Simulation of the ductile machining mode of silicon

SPH was introduced 1977 in astrophysics for the calculation of a smoothed density from point clouds [3]. A simple derivation of the method is based on the partition of unity [4], where a field value at a spatial location x can be determined as: The Dirac-delta function δ(x) in Eq. 1 has two important properties: and

Replacing the Dirac-delta function δ with a smoothing function, the so-called Kernel, W(x − x,h), e.g. the Gauß-function, with h being a smoothing length, the behaviour of the Dirac-delta can be reproduced for the limit: Inserting (4)–(3) gives an approximation of the function value f(x) at x:

This can be integrated within a discrete neighborhood using a Riemann-sum: with the point index i at which the function value is to be approximated by its neighbor points j, x_ij is the spatial distance between point i and j and ΔV_j being an integration weight of point j. Computation of the function’s derivative leads to where only the derivative of the Kernel W(x_ij,h) is required. In this way, derivatives of values given at point cloud locations can be computed without the requirement of a functional description or a mesh-based relation between these points (particles). With this meshfree approximation derivatives in continuum mechanics equation can be computed by sums of discrete values in the particles neighborhood. Meshfree techniques were adopted in early 1990s to structural simulations [5] and for numerical cutting simulations first by [6]. At IWF of ETH Zürich the software mfree_iwf was developed in the past years for SPH based cutting simulations [7]. The software is capable of performing CPU as well as GPU-enhanced computations of metal cutting simulations. It is establishing the state-of-the-art in SPH simulations as it facilitates the most recent correctors and stabilisation measures for mechanical [8] and thermal simulations [9]. The software successfully demonstrated computationally highly efficient metal cutting simulations [10], identification of friction parameters [11], and its ability to adaptive refinements [12]. Furthermore, single grain grinding simulations of engineered grinding tools [13] were successfully demonstrated.

In the presented investigation, the software is adopted to simulate the cutting of silicon with a single diamond grain assuming isotropic behaviour of silicon with the Young’s modulus of Silicon in the 〈100〉 directions [14]. A flow stress model according to Johnson and Cook [15] came to application. The model is commonly used to describe metal plasticity within machining simulations and is given as:

with A, B, C, m, and n being material parameters, ε_pl the current plastic strain, \(\dot \varepsilon _{pl}\) the current plastic strain rate, and T the current temperature. T_f is the melting temperature, T_ref is the reference temperature, and \(\dot \varepsilon _{pl}^{0}\) the reference plastic strain rate. The first two terms describe hardening due to plastic strain and plastic strain rate, respectively. The third term controls thermal softening upon increasing temperature. In this investigation, material parameters for silicon according to [16, 17] were used.

Important contributions to the understanding of cutting of hard and brittle materials and the damage associated were first introduced by Hamilton and Goodman [18] and Lawn [19] who presented analytical models of the pressure field under a sliding indenter. The mechanics and mechanisms of plastic deformation and fracture of brittle materials upon indentation have been studied by Lawn et al. They introduced the median-radial crack system [20], as well as the lateral crack system [21]. Bifano et al. introduced the possibility of machining brittle materials in ductile cutting regime [22] and presented a model describing the critical depth of cut h_cu,crit which presents a threshold above which brittle fracture is introduced. Based on these concepts, the ductile machining of silicon has been extensively studied taking into account the effects of various aspects. Larger cutting edge radii lead to higher normal and cutting forces [23], increase the critical depth of cut [24], and lead to horseshoe-like cracks in the grooves rather than radial-chevron cracks [25]. The critical depth of cut is very sensitive to the crystallographic plane and direction and varies between 0.112μm in (001)[010] and 1.270μm in \((110)[1\bar {1}\bar {1}]\) [26]. High cutting speed, as present in DWS of silicon, leads to amorphisation of silicon which is accompanied by a volumetric expansion and consequently lower residual cutting depth [27]. Furthermore, phase transitions happen due to high contact pressure [28‐30], which affect the mechanical properties of the material and therefore the material removal behaviour. Recently, modelling the elastic stress field has gained attention again to predict damage [31‐33] with good success in predicting the direction of cracks, their frequency, and their length.

With exception of Bifano’s work, all studies used idealised grain shapes to study the respective phenomena, presenting a limitation to their applicability towards grinding processes. In order to exploit the freedom of geometry presented by the SPH method, the experiments conducted in this study use a non-idealised grain shape.

The grain chosen for the scratch test is shown in Fig. 2. The grain is embedded on a commercial diamond wire; the metallic matrix covering it has been carefully removed with sand paper. The wire is bent over a pin in such a way that the selected grain is isolated on the highest point.

Prior to the scratch test, this diamond grain was optically measured with an Alicona microscope and the surface data was retrieved as a point cloud. This point cloud was triangulated in MATLAB and a 3D mesh with tetrahedrons was created from the top part of the diamond as the cutting takes place in this region only. The surface mesh is shown in Fig. 3, the geometry part above the black plane indicates the relevant part for the simulations conducted.

Due to the geometric properties of fractured diamond grains used on electroplated diamond wires, only some of the parameters typically used to describe the tool geometry in micro machining can be applied: The projected cross-section of the grain is approximately trapezoidal with a slight tilt of the flat as shown in Fig. 2. The radii at the grain flanks are approximately 3μm on both sides, while the left side in direction of cut (right side in Fig. 2) is only engaged at large depth of cut of approximately 3μm. The cutting edge radius varies between 1.4 and 6.7 μm over the cutting edge and is smallest at the sides and larger in the centre of the grain. Rounding of the edge due to wear leads to an increase with increasing scratch length. The width-to-height ratio r of the grain is evaluated for each simulated depth of cut h_cu separately: \(r_{h_{cu}=0.2 \mu m} \approx 27\), \(r_{h_{cu}=0.5 \mu m} \approx 17\), \(r_{h_{cu}=1 \mu m} \approx 13\), and \(r_{h_{cu}=1.5 \mu m} \approx 11\).

The cutting tests were performed on a Fehlmann Picomax Versa 825 5-axis milling machine. The set-up is shown in Fig. 4.

A polished silicon surface is attached to and rotated with the spindle with angular velocity ω. The diamond grain on the wire is isolated by bending it over a pin and mounted onto a three-component force dynamometer (Kistler 9256C, eigenfrequency f ≈ 6kHz) fixed to the stationary machine table. The force measurement set-up is completed by a Kistler Type 5080A charge amplifier with active drift compensation and 1000-Hz low-pass filter and a National Instruments NI 9222 A/D converter, sampling each channel at 211.64kS/s. A pyrometer fibre was furthermore installed and connected to a Fire-3 two-colour fibre optic pyrometer for the measurement of flash temperatures. The diamond grain is kept stationary while the rotating spindle is fed vertically into the grain with normal feed speed v_f and horizontally across the grain with lateral feed speed v_p in such a way that the vertical displacement is equal to 3μm and the horizontal displacement is equal to 1000μm over 20 spindle rotations. The angular velocity ω is chosen in such a way that the resulting cutting speed v_c equals 10m/s. The motion results in spiral-shaped contact path; due to a small but relevant perpendicularity error of the Si surface with the spindle axis, the cut is interrupted and the engagement length equals roughly 4cm per revolution.

The topography of the scratches is optically evaluated at their centre, where the engaged depth is largest. The residual surface aligned with the resulting topography profile line is shown in the bottom of Fig. 5. The residual profile is generated by plotting the median profile height for all data points that lie on a line in cutting direction at a given radial distance x. The median value has proven to be closer to the profile depth of a manually chosen representative section of each scratch than the arithmetic mean.

The residual scratch depth h_res for each pass is identified as the vertical distance from a reference height determined in the unscratched area of the workpiece, shown as a dashed line “assumed surface” in Fig. 5. Cutting force F_c and normal force F_n are identified from the peaks in the respective signals and also aligned with the surface topography. All experimentally determined values are compiled in Table 1.

The determination whether the removal mode is mainly ductile, brittle, or in the transition is subjective and based on the optical appearance of the scratch. Appearance of cracks and pittings in the bottom of scratch is used as a transition criterion; however, also scratches classified as ductile show some fracture on the side.

Cracks on the side of the scratch appear first on the side with the larger engagement (left side in the bottom of Fig. 2). Their appearance is furthermore dominant in the regions of the grain where the cutting edge radius is small in agreement with [24].

It is noteworthy that scratches with large proportions of material removed in ductile cutting mode are observed with residual depths much larger than the critical depth of cut proposed by Bifano et al. and determined to be in the vicinity of 60nm for silicon [34], that is, in spite of elastic recovery which happens after the cut and renders the engaged depth deeper than the residual depth. The observation is, however, in agreement with findings published in [26], where the critical depth of cut for ductile removal of silicon with a cone shaped indenter with a cutting edge radius of 8μm was determined to lay between 0.122 − 1.270μm depending on the crystallographic plane and orientation.

It is furthermore observed, that the scratches 16–18 show a ductile bottom of the scratch with comparably few breakouts on the edges while the scratches before and especially the scratches after them show significant brittle fracture also on the bottom of the scratch. The residual scratch depth decreases over the last scratches in spite of the increasing in-feed (evident in the increasing normal force), which might be attributed to increased amorphisation and associated volumetric expansion [27] and increased elastic recovery.

It was further attempted to quantify the elastic recovery and the undeformed chip thickness from the experimentally determined residual depths by applying models found in literature. For ductile removal, a reasonable undeformed chip thicknesses that can be used in this context could not be determined, which is mainly due to the large deviations between the idealised indenter geometries used to develop the model and the real grain geometry used in this study. The model has to be developed further which goes beyond the scope of this study. For this reason, comparison between experimental and stimulative results is based on residual scratch depth only. The models evaluated are only described briefly and can be referred to in the references mentioned.

For ductile material removal, indentation theory proposed by Oliver and Pharr [35] and a model developed and validated for single grain scratching of silicon by Huang et al. [36] is applied to consider elastic recovery, with the assumption that the grain tip is spherical. For a spherical indenter, the actively engaged tip radius can be determined from the residual scratch width and the contact depth. With a known tip radius and the assumption that the residual scratch width is equal to the width of the tool tip in engagement [35], it is possible to solve the equations for the undeformed chip thickness .

In case of brittle material removal, a model developed by Marshall et al. [21] can be applied to accommodate spalling of material due to brittle fracture. Since both the scratch width and depth are affected by brittle removal, consideration of forces and material parameters is required to determine the undeformed chip thickness. The determination requires the assumption that the depth of the material removed equals the depth of the plastic zone [21] and that material on the side of the groove is removed due to the propagation of lateral cracks to the surface [37]. The plastic zone extends equally in half width and depth and equals the residual, measured depth of the scratch, while the length of the lateral cracks equals half of the measured width of the residual scratch. The engaged depth equals the undeformed chip thickness [21]. Details of the application of this approach to real grain geometries can be found in [38].

For the generation of the residual profile and estimation of the elastic recovery, the SPH particle positions at the simulation end are triangulated with Open3D [40] and the depth coordinates are mapped from the particles to the mesh. Figure 11 shows the residual depth after the cut.

The numerical simulation predicts for the lowest depths of cut h_cu = 200nm and h_cu = 500nm no chip or debris formation, while for cut depths of h_cu = 1000nm and h_cu = 1500nm, a chip develops in front of the tool as well as smaller breakouts and debris sideways of the tool are created. These breakouts are much smaller than those observed in the experiments

The first two simulated scratches with h_cu = 200nm and h_cu = 500nm result in very shallow groves of h_res = 10nm and h_res = 100nm respectively which cannot be matched up with experimental results as the lowest experimentally determined residual depth is 550nm and therefore much larger. The 3rd and 4th simulated scratches produce groves of a residual depth of h_res = 800nm and h_res = 1300nm respectively, which lays in the vicinity of the experimental scratches number 3 to 6, for which an enlarged view is provided in Fig. 12.

The general appearance of the simulated scratches agrees with that of the experimentally cut ones: the left flank in direction of cut is steeper with “break-outs” which are significantly deeper than those on the right flank. The bottom of the grooves gets rougher with increasing residual depths in both cases.

For the lowest simulated depth of cut of h_cu = 200nm, the imprint has a maximum depth of around 10nm which implies an elastic recovery depth h_rec of 190nm. At h_cu = 500nm, the elastic recovery is around 400nm and for h_cu = 1000nm and h_cu = 1500nm about 200nm. In comparison with experimental results obtained from grinding of silicon with grains of similar average size (≈ 25.3μm) and cutting speed of v_c = 44m/s [36], the simulated recovery depth is higher for small depth of cut and lower for high depth of cut. Recovery depth of \(h_{rec, h_{cu} = 200nm} \approx 100 nm\), \(h_{rec, h_{cu} = 500nm} \approx 200 nm\), and \(h_{rec, h_{cu} = 800nm} \approx 300 nm\) were observed in the grinding experiments.

In comparison to the experimental values from Table 1, the predicted process forces are generally higher. The errors in the prediction become smaller towards higher depth of cut h_cu. A comparison of the experimentally and numerically determined forces is provided with Fig. 13 where forces are plotted over the residual depth h_res.

While the numerical cutting force is in an acceptable agreement to the experimental values, the simulated normal force tends to be higher. The reason for the over-estimation of the normal forces may lie with the very low fracture toughness of silicon of < 1 MPam^1/2 [16] which is not considered with the modelling approach here. A small flaw in the workpiece initiates crack growth and brittle fracture with low energy intake. The cracks will not lead to breakouts as long as the hydrostatic stress is compressive; however, the residual hydrostatic stresses in the wake of the tool are partially in tension (refer to chapter Section 4.4) and indicate that the breakouts may likely occur. The ductile deformation and shearing of the material which are simulated require significantly more energy leading to higher forces since fracture is not modelled.

The simulated cutting and normal force characteristics during the cut are shown for all simulations in Fig. 14. At increased depth of cut, the process forces are slightly varying due to formation of chips and small debris as well as torn out material.

The hydrostatic pressure is investigated below the cutting tool where positive values are in compression. The pressure distributions are displayed in Fig. 15 with the tool shown transparent to visualise the hydrostatic pressure distribution below the tool as well. A video of this simulation is uploaded to https://​youtu.​be/​nMTaoP0XIj4.

The increasing contact area with increasing depth of cut can be seen. The region of maximum hydrostatic pressure shifts towards the right side (in cutting direction) of the grain. At the lowest simulated depth of cut h_cu = 200nm, the maximum hydrostatic pressure is in the order 4GPa, at h_cu = 500nm increasing to 13 GPa while with further increasing cut depth the area of highly compressed zones increased steadily and surpassed in some spots even 60GPa. The levels of hydrostatic pressures found in the numerical prediction give rise to the assumption that phase transformation in the workpiece will occur, as for example described in [28] and simulated and validated in [27]. At the lowest cut depth, very small hydrostatic pressures in compression remain in the wake of the tool, while at h_cu = 500nm they are in the order of 3 − 4GPa. At both larger depths of cut of h_cu = 1000nm and h_cu = 1500nm, the residual hydrostatic pressures are lower in the cut surface compared to h_cu = 500nm but exceed 5 GPa locally (green to red in Fig. 15). On the other hand, they turn partially into tension (dark-blue in Fig. 15). This observation is in general agreement with a SPH simulation presented in [41] where cutting of KDP crystal was modelled by means of constitutive model based on the relative length of crack.

Figure 16 shows the stress distributions for the four depths of cut . The equivalent stress reaches up to 5.5 GPa under the tool at h_cu = 200nm and increases locally up to 10–11 GPa for the three higher depths of cut of h_cu = 500nm, h_cu = 1000nm, and h_cu = 1500nm. A video of the simulations is uploaded to https://​youtu.​be/​QZw88fNM1CA.

In the wake of the tool, residual stresses form which, when overlayed with tensile hydrostatic pressures, potentially lead to opening of cracks and breakouts. The cracks are not accommodated for in the Johnson-Cook flow stress model but are visible the bottom of the experimentally generated scratches in Fig. 12.

The predicted peak temperatures in the workpiece strongly depend on the cut depth h_cu and increase from around T = 310K (h_cu = 200nm) up to the melting temperature T = 1700K for h_cu = 1000nm and h_cu = 1500nm, see Table 3 and Fig. 17. The temperatures at the higher depths of cut are in line with observations for experiments conducted in the same manner as the one presented in this study [38] where the flash temperature was measured to lie in the range around 1500K; however, the experimental results show that the temperature decreases with increasing depth of cut and brittle behaviour. At lower cut depths of h_cu = 200nm and h_cu = 500nm, the temperatures are therefore well below the experimental values. For this reason, the accuracy of the temperature prediction is questionable. A video of the simulations is uploaded to https://​youtu.​be/​Vw4zPtsh3Dohttps://​youtu.​be/​Vw4zPtsh3Do.