**3. The chip separation criterion on different materials used in the FEM**

Presently, two FE methods exist for analyzing the precision machining process. In the first method, it is assumed that the chip formation is continuous and the shape of the chip is known in advance. Thus, the process is analyzed as a steady-state process. This method is called Eulerian method. In this method, a chip separation criterion is not required. In the second method, the process is analyzed from the beginning to the steady state chip formation. This is called Updated Lagrangian Formulation. In this method, a chip separation criterion is required to predict the chip geometry. Early applications of finite element method to the machining process were mainly Eulerian method. The main objective of many of these studies was to predict the temperature distribution and therefore, the determination of deformation and stress fields was only an intermediate step. These studies considered the machined material as rigid-plastic. But, later applications of Eulerian formulation to machining process also included viscoplastic effects. All of these applications have considered only orthogonal machining. The first finite element study of the machining process using an modified Lagrangian Formulation was made by Strenkowski and Carrol[8]. A critical value of the equivalent plastic strain was used to model the separation of a chip. Later on, several researchers used the Updated Lagrangian Formulation for analyzing twoand three-dimensional machining processes. The criterion used for chip separation has been based on controlled crack propagation or some geometrical considerations. Remeshing technique has been used to simulate the chip formation.

As the size of the material removed decreases in the precision machining, the probability of encountering a stress-reducing defect decreases. There are some new disciplines dominate the chip separation process. The metal cutting process is different from general metal forming process as there are always accompanied with chip separation or materials removal phenomenon. The separation of chip is of utmost important about numerical simulation of precision machining. The simulation results can only be meaningful only if the reasonable chip separation criteria which can reflect materials mechanical and physical property (such as morphology of chip, force, temperature and the residual stress etc.) were applied in the simulation model. Besides, the criterion for chip separation should be invariant for definite materials but not change with the different working conditions. In the metal cutting process, some kinds of materials may generate continuous chip while others may generate saw-like chip thus different materials fracture criteria should be included in the finite element model.

Presently, there are two kinds of chip separation criteria, namely, the geometric criterion and the physical criterion. Materials removal (chip separation) using geometric criterion is realized through the variation of size of deformable body. On the other hand, the physical criterion is based on if some key physical parameters approached the critical value, these physical criterion includes effective plastic strain criterion, strain energy density criterion and the fracture stress criterion and so on.

### **3.1 Fracture mechanics criterion**

### **3.1.1 Stress intensity factor**

108 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

called primarily deformation zone where the complete plastic deformation of the work materials takes place can hardly exceed 250 oC. It is understood that the mechanical properties of the work material obtained at room temperature are not affected by this temperature so metal cutting is a cold working process, although the chip appearance can be cherry-red. Fourth, it is completely unclear how to correlate the properties of the work materials obtained in SHPB uniaxial impact testing with those in metal cutting with a strong

**3. The chip separation criterion on different materials used in the FEM** 

technique has been used to simulate the chip formation.

should be included in the finite element model.

Presently, two FE methods exist for analyzing the precision machining process. In the first method, it is assumed that the chip formation is continuous and the shape of the chip is known in advance. Thus, the process is analyzed as a steady-state process. This method is called Eulerian method. In this method, a chip separation criterion is not required. In the second method, the process is analyzed from the beginning to the steady state chip formation. This is called Updated Lagrangian Formulation. In this method, a chip separation criterion is required to predict the chip geometry. Early applications of finite element method to the machining process were mainly Eulerian method. The main objective of many of these studies was to predict the temperature distribution and therefore, the determination of deformation and stress fields was only an intermediate step. These studies considered the machined material as rigid-plastic. But, later applications of Eulerian formulation to machining process also included viscoplastic effects. All of these applications have considered only orthogonal machining. The first finite element study of the machining process using an modified Lagrangian Formulation was made by Strenkowski and Carrol[8]. A critical value of the equivalent plastic strain was used to model the separation of a chip. Later on, several researchers used the Updated Lagrangian Formulation for analyzing twoand three-dimensional machining processes. The criterion used for chip separation has been based on controlled crack propagation or some geometrical considerations. Remeshing

As the size of the material removed decreases in the precision machining, the probability of encountering a stress-reducing defect decreases. There are some new disciplines dominate the chip separation process. The metal cutting process is different from general metal forming process as there are always accompanied with chip separation or materials removal phenomenon. The separation of chip is of utmost important about numerical simulation of precision machining. The simulation results can only be meaningful only if the reasonable chip separation criteria which can reflect materials mechanical and physical property (such as morphology of chip, force, temperature and the residual stress etc.) were applied in the simulation model. Besides, the criterion for chip separation should be invariant for definite materials but not change with the different working conditions. In the metal cutting process, some kinds of materials may generate continuous chip while others may generate saw-like chip thus different materials fracture criteria

Presently, there are two kinds of chip separation criteria, namely, the geometric criterion and the physical criterion. Materials removal (chip separation) using geometric criterion is realized through the variation of size of deformable body. On the other hand, the physical criterion is based on if some key physical parameters approached the critical value, these

degree of stress triaxiality.

In reality, chip separation process can be assumed as the formation and development of crack. Under what conditions and what manners can the materials be cut off is closely related with the fracture criterion[2]. Consider plane crack extending through the thickness of flat plane. There are three independent kinematic movements of the upper and lower crack surfaces with respect to each other. These three basic modes of deformation are illustrated in figure 1, which presents the displacements of the crack surface of a local element containing the crack front. Any deformation of the crack surface can be viewed as a superposition of these basic deformation modes, which are defined as follows:


Fig. 1. Three basic modes of crack extension (i) Opening mode; (ii) Sliding mode; (iii) Tearing mode

The stress and deformation fields associated with each of these three deformation modes will be determined in the sequel for the case of plane strain and generalized plane stress. Solid materials is defined to be in a state of plane strain parallel to the plane xy if

$$u = u(\mathbf{x}, y), \; v = v(\mathbf{x}, y), \; w = 0 \tag{2}$$

where *u, v, w* denote the displacement components along the axes *x*, *y* and *z*. Chip separation originated from crack while the static, stable or extension of the crack are all closely related with the distribution of stress field around the crack. The study of stress field near the crack tip is of great important as this field govern the fracture process that takes place at the crack tip.

#### a. Opening mode

Infinite plate with a crack of length 2a subjected to equal stresses at infinity is give by

$$Z\_{l}(z) = \frac{\sigma z}{\sqrt{z^2 - a^2}}\tag{3}$$

Analysis Precision Machining Process Using Finite Element Method 111

Following the same procedure in the previous case, and recognizing the general applicability of the singular solution for all sliding mode crack problems, the following



The stress intensity factor is a fundamental quantity that governs the stress field near the crack tip. Several methods have been used for the determination of stress intensity factors as

a. Theoretical method (Westergaard semi-inverse method and method of complex

b. Numerical method (Green's function, weight functions, boundary collocation, alternating method, integral transforms, continuous dislocations and finite element

Theoretical method is generally restricted to plates of infinite extent with simple geometrical configurations of cracks and boundary conditions. For more complicated situations one

The stress intensity factor is one of the key parameters for characterizing stress field around

here *KIC* , *KIIC* , *KIIIC* are the fracture toughness of I, II and III modes separately, which is

<sup>3</sup> sin 2 cos cos 2 2 22

<sup>3</sup> sin cos cos 2 22 2

<sup>3</sup> cos 1 sin sin 2 2 22

  

 

> 

(14)

(15)

(16)

(17)

(18)

*K K I IC* , *K K II IIC* , *K K III IIIC* (19)

b. Sliding mode

c. Tearing mode

listed following:

potentials)

method)

1. Single mode criterion

also the inherent property of materials.

equations for stresses and displacements are obtained:

*II <sup>x</sup> K r*

*y*

*xy*

c. Experimental method (photoelasticity, holography, caustics)

crack, which can be used as the criterion for crack extension.

The single mode criterion can be expressed as follows:

must result to numerical or experimental methods.

*II*

*K r*

*II*

The KII is the sliding mode stress intensity and can be obtained as following

*K r*

If we place the origin of the coordinate system at the crack tip z=a through the transformation

$$
\varphi = z - a \tag{4}
$$

Then the equation (3) takes the form

$$Z\_I = \frac{\sigma(\zeta + a)}{\sqrt{\varepsilon(\zeta + 2a)}}\tag{5}$$

using polar coordinates, *r* and we have

$$
\varphi = r e^{i\theta} \tag{6}
$$

the stress near the crack tip can be derived as follows:

$$
\sigma\_x = \frac{K\_l}{\sqrt{2\pi r}} \cos\frac{\theta}{2} \left(1 - \sin\frac{\theta}{2} \sin\frac{3\theta}{2}\right) \tag{7}
$$

$$
\sigma\_y = \frac{K\_l}{\sqrt{2\pi r}} \cos\frac{\theta}{2} \left( 1 + \sin\frac{\theta}{2} \sin\frac{3\theta}{2} \right) \tag{8}
$$

$$\tau\_{xy} = \frac{K\_l}{\sqrt{2\pi r}} \cos\frac{\theta}{2} \sin\frac{\theta}{2} \cos\frac{3\theta}{2} \tag{9}$$

$$
\mu = \frac{K\_l}{4G} \sqrt{\frac{r}{2\pi}} \left[ (2\beta - 1) \cos\frac{\theta}{2} - \cos\frac{3\theta}{2} \right] \tag{10}
$$

$$v = \frac{K\_l}{4G} \sqrt{\frac{r}{2\pi}} \left[ (2\beta + 1) \sin \frac{\theta}{2} - \sin \frac{3\theta}{2} \right] \tag{11}$$

$$w = 0\tag{12}$$

here *<sup>x</sup>* , *<sup>y</sup>* and *xy* are the stress component, *u*, *v* and *w* are the displacement component, *G* is the shear modulus, is the poisson ratio, 3 4 . The *KI* is the stress intensity factor and expresses the strength of the singular elastic stress field. As put forward by Irwin[9], equation (7) ~ (9) applies to all crack tip stress fields independently of crack/body geometry and the loading conditions. The stress intensity factor depends linearly on the applied load and is a function of crack length and the geometrical configuration of the cracked body.

$$K\_I = \lim\_{|\boldsymbol{\xi}| \to 0} \sqrt{2\pi\xi} Z\_I \tag{13}$$

Equation (13) can be used to determine the *KI* stress intensity factor when the *ZI* is known.

#### b. Sliding mode

110 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

2 2 ( ) *<sup>I</sup> <sup>z</sup> Z z*

If we place the origin of the coordinate system at the crack tip z=a through the transformation

 

 

> *i re*

42 2 2

42 2 2

0 *w* (12)

factor and expresses the strength of the singular elastic stress field. As put forward by Irwin[9], equation (7) ~ (9) applies to all crack tip stress fields independently of crack/body geometry and the loading conditions. The stress intensity factor depends linearly on the applied load and is a function of crack length and the geometrical configuration of the

> || 0 lim 2 *K Z I I*

Equation (13) can be used to determine the *KI* stress intensity factor when the *ZI* is known.

<sup>3</sup> cos 1 sin sin 2 2 22

<sup>3</sup> cos 1 sin sin 2 2 22

> <sup>3</sup> cos sin cos 2 22 2

> > <sup>3</sup> 2 1 cos cos

<sup>3</sup> 2 1 sin sin

3 4

 are the stress component, *u*, *v* and *w* are the displacement component,

(13)

 

 

 

(9)

the stress near the crack tip can be derived as follows:

we have

*<sup>I</sup> <sup>x</sup> K r*

*I*

*I*

*K r*

is the poisson ratio,

*K r*

*y*

*xy*

*K r <sup>I</sup> <sup>u</sup> G*

*K r <sup>I</sup> <sup>v</sup> G*

Then the equation (3) takes the form

using polar coordinates, *r* and

here *<sup>x</sup>* , 

cracked body.

 *<sup>y</sup>* and *xy* 

*G* is the shear modulus,

*z a* 

( ) ( 2) *<sup>I</sup> <sup>a</sup> <sup>Z</sup>*

*a*

(3)

(5)

*z a* (4)

(6)

(7)

(8)

(10)

(11)

. The *KI* is the stress intensity

Following the same procedure in the previous case, and recognizing the general applicability of the singular solution for all sliding mode crack problems, the following equations for stresses and displacements are obtained:

$$\sigma\_x = -\frac{K\_{ll}}{\sqrt{2\pi r}} \sin\frac{\theta}{2} \left(2 + \cos\frac{\theta}{2} \cos\frac{3\theta}{2}\right) \tag{14}$$

$$
\sigma\_y = \frac{K\_{ll}}{\sqrt{2\pi r}} \sin\frac{\theta}{2} \cos\frac{\theta}{2} \cos\frac{3\theta}{2} \tag{15}
$$

$$\tau\_{xy} = \frac{K\_{\text{II}}}{\sqrt{2\pi r}} \cos\frac{\theta}{2} \left(1 - \sin\frac{\theta}{2} \sin\frac{3\theta}{2}\right) \tag{16}$$

The KII is the sliding mode stress intensity and can be obtained as following

$$K\_{\rm II} = \lim\_{|\boldsymbol{\xi}| \to 0} i \sqrt{2\pi \boldsymbol{\xi}} Z\_{\rm II} \tag{17}$$

c. Tearing mode

$$K\_{\rm III} = \lim\_{|\xi| \to 0} \sqrt{2\pi\xi} Z\_{\rm III} \tag{18}$$

The stress intensity factor is a fundamental quantity that governs the stress field near the crack tip. Several methods have been used for the determination of stress intensity factors as listed following:


Theoretical method is generally restricted to plates of infinite extent with simple geometrical configurations of cracks and boundary conditions. For more complicated situations one must result to numerical or experimental methods.

The stress intensity factor is one of the key parameters for characterizing stress field around crack, which can be used as the criterion for crack extension.

1. Single mode criterion

The single mode criterion can be expressed as follows:

$$K\_I \ge K\_{I\mathbb{C}} \,, \ K\_{II} \ge K\_{II\mathbb{C}} \,, \ K\_{III} \ge K\_{III\mathbb{C}} \tag{19}$$

here *KIC* , *KIIC* , *KIIIC* are the fracture toughness of I, II and III modes separately, which is also the inherent property of materials.

#### 2. Mixed mode criterion

The mixed mode criterion can be acquired using Ellipsoid Criterion:

$$\left(\frac{\mathcal{K}\_{\text{I}}}{\mathcal{K}\_{\text{IC}}}\right)^{2} + \left(\frac{\mathcal{K}\_{\text{II}}}{\mathcal{K}\_{\text{IIC}}}\right)^{2} + \left(\frac{\mathcal{K}\_{\text{III}}}{\mathcal{K}\_{\text{IIIC}}}\right)^{2} \ge \mathbf{1} \tag{20}$$

Analysis Precision Machining Process Using Finite Element Method 113

inside of the contour, *Ti* would define the normal stress acting at the boundaries. The

*T n i i* 

 1 1 22 2 8 2 *I II III J KK K G*

As for nonlinear elastic materials, the system potential enclosed by curve can be

*p u d*

*a*

1 *m* 1

1

 

*m m*

( )

<sup>1</sup> ( )

here *a* is the crack length. The *J* integral is essentially variation rate of system potential energy which is mainly transform into irreversible plastic work. If the work needed to extend crack a unit length is a constant, then the *J* integral based elastic-plastic fracture criterion can be deduced. It is because the *J* integral can be used to characterize the elastic plastic stress field solved by deformation theory that the *J* integral is selected as elastic plastic fracture criterion. In 1968, Hutchinson[11], Rice and Rosengren[12] investigated the elastic plastic stress field

> *ij ij Y Y J a Ir*

1

*ij Y ij Y Y*

*<sup>m</sup> <sup>m</sup> <sup>m</sup> <sup>m</sup> i i*

*<sup>i</sup> u* . The numerical method and the energy method are the two practical solutions. The numerical method mainly makes use of elastic-plastic finite element method and integrates along several paths around crack tip and acquires the *J* integral. The final J integral can be

 

*a Ir*

 

 

*<sup>J</sup> <sup>a</sup>*

*Y Y <sup>J</sup> <sup>u</sup> r u a Ir*

 

, *ui* is a function of

integral using equation (27) ~ (29) because of the complex regular expression of

 *J*

around crack using deformation theory and acquired singular solution as follows:

 

*j j* (23)

(24)

(25)

(26)

(27)

(28)

(29)

 *ij* , *ij* and

. In reality, it is difficult to solve the *J*

components of the traction vector are given by:

computed as follows:

here *I* is definite integral of

computed as follows:

Therefore

here *nj* is the component of the unit vector normal to .

() *W dA j j*

As for linear elastic materials, there some relationship as follows:

#### **3.1.2** *J***-integral theory**

The stress intensity factor can only be applied to small yield around crack tip, other appropriate parameters should be developed to evaluated the large fracture strength. Rice[10] introduced path independent line integral as the elastic-plastic parameter for characterizing the status of crack which also named as *J*-integral. Hutchinson[11] and Rice and Rosengren[12] showed that *J* uniquely characterizes crack tip stress and strains in nonlinear materials. Thus the *J* integral can be viewed as both an energy parameter and a stress intensity parameter. After that, many researchers investigate the *J*-integral which establish the theoretical foundation of the path independent *J*-integral and its use as a fracture criterion. Presently, the main efforts in the study of elastic-plastic fracture mechanics is building up the evaluating method on fracture strength using *J*-integral while the yield materials around crack tip can be considered as non-linear elastic materials.

As for crack in the nonlinear elastic continuum medium, Rice[10] found that the integral around crack tip is path independent and is given by:

$$J = \oint\_{\Gamma} \left( wdy - T\_i \frac{\partial u\_i}{\partial \mathbf{x}} ds \right) \tag{21}$$

here *w* is the strain energy density, *Ti* is the component of the traction vector, *ui* is the displacement vector component and *ds* is a length increment along the contour . The stress energy density is defined as:

0 *ij w dij ij* (22) 

Fig. 2. Arbitrary contour around the tip of a crack

here *ij* and *ij* are the stress and strain tensors separately. The traction is a stress vector normal to the contour. That is, if we were to construct a free body diagram on the material inside of the contour, *Ti* would define the normal stress acting at the boundaries. The components of the traction vector are given by:

$$T\_i = \sigma\_{i\dagger} n\_j \tag{23}$$

here *nj* is the component of the unit vector normal to .

As for linear elastic materials, there some relationship as follows:

$$J = \frac{\kappa + 1}{8\mu} \left( K\_I^2 + K\_{II}^2 \right) + \frac{1}{2\mu} K\_{III}^2 = G \tag{24}$$

As for nonlinear elastic materials, the system potential enclosed by curve can be computed as follows:

$$
\Pi = \int \mathcal{W}(\mathcal{E}) dA\_{\Gamma} - \int\_{\Gamma} p\_{j} u\_{j} d\Gamma \tag{25}
$$

Therefore

(20)

112 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

22 2 1 *<sup>I</sup> II III IC IIC IIIC KK K KK K* 

The stress intensity factor can only be applied to small yield around crack tip, other appropriate parameters should be developed to evaluated the large fracture strength. Rice[10] introduced path independent line integral as the elastic-plastic parameter for characterizing the status of crack which also named as *J*-integral. Hutchinson[11] and Rice and Rosengren[12] showed that *J* uniquely characterizes crack tip stress and strains in nonlinear materials. Thus the *J* integral can be viewed as both an energy parameter and a stress intensity parameter. After that, many researchers investigate the *J*-integral which establish the theoretical foundation of the path independent *J*-integral and its use as a fracture criterion. Presently, the main efforts in the study of elastic-plastic fracture mechanics is building up the evaluating method on fracture strength using *J*-integral while the yield materials around

As for crack in the nonlinear elastic continuum medium, Rice[10] found that the integral

here *w* is the strain energy density, *Ti* is the component of the traction vector, *ui* is the displacement vector component and *ds* is a length increment along the contour . The stress

0

normal to the contour. That is, if we were to construct a free body diagram on the material

 

*ij w dij ij* 

*i i <sup>u</sup> J wdy T ds x* 

are the stress and strain tensors separately. The traction is a stress vector

(21)

(22)

The mixed mode criterion can be acquired using Ellipsoid Criterion:

crack tip can be considered as non-linear elastic materials.

around crack tip is path independent and is given by:

Fig. 2. Arbitrary contour around the tip of a crack

2. Mixed mode criterion

**3.1.2** *J***-integral theory** 

energy density is defined as:

here *ij* and *ij* 

$$J = -\frac{\partial \Pi}{\partial a} \tag{26}$$

here *a* is the crack length. The *J* integral is essentially variation rate of system potential energy which is mainly transform into irreversible plastic work. If the work needed to extend crack a unit length is a constant, then the *J* integral based elastic-plastic fracture criterion can be deduced. It is because the *J* integral can be used to characterize the elastic plastic stress field solved by deformation theory that the *J* integral is selected as elastic plastic fracture criterion.

In 1968, Hutchinson[11], Rice and Rosengren[12] investigated the elastic plastic stress field around crack using deformation theory and acquired singular solution as follows:

$$
\sigma\_{ij} = \left(\frac{I}{a\varepsilon\_Y \sigma\_Y Ir}\right)^{\frac{1}{m+1}} \tilde{\sigma}\_{ij} \tag{27}
$$

$$\varepsilon\_{\vec{ij}} = a \varepsilon\_Y \left( \frac{I}{a \varepsilon\_Y \sigma\_Y Ir} \right)^{\frac{m}{m+1}} \tilde{\varepsilon}\_{\vec{ij}}(\theta) \tag{28}$$

$$\mu\_i = \left(\frac{J}{a\varepsilon\_Y \sigma\_Y Ir}\right)^{\frac{m}{m+1}} r^{\frac{m}{m+1}} \tilde{u}\_i(\theta) \tag{29}$$

here *I* is definite integral of , *ui* is a function of . In reality, it is difficult to solve the *J* integral using equation (27) ~ (29) because of the complex regular expression of *ij* , *ij* and *<sup>i</sup> u* . The numerical method and the energy method are the two practical solutions. The numerical method mainly makes use of elastic-plastic finite element method and integrates along several paths around crack tip and acquires the *J* integral. The final J integral can be computed as follows:

$$J = \sum \frac{J\_i}{m} \tag{30}$$

Analysis Precision Machining Process Using Finite Element Method 115

The depth of cut in the precision machining is very small, chips are formed at very narrow regions. The work material is subjected to extremely high plastic deformation and the strain rates can reach the values of about 105 s-1. The large strain and high strain rate plastic deformation evolves out of hydrostatic pressure that travels ahead the tool as it pass over. The zone has, like all plastic deformations an elastic compression region that becomes the plastic compression region as the field boundary is crossed. The plastic compression generates dense dislocation tangles and networks which lead to the materials shear after the materials experience fully work hardened. The theory of micro-plasticity, which mathematically describes the stress and strain at small scale, is adopted to calculate the distributions of stress

**4.1 Plastic deformation and chip formation in the precision machining titanium alloy**  The numerical analysis method applied to materials cutting process can be divided into two categories, namely, the elastic-plastic FEM and the rigid-plastic FEM. Furthermore, thermoelastic FEM and the thermo-rigid FEM are introduced if the temperature and the velocity are considered in the materials processing technology. The simulation results are almost same whether the problem analysed by either elastic-plastic FEM or rigid-plastic FEM if the size of the workpiece and the amount of discreted element are same for these two methods. The elastic-plastic FEM mainly applied to solve the residual stress and the elastic recovery while the rigid-plastic FEM cannot solve this type of problems as it ignored elastic deformation

In this research work, the commercial finite element analysis package (Advantedge®) is utilized to gain good understanding of the materials deformation behavior underlying machining of titanium alloy. Among the different alloys of titanium, Ti-6Al-4V is by far the most popular with its widespread use in the chemical, surgical, ship building and aerospace industry. The primary reason for wide applications of this titanium alloy is due to its high strength-to-weight ratio that can be maintained at elevated temperatures and excellent corrosion and fracture resistance. On the other hand, Ti-6Al-4V is notorious for poor machinability due to its low thermal conductivity that causes high temperature on the tool face, strong chemical affinity with most tool materials, which leads to premature tool failure, and inhomogeneous deformation by catastrophic shear that makes the cutting force

Fig. 4. FE simulation based on geometrical separation criterion

and thus it has higher solution efficiency.

**4. Materials deformation behavior in the precision machining** 

and strain in the distorted bodies.

Separation line

here *Ji* is the *J* integral corresponding to path *<sup>i</sup>* , *n* is the number of integrate path. The integrate path is generally continuous smooth curve which can reduce the error resulted by the discontinuous surface force.

#### **3.2 Geometrical criterion**

The geometrical criterion mainly takes effect through judging if the geometrical size of materials exceeding the criterion. Figure 3 shows the geometrical model in which a separation line is defined. The nodes at the chip side and the nodes at workpiece side are overlapped at the beginning. But the separation of two nodes occurs when the distance *D* between the tool cutting edge (point *d*, in Figure 3) and the node immediately ahead (node *a*) becomes less than a predefined critical value thus the machined surface and the chip bottom are generated.

Fig. 3. Geometrical criterion model

Usui and Shirakashi[13] first put forward the geometrical criterion and found it is a stable criterion. Komvopoulos and Erpenbeck[14] pointed that there should be enough distance between tool tip and the overlap point to prevent the convergence problem resulted by the excessive distortion of finite element mesh. Zhang and Bagchi[15] brought forward that the geometrical distance should be less than 30 percent to 50 percent of element length. Furthermore, they also put up a new geometrical separation criterion which is based upon the ratio of geometrical distance to depth of cut which is equivalent to the microscopic fracture mechanics criterion.

The geometrical criterion is simple to be used in the FE computation. However, the distance (*D*) between tool tip and the separation point is closed to zero which result in the difference between the set value of *D* with the reality. The selection value of *D* will have a great influence upon the convergence of FE simulation and only the experienced researcher can deduce appropriate valuable critical value. In addition, the separation line which separates the mesh of chip and that of the workpiece should be built up in advance. Figure 4 shows the FE simulation of precision machining process based on geometrical separation criterion.

*<sup>i</sup> <sup>J</sup> <sup>J</sup>*

here *Ji* is the *J* integral corresponding to path *<sup>i</sup>* , *n* is the number of integrate path. The integrate path is generally continuous smooth curve which can reduce the error resulted by

The geometrical criterion mainly takes effect through judging if the geometrical size of materials exceeding the criterion. Figure 3 shows the geometrical model in which a separation line is defined. The nodes at the chip side and the nodes at workpiece side are overlapped at the beginning. But the separation of two nodes occurs when the distance *D* between the tool cutting edge (point *d*, in Figure 3) and the node immediately ahead (node *a*) becomes less than a predefined critical value thus the machined surface and the chip

Usui and Shirakashi[13] first put forward the geometrical criterion and found it is a stable criterion. Komvopoulos and Erpenbeck[14] pointed that there should be enough distance between tool tip and the overlap point to prevent the convergence problem resulted by the excessive distortion of finite element mesh. Zhang and Bagchi[15] brought forward that the geometrical distance should be less than 30 percent to 50 percent of element length. Furthermore, they also put up a new geometrical separation criterion which is based upon the ratio of geometrical distance to depth of cut which is equivalent to the microscopic

Workpiece

Chip

The geometrical criterion is simple to be used in the FE computation. However, the distance (*D*) between tool tip and the separation point is closed to zero which result in the difference between the set value of *D* with the reality. The selection value of *D* will have a great influence upon the convergence of FE simulation and only the experienced researcher can deduce appropriate valuable critical value. In addition, the separation line which separates the mesh of chip and that of the workpiece should be built up in advance. Figure 4 shows the FE simulation of precision machining process based on geometrical

the discontinuous surface force.

**3.2 Geometrical criterion** 

bottom are generated.

Fig. 3. Geometrical criterion model

fracture mechanics criterion.

separation criterion.

*<sup>n</sup>* (30)

Separation line
