A novel surface roughness model for potassium dihydrogen phosphate (KDP) crystal in oblique diamond turning

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A novel surface roughness model for potassium dihydrogen phosphate (KDP) crystal in oblique diamond turning Shuo Zhang , Wenjun Zong PII: DOI: Reference:

S0020-7403(19)33577-5 https://doi.org/10.1016/j.ijmecsci.2020.105462 MS 105462

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

21 September 2019 22 December 2019 19 January 2020

Please cite this article as: Shuo Zhang , Wenjun Zong , A novel surface roughness model for potassium dihydrogen phosphate (KDP) crystal in oblique diamond turning, International Journal of Mechanical Sciences (2020), doi: https://doi.org/10.1016/j.ijmecsci.2020.105462

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Highlights 

A surface roughness model is developed for oblique cutting of KDP crystal.



Hydrostatic pressure is considered in modeling relative length of crack (RLC).



Roughness component of cracks effect Rc is modeled based on RLC.



Rc is a non-negligible factor in predicting surface roughness.

1 / 41

A novel surface roughness model for potassium dihydrogen phosphate (KDP) crystal in oblique diamond turning Shuo Zhang, Wenjun Zong* Center for Precision Engineering, Harbin Institute of Technology, Harbin, 150001, China

Abstract. KDP crystal is a typical soft-brittle material, and the challenge of predicting its surface roughness at a wide range of feed rates in oblique cutting invokes the comprehensive consideration of various factors, e.g. the plastic behavior in ductile mode and the cracks effect in brittle mode. In this work, a novel theoretical roughness model, into which the components of kinematics, plastic side flow, materials elastic recovery and cracks effect are integrated, is proposed to reveal the underlying mechanisms of surface roughness variation in oblique diamond turning of KDP crystal. In modeling, the duplication effect of the active cutting edge contour is successfully used to calculate the component of kinematics, and an empirical expression with consideration of equivalent cutting edge radius in oblique cutting is reconstructed to estimate the materials elastic recovery. For the component of plastic side flow, the effect of the volume of unremoved materials and the scale coefficient of plastic side flow determined by feed rate and tool inclination angle are taken into account. Moreover, the component of cracks effect is quantified based on the relative length of crack (RLC) model that considers the material properties, tool geometries, process parameters and the suppression of hydrostatic pressure in the cutting area. In experiments, a method of oblique fly-cutting is employed to reduce the impact of material anisotropy on surface roughness. The results show that the RLC model can well predict the distribution of cracks, and the surface roughness model considering these four components has a satisfactory prediction accuracy. Keywords: Oblique diamond turning; KDP crystal; Surface roughness; Cracks effect; RLC model.

1 Introduction Owing to the excellent nonlinear optical characteristics, KDP crystal is widely employed as the frequency converters and optical switches in the inertial confinement fusion program [1]. However, due to the inherent material properties, e.g. soft texture, extreme brittleness, and thermal sensitivity [2], it is prone to generate micro defects such as cracks and pits during the manufacturing process [3]. * Corresponding author at: Center for Precision Engineering, Harbin Institute of Technology, Harbin, 150001, China. E-mail: [email protected] (W.J. Zong) 2 / 41

To obtain a smooth optical surface, Fuchs et al. [4] firstly employed the single point diamond turning (SPDT) technique to directly finish the KDP optical components without an additional surface polishing. Subsequently, Namba et al. [5] carried out various SPDT experiments for higher productivity. A smooth surface with the roughness of 0.844 nm Ra was obtained by using a diamond tool with a rake angle of -25° and a nose radius of 1 mm at specific process parameters. In fact, the machining of brittle materials heavily relies on the study of plastic deformation of brittle materials. All materials exhibit the plasticity behavior no matter how brittle they are [6]. King et al. [7] firstly reported the plastic deformation in the rock-salt scratching test. They found that the brittle fracture is prevented when the material in the contact region is under high hydrostatic pressure. Subsequently, the indentation test was widely employed for evaluating the ductility of brittle materials. When the load is small enough that is below a certain critical force, brittle materials will exhibit plastic deformation characteristic in indentation tests [8]. Moreover, Clarke et al. [9] claimed that the ductile behavior of brittle materials below the indenter is related to the phase transformation under the impact of hydrostatic pressure. For the ductile-mode cutting of brittle materials, Liu et al. [10] systematically summarized the removal mechanism, characteristics, modeling and simulation method. Up to now, the discovery of the mechanism of brittle-ductile transition has become a hot topic. In practical machining, the ductile and brittle removal modes usually coexist, in which the cracks would generate in the brittle removal region. Thus, the produced cracks would have a great effect on the machined surface quality, especially for the initial crack generated at the brittle-ductile transition (BDT) depth. Taking account of material properties and cutting parameters, Nakasuji et al. [11] proposed a model for the BDT in diamond turning of brittle materials. Bifano et al. [12] proposed a model of the critical undeformed chip thickness (CUCT) which is defined as a function of the material properties. Furthermore, the 3 / 41

importance of hydrostatic pressure produced by negative tool rake angle was also concerned by previous works. For instance, Blackley et al. [13] developed a quantitative method to determine the machinability of a brittle material with respect to the rake angle in SPDT. They claimed that at large negative rake angles the hydrostatic stress is increased within the plastic deformation zone ahead of the tool cutting edge, which reduces the tendency for fractures to occur. Fang et al. [14] analyzed the effect of cutting edge radius on the removal of brittle materials, and found that the ductile cutting can also be achieved by using a diamond tool with 0° rake angle when the depth of cut is comparable to the cutting edge radius. Because the effective rake angle is actually negative which can also produce the necessary hydrostatic pressure to promote the plastic deformation. Moreover, Lee et al. [15] developed a method of solidified coating on the workpiece surface before machining to enhance the ductile-mode cutting of calcium fluoride (CaF2) single crystal. They found that the critical uncut chip thickness along different cutting directions of the (111) CaF2 is increased with the application of solidified coating and the subsurface damage is also reduced, which benefits from the compressive stress applied by the solidified coating in the cutting area. Moreover, Zong et al. [16] achieved a crack-free surface finished in a brittle-ductile coupled mode and found that the oblique cutting process is an effective approach to improve the surface quality of Cleartran ZnS crystal. Unfortunately, they only used the method of topography observation to illustrate the improvement of surface quality. The surface roughness model of oblique cutting had not been established to quantify such improvement. Therefore, two interesting questions are raised but not answered: whether the effect of tool inclination angle on the surface quality is related to the hydrostatic pressure in the cutting area? Furthermore, whether the surface roughness model in oblique cutting with consideration of the effect of cracks on the machined surface can be quantitatively established? 4 / 41

According to the previous works on the ductile mode machining of brittle materials as outlined above, there are sufficient reasons to give a definitive answer to the first question. The presence of inclination angles will cause the uncut materials to be subjected to the pressure in tool feeding direction, i.e. the feed cutting force produced in oblique cutting, which naturally leads to the change of the hydrostatic pressure in the cutting area. As for the establishment of surface roughness model in oblique cutting, because of the soft-brittle characteristic of KDP crystal, it is necessary to consider not only the effect of cracks on the machined surface but also the influencing factors of ductile behaviors in surface formation, being similar to that considered in the surface roughness modeling for ductile materials. With regard to the theoretical roughness modeling of ductile materials, Grzesik et al. [17] proposed a revised roughness model, in which an assumption has to be made regarding the determination of an accurate value of the minimum undeformed chip thickness related to the cutting edge radius. Zong et al. [18] established a comprehensive model that considers the influences of kinematics, minimum undeformed chip thickness, plastic side flow and materials elastic recovery to predict the surface roughness of pure copper in SPDT. He et al. [19] proposed a theoretical roughness model to reveal the underlying mechanisms for the size effect of surface roughness in diamond turning of aluminium alloy, in which the kinematics, plastic side flow, material spring back and random factors are considered. With respect to the prediction model of surface roughness for brittle materials, Jiang et al. [20] developed a brittle material removal fraction (BRF) model by analyzing the characteristics of a micro-topography image of the machined surface, which is a reasonable manner for estimating surface roughness during precision-grinding of optical glass. Fadin et al. [21] developed an approach to estimate the surface roughness of brittle materials based on the measurement of friction-induced acoustic emission signals. 5 / 41

Although these interesting works outlined above have made great progress in modeling the surface roughness for ductile materials or brittle materials, some problems remain unresolved for the soft-brittle materials in oblique cutting. Firstly, the influencing factors of ductile behaviors in surface formation has been widely investigated in formulating the surface roughness model for ductile materials. However, the effect of tool inclination angle on these influencing factors is rarely mentioned. Secondly, little attention has been paid to the theoretical modeling of the effect of tool inclination angle on surface quality by calculating the crack length under the hydrostatic pressure in the cutting area. Thirdly, in modelling the surface roughness for brittle materials, limited researchers directly explore the effect of cracks on surface roughness. They usually employed an intermediary means to predict the roughness, including the calculation of BRF model by analyzing the topography image of machined surface and the measurement of acoustic emission signals during processing. Therefore, in this work, a novel theoretical roughness model which is composed of four components, i.e. kinematics, plastic side flow, materials elastic recovery and cracks effect, is established to predict the surface roughness of KDP crystal in oblique diamond turning. In particular, the roughness component of cracks effect is modeled on the basis of the RLC model that estimates the relative length of crack with full consideration of material properties, tool geometries, process parameters and the suppression of hydrostatic pressure in the cutting area. 2 Theoretical modeling 2.1 Analysis of surface roughness for the soft-brittle materials For the diamond turned surface of aluminium alloy, the theoretical model of surface roughness accounting for the effect of multiple factors had been established by He et al. [19], which is given by Rth  Rtew  s  w

(1)

where Rth represents the theoretical peak-valley surface roughness. Rtew is the kinematic component 6 / 41

caused by the duplication effect of tool cutting edge. s and w represent the materials elastic recovery and plastic side flow, respectively. However, Eq. (1) is no longer applicable for the ductile-mode machining of soft-brittle materials in oblique turning. The roughness components of kinematics, elastic recovery and plastic side flow will separately change due to the presence of tool inclination angle. More importantly, the effect of tool inclination angle on cracks in the brittle-mode machining must be taken into account in modeling the surface roughness. For the soft-brittle materials like KDP crystal, owing to their inherent characteristics such as low hardness and high brittleness, the surface roughness in oblique diamond turning at a wide range of feed rates should be formulated in terms of both the plastic behavior in ductile mode and the cracks effect in brittle mode. Therefore, the roughness model in this work is divided into four components, i.e. kinematics, plastic side flow, materials elastic recovery and cracks effect. The formulated expression is written as Ra  Ride  k0  p  e   Rc

(2)

where Ra represents the average surface roughness. Ride is the roughness component of ideal surface profile caused by the duplication effect of the active cutting edge. p and e are the roughness components of plastic side flow and materials elastic recovery in oblique cutting, respectively. k0 is a constant conversion coefficient for the roughness components of plastic side flow and materials elastic recovery that converts the peak-valley roughness to the average roughness, which is empirically configured as 0.2566 [18]. Rc is the roughness component of cracks effect for the surface achieved in brittle mode at relatively large feed rate. 2.2 Roughness components of kinematics, plastic side flow and materials elastic recovery In oblique cutting, the duplication effect of the active cutting edge on surface forming is not only 7 / 41

related to the feed rate, but also to the tool inclination angle. As depicted in Fig. 1, one cycle of the ideal surface profile is taken out to analyze the roughness component of kinematics Ride in oblique cutting. The red curve represents the active cutting edge that participates in surface forming, which is oriented with an inclination angle of λs. The blue curve represents the resultant ideal surface profile formed on the plane perpendicular to the cutting velocity, which is used for the calculation of Ride. As marked in Fig. 1, two corresponding points g and g´ are located on the ideal surface profile and the cutting edge of the used tool with a nose radius of rε, respectively. Through general geometric derivation, the function of the resultant ideal surface profile can be expressed as  xi  f zi  r  r      cos  s  2cos  s  

2

2

(3)

According to the definition of average surface roughness, the roughness component of kinematics can be formulated as Ride  1 f   z ( x)  z dx f

(4)

0

where

z represents the mean line of the ideal surface profile. Z

f v

ap

X´

rε

zi O

g´ ． g ． x´

λs

i

xi

f/2

f

X

Fig. 1. Geometrical analysis for one-cycle ideal surface profile in oblique cutting.

As revealed in previous works, plastic side flow has a significant impact on the surface roughness in diamond turning. Kong et al. [22] pointed out that the materials at the trailing edge of the tool inevitably flow to the side to relieve the stress concentrated ahead of the cutting edge due to the material swelling effect. He et al. [19] declared that the formation of plastic side flow heavily depends 8 / 41

on the unremoved material volume, which is mainly affected by the minimum undeformed chip thickness hmin along the active cutting edge. In oblique cutting, however, the plastic side flow is complicated because it is influenced by more factors such as feed rate, tool inclination angle, minimum undeformed chip thickness hmin, etc. Therefore, in this work, the roughness component of plastic side flow is divided into two parts as follows p  Vb k p

(5)

where Vb is the volume of the unremoved materials per unit cutting length, which will be burnished and rubbed by the tool flank face and form the plastic side flow. kp is the scale coefficient of plastic side flow, which is determined by feed rate and inclination angle. As presented in Fig. 2 (a), a diamond tool with a negative rake angle is employed in oblique cutting. Purple part denotes the undeformed chip ahead of the active cutting edge. γ is the nominal rake angle of the tool. Vc represents the chip flow direction on the tool rake face with a chip flow angle εc. To express the geometric relationship in oblique cutting more clearly, three coordinate systems are established in this work. The first is the workpiece coordinate system C0: O-X0Y0Z0, which is consisted of the Z0 axis perpendicular to the machined surface, and the X0 and Y0 axes along the directions of feed rate f and cutting velocity v, respectively. The second is the oblique coordinate system C1: O-X1Y1Z1, which is derived from C0 that rotated anticlockwise around Z0 axis by the angle of λs. The third is the rake face coordinate system C2: O-X2Y2Z2, which is derived from C1 that rotated anticlockwise around X1 axis by the angle of γ. In the following text, all the modeling and corresponding analyses are performed in these uniform coordinate systems. Pobl is the tool base plane in the O-X1Z1 plane. Fig. 2 (b) illustrates the formation of the roughness components of plastic side flow and materials elastic recovery, which is taken out from the machined surface as marked in Fig. 2 (a). The 9 / 41

combination of orange and green areas represents the contact area of tool and workpiece in oblique cutting. The orange curve is the ideal surface profile along X1 axis without consideration of plastic side flow and materials elastic recovery. The green area with a thickness of hmin will be burnished and rubbed by the tool flank face, which eventually leads to the side flow to relieve the stress concentration. (a) Z0 (Z1)

f

Z2

γ

ap

v

Pobl Rake face

Diamond tool

X1 (X2)

ε c Vc

λs O

X0 Undeformed chip Y1

Y0

Y2

Workpiece

(b)

Z0

O2

J hmin

f v

ap

O1

fobl

rε ζ

Region II

e

X1

p

Region I B

f

λs

A X0

10 / 41

(c) f

v ap

Chip Diamond tool S

ap

Machined surface

robl

hmin e Workpiece Fig. 2. Oblique diamond cutting model: (a) schematic 3D model of oblique cutting; (b) formation of the roughness components of plastic side flow and materials elastic recovery taken from the machined surface as marked in (a); (c) illustration for the minimum undeformed chip thickness and materials elastic recovery in an orthogonal plane.

According to the tool tip of point B as marked in Fig. 2 (b), the green area is divided into two regions, i.e. the region I surrounded by yellow dotted lines and the region II on the right side of point B. In oblique cutting, it can be assumed that only the materials in region I flow to the side of the machined surface and accumulate at point A to form the plastic side flow. In contrast, the materials in region II flow to the side of the unmachined surface and accumulate at point J, which is not considered in this work because it will be removed by next cut. Consequently, the volume of the materials in region I per unit cutting length, i.e. Vb, can be expressed as Vb  r  hmin    f  1   sin  2r cos    s   

(6)

where ζ is the central angle of arc AB as shown in Fig. 2 (b). Moreover, hmin is heavily dependent on the cutting edge radius rn, which is formulated as hmin  crn

(7)

where c is a coefficient equal to 0.35 [19]. According to the direction of tool cutting velocity, the cutting edge radius plays a key role in the formation of hmin as illustrated in Fig. 2 (c). It can be seen that there is a critical point S on the equivalent cutting edge, which defines hmin. For the undeformed materials with a thickness of less than 11 / 41

hmin, they will be burnished and rubbed by the tool flank face rather than forming chips. Therefore, in oblique cutting, the equivalent cutting edge radius robl calculated in the orthogonal plane, i.e. robl=rn/cos(λs), can be used instead of rn in Eq. (7) to calculate the actual value of hmin in this work. In addition, about the scale coefficient of plastic side flow in oblique cutting, it is a composite factor influenced by feed rate and inclination angle, which is expressed as k p  kd kobl k f

(8)

where kd is a constant coefficient accounting for the material properties and configured as 0.001 nm-1 in this work. kobl and kf represent the effect of inclination angle and feed rate, respectively. In light of previous findings [18], the decrement of feed rate strengthens the stress concentration ahead of the active cutting edge, which leads to an enlargement of materials swelling. As for the effect of inclination angle on kp in this work, a large inclination angle promotes the efficiency of chip removal and relieves the materials swelling towards the machined surface because of the existence of chip flow angle in oblique cutting. Thus, a formula consisting of two exponential functions is constructed to consider the effects of feed rate and inclination angle as follows kobl  a0 exp  b0 s   c0 exp  d0 s     k f  a1 exp  b1 f   c1 exp  d1 f 

(9)

where a0, b0, c0 and d0 are the fitted coefficients for inclination angle, which are empirically recommended to be configured as 1.139, -0.4524, -0.005345 and 3.665, respectively. a1, b1, c1 and d1 are the fitted coefficients for feed rate, which are empirically suggested to be set to 1.645, -0.1161, 5.121 and -1.22, respectively. By using Eqs. (2), (4) and (10) and considering that the roughness component of cracks effect Rc is zero, the above coefficients are fitted by the experimental results of crack-free surface at small feed rates (f ≤ 6 μm/r) as presented later. As discussed above, the plastic side flow is partly responsible for the rise of roughness. As for the 12 / 41

materials elastic recovery, it is known that it reduces the roughness of turned surface significantly. Thus, as shown in Fig. 2 (b), the red curve represents the surface profile after the plastic side flow and materials elastic recovery in oblique cutting. The maximum recovery height e at the bottom of cutting edge, i.e. point B, is used to calculate the materials elastic recovery. Assumed that the materials elastic recovery is a function of H/E and depends linearly on the cutting edge radius [23], which is also replaced with the equivalent cutting edge radius as shown in Fig. 2 (c), the empirical model of materials elastic recovery in oblique cutting can be rewritten as e  k1

H robl E

(10)

where H and E are the hardness and Young’s modulus of KDP crystal, respectively. k1 is a constant coefficient related to tool geometries, which is configured as 8.1 in this work. k1H/E is calculated as an overall coefficient to satisfy the relationship between the materials elastic recovery and the equivalent cutting edge radius [19], i.e. e=0.32robl. 2.3 Roughness component of cracks effect In order to establish the roughness component of cracks effect, two issues should be addressed in this section. One is to quantify the cracks length produced in processing, the other is to quantify the relationship between the crack length and surface roughness. Therefore, a novel crack quantification method that considers the cutting force and the suppression of hydrostatic pressure is proposed by modeling the relative length of crack (RLC) at the BDT depth. Afterwards, the roughness component of cracks effect is successfully established based on the RLC model. 2.3.1 Equivalent rake angle As shown in Fig. 2 (a), the chip flow angle εc is defined on the tool rake face as the angle between Z2 axis and the direction of chip flow in oblique cutting. In light of the Stabler’s rule [24], the chip flow angle can be assumed to be equal to the inclination angle, i.e. εc=λs. 13 / 41

In oblique cutting, the equivalent rake angle appears because the chip no longer flows in the direction perpendicular to the cutting edge. Taking an infinitesimal element of the cutting edge as an example, the rounded cutting edge can be simplified into a straight edge as illustrated in Fig. 3. Ps and Pobl are the cutting plane and tool base plane, respectively. The dotted orange lines represent the normal plane Pn, which is vertical to Ps and Pobl. The nominal rake angle γ and shear angle  are both defined on Pn. Psh is the shear plane in green. The red plane represents the rake face of diamond tool, on which the chip flows. To calculate the equivalent rake angle, an equivalent plane Pe is established, which is determined by Vc and Y1 axis. On plane Pe, the angle between the rake face and Pobl is defined as the equivalent rake angle in oblique cutting, which is marked as γe in Fig. 3. Based on analytic geometry, the equivalent rake angle γe can be calculated as 

 Vc 1  Y1     = arccos   sin  cosc   V Y  2 2 1  c 1 

 e = arccos 

(11)

where Y1 is the unit vector of Y1 axis, i.e. Y1   0 1 0 T . Vc 1 is the unit vector of chip flow in C1, i.e. Vc 1   sinc  sin  cosc cos  cosc T . Z2 P e

Z1

Rake face

Pobl

Pn γ

O

γe εc

Vc

X1(X2)

Psh

 Ps Y1 Y2

Fig. 3. Schematic diagram of the oblique cutting model for an infinitesimal element of the cutting edge.

In this work, the chip formation analysis here and the cutting force calculation presented later are both performed on the basis of infinitesimal elements, which can be considered as countless orthogonal cuttings with different depths. Therefore, the shear angle  can be defined on the normal plane Pn, which is given by [25] 14 / 41

=

 4



 2





(12)

2

where β is the friction angle, which is a function of friction coefficient μ, i.e. β=arctan(μ). 2.3.2 Cutting force model Fig. 4 illustrates the undeformed chip, i.e. the purple part colored in Fig. 2 (a). It can be seen that the UCT gradually increases from zero at the tool tip to a maximum value at the top of uncut shoulder. Considering the BDT behavior of brittle materials during processing, there is a critical UCT equaling to dc, which separates the cutting area into two regions, i.e. the yellow region I in ductile mode and the blue region II in brittle mode. In terms of geometries, the UCT d(x) along the X1 axis can be estimated as  f obl  r x   d ( x)    xl ( xl  f obl  x)  r 

0  x  xl

(13-1) xl  x  xl  f obl

xl  r 2   r  a p   2

f obl 2

(13-2)

where ap is the depth of cut. fobl is the equivalent feed rate in oblique cutting, which is equal to f/cos(λs). f

Z1 (Z0)

v f

O2

ap

φ Fc-1

fobl

T

Fr-0

ap

II UCT

C dc

cracks

S X1 xl+fobl

O1

xl

Y1 Fc-0

Ft-1 X0 I

xc

zc

λs

Y0

O

Fig. 4. Illustration of the gradually increasing UCT and the cutting force model in oblique cutting.

In region I, the undeformed materials undergo plastic flow along the shear plane rather than fractures. The formation of continuous chips in ductile mode makes it convincing that the cutting 15 / 41

forces can be calculated by the classical force model in metal cutting. Therefore, the principal cutting force Fc and thrust cutting force Ft per unit cutting width corresponding to an infinitesimal increment dx along X1 axis can be expressed as [26] ks d ( x) w cos(    e )  dFc  sin  cos(     ) dx  e  k d ( x ) w sin(    e) dF  s dx t  sin  cos(     e )

where ks is the shear stress flowing on the shear plane, which is empirically equal to

(14)

H 3 3

[26]. w is

the unit cutting width. γe and  represent the equivalent tool rake angle and shear angle in oblique cutting, respectively, which are elaborated in last section. The principal cutting force in region I, i.e. Fc-1-I, can be treated as the sum of cutting force components at different depths of cut from zero to dc along the active cutting edge, which is formulated as Fc 1 I  

xc

0

ks cos(    e ) fx dx sin  cos(     e )r cos(s )

(15)

where xc denotes the abscissa of point C as marked in Fig. 4, which is equal to rεdccos(λs)/f. dc is the BDT depth related to material properties, which is configured as 0.198 μm [27]. As illustrated in Fig. 4, when the UCT exceeds dc, the materials in region II are removed in brittle mode. In light of previous findings [27], the fracture energy Ef required for new surfaces generated by crack propagation can be expressed as  7 E f  Ac s  2   lTS L s   

(16)

where Ac is the area of new surfaces generated by cracks. γs is the specific surface energy of KDP crystal, which is equal to 0.6 J/m2 [28]. lTS represents the length of the uncut shoulder arc between point T and point S as marked in Fig. 4. L is the unit cutting length. Marshall et al. [29] reported that in indentation test the plastic deformation still exists although the 16 / 41

crack system appears during indenter loading. In addition, Wang et al. [30] also observed the plastic deformation related continuous chips on a brittle-mode machined surface. They integrated the force for cracks generation into the principal cutting force model, which yields a satisfied prediction accuracy. Thus, the principal cutting force in region II can be divided into two components. One is the force to promote plastic deformation, the other is the force required to generate cracks. As a result, considering the variability of d(x) in region II, the principal cutting force can be expressed as Fc 1 II  

xl

xc

xl  f /cos( s ) k cos(    ) x [ x  f / cos( )  x] ks cos(    e ) fx A s e l l s dx   dx  c s xl sin  cos(     e )r cos(s ) sin  cos(     e ) r L

(17)

Finally, by summing Eqs. (15) and (17) together, the principal cutting force Fc-1 calculated in C1 is given by Fc 1  Fc 1I  Fc 1II

(18)

where the direction of Fc-1 is consistent with the positive direction of Y1 axis. As illustrated in Fig. 4, the principal cutting force Fc-1 in C1 can be decomposed into two separate forces in C0, i.e. Fc-0 along the positive direction of Y0 axis and Fr-0 along the positive direction of X0 axis. Thus, the oblique cutting forces in the workpiece coordinate system C0, including the principal cutting force Fc-0, feed cutting force Fr-0 and thrust cutting force Ft-0, can be written as  Fc 0  Fc 1 cos(s )   Fr  0  Fc 1 sin(s ) F  F t 1  t 0

(19)

In Eq. (19), the directions of Fr-0 and Ft-0 are both perpendicular to the cutting velocity, which produce the compression onto the undeformed materials and is responsible for the generation of hydrostatic pressure ahead of the active cutting edge. The direction of Fc-0 is along the cutting velocity, which plays the role of removing chips with cracks. 2.3.3 Modeling of RLC at the BDT depth The cutting force analysis discussed above indicates that the materials beneath and ahead of the 17 / 41

active cutting edge are subjected to a compressed state owing to the extrusion of thrust cutting force Ft-0 and feed cutting force Fr-0. As illustrated in Fig. 5 (a), it is a diagrammatic decomposition of the cutting edge in oblique cutting. The light blue gap beneath the cutting edge tip represents the initial crack at the BDT depth. The green curve denotes the nominal cutting edge. The red curve and blue curve represent the equivalent cutting edges in the directions of cutting velocity v and feed rate f, respectively. Fig. 5 (b) illustrates the material removal at dc with the red equivalent cutting edge in the O-Y0Z0 plane. The area surrounded by the yellow dotted curve represents the hydrostatic pressure zone. Fig. 5 (c) shows the stress distribution on the tool face corresponding to Fig. 5 (b). (a)

(b)

X0

Z0 (Z1)

f

Rake face

v

v Z0 Diamond tool

Y1

O

ap

Y0

robl

Cutting edge λs Flank face Initial crack

(c)

dc

G Initial crack

O

Ft-0-dc

y

(d)

γe

Hydrostatic pressure zone

Workpiece

Y0

G

v Z0

O

x  xy

Fr-0-dc X0

ar

Diamond tool

 yx

dc G Initial crack

G´

Hydrostatic pressure zone

Fig. 5. Illustration of stress analysis at the BDT depth: (a) 3D diagrammatic decomposition of the cutting edge at the BDT depth in oblique cutting; (b) 2D schematic of material removal process in the O-Y0Z0 plane; (c) stress distribution on the tool face corresponding to (b); (d) 2D schematic of material removal process and stress analysis in the O-X0Z0 plane.

Assuming a Hertzian contact condition [31], the contact stress PG´-ver acting at the lowest point of the active cutting edge, i.e. G´ marked in Fig. 5 (c), can be formulated as PG´ ver 

3Ft 0 dc 2 wrobl cos  e

(20)

where w and δ denote the unit cutting width and the ability of materials elastic recovery that 18 / 41

configured as π in this work [31], respectively. Ft-0-dc is the thrust cutting force calculated at dc in C0, which can be estimated by Eq. (14). In terms of the relationship between interaction forces, the contact stress PG-ver applied to the material beneath the cutting edge tip, i.e. point G marked in Fig. 5 (b), is equal to PG´-ver, and the direction of which is vertically downward. Fig. 5 (d) illustrates the material removal at dc with the blue equivalent cutting edge in the O-X0Z0 plane. Obviously, the feed cutting force is perpendicular to the cutting velocity, which has no direct contact stress loading at point G. However, the feed cutting force always squeezes the uncut materials. As a result, the indirect effect of feed cutting force on the stress PG-hor at point G can’t be ignored, which can be estimated in the form of PG-ver formulated above, i.e. PG  hor 

3k2 Fr 0 dc 2 wrobl cos  e

(21)

where the direction of PG-hor is horizontal along the X0 axis. k2 is a proportional coefficient that determines the degree of influence, which is configured as 0.42 in this work. Fr-0-dc is the feed cutting force calculated at dc in C0, which can be estimated by Eqs. (14) and (19). An infinitesimal element nearby point G is taken out and simplified into a plane stress distribution as presented in the close-up of Fig. 5 (d). The stress conditions can be expressed as  x   PG  hor   y   PG ver   xy   yx

(22)

According to the knowledge of material mechanics, the hydrostatic pressure ζh nearby point G can be formulated as h  

1 1  x   y    PG hor  PG ver   3 3

(23)

where the positive value of which represents the hydrostatic compressive stress. It reduces the volume of deformed materials with no shape change and suppresses the crack propagation. 19 / 41

In modeling, an assumption should be made in advance that the crack length can be calculated by the model of median crack in Vickers indentation, in which the indentation force is replaced by the principal cutting force. Considering the influence of hydrostatic pressure on the fracture toughness KIC [32], the relation between the median crack length and the principal cutting force Fc-0 in oblique cutting can be formulated as 12

12

Fc 0 C  E K IC  0    2  mh  32 H C      mh



Cmh

0

C

mh

h 2

 x2 

12

dx

(24)

where ε0 is a coefficient related to the dynamic effect, which is empirically set as 0.006±0.0005 [32]. Cm-h denotes the median crack length at dc under the inhibition of hydrostatic pressure. Moreover, in order to model the relative length of crack, the direction of crack propagation φ should be determined in triangle O1O2C as presented in Fig. 4. The cosine law is fulfilled as follows

 r  dc  cos   

2

 f obl 2  r 2

2 f obl  r  d c 

(25)

By simple geometric derivation, the RLC model can be given by Cre  zc   Cm  h  dc  sin   2   d c r f obl  2   zc  r  r    2   f obl 

(26)

where Cre represents the relative length of the median crack at dc. zc is the ordinate value of point C in coordinate system C1 as labeled in Fig. 4. In Eq. (26), the absolute value of Cre represents the distance between the crack propagation tip and the finally machined surface. When Cre is larger than zero, it indicates that the crack will not extend to the finally machined surface, which corresponds to the smooth surface without cracks. Otherwise, the crack will extend into the machined surface and destroy the surface quality. 2.3.4 Modeling of Rc based on RLC In this section, the relationship between Rc and Cre is further quantified. In RLC model, the absolute 20 / 41

value of negative Cre represents the distance between the crack propagation tip and the machined surface, or in other words the depth of cracks extending below the machined surface after processing. As is well known, the deeper the cracks extended, the worse the surface quality obtained. Therefore, a coupled function including the exponential equation and quadratic without constant term is formulated to describe the roughness component of cracks effect, which ensures that Rc is equal to zero when Cre is zero. Moreover, to remove the effect of cracks on roughness when Cre is positive, a piecewise function is established as 2   R  exp  M  Cre  N  Cre   1 Rc    0

Cre  0 Cre  0

(27)

where R, M and N are fitted coefficients in this work. By using of Eqs. (2), (4), (5) and (10), the experimental results presented later suggest that these three coefficients can be empirically set to 0.40467, -0.20266 and -3.30368, respectively. Through the analysis of variance, the order of sensitivities for these three coefficients is R, N and M. As an important part, the component of cracks effect Rc plays multiple roles in predicting the surface roughness of soft-brittle materials. Firstly, it predicts the surface condition, i.e. smooth or cracked. Secondly, for the cracked surface, it works together with the other three roughness components, i.e. kinematics, plastic side flow and materials elastic recovery, to predict the surface roughness. In contrast, it equals to zero to remove the effect of cracks on the surface roughness for a smooth surface. 3 Experiments and Methodologies In order to reduce the impact of material anisotropy on the machined surface quality, a special method named oblique fly-cutting technique was employed in this work. Fig. 6 (a) illustrates the experimental setup that performed on a commercial available ultra-precision machine (Nanotech 21 / 41

350FG, supplied by Moore Tools). Diamond tool is fixed on the indexing fixture which can provide high-precision rotation to obtain the tool inclination angle. A round jig with a radius of 75 mm is attracted to the vacuum chuck, on which there are two symmetrical parts fixed in the slots, i.e. the KDP workpiece and the balancing weight. The tool marks left on KDP surfaces are just 7.6 degree arcs with a radius around 75 mm, which can be seen approximately as cutting along the same crystal orientation. As presented in Fig. 6 (b), the dimensions of each workpiece are 15mm in width, 10mm in length and 12mm in thickness, which were cut from a type II KDP bulk by diamond wire cutting machine (STX-202A, supplied by KJ Group). (b)

(a) Spindle

KDP workpiece Wire cutting

Diamond tool

(c) Indexing fixture

1st cut 2nd cut 3 cut rd

4th cut

Length

Width

Height

Round jig

Balancing weight Fig. 6. Experimental setup and design employed in oblique fly-cutting of KDP crystal: (a) experimental setup; (b) workpieces preparation; (c) sketch map of step cutting method.

Before cutting experiment, the top surface of the KDP crystal substrate was flattened by fly-cutting to remove the preexisting cracks as more as possible. In order to investigate the surface roughness of KDP crystal in oblique cutting at a wide range of feed rates, a lot of experiments corresponding to different process parameters were performed. To increase the experiment efficiency, a method named step cutting was employed to assemble four surfaces with different process parameters onto one workpiece. The surface widths of four ordered cuts with different feed rates decreased in the order of 15 mm, 11.5 mm, 8 mm and 4.5 mm, which results in the step surface as shown in Fig. 6 (c). As listed in Table 1, each workpiece corresponds to a set of specified process parameters. The depth of cut is 22 / 41

uniformly configured as 3 μm in this work. The inclination angle increases from 0° to 60° with an increment of 15°. Each inclination angle corresponds to 11 sets of feed rates ranging from 1 μm to 20 μm. In Fig. 7, the tool nose radius and cutting edge radius of the employed diamond tool were measured by DTRC microscope system and AFM, respectively. Machining conditions configured in this work are summarized in Table 2, including tool geometries and process parameters. After experiments were completed, three different sampling points on each step surface were selected to measure for the average result of surface roughness by AFM. Table 1 Process parameters configured in oblique cutting experiments. ap [μm] 3

λs [°] 0

15

30

45

60

Workpiece No. A B C D E F G H I J K L M N O

Feed rate f [μm/r] Step 1 Step 2 1 2 6 8 14 16 1 2 6 8 14 16 1 2 6 8 14 16 1 2 6 8 14 16 1 2 6 8 14 16

Step 3 4 10 18 4 10 18 4 10 18 4 10 18 4 10 18

Step 4 12 20 12 20 12 20 12 20 12 20

(b)

(a)

rε

Z [nm]

rn

Y [nm] X [nm]

Fig. 7. Evaluation for the employed diamond tool: (a) tool nose radius rε; (b) cutting edge radius rn.

23 / 41

Table 2 Machining conditions configured in oblique cutting experiment. Machining condition

Parameter configuration

Workpiece material Tool material Tool nose radius rε Tool cutting edge radius rn Nominal rake angle γ Relief angle Depth of cut ap Spindle speed n Feed rate f Tool inclination angle λs Cutting fluid

Type II KDP crystal Single crystal diamond 0.989 mm 40.0 nm (35.0-42.8) –25° 33° 3 μm 1200 rpm 1-20 μm/r 0°, 15°, 30°, 45°, 60° None

4 Experimental results and discussion 4.1 Critical feed rate for the judgment of surface condition The judgment of surface condition, i.e. smooth surface without cracks or cracked surface generated by brittle removal, is an important step in modeling the surface roughness for soft-brittle materials, which is quantified by the RLC model as illustrated in Fig. 8. Obviously, the trends are roughly similar at different inclination angles, i.e. decreasing sharply at small feed rates and slowly at large feed rates. Considering that the depth of cut is uniformly configured as 3 μm in this work, the Cre larger than that in Fig. 8 has no practical meaning. Because the cracks don’t exist in this case due to the ductile-mode removal in the whole cutting area. However, this is an effective mathematical method to describe the state of cracks in oblique cutting of brittle materials.

Finally machined surface

RLC Cre [μm]

0° 15° 30° 45° 60°

Feed rate f [μm/r] Fig. 8. Relative length of median crack Cre at different inclination angles.

The critical feed rate in this work is defined as the feed rate when cracks appearing on the machined 24 / 41

surface. As presented in the close-up of Fig. 8, the purple dotted line represents the finally machined surface with the constant value of zero. The critical feed rates calculated by the RLC model at different inclination angles are listed in Table 3. For each inclination angle, if the configured feed rate is larger than the critical threshold, the cracks would extend to the finally machined surface and propagate into the substrate. Table 3 Critical feed rates calculated by the RLC model at different inclination angles. Inclination angle [°] 0 15 30 45 60

Critical feed rate [μm/r] 6 12.8 14.2 12.2 8.1

Fig. 9 presents the surface topographies around the critical feed rate in oblique cutting. When the configured feed rate is less than the critical feed rate at each inclination angle, the value of Cre calculated from the RLC model is positive. Consequently, the machined surface is smooth without cracks. If only the configured feed rate is more than the critical threshold, visible cracks appear on the machined surface at each inclination angle. The experimental observations prove that the RLC model can effectively predict the condition of machined surface in oblique cutting.

25 / 41

Inclination angle λs

Achieved in ductile mode

Achieved in brittle mode

f=6 μm/r Cre=0 μm

f=8 μm/r Cre=-0.42 μm

f=12 μm/r Cre=0.14 μm

f=14 μm/r Cre=-0.21 μm

f=14 μm/r Cre=0.03 μm

f=16 μm/r Cre=-0.28 μm

f=12 μm/r Cre=0.04 μm

f=14 μm/r Cre=-0.29 μm

f=8 μm/r Cre=0.03 μm

f=10 μm/r Cre=-0.36 μm

0°

15°

30°

45°

60°

Fig. 9. Surface topographies around the critical feed rate for specified inclination angles.

4.2 Verification of the proposed surface roughness model According to the process parameters listed in Table 1, the calculated results of roughness components are summarized in Table 4 (Table 4-1 and Table 4-2). It is apparent that the roughness 26 / 41

component of kinematics Ride gradually increases with the increment of feed rate for each inclination angle. Moreover, the Ride at different inclination angles is plotted in Fig. 10. In addition to the impact of feed rate, an interesting phenomenon is found that the influence of inclination angle becomes more significant as it increases greater than 45°, which will dramatically increase the total surface roughness. As for the Vb listed in Table 4, i.e. the volume of unremoved materials, it increases with the increments of feed rate and inclination angle, because the active cutting edge participating in cutting becomes longer. However, kp, i.e. the scale coefficient of plastic side flow, decreases with the increment of feed rate and inclination angle. Owing to the different trends of Vb and kp, the plastic side flow dependent roughness k0p as listed in Table 4 varies complexly in oblique cutting. As the feed rate increases, k0p increases first and then decreases. As for the influence of tool inclination angle, k0p has a minimum value when the inclination angle is 15°. Moreover, the materials elastic recovery related roughness k0e only varies with the inclination angle due to the change of equivalent cutting edge radius in oblique cutting. Moreover, the roughness component of cracks effect Rc exhibits obvious segmentation characteristic at the critical feed rate according to the Cre. In order to reveal the effect of cracks on surface roughness, the predicted roughness Rd without the roughness component Rc can be simplified as Rd  Ra  Rc  Ride  k0  p  e 

27 / 41

(28)

Table 4-1 Calculated results of different roughness components, λs=0°, 15°, 30°. λs [°] 0

15

30

f [μm/r] 1 2 4 6 8 10 12 14 16 18 20 1 2 4 6 8 10 12 14 16 18 20 1 2 4 6 8 10 12 14 16 18 20

Ride [nm] 0.0328 0.1312 0.5245 1.1799 2.0974 3.2770 4.7187 6.4226 8.3885 10.6166 13.1069 0.0352 0.1406 0.5621 1.2646 2.2480 3.5123 5.0575 6.8837 8.9908 11.3789 14.0479 0.0438 0.1749 0.6993 1.5732 2.7965 4.3694 6.2917 8.5635 11.1848 14.1556 17.4759

Vb [103 nm2] 7.0 14.0 28.0 42.0001 56.0002 70.0003 74.0005 98.0008 112.0012 126.0018 140.0024 7.5026 15.0052 30.0103 45.0155 60.0208 75.0261 90.0315 105.0370 120.0427 135.0484 150.0543 9.3333 18.6667 37.3334 56.0001 74.6669 93.3339 112.0009 130.6682 149.3355 168.0032 186.6710

kp [10-3 nm-1] 3.3730 1.9836 1.2157 0.9327 0.7367 0.5839 0.4629 0.3670 0.2909 0.2307 0.1829 2.9688 1.7459 1.0700 0.8209 0.6485 0.5139 0.4074 0.3230 0.2561 0.2030 0.1610 2.5657 1.5089 0.9247 0.7095 0.5604 0.4441 0.3521 0.2791 0.2213 0.1755 0.1391

k0p [nm] 6.0585 7.1258 8.7344 10.0519 10.5868 10.4875 9.9771 9.2283 8.3616 7.4579 6.5697 5.7155 6.7224 8.2399 9.4828 9.9874 9.8937 9.4122 8.7058 7.8882 7.0356 6.1977 6.1448 7.2273 8.8587 10.1950 10.7375 10.6368 10.1192 9.3597 8.4806 7.5640 6.6632

k0e [nm] 3.3707 3.3707 3.3707 3.3707 3.3707 3.3707 3.3707 3.3707 3.3707 3.3707 3.3707 3.4896 3.4896 3.4896 3.4896 3.4896 3.4896 3.4896 3.4896 3.4896 3.4896 3.4896 3.8921 3.8921 3.8921 3.8921 3.8921 3.8921 3.8921 3.8921 3.8921 3.8921 3.8921

Cre [μm] 0 0 0 0 -0.4227 -0.6698 -0.8375 -0.9629 -1.0627 -1.1458 -1.2171 0 0 0 0 0 0 0 -0.2136 -0.4691 -0.6624 -0.8154 0 0 0 0 0 0 0 0 -0.2950 -0.5184 -0.6931

Rc [nm] 0 0 0 0 1.1724 2.9730 5.1798 7.6683 10.3707 13.2571 16.3054 0 0 0 0 0 0 0 0.4073 1.4183 2.8981 4.8252 0 0 0 0 0 0 0 0 0.6490 1.7197 3.2199

Table 4-2 Calculated results of different roughness components, λs=45°, 60°. λs [°] 45

60

f [μm/r] 1 2 4 6 8 10 12 14 16 18 20 1 2 4 6 8 10 12 14 16 18 20

Ride [nm] 0.0657 0.2624 1.0490 2.3598 4.1948 6.5541 9.4375 12.8453 16.7773 21.2336 26.2143 0.1313 0.5248 2.0979 4.7196 8.3896 13.1082 18.8754 25.6911 33.5555 42.4687 52.4307

Vb [103 nm2] 14.0 28.0 56.0001 84.0003 112.0006 140.0012 168.0021 196.0033 224.0050 252.0071 280.0097 28.0 56.0 112.0003 168.0011 224.0025 280.0049 336.0084 392.0134 448.0199 504.0284 560.0390

kp [10-3 nm-1] 2.0925 1.2306 0.7542 0.5786 0.4571 0.3622 0.2872 0.2277 0.1805 0.1431 0.1135 1.3715 0.8066 0.4943 0.3793 0.2996 0.2374 0.1882 0.1492 0.1183 0.0938 0.0744

28 / 41

k0p [nm] 7.5171 8.8413 10.8372 12.4718 13.1355 13.0123 12.3791 11.4501 10.3747 9.2534 8.1515 9.8541 11.5901 14.2064 16.3493 17.2194 17.0579 16.2280 15.0101 13.6004 12.1306 10.6861

k0e [nm] 4.7669 4.7669 4.7669 4.7669 4.7669 4.7669 4.7669 4.7669 4.7669 4.7669 4.7669 6.7414 6.7414 6.7414 6.7414 6.7414 6.7414 6.7414 6.7414 6.7414 6.7414 6.7414

Cre [μm] 0 0 0 0 0 0 0 -0.3159 -0.5471 -0.7233 -0.8634 0 0 0 0 0 -0.3574 -0.6077 -0.7846 -0.9189 -1.0260 -1.1148

Rc [nm] 0 0 0 0 0 0 0 0.7214 1.9166 3.5656 5.6247 0 0 0 0 0 0.8796 2.3911 4.3666 6.6942 9.2898 12.1036

Ride [nm]

λs=0° λs=15° λs=30° λs=45° λs=60°

f [μm/r] Fig. 10. Roughness component of kinematics Ride at different feed rates f and inclination angles λs.

In addition, the prediction errors can be calculated by comparing the absolute residual error with the measured result, which is expressed as  d R  Rd  a  mea  d  Ra  mea Ra  mea      a  Ra  mea  Ra  a R Ra  mea a  mea 

(29)

where δd and δa represent the prediction errors of Rd and Ra, respectively. Ra-mea is the average result of the measured surface roughness by AFM. εd and εa represent the residual errors of Rd and R a, respectively. Table 5 summarizes the output results, including the predicted roughness, the measured roughness and corresponding standard deviations Sd, residual errors and prediction errors. It can be seen that the prediction error δd is not well controlled at large feed rates after the corresponding critical feed rate for each inclination angle. The maximum prediction error reaches as much as 49.15%, which indicates that the predicted roughness Rd without consideration of cracks effect has a poor consistency with the experimental results. In contrast, with the additional consideration of cracks effect, the prediction error δa of the predicted roughness Ra is always at a low level with a satisfactory accuracy, in which the maximum prediction error is only 12.91%. Such observations validate that the roughness component of cracks effect is a non-negligible factor in predicting the surface roughness in oblique cutting of 29 / 41

soft-brittle materials. In addition, the residual error εa as listed in Table 5 varies around zero with no obvious regularity. The method of Kolmogorov-Smirnov test was further employed in this work to reveal the variation rule of εa. The result shows that εa follows a normal distribution, i.e. εa~N (μ, ζ2), in which the expectation value μ and the variance ζ are 0.1143 nm and 0.8887 nm, respectively. It indicates that the residual error εa is caused by random factors and the roughness model Ra is established correctly. Table 5-1 Predicted roughness, measured roughness, prediction errors and residual errors, λs=0°, 15°, 30°. λs [°] 0

15

30

f [μm/r] 1 2 4 6 8 10 12 14 16 18 20 1 2 4 6 8 10 12 14 16 18 20 1 2 4 6 8 10 12 14 16 18 20

Rd [nm] 2.7207 3.8863 5.8882 7.8611 9.3135 10.3938 11.3252 12.2802 13.3794 14.7038 16.3059 2.2611 3.3734 5.3124 7.2578 8.7457 9.9164 10.9802 12.0999 13.3894 14.9249 16.7560 2.2964 3.510 5.6659 7.8760 9.6419 11.1140 12.5187 14.0310 15.7733 17.8275 20.2470

Ra [nm] 2.7207 3.8863 5.8882 7.8611 10.4859 13.3668 16.5050 19.9485 23.7501 27.9609 32.6113 2.2611 3.3734 5.3124 7.2578 8.7457 9.9164 10.9802 12.5073 14.8077 17.8230 21.5813 2.2964 3.510 5.6659 7.8760 9.6419 11.1140 12.5187 14.0310 16.4222 19.5472 23.4669

Ra-mea [nm] 2.7679 3.8105 6.2223 9.0264 9.8849 12.8043 16.1730 18.9130 21.650 27.2337 32.0680 2.0817 3.3882 5.4450 7.1571 8.8659 9.7037 10.0235 13.5747 15.5333 18.3040 21.9937 2.3233 3.6189 6.1090 8.7680 9.4399 11.3383 12.3476 13.7520 17.7340 20.8700 24.4847

Sd [nm] 0.2604 0.0633 0.3425 0.1006 0.3112 0.7561 1.4150 2.5238 1.9761 3.7505 6.0286 0.0790 0.0675 0.4608 0.0946 0.0458 0.1861 0.1509 0.4867 0.9898 1.1988 1.4671 0.0616 0.0998 0.0417 0.1850 0.1608 0.3446 0.2269 0.2618 0.8440 0.6728 1.5340

δd [%] 1.7042 1.9892 5.3699 12.9099 5.7805 18.8259 29.9746 35.0701 38.2014 46.0087 49.1521 8.6164 0.4368 2.4358 1.4070 1.3558 2.1923 9.5442 10.8641 13.8021 18.4610 23.8144 1.1566 3.0100 7.2537 10.1737 2.1395 1.9785 1.3857 2.0288 11.0562 14.5783 17.3074

δa [%] 1.7042 1.9892 5.3699 12.9099 6.0800 4.3928 2.0528 5.4751 9.7002 2.6703 1.6942 8.6164 0.4368 2.4358 1.4070 1.3558 2.1923 9.5442 7.8630 4.6714 2.6278 1.8749 1.1566 3.0100 7.2537 10.1737 2.1395 1.9785 1.3857 2.0288 7.3971 6.3383 4.1568

εa [nm]

0.0472 -0.0758 0.3341 1.1653 -0.601 -0.5625 -0.332 -1.0355 -2.1001 -0.7272 -0.5433 -0.1794 0.0148 0.1326 -0.1007 0.1202 -0.2127 -0.9567 1.0674 0.7256 0.481 0.4124 0.0269 0.1089 0.4431 0.892 -0.202 0.2243 -0.1711 -0.279 1.3118 1.3228 1.0178

Furthermore, the comparisons between the measured roughness and the predicted roughness are pictorially presented in Fig. 11. Obviously, when the feed rate is smaller than the corresponding critical feed rate at each inclination angle, the curves of Rd and Ra are completely coincident. In this 30 / 41

case, the feed rate is small enough that the value of Cre is larger than zero and there are no cracks left on the smooth surface as machined. However, when the assigned feed rate is larger than the critical feed rate, the difference between Rd and Ra begins to appear and increases with the increment of feed rate. In this case, the predicted roughness Rd is visibly smaller than the measured roughness. Whereas, the predicted roughness Ra has a good consistency with the measured results due to the additional consideration of roughness component Rc. Table 5-2 Predicted roughness, measured roughness, prediction errors and residual errors, λs=45°, 60°. λs [°] 45

60

f [μm/r] 1 2 4 6 8 10 12 14 16 18 20 1 2 4 6 8 10 12 14 16 18 20

Rd [nm] 2.8159 4.3368 7.1193 10.0647 12.5634 14.7995 17.0498 19.5285 22.3851 25.7202 29.5988 3.2440 5.3734 9.5630 14.3275 18.8676 23.4248 28.3620 33.9599 40.4146 47.8579 56.3754

Ra [nm] 2.8159 4.3368 7.1193 10.0647 12.5634 14.7995 17.0498 20.2499 24.3017 29.2858 35.2235 3.2440 5.3734 9.5630 14.3275 18.8676 24.3044 30.7531 38.3265 47.1088 57.1477 68.4791

Ra-mea [nm] 2.7124 4.2243 7.1620 10.4710 12.1470 13.9273 16.9323 21.8967 24.9753 30.1847 36.2957 3.1733 5.4965 9.4875 14.1250 17.3977 24.9863 32.0017 38.5637 46.2790 55.4577 71.9237

Sd [nm] 0.1217 0.0369 0.1647 0.3792 0.6306 0.5801 0.5320 0.6133 0.9933 3.1926 4.5602 0.0705 0.4275 0.1994 0.2623 0.3108 1.2879 0.4835 1.1928 2.0568 3.2961 7.3248

δd [%] 3.8170 2.6632 0.5966 3.8802 3.4280 6.2623 0.6938 10.8152 10.3712 14.7905 18.4509 2.2270 2.2401 0.7961 1.4336 8.4490 6.2495 11.3734 11.9381 12.6718 13.7037 21.6177

δa [%] 3.8170 2.6632 0.5966 3.8802 3.4280 6.2623 0.6938 7.5206 2.6972 2.9779 2.9540 2.2270 2.2401 0.7961 1.4336 8.4490 2.7292 3.9016 0.6150 1.7930 3.0474 4.7892

εa [nm]

-0.1035 -0.1125 0.0427 0.4063 -0.4164 -0.8722 -0.1175 1.6468 0.6736 0.8989 1.0722 -0.0707 0.1231 -0.0755 -0.2025 -1.4699 0.6819 1.2486 0.2372 -0.8298 -1.69 3.4446

As listed in Table 3, when tool inclination angle is 0°, the corresponding critical feed rate reaches the smallest. Thus, as shown in Fig. 11 (a), the difference between Rd and Ra-mea caused by the cracks effect is the biggest compared to others presented in Fig. 11 (b) - (e). Especially at the inclination angles of 15°, 30° and 45°, as shown in Fig. 11 (b) - (d), the difference is small because the cracks effect at these three inclination angles is well suppressed. Fig. 12 illustrates the surface topographies acquired by AFM in response to different inclination angles at the same feed rate, i.e. f=20 μm/r. Left is the 3D topography, right is the corresponding 2D view. Obviously, there are more large-size cracks 31 / 41

left on the machined surfaces at tool inclination angles of 0° and 60°. On the contrary, when tool inclination angles are 15°, 30° and 45°, less small-size cracks are left on the machined surfaces. As confirmed by the observations in Fig. 12, it can be concluded that the difference in cracks condition is the dominant reason causing the great errors between Rd and Ra-mea. However, the roughness component Rc can well remove such large errors, which results in that the predicted roughness Ra presents a satisfactory prediction accuracy as revealed in Fig. 11. Measured Ra-mea Predicted without cracks Rd Predicted with cracks Ra

Critical feed rate

(b) Surface roughness [nm]

Surface roughness [nm]

(a)

Feed rate f [μm/r]

Measured Ra-mea Predicted without cracks Rd Predicted with cracks Ra

Feed rate f [μm/r]

(d) Surface roughness [nm]

Surface roughness [nm]

(c)

Measured Ra-mea Predicted without cracks Rd Predicted with cracks Ra

Feed rate f [μm/r]

Measured Ra-mea Predicted without cracks Rd Predicted with cracks Ra

Feed rate f [μm/r]

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Surface roughness [nm]

(e)

Measured Ra-mea Predicted without cracks Rd Predicted with cracks Ra

Feed rate f [μm/r] Fig. 11. Measured roughness Ra-mea, predicted roughness without cracks Rd and predicted roughness with cracks Ra at different inclination angles: (a) λs=0°; (b) λs=15°; (c) λs=30°; (d) λs=45°; (e) λs=60°.

(a)

(b)

(c)

(d)

(e)

Fig. 12. Surface topographies measured by AFM in 3D and 2D views at the feed rate of 20 μm/r in response to different tool inclination angles: (a) λs=0°; (b) λs=15°; (c) λs=30°; (d) λs=45°; (e) λs=60°.

Moreover, as illustrated in Fig. 11, the surface roughness at the inclination angle of 60° is significantly larger than others, which is attributed to the sharp rise of Ride, as demonstrated in Fig. 10. When tool inclination angles are 15° and 30°, the surface roughness is relatively small compared with others. However, there are minute differences between these two inclination angles. When the surface is achieved in ductile mode, the observation that the surface roughness at the inclination angle of 15° 33 / 41

is slightly smaller than the one achieved at 30° can be attributed to one of the important rules in oblique cutting, i.e. the plastic side flow has the minimum value at the inclination angle of 15°, as revealed in Table 4. When the surface is achieved in brittle mode, the surface roughness finished at the inclination angle of 30° increases more slowly than that achieved at 15°, which is due to the effective suppression for cracks under the high enough hydrostatic pressure in cutting area. In the established roughness model Ra, a small increment of roughness component Rc also can be observed at the inclination angle of 30° as listed in Table 4. More interestingly, as demonstrated in Fig. 11, after the corresponding critical feed rate at every inclination angle, the standard deviation of roughness becomes visible due to the ever-decreasing inhibition for cracks. 5 Conclusions In order to predict the surface roughness of KDP crystal at a wide range of feed rates in oblique diamond turning, a novel roughness model considering four roughness components, i.e. kinematics, plastic side flow, materials elastic recovery and cracks effect based on the RLC model, is established and validated in this work. Finally, some important conclusions can be drawn as follows. 1) Theoretical prediction reveals that the roughness component of kinematics increases sharply at large inclination angles, especially for 60°, which would significantly increase the total surface roughness. The roughness component of plastic side flow is a complex variable derived from the competition between the unremoved materials volume and the scale coefficient, which outputs the smallest value at an inclination angle of 15°. The roughness component of materials elastic recovery is primarily related to the inclination angle due to the presence of equivalent cutting edge radius in oblique cutting, and a large inclination angle corresponds to a large materials elastic recovery. The roughness component of cracks exhibits segmented characteristics at the critical feed rate, which has a great effect on the total surface roughness in the brittle-mode machining. 34 / 41

2) As revealed by experimental observations, the RLC model can well predict the distribution of cracks, which can be effectively used to determine the critical feed rate for achieving a crack-free surface of KDP crystal in oblique cutting. As the inclination angle increases, the corresponding critical feed rate increases first and subsequently decreases. When the inclination angle is 30°, the maximum critical feed rate, i.e. 14.2 μm/r, is reached. However, when the inclination angles are set to 0° and 60°, small critical feed rates are obtained, i.e. 6 μm/r and 8.1 μm/r respectively. This is attributed to the reduced hydrostatic pressure that loses the sufficient ability to suppress the crack propagation. 3) Compared with the experimental results, the predicted surface roughness without consideration of cracks effect has a poor prediction accuracy when the feed rate is larger than the corresponding critical feed rate. In contrast, after considering the roughness component of cracks effect, the predicted surface roughness has a satisfactory consistency with the measured results in experiments. Therefore, the roughness component of cracks effect is a non-negligible factor in predicting the surface roughness in oblique diamond turning of soft-brittle materials, which can be effectively quantified by the RLC model to improve the prediction accuracy for the brittle-mode machining. Acknowledgments The authors would like to thank the National Natural Science Foundation of China (No. 51675133) and the Science Challenge Project (No. TZ2018006-0202-02) for the support of this work.

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Relevant declarations Declarations of interest: none

Submission declaration and verification None of the material has been published or is under consideration for publication elsewhere. Furthermore,

if accepted, it will not be published elsewhere in the same form, in English or in any other language, including electronically without the written consent of the copyright-holder.

Contributors Zhang and Zong designed the theoretical and experimental framework for this work. Zhang carried out the experiments, data acquisition as well as data analysis and drafted the manuscript. Zhang and Zong analyzed the results and wrote the manuscript.

Author agreement All authors confirm that this manuscript is the authors’ original work. All authors checked the manuscript and agreed to its submission. Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Graphical abstract

1. Modeling

Z0 (Z1) γ Z

Kinematics

2. Experiment

Z2

KDP workpiece

Diamond tool X1 (X2) λs X´

rε

zi O

Diamond tool

X0 Indexing fixture

g´ ． g ．x ´

Workpiece

λs

i

xi

Round jig

O

Y1 f/2

Y0

X

f

Plastic side flow & elastic recovery Z0

O2

Balancing weight

Y2

Cracks

J

3. Results

O1

Diamond tool

rε ζ

e

B

robl

X1

p

dc

G

f

λs

Initial crack

A

Workpiece

X0

Hydrostatic pressure zone

Surface roughness [nm]

hmin

Measured Ra-mea Predicted without cracks Rd Predicted with cracks Ra

Critical feed rate

Feed rate f [μm/r]

Nomenclature Ac Cm-h

Area of the new surface generated by cracks Length of median crack under the inhibition of hydrostatic pressure

Cre

Relative length of the median crack

C0

Workpiece coordinate system

C1

Oblique coordinate system

C2

Rake face coordinate system

E

Young’s modulus

Ef

Fracture energy for new surface generation

Fc-0

Principal cutting force in C0

Fr-0

Feed cutting force in C0

Ft-0

Thrust cutting force in C0

Fc-1

Principal cutting force in C1

KIC

Fracture toughness

Pe PG-ver

Equivalent plane Vertical contact stress at point G 40 / 41

PG-hor Pn Pobl Ps Psh R, M, N

Horizontal contact stress at point G Normal plane Tool base plane Cutting plane Shear plane Fitted coefficients for Rc

Ra

Predicted surface roughness

Ra-mea

Measured surface roughness

Rc

Roughness component of cracks effect

Ride

Roughness component of kinematics

Vb

Volume of the unremoved materials

a0, b0, c0, d0

Fitted coefficients for inclination angle

a1, b1, c1, d1

Fitted coefficients for feed rate

ap

Depth of cut

c, k1, k2, kd

Coefficients

dc

BDT depth

e

Roughness component of materials elastic recovery

f

Feed rate

fobl hmin

Equivalent feed rate Minimum undeformed chip thickness

k0

Conversion coefficient

kp

Scale coefficient of plastic side flow

p

Roughness component of plastic side flow

rε

Tool nose radius

rn

Cutting edge radius

robl

Equivalent cutting edge radius

γ

Nominal rake angle

γe

Equivalent rake angle

γs

Specific surface energy of material

λs

Tool inclination angle

εc

Chip flow angle



Shear angle

β

Friction angle

ζh

Hydrostatic pressure

φ

Direction of the median crack at dc

41 / 41