1 Introduction

KDP (KH2PO4, potassium dihydrogen phosphate) crystal is an important functional crystal material. As its properties of large laser damage threshold [1], KDP crystal is widely used in the fields of laser frequency conversion and photoelectric switch [2]. Slicing is the first process of machining [3]. KDP crystal would be sliced to slices by the diamond wire saw [4, 5]. Schematic of KDP crystal sliced by the diamond wire saw is illustrated in Fig. 1.

Fig. 1
figure 1

Schematic of KDP crystal sliced by diamond wire saw

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It can be observed in Fig. 1 that the diamond wire saw moves in the Y direction, while the feed direction of KDP crystal is against the X direction.

As KDP crystal is an anisotropic material, properties with different crystal orientation would be very different. The KDP crystal contacts with the diamond wire saw in slicing process would have different crystal orientation and properties. The anisotropic properties of KDP crystal would lead to the deviation of the diamond wire saw in the thickness direction and form the surface shape deviation of the slice.

The slices should meet specific crystal orientation requirements. The (001) crystal plane, the double-frequency crystal plane, and the triple-frequency crystal plane are the crystal planes that should be processed of KDP crystal. All the three crystal planes have precise requirements for crystal orientation. The surface shape deviation caused by the anisotropic properties of KDP crystal would affect the positioning accuracy of the slices. Besides, the surface shape deviation of the slices would determine the amount of material to be removed in subsequent processes and thus affects the material utilization rate. So the effect of anisotropy on surface shape deviation in diamond wire saw slicing of KDP crystal should be investigated.

The anisotropic properties of KDP crystal have been studied by many scholars. Fang [6] studied the mechanical parameters of KDP crystals under several different conditions. Zhang [7] measured the elastic modules of two different crystal planes of KDP crystal by the uniaxial compression test. Guin [8] studied the plastic strain of different crystal planes of KDP crystal by the method of indentation.

In the area of the influence of the anisotropy of KDP crystals on processing, Chen [9] conducted an experimental study on the influence of crystal orientation of KDP crystal on cutting force and machining quality in the single point diamond turning. To explore the anisotropic properties and deformation mechanisms of KDP crystal at the atomic scale, Yang [10] established a potential function for the molecular dynamics (MD) simulations and found that the deformation mechanisms would be affected by the anisotropy.

To study the effect of anisotropy on surface shape deviation in diamond wire saw slicing of KDP crystal, the sawing force on the diamond wire saw needs to be studied. The sawing force has been studied by many researchers by methods of experimental measurements and simulation. Clark [11, 12] measured the sawing force of diamond wire saw and studied the force ratio of wood and foam ceramics. The sawing force of SiC [13, 14] is also measured by some researchers. Liu [15] simulated the sawing force of silicon by the diamond wire saw. Wang [16] simulated the sawing force for silicon carbide with the abrasive diamond wire saw. But none of them considered the influence anisotropy to the sawing force. Huang [17] measured the sawing forces of two different crystal planes of sapphire in slicing by the diamond wire saw and found that the forces were different. Though the anisotropy properties of KDP crystals and the influence of anisotropy on processing have been partially studied, the influence of anisotropy on slicing process has not been studied yet.

In this paper, influence of anisotropy of KDP crystal on the surface shape deviation of the slice by the diamond wire saw is studied. The (001) crystal plane, the double-frequency crystal plane, and the triple-frequency crystal plane are studied, respectively, in this paper.

As the anisotropic properties of KDP crystal are determined by the crystal orientation, crystal orientation expressions of KDP crystal in slicing of the three crystal planes with different feed angles of diamond wire saw are studied through coordinate transformation. Then a model of diamond wire saw considering the anisotropic properties of KDP crystals is established. And the sawing forces of the diamond wire saw during slicing are obtained by the model. Finally, the surface shape deviation is obtained by applying sawing forces to the diamond wire saw in ABAQUS.

In this paper, for the (001) crystal plane, the double-frequency crystal plane, and the triple-frequency crystal plane, variation rules of surface shape deviation of slice with the feed angle of diamond wire saw are obtained. Also, change rule of surface shape deviation with the tension is also obtained.

According to the results in this paper, the surface shape deviation of slice caused by anisotropy of KDP crystal can be reduced by selecting appropriate feed angles of diamond wire saw.

2 Anisotropy of KDP crystal in slicing by diamond wire saw

To study the influence of anisotropy on the diamond wire saw slicing, the material properties of the KDP crystal that interacts with the diamond wire saw during slicing should be determined. Anisotropic properties of KDP crystals mainly include elastic modulus, hardness, and critical depth of cut. The elastic modulus of the KDP crystal can be obtained according to its crystal orientation expressions.

The KDP crystal has a tetragonal structure at room temperature. It has a fourth-order symmetry axis and a second-order symmetry axis perpendicular to it. The ideal outline drawing of KDP crystal is shown in Fig. 2.

Fig. 2
figure 2

Ideal outline drawing of KDP crystal

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Because of the symmetry of the crystal structure, the elastic modulus corresponding to any crystal direction of the KDP crystal can be obtained by Eq. (1) based on the theory of elasticity.

$$ {\displaystyle \begin{array}{c}E=1/\left({s}_{11}\left({\alpha}_0^4+{\beta}_0^4\right)+{s}_{33}{\gamma}_0^4+\left(2{s}_{13}+{s}_{44}\right)\left({\alpha}_0^2+{\beta}_0^2\right){\gamma}_0^2+\left(2{s}_{12}+{s}_{66}\right){\alpha}_0^2{\beta}_0^2\right)\\ {}\mathrm{While},\left\{\begin{array}{c}{\alpha}_0=h/\left({h}^2+{k}^2+{l}^2\right)\\ {}{\beta}_0=k/\left({h}^2+{k}^2+{l}^2\right)\\ {}{\gamma}_0=l/\left({h}^2+{k}^2+{l}^2\right)\end{array}\right.\end{array}} $$
(1)

In Eq. (1), sij is the flexibility constant of KDP crystal. [hkl] is the expression of crystal orientation. While α0 is the cosine value of the crystal direction and the X axis, β0 is the cosine value of the crystal direction and the Y axis, and γ0 is the cosine value of the crystal direction and the Z axis. The flexibility coefficient of KDP crystal is listed in Table 1 [18].

Table 1 Flexibility coefficient of KDP crystal
Full size table

According to Eq. (1), the elastic modulus of any crystal orientation can be calculated as long as the expression of the crystal orientation is determined. Then, distribution of elastic modulus with crystal orientation for KDP crystal can be obtained as shown in Fig. 3.

Fig. 3
figure 3

Distribution of elastic modulus with crystal orientation

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Therefore, to get the required elastic modulus of KDP crystal used in slicing, expression of the crystal orientation should be determined.

2.1 The angular relationship of the three crystal planes

The (001) crystal plane, the double-frequency crystal plane, and the triple-frequency crystal plane are the three commonly used crystal planes of KDP crystal. The schematic of the three crystal planes is shown in Fig. 4.

Fig. 4
figure 4

Schematic of the three crystal planes of KDP crystal

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The (001) crystal plane is the plane perpendicular to the [001] axis. According to the geometric relationship, the double-frequency crystal plane can be obtained by rotating the (001) crystal plane around the [100] axis by 41° and then rotating this plane around the [001] axis by 45°. The triple-frequency crystal plane can be obtained by rotating the (001) crystal plane around the [100] axis by − 59.5°.

2.2 Derivation of coordinate change of the (001) crystal plane

The schematic of coordinate system conversion of the (001) crystal plane during slicing is shown in Fig. 5.

Fig. 5
figure 5

Schematic of (001) crystal plane conversion

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In Fig. 5, the coordinate system of the KDP crystal is OXYZ, while O11X11Y11Z11 is the coordinate system of the (001) crystal plane. The initial feed direction of the diamond wire saw is X11O11. The angle γ1 is the feed angle of the diamond wire saw for the (001) crystal plane with the range of 0°–360°.

The change matrix from the coordinate system OXYZ to the (001) crystal plane coordinate system O11X11Y11Z11 is T11. The coordinate system O12X12Y12Z12 can be obtained by rotating the coordinate system O11X11Y11Z11 along the Z11 axis by the angle γ1, and its change matrix is T12. T11 and T12 can be deduced by the matrix transformation according to the geometric relationship.

$$ {T}_{11}=\left[\begin{array}{l}1\ 0\ 0\\ {}0\ 1\ 0\\ {}0\ 0\ 1\end{array}\right] $$
(2)
$$ {T}_{12}=\left[\begin{array}{l}\cos\ {\gamma}_1-\sin\ {\gamma}_1\ 0\\ {}\sin \kern0.45em {\gamma}_1\kern0.45em \cos\ {\gamma}_1\ 0\\ {}\kern0.5em 0\kern2.5em 0\kern1.75em 1\end{array}\right] $$
(3)

To study the effect of anisotropy, elastic modulus at the contact position of the KDP crystal and the diamond wire saw in slicing need to be obtained. Distribution of diamond grits on the section of the diamond wire saw is shown in Fig. 6.

Fig. 6
figure 6

Distribution of diamond grits on the section of diamond wire saw

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In Fig. 6, the KDP crystal feeds in the X12 axis, and the movement direction of the diamond wire saw is the Y12 axis, while θL is the location angle of the diamond grits on the section of diamond wire saw, whose range is − 90° to 90°. In the coordinate system O12X12Y12Z12, expression of the crystal orientation corresponding to the diamond grits with a position angle of θL is shown in Eq. (4).

$$ {A}_{12}=\left[\begin{array}{l}-\cos\ {\theta}_L\\ {}\kern2em 0\\ {}\kern0.5em \sin\ {\theta}_L\end{array}\right] $$
(4)

A1 is the expression of the crystal orientation in coordinate system OXYZ. It can be obtained by matrix transformation as Eq. (5).

$$ {A}_1={T}_{12}{T}_{11}{A}_{12}=\left[\begin{array}{c}-\cos {\gamma}_1\cos {\theta}_L\\ {}-\sin {\gamma}_1\cos {\theta}_L\\ {}\ \sin {\theta}_L\end{array}\right] $$
(5)

By substituting Eq. (5) into Eq. (1), elastic modulus of KDP crystal in slicing of (001) crystal plane can be obtained in Eq. (6).

$$ {\displaystyle \begin{array}{c}E=1/\Big({s}_{11}\left({\cos}^4{\gamma}_1+{\sin}^4{\gamma}_1\right){\cos}^4{\theta}_L+{s}_{33}{\sin}^4{\theta}_L+\left(2{s}_{13}+{s}_{44}\right){\cos}^2{\theta}_L{\sin}^2{\theta}_L\\ {}+\left(2{s}_{12}+{s}_{66}\right){\cos}^2{\gamma}_1{\sin}^2{\gamma}_1{\cos}^4{\theta}_L\Big)\end{array}} $$
(6)

2.3 Derivation of coordinate change of the double-frequency crystal plane

The schematic of coordinate system conversion of the double-frequency crystal plane during slicing is shown in Fig. 7.

Fig. 7
figure 7

Schematic of the double-frequency crystal plane conversion

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In Fig. 7, the coordinate system of the KDP crystal is OXYZ, while O21X21Y21Z21 is the coordinate system of the double-frequency crystal plane. The initial feed direction of the diamond wire saw is X21O21. The angle γ2 is the feed angle of the diamond wire saw for the double-frequency crystal plane with the range of 0°–360°. While the feed angle of the diamond wire saw is γ2, the coordinate system O22X22Y22Z22 can be obtained by rotating O21X21Y21Z21 around O21Z21 by γ2.

According to the geometric relationship, the double-frequency crystal plane coordinate system O21X21Y21Z2 is obtained by rotating the coordinate system OXYZ around the X axis by 41° and then rotating it around the Z axis by 45°. The change matrix from the coordinate system OXYZ to O21X21Y21Z21 is T21.

The coordinate system O22X22Y22Z22 is obtained by rotating the coordinate system O21X21Y21Z21 around the Z21 axis by γ2, and its change matrix is T22. The T21 and T22 can be deduced by the matrix transformation according to the geometric relationship. The expressions are shown in Eqs. (7) and (8).

$$ {T}_{21}=\left[\begin{array}{ccc}\cos {45}^{{}^{\circ}}& -\sin {45}^{{}^{\circ}}\cos {41}^{{}^{\circ}}& \sin {45}^{{}^{\circ}}\sin {41}^{{}^{\circ}}\\ {}\sin {45}^{{}^{\circ}}& \cos {45}^{{}^{\circ}}\cos {41}^{{}^{\circ}}& -\cos {45}^{{}^{\circ}}\sin {41}^{{}^{\circ}}\\ {}0& \sin {41}^{{}^{\circ}}& \cos {41}^{{}^{\circ}}\end{array}\right] $$
(7)
$$ {\displaystyle \begin{array}{c}{T}_{22}=\left[\begin{array}{l}{a}_2^2+\left(1-{a}_2^2\right)\cos {\gamma}_2\kern0.5em {a}_2{b}_2\left(1-\cos {\gamma}_2\right)-{c}_2\sin {\gamma}_2\kern0.5em {a}_2{c}_2\left(1-\cos {\gamma}_2\right)+{b}_2\sin {\gamma}_2\\ {}{a}_2{b}_2\left(1-\cos {\gamma}_2\right)+{c}_2\sin {\gamma}_2\kern0.5em {b}_2^2+\left(1-{b}_2^2\right)\cos {\gamma}_2\kern0.5em {b}_2{c}_2\left(1-\cos {\gamma}_2\right)-{a}_2\sin {\gamma}_2\\ {}\begin{array}{ccc}{a}_2{c}_2\left(1-\cos {\gamma}_2\right)-{b}_2\sin {\gamma}_2& {b}_2{c}_2\left(1-\cos {\gamma}_2\right)+{a}_2\sin {\gamma}_2& {c}_2^2+\left(1-{c}_2^2\right)\cos {\gamma}_2\end{array}\end{array}\right]\\ {}\mathrm{While},\left[\begin{array}{c}{a}_2\\ {}{b}_2\\ {}{c}_2\end{array}\right]=\left[\begin{array}{c}\sin 45{}^{\circ}\sin 41{}^{\circ}\\ {}-\cos 45{}^{\circ}\sin 41{}^{\circ}\\ {}\cos 41{}^{\circ}\end{array}\right]\end{array}} $$
(8)

Similar to the (001) crystal plane, for the double-frequency crystal plane, expression of the crystal orientation corresponding to the diamond grits with a position angle of θL in the coordinate system O22X22Y22Z22 is A22. Its expression is shown in Eq. (9).

$$ {A}_{22}=\left[\begin{array}{c}-\cos {\theta}_L\\ {}\ 0\\ {}\ \sin {\theta}_L\end{array}\right] $$
(9)

A2 is the expression of the crystal orientation in the coordinate system OXYZ. It can be obtained by matrix transformation. Its expression is shown in Eq. (10).

$$ {\displaystyle \begin{array}{c}{A}_2={T}_{22}{T}_{21}{A}_{22}\\ {}=\left[\begin{array}{c}\left(\sqrt{2}/2\right)\left(-\sin {\gamma}_2\cos {\theta}_L+\left(\sin 41{}^{\circ}\sin {\theta}_L-\cos 41{}^{\circ}\cos {\gamma}_2\cos {\theta}_L\right)\right)\\ {}\left(\sqrt{2}/2\right)\left(-\sin {\gamma}_2\cos {\theta}_L-\left(\sin 41{}^{\circ}\sin {\theta}_L-\cos 41{}^{\circ}\cos {\gamma}_2\cos {\theta}_L\right)\right)\\ {}\ \sin 41{}^{\circ}\cos {\gamma}_2\cos {\theta}_L+\cos 41{}^{\circ}\sin {\theta}_L\end{array}\right]\end{array}} $$
(10)

Substituting Eq. (10) into Eq. (1), elastic modulus of KDP crystal in slicing of double-frequency crystal plane can be obtained through Eq. (11).

$$ {\displaystyle \begin{array}{c}E=1/\left({s}_{11}\left({\alpha}^4+{\beta}^4\right)+{s}_{33}{\gamma}^4+\left(2{s}_{13}+{s}_{44}\right)\left({\alpha}^2+{\beta}^2\right){\gamma}^2+\left(2{s}_{12}+{s}_{66}\right){\alpha}^2{\beta}^2\right)\\ {}\mathrm{While},\left\{\begin{array}{c}\alpha =\left(\sqrt{2}/2\right)\left(-\sin {\gamma}_2\cos {\theta}_L+\left(\sin 41{}^{\circ}\sin {\theta}_L-\cos 41{}^{\circ}\cos {\gamma}_2\cos {\theta}_L\right)\right)\\ {}\beta =\left(\sqrt{2}/2\right)\left(-\sin {\gamma}_2\cos {\theta}_L-\left(\sin 41{}^{\circ}\sin {\theta}_L-\cos 41{}^{\circ}\cos {\gamma}_2\cos {\theta}_L\right)\right)\\ {}\gamma =\sin 41{}^{\circ}\cos {\gamma}_2\cos {\theta}_L+\cos 41{}^{\circ}\sin {\theta}_L\end{array}\right.\;\end{array}} $$
(11)

2.4 Derivation of coordinate change of the triple-frequency crystal plane

The schematic of coordinate system conversion of the triple-frequency crystal plane during slicing is shown in Fig. 8.

Fig. 8
figure 8

Schematic of the triple-frequency crystal plane conversion

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In Fig. 8, the coordinate system of the KDP crystal is OXYZ, while O31X31Y31Z31 is the coordinate system of the triple-frequency crystal plane. The initial feed direction of the diamond wire saw is X31O31. The angle γ3 is the feed angle of the diamond wire saw for the triple-frequency crystal plane with the range of 0°–360°. While the feed angle is γ3, the coordinate system O32X32Y32Z32 can be obtained by rotating O31X31Y31Z31 along O31Z31 by an angle of γ3.

According to the geometric relationship, the triple-frequency crystal plane coordinate system O31X31Y31Z31 can be obtained by rotating the coordinate system OXYZ around the X axis by − 59.5°. The change matrix from the coordinate system OXYZ to O31X31Y31Z31 is T31. The coordinate system of O32X32Y32Z32 is obtained by rotating the coordinate system O31X31Y31Z31 along the O31Z31 axis by γ3, and its change matrix is T32. T31 and T32 can be deduced by the matrix transformation according to the geometric relationship. The expressions are shown in Eqs. (12) and (13).

$$ {T}_{31}=\left[\begin{array}{l}\kern0.85em 1\kern2em 0\kern3.1em 0\ \\ {}0\ \cos \kern0.15em {59.5}^{\circ}\kern0.35em \sin\ {59.5}^{\circ}\\ {}0-\sin \kern0.15em {59.5}^{\circ}\cos\ {59.5}^{\circ}\end{array}\right] $$
(12)
$$ {\displaystyle \begin{array}{c}{T}_{32}=\left[\begin{array}{c}{a}_3^2+\left(1-{a}_3^2\right)\cos {\gamma}_3{a}_3{b}_3\left(1-\cos {\gamma}_3\right)-{c}_3\sin {\gamma}_3{a}_3{c}_3\left(1-\cos {\gamma}_3\right)+{b}_3\sin {\gamma}_3\\ {}{a}_3{b}_3\left(1-\cos {\gamma}_3\right)+{c}_3\sin {\gamma}_3{b}_3^2+\left(1-{b}_3^2\right)\cos {\gamma}_3{b}_3{c}_3\left(1-\cos {\gamma}_3\right)-{a}_3\sin {\gamma}_3\\ {}{a}_3{c}_3\left(1-\cos {\gamma}_3\right)-{b}_3\sin {\gamma}_3{b}_3{c}_3\left(1-\cos {\gamma}_3\right)+{a}_3\sin {\gamma}_3{c}_3^2+\left(1-{c}_3^2\right)\cos {\gamma}_3\end{array}\right]\\ {}\mathrm{While},\left[\begin{array}{c}{a}_3\\ {}{b}_3\\ {}{c}_3\end{array}\right]=\left[\begin{array}{c}0\\ {}\sin 59.5{}^{\circ}\\ {}\cos 59.5{}^{\circ}\end{array}\right]\end{array}} $$
(13)

Similar to the condition of the (001) crystal plane, for the triple-frequency plane, expression of the crystal orientation corresponding to the diamond grits with a position angle of θL in the coordinate system O32X32Y32Z32 is A32. Its expression is shown in Eq. (14).

$$ {A}_{32}=\left[\begin{array}{l}-\cos\ {\theta}_L\\ {}\kern1.9em 0\\ {}\kern0.5em \sin\ {\theta}_L\end{array}\right] $$
(14)

A3 is the expression of the crystal orientation in the coordinate system OXYZ. It can be obtained by matrix transformation. Its expression is shown in Eq. (15).

$$ {A}_3={T}_{32}{T}_{31}{A}_{32}=\left[\begin{array}{c}-\sin {\gamma}_3\cos {\theta}_L\\ {}\sin 59.5{}^{\circ}\sin {\theta}_L+\cos 59.5{}^{\circ}\cos {\theta}_L\cos {\gamma}_3\\ {}\cos 59.5{}^{\circ}\sin {\theta}_L-\sin 59.5{}^{\circ}\cos {\theta}_L\cos {\gamma}_3\end{array}\right] $$
(15)

Substituting Eq. (15) into Eq. (1), the elastic modulus of KDP crystal used in slicing of triple-frequency crystal plane can be obtained through Eq. (16).

$$ {\displaystyle \begin{array}{c}E=1/\left({s}_{11}\left({\alpha}^4+{\beta}^4\right)+{s}_{33}{\gamma}^4+\left(2{s}_{13}+{s}_{44}\right)\left({\alpha}^2+{\beta}^2\right){\gamma}^2+\left(2{s}_{12}+{s}_{66}\right){\alpha}^2{\beta}^2\right)\\ {}\mathrm{While},\left\{\begin{array}{c}\alpha =-\sin {\gamma}_3\cos {\theta}_L\\ {}\beta =\sin 59.5{}^{\circ}\sin {\theta}_L+\cos 59.5{}^{\circ}\cos {\theta}_L\cos {\gamma}_3\\ {}\gamma =\cos 59.5{}^{\circ}\sin {\theta}_L-\sin 59.5{}^{\circ}\cos {\theta}_L\cos {\gamma}_3\end{array}\right.\end{array}} $$
(16)

Hardness of KDP crystal in any crystal direction can be obtained from Eq. (17):

$$ H=0.0226E+0.7555 $$
(17)

where H is the hardness of KDP crystal and E is the elastic modulus of KDP crystal.

The fracture toughness of KDP crystal is Kc. It can be obtained by Eq. (18):

$$ {K}_c=k{\left(E/H\right)}^nP/{c}^{1.5} $$
(18)

where k is 0.016, n is 0.5, E is the elastic modulus, H is the hardness, P is the nominal load of indentation, and P/c1.5 is a constant value which can be calculated from the known data.

The critical depth of KDP crystal is dc. It can be obtained by Eq. (19).

$$ {d}_c=\xi {\left({K}_c/H\right)}^2\left(E/H\right) $$
(19)

While ξ is 0.3717, it is a constant that depends on the processing condition.

Therefore, the anisotropic properties of KDP crystal in slicing by the diamond wire saw can be obtained.

3 Simulation of the surface shape deviation

To obtain the surface shape deviation in slicing, the sawing force should be determined firstly. A model of the diamond wire saw is established to calculate the sawing force in the slicing process considering the anisotropic properties of KDP crystal.

3.1 Simulation of the sawing force

3.1.1 Establishing a diamond wire saw model

The SEM image of a diamond wire saw is shown in Fig. 9. A diamond wire saw model is established as Fig. 10.

Fig. 9
figure 9

SEM image of a diamond wire saw

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Fig. 10
figure 10

Schematic of diamond grit distribution on diamond wire saw

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In Fig. 10, the diameter of the diamond wire saw is D, and the diameter of the diamond grit is d. The density of diamond grit distribution is η. Diamond grits are assumed to be distributed randomly on the diamond wire saw surface and do not overlap each other. The diameter of the diamond grit is normally distributed [19], while the average diameter is d0, and its standard deviation σ is 3.33. Then the probability of diamond grit with diameter d is in Eq. (20).

$$ p(d)=\exp \left(-{\left(d-{d}_0\right)}^2/\left(2{\sigma}^2\right)\right)/\left(\sqrt{2\pi}\sigma \right) $$
(20)

3.1.2 Sawing force and removed material of a single diamond grit

The material removal mechanism of the diamond wire saw is similar to the indentation and scratching. The diamond grits are generally considered as rigid bodies [20]. Diamond grits are usually simplified into the sphere [21, 22] or the cone [15] shape in simulation. According to the observation of the tip morphology of diamond grits [23], the tip of the abrasive particle is mostly triangular pyramid. To simplify the analysis, the shape of the abrasive grains is equivalent to a cone in this paper. Indentation fracture mechanics can be utilized to analyze the force on a single diamond grits from a micro perspective.

Though the mechanism between the diamond grits and the workpiece is complicated, the normal force of a diamond grit can be calculated by Eq. (21) as long as the cutting depth is known [24]:

$$ {f}_{nij}=\pi {Hh}_{ij}^2{\tan}^2{\theta}_{ij}/2 $$
(21)

where fnij is the normal force, hij is the cutting depth, and θij is the half vertex angle of diamond grit.

The material removal method depends on the cutting depth of the diamond grit. When the cutting depth is less than the critical cutting depth, the material is removed in the plastic mode. While the cutting depth is larger than the critical cutting depth, the material would be removed in the brittle mode [25]. When the material is removed in brittle mode, the schematic of the cracks is shown in Fig. 11.

Fig. 11
figure 11

Schematic of cracks during brittleness removal

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In Fig. 11, Cl is the length of lateral crack [26].

$$ {C}_l={\alpha}_l{\left({\left(\cot \varphi \right)}^{5/6}{\left(E/H\right)}^{3/4}/\left({K}_c{H}^{1/4}\right)\right)}^{1/2}{P}^{5/8} $$
(22)

αl is a constant independent of the indenter and material of workpiece whose value is 0.226.

The depth of plastic deformation zone can be considered the depth of lateral cracks [27]. It can be obtained by Eq. (23):

$$ {h}_l={\alpha}_l{\left(\cot \varphi \right)}^{1/3}{E}^{1/2}{P}^{1/2}/H $$
(23)

where hl is the depth of lateral cracks.

According to the research of Chung [28], the tangential force of a diamond grit can be calculated by Eq. (24).

$$ {f}_t=\left\{\begin{array}{l}{H}_1{h}_{ij}^2\tan {\theta}_{ij}\ \left(0<h<{d}_c\right)\\ {}2{H}_1{C}_l{h}_l\kern0.5em \left({d}_c\le h<{d}_c\right)\end{array}\right. $$
(24)

The feed distance of diamond wire saw is defined as s. It can be obtained by comparing the amount of material removed from the macro and micro perspectives. The cutting depths of the diamond grits can be obtained by Eq. (25) according to the geometric relationship in a diamond wire saw section.

$$ {h}_{ij}=s\cos {\theta}_{Lij}-\left({d}_{\mathrm{max}}-{d}_{ij}\right) $$
(25)

While dij is the exposed height of the diamond grits, dmax is the largest value in the diamond wire saw section.

The material removed by a diamond grit is shown in Eq. (26).

$$ {V}_{ij}=\left\{\begin{array}{l}{v}_s{th}_{ij}^2\tan {\theta}_{ij}\ \left({h}_{ij}<{d}_c\right)\\ {}2{v}_s{tc}_{lij}{h}_{cij}\ \left({h}_{ij}\ge {d}_c\right)\end{array}\right. $$
(26)

From the micro perspective, the total volume removed by the diamond grits can be obtained by Eq. (27).

$$ {V}_1=\sum {V}_{ij} $$
(27)

From the macro perspective, the volume of material removed within time t is V2. It can be obtained in Eq. (28):

$$ {V}_2={Dl}_2{v}_ft $$
(28)

where l2 is the length of the diamond wire saw contacts with KDP crystal.

According to Eqs. (27) and (28), s can be obtained by iterative calculation in Matlab. Substituting the obtained s into Eqs. (21) and (24), the sawing forces of the diamond grits can be obtained. The process of simulation of the sawing force is shown in Fig. 12.

Fig. 12
figure 12

Process of simulation of the sawing force

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Some parameters used in the simulation of sawing force are shown in Table 2.

Table 2 Parameters used in the simulation of sawing forces
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At last, the sawing forces of the three crystal planes in KDP crystal with different feed angles of the diamond wire saw are got by simulation.

3.2 Calculation of the surface shape deviation

The schematic of deformation of the diamond wire saw in slicing is shown in Fig. 13.

Fig. 13
figure 13

Schematic of deformation of diamond wire saw in slicing

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In Fig. 13, the surface CDEF is the ideal plane to be formed. The line AGHB is the trajectory formed by the diamond wire saw while the diamond wire saw has an offset in the Z direction.

The force analysis of the diamond wire saw in slicing is shown in Fig. 14. The length of the diamond wire saw between the two guide wheels is l1. While l2 is the length of the diamond wire saw contacts with KDP crystal, it is the red part in Fig. 14.

Fig. 14
figure 14

Schematic of the forces in slicing

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In Fig. 14, T and T2 are tension forces, fn is the normal force, and ft is the tangential force. GH is a part of the diamond wire saw contacts with the KDP crystal, where the fn and ft are distributed. The direction of fn is perpendicular to the wire. For a point I on GH, fn can be broken down into three component forces fnX, fnY, and fnZ in the X, Y, and Z directions. The direction of ft is parallel to the axis of the diamond wire saw.

The slice sliced by the diamond wire saw can be obtained by moving the trajectory curve GH along the X direction. The schemic of a slice of KDP crystal is shown in Fig. 15.

Fig. 15
figure 15

Schemic of a slice of KDP crystal

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The diamond wire saw would deform in the Z direction due to the fnZ. As shown in Fig. 15, deviation of the diamond wire saw in the Z direction is recorded as DZ. In the length of l1, the maximum deviation of the diamond wire saw in the Z direction is recorded as DZmax. In the length of l2, the maximum deviation of the diamond wire saw in the Z direction is recorded as dZmax. The dZmax is the maximum value of the surface shape deviation of a slice, and dZmax is the object that should be studied in this paper.

To obtain the surface shape deviation of slice, the sawing forces are applied to an ABAQUS model to calculate the deformation of the diamond wire saw. After simulation of the deformation of the diamond wire saw is done, the coordinate values of each node on the diamond wire saw are achieved in the post-processing stage. Also, the trajectory expression of the diamond wire saw can be obtained by curve fitting. As a result, the surface shape deviation of slice by diamond wire saw can be obtained.

4 Results and discussion

4.1 Distribution of elastic modulus on the cross section of diamond wire saw

The schematic of the forces on the section of the diamond wire saw during slicing is shown in Fig. 16.

Fig. 16
figure 16

Schematic of forces on the section of the diamond wire saw

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The feed direction of the diamond wire saw is -X. The thickness direction of the slice is the Z axis. The fnij is the normal force of the diamond grit. On the section of the diamond wire saw, the resultant force of fnij is fn. The component of fn in the Z direction is fnZ. While the elastic modulus on both sides of the feed direction on the saw wire section is symmetrical, the fnZ would be zero. In this condition, the diamond wire saw would have no deviation in the Z direction. As a result, the anisotropy of KDP crystal would have no influence on the surface shape deviation of the slice.

4.1.1 Distribution of elastic modulus in slicing of the (001) crystal plane

In slicing of the (001) crystal plane with the feed angle γ1, elastic modulus of KDP crystal can be obtained according to Eq. (6). If the properties of the KDP crystal are symmetric about the feeding direction, Eq. (29) needs to be established.

$$ E\left({\theta}_L\right)=E\left(-{\theta}_L\right) $$
(29)

The result can be obtained by substituting Eq. (6) into Eq. (29). It can be noticed that Eq. (29) can be satisfied for all feed angles of (001) crystal plane. In slicing of the (001) crystal plane, variation of the elastic modulus with position angle θL can be drawn according to Eq. (6), as shown in Fig. 17.

Fig. 17
figure 17

Variation of E with position angle θL for (001) crystal plane

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It can also be noticed from Fig. 17 that the elastic modules are symmetrical about the 0 degree while the feed angle γ1 has different values. For any feed angle of the diamond wire saw, the elastic modulus in the diamond wire saw section is symmetrical about the feed direction. According to the Neumann principle in crystal physics, the symmetry of crystal physical properties is contained in its microstructure. Once the elastic modulus is symmetrical, other physical properties of the KDP crystal would also be symmetrical.

In conclusion, the anisotropic properties of KDP crystal have no effect on the surface shape deviation of the (001) crystal plane in slicing by the diamond wire saw.

4.1.2 Distribution of elastic modulus in slicing of the double-frequency crystal plane

The elastic modulus of KDP crystal in slicing of the double-frequency plane with the feed angle γ2 can be obtained according to Eq. (11). If the properties of the KDP crystal are symmetric about the feeding direction, Eq. (29) needs to be established.

The result can be got by substituting Eq. (11) into Eq. (29) that Eq. (29) can be satisfied while the γ2 is 90° or 270°.

In slicing of the double-frequency crystal plane, variation of the elastic modulus with position angle θL can be drawn according to Eq. (11), as shown in Fig. 18.

Fig. 18
figure 18

Variation of E with θL for the double-frequency crystal plane

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It can be seen from Fig. 18 that the distribution of E on the cross section of the diamond wire saw is symmetric about the 0 degree while γ2 has a value of 90°. The distribution of E would be asymmetric while the γ2 has other values.

That is to say, anisotropic properties of KDP crystal would affect the surface shape deviation of the double-frequency plane in slicing by the diamond wire saw. The degree of influence would be different while γ2 has different values.

4.1.3 Distribution of elastic modulus in slicing of the triple-frequency crystal plane

In slicing of the triple-frequency crystal plane with the feed angle γ3, elastic modulus of KDP crystal can be got according to Eq. (16). If the properties of the KDP crystal are symmetric about the feeding direction, Eq. (29) needs to be established.

The result can be got by substituting Eq. (16) into Eq. (29) that Eq. (29) can be satisfied while γ3 has values of 90° or 270°.

In slicing of the triple-frequency crystal plane, variation of the elastic modulus with position angle θL can be drawn according to Eq. (16), as shown in Fig. 19.

Fig. 19
figure 19

Variation of E with θL for the triple-frequency crystal plane

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It can be seen from Fig. 19 that the distribution of E on the cross section of the diamond wire saw is symmetric about the 0° while the γ3 has a value of 90°. The distribution of E would be asymmetric while the γ3 has other values.

That is to say, the anisotropic properties of KDP crystal would affect the surface shape deviation of the triple-frequency plane in slicing by the diamond wire saw. The degree of influence would be different while γ3 has different values.

4.1.4 Degree of anisotropic properties of KDP crystal

To study the degree of anisotropic properties of KDP crystal, the fnz needs to be considered. The fnz is a component of the normal force in the Z direction, and it can be got by Eq. (30).

$$ {f}_{nz}=\pi {H}_1{h}_{ij}^2{\left(\tan {\theta}_{ij}\right)}^2\sin {\theta}_L/2 $$
(30)

The sine values of the elastic modulus at both sides of the feed direction are integrated respectively, and the difference of the integration results is defined as the degree of anisotropy Ed of the elastic modulus.

$$ {E}_d=\left(\left|{\int}_{-\pi /2}^0E\sin {\theta}_Ld{\theta}_L\right|-\left|{\int}_0^{\pi /2}E\sin {\theta}_Ld{\theta}_L\right|\right)/\left|{\int}_{-\pi /2}^{\pi /2}E\sin {\theta}_Ld{\theta}_L\right| $$
(31)

As shown in Eq. (31), Ed can be regarded as a degree of anisotropy of the KDP crystal. The Ed is used to characterize the degree of influence of anisotropy on the slice. While Ed has small values, the anisotropy of KDP crystal would have a small effect on the slice and vice versa. According to Eq. (31), variation of Ed with the feed angle of the diamond wire saw of three crystal planes can be obtained, as shown in Fig. 20.

Fig. 20
figure 20

Variations of Ed with γ of three crystal planes

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It can be observed from Fig. 20 that, for the (001) crystal plane, the Ed would be zero regardless of the feed angle of the diamond wire saw.

For the double-frequency crystal plane, the Ed would have peak values while the feed angle of diamond wire saw has values of 0° or 180°. While the feed angle is in the range of 60° to 120° and 240° to 300°, the corresponding Ed is close to zero.

For the triple-frequency crystal plane, the Ed would have peak values while the feed angle of diamond wire saw has values of 30°, 150°, 210°, and 330°. While the feed angle is 90° or 270°, the corresponding Ed would be zero. At this time, the anisotropy would have no effect on the surface shape deviation.

4.2 Influence of γ on surface shape deviation

For the (001) crystal plane, material properties are symmetrical to the feed direction of the diamond wire saw, as analyzed in Section 4.1. So anisotropy has no effect on the surface shape deviation of the slice.

For the double-frequency crystal plane and the triple-frequency crystal plane, anisotropy would have an effect on the surface shape deviation of the slice. The influence rules of anisotropic properties of KDP crystal on the surface shape deviation of slice for the double-frequency crystal plane and the triple-frequency crystal plane are analyzed.

4.2.1 Influence of γ 2 on surface shape deviation for the double-frequency crystal plane

For the double-frequency crystal plane, variations of the maximum value of the surface shape deviation of slice dZmax with the cutting angle of the diamond wire saw γ2 are shown in Fig. 21.

Fig. 21
figure 21

Variations of dZmax with γ2 for the double-frequency crystal plane

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It can be seen from Fig. 21 that the change rule of dZmax with γ2 is similar to that of Ed in Fig. 20. For the double-frequency crystal plane, the change rule of dZmax with γ2 is as follows.

The dZmax has large values while γ2 has values of 0° or 180°. At this time, the anisotropy of KDP crystal has the largest effect on the surface shape deviation. The dZmax would have small values close to zero, when the γ2 is in the range of 60°–120° or 240°–300°. At this time, the anisotropy of KDP crystal would have a small effect on the surface shape deviation. It can also be observed that the dZmax would decrease with the increase of the tension force. While for a certain crystal plane, the change rule of the dZmax with the feed angle of the diamond wire saw is the same regardless of the other parameters, such as the size of KDP crystal and the tension force of the diamond wire saw.

4.2.2 Influence of γ 3 on the surface shape deviation for the triple-frequency crystal plane

For the triple-frequency crystal plane, variations of the maximum value of the surface shape deviation of slice dZmax with the cutting angle of the diamond wire saw γ3 are shown in Fig. 22.

Fig. 22
figure 22

Variations of dZmax with γ3 for the triple-frequency crystal plane

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It can also be seen from Fig. 22 that the change rule of dZmax with γ3 is similar to that of Ed in Fig. 20. For the triple-frequency crystal plane, the change rule of dZmax with γ3 is as follows.

When γ3 is about 30°, 150°, 210°, or 330°, the dZmax would have peak values. At this time, the anisotropy of KDP crystal has the largest effect on the surface shape deviation. The dZmax would be zero, when the γ3 has values of 90° or 270°. At this time, the anisotropy of KDP crystal would have no effect on the surface shape deviation. It can also be observed that the dZmax decreases with the increase of tension force while the change rule of the dZmax with the γ3 is the same regardless of the tension force.

5 Conclusion

In this paper, the anisotropy of KDP crystal in slicing by the diamond wire saw is analyzed through coordinate changes. A model of diamond wire saw considering the anisotropy of KDP crystal is established to obtain the sawing force. Then the effect of anisotropy on surface shape deviation of the slice is studied. Conclusions can be summarized as follows:

1 Distributions of elastic modulus in slicing of the (001) crystal plane, the double-frequency crystal plane, and the triple-frequency crystal plane are obtained. The difference of integration values of the elastic modulus at both sides of the feed direction is chosen to characterize the degree of influence of anisotropy on the surface shape deviation of slice.

2 In slicing of KDP crystal by the diamond wire saw, the sawing force is obtained by simulation while the anisotropic properties are considered. The surface shape deviation of slice is obtained by the sawing force. The effect of anisotropy of KDP crystal on surface shape deviation of slice is got.

3 For the (001) crystal plane, the anisotropic properties of KDP crystal have no effect on the surface shape deviation of slice, regardless of the feed angle of the diamond wire saw.

4 For the double-frequency crystal plane, the anisotropic properties would affect the surface shape deviation of slice. The surface shape deviation of slice would have small values close to zero, when the feed angle of the diamond wire saw is in the range of 60°–120° and 240°–300°.

5 For the triple-frequency crystal plane, the anisotropic properties would affect the surface shape deviation of slice. While the feed angle of the diamond wire saw is 90° or 270°, the surface shape deviation of slice would have the smallest value.