Strain, strain rate and velocity fields determination at very high cutting speed

Journal of Materials Processing Technology 213 (2013) 693–699

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Strain, strain rate and velocity fields determination at very high cutting speed G. List a,∗ , G. Sutter a , X.F. Bi b , A. Molinari a , A. Bouthiche a a b

LEM3, UMR C.N.R.S. n◦ 7239, Université de Lorraine – Ile du Saulcy, 57045 Metz Cedex 1, France Shanghai-Hamburg College, University of Shanghai for Science and Technology, 200093 Shanghai, China

a r t i c l e

i n f o

Article history: Received 22 May 2012 Received in revised form 21 November 2012 Accepted 24 November 2012 Available online 3 December 2012 Keywords: High speed cutting Streamlines model FEM

a b s t r a c t High speed imaging is used for the purpose of examining the strain and strain rate variations in the primary shear zone at cutting speed of 1020 m/min. Experimental investigation focused on flow pattern describing the severe plastic deformation zone where a general streamline model is employed to investigate the distribution of velocity. Strain and strain rate distribution are directly deduced from the experimental observation of a cross-section of chip obtained through a high-speed camera system. The strain and strain rate gradients were analyzed along several streamlines. A finite element method model based on a Lagrangian formulation has been used to corroborate the conclusion of the streamline model. It has been found that the simulation results are similar to the experimental observations with regards to the magnitude of the equivalent strain rate and cumulative plastic strain but slightly differ in the geometry of flow pattern. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Metal machining is characterized by very large strain (>1) and strain rate that can be quite important (>104 s−1 ). Information about the magnitude of these quantities is essential for modeling the machining process with the finite element method to predict temperature rise and cutting forces with accuracy. Experimental measurements of strain field in metal forming have generally used the grid technique where the distortion of the grid is examined after imposition of the deformation. Grid deformation approaches have also been applied to characterize the severe plastic deformation in metal machining. The research of chip formation mechanisms has made significant progress by the employment of this technique coupled with quick-stop devices, which provide a method to freeze the chip. However, an important uncertainty in accuracy of observation remains about the measurement of deformation. An alternative of the quick-stop technique is the photography of the chip cross-section during its formation by the employment of a camera. With a dedicated experimental arrangement including a special cutting machine, a microscope and a lighting installation, Warnecke (1977) has observed the process of chip formation and particularly the built-up edge formation for steel at cutting speeds up to 60 m/min. With two “snapshots” of the deformed grid taken at different times, Childs (1971) analyzed with accuracy the velocity field in the deformation zone thanks to double exposure

∗ Corresponding author. Tel.: +33 3 87 31 72 49; fax: +33 3 87 31 53 66. E-mail address: [email protected] (G. List). 0924-0136/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmatprotec.2012.11.021

photograph at 0.025 m/min. This technique improves the determination of the velocity gradient but the method can only be applied to very slow cutting speed. Pujana et al. (2008) utilized laser printing in combination with high-speed photography techniques and managed to get a picture of deformation grid during machining with a relatively high cutting speed of 300 m/min. The use of grid method at higher cutting speed makes it the record of images very difficult and only one snapshot of the chip is expected. Grid offers direct measurement of the strain but it is necessary to obtain recorded images sufficiently clear to analyze the deformed squares. Direct applications of particle image velocimetry (PIV) or digital image correlation (DIC) have been also recently reported. Lee et al. (2006) have analyzed the velocity distributions in the primary and secondary shear zones using the PIV method in the case of the orthogonal cutting of copper, lead and an aluminum alloy at the low cutting speed of 0.6 m/min. Their results have shown that at this velocity, the deformation is not confined to an infinitesimally thin zone. The maximum plastic shear strain reached high values ranging from 0.85 to 4.5 whereas the strain rates were relatively weak ranging from 150 s−1 to 300 s−1 . Shankar et al. (2006) also used the PIV method to study the severe plastic deformation during the plain strain machining of a pure titanium piece at 0.6 m/min. From the observations, the authors could distinguish two distinct zones from the primary shear deformation: a fan-shaped deformation zone that extends ahead of the cutting tool with relatively small values of strain rate and accumulated plastic strain, and a zone of localized severe deformation with larger strain rates and strains. Significant change of material microstructure by refinement also accompanies the plastic deformation. The use of the PIV method for machining process demonstrated the ability to analyze the variation

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of the strain rate and cumulative plastic deformation along a flow line path. However, this technique requires a sequence of images to follow the movement of particles. Typically, with a system composed of one single intensified digital camera, only the slow velocity domain can be explored because of the limitation of the frame transfer time in two successive frames. In order to capture sequences at ultra high speed for higher cutting speed, multiple cameras sharing the same viewing axis combined into multichannel systems is necessary. Hijazi and Madhavan (2008) analyzed strain rate and velocity fields at 200 m/min using the DIC method thanks to a complex special device composed of four nonintensified digital cameras. The determination of the strain field by these methods also concerns low or moderate cutting speeds. In metal machining, the analysis of grid or particle movement can also be replaced by the analysis of flow lines. Goriani and Kobayashi (1967) proposed a method based on experimentally determined streamlines to calculate strain and strain rate in the deformation zone. A mechanical scratching and a camera are used to obtain photograph of streamlines during a moderate cutting speed of 7.5 m/min. Leopold (2003) analyzed the deformation zones by a visio-plasticity approach including experimental streamlines for cutting speeds up to 300 m/min. The chip formation is nevertheless not directly observed by a camera system but using a special quick-stop device. The present study describes a method of investigation in which experimentally determined streamlines are also employed for computing the strain rate at very high cutting speed exceeding 1000 m/min. The velocity, strain rates and strain distributions are calculated from the mathematical expression of the streamline functions. An ultra-rapid intensified CCD camera fixed on a specific device is employed to obtain in real time a picture of the chip in a plane strain configuration. In order to validate the approach, results are confronted to those obtained by the finite element method following a coupled thermo-viscoplastic model.

2. Experimental Mild steel AISI 1018 in parallelepiped form was subjected to machining in plane strain configuration. Cutting tests were carried out on a ballistic apparatus designed by Sutter et al. (1998) (see Fig. 1), reproducing the condition of orthogonal cutting at very high cutting speed. The specimen is fixed on a projectile, which is propelled inside the launch tube. Sutter (2005) used a tube sufficiently long to ensure a constant speed for the projectile. Two tools symmetrically mounted at the entry of the receiving tube perform the cutting process. All cutting tests were carried out with uncoated carbide inserts (H13A-P15, 6% Co and 94% WC) without chip-breaker, with a rake angle ˛ = 0◦ and a clearance angle of 7◦ . The geometrical dimensions of the workpiece define the feed t1 , the width of cut w (10 mm) and the cutting length L of 12 mm. The high rigidity of the device limits the apparition of vibration and no variable thickness chip is formed. The good accessibility of the chip cross-section allows the arrangement of an intensified CCD camera system to observe the cutting phenomenon in real time. However, due to high cutting speed, the exposure times are very much reduced and considerable attention and practice are required to obtain high quality pictures. In order to obtain high contrast and to get more details, the optical magnification (10:1) is obtained by means of an optical of microscope directly attached on the camera. The maximum resolution of 1024 × 1024 pixels was used for a size of the specific area to be observed of around 3.5 mm2 . Such a magnification imposes to reduce the distance between the lens and the object at a value determined by the focal of the lens. This distance of about 30 mm must be taken into account in the lighting of the process. In other respects, high cutting speed reduces the duration

of the process, in our case 0.7 ms. The phenomenon duration time to be filmed requires a very short exposure time (of the order of a few microseconds) so a significant “luminous intensity” is needed, which is assured by two flashes of high power. The two light sources are balanced to overcome shadows and a perfect synchronization of the trigger mechanism with the photographic recording system is required. To guarantee an optimum intensity of light, the times of response of the different electronic components are taken into account by a post-synchronization of the trigger. In order to design the flow line pattern, the mechanical scratching method was used to draw the lines parallel to the cutting direction on the side of the workpiece before machining. To ensure a sufficient spatial resolution around the primary shear zone, a high feed t1 of 0.84 mm was selected. Fig. 2(a) shows the deformed flow lines during the cutting process at 1020 m/min. The sharpness of the picture is sufficient to analyze the geometry of the streamlines with accuracy. 3. Streamlines model 3.1. Cinematically admissible velocity field Streamlines analysis is the selected method to determine the plastic strain and strain rate in the primary shear zone. Taking into account the complex variation of the streamline shape, a general stream function is considered. A similar method was used by Hasani et al. (2008) to describe the severe deformation of the equal channel extrusion process and was applied by Bi et al. (2009) in order to validate the method in the domain of conventional machining by confrontation with experimental results from Stevenson and Oxley (1969). In the present paper, the flow line is mathematically described by the flow function ϕ: ϕ(x, y) = x + y

(1)

The parameters  and  control the shape of the streamlines. Each flow line is described by a mathematical implicit equation which depends to the planar coordinates x and y: 

x + y = x0

(2)

where x0 defines the incoming position of the flow line. Before the deformation, at the position (x0 , 0), the incoming cutting velocity Vc gives the horizontal velocity component: y (x0 , 0) = Vc

(3)

To ensure plastic flow incompressibility the velocity gradients are defined from the flow function by: x (x, y) = k

∂ϕ ∂y

y (x, y) = −k

(4)

∂ϕ ∂x

(5)

The parameter k is deduced from Eq. (3) given the expression: k=

−Vc

(6)

−1

x0

From the differentiation of Eqs. (4) and (5):

vx (x, y) = − vy (x, y) =

Vc −1

x0

Vc −1

x0

y−1

x−1

(7)

(8)

Finally, by including Eq. (2), we can re-express the velocity components:

vx (x, y) = −Vc y−1 (x + y )(1−)/

(9)

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Fig. 1. Ballistic set-up used to produce orthogonal machining at high cutting speed.

vy (x, y) = Vc x−1 (x + y )(1−)/

(10)

3.2. Strain rates and strain fields The plane strain condition being assumed, the strain rate tensor components are expressed as following: ε˙ xx =

∂vx ∂x

(11)

ε˙ yy =

∂vy ∂y

(12)

ε˙ xy = ε˙ yx

1 = 2



∂ vy ∂vx + ∂y ∂x

 (13)

The equivalent strain rate in the sense of Von Mises is given by:



ε˙ =

2 ε˙ ε˙ 3 ij ij

(14)

The cumulative plastic strain is obtained by the time integration of the equivalent strain rate as given below:

 ε¯ =

t

˙ ε(u)du 0

(15)

3.3. Determination of the flow lines parameters According to the flow lines observed in Fig. 2, the parameters of the streamline model can be fitted by regression analysis. First of all, an appropriate Cartesian coordinate system (O, x , y ) should be defined. The axis x is taken perpendicular to the direction of the cutting velocity and to the streamlines in the non-machined part of the workpiece as shown in Fig. 2(b). Some locations on flow lines were extracted from the experimental picture. Based on these discrete points represented by asterisk symbols in Fig. 3, a nonlinear fitting program was implemented to determine the parameters  and  in the flow function. The results of fitting are shown in Fig. 3, where continuous lines are the fitting results. The shape of flow lines located in the primary shear zone varies with the depth h as defined in Fig. 2(b). Accordingly the parameters  and  vary with the different flow lines. 3.4. Results The angular variable  (◦ ) is introduced, as defined in Fig. 2(b), to compute the position along a streamline. The distribution of the equivalent strain rate ε¯˙ over four flow lines is shown in Fig. 4(a). The maximum, value of the strain rate is obtained for the streamline located closer to the tool tip (line 1, h = 0.12 mm). The value exceeds 105 s−1 but it is probably higher at the neighborhood of the cutting

Fig. 2. Orthogonal cutting configuration. (a) Definition of the chip cross-section observed during the test with the flow lines design. (b) Experimental observation at Vc = 1020 m/min.

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y (mm) 1

0.8

0.6

0.4

0.2

0

0

line 4

0.4

line 3

0.6

line 2

0.8

line 1

1

x (mm)

0.2

line 1 2 3 4

h (mm) 0.12 0.31 0.53 0.63

x0 (mm) 1.028 0.85 0.644 0.522

1/

1 1.3 2.5 1.8

40 16 11 5.5

Fig. 3. Determination of the flow lines parameters.

edge, although it cannot be demonstrated here. The maximum is also located at the angular position  = 45◦ . It is interesting to note that this value exactly corresponds to that is classically used in Merchant’s model (Merchant, 1945) to estimate the shear angle by Eq. (16):

 = tan−1

(t1 /t2 ) cos ˛ 1 − (t1 /t2 ) sin ˛

 (16)

where ˛ is the rake angle (here ˛ = 0). In the present results, the chip thickness t2 is equal to the uncut chip thickness t1 . The line AB in Fig. 2(b) could be approximately associated with the definition of the shear plane in Merchant’s model. However, the observation of the location, where the change of velocity is the most intensive, is slightly deviated with respect to AB. The deviation is increased for the streamlines far away from the tool tip. The evolution of the cumulative equivalent strain along the four streamlines is shown in Fig. 4(b). A non-uniform strain distribution is observed in the primary shear. The strain increases rapidly over the primary shear zone and stabilizes gradually after. This reveals that the primary shear zone has approximately a triangular shape opening from tool edge up to free surface. The bold lines in Fig. 3 define the onset and the saturation of the plastic deformation. The maximum plastic strain is similar to the classical

equation found in the shear plane theory (¯εmax = 1.154) calculated by Eq. (17): 1 cos(˛) ε¯ max = √ 3 sin( ) cos( − ˛)

(17)

Previous studies have proposed some analytical streamline models in metal cutting. Spaans (1971) employed a bounded parabola equation for the stream function, the simplicity of the model led however to a constant strain rate within a parallel-sided shear zone, which is not consistent with experiments. Tay et al. (1974) proposed to approximate the streamlines by hyperbolae, which have asymptotes for deformed and undeformed chip velocity vectors. The model depicts a more complex distribution of the strain rate field, but the maximum shear strain along the streamlines is assumed to be maximal in the direction of the shear plane AB described by the predefined shear angle . The present work provides an appropriate way to describe the complexity of the strain and strain rate fields without assumption about the shear angle or the shape of the primary shear zone. Since the velocity components vx and vy of chip material along flow lines are calculated using Eqs. (9) and (10), the resultant velocity V can also be obtained by:



V (x, y) = vx (x, y)2 + vy (x, y)2

1/2

(18)

The evolution of V along the flow lines is presented in Fig. 5. The velocity decreases to a minimum value after entering into the

Fig. 4. Distributions of strain and strain rate over the flow lines calculated from the streamline model. (a) Equivalent plastic strain rate. (b) Cumulative plastic strain.

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697

Table 1 Parameters for the constitutive laws. A (MPa)

B (MPa)

n

C

m

d1

d2

d3

520

269

0.282

0.0476

0.53

0.24

1.1

1.5

(2008). The failure of an element is effective when the critical the damage parameter ω exceeds the value of 1: ω=

¯εp

(20)

ε¯ f

¯εp is the increment of plastic deformation during a cycle of integration and the failure strain under the condition of triaxiality:

 m

ε¯ f = d1 + d2 exp d3

f = min(p, crit ) primary shear zone, and then increases to a stable value after flowing out the primary shear zone. It is interesting to observe that the velocity in the bulk chip is not homogeneous whereas it is generally assumed to be constant in most of chip formation models. Due to the steeper transition of occurring with a maximum shear angle at 45◦ , the velocity in the chip for the line 4 (h = 0.12 mm) is approximately equal to the cutting velocity Vc (17 m/s). However, the velocity in the bulk chip decreases from line 1 to line 4. The chip material close to the rake face moves faster than outermost chip material. This velocity variation of chip material probably can be regarded as the mean reason for chip curling. 4. Finite element approach 4.1. Chip formation modeling The metal cutting is modeled by the finite element method thanks to the code ABAQUS/Explicit in version 6.10. Details of the modeling approach can be found in the work of List et al. (2012) who explained how to select the contact and friction parameters regarding the cutting conditions and the mean interfacial temperature. The 2D model (see Fig. 6(a)), with four nodes elements, follows a Lagrangian scheme to study the evolution of chip formation from its initial phase until the steady state. The tool is considered as an elastic rigid body while the workpiece material follows a thermo-viscoplastic behavior described by the Johnson–Cook’s law (Johnson and Cook, 1983), with parameters calibrated by Sasso et al. (2008) in order to represent the response of mild steel AISI 1018:









1 + C ln

ε¯˙ ε˙ 0

 

1−

 T − T m  0 Tm − T0

(21)

where m is the hydrostatic pressure and d1 , d2 , d3 are the damage material constants reported in Table 1. To calculate the friction stress f the following relationship is used:

Fig. 5. Distributions of the resultant velocity V along the streamlines.

¯ = A + Bε¯ n

¯

(19)

Here ¯ is the effective flow stress of the material, ε¯ is the cumulative plastic strain, ε˙ the effective plastic strain rate, ε˙ 0 the reference plastic strain-rate of 1 s−1 , T0 the room temperature and Tm (1520 ◦ C) the melting temperature. A, B, n are the material constants for the strain hardening, C is the constant for the strain rate dependence and m is the thermal softening exponent. The values of the parameters are given in Table 1. A damage criterion was also applied to the workpiece elements allowing the material separation following the Johnson and Cook’s damage law (1985) with the parameters determined by Goto et al.

(22)

where  is the Coulomb’s friction factor, p is the contact pressure at the tool–chip interface and crit is the critical shear stress that the fiction stress cannot exceed. The value of crit is generally regarded as the shear flow stress in the material which can depend of the mean temperature at the interface. In the present simulation, according to the previous work of List et al. (2012), crit is equal to 62 MPa and the value of  was selected to 0.12 based on experimental results from friction tests at high sliding velocities. Before comparing with the experimental results, the influence of the element size must be considered. Preliminary calculations were carried out with different mesh sizes: mesh (A), (B), (C), (D) and (E) at respectively 0.04 mm, 0.02 mm, 0.015 mm, 0.012 mm and 0.01 mm. As shown in Fig. 6(b), for the streamline with label h = 0.12 mm, the smaller is the element size, the higher is the strain rate. Moreover, the location of the maximal strain rate along a streamline changes with the mesh size. Mesh convergence is reached for element size around 0.01 mm when the maximum is located at the angular position around 45◦ . 4.2. Results Fig. 7 illustrates the distribution of equivalent strain rate and strain along the four streamlines obtained by FEM with element size selected at 0.01 mm. The simulation results present the same trends as the experimental data displayed in Fig. 4. For the streammax line labeled h = 0.12 mm, the maximum strain rate ε˙ predicted by simulation is very close to those measured in experiment max decreases when approaching (∼1.5 × 105 s−1 ). It appears that ε˙ the free surface. max However, the rate of decreasing of ε˙ is faster in experiment max than in simulation: ε˙ is divided by a factor 2 in the simulation when comparing streamlines h = 0.12 mm and h = 0.63 mm and by a factor 3.5 in experiment (see Fig. 8). This is an indication that the width of the primary shear zone is more uniform in simulation than in experiment. Moreover, a difference lies in the fact that the maximum of strain rates is always located at the same angle  for the simulation while there is a slight deviation in the experiment. A variance with the cumulative plastic strain ε¯ can be also noted. In particular, for the flow line h = 0.12 mm, a higher value is observed. This trend could be explained by the contribution of the secondary shear zone at the proximity of the tool–chip interface. This evolution is less pronounced in the experimental observation.

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Fig. 6. FEM simulation of orthogonal cutting, Vc = 1020 m/min, t1 = 0.84 mm. (a) Visualization of the flow lines with the deformed mesh for element size of 0.04 mm. (b) Influence of element size on the strain rate along the streamline with label h = 0.12 mm.

Fig. 7. Distributions of strain and strain rate over the flow lines calculated from the finite element model. (a) Equivalent plastic strain rate. (b) Cumulative plastic strain.

Fig. 8. Comparison between distributions of strain and strain rate over the two extreme flow lines (h = 0.12 and 0.63 mm) calculated by using the finite element model and the streamline model.

5. Conclusion An original approach based on the high speed photography coupled with a general streamline model was used to study the

material flow in metal cutting at very high cutting speed. Orthogonal cutting test for mild steel at cutting speed of 1020 m/min was used to study the velocity field in the primary shear zone. Compared to methods available in conventional machining, the present

G. List et al. / Journal of Materials Processing Technology 213 (2013) 693–699

approach has the advantage of not using grid pattern since only the streamline shape is exploited. A single snapshot of the cross-section of the chip is necessary to study the material flow. The parameters controlling flow line shape were fitted based on a stream function and experimental data. According to the velocity field found along the streamline, the equivalent strain rate and cumulative plastic strain were obtained. The complexity of their distribution can be studied in detail. In particular, the results illustrate that the primary shear zone has a triangle in shape with the maximum shear strain rate increasing significantly toward the tool cutting edge. The results were also compared with a simulation of chip formation based on the finite element method. Experimental observations indicate that the distribution of the strain rate slightly differs from the finite element models but the global trends are preserved. It was also demonstrated that the size of the finite elements is crucial to obtain results similar to the experiments. The results globally validate the streamline model presented in this paper and show that this approach is effective for investigating the material flow in metal cutting at very high cutting velocities. The observations can be used to validate or disqualified the results obtained by finite element models or analytical model of chip formation. References Bi, X.F., List, G., Liu, Y.X., 2009. Calculation of material flow in orthogonal cutting by using streamline model. Key Engineering Materials 407, 490–493. Childs, T.H.C., 1971. A new visio-plasticity technique and a study of curly chip formation. International Journal of Mechanical Sciences 13, 373–374. Goriani, V.L., Kobayashi, S., 1967. Strain and strain-rate distributions in orthogonal metal cutting. CIRP Annals, 425–431. Goto, D., Becker, R., Orzechowski, T., Springer, H., Sunwoo, A., Syn, C.K., 2008. Investigation of the fracture and fragmentation of explosively driven rings and cylinders. International Journal of Impact Engineering 35, 1547–1556. Hasani, A., Lapovok, R., Tóth, L.S., Molinari, A., 2008. Deformation field variations in equal channel angular extrusion due to back pressure. Scripta Materialia 58, 771–774.

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