Int J Adv Manuf Technol (2007) 32: 666–674 DOI 10.1007/s00170-005-0388-z
ORIGINA L ARTI CLE
W. J. Zong . D. Li . K. Cheng . T. Sun . Y. C. Liang
Finite element optimization of diamond tool geometry and cutting-process parameters based on surface residual stresses Received: 4 May 2003 / Accepted: 6 October 2005 / Published online: 2 November 2006 # Springer-Verlag London Limited 2006
Abstract In this paper, a coupled thermo-mechanical plane-strain large-deformation orthogonal cutting FE model is proposed on the basis of updated Lagrangian formulation to simulate diamond turning. In order to consider the effects of a diamond cutting tool’s edge radius, rezoning technology is integrated into this FE based model. The flow stress of the workpiece is modeled as a function of strain, strain rate, and temperature, so as to reflect its dynamic changes in physical properties. In this way, the influences of cutting-edge radius, rake angle, clearance angle, depth of cut, and cutting velocity on the residual stresses of machined surface are analyzed by FE simulation. The simulated results indicate that a rake angle of about 10° and a clearance angle of 6° are the optimal geometry for a diamond tool to machine ductile materials. Also, the smaller the cutting edge radius is, the less the residual stresses become. However, a great value can be selected for cutting velocity. For depth of cut, the ‘size effect’ will be dependent upon it. Residual stresses will be reduced with the decrement of depth of cut, but when the depth of cut is smaller than the critical depth of cut (i.e., about 0.5 μm according to this work) residual stresses will decrease accordingly. Keywords FE simulation . Diamond turning . Residual stresses . Optimization . Cutting-edge radius
1 Introduction The metal cutting process is designed to remove unwanted material of a component based on the geometric shape and W. J. Zong (*) . D. Li . K. Cheng . T. Sun . Y. C. Liang Harbin Institute of Technology, P.O. Box 413, Harbin, 150001, China e-mail:
[email protected] K. Cheng School of Engineering and Design, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK
dimensional accuracy that is required. This process brings with it large plastic deformation and mechanical effects, and some strains or stresses will remain within the machined surface layer, which are called “residual strains” or “residual stresses”. Tensile residual stress leads the cracks initializing on the machined surface, degrades the resistance of the component to fatigue and wear, and decreases the component’s service life. Also, while the machined component is removed from the chucking system, these inherent residual stresses will provoke some additional deformation so as to magnify the component’s defective index. Therefore, how to reduce the residual stresses of a machined surface is always a difficult but important problem, especially for diamondturned components. In 1973, Klamercki introduced the finite element method (FEM) into the manufacturing field [1]. By far, FEM is considered as one of the best methods to investigate various problems regarding metal machining, such as chip formation, material removal, specific machining processes, effects of tool geometry, process parameters, thermal effects, residual stresses, machine-tool control, tool wear, and failure, etc. [2–4]. In general, FE simulations can be classified into two approaches: the Lagrangian approach [5] and the Eulerian approach [6]. The Lagrangian approach needs a chip separation criterion for FE model, either the geometrical ones, including node distance, fracture initiation or damage and elements rezoning, or the physical ones, such as equivalent plastic strain, strainenergy density and normal failure stress. Of course, the geometrical or physical criterion has its merits and demerits and is adopted according to different conditions [7, 8]. The Eulerian approach has no need for chip separation criterion, but the chip shape must be foreknown in order to control its flow. In 1988, Carroll et al. were the first to investigate the diamond turning process by FE simulation [9]. At that time, they studied the diamond turning process with both the Lagrangian approach and the Eulerian approach and also compared the different simulation results of these two approaches. Up to now, many important works have been done in FE simulation for diamond turning [10–12, 14–16].
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In [12] and [14], Lin et al. investigated the effects of cutting velocity, depth of cut, and tool flank on the residual stresses within the machined surface of an Ni-P alloy, and in [15], Lo also studied the influence of rake angle on the residual stresses within machined surface of OFHC copper. However, in [12–16], node distance or strain-energy density was regarded as chip separation criterion in FE models and diamond tool was assumed very sharp. As a result, the effects of the cutting-edge radius weren’t considered in these FE models. In [9–11], either the temperature’s effects were not considered for the workpiece’s material model or only the heat conduction was analyzed. Above all, these researching works paid little attention to optimal selection for a diamond tool’s geometry or process parameters in reducing the residual stresses within the machined surface. Therefore, in this FE-based model, a remeshing chip separation criterion is adopted based on the effects of the cutting-edge radius, and both heat conduction and convection are carried out for thermal analyses. According to this FE-based model, the influences of some important parameters on the residual stresses are analyzed, including the cutting-edge radius, rake angle, clearance angle, depth of cut, and cutting velocity. Finally, the diamond cutting tool’s geometry and processing parameters are optimized aiming at the minimal surface residual stresses.
2 Finite element model and boundary conditions As is known, width of cut is at least five times greater than depth of cut in orthogonal cutting, which is the same in diamond turning. So the realistic 3-D diamond turning process can be simplified into a 2-D cutting model. According to this assumption, a plane-strain finite element model based on an updated Lagrangian approach is presented. The initial meshing of finite elements is shown in Fig. 1. In this figure, Vr designates cutting velocity and rn denotes cutting-edge radius. The workpiece consists of a four-node isoparametric quadrilateral plane strain coupled elements. Like a chucking system, the vertical and horizontal freedoms of the nodes at the bottom of the Fig. 1 Initial meshing finite elements
workpiece are configured as zero, and the horizontal displacements of the nodes on the left of the workpiece are also set as zero. The workpiece dimensions are 40 μm× 12 μm, and assumed large enough in both directions x and y. Therefore, the boundary conditions have no influences on the whole cutting process in simulation. The four-node heat transfer planar quadrilateral elements make up the diamond tool and since it is assumed to be rigid, only heat transfer analyses are carried out. In addition, the diamond tool keeps a horizontal velocity Vr along the predefined cutting path and has no displacements in direction y. The details of the cutting-edge radius are shown in magnified section of Fig. 1.
3 Modeling on the workpiece material Modeling on the material is quite important to FE simulation. The material model must represent the actual changes of the material’s physical properties during machining. In this paper, OFHC copper is employed as the workpiece material and it is sensitive to strain, strain rate, and temperature. For strain and strain rate, they will harden the material in deformation regions. For high temperature, it will soften the material. Therefore, the ideas for stress calculation in Oxley’s cutting model are introduced, i.e., that the flow stress is assumed to be a function of strain, strain rate, and temperature, whose Johnson-Cook’s constitutive equation is expressed as [17] h . i m (1) 1T σ ¼ ða þ b"n Þ 1 þ c ln " "0
where σ is the flow stress; " the equivalent total strain; "
the equivalent total strain rate; "0 the initial strain rate and
"0 ¼ 1; a, b, c, n, and m are constant factors and a=90 Mpa, b=292.8 Mpa, c=0.025, n=0.31, m=1.09; T =(Tt−Troom)/ (Tmelt−Troom); Tt the transient temperature and Tmelt=1,083°C, the molten temperature of OFHC copper; Troom=20°C, the ambient temperature. The other physical properties, such as Young’s modulus E, Poisson’s ratio ν, specific heat C,
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material density ρ, thermal conductivity κ and thermal expansion coefficient λ of OFHC copper and diamond tool, are listed in Table 1.
α
Xn
4 Chip separation criterion
X est n+2
X n+1
Fig. 2 Changing of workpiece elements’ internal angle
In this work, the cutting-edge radius of a diamond tool is modeled. Like the realistic diamond turning process, the severe extruding, scratching and burnishing between the chip-workpiece and the tool edge or rake face follow the whole FE simulation. That is to say, the deformation between them is quite large. In order to consider these large deformations, in this work, chip formation is simulated by the remeshing chip separation criterion [18]. During the simulation process, if the workpiece’s elements suffer excessive geometrical deformation, then the meshes are regenerated, and the state variables are transferred from old elements to new elements. In the present cutting model, the criteria for estimating element distortion are the internal angle’s changing of workpiece elements and contact penetration between tool elements and workpiece elements. Firstly, each internal angle of workpiece elements is checked at the final iteration of previous increment and the changes of internal angle are estimated in the next increment. Presuming that Xn is the initial coordinate of the current increment, and ΔUn is the displacement in current increment, then X nþ1 ¼ X n þ ΔU n ; X est nþ2 ¼ X nþ1 þ ΔU n
(2)
The internal angle’s changes of workpiece elements are shown in Fig. 2. If cosα>0.8 and cosβ>0.9 or cosα>0.9 and cosβ>cosα, remeshing takes place and new elements are recreated, and the chip is formed step by step. The penetration between tool elements and workpiece elements is examined at the end of current increment in the followed step. The threshold of the penetration variable is set according to the realistic cutting case. During diamond turning, the penetration between the diamond cutting tool and the workpiece could not take place. So a suitable value, zero, is recommended in this orthogonal cutting model. If the penetration in simulation is more than this threshold, then remeshing action is carried out and new elements are regenerated to avoid the overlaps between tool elements and chip elements.
5 Friction model Friction effects in diamond turning are very complicated and could not be ignored. Experimental observations Table 1 Physical properties of OFHC copper and diamond tool
β
OFHC copper Diamond
revealed that both sticking and sliding would happen at the interface between tool and chip. In order to consider both sticking and sliding friction, a modified Coulomb sticking-sliding frictional model is introduced. The frictional stress σfr at the tool-chip’s interface is expressed as 2 Vr σfr μσn arctan i (3) π V cnst where μ is the frictional coefficient and assumed to a constant of 0.3; σn the normal stress at the tool-chip’s interface; Vr the cutting velocity or relative sliding velocity between tool and workpiece; Vcnst the critical velocity for sticking and sliding transition, about 1∼10% of Vr , i.e., that below this threshold, sticking takes place; i=Vr/∣Vcnst∣, the tangent unit vector of Vr .
6 Heat transfer Elevated temperature is a significant problem in metal cutting because it has visible influences on the physical property changes of the workpiece in cutting and the machined surface quality. Therefore, two heat sources are considered in this work; the friction heat and the plastic work. For the heat transfer analyses, both conduction and convection are considered but radiation is neglected. In 2-D coupled thermo-mechanical analyses, the global distribution of transient temperature T is written as 2 @ T @2 T @T Q κ (4) þ þ ¼ ρC @x2 @y2 @t
where κ is the thermal conductivity; Q the comprehensive rate of specific volumetric heat flux and t the time. The heat flux qp, which converts from plastic work due to shearing deformation, is given by qp ¼
Mfp Wp ρ
(5)
where M is the mechanical equivalent of heat; fp the percentage of plastic work converting into heat, about 0.9 and Wp the plastic work.
E (GPa)
ν
C (J/kg/°C)
ρ (kg/m3)
κ (W/m/°C)
λ (10−6/°C)
128 1,050
0.3 0.1
385.5 420
8,960 3,520
393.6 1,000
16.5 2.5
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The heat flux qf, which is generated due to the contact friction, is calculated as (6) qf ¼ Mfc Ff Vr Sc where fc is the assignment operator of contact heat accounting for the percentage of contact heat into workpiece-chip or tool, which is about 28% for the workpiecechip and 72% for the tool according to their thermal conductivity; Ff the contact frictional force and Sc the contact area. The heat flux qc, which is consumed by heat convection on the tool or the workpiece’s topmost surfaces, is expressed as qc ¼ hc ðTt Troom Þ
threshold, the residual stress becomes tensile again. Giving an obvious distinction with the residual stress in direction x, residual stress in direction y is compressive on the topmost machined surface and changes from compressing to stretching with the increment of sublayer depth. If the sublayer depth increases gradually, then the residual stress turns into compressive slowly. Finally, the residual stresses in both directions drop to zero at the same depth, i.e., at about 1.4 μm. It can be clearly seen from this figure that the maximal residual stresses in direction x and in direction y are on the topmost machined surface. In order to enable the analyses results having comparability and consistency, the maximal residual stresses are used to analyze the influences of tool geometry and process parameters in the following sections.
(7) 7.1 Cutting-edge radius
where hc is the convection coefficient of heat transfer; Tt the transient temperature on tool or workpiece’s surface. Furthermore, the initial temperature of the workpiece or tool is set as 20°C.
7 Factors influencing the residual stresses From the viewpoint of tool geometry and cutting process, the factors influencing residual stresses include cuttingedge radius rn, rake angle γo, clearance angle αo, depth of cut ap, cutting velocity Vr , and feed rate f. Considering that a 2-D cutting model is used, feed rate f is a normalized factor. So it could not be analyzed and assumed to be one unit in this paper. Figure 3 is a schematic presentation of residual stresses distribution within the machined subsurface layer under selected machining parameters as listed in the chart. As shown in this figure, the residual stress in direction x and that in direction y change with the variation of sublayer depth. The residual stress in direction x is tensile on the topmost machined surface but changes from stretching to compressing with the increment of sublayer depth. When the subsurface depth reaches a certain Fig. 3 Distribution of residual stresses within machined subsurface
In this section, the simulation parameters are configured as: a rake angle γo of 0°, a clearance angle αo of 6°, a cutting velocity Vr of 6 m/min and a depth of cut ap of 2 μm. When the cutting path length reaches to 24 μm, the maximal residual stresses on the surface machined with different cutting edge radiuses rn are shown in Fig. 4. As the curve plotted in this figure, the maximal residual stresses in direction x and y within the machined surface will both increase when the cutting-edge radius enlarges. This is because when the cutting-edge radius increases gradually, the extruding action of the tool edge to chip root aggravates so that the chip deformation enhances. That is to say, plastic deformation or shear deformation will increase, and then residual stresses will rise consequently. Meanwhile, the rubbing, extruding, and burnishing between the machining surface and tool edge exacerbate with the increment of cutting-edge radius. So the cutting temperature enlarges step by step, as described in Fig. 5. As is well known in metal cutting, the higher the cutting temperature is, the more the thermal expansion and cold shrinkage in the cutting region, i.e., that the more the deformation of the machined surface is, and a larger surface deformation leads to more residual stresses. From the viewpoint of the
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Fig. 4 Maximal residual stresses with different cutting-edge radii rn
Fig. 6 Maximal residual stresses with different rake angles γo
residual stresses within the machined surface, the sharper the cutting-edge radius is, the less the tensile or compressive residual stresses of the machined surface.
In this part, the simulation parameters are: a cutting-edge radius rn of 200 nm, a clearance angle αo of 8°, a cutting velocity Vr of 120 m/min, and a depth of cut ap of 2 μm. Rake angles γo are set as −10°, −5°, 0°, 5°, 8°, 10°, and 15°, respectively. The maximal residual stresses and cutting temperature simulated with different rake angles are shown in Figs. 6 and 7, respectively. In Fig. 6, the curve slope indicates that when the rake angle changes from negative to positive, the maximum residual stress in directions x and y will also reduce linearly. This is because the enlargement of the rake angle leads to a decrement in shearing effects and chip deformation. At the same time, the extruding action of the tool to the chip weakens. So the
maximal residual stresses decrease when rake angle changes from negative to positive, but it could not be concluded that the bigger the rake angle is, the smaller the residual stresses are. For the tensile residual stress in direction x, when the rake angle increases to 10°, tensile residual stress reaches its smallest level. If the rake angle is more than 10°, then the residual stress enlarges immediately. These phenomena are also related to the cutting temperature. Figure 7 shows the changes of maximal cutting temperature in the cutting region with a cutting path length of 24 μm. It can be clearly seen from this figure that the changing law is different from that in conventional metal cutting. This is because in diamond turning, the plastic work becomes a dominant heat source to the whole thermal energy in cutting region [19] and the heat diffusion of the diamond tool degrades with an increase in the rake angle. Obviously, the increment of temperature results in more and more thermal expansions of the workpiece. As shown in Fig. 7, the ascending curve of cutting temperature levels off when rake angle is more than 10°. At this time,
Fig. 5 Maximal cutting temperature in cutting regions with different cutting edge radii rn
Fig. 7 Maximal cutting temperature in cutting regions with different rake angles γo
7.2 Rake angle
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the increment of thermal residual stress due to the climbing of the cutting temperature is dominant in the whole increment of tensile residual stress, which is more than the decrement of tensile residual stress due to the weakening of chip deformation. As a result, the maximal residual stress rises. The compressive residual stress declines to its smallest level while the rake angle is close to 5°. If rake angle is more than this critical threshold, the compressive residual stress will also increase. Based on the analyses above, the optimal rake angle of a diamond tool is 10° when plastic materials are machined. 7.3 Clearance angle In this section, the identical parameters of the following simulations are: a cutting edge radius rn of 200 nm, a rake angle γo of 0°, a cutting velocity Vr of 6 m/min and a depth of cut ap of 2 μm. The clearance angle is set as 2°, 4°, 6°, 8° and 10°, respectively. The simulated maximal residual stresses of a machined surface with different clearance angles are shown in Fig. 8. In this picture, the residual stresses in direction x and in direction y both reduce with the decrement of clearance angle. For the maximal tensile residual stress in direction x, the minimum clearance angle is 6°. If the angle exceeds 6°, tensile residual stress then increases. The maximal compressive residual stress in direction y has the same laws, but its critical angle is only 4°. These phenomena may be due to two causes: one is the change of cutting temperature and the other is rubbing, extruding, and burnishing between the machining surface and tool frank. When the length of the cutting path reaches 24 μm, the variations of maximal cutting temperature with different clearance angles are presented in Fig. 9. As the plotted curve in this figure, it should be noted the cutting temperature reaches the curve valley while the clearance angle is 6°. Provided that the clearance angle is more than this critical threshold, then the capability of the diamond tool for heat diffusion depresses so as to result in an increase in cutting temperature and provoking more
Fig. 8 Maximal residual stresses with different clearance angles αo
Fig. 9 Maximal cutting temperature in cutting regions with different clearance angles αo
thermal deformation. On the other hand, assuming that the elastic recovery of the machined surface is a constant, when the clearance angle reduces from 6°, the smaller the clearance angle is, the more the contact areas generate between machining surface and tool flank. As considered, an enlargement of the contact area will aggravate the extruding and rubbing actions and lead to a higher cutting temperature directly, which can be found in Fig. 9. Finally, due to the aggravation in extruding, rubbing or burnishing between the machining surface and tool flank, the cutting temperature increases. As a result, the residual stresses ascend. In conclusion, the clearance angle of a diamond tool should be adjusted to 6° when ductile materials are machined. 7.4 Depth of cut In this part, the identical parameters of all the simulations are: a cutting edge radius rn of 200 nm, a rake angle γo of 0°, a clearance angle αo of 6° and a cutting velocity Vr of 120 m/min. The simulated results with different depths of cut are plotted in Fig. 10. In this figure, when the depth of cut decreases from 4 μm, the maximal residual stresses in direction x and in direction y both reduce. This is because a decrement of depth of cut is equivalent to a diminution of thickness of cut. As a result, chip can flow out easily and then the plowing of tool edge weakens at the same time, which introduces a reduction in normal compressive stress between diamond tool and chip root, as shown in Fig. 11. So the residual stresses, including tensile and compressive, reduce with the decrement of depth of cut. When the depth of cut falls into the same order of magnitude as the cutting edge radius, however, any additional decrease in depth of cut will cause an increase in the residual stresses, as shown in Fig. 10. At this time, the extruding of tool edge to chip root and the burnishing as well as rubbing between machining surface and tool flank become more and more evident. The tool’s cutting
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Fig. 10 Maximal residual stresses with different depth of cut ap
Fig. 12 Maximal residual stresses with different cutting velocity Vr
process likes a rough plowing action, and, as a result, the cutting state near the tool tip deteriorates and the extruding effects within the contact area enlarge, as exhibited in Fig. 11. Obviously there is a critical depth of cut acp over the relationship between depth of cut and residual stresses, i.e., about 0.5 μm or 2.5 rn in this work. This is the so-called “size effect”. That is to say in diamond turning, there is no conclusion that the smaller the depth of cut, the better the machined surface quality. A given tool cutting edge radius has a smallest depth of cut for the best machining quality.
As can be seen from this figure, the decrement of cutting velocity gives a minute influence to the residual stress in direction y, but leads to a visible reduction of the residual stress in direction x, and Fig. 13 is a pictorial presentation of the cutting temperature in all simulations when length of cutting path reaches 24 μm. As shown in Fig. 13, the cutting temperature rises incrementally with the enlargement of the cutting velocity. As expected, the increasing cutting temperature within the cutting regions softens more and more being machined workpiece’s materials, descends chip deformation, and releases the extruding effects of tool edge to chip root. In Fig. 14, the decrement of chip deformation can be seen clearly while cutting velocity changes from 6 to 120 m/min, which is characterized with a lessening thickness of chip. The maximal equivalent stress within the shear zone also reduces from 621.0 to 548.6 Mpa, and its location shifts from the inner to the outer side of the chip. On the other side, although the cutting temperature ascends while cutting velocity increases, the rate of chip deformation also accelerates at the same time. The increasing rate of chip
7.5 Cutting velocity In this section, the simulation parameters are set as: a cutting edge radius rn of 200 nm, a rake angle γo of 5°, a clearance angle αo of 8° and a depth of cut ap of 2 μm. Cutting velocity is selected as 6, 30, 60, and 120 m/min, respectively. Figure 12 presents the final simulated results.
Fig. 11 Maximal compressive stresses at chip root with different depth of cut ap
Fig. 13 Maximal cutting temperature in cutting regions with different cutting velocity Vr
673 Fig. 14 Influences of cutting velocity on the chip deformation
deformation makes an increasing amount of thermal energy converting from plastic work and frictional work congregate in the first and second deformation regions, i.e., the contact region between rake face and chip and the shear deformation region, however, a majority of this thermal energy is removed frequently by chip flows. Therefore, the increment of cutting temperature has only minute effects on the thermal deformation of the third deformation region, i.e., the contact region between the machined surface and flank face. That is to say, within the third deformation region, the increment of residual thermal stress due to the enlargement of cutting temperature is much less than the decrement of residual tensile stress due to chip deformation weakening. As a result, tensile residual stress will slowly decrease with the increment of cutting velocity. Like the analyses above, within the third deformation region, the increment of residual thermal stress in direction y due to the increment of cutting temperature is equal to the decrement of residual stress due to chip deformation weakening, so the compressive residual stress varies little when cutting velocity increases. Therefore in realistic diamond turning, a greater cutting velocity can be adopted.
Cutting-edge radius has significant influence on residual stresses, either tensile residual stress or compressive residual stress. When the cutting-edge radius increases from 100 to 300 nm, the smaller the cutting edge radius is, the less the tensile residual stress or the lower the compressive residual stress. In diamond turning of ductile materials, a rake angle of about 10° and a clearance angle of 6° are the optimum geometry for the diamond cutting tool from the viewpoint of tensile residual stresses. There is a “size effect” among the relationship between the depth of cut and residual stresses. If the depth of cut is equal to the critical threshold, i.e., about 0.5 μm or 2.5rn as presented in this work, the residual stresses (both tensile and compressive residual stresses) are reduced to their smallest level. Ranging from 6 to 120 m/min, cutting velocity has a visible affect on the tensile residual stress. However, it has a minute influence on the compressive residual stress. Cutting temperature provides prominent effects to the tensile and compressive residual stresses by the role of thermal deformation, which cannot be neglected in diamond turning.
8 Conclusions 9 Nomenclature In this paper, with the finite element method, a coupled thermo-mechanical plane strain large deformation orthogonal cutting model is developed based on the updated Lagrangian approach to simulate diamond turning. Therefore, the influences of diamond tool geometry and cuttingprocess parameters on the residual stresses within a machined surface can be analyzed, such as cutting-edge radius, rake angle, clearance angle, depth of cut, and cutting velocity. According to the analyses above, some important conclusions can be drawn as follows:
T Tmelt Tt Troom E C Ff
Q M
Transient temperature of workpiece Molten temperature of workpiece Transient temperature of workpiece or tool surface Ambient temperature Young’s modulus Specific heat Contact frictional force Rate of specific volumetric heat flux Mechanical equivalent heat
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Wp ap acp Vr Vcnst i t fp fc qp qf qc hc rn γo αo "
"
"0 ν ρ κ λ σ σfr
Plastic work Depth of cut Critical depth of cut Cutting velocity between tool and workpiece Critical velocity between sticking and sliding Tangent unit vector in the same direction of Vr Time Percentage of plastic work converted into heat Contact heat transfer coefficient used to account for contact heat dissipating Heat flux converting from plastic work Heat flux generating due to contacting friction on rake face Heat flux consuming by convection Convection coefficient of heat transfer Cutting-edge radius Rake angle Clearance angle equivalent total strain equivalent plastic strain rate Initial plasitc strain rate Poisson’s ratio Material density Thermal conductivity Thermal expansion coefficient Flow stress Frictional stress
Acknowledgement The authors acknowledge the financial support provided by the National Natural Science Foundation of China (No.50175022) for this research work.
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