**3. Experiment Setup for Model Validation**

Orthogonal cutting is the basic cutting configuration for all machining processes. The essential geometrical parameters include rake angle, clearance angle and the depth of cut. In this experiment, the solid foams, the 30 and 40 pcf, are sectioned to a 20 × 30 × 3 mm testing sample. Each sample is hand-polished with the same grit size to ensure a smooth surface and uniform depth of cut. Figure 6 illustrates the experimental setup for the orthogonal cutting setup which consists of two linear actuators and a dynamometer for force measurement. The cutting tool is attached to the vertical linear actuator through a customized tool holder. The linear actuator (L70, Moog Animatics, Milpitas, CA, USA) is driven by a high-torque servo-motor to maintain a constant feed rate during cutting. The force dynamometer (Model 9272, Kistler, Winterthur, Switzerland) is used to capture high-speed or high-frequency force data up to 5 kHz. Data collection is performed via an amplifier, a shielded connector block, and a data acquisition device (PCle-6321, National Instruments, Austin, TX, USA), along with a data recorder, LabVIEW, at 2 kHz sampling rate. The workpiece is fixed by a clamping system on the top of the dynamometer which is placed on the other linear slider to control the depth of cut for each test.

The cutting tool has a tungsten carbide substrate and a polycrystalline diamond (PCD) insert as a cutting edge, provided by Sandvik (Model TCMW16T304FLP-CD10). This PCD insert is extremely hard and minimizes any possible deformation or wear at the cutting edge. This cutting tool has a zero-rake angle and a clearance angle of 7◦. The cutting edge radius is 11 μm, measured by a high-definition surface profiler (Alicona InfiniteFocus G4, Graz, Austria).

In this experiment, two depths of cut, 0.1 mm and 0.3 mm, are used to present common chip loads for a machining process. The cutting tool is moved at a constant velocity of 10 m/min to represent a machining condition. These parameters are applied to two specimens and repeated for four times each.

**Figure 6.** Schematic of the orthogonal cutting setup for model validation.

#### **4. Simulation and Experiment Results**

The simulation results are compared to the experiments in different depths of cut and material properties (30 pcf and 40 pcf) in this section.

### *4.1. Chip Formation*

To find an appropriate scaling factor, a qualitative comparison of chip formation behavior against the experiment is conducted. In brittle materials, the chip can be generated in various forms, including dusty debris, fragmented and irregular pieces, or equal-sized small chips. Different scaling factors are tested until a similar chip behavior to the experiment is achieved or no obvious behavior difference can be observed. For this purpose, the initial guess for the scaling factor is recommended to be half of the element size (i.e., δc/*d* = 0.5) to ensure the material brittleness. Then, a binary search method is used. If the current *f* does not show a good match, half of the value (*f*/2) will be investigated until the best fit is found or further improvement is not distinguishable.

Following the aforementioned procedure, the model calibration is performed for 30 pcf and DOC = 0.1 mm. Figure 7a shows the corresponding simulation results with a selected *f* = 0.02, which has a similar chip formation to that of the experiment. The simulation can capture the irregular chips of different sizes generated from the cutting zone. Then this scaling factor is also used to simulate the case of 0.3 mm DOC. The result is shown in Figure 7b. A larger DOC tends to generate bigger chips surrounded by small debris as compared to the case of 0.1 mm DOC. Consistently, the experiment also sees much bigger or clustered pieces when DOC increases to 0.3 mm. The results of 30 pcf with the selected scaling factor show qualitative agreement between the model and experiment in terms of chip behavior. Chip sizes of simulation and experiment do not match exactly due to the limited observation window and material uncertainty, but the difference is in the order of sub-mm.

**Figure 7.** Simulated and experimentally measured chip formation of the 30 pcf with (**a**) depth of cut (DOC) = 0.1 mm and (**b**) DOC = 0.3 mm.

For the 40 pcf, the same scaling factor of 0.02 is used, which corresponds to a maximum of 0.00294 mm deformation (29.4% elongation). This value also makes the workpiece more ductile than the 30 pcf (12.8% elongation). The simulation result of 40 pcf at 0.1 mm DOC and corresponding experimental observations are shown in Figure 8a. Different from 30 pcf at 0.1 mm DOC, bigger and similarly-sized chips are generated with dusty debris around. This phenomenon also indicates a more ductile behavior as tested in Figure 4.

When the same scaling factor is applied to the case of 0.3 mm DOC, the simulation of the cutting process starts to show unstable chip formation, as shown in Figure 8b,c at different time steps. Cracks can propagate ahead of the cutting tool motion to generate large chips and sudden fracture along the cutting direction to shear the chip layer. This phenomenon is also seen in the experiment, though the unstable cracks into the workpiece could not really be captured due to the material uncertainty and the randomness of cracks.

**Figure 8.** Simulated and experimentally measured chip formation of the 40 pcf with (**a**) DOC = 0.1 mm, (**b**) DOC = 0.3 mm, and (**c**) DOC = 0.3 mm at a later time step with a sudden crack propagation.

#### *4.2. Cutting Force*

Figure 9a shows the cutting forces measured from four repeated tests for 30 pcf at DOC = 0.1 mm. Force profiles are oscillating due to the brittle nature of the material. The system noise is assumed minimal considering the system rigidity. During a roughly 0.14 s cutting period, the cutting forces can reach and stay at a certain level, namely the steady cutting, and then drop toward the end. That said, the simulation length of about 0.01 s is enough to reach the steady cutting to extract the force. According to the scaling factor *f* = 0.02 used in these simulations, the simulated force is scaled by 50 times (1/*f*) and overlaid on Test 4, shown by the comparison in Figure 9b. Since the simulation ran at every 0.00006 s increment, the sampling frequency is equivalent to 16.7 kHz as opposed to 2 kHz of the experiment. The averaged force of simulation is 12.5 N, and the experimental average across the steady cutting is about 9 N. Although the forces are at a similar magnitude, the simulated force is oscillating much more significantly (0 to 35 N). These discrepancies may be attributed to the fact that embedded CZ elements have a different property from the main elements and less deformability. Such an oscillating profile is seen in all simulation cases of 30 and 40 pcf at 0.1 and 0.3 mm DOCs.

**Figure 9.** (**a**) Experimentally measured cutting forces of 30 pcf at DOC = 0.01 mm and (**b**) the comparison between the experiment and the simulated, scaled cutting force.

Figure 10 compares all simulated cases with the corresponding experiments in terms of the average force of cutting, where the error bars stand for one standard deviation from the four replicated tests. The overall trend of model prediction agrees with the experiments in different materials and depths of cut. However, the simulated forces are always higher by 30% to 50%, likely due to an over-estimated fracture energy or non-linearity of the cutting force to the cutting energy. The causes of oscillating and overestimated force will be elaborated more in the discussion section. Nonetheless, based on the

results, the concept of ECZ–FEM is considered viable to approximate the magnitude of cutting force and to predict the changes of cutting force and chip behavior in different brittle cutting scenarios.

**Figure 10.** Comparisons between all simulated cutting forces and experimentally measured cutting forces (averaged).

#### **5. Discussion**

In ECZ–FEM, the key to a successful simulation is choosing an appropriate scaling factor by calibrating the model behavior with an experiment. As mentioned, CZ elements are determined by a traction–displacement relationship. When CZ elements are embedded in the workpiece, their allowable deflection can change the material ductility, and thus it must be limited. Figure 3 has shown how different scaling factors can change the chip formation from very ductile to brittle. Although limiting CZ deflection inevitably changes the material property (*Gf*), the effect on cutting force can be assumed linearly scaled under the assumption of 100% cutting energy conversion. This is reasonable because most of the brittle materials do not plastically deform and do not produce significant friction and frictional heat due to discontinuous chip formation.

The model predicts the relative behavior well among different materials and depths of cut, but the calculated cutting forces are always higher. One explanation is that it is due to the oscillating force profile, but it can also be caused by an overestimated fracture energy. The over-estimation can be from the difference between the static and dynamic fracture toughness, *Kc*. The fracture energy *Gf* is determined by the material toughness *Kc*, which is measured from a quasi-static test. Thus, the obtained *Kc* is the static fracture toughness while the actual dynamic toughness may be much lower, as reported in the literature [20,21]. However, it is technically challenging to measure a dynamic toughness at a comparable speed of cutting (10 m/min or 167 mm/s).

Another issue is the significant oscillating force profile as shown in Figure 9. This is because the model consists of embedded CZ elements which have different material properties and fewer degrees of freedom than those of the main elements. Therefore, the force can change drastically when the cutter makes a pass and the workpiece experiences deformation and damage. Another reason could be a non-self-contact definition of the main elements. This may result in intermittent contact between the tool and material and thus significant force changes. A much finer mesh with full contact definition can mitigate the problem at the cost of computational time.
