3.1. Analysis of the cutting state of the grinding tool and workpiece
The grinding process in the tool-workpiece system can be considered as the combination of the cutting action of the whole diamond grains. A clear understanding of the interaction between the diamond grains and workpiece can be obtained by analyzing the cutting state and dynamic trajectory of a single diamond.
The schematic diagram of side grinding is shown in Fig. 3(a). The directions of feeding, cutting, and the tool axis are set as the x-axis, y-axis, and z-axis, respectively. Fx and Fy are the average forces measured by the dynamometer in the x and y directions in the grinding process, respectively; Fm is the combined force of these two forces used to evaluate the cutting forces in the entire grinding process; vf is the feed rate of the tool, mm/s; ω is the angular velocity of the spindle, rad/s. ω = 2πn, and n is the spindle rotation speed, r/min. The thickness of the workpiece is denoted as hm, mm. The diamond grains are considered octahedrons to simplify the modelling, as shown in Fig. 3(b). θ is the semi-angle between two opposite edge of the diamond grains. Sa is the edge length of the diamond grains.
However, the diamond grains on the tool are randomly distributed, and the protruding height is not uniform, as shown in Fig. 4(a). O represents the tool center, and Rt is the radius of the tool substrate. The random protrusion height of diamond grains is δgi. Fig. 4(b) shows the plane expansion of the diamond grains on the tool surface. The grain height is distributed around the average height. Therefore, the probability density function is required to describe the random protruding height characteristics of the diamond grains to better analyse the interaction between the diamond grains and workpiece and conveniently establish the cutting force model. The protruding height of diamond grains on the tool surface conforms to Rayleigh distribution [3]:
\(f\left( {{\delta _{gi}}} \right)=\left\{ {\begin{array}{*{20}{l}} {\left( {{\delta _{gi}}/{\beta ^2}} \right){e^{ - \left( {\delta _{{gi}}^{2}/2{\beta ^2}} \right)}},{\delta _{gi}} \geqslant 0} \\ {{\text{ }}0{\text{ , }}{\delta _{gi}}<0} \end{array}} \right.\)
(14)
where β is parameter defined by the probability density function.
The average protrusion height of diamond grains is δga, as shown in Fig. 4(c). Ra is the average tool radius used to calculate the diamond grains trajectory. Ra =Rt + δga; δga can be determined by the expectation of the Rayleigh function. According to the probability statistics method, the expected and variance can be obtained by the following equations:
\(\begin{gathered} E\left( {{\delta _{gi}}} \right)=\beta \sqrt {\frac{\pi }{2}} \approx {\text{1}}{\text{.253}}\beta {\text{ }} \hfill \\ Var\left( {{\delta _{gi}}} \right)=\left( {2 - \frac{\pi }{2}} \right){\beta ^2} \approx 0.429{\beta ^2} \hfill \\ E\left( {\delta _{{gi}}^{2}} \right)=\int_{0}^{\infty } {\delta _{{gi}}^{2}} f\left( {{\delta _{gi}}} \right)d{\delta _{gi}} \hfill \\ \end{gathered}\)
(15)
Substituting Eq. (14) into Eq. (15), it is given as follows:
\(E\left( {\delta _{{gi}}^{2}} \right)=\left[ {{e^{ - \delta _{{gi}}^{2}/2{\beta ^2}}}\left( { - 2{\beta ^2} - \delta _{{gi}}^{2}} \right)} \right]_{0}^{\infty }=2{\beta ^2}\)
(16)
The total removal volume of all diamond grains should be equal to the removal volume of the grinding process, which can be expressed as follows:
\(E\left( {\delta _{{gi}}^{2}} \right)=\frac{{2{a_e}{v_f}}}{{\pi {C_e}{l_c}{v_s}}}\)
(17)
where Ce is the number of active diamond grains per square millimetre, Ce = 5 in this paper; vs is the cutting speed of a single diamond grain, mm/s; ae is the cutting width, mm; lc is grinding tool/workpiece arc length of contact, which can be expressed as follows:
\({l_c}=\sqrt {2{R_a} \cdot {a_e}}\)
(18)
Therefore, δga can be obtained by the following expression:
\({\delta _{ga}}=E\left( {{\delta _{gi}}} \right)=\sqrt {\frac{{{a_e}{v_f}}}{{2{C_e}{l_c}{v_s}}}}\)
(19)
As shown in Fig. 4(d), after the average of the random protruding height, the randomly distributed diamond grains on the tool have a uniform height, consistent with the height of the average height plane, thus simplifying the grinding behaviour generated by the random protruding height, beneficial to the complete description of the processed surface.
The cutting mechanism of a single diamond grain is shown in Fig. 5(a1). Based on research on indentation and scratch experiments of hard and brittle materials, there are three cutting stages with a gradual increase of the normal cutting force Fn of a single diamond grain: ductile stage (stage I), ductile-to-brittle transition stage (stage II), and brittle stage (stage III). The normal and tangential cutting forces are represented by Fn and Ft, respectively. Fp is the resultant of these two forces, the theoretical cutting force, as shown in Fig. 5(a2).
According to the definition of Vickers hardness, the normal cutting force can be obtained by the following equation [27]:
\({F_n}=\xi H{a^2}\)
(20)
where ξ is the geometrical factor of the indenter, ξ ≈ 1.885; H denotes Vickers hardness, which is 20 GPa; a denotes the indentation size, a = hgtanθ, where hg is the cutting depth of a single diamond grain.
Therefore, the three cutting stages depend on hg, influencing the local contact deformation and material removal mechanism.
When 0<hg1<hgp, the cutting stage is the ductile stage (stage I), as shown in Fig. 5(b1). Here, the contact area between the diamond grain and workpiece is mainly the plastic deformation area caused by the pressure of the diamond grain, and no obvious cracks are present. Therefore, the materials are removed through plastic flow, and the tangential direction of the diamond grain is mainly affected by the rubbing force. The critical state of the ductile-brittle transition refers to the state in which the crack generated by the last diamond grain is immediately removed by the next grain. The critical depth can be expressed as follows [28]:
\({h_{gp}}=\tau {h_{gc}}\)
(21)
where hgp is the critical depth between stage I and stage II; hgc is the critical depth between stage II and stage III; τ is coefficients of the ductile stage, which is 0.25 in this paper.
When hgp<hg2<hgc, the cutting stage is the ductile-brittle transition stage (stage II), as shown in Fig. 5(b2). At this stage, the plastic zone beneath the diamond gradually expands. The median crack begins to appear beneath the plastic zone, which is usually related to strength degradation. The median crack occurs in the loading and unloading process. Unloading and tool wear result in uneven local stress distribution along the grinding path, and lateral cracks occurred during unloading. The residual stress component is the main source of crack propagation, and the tangential direction of the diamond grain is mainly subjected to ploughing force. The critical depth between stage II and stage III is related to the material properties, which can be expressed as follows [12]:
\(h_{{gc}}^{ * }=\psi \left( T \right)\left( {\frac{{{E_e}}}{H}} \right){\left( {\frac{{{K_{IC}}}}{H}} \right)^2}\)
(22)
where KIC is the static fracture toughness of the material, which is 15.5 MPa/m1/2; ψ(T) is a function of temperature, which is given as follows:
\(\psi \left( T \right)=0.52+0.85\exp \left( { - T/{T_0}} \right){\text{ }}{T_0}=251.1^\circ C\)
(23)
where T is ambient temperature, ℃.
However, based on previous reports, using static fracture toughness in dynamic processing is not appropriate. The dynamic fracture toughness KID is approximately 30% of KIC [29]. Therefore, substituting KID for KIC can better conform to the actual grinding process and provide more accurate theoretical guidance for modeling, and the critical depth hgc can be written as follows:
\({h_{gc}}=\psi \left( T \right)\left( {\frac{{{E_e}}}{H}} \right){\left( {\frac{{{K_{ID}}}}{H}} \right)^2}\)
(24)
When hgc<hg3, the cutting stage is the brittle transition stage (stage III), as shown in Fig. 5(b3). At this stage, the plastic zone beneath the diamond grain expands further. Continuous crack branches occur beneath the plastic zone, and obvious transverse cracks occur in the workpiece and extend to the ground surface, resulting in material detritus and a large amount of removal. The diamond grains are subjected to tangential and normal loads. The tangential cutting force causes expansion of transverse cracks, reducing the surface quality of the workpiece and improving the removal rate. The lengths of the median crack, lateral crack, and plastic zone, denoted as Cm, Cl and Ch, respectively, can be expressed as follows [30]:
\(\begin{gathered} {C_m}=\eta _{1}^{{2/3}}{\left( {\frac{{{E_e}}}{H}} \right)^{1/3}}{\cos ^{4/9}}\theta {\left[ {\frac{{{\chi _e}}}{{{\chi _r}}}\left( {\frac{{{F_{t3}}}}{{{K_{ID}}}}} \right)+\left( {\frac{{{F_{n3}}}}{{{K_{ID}}}}} \right)} \right]^{2/3}} \hfill \\ {C_l}={\eta _2}{\cot ^{5/12}}\theta {\left[ {\frac{{E_{e}^{{3/4}}}}{{H{K_{ID}}{{\left( {1 - \upsilon _{e}^{2}} \right)}^{1/2}}}}} \right]^{1/2}}F_{{n3}}^{{5/8}} \hfill \\ {C_h}={\eta _3}{\cot ^{1/3}}\theta \frac{{E_{e}^{{1/2}}}}{H}F_{{n3}}^{{1/2}} \hfill \\ \end{gathered}\)
(25)
where χe and χr are the indentation coefficients of the elastic stress field and residual stress field, respectively (χe = 0.032, χr = 0.026); η1, η2 and η3 are the dimensional constants (η1 = 0.0366, η2 = η3 = 0.226).
The cutting depth increases gradually from the time when a single diamond grain touches the workpiece to when it is removed. The maximum undeformed chip thickness hgm can be obtained by the following expression [31]:
\({h_{gm}}=2{\delta _{ga}}{\left( {\frac{{{a_e}}}{{2{R_a}}}} \right)^{1/4}}\sqrt {\frac{{{v_f}}}{{{v_s}{\eta _c}f}}{{\left( {\frac{{4\pi }}{{3{V_d}}}} \right)}^{2/3}}}\)
(26)
where ηc is the ratio of chip width to average undeformed chip thickness, ηc = 10; f is the fraction of diamond grains that actively cut during grinding, f = 0.5; The grinding tool used in this study has a concentration of 100 or volume fraction of Vd = 0.25.
3.3. Cutting force model of a single diamond grain under three cutting stages
3.3.1. Kinematical analysis of a single diamond grain
The dynamic trajectory of a single diamond grain was analysed dynamically to clarify the side grinding process, and its position can be expressed by machining parameters. The three cutting stages of the ground region from A to A’ is shown in Fig. 6. The position and velocity of a single diamond grain in the grinding process can be expressed as follows:
\(\left\{ {\begin{array}{*{20}{l}} {x={R_a}\sin (2\pi nt)+{v_f}t} \\ {y={R_a}\cos (2\pi nt)} \\ {z=0} \end{array}} \right.{\text{ }}\left\{ {\begin{array}{*{20}{l}} {{v_x}=2\pi n{R_a} \cdot \cos (2\pi nt)+{v_f}} \\ {{v_y}= - 2\pi n{R_a} \cdot \sin (2\pi nt)} \\ {{v_z}=0} \end{array}} \right.\)
(27)
where t is the cutting time, s.
The moment when a single diamond grain touches the workpiece is t0, t0=0 for convenience. At stage I, the tool center moves from O to O1, and the corresponding cutting time is t0-t1. The corresponding cutting time is t1-t2 and t2-t3 when the tool center moves from O1 to O2 at stage II and from O2 to O3 at stage III, respectively. The cutting lengths of a single diamond grain during the grinding process can be expressed as follows:
\(\begin{gathered} {l_1}=\int_{{{t_0}}}^{{{t_1}}} {\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}} } dt \hfill \\ {l_2}=\int_{{{t_1}}}^{{{t_2}}} {\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}} } dt \hfill \\ {l_3}=\int_{{{t_2}}}^{{{t_3}}} {\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}} } dt \hfill \\ \end{gathered}\)
(28)
where l1, l2, and l3 are the cutting lengths at stage I, stage II, and stage III, respectively.
The feeding distance and cutting time of a single diamond grain can be obtained by the following expression:
\(\begin{array}{*{20}{c}} {{x_1}={v_f}{t_1}{\text{ }}}&{{x_2}={v_f}{t_2}}&{{x_3}={v_f}{t_3}} \\ {{\gamma _1}=\omega {t_1}}&{{\gamma _2}=\omega {t_2}}&{{\gamma _3}=\omega {t_3}} \\ {{t_1}=\frac{{{h_{gp}}}}{{{v_f}\sin {\gamma _1}}}}&{{t_2}=\frac{{{h_{gc}}}}{{{v_f}\sin {\gamma _2}}}}&{{t_3}=\frac{{{h_{gm}}}}{{{v_f}\sin {\gamma _3}}}} \end{array}\)
(29)
where x1, x2, and x3 are the feeding distance at stage I, stage II, and stage III, respectively. γ1, γ2, and γ3 are the rotation angle of the tool and the corresponding cutting time.
3.3.2. Cutting force model of a single diamond grain at the ductile stage
As the cutting depth hg1 of a single diamond grain is constantly changing with time at the ductile stage, the average cutting depth needs to be determined. The material removal volume of a single diamond grain at stage I, denoted as Vg1, can be considered as a triangular pyramid volume with the following expression:
\({V_{g1}}=\frac{1}{3}{l_1} \times {h_{gp}} \times {h_{gp}}\tan \theta =\frac{1}{3}{l_1}h_{{gp}}^{2}\tan \theta\)
(30)
The equivalent removal volume of a single diamond grain at the ductile stage can be idealized as a triangular prism, denoted as Vga1 with the following expression:
\({V_{ga1}}={l_1} \times {h_{ga1}} \times {h_{ga1}}\tan \theta ={l_1}h_{{ga1}}^{2}\tan \theta\)
(31)
Then Vg1 = Vga1, and the average cutting depth at stage I, denoted as hga1, can be expressed as follows:
\({h_{ga1}}=\frac{{\sqrt 3 }}{3}{h_{gp}}\)
(32)
The connection between the average normal cutting force Fn1 and average cutting depth hga1 at stage I can be expressed as follows [32]:
\({F_{n1}}=2Hh_{{ga1}}^{2}\tan \theta {\left( {2+{{\tan }^2}\theta } \right)^{1/2}}\)
(33)
Substituting Eqs. (21) and (32) into Eq. (33), the average normal cutting force Fn1 can be expressed as follows:
\({F_{n1}}=\frac{{Hh_{{gc}}^{2}\tan \theta }}{{24}}{\left( {2+{{\tan }^2}\theta } \right)^{1/2}}\)
(34)
At stage I, plastic deformation of the workpiece begins at yield criterion point with the increase of cutting depth, and the single diamond grain is mainly subjected to the rubbing force Ft1 in the tangential direction, which can be expressed as follows:
\({F_{t1}}=\mu {F_{n1}}\)
(35)
where µ is the friction coefficient.
The friction coefficient of the diamond grain can be approximated to the ratio of the projected areas in the cutting and tangential directions [33], which is given as follows:
\(\mu =\frac{{{S_{t1}}}}{{{S_{n1}}}}\)
(36)
where St1 and Sn1 are the tangential and normal projection areas of the diamond grain at stage I, respectively, and they can be expressed as follows:
\(\begin{gathered} {S_{t1}}=\frac{1}{2} \times 2 \times {h_{g1}}\tan \theta \times {h_{g1}}=h_{{g1}}^{2}\tan \theta \hfill \\ {S_{n1}}={\left( {\sqrt 2 \times {h_{g1}}\tan \theta } \right)^2}=2h_{{g1}}^{2}{\tan ^2}\theta \hfill \\ \end{gathered}\)
(37)
Substituting Eqs. (35), (36) and (37) into Eq. (34), the average rubbing force Ft1 can be expressed as follows:
\({F_{t1}}=\frac{{Hh_{{gc}}^{2}\tan \theta }}{{48}}{\left( {2+{{\tan }^2}\theta } \right)^{1/2}}\)
(38)
Finally, the total average cutting force at the ductile stage can be expressed as follows:
\({F_1}=\sqrt {F_{{n1}}^{2}+F_{{t1}}^{2}} {\text{=}}\frac{{\sqrt 5 Hh_{{gc}}^{2}\tan \theta }}{{48}}{\left( {2+{{\tan }^2}\theta } \right)^{1/2}}\)
(39)
3.3.3. Cutting force model of a single diamond grain at the ductile-brittle transition stage
The material removal volume of the single diamond grain at stage II, denoted as Vg2, can be obtained by the following expression:
\({V_{g2}}=\frac{1}{3}\left( {{l_1}+{l_2}} \right) \times {h_{gc}} \times {h_{gc}}\tan \theta - \frac{1}{3}{l_1} \times {h_{gp}} \times {h_{gp}}\tan \theta\)
(40)
The removed volume of this stage is equivalent to the volume of the triangular prism, denoted as Vga2, can be expressed as follows:
\({V_{ga2}}={l_2} \times {h_{ga2}} \times {h_{ga2}}\tan \theta ={l_2}h_{{ga2}}^{2}\tan \theta\)
(41)
Then Vg2 = Vga2, and the average cutting depth at stage II, denoted as hga2, can be expressed as follows:
\({h_{ga2}}={h_{gc}}{\left( {\frac{{15{l_1}+16{l_2}}}{{48{l_2}}}} \right)^{1/2}}\)
(42)
Under the same normal load, the crack depth of multiple diamond grinding is approximately half of that of single diamond grinding. The connection between the average normal cutting force Fn2 and average cutting depth hga2 at stage II can be expressed as follows [18]:
\({F_{n2}}=\frac{1}{2}\xi Hh_{{ga2}}^{2}{\tan ^2}\theta\)
(43)
Substituting Eq. (42) into Eq. (43), the average normal cutting force Fn2 can be expressed as follows:
\({F_{n2}}=\frac{1}{2}\xi Hh_{{gc}}^{2}\left( {\frac{{15{l_1}+16{l_2}}}{{48{l_2}}}} \right){\tan ^2}\theta\)
(44)
The single diamond grain is mainly subjected to the ploughing force Ft2 in the tangential direction at this stage, which can be expressed as follows:
\({F_{t2}}={\sigma _s}{S_{t2}}\)
(45)
where σs is the compressive yield stress at the contact area, which is defined as follows [34]:
\({\sigma _s}={\left( {\frac{{{H^4}}}{{{E_e}}}} \right)^{1/3}}\)
(46)
and St2 is the projected area of the tangential direction of the diamond grain, which is related to the cutting depth. The projected area can be calculated by the average cutting depth as follows:
\({S_{t2}}=\frac{1}{2} \times 2{h_{ga2}}\tan \theta \times {h_{ga2}}=h_{{ga2}}^{2}\tan \theta\)
(47)
Substituting Eqs. (42), (46), and (47) into Eq. (45), the average ploughing force can be expressed as follows:
\({F_{t2}}=\frac{{h_{{gc}}^{2}\left( {15{l_1}+16{l_2}} \right)\tan \theta }}{{48{l_2}}}{\left( {\frac{{{H^4}}}{{{E_e}}}} \right)^{1/3}}\)
(48)
Finally, the total average cutting force at the ductile-brittle transition stage is given as follows:
\({F_2}=\sqrt {F_{{n2}}^{2}+F_{{t2}}^{2}} {\text{=}}\frac{{h_{{gc}}^{2}H\tan \theta \left( {15{l_1}+16{l_2}} \right)\sqrt {4{H^{2/3}}+{\xi ^2}E_{e}^{{2/3}}{{\tan }^2}\theta } }}{{96{l_2}E_{e}^{{1/3}}}}\)
(49)
3.3.4. Cutting force model of a single diamond grain at the brittle stage
The material removal volume of the single diamond grain at stage III, denoted as Vg3, can be obtained by the following expression:
\({V_{g3}}=\frac{1}{3}{l_c} \times {h_{gm}} \times {h_{gm}}\tan \theta - \frac{1}{3}\left( {{l_1}+{l_2}} \right) \times {h_{gc}} \times {h_{gc}}\tan \theta\)
(50)
The removed volume of this stage is equivalent to the volume of the triangular prism, denoted as Vga3, can be expressed as follows:
\({V_{ga3}}={l_3} \times {h_{ga3}} \times {h_{ga3}}\tan \theta ={l_3}h_{{ga3}}^{2}\tan \theta\)
(51)
Then Vg3 = Vga3, and the average cutting depth at stage III, denoted as hga3, can be expressed as follows:
\({h_{ga3}}={\left( {\frac{{h_{{gm}}^{2}{l_c} - h_{{gc}}^{2}\left( {{l_1}+{l_2}} \right)}}{{3{l_3}}}} \right)^{1/2}}\)
(52)
The average normal cutting force Fn3 and tangential cutting force Ft3 of a single diamond grain at stage III can be obtained by the following expression [35]:
\(\begin{gathered} {F_{n3}}=\frac{1}{{{\eta _2}}}h_{{ga3}}^{2}{H^2}{\tan ^{8/3}}\theta \left[ {\frac{{3\left( {1 - 2{\upsilon _e}} \right)}}{{{E_e}\left( {5 - 4{\upsilon _e}} \right)}}+\frac{{2\sqrt 3 \cot \theta }}{{\pi \left( {5 - 4{\upsilon _e}} \right){\sigma _s}}}} \right] \hfill \\ {F_{t3}}=\frac{{{C_l}}}{{{\eta _2}{C_h}}}h_{{ga3}}^{2}{H^2}{\tan ^{8/3}}\theta \left[ {\frac{{3\left( {1 - 2{\upsilon _e}} \right)}}{{{E_e}\left( {5 - 4{\upsilon _e}} \right)}}+\frac{{2\sqrt 3 \cot \theta }}{{\pi \left( {5 - 4{\upsilon _e}} \right){\sigma _s}}}} \right] \hfill \\ \end{gathered}\)
(53)
The total average cutting force at the brittle stage can be expressed as follows:
\({F_3}=\sqrt {F_{{n3}}^{2}+F_{{t3}}^{2}}\)
(54)
Substituting Eqs. (52) and (53) into Eq. (54), the total average cutting force F3 is given as follows:
\({F_3}=\frac{{{H^2}{{\tan }^{8/3}}\theta \sqrt {C_{l}^{2}+C_{h}^{2}} }}{{3{\eta _2}{l_3}{C_h}}}\left[ {h_{{gm}}^{2}{l_c} - h_{{gc}}^{2}\left( {{l_1}+{l_2}} \right)} \right]\left[ {\frac{{3\left( {1 - 2{\upsilon _e}} \right)}}{{{E_e}\left( {5 - 4{\upsilon _e}} \right)}}+\frac{{2\sqrt 3 \cot \theta }}{{\pi \left( {5 - 4{\upsilon _e}} \right){\sigma _s}}}} \right]\)
(55)
3.4. Final cutting force model of side grinding of orthogonal laminated SiCf/SiC composites
The final average cutting force Fs is a combination of the total average cutting force at the three cutting stages, which can be obtained by the following expression:
\({F_s}=\frac{{{l_1}{F_1}+{l_2}{F_2}+{l_3}{F_3}}}{{{l_c}}}\)
(56)
The total theoretical removal volume rate of a single diamond in the grinding process, denoted as Vs, depends on the amount of interference between the diamond and workpiece at stage I and stage II and the propagation of transverse cracks at stage III; Vs can be expressed as follows:
\({V_s}={l_1}h_{{ga1}}^{2}\tan \theta +{l_2}h_{{ga2}}^{2}\tan \theta +2{l_3}{C_h}{C_l}\)
(57)
The total material removal volume rate in a rotation period, denoted as Vm, can be expressed as follows:
\({V_m}={l_c}{h_m}{h_{gm}}\)
(58)
Therefore, the number of theoretical contact diamond grains between the tool and workpiece in the grinding process, denoted as Ca, can be expressed as follows:
\({C_a}=\frac{{{V_m}}}{{{V_s}}}=\frac{{3{h_m}{h_{gm}}{l_c}}}{{h_{{gc}}^{2}\tan \theta \left( {{l_1}+{l_2}} \right)+6{C_h}{C_l}{l_3}}}\)
(59)
However, the random distribution of diamond grains and the overlap and interference of different diamond grains will affect the total material removal volume and cutting force. Therefore, the correction coefficient λ is introduced. The final theoretical cutting force in the grinding process is given as follows:
\({F_p}=\lambda {C_a}{F_s}=\frac{{3\lambda {h_m}{h_{gm}}\left( {{l_1}{F_1}+{l_2}{F_2}+{l_3}{F_3}} \right)}}{{h_{{gc}}^{2}\tan \theta \left( {{l_1}+{l_2}} \right)+6{C_h}{C_l}{l_3}}}\)
(60)
Substituting Eqs. (39), (49), (55), and (59) into Eq. (60), the final theoretical cutting force Fp is given as follows:
\({F_p}=\frac{{\lambda {h_m}{h_{gm}}H\left[ {0.333h_{{gc}}^{2}{l_1}{C_h}+0.298h_{{gc}}^{2}{l_2}{C_h}+0.684\sqrt {C_{h}^{2}+C_{l}^{2}} \left[ {h_{{gm}}^{2}{l_c} - h_{{gc}}^{2}\left( {{l_1}+{l_2}} \right)} \right]} \right]}}{{{C_h}\left[ {\sqrt 3 h_{{gc}}^{2}\left( {{l_1}+{l_2}} \right)+6{C_h}{C_l}{l_3}} \right]}}\)
(61)