, the workpiece material is constrained so that no displacement occurs at its boundaries, and the tool tip may penetrate the workpiece into the chip and machined surface along the parting line as the tool advances. A material failure criterion is set to allow the separation between the chip and workpiece to occur when a prescribed critical effective strain value is reached in the elements at the tool tip. In this case, a layer from the workpiece can be separated by the tool tip to form a chip. The workpiece material is assumed to obey a strain-hardening thermal-softening model, which results into an effective stress–strain curve that exhibits strain hardening followed by strain (thermal) softening. The transition from the hardening region to the softening region occurs when the stress reaches a maximum value at which material instability occurs, leading to adiabatic shear band development.

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2. Governing equations and material models 2.1. Governing equations The finite element method has been successfully implemented to study large deformation problems in metal forming. Metal cutting is similar to metal forming where large plastic deformation occurs. However, several modifications must be made in order to use the finite element technique in metal cutting to overcome the above mentioned complications. The basic idea of using the finite element method is to seek a solution to the momentum equation

ij,j + bi = ˙ i (i,j = 1,2,3),

(1)

where ij is the Cauchy stress tensor, is the current mass density, bi is the body force, i is the particle velocity in Cartesian coordinates, ˙ i is the acceleration, and j indicates partial differentiation with respect to xj. For elastic–plastic deformations, the strain rate tensor ⑀˙ ij is usually decomposed into an elastic part ⑀˙ eij and a plastic part ⑀˙ pij such that

⑀˙ ij =

1 ( + j,i), ⑀˙ ij = ⑀˙ eij + ⑀˙ pij . 2 i,j

(2)

From Hook’s Law we have ⴰ

⑀˙ eij = Ceijkl⫺ 1 kl; Ceijkl = 2G ␦ik ␦jl +

(3) 2G ␦ ␦ , 1 ⫺ 2 ij kl

where Ceijkl is the elasticity tensor, G and are the shear modulus and Poison’s ratio respectively, ⴰ and ij is the corotational stress rate given by ⴰ

ij = ˙ ij ⫺ ikjk + ij ik

(4)

with ij being the material spin. Using the von Mises flow rule, ⑀˙ pij is obtained as [26] ·

⑀˙ pij =

␥¯ sij ,

2¯

(5) ·

in Eq. (5), sij is the deviatoric stress, ␥¯ = (2⑀˙ pij ⑀˙ pij )1/2 is the effective strain rate, and

¯ =

冉 冊 1 s s 2 ij ij

1/2

is the effective shear stress. ·

The effective strain rate ␥¯ is determined from the yield criterion for rate-independent materials. For von Mises materials, the yield criterion is given by

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f = √J2 ⫺ ¯ (␥¯ ) = 0,

(6)

where J2 is the second invariant of sij. Then, from the consistency condition ˙f = 0 one can obtain the relation ⴰ

sij sij ∂¯ , ␥¯ = , h= 2√J2h ∂␥¯ ·

(7)

where h is the strain hardening/softening modulus. Combining Eqs (2), (3), (5) and (7) we obtain ⴰ

ij = Cijkl ⑀˙ ij ,

(8)

where Cijkl is the elastic–plastic incremental tensor and is given by Cijkl = Ceijkl ⫺ Cpijkl,

Cpijkl =

G2 s s . ¯ 2(G + h) ij kl

(9)

·

For rate-dependent materials the flow stress ¯ = K(␥¯ ,␥¯ ) and Eqs (4) and (5) yield

ⴰ

冉

ij = Ceijkl ⑀˙ kl ⫺

·

␥¯ 2¯

冊

skl .

(10)

These equations are implemented in the finite element modeling program. In the program, the convergence of the equilibrium iterations in the finite element analysis is determined by examining both the displacement norm 兩兩{⌬u}i兩兩 ⱕ ed, umax

(11)

and the energy norm: ({⌬u}i)T{Q}in + 1 ⱕ ee. ({⌬u}0)T{Q}0n + 1

(12)

In Eqs (11) and (12), {⌬u}0 and {⌬u}i are the increments in displacement in which the superscript denotes iteration number, umax is the maximum displacement norm over all of the n steps including the current iteration, {Q}n + 1 is the residual, ed and ee are tolerances that are typically 10⫺2 to 10⫺3 or smaller and usually are adjustable for different problems. If convergence is not attained and the solution is not divergent, the displacement is updated using a line search scheme and iterations are then continued. If the solutions are determined to be divergent, or convergence fails to occur within an assigned number of iteration, the stiffness matrix [K] is reformed using the current estimation of geometry before continuing equilibrium iteration [6].

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2.2. Material model for the flow stress In metal cutting, large plastic deformations take place accompanied by excessive heat generation. Both high temperature and high strain rate effects (among other factors), on the material behavior need to be considered when simulating the chip formation, especially for modeling the flow localization. Correct determination of the material constitutive model is the key to successfully simulate the formation of shear bands in metal cutting. The material behavior during the formation of the shear localized chip is very complicated and there has been no confirmed model to describe the behavior of workpiece material during the machining process to date. The majority of the energy dissipated in cutting is converted into heat in a very small zone, which makes the temperature distribution in the deformation zone complex. The appearance of the shear localization in the chip involves several mechanical, physical and thermophysical properties. Numerous experiments have shown that temperature plays an important role in the flow instability in chips. The high plastic deformation rate and severe tool–workpiece friction can quickly increase the rate of heat generation. The adiabatic or quasi-adiabatic condition may be reached resulting from the high net accumulation of heat. The temperature can become very high locally to result in further thermal softening in the workpiece material. In this case, the material strain-hardening capacity is reduced so that the instability may take place in a narrow band of the chip. It is believed that shear instability is directly caused by the material flow in the shear bands, and the workpiece material behavior of plastic deformation in cutting process is quite different from the material behavior in ordinary plastic deformation. Although various studies have been made to investigate the material behavior in the machining process, there is still no satisfied materials constitutive model as a function of strain, strain-rate, temperature, etc. to properly describe the material behavior during cutting process. In this investigation, the cutting tool is assumed to be rigidly elastic material with no plastic deformation, and the workpiece material is assumed to be elastoplastic as given by Eq. (9). An isotropic, non-linear hardening law is specified by defining the relationship between yield stress and effective plastic strain as a piecewise curve. This gives some flexibilities in specifying the material model for the flow stress to describe the effects of the thermal softening as a result of quasi-adiabatic heating from large and rapid plastic deformation, which occurs during chip formation. Thus, a mechanical model for the flow stress that exhibits strain hardening, which is followed by strain softening caused by thermal softening, is assumed. Figure 1 shows a schematic of the strain-hardening and thermal-softening model. It is known that plastic deformation takes place when the shear stress on the workpiece material reaches the yield point A in Fig. 1. Then, strain hardening begins as the plastic strain further increases from point A to point B. In this case, the stress at the yield point A is 802.69 MPa and the stress at the point B is 931.47 MPa [8]. Because of large plastic deformations and severe friction in metal cutting, a large amount of heat is generated in the deformation or shear zone of the workpiece material during the process of chip formation. Both the temperature and plastic deformation work in the shear zone do not have enough time to escape due to very high strain rates. Consequently, the temperature in the shear zone increases rapidly to elevated temperatures. This high temperature causes thermal softening in the shear zone which, in return, results in an increase of shear strain in the shear zone with a decreasing stress. This may explain the occurrence of shear localization in metal cutting under certain conditions.

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Fig. 1. Schematic of strain-hardening and thermal-softening material model.

Therefore, the strain-hardening and thermal-softening material model can properly simulate the material behavior in the metal cutting process. In this investigation, the workpiece material property up to point B is taken from a materials handbook. The thermal-softening part of the material property is arbitrarily composed so that the stress will decrease 50% from point B to point C; the strain at point C is twice that of strain at point B. 3. FEA simulation of orthogonal cutting process 3.1. Machining model and boundary conditions The linear quadrilateral elements are used for both the workpiece and the tool materials. Fig. 2 illustrates a schematic of the finite element modeling of an orthogonal cutting process. The top edge of the cutting tool is fixed in the vertical direction and the tool can move horizontally. The bottom edge of the workpiece is constrained in both directions and its left edge is fixed in the horizontal direction. The mesh density of the elements is determined with a convergence error analysis by running some pre-testing models. In this study, the workpiece is divided into 1764 elements and the cutting tool into 30 elements. The cutting process is eventually produced by applying a horizontal displacement boundary condition along the right edge (far side of the cutting edge) of the cutting tool (Fig. 2). The cutter starts from the outside of the workpiece and moves into it at a given cutting speed. The cutting speed can be changed by assigning different displacement boundary conditions. As the cutting tool advances into the workpiece and the chip separates from it, there exist frictions between the tool rake face and chip bottom surface, and between the tool flank face and the machined surface on the workpiece. Some corresponding frictional coefficients can be assigned

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Fig. 2. Finite element model of 2-D machining process.

to these contacting surfaces based on cutting conditions, especially cutting speed and the tool and workpiece materials. These frictions are quite complicated because both sliding and sticking frictions may be involved, and there is still no accurate model to describe the friction in metal cutting. The widely used friction model is the chip–tool friction which is a combination of sticking and sliding regions [2,4,18]. Sticking friction usually takes place in the region near the tool tip, while sliding friction takes place in the region near the point where the chip loses contact with the tool face. In this investigation, the coefficient of friction along the tool–chip interface is assigned based upon the assumption that sticking friction takes place over a distance of 0.01 mm from the tool tip and sliding friction takes place over the remaining part along the tool–chip interface. The friction between the tool flank face and the machined surface is assumed to be sliding friction. 3.2. Simulation of the chip formation To simulate the formation of a chip in the metal cutting process using the finite element method, a tie-break type slideline consisting of two sets of nodes on both surfaces simultaneously between the chip to be formed and the machined workpiece surface to be generated is specified. These two sets of nodes are initially tied together and are allowed to separate (break) when a specified criterion is reached. Although this slideline is the anticipated parting line between the chip and the machined workpiece surface, it does reflect the reality of the actual formation process of the

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chip and the machined surface. A criterion needs to be specified to the slideline so that the chip may separate from the workpiece along the slideline when the criterion is reached. The separation criterion used is the effective plastic strain in the two elements adjacent to the tool tip region. The effective plastic strain ⑀¯ (p) based on the von Mises criterion is calculated for every node at each time step. When the value of the effective plastic strain at a nodal point reaches a prescribed critical value ⑀¯ (p) 0 , that is

⑀¯ (p) ⱖ ⑀¯ (p) 0 ,

(13)

the ‘tie’ at this nodal point is broken and material separation occurs. In this case, the material which is originally tied together as one part is separated into the chip and the machined workpiece surface. As the tool advances into the workpiece, the nodes ahead of the tool tip continue to separate when the effective plastic strains at those nodes reach the separation criterion, and the chip formation continues. Figure 3 shows the chip formation process as the tool advances into the workpiece. Generally, the value of the separation criterion may be between 0.3 and 0.7. However, the separation criterion varies slightly for different workpiece materials and different cutting conditions. Different values of the separation criterion may not significantly affect chip geometry and stress distributions in the chip, but may affect the stress distributions in the machined surface and the effective plastic strain distributions both in the chip and in the machined surface. Therefore,

Fig. 3. Chip formation with localized shear band modeled by FEA.

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some experience may be needed in order to assign a reasonable value for the critical effective plastic strain. In this study, ⑀¯ (p) 0 = 0.5 was used. 4. Results and discussions 4.1. Experimental tests Several orthogonal cutting tests were carried out to verify the results from the finite element model. The orthogonal machining process was attained by cutting a cylindrical workpiece tube on one end on a CNC lathe [23]. This was to ensure that a plane strain condition could prevail during the process of chip formation. The cutting tool was the cemented carbide cutting tip which was installed on a tool holder. After installation, the cutting tool had a rake angle, ␣n, of 8° and flank clearance angle, n, of 7°. In this investigation, the sharp tool without flank wear was assumed and used for all the cutting tests. The workpiece material used in the cutting tests was Ti-6Al-4V titanium alloy. Some preliminary cutting tests were conducted to determine the range of cutting conditions such that the machining process was smooth to ensure the absence of chatter or built-up edge. The cutting speed was set at 5.0 m s⫺1 and the feed rate was set at 0.3 mm rev⫺1. All these are in accordance with the finite element machining model. The cutting forces were recorded using a Kistler piezoelectric dynamometer through a computer controlled data acquisition system. 4.2. Simulation of chip formation and shear banding The chips from cutting tests were collected to study the chip formation and flow localization. The chip specimens were mounted, polished metallographically, and etched. The specimens were then examined and photographed on a microscope. Figure 4 is a photograph of a shear localized chip with shear bands. Figure 5(a) shows a chip formation process with localized shear banding modeled by the finite element analysis. From Figs 4 and 5(a), it is apparent that the chip formation and shear banding modeled by FEA are very similar to those of chips experimentally formed. In metal cutting, the shear band forms along an angle usually called the shear angle. It can be seen from either Fig. 4 or Fig. 5(a) that localized plastic deformation takes place in a very narrow zone rather than in a plane as is often assumed when analysing the orthogonal machining process. The width of the shear band predicted by the finite element analysis is found to be about 0.03 mm which falls into within the experimentally measured values of 0.008 mm to 0.031 mm. It can be noted that the predicted width of the shear banding is not significantly affected by the element size. However, in order to correctly model the shear band width, different constitutive models that include size effects might be considered as suggested by Zbib and Aifantis (1988)[26] and by Zbib (1992)[27]. Chip formation, in practice, is an extremely quick process, especially at high cutting speeds. In metal machining, the cutting tool advances into the workpiece material and tool tip cuts the material off, accompanied with some large plastic deformation, as shown in Fig. 3. Certain strainhardening may take place in the workpiece material in this stage. As the cutting process continues, the tool tip and rake face further cut and press the workpiece material in the area ahead of the

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Fig. 4. Shear localized chip with shear bands (Ti-6Al-4V).

tool tip and cause more plastic deformation. The tremendous pressure as well as frictions, plastic deformation and material fracture near the tool tip, may all contribute to a very high temperature at the tool tip and its nearby area. Because of the very high speed of the cutter movement, the strain rate may reach as high as 105 or even higher, so that the workpiece material in front of the tool tip will be squeezed severely before it can flow along the tool rake face, as shown in Fig. 5(b). The tool–workpiece friction becomes more severe which will then generate more heat. The adiabatic or quasi-adiabatic condition may be reached due to the very high net accumulation of heat and high strain rate. In this case, the material near the tool tip continues to deform and the temperature can be very high locally in some areas of the workpiece, resulting in further thermal softening. This thermal softening reduces the material’s strain-hardening capacity so the instability takes place in a narrow band of the chip and finally forms an intense shear band. The very high pressure near the tool tip also causes sticking and sliding frictions between the tool rake face and the chip. The sticking of the workpiece material on the tool rake face is one of the reasons that form the built-up edge (BUE) on cutting tools. 4.3. Strain and stress The finite element analysis provides an effective method to numerically predict the stress and strain in the deformation area and cutting forces. Figure 6(a) and (b) show the contours of effective plastic strain for different tool positions. At the initial stage of cutting, plastic strains are relatively small and concentrate around the tool tip in both the chip and the machined surface, as shown in Fig. 6(a). As the cutting process continues until the shear band is fully formed, the plastic

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Fig. 5. The chip formation process as tool advances into the workpiece. (a) Tool is advancing into the workpiece. (b) Further deformed chip formation.

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Fig. 6. Contours of effective plastic strain. (a) One stage during cutting process. (b) When shear band is fully formed.

strains increase and spread out into a narrow zone where shear banding takes place as can be seen in Fig. 6(b). The maximum effective plastic strain reaches 2.3. These results explain some of the material instability, in particular the formation of shear banding in the primary deformation zone. Figure 6(a) and (b) indicate that high plastic strains at different time steps of cutting process

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occur near the tool tip; and the chip material above the shear zone does not have significant plastic deformation. This also illustrates the catastrophic plastic deformation that occurs in a limited shear zone. The contours of maximum shear stress at two different cutting stages are plotted in Fig. 7(a) and (b). By comparing the results in Fig. 7(a) and (b) with the results in Fig. 6(a) and (b), it can be seen that the field of the maximum shear stress is around the plastic deformation zone and approximately matches with the areas of the primary and secondary deformation zones. Unlike the effective plastic strain, the maximum shear stress is not confined to a narrow zone, although the shear instability does occur in a narrow area. As the cutting tool further advances into the workpiece material, the total area of the maximum shear stress expands as can be seen in Fig. 7(b), though the maximum shear stress keeps the same magnitude of approximately 480 N mm⫺2 for different cutting stages. This means that the maximum shear stress of the workpiece material is independent of the cutting stages. Shear instability will occur as long as the maximum shear stress is reached. This agrees with the result of ordinary materials test. Fig. 7(b) clearly shows the initiation of the shear banding in front of the tool tip. It can also be noted from Fig. 7(b) that the shear stress in the shear band just ahead of the tool tip is lower than that in the nearby area. This is the indication of shear band initiation due to material’s shear instability, because the low shear stress near the tool tip is apparently caused by the thermal softening of the workpiece material, as mentioned above. This also shows that in metal cutting, the chip does start to shear from the position of tool tip. Figure 8 plots the distribution of effective stress on the tool rake face predicted by the finite element analysis. Figure 9 is the relationship between the effective stress on tool face and the distance from tool tip. It can be seen that the effective stress has the maximum value at the tool tip and gradually diminishes to a negligible value at the position where the chip is no longer in contact with the tool rake face. This agrees qualitatively with the results by Komvopoulos et al. (1990) [11] and Usui et al. (1984) [22]. In addition, Figs 6 and 7 also show the existence of shear stress and plastic strain beneath the machined surface. It can be seen that the maximum stress induced at some distance beneath the machined surface, instead of on top of the machined surface. This is in good agreement with the earlier findings by the authors in characterizing the machined surface integrity [3,24,25]. This indicates that the finite element analysis may effectively be used as a tool to simulate and evaluate the stress, strain and integrity of the machined surface. 4.4. Cutting forces Figure 10 shows the predicted cutting force per unit width of cut. The Cutting force Fc is determined by integrating the stresses acting on both the tool rake face/chip and the flank face/machined interfaces. The actual cutting force can then be calculated by multiplying the cutting forces per unit width of cut by the actual width of cut. It can be seen from Fig. 10 that the cutting force increases from zero linearly to a peak value, then decreases and finally gradually approaches steady value. The steady value is the average cutting force usually measured in cutting tests. It is apparent that the finite element analysis can show more dynamic details of the predicted instantaneous cutting forces, while the ordinary cutting tests measure the average cutting forces. In this study, the width of cut was 5 mm. The steady value of the predicted cutting force per

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Fig. 7. Contours of effective shear stress (105 N mm ⫺ 2). (a) At an early stage of cutting process. (b) Indication of the shear banding initiation.

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Fig. 8. Distribution of effective stress on the tool rake face (105 N mm ⫺ 2).

Fig. 9.

Effective stress on the tool face versus distance from tool tip.

unit width of cut is used to calculate the predicted main cutting force, Fc. The measured average Fc is 921 N while the predicted Fc is 865 N. The predicted value of Fc is in good agreement with the experimentally measured value with an error of 6% in this case. It should be pointed out that there might be a difference between the actual machining process

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Fig. 10. Cutting forces predicted by finite element analysis.

and the finite element modeling of machining process. In actual machining, there is a ‘thrusting’ force which is induced by the feed rate. The thrusting force heavily depends on the magnitude of the feed rate and workpiece material, and is usually the dominant component in the measured thrust cutting force. Without a proper feed rate to induce a thrusting movement on cutting tool, the real machining process will not take place. So the experimentally measured cutting and thrust forces always include the thrusting force component. On the other hand, in the finite element modeling, such a thrusting force is not needed to enforce a feed rate in order to accomplish the cutting process. Therefore, such a thrusting force is not exerted when the boundary conditions were assigned. The lack of this thrusting force would largely reduce the predicted value of thrust force and slightly reduce the predicted cutting force. 4.5. Effects of tool rake angle A series of finite element simulated machining tests with different tool rake angles ranging from ⫺16° to 20° was carried out to study the effects of rake angle on shear band angle and cutting forces. Figure 11 shows a machining process modeled with a negative rake angle ␣n = ⫺8°. Fig. 12 shows the effect of tool rake angle on the shear angle as predicted by the FEA. In the practical range of tool rake angle from ⫺16° to 20° the shear band angle increases from about 36° to 55°. It can be seen that as the rake angle increases, the increase of shear angle slows down and tends to approach to a constant value; as the magnitude of rake angle decreases negatively, the decrease of shear angle also slows down and tends to approach to a constant value. These are in agreement with previously observed experimental results [4,14,20]. Figure 13 shows the effect of the rake angle on the cutting force. As the rake angle decreases from 20° to ⫺16°, the main cutting force, Fc, increases. This is apparent because decreasing the rake angle raises the force component in the direction of Fc. The trend of the effects of rake

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Fig. 11. Machining process modeled with a negative rake angle.

Fig. 12. Relationship between tool rake angle and shear banding angle.

angle on the main cutting force Fc agrees well with experimentally obtained results [12,15,20]. This, therefore, gives this FEA model an obvious advantage in machine tool, cutting tool and tool holder/fixture designs, because the tool geometry and other factors can be simulated and optimized in the computer without running a cutting test. In the real world, though, almost all machinings are three dimensional. It, therefore, may require

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Fig. 13. Effects of tool rake angle on cutting forces.

3-D modeling of the machining process in order to predict the results more accurately when FEA is applied in cutting tool design. 4.6. Material behavior modeling In this study, the material behavior in metal cutting was modeled and evaluated using a commercial FEA code, ABAQUS. Two workpiece materials with different material properties and constitutive relations were defined in the software’s user-defined subroutine. The intention here was to qualitatively illustrate the possibility of numerically simulating varied chip formations for different workpiece materials. So less emphasis was put on the accuracy of matching material’s constitutive models with real materials. One material was assumed as the Ti-6Al-4V titanium alloy with ductile failures which is defined by the modified Gurson constitutive model [19], in which the local failure between individual voids combined with the original Gurson constitutive relations could provide a means of tracking deformation and failure of a continuum element. Another material was assumed as an ordinary elastic–plastic low-carbon steel, AISI 1020, with the strain-hardening thermal-softening law, as described earlier in this paper. These two materials were modeled as both have the same cutting conditions. The cutting tool was modeled as a rigid body with the chip breaker. Figures 14 and 15 show the FEA modeled chip formations of titanium alloy and the low-carbon steel, respectively. It can be seen that the discontinuous chips were formed for the titanium alloy which is a typical material that shear localization usually takes place during machining. However, a continuous chip was formed for the low-carbon steel, as expected. Both results are in accordance with experimental observations. This shows that FEA can be used as an efficient numerical method to test the materials behaviors and/or chip formations for different workpiece materials without running cutting tests. This is especially convenient and cost-saving when only relative comparisons between

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Fig. 14.

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Discontinuous chips modeled by FEA.

Fig. 15. Continuous chip modeled by FEA.

different materials are needed. Chips could be produced, analysed and compared on the computer instead of on the real machine tools. This could be very helpful in cutting tool design optimization and in determining optimal machining process. Meanwhile, this also suggests that more accurate and complete material constitutive models are needed in order to correctly simulate the machining processes using finite element analysis. 5. Conclusions

1. A quasi-static finite element modeling of chip formation and shear banding in orthogonal metal cutting has been presented. The updated Lagrangian formulation for plane strain condition is used in this investigation. The ‘tiebreak’ slideline was used to separate the newly formed chip from the workpiece surface. The effective plastic strain is used as the material failure criterion.

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In order to model the shear banding during chip formation, a strain-hardening thermal-softening model for the flow stress is used. 2. The FEA simulations of chip formation and shear banding show that the finite element analysis can successfully model the metal machining process. In addition, cutting forces, distribution of effective stress on tool rake face, maximum shear stress and plastic strain fields in the workpiece material can also be predicted. 3. The simulated results show that the effective plastic strain is within a narrow area along the shear zone angle, and the field of the maximum shear stress matches with the area of the primary and secondary deformation zones. This finite element model predicted the detailed deformation in front of the tool tip and the initiation of the shear band. The comparison between predicted and experimentally measured cutting forces indicates that cutting forces, among others, can be reasonably simulated with the finite element analysis. However, the lack of thrusting force in FEA modeling may bring in some errors to the predicted cutting forces, especially to the thrust cutting force. The effect of tool rake angle on the shear band angle and cutting forces is also predicted by the FEA modeling. This implies that FEA could be an important part of a computer integrated manufacturing (CIM) system since several machining parameters can be simulated and optimized without running experiments. 4. The material behavior in metal cutting was successfully modeled by FEA. Both discontinuous and continuous chips were simulated and the computer modeled results are in accordance with experimental observations. Therefore, FEA can be used as an efficient numerical method to test the materials behaviors and/or chip formations for different workpiece materials without running cutting tests. Chips could be simulated, analysed and compared on the computer instead of on the real machine tools. This study also suggests that more accurate and complete material constitutive models are needed in order to correctly simulate the machining processes using finite element analysis.

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[10] B.E. Klamecki, Incipient chip formation in metal cutting—A three dimension finite element analysis, Ph.D. dissertation, University of Illinois at Urbana–Champaign, 1973. [11] K. Komvopoulos, S.A. Erpenbeck, Finite element modeling of orthogonal metal cutting, in: Computer Modeling and Simulation of Manufacturing Processes, ASME Winter Annual Meeting, MD-vol. 20, PED-vol. 48, November 1990, p. 1. [12] D. Lee, W.W. Wilkening, Material modeling and high speed machining process, in: Annual Technical Report to Advanced Machining Research Program (AMRP), General Electric Company, August 1982. [13] T.D. Marusich, M. Ortiz, Modeling and simulation of high-speed machining, International Journal for Numerical Methods in Engineering 38 (1995) 3675. [14] R.T. Sedgwick, H.E. Read, D.E. Wilkins, Numerical modeling of high-speed machining processes, in: Annual Technical Report to Advanced Machining Research Program (AMRP). General Electric Company, August 1982. [15] M.C. Shaw, Metal Cutting Principles, Clarendon Press, Oxford, 1984. [16] A.J.M. Shih, S. Chandrasekar, H.T.Y. Yang, Finite element simulation of metal cutting process with strain-rate and temperature effects, in: Fundamental Issues in Machining, ASME-WAM, PED-vol. 43, Dallas, Texas, November 1990, p. 11. [17] J.S. Strenkowski, J.T. Carroll III, A finite element model of orthogonal metal cutting, Trans. ASME J. of Eng. Ind. 107 (1985) 349. [18] E.M. Trent, Metal Cutting, 3rd ed., Butterworth–Heinemann Ltd., London, 1991. [19] V. Tvergaard, A. Needleman, Analysis of the cup–cone fracture in a round tensile bar, Acta Metall. 32 (1984) 157. [20] E. Usui, A. Hirota, Analytical prediction of three dimensional cutting process, Part 2, ASME J. of Eng. Ind., May 100 (1978) 229. [21] E. Usui, T. Shirakashi, Mechanics of machining—From ‘Descriptive’ to ‘Predictive’ theory, in: On the Art of Cutting Metals—75 Years Later, ASME Publication PED, vol. 7, 1982, p. 13. [22] E. Usui, T. Shirakashi, T. Kitagawa, Analytical prediction of cutting tool wear, Wear 100 (1984) 129. [23] Q. Xie, An analytical and finite element analysis of chip formation and shear banding in metal cutting process, Ph.D. dissertation, Washington State University, 1993. [24] Q. Xie, A.E. Bayoumi, L.A. Kendall, G.L. Sheldon, Study on residual stresses and tool wear induced by machining processes, in: Trans. NAMRC XVII, 1989, p. 170. [25] Q. Xie, A.E. Bayoumi, L.A. Kendall, Tool wear and its effects on the machined surface integrity, J. of Materials Shaping Tech. 8 (4) (1990) 255. [26] H.M. Zbib, E.C. Aifantis, On the localization and postlocalization behavior of plastic deformation, I. On the initiation of shear bands, Res Mechanica 23 (1988) 261. [27] H.M. Zbib, The strain gradient phenomenon in viscoplasticity: Theory and application to shear instability, in: S.I. Anderson et al. (Eds.), Modeling of Plastic Deformation and Its Engineering Application, Reso National Lab, Denmark, 1992, p. 525. [28] B. Zhang, A. Bagchi, A study of chip separation and its approximation in finite element simulation of continuous chip formation, in: D.A. Stephenson, R. Stephenson (Eds.), Materials Issues in Machining—II and the Physics of Machining Processes—II, Winter Annual Meeting of the ASME, Chicago, 1994, p. 157.

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