List of Symbols
Step 3: Step 3: Derive the element characteristic (stiffness and mass) matrices and characteristic (load) vector. The strain can be expressed as
4.5 Results and Discussion
The main objective of this present work is to determine the ductile to brittle transition through the plunge cutting simulation of brittle material. It is believed that the critical depth of cut obtained during the simulation can be utilized to obtain crack free surface by keeping the depth below the critical depth of cut. In the current simulation, surface profile, cutting force and specific cutting energy are analyzed to find out the transition point between ductile and brittle. The approach employed in the present work is to obtain critical depth of cut.
4.5.1 Determination of Critical Depth of Cut
After the study of formation of chips, it was decided to determine the critical depth of cut (CDC). In this work, three different methods were selected for the determination of CDC.
These are elaborated one by one in the following sub sections.
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Figure 4.11 shows the profile generated from the plunge cut simulation using 0º rake angle tool when the depth of cut gradually increases from 0 to 600 nm. The surface profile is carefully observed and it is noted that there is formation of first brittle fracture. This was occurred at 211.4 nm depth in case of 0º rake angle tool, which was considered to be the critical depth of cut for the chosen process condition.
Figure 4.11 Plunge cut with 0 rake angle
Two more sets of simulations of plunge cut were carried out by taking −25º and −30º rake angles to analyze the effect of rake angle on critical depth of cut or DBT. The surface profiles obtained during these simulations are shown in Figure 4.12 and 4.13.
Figure 4.12 Plunge cut with −25 rake angle
Figure 4.13 Plunge cut with −30 rake angle TH-2306_10610325
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The critical depths of cut evaluated from the Figures 4.11, 4.12 and 4.13 using visual inspection method are tabulated in Table 4.1. These are based on the first visible micro cracks on the machined surface.
Table 4.1 Critical depth of cut for 0º, −25º and −30º rake angle from visual inspection Rake angle () X co-ordinate Y co-ordinate Critical thickness (nm)
0 0.0038453965 0.0017886002 211.40
–25 0.0037578676 0.0017745867 225.41
–30 0.0036918412 0.0017568420 243.16
The coordinates of the first visible micro crack were determined using plot digitizer.
The plot digitizer takes the original dimension of the workpiece (5 µm × 2 µm) and then displays the coordinates, i.e., x and y length based on the position of the computer-mouse.
After determining the position of the first visible micro crack by clicking the mouse on the crack, the critical depths of cut was calculated by subtracting the y coordinate length from the original height of the workpiece. The critical depths of cut obtained from visual inspection are listed in Table 4.1 for 0º, −25º and −30º rake angle tools and y direction speed of 1000 mm/s.
From the table, it can be seen that, the critical depths of cut are 211.4, 225.41, 243.16 nm for 0º, −25º and −30º rake angle tools respectively. From the table, it can also be observed that, with the increase in rake angle, the critical depth of cut increases. This is due to the fact that, with higher rake angle tool, more material in front of the cutting edge is downward suppressed and the compressive stress component becomes predominant. This provides a stress state similar to the hydrostatic stress field. From the theory of plasticity, hydrostatic pressure determines strain at fracture, which in turn determines material ductility or brittleness. The high hydrostatic pressure induced by the extremely negative rake angle is reported to cause phase transformations to metallic phases, and prevent the initiation or propagation of cracks. Therefore, with increase in negative rake angle tool, the critical depth of cut during plunge cut also increases.
4.5.1.2Analysis of Machining Forces
In this present work, cutting force, thrust force and coefficient of friction during plunge cutting simulation are extracted and carefully examined to estimate the ductile to brittle transition. Figure 4.14 presents the variation in cutting force and thrust force in a single plunge cut with a 0º rake angle tool.
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tool for continuously varying depth of cut
From the figure 4.14, it can be seen that as the depth of cut increases from 0 to 78 nm, both the cutting force and the thrust force increase smoothly. The fluctuations of the forces are noted to be more prominent when the depth of cut increases beyond 138 nm. These observations can be inferred that, below 78 nm depth of cut, the cutting process is completely in ductile mode similar to metal machining through plastic deformation. When the depth of cut increases beyond 78 nm, micro-fractures began to form at the bottom of the groove. Thus the machining mode switches into a partially brittle mode in which the brittle chips form. Due to the formation of brittle micro-fractures, the force values start fluctuating. Thus the zone of 78 nm to 138 nm can be considered as transition zone. The transition from the ductile to brittle cut can also distinguished from the variation of cutting force and thrust force as suggested by Yan et al. (2009b). As per Yan et al. (2009b), when the cutting process changes from ductile to brittle, cutting force starts dominating the thrust force. This is due to the fact that, in ductile mode cutting, depth of cut is smaller than cutting edge radius and the effective rake angle is highly negative, whereas, in the brittle region, i.e., beyond 78 nm depth of cut, the cutting force becomes higher than thrust force. This finding is also depicted from the co- efficient of friction (Fc/Ft) curve of Figure 4.14 that shows below 78 nm depth of cut, the cutting force is lower than the thrust force. When the depth of cut reaches the transition point, the cutting force and thrust force becomes equal. Beyond transition point, as the depth of cut increases beyond 78 nm, the cutting force becomes higher than the thrust force and the co- efficient of friction value becomes higher than unity. Thus, the machining force can be classified into three distinct zones, i.e. ductile-mode, transitional-mode and brittle-mode.
These observations were also observed during the simulations with −25º and −30º rake angles. Figures 4.15 and 4.16 show the variation of cutting and thrust forces that generated TH-2306_10610325
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during simulations with −25º and −30º rake angles respectively. The transition depths were found to be 84 nm and 108 nm for the rake angles of −25º and −30º respectively. These values are quite closer to the results reported by Yan et al. (2009b) and Wang and Liao (2008). Yan et al. (2009b) reported 120 nm as the transition depth during the experimental study using
−30º rake angle tool. Similarly, Wang and Liao (2008) reported a transition depth between 100 nm to 500 nm. From the present analysis it was noted that the values of transition depths estimated using proposed force analysis approach fairly match with the reported experimental values. However, some of the researchers have used specific cutting energy concept to determine the transition depth. In the next section, determination of ductile to brittle transition depth using specific cutting energy is discussed.
Figure 4.15 Variations of cutting and thrust forces during a single plunge cut using −25º rake angle tool
Figure 4.16 Variations of cutting and thrust forces during a single plunge cut using −30º rake angle tool
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Literature reports some important studies on the concept of characterization of the mode of material removal in machining and grinding of brittle material using specific cutting energy approach. According to these studies, the change in the specific cutting energy can be used to determine the ductile to brittle transition phenomenon of the plunge cut process. The term specific cutting energy (SCE) is defined as the energy needed to remove or displace a unit volume of work material [An et al. (2015), Wang and Liao (2008)]. It is given by the ratio of energy per unit time to the volume removed per unit time. Thus, specific cutting energy is dependent upon the cutting load and the contact area between the tool rake face and the chip, i.e., SCE changes with change in cutting depth. Therefore, the change of specific cutting energy can be used as a criterion for estimating the critical depth of cut.
In the present work, the specific cutting energy values were computed and based on them the ductile to brittle transition depths were identified. The specific cutting energy can be expressed as: