**2. The flow stress characteristics of the workpiece materials**

The flow stress characteristics are an important issue in the numerical analysis which is directly affects the loads and stresses in the precision machining. The flow stress is generally

hardening, strain rate sensitivity, temperature dependence, chip formation and the chip-tool interface behaviors are not fully accounted for by the analytical methods. Experimental studies on precision machining are expensive and time consuming. Moreover, their results are valid only for the experimental conditions used and depend greatly on the accuracy of calibration of the experimental equipment and apparatus used. Advanced numerical techniques such as Finite Element Method is a potential alternative for solving precision machining problems.

Finite Element Method (FEM) which is originated from continuum mechanics, has already been justified as successful method in analyzing complicated engineering problem[3-8]. There are many advantages of using FEM to investigate machining: multi-physical machining variables output can be acquired (cutting force, chip geometry, stress and temperature distributions), improving precision and the efficiency comparing with Try-Out-Method and so on. In the last three decades, FEM has been progressively applied to metal cutting simulations. Starting with two-dimension simulations of the orthogonal cutting more than two decades ago, researches progressed to three-dimensional FEM models of the oblique cutting, which capable of simulating metal cutting processes such as turning and milling. Increased computation power and the development of robust calculation algorithms (thus widely availability of FEM programs) are two major contributors to this progress. Unfortunately, this progress was not accompanied by new developments in precision machining theory so the age-old problems such as the chip formation mechanism and tribology of the contact surfaces are not modeled properly. Further, even at a moderate cutting speed, the strain rates are quite high, almost of the order of 104 per second and the temperature rise is also quite large. As a result, the visco-plasticity and temperature-softening effects become more important compared to strain-hardening. Therefore, the material properties associated with these two effects should be known for a range of strain rates and temperatures occurring in typical machining processes. Additionally, to incorporate the temperature rise in the analysis, one needs to solve the heat transfer equation governing the temperature field in conjunction with the usual three equations governing the deformation field. For plastic deformation, these

In material removal processes at the precision scale, the undeformed chip thickness can be on the order of a few microns or less, and can approach the nanoscale in some cases. At these length scales, the surface, subsurface, and edge condition of machined features and the fundamental mechanism for chip formation are much more intimately affected by the material properties and microstructure of the workpiece material, such as ductile/brittle behavior, crystallographic orientation of the material at the tool/chip interface, and microtopographical features such as voids, secondary phases, and interstitial particulates. Characterizing the surface, subsurface, and edge condition of machined features at the precision scale in the FEM analysis are of increasing importance for understanding, and controlling the manufacturing process. There are still many challenges in the investigation

As mentioned above, this chapter will give some key factors on numerical modeling of

The flow stress characteristics are an important issue in the numerical analysis which is directly affects the loads and stresses in the precision machining. The flow stress is generally

**2. The flow stress characteristics of the workpiece materials** 

equations are coupled, and hence difficult to solve.

of precision machining by means of FEM.

precision machining and current advancements.

considered as function of strain, strain rate and temperature. Many research works justify that the influence of strain rate on flow stress become more important when the temperature becomes higher. It is important to build the appropriate flow stress models fit for different working conditions.

Accuracy and reliability of the predictions heavily depend on the materials flow stress at cutting areas such as high deformation rates and temperatures and variable friction characteristics at tool-chip interface which are not completely understood and need to be determined. Materials property at local shear band is very complex in the precision machining which makes it difficult to build up real robust flow stress model fitting for manufacturing process. Most of the energy consumption limited to local cutting area and transformed into heat which complicated the distribution of temperature at the local deformation area. The temperature plays an important role in the unstable chip flow. Larger plastic deformation rate and the intense friction at the tool-chip interface increase the heat generation rate and lead to the material softening thus decreasing the strain hardening ability and instability of materials flow. Therefore, the instability of shear behavior is directly induced by materials flow. Presently, researchers can't build up reasonable materials consititutive relationship which can characterize strain rate and the temperature and reflect the variation of materials property in the precision machining process.

Sound theoretical models based on atomic level material behavior are far from being accomplished. Semi-empirical constitutive models are widely utilized. Several material constitutive models are used in Finite Element (FE) simulation of metal cutting, including rigid-plastic, elasto-plastic, viscoplastic, elasto-viscoplastic and so on. These models take into account the high strains and temperatures reportedly found in metal cutting. Among others, the most widely used is the Johnson and Cook[7] (*JC*) model which is a thermo-elastovisco-plastic material constitutive model expressed as follows:

$$\overline{\sigma} = \left[ A + B \left( \overline{\varepsilon} \right)^{n} \right] \left[ 1 + C \ln \left( \frac{\dot{\overline{\varepsilon}}}{\overline{\dot{\overline{\varepsilon}}\_{0}}} \right) \right] \left[ 1 - \left( \frac{T - T\_{0}}{T\_{m} - T\_{0}} \right)^{m} \right] \tag{1}$$

here *A* is the initial yield stress of the material at the room temperature, strain rate 1/s and represents the equivalent plastic strain. The equivalent plastic strain rate is normalized with a reference strain rate 0 . Temperature term in *JC* model reduces the flow stress to zero at the melting temperature of the work materials, *Tm*, leaving the constitutive model with no temperature effect. In general, the parameters *A*, *B*, *C*, *n*, and *m* of the model are fitted to the data obtained by several material test conducted at low strains and strain rates and at room temperature as well as Split Hopkinson Pressure Bar (SHPB) test at strain rates up to 1000/s and at temperatures up to 600 °C. *JC* model provides good fit for strainhardening behavior of metals and it is numerically robust and can easily be used in FE simulation models.

Besides, there are two major problems with the use of the discussed model and its method of the determination of its constants. First, only few laboratories and specialist in the world can conduct SHPB testing properly, assuring the condition of dynamic equilibrium. None of the known tests in metal cutting was carried out in these laboratories. Second, the high strain rate in metal cutting is rather a myth than reality. Third, the temperature in the so-

Analysis Precision Machining Process Using Finite Element Method 109

physical criterion includes effective plastic strain criterion, strain energy density criterion

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

1. Opening mode, the crack surfaces separate symmetrically with respect to the planes xy

2. Sliding mode, the crack surfaces slide relative to each other symmetrically with respect

3. Tearing mode, the crack surfaces slide relative to each other skew-symmetrically with

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

Solid materials is defined to be in a state of plane strain parallel to the plane xy if

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.

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.

*u*=*u*(*x*,*y*), *v*=*v*(*x*,*y*), *w*=0 (2)

at infinity is give by

Z

Y

X

superposition of these basic deformation modes, which are defined as follows:

to the planes xy and skew-symmetrically with respect to plane xz

(i) (ii) (iii)

Infinite plate with a crack of length 2a subjected to equal stresses

and the fracture stress criterion and so on.

respect to both planes xy and xz.

**3.1 Fracture mechanics criterion**

**3.1.1 Stress intensity factor** 

and xz

Tearing mode

a. Opening mode

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 degree of stress triaxiality.
