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Experimental investigations on deformation of soft rock during cutting
International Journal of Rock Mechanics and Mining Sciences 105 (2018) 123–132
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International Journal of Rock Mechanics and Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Experimental investigations on deformation of soft rock during cutting a
b
Shwetabh Yadav , Christopher Saldana , Tejas G. Murthy a b
a,⁎
T
Indian Institute of Science, Bangalore, India The Georgia Institute of Technology, Atlanta, GA, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: Orthogonal cutting Porous brittle solids Image analysis
An experimental study has been made on the deformation of model soft rocks during orthogonal cutting. Through high resolution imaging of orthogonal cutting experiments under 2-D plane-strain conditions, and consequent image analysis, using particle image velocimetry (PIV), we present the evolution of deformation during the cutting process. In addition to examining the deformation fields, and by analysing force signatures, we examine the effect of tool geometry and the depth of cut on the mechanics of cutting in porous brittle solids. In tracking the deformation continuously, we identify the process of fracture initiation and propagation, as well as the corresponding formation of chips. The cutting force was influenced by depth of cut in far more significant way at negative rake angles. We further demonstrate two types of cutting mechanisms occurring with change in tool geometry and draw correspondence between initiation and propagation of fractures and the cyclic nature of force signatures. In the case of machining at negative rake angle, in addition to cyclic increase and decrease of the cutting force, a clear development of a triangular dead zone at the tool tip is measured and the size of this zone also varies cyclically. The volume change occurring in this dead zone formed at the tool corner during negative rake angle cutting is also tracked. By using image analysis, and isolating the geometry of the crack developed during the experiments, we find that the length of crack propagated during cutting of porous brittle solids is a function of the deformation geometry. The effect of rake angle being far more prominent than the depth of cut. The experiments also show that the extent of surface and subsurface deformation during cutting is limited, and the deformation is localized.
1. Introduction Drilling, especially in brittle solids such as rock is often encountered in the petroleum and mining sectors in extraction of hydrocarbons, in pile driving in the infrastructure sector and in biomedical sectors in drilling of hard tissue (e.g., teeth, bone). The deformation in a porous brittle solid is complex because of the propensity for localization, fracture and simultaneously occurring volume change during large deformations. Optimal and efficient execution of drilling necessitates a deep understanding of interaction of various dynamic and kinematic variables, which include constitutivity of the solid, geometry of the drill, weight and torque on the drill bit, among others. The advent of polycrystalline diamond compact (PDC) cutters used in drilling of rocks has only underscored the need for a deeper understanding of the interaction between an individual cutter and the solid. An oft-used approach is to decouple the mechanics of drilling into cutting and indentation.1 Such understanding of the mechanics of cutting and indentation have contributed to optimizing drilling across a range of solids.
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Corresponding author. E-mail address: [email protected] (T.G. Murthy).
https://doi.org/10.1016/j.ijrmms.2018.03.003 Received 28 April 2017; Received in revised form 8 February 2018; Accepted 7 March 2018 1365-1609/ © 2018 Elsevier Ltd. All rights reserved.
The mechanics of severe plastic deformation processes of indentation, cutting and drilling in metals2–4 have provided the necessary theoretical backing for furthering the understanding of deformation processes in a variety of solids. Porous brittle solids (e.g., soft rocks), due to their complex constitutivity, offer further challenges such as fracture, pore collapse in addition to severe plastic deformation. The orthogonal cutting configuration, especially in metals, has offered an ideal platform for a comprehensive understanding of the mechanics of severe plastic deformation in a 2-D platform, affording a construct to study the effect of strain, strain rate and the dissipation of temperature.5 Most of the understanding gained on the mechanics of severe plastic deformation in orthogonal machining of metals has been made by post-mortem analysis of the chip. More recently, utilization of image-based measurement techniques, along with particle image velocimetry (PIV), has opened new doors of understanding6,7 regarding the mechanics of severe plastic deformation in metals and other solids. Additionally, dynamics of the material-tool interaction, quantification of the deformation, chip characteristics, shear band properties and fracture induced in the material through direct observations have all
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that the crack initially moves parallel to the surface of the material and as the bending moment on the crack tip increases, the crack trajectory moves towards the free surface resulting in removal of the chip. Mishnaevsky13 developed a mathematical model using the concept of the energy required for spalling of a chip element, for cutting of brittle materials and derived the expressions for cutting force, depth of damage zone and energy required for the spalling of the chip. This model was backed by experimental observations made during cutting of brittle materials. More recently, analytical and experimental work on brittle solids, such as glass fiber reinforced composites and bulk metallic glasses, have also been studied through cutting experiments. Soldani et al.14 and Arola and Ramulu15 reported a series of 2-D finite element (FE) simulations for orthogonal cutting of Long Fiber Reinforced Polymer (LFRP) to determine the effect of numerical parameters, with validation by experimental results on orthogonal cutting of unidirectional graphite epoxy composite. The effect of numerical parameters (e.g., mesh orientation, energy required for breakage of a mesh element) and tool geometry (e.g., rake angle, cutting edge radius) were analyzed in these studies. From these results, thrust and cutting forces increased with decrease in rake angle and increase in cutting radius. Further, these FE simulations showed that the chip morphology was influenced by rake angle and energy needed for the breakage of an element. Rocks, which are another important class of porous brittle solids, also show a transition from ductile to brittle response when subjected to large deformations, such as that occurring in cutting. FE analysis of rock cutting has revealed some very interesting physics of the response, including transition from ductile to brittle response and the role of individual process parameters. The rake angle and depth of cut used in rock cutting have been shown to have a significant influence on cutting force and chip formation.16,17 In that, cutting forces increased with increase in depth of cut and decrease in rake angle, while the velocity of cut did not have a primary influence on the cutting force. Similar FE modelling was performed to study the cutting by a single PDC cutter18 and has also been extended to 3D using both FE and DEM simulations.19–21 The change in behaviour from ductile to brittle response in rocks depends on the depth of cut of the material and is related to the characteristic length l=KIC/σ where KIC is fracture toughness and σ is the uniaxial compressive strength of the material.22–24 Grima et al.25 studied the effect of hyperbaric pressure on the material removal mechanism of rocks. It was observed that at higher ambient pressure, the chip formation was dominated by shear failure and at low ambient pressure the chip formation was dominated by tension cracks. Rock fragmentation in orthogonal cutting is affected by the rake angle and depth of cut, and is independent of cutting speed.17 The cutting force increases with increase in depth of cut and decrease in the rake angle. Various experimental studies have focused on understanding the effect of cutting tool geometry,26–28 friction between
been observed through image-based measurement techniques. The mechanics of cutting in rocks, representative of a class of natural porous brittle solids, have been analyzed mostly through empirical relationships specifically for certain rock formations. Rigorous analytical and experimental studies have been few and far between. This paper focuses on establishing a physical understanding of the mechanics of cutting in porous brittle solids using image-based experimental measurements. We report a series of orthogonal cutting experiments on a model porous brittle solid. The experiments were conducted by varying generalizable processing parameters in terms of tool geometry and the depth of cut. Images of the region around the tool were captured in time and were analyzed using a particle image analysis algorithm. We further report occurrence of various mechanisms of material removal when the geometry of the cutting configuration is modified. 2. Background The physics and mechanics of metal cutting since the days of its inception have focused around ex situ analyses, including observations of segmented chips, surface folds, wrinkled surfaces, among others. Traditionally, grid-deformation techniques have been used to measure strain in the metal undergoing orthogonal cutting. Parallel marker lines or circles were inscribed on the surface of the metal normal to the cutting direction and the strain was measured by measuring the deflection of the marked lines or the deformation of the circles into ellipses.8–10 These techniques provided quantifiable measures of the final/overall deformation with very little recourse for elucidating the history of the deformation. More recently, the use of image-based measurement techniques have facilitated accurate, in situ measurement at high temporal and spatial resolution of localization features such as surface folds and regions of intense strain rate.6,7 These developments in metal cutting have the potential for informing a high fidelity understanding of the mechanics of cutting in more complex materials, such as that occurring in glass fiber reinforced composites, glass, ceramics, porous brittle solids (e.g., rocks). Glass is often used as a model system for studying brittle solids. In the cutting or machining of glass, the depth of cut employed governs whether the fundamental material removal mechanism occurs in a ductile or brittle mode. In a series of orthogonal cutting experiments on glass,11 the depth of cut was continually increased by tilting workpiece specimens by a small angle. It was observed that at lower depths of cut, the material removal occurred predominantly through plastic deformation of the workpiece (ductile mode) and at higher depths of cut, the material spalled out of the glass through brittle fracture (brittle mode). Complimentarily, Chiu et al.12 have also used an analytical model for a crack propagating along an interface parallel to the free surface during cutting. These finite element analyses have also shown
Fig. 1. Stress-strain curve for 52% porosity samples showing (a) compressive, (b) tensile stress. 124
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cutting tool and the material29 and hydrostatic pressure30 on the cutting force in rock cutting. The application of hydrostatic force increases the cutting force and the causes the fundamental material removal mechanism to change from brittle to ductile-brittle response.30 While these studies have provided extensive empirical evidence of the effect of variables such as tool geometry, friction, temperature and material properties on the cutting response, a comprehensive understanding of the mechanics of cutting in rocks and porous brittle solids has yet to emerge. In this regard, use of image-based measurement techniques (PIV) can provide in situ measurements of velocity field and strain-rate field at both high spatial and temporal resolution, as well as development of fracture and crack length. In the present study, the effects of rake angle and depth of cut on the deformation field in orthogonal cutting of a model porous solid are explored. Images at high spatial and temporal resolution were captured during cutting, and analyzed using an image analysis algorithm. From the observations made on these experiments, mechanisms are proposed through which the severe plastic deformation of cutting occurs in model porous solids. The initiation and propagation of “fracture” through the solid workpiece and the complex interaction of the presence of porosity in the development of various localization features are controlled by the boundary conditions during cutting as discussed in the ensuing.
around the cutting tool were captured using a high-speed camera (PCO1200HS) and microscope (Olympus SZX7) assembly. The images were captured at a constant rate of 4 frames per minute having spatial resolution of 0.010 mm. Representative images for three rake angles and depth of cut of 3.0 mm are shown in Fig. 3. The captured images were analyzed using a PIV algorithm to obtain time varying velocity field maps of the region surrounding the cutting tool. A detailed description of the image analysis algorithm PIV technique is given in Ref..35,36 4. Results We report the results of a series of orthogonal cutting experiments conducted on model gypsum cast at a porosity of 52% with varying rake angles (α) and depth of cut (d) at constant cutting velocity of 1.67 × 10−3 mm/s. The images of the region around the cutting tool are captured using a high-speed camera and microscope assembly. The cutting tool was kept stationary with respect to the camera assembly so that a constant field of view was maintained through the experiment. The images captured during the experiment were analyzed using PIV algorithm and the velocity field contour map of the region surrounding the cutting tool was obtained. 4.1. Positive rake angle
3. Experimental
Fig. 4 and electronic supplementary material, movie M1, shows velocity field contour maps for the case of d = 3.0 mm and α = 30° cutting tool. Velocity field maps captured at 9 different time instances are presented in the figure. A quiver plot was superimposed on the contour maps to identify the direction of material traverse. The horizontal and vertical axis in the velocity field map was normalized with respect to the depth of cut - d. From the figure, two distinct regions can be identified in the velocity field. The first region predominantly occurs below the horizontal level of the cutting tool tip and exhibits a constant velocity in the horizontal direction that is equal to the cutting velocity. The second region that can be clearly differentiated is the chip, which is removed from the material and has velocity higher than the cutting velocity. In addition to in situ imaging of the deformation field, cutting forces were also measured to understand the nature of the loading on the cutting tool. Fig. 5 summarizes the force data during the machining process, measured in the direction of cutting velocity. From the figure, the cutting force oscillates as a function of time, this ranging between 50 N and 180 N throughout the process. To facilitate mapping of these two data sets, time instances corresponding to the velocity field contour maps shown in Fig. 4 are marked as points on the force-time trace in Fig. 5. Using this data, the physics of the cutting process is presented in the ensuing. Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.ijrmms.2018.03.003. The cutting mechanism for the model brittle solid can be classified into three distinct stages. In the first stage, the cutting tool engages with the material as shown in Fig. 4a. When the tool engages with the workpiece, correspondingly, the cutting force reaches a peak value of 164 N. This peak value of the cutting force, marked in Fig. 5 as point ‘a′, showing a local maxima. With progress in the cutting, a second stage in the cutting mechanism can be identified wherein a crack initiates from the tool cutting edge and propagates towards the free surface of the material, resulting in formation of a chip (i.e. removal of a chip from the bulk of the material). This is shown in Fig. 4b where a chip separates from the bulk of the material at a higher velocity than the cutting velocity. This chip moves along the rake face of the cutting tool upon disengaging from the bulk of the material. At this stage, the cutting force drops significantly to 31 N and reaches a local minima, also highlighted in Fig. 5 as point ‘b′. Thirdly, the tool scratches the work material surface until it re-engages as shown in Fig. 4c. At this stage the cutting force increases gradually to 87 N as can be seen from point ‘c′ in
Plane strain orthogonal cutting experiments on model soft rock, gypsum,31–34 were carried out in this study. Samples of 135 mm × 25 mm × 15 mm were prepared by hydration of calcium sulphate hemihydrate (> 97% purity, Sigma Aldrich). Initial water content of 70% by weight of calcium sulphate hemihydrate powder was used to cast samples which resulted in final porosity of 52%. Detailed description about model material and sample preparation technique has been given in.35 The resulting porosity of the as cast solid was measured through mercury intrusion porosimetry (52%). For the given porosity, compression and tension tests have been performed as per ASTM C47299 and ASTM D2936-08 respectively. Fig. 1 shows mean compressive and tensile stress vs strain plot along with standard deviation (surrounding grey region), for three tests each of gypsum samples tested under uniaxial tension conditions. The peak compressive strength was found to be 8.19 MPa with Young's modulus of 0.49 GPa while the peak tensile strength and Young's modulus was significantly lower at 1.42 MPa and 0.025 GPa respectively. The experiments presented here have been performed in orthogonal cutting configuration as shown in Fig. 2, wherein the tool is placed stationary while the work piece was moved through a cutting die. This steel cutting die was designed with rake angles of 30°, 60° and −30° and a relief angle of 5°. 2D plane strain conditions were ensured by pressing a soft tempered glass on the workpiece so as to eliminate out of plane deformation. The experiments were performed at two different depths of cut (d: 3.0 mm and 1.5 mm). The width of cut was 15 mm. A constant cutting speed of 1.67 × 10−3 mm/s was used. The cutting force was measured at a frequency of 10 Hz. The images of the region
Fig. 2. Schematic diagram showing orthogonal cutting. 125
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Fig. 3. Sample images of cutting experiments with (a) −30°, (b) 30° and (c) 60° rake angle cutting tool.
in Fig. 7. With further progress of cutting, some material gets removed from the ‘dead zone’ as well, resulting in the reduction of the size of the ‘dead zone’. Figs. 6f and 6g show that the gradual reduction in the size of the ‘dead zone’. The corresponding points marked in Fig. 7 as point ‘f′ and point ‘g′ show that the size of the dead zone reduces to 1.5 ‘d′ and ‘d′ respectively. With further progress of cutting, the material again starts depositing at the tool cutting edge as shown in Fig. 6h, resulting in the increase of the size of ‘dead zone’ to 2 times ‘d′ (point ‘g′ in Fig. 7). This cycle of increase and decrease in the size of the ‘dead zone’ is repeated with the progress of cutting. The two cycles showing increase and decrease of the size of the ‘dead zone’ is shown in Fig. 6d - 6f and Fig. 6g - 6i. Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.ijrmms.2018.03.003. The size of the dead zone can be elucidated through maps showing velocity discontinuity. Velocity discontinuity maps were obtained by spatial differentiation of the velocity field maps, as given by Eq. (4.2).
Fig. 5. These three distinct stages occur cyclically throughout the cutting process. Further, two cycles of cutting are shown in Fig. 4d-4f and Fig. 4g-4i. The corresponding time instances marked in Fig. 5 exhibit the corresponding cyclic rise and drop in the cutting force values. At the initiation of the cutting, (i.e. the formation of the first chip), we find that the characteristics of the chip are significantly different from the chips formed beyond the first cycle. The chips formed in the first cycle were usually larger than the rest of the chips. Therefore, we refer to the first cycle as a transient stage and is highlighted in Fig. 4a - 4c. 4.2. Negative rake angle Fig. 6 presents the velocity field contour maps at nine time instances for experiments performed with cutting tool of α = −30° and d = 1.5 mm. These velocity field maps can also be seen in the electronic supplementary material, movie M2. In this velocity field map, a third stage can be identified, in addition to the two identified in Fig. 4 for a d = 3.0 mm and an α = 30° cutting tool. This third stage occurs in the vicinity of the cutting tool edge, where the velocity of the work material becomes zero. This is indicative of material in this region forming a ‘dead zone’, which adheres to the tool as the cutting progresses. The normalized size of the ‘dead zone’ in the horizontal direction was calculated at each time instant and plotted in Fig. 7, along with corresponding labeled points for the velocity field maps of Fig. 6. From the figure, the size of this ‘dead zone’ in the horizontal direction varies with time and was always found to be between 1 and 2 times d. The variation of the size of the ‘dead zone’ occurred cyclically, with the progress of the cutting process. At the start of the cutting process, a transient phase is present wherein the size of the ‘dead zone’ increased up to approximately 1.2 times d (e.g., point ‘a′). This transient formation of the dead zone is also clearly shown in Figs. 6a - 6c. As the cutting progresses, the size of ‘dead zone’ increased up to 2 times d as can be seen from point ‘e′
̇ = εxx
̇ = εeff
∂u , ∂x
̇ = εyy
∂v , ∂y
γ ̇ = 2ε ̇xy =
∂v ∂u + , ∂x ∂y
4 1 3 2 2 ⎛ [(εxx ̇ − εyy ̇ )2 + εxx ̇ + ε yy ̇ ] + γ ̇2 ⎞ 9 ⎝2 4 ⎠
(4.1)
(4.2)
Fig. 8 shows the velocity discontinuity maps obtained by spatial differentiation of the velocity field maps (as presented in Fig. 6). The regions showing high values of velocity discontinuity in the order of 5.0 × 10−3 s−1 indicate a drastic jump in the velocity values through a small spatial region. The region of high velocity discontinuity magnitudes surrounding cutting edge of the tool forms the boundary of the ‘dead zone’. Fig. 8a - 8c show a transient stage where initially the boundary of the ‘dead zone’ is diffuse (Fig. 8a), this boundary becomes sharper as the ‘dead zone’ develops (Fig. 8c). Fig. 8d - 8i show that the 126
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Fig. 4. Velocity field contour map showing different stages of cutting with a 30° rake angle tool.
4.3. Cutting force Fig. 9 shows the change in cutting force as a function of the length of traverse during the machining operation (cutting length) plot for three values of α = [−30°, 30°, 60°] and two values of d [1.5 mm, 3 mm]. Figure shows cyclic rise and drop in the force values for these different rake angles. For the case of d = 1.5 mm and α = −30° (Fig. 9a), the maximum cutting force is around 250 N and minimum is about 80 N. A reduction or drop in the cutting force corresponds to initiation of crack in the solid and the gain in the cutting force corresponds to the re-engaging of the cutting tool after the removal of the chip. The distance between two peak values of cutting forces is related to the length of the chip removed from the material. For the case shown in Fig. 9a, the distance between two peaks is around 6 times d. This distance for any particular α and d is not constant due to the porosity and heterogeneity of the material. However, average distance between two peak values of the cutting force increases with increase in depth of cut and with decrease in rake angle.
Fig. 5. Force vs Normalized Cutting Length plot for a 30° rake angle tool.
‘dead zone’ is triangular in shape with one of the edge coinciding with the rake face of the tool and one vertex at the tool cutting edge. The edge opposite the tool tip forms an approximately 45° angle from the vertical direction. The normalized size of the base of the triangle is here taken to represent the characteristic size of the ‘dead zone’. The high velocity discontinuity region adjacent to the rake face is the boundary of the chip climbing up the rake face of the tool.
5. Discussion A series of 2-D orthogonal cutting experiments were performed to understand the mechanics of cutting in a model porous brittle solid as a 127
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Fig. 6. Velocity field contour map showing different stages of cutting with a −30° rake angle tool.
angle, the orthogonal cutting process involved removal of chip from the material predominantly through fracture. In this case, fracture initiates from the tool cutting edge and propagates towards the free surface of the material. The cutting tool scratches the work surface until it reengages with the workpiece and the removal of a subsequent chip occurs. Correspondingly, the cutting force also reaches local minima when the chip is removed and local maxima when the cutting tool re-engages with the material. This cyclic process is depicted in Figs. 4 and 5. For the case of negative rake angle, the cutting process involves crushed rock ahead of the face of the cutting tool. This mode of failure has been described as ductile mode of failure by Huang et al.22 It can also be observed from the present experiments that with an increase in the rake angle, the effect of the depth of cut on cutting force diminishes. This is also clear in Fig. 10 where the cutting force averaged over the cutting length is summarized as a function of the rake angle and depth of cut. The average cutting force for α = −30° is 154.1 N and 355.8 N for d = 1.5 mm and d = 3.0 mm, respectively. As α increased to α = 60°, the average cutting force reduces to 53.3 N and 54.8 N for d = 1.5 mm and
Fig. 7. Normalized size of ‘dead zone’ vs Normalized Cutting Length plot for a −30° rake angle tool.
function of tool rake angle and depth of cut. From our experiments, we observe that the cutting mechanism changes from a ductile to brittle type response as a function of rake angle. For the case of a positive rake 128
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Fig. 8. Contour map showing velocity discontinuity during different stages of cutting with a −30° rake angle tool.
be observed that the horizontal spacing between the points in the bottom three rows, which corresponds to the region of the material which lies below the horizontal level of the cutting tool, remains unchanged. This highlights the fact that there is no deformation in the material below the horizontal level of the cutting tool, unlike observations made of surface deformation in metals.7 However, the horizontal distance between the marker points in the row designated B reduces as the material moves towards to the cutting tool, evident in Fig. 11d - 11f. This reduction in the marker distance is an indicator of localized compression and corresponding densification in the material as the solid moves towards the cutting tool. In this region, the velocity tends towards zero in the vicinity of the cutting tool, indicating the formation of a ‘dead zone’. The first row of markers (row A) of Fig. 11 also shows the reduction in the horizontal distance as the material moves towards the cutting tool, however the trajectory is significantly different. This triangular region between the points taking curved path in the first row and the points in the second row has been marked as ‘dead zone’ in Fig. 11f. The existence of a dead zone is also suggested
d = 3.0 mm, respectively. 5.1. Evolution of dead zone The present experiments also reveal significant differences in the mechanism of cutting in case of a negative α versus a positive α tool. Apart from the cyclic process of chip formation-scratching, development of a ‘dead zone’ also occurs at the tool cutting edge, as shown in Fig. 11f. Fig. 11a shows an image captured during orthogonal cutting experiment with α = −30° cutting tool and d = 1.5 mm. 115 points (markers) are established in the image in 5 horizontal rows. The spacing between each row is 50 pixels (0.5 mm) and between each point the spacing is 10 pixels (0.1 mm). With the progress of cutting, the material traverse throughout the field is tracked by the marker points using the local velocity of the material obtained from the experimental measurements. Fig. 11 shows six of these time instances captured after every 0.6 mm traverse of the material. These grid deformation maps can also be seen in the electronic supplementary material, movie M3. It can 129
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Fig. 9. Cutting force vs. Normalized length of traverse during cutting for (a) α = −30°, d = 1.5 mm, (b) α = 30°, d = 1.5 mm, (c) α = 60°, d = 1.5 mm, (d) α = −30°, d = 3.0 mm, (e) α = 30°, d = 3.0 mm, (f) α = 60°, d = 3.0 mm.
depends primarily on the amount of strain imposed by a given α of the cutting tool. In the current experiments the size of the ‘dead zone’ remains constant for a particular α = −30° cutting tool (Fig. 12). Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.ijrmms.2018.03.003. 5.2. Effect of α and d on crack length The mechanics of cutting in porous brittle solids is dominated by initiation and propagation of fracture. The crack initiates from the tip of the cutting tool and propagates towards the free surface of the material. The removal of the chip by propagation of cracks occurs in cyclic manner as explained above. Variations in the length of the crack in each cycle can be linked to the inhomogeneous distribution of the pores in the material, which serve as nodes of crack initiation and facilitate crack propagation.23 The extent of the crack normalized with respect to the depth of cut is shown in Fig. 13 by mapping the crack path from the PIV. Each panel in Fig. 13 shows the crack path map for multiple trials of the orthogonal cutting experiment for a particular rake angle and depth of cut. In order to obtain a crack path map, an image corresponding to the time instant when the crack propagates to the material surface is considered. A corresponding binary image is obtained by highlighting the crack in black while the rest of the solid work piece is rendered in white. A series of maps corresponding to the time instant when the crack propagates to the material surface for similar depth of cut and rake angle were then overlapped to obtain crack path maps for specific conditions, as shown in Fig. 13. Slight variation in the length of the crack is observed in each case which can be attributed to the stochastic distribution of pores in the material. The crack path for the case of α = 60° is presented in Figs. 13c and 13f. The crack path for this case is initially parallel to the material surface and curves towards the free surface of the material. In contrast, the path of the crack for the case of lower α = [300] sharply tends towards the surface of the solid. A stark difference in the extent of crack in the horizontal direction can be seen when α changes from positive to negative. The crack extent in the horizontal direction changes from around 3 times of d to 7 times of d with the change in α from 30° to −30°, respectively. This signifies that the geometry of the problem has predominant effect on the extent of the
Fig. 10. Average cutting force as a function of α and d.
in16 as a region of crushed material which adheres to the cutting tool and which is similar to the ‘built-up edge (BUE)’ in metal cutting. The normalized size of this ‘dead zone’ in the horizontal direction is calculated with the help of velocity discontinuity maps as explained in Section 4(b). The size of the ‘dead zone’ reduces when the chip is removed from the material because a portion of the ‘dead zone’ is removed along with the chip. The size of the ‘dead zone’ again increases when the cutting tool starts re-engaging the work material and material accumulates near the tool cutting edge. The cyclic character of the ‘dead zone’ size follows variation between 1 and 2 times d for d = 1.5 mm and 0.5–1.0 times d for d = 3.0 mm. We conjecture that for a cutting tool of a given rake angle and friction condition, the size of the ‘dead zone’ remains independent of d. This is based on direct observations of deformation zone size in surface generation processes. For example, Madhavan et al.37 compared the strain imposed in work materials through post process examination of textural lines in metals for the case of wedge indentation and orthogonal machining. They further propose that cutting is mechanistically equivalent to an asymmetric wedge indentation. Additionally, Yadav et al.35 performed wedge indentation experiments on porous brittle solids and showed that the extent of deformation zone depends on the amount of strain applied to the material and it increases with the increase of wedge angle. In the case of the present study, the extent of the ‘dead zone’ 130
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Fig. 11. Particle traverse with progress of cutting for α = −30° and d = 1.5 mm A quantitative assessment of the amount of densification in the ‘dead zone’ is been made by calculating the volume change in the ‘dead zone’. The volume change map has been obtained by the method described in Yadav et al.35 Fig. 12 shows a typical volume change map for α = −30° and d = 1.5 mm. A threshold has been applied to highlight the regions having volume change above 50%. The volume change map shows the rock in the dead zone marked in Fig. 11 is crushed and the voids are reduced leading to the densification of the material in the dead zone.
6. Conclusions The cutting mechanism for porous brittle solids is presented here through a series of orthogonal cutting experiments on a model porous brittle solid for a series of boundary conditions of both positive and negative tool rake angles. In the case of machining with a tool of positive rake angle, the cutting mechanism is dominated by fracture and the initiation of the crack occurs from the cutting edge of the tool. There is a cyclic increase and decrease in the cutting force which corresponds to the engaging of the tool in the material and crack propagation, respectively. The velocity vectors in the velocity field suggest that the material removal occurs due to shear as proposed by Nishimatsu.16 The cutting force for the case of positive rake angle tool was estimated using Nishimatsu's16 model. The theoretical and experimental values of the cutting force were comparable at lower values of α and d whereas for the case of higher values of α and d, a significant deviation of the theoretical estimate from the experimentally obtained cutting force values was observed. In case of cutting with a negative rake angle, a triangular shaped dead zone formed at the cutting edge of the tool and the size of this dead zone changed cyclically with the progress of cutting. The cutting force was also found to be more sensitive to changes in depth of cut for negative rake angle tools. Finally, the length of the crack in this process was found to increase with increase of depth of cut and with reduction in rake angle.
Fig. 12. Volume change map for the ‘dead zone’ observed in cutting with α = −30° and d = 1.5 mm.
crack. However, no significant difference in the crack length in the horizontal direction is observed for α = 30° and 60° cutting tools. For a particular α, the normalized crack length remains almost constant with the change in d, suggesting that the absolute length of the crack changes in the same order of magnitude as d. Lastly, the changes in porosity and friction condition of the material will also significantly affect the cutting force required for severe plastic deformation of the solid. Yadav et al.38 observed indentation force to increase with a decrease in the porosity of similar porous brittle solid. The overall cutting mechanism remained by and large unchanged with change in material porosity. However, the extent of the ‘dead zone’ and extent of the crack path are functions of the material porosity.
Funding statement This work was supported in part by DST grant no. SR-CE-0057-2010 (Indian Institute of Science) and NSF grant nos. CMM I1130852/ 1254818 (The Georgia Institute of Technology).
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Fig. 13. Crack path map for (a) α = −30°, d = 1.5 mm, (b) α = 30°, d = 1.5 mm, (c) α = 60°, d = 1.5 mm, (d) α = −30°, d = 3.0 mm, (e) α = 30°, d = 3.0 mm, (f) α = 60°, d = 3.0 mm.
References
20. Rojek J, Oñate E, Labra C, Kargl H. Discrete element simulation of rock cutting. Int J Rock Mech Min Sci. 2011;48(6):996–1010. 21. Su O, Akcin NA. Numerical simulation of rock cutting using the discrete element method. Int J Rock Mech Min Sci. 2011;48(3):434–442. 22. Huang H, Lecampion B, Detournay E. Discrete element modeling of tool-rock interaction I: rock cutting. Int J Numer Anal Methods Geomech. 2013;37(13):1913–1929. 23. Richard T, Dagrain F, Poyol E, Detournay E. Rock strength determination from scratch tests. Eng Geol. 2012;147:91–100. 24. Zhou Y, Lin JS. Modeling the ductile-brittle failure mode transition in rock cutting. Eng Fract Mech. 2014;127:135–147. 25. Grima MA, Miedema SA, van de Ketterij RG, Yenigül NB, van Rhee C. Effect of high hyperbaric pressure on rock cutting process. Eng Geol. 2015;196:24–36. 26. Appl FC, Wilson CC, Lakshman I. Measurement of forces, temperatures and wear of PDC cutters in rock cutting. Wear. 1993;169(1):9–24. 27. Che D, Zhang W, Ehmann KF. Rock Cutter Interactions in Linear Rock Cutting. In: ASME 2016 11th International Manufacturing Science and Engineering Conference. Blacksburg; June 27-July1 2016: V001T02A071. 28. Zhao XB, Yao XH, Gong QM, Ma HS, Li XZ. Comparison study on rock crack pattern under a single normal and inclined disc cutter by linear cutting experiments. Tunn Undergr Space Technol. 2015;50:479–489. 29. Yahiaoui M, Paris JY, Delbé K, Denape J, Gerbaud L, Dourfaye A. Independent analyses of cutting and friction forces applied on a single polycrystalline diamond compact cutter. Int J Rock Mech Min Sci. 2016;85:20–26. 30. Kaitkay P, Lei S. Experimental study of rock cutting under external hydrostatic pressure. J Mater Process Technol. 2005;159(2):206–213. 31. Bobet A, Einstein HH. Fracture coalescence in rock-type materials under uniaxial and biaxial compression. Int J Rock Mech Min Sci. 1998;35(7):863–888. 32. Reyes O, Einstein HH. Failure mechanisms of fractured rock-a fracture coalescence model. In: 7th ISRM Congress International Society for Rock Mechanics. Aachen; 1620 September 1991: 333–340. 33. Shen B, Stephansson O, Einstein HH, Ghahreman B. Coalescence of fractures under shear stresses in experiments. J Geophys Res Solid Earth. 1995;100(B4):5975–5990. 34. Vekinis G, Ashby MF, Beaumont PW. Plaster of Paris as a model material for brittle porous solids. J Mater Sci. 1993;28(12):3221–3227. 35. Yadav S, Saldana C, Murthy TG. Deformation field evolution in indentation of a porous brittle solid. Int J Solids Struct. 2015;66:35–45. 36. Adrian RJ, Westerweel J. Particle Image Velocimetry. Cambridge University Press; 2011. 37. Madhavan V, Chandrasekar S, Farris TN. Machining as a wedge indentation. J Appl Mech. 2000;67(1):128–139. 38. Yadav S, Saldana C, Murthy TG. Porosity and geometry control ductile to brittle deformation in indentation of porous solids. Int J Solids Struct. 2016;88:11–16.
1. Fairhurst C, Lacabanne WD. Hard rock drilling techniques. Mine Quarry Eng. 1957;23:151–161. 2. Atkins T. The Science and Engineering of Cutting: The Mechanics and Processes of Separating, Scratching and Puncturing Biomaterials, Metals and Non-metals. Butterworth-Heinemann; 2009. 3. Bhattacharyya A. Metal Cutting: Theory and Practice. Jamini Kanta Sen of Central Book Publishers; 1984. 4. Shaw MC. Metal Cutting Principles. New York: Oxford university press; 1984. 5. Brown TL, Saldana C, Murthy TG, et al. A study of the interactive effects of strain, strain rate and temperature in severe plastic deformation of copper. Acta Mater. 2009;57(18):5491–5500. 6. Mahato A, Guo Y, Sundaram NK, Chandrasekar S. Surface folding in metals: a mechanism for delamination wear in sliding. Proc. R. Soc. A. 2014;470:0297. 7. Guo Y, Compton WD, Chandrasekar S. In situ analysis of flow dynamics and deformation fields in cutting and sliding of metals. Proc. R. Soc. A. 2015;471:0194. 8. Nakayama K. The formation of saw-toothed chip in metal cutting. in: Proceedings of International Conference on Production Engineering. Tokyo; 1974. 1974: 572–577. 9. Okushima K, Hitomi K. An analysis of the mechanism of orthogonal cutting and its application to discontinuous chip formation. J Eng Ind. 1961;83(4):545–555. 10. Dautzenberg JH, Zaat JH. Quantitative determination of deformation by sliding wear. Wear. 1973;23(1):9–19. 11. Chiu WC, Endres WJ, Thouless MD. An experimental study of orthogonal machining of glass. Mach Sci Technol. 2000;4(2):253–275. 12. Chiu WC, Thouless MD, Endres WJ. An analysis of chipping in brittle materials. Int J Fract. 1998;90(4):287–298. 13. Mishnaevsky LL. Investigation of the cutting of brittle materials. Int J Mach Tools Manuf. 1994;34(4):499–505. 14. Soldani X, Santiuste C, Muñoz-Sánchez A, Miguélez MH. Influence of tool geometry and numerical parameters when modeling orthogonal cutting of LFRP composites. Compos Part A: Appl Sci Manuf. 2011;42(9):1205–1216. 15. Arola D, Ramulu M. Orthogonal cutting of fiber-reinforced composites: a finite element analysis. Int J Mech Sci. 1997;39(5):597–613. 16. Nishimatsu Y. The mechanics of rock cutting. Int J Rock Mech Min Sci Geomech Abstr. 1972;9(2):261–270. 17. Menezes PL, Lovell MR, Avdeev IV, Higgs CF. Studies on the formation of discontinuous rock fragments during cutting operation. Int J Rock Mech Min Sci. 2014;71:131–142. 18. Jaime MC, Zhou Y, Lin JS, Gamwo IK. Finite element modeling of rock cutting and its fragmentation process. Int J Rock Mech Min Sci. 2015;80:137–146. 19. Huang J, Zhang Y, Zhu L, Wang T. Numerical simulation of rock cutting in deep mining conditions. Int J Rock Mech Min Sci. 2016;84:80–86.
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Contents lists available at ScienceDirect
International Journal of Rock Mechanics and Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Experimental investigations on deformation of soft rock during cutting a
b
Shwetabh Yadav , Christopher Saldana , Tejas G. Murthy a b
a,⁎
T
Indian Institute of Science, Bangalore, India The Georgia Institute of Technology, Atlanta, GA, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: Orthogonal cutting Porous brittle solids Image analysis
An experimental study has been made on the deformation of model soft rocks during orthogonal cutting. Through high resolution imaging of orthogonal cutting experiments under 2-D plane-strain conditions, and consequent image analysis, using particle image velocimetry (PIV), we present the evolution of deformation during the cutting process. In addition to examining the deformation fields, and by analysing force signatures, we examine the effect of tool geometry and the depth of cut on the mechanics of cutting in porous brittle solids. In tracking the deformation continuously, we identify the process of fracture initiation and propagation, as well as the corresponding formation of chips. The cutting force was influenced by depth of cut in far more significant way at negative rake angles. We further demonstrate two types of cutting mechanisms occurring with change in tool geometry and draw correspondence between initiation and propagation of fractures and the cyclic nature of force signatures. In the case of machining at negative rake angle, in addition to cyclic increase and decrease of the cutting force, a clear development of a triangular dead zone at the tool tip is measured and the size of this zone also varies cyclically. The volume change occurring in this dead zone formed at the tool corner during negative rake angle cutting is also tracked. By using image analysis, and isolating the geometry of the crack developed during the experiments, we find that the length of crack propagated during cutting of porous brittle solids is a function of the deformation geometry. The effect of rake angle being far more prominent than the depth of cut. The experiments also show that the extent of surface and subsurface deformation during cutting is limited, and the deformation is localized.
1. Introduction Drilling, especially in brittle solids such as rock is often encountered in the petroleum and mining sectors in extraction of hydrocarbons, in pile driving in the infrastructure sector and in biomedical sectors in drilling of hard tissue (e.g., teeth, bone). The deformation in a porous brittle solid is complex because of the propensity for localization, fracture and simultaneously occurring volume change during large deformations. Optimal and efficient execution of drilling necessitates a deep understanding of interaction of various dynamic and kinematic variables, which include constitutivity of the solid, geometry of the drill, weight and torque on the drill bit, among others. The advent of polycrystalline diamond compact (PDC) cutters used in drilling of rocks has only underscored the need for a deeper understanding of the interaction between an individual cutter and the solid. An oft-used approach is to decouple the mechanics of drilling into cutting and indentation.1 Such understanding of the mechanics of cutting and indentation have contributed to optimizing drilling across a range of solids.
⁎
Corresponding author. E-mail address: [email protected] (T.G. Murthy).
https://doi.org/10.1016/j.ijrmms.2018.03.003 Received 28 April 2017; Received in revised form 8 February 2018; Accepted 7 March 2018 1365-1609/ © 2018 Elsevier Ltd. All rights reserved.
The mechanics of severe plastic deformation processes of indentation, cutting and drilling in metals2–4 have provided the necessary theoretical backing for furthering the understanding of deformation processes in a variety of solids. Porous brittle solids (e.g., soft rocks), due to their complex constitutivity, offer further challenges such as fracture, pore collapse in addition to severe plastic deformation. The orthogonal cutting configuration, especially in metals, has offered an ideal platform for a comprehensive understanding of the mechanics of severe plastic deformation in a 2-D platform, affording a construct to study the effect of strain, strain rate and the dissipation of temperature.5 Most of the understanding gained on the mechanics of severe plastic deformation in orthogonal machining of metals has been made by post-mortem analysis of the chip. More recently, utilization of image-based measurement techniques, along with particle image velocimetry (PIV), has opened new doors of understanding6,7 regarding the mechanics of severe plastic deformation in metals and other solids. Additionally, dynamics of the material-tool interaction, quantification of the deformation, chip characteristics, shear band properties and fracture induced in the material through direct observations have all
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that the crack initially moves parallel to the surface of the material and as the bending moment on the crack tip increases, the crack trajectory moves towards the free surface resulting in removal of the chip. Mishnaevsky13 developed a mathematical model using the concept of the energy required for spalling of a chip element, for cutting of brittle materials and derived the expressions for cutting force, depth of damage zone and energy required for the spalling of the chip. This model was backed by experimental observations made during cutting of brittle materials. More recently, analytical and experimental work on brittle solids, such as glass fiber reinforced composites and bulk metallic glasses, have also been studied through cutting experiments. Soldani et al.14 and Arola and Ramulu15 reported a series of 2-D finite element (FE) simulations for orthogonal cutting of Long Fiber Reinforced Polymer (LFRP) to determine the effect of numerical parameters, with validation by experimental results on orthogonal cutting of unidirectional graphite epoxy composite. The effect of numerical parameters (e.g., mesh orientation, energy required for breakage of a mesh element) and tool geometry (e.g., rake angle, cutting edge radius) were analyzed in these studies. From these results, thrust and cutting forces increased with decrease in rake angle and increase in cutting radius. Further, these FE simulations showed that the chip morphology was influenced by rake angle and energy needed for the breakage of an element. Rocks, which are another important class of porous brittle solids, also show a transition from ductile to brittle response when subjected to large deformations, such as that occurring in cutting. FE analysis of rock cutting has revealed some very interesting physics of the response, including transition from ductile to brittle response and the role of individual process parameters. The rake angle and depth of cut used in rock cutting have been shown to have a significant influence on cutting force and chip formation.16,17 In that, cutting forces increased with increase in depth of cut and decrease in rake angle, while the velocity of cut did not have a primary influence on the cutting force. Similar FE modelling was performed to study the cutting by a single PDC cutter18 and has also been extended to 3D using both FE and DEM simulations.19–21 The change in behaviour from ductile to brittle response in rocks depends on the depth of cut of the material and is related to the characteristic length l=KIC/σ where KIC is fracture toughness and σ is the uniaxial compressive strength of the material.22–24 Grima et al.25 studied the effect of hyperbaric pressure on the material removal mechanism of rocks. It was observed that at higher ambient pressure, the chip formation was dominated by shear failure and at low ambient pressure the chip formation was dominated by tension cracks. Rock fragmentation in orthogonal cutting is affected by the rake angle and depth of cut, and is independent of cutting speed.17 The cutting force increases with increase in depth of cut and decrease in the rake angle. Various experimental studies have focused on understanding the effect of cutting tool geometry,26–28 friction between
been observed through image-based measurement techniques. The mechanics of cutting in rocks, representative of a class of natural porous brittle solids, have been analyzed mostly through empirical relationships specifically for certain rock formations. Rigorous analytical and experimental studies have been few and far between. This paper focuses on establishing a physical understanding of the mechanics of cutting in porous brittle solids using image-based experimental measurements. We report a series of orthogonal cutting experiments on a model porous brittle solid. The experiments were conducted by varying generalizable processing parameters in terms of tool geometry and the depth of cut. Images of the region around the tool were captured in time and were analyzed using a particle image analysis algorithm. We further report occurrence of various mechanisms of material removal when the geometry of the cutting configuration is modified. 2. Background The physics and mechanics of metal cutting since the days of its inception have focused around ex situ analyses, including observations of segmented chips, surface folds, wrinkled surfaces, among others. Traditionally, grid-deformation techniques have been used to measure strain in the metal undergoing orthogonal cutting. Parallel marker lines or circles were inscribed on the surface of the metal normal to the cutting direction and the strain was measured by measuring the deflection of the marked lines or the deformation of the circles into ellipses.8–10 These techniques provided quantifiable measures of the final/overall deformation with very little recourse for elucidating the history of the deformation. More recently, the use of image-based measurement techniques have facilitated accurate, in situ measurement at high temporal and spatial resolution of localization features such as surface folds and regions of intense strain rate.6,7 These developments in metal cutting have the potential for informing a high fidelity understanding of the mechanics of cutting in more complex materials, such as that occurring in glass fiber reinforced composites, glass, ceramics, porous brittle solids (e.g., rocks). Glass is often used as a model system for studying brittle solids. In the cutting or machining of glass, the depth of cut employed governs whether the fundamental material removal mechanism occurs in a ductile or brittle mode. In a series of orthogonal cutting experiments on glass,11 the depth of cut was continually increased by tilting workpiece specimens by a small angle. It was observed that at lower depths of cut, the material removal occurred predominantly through plastic deformation of the workpiece (ductile mode) and at higher depths of cut, the material spalled out of the glass through brittle fracture (brittle mode). Complimentarily, Chiu et al.12 have also used an analytical model for a crack propagating along an interface parallel to the free surface during cutting. These finite element analyses have also shown
Fig. 1. Stress-strain curve for 52% porosity samples showing (a) compressive, (b) tensile stress. 124
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cutting tool and the material29 and hydrostatic pressure30 on the cutting force in rock cutting. The application of hydrostatic force increases the cutting force and the causes the fundamental material removal mechanism to change from brittle to ductile-brittle response.30 While these studies have provided extensive empirical evidence of the effect of variables such as tool geometry, friction, temperature and material properties on the cutting response, a comprehensive understanding of the mechanics of cutting in rocks and porous brittle solids has yet to emerge. In this regard, use of image-based measurement techniques (PIV) can provide in situ measurements of velocity field and strain-rate field at both high spatial and temporal resolution, as well as development of fracture and crack length. In the present study, the effects of rake angle and depth of cut on the deformation field in orthogonal cutting of a model porous solid are explored. Images at high spatial and temporal resolution were captured during cutting, and analyzed using an image analysis algorithm. From the observations made on these experiments, mechanisms are proposed through which the severe plastic deformation of cutting occurs in model porous solids. The initiation and propagation of “fracture” through the solid workpiece and the complex interaction of the presence of porosity in the development of various localization features are controlled by the boundary conditions during cutting as discussed in the ensuing.
around the cutting tool were captured using a high-speed camera (PCO1200HS) and microscope (Olympus SZX7) assembly. The images were captured at a constant rate of 4 frames per minute having spatial resolution of 0.010 mm. Representative images for three rake angles and depth of cut of 3.0 mm are shown in Fig. 3. The captured images were analyzed using a PIV algorithm to obtain time varying velocity field maps of the region surrounding the cutting tool. A detailed description of the image analysis algorithm PIV technique is given in Ref..35,36 4. Results We report the results of a series of orthogonal cutting experiments conducted on model gypsum cast at a porosity of 52% with varying rake angles (α) and depth of cut (d) at constant cutting velocity of 1.67 × 10−3 mm/s. The images of the region around the cutting tool are captured using a high-speed camera and microscope assembly. The cutting tool was kept stationary with respect to the camera assembly so that a constant field of view was maintained through the experiment. The images captured during the experiment were analyzed using PIV algorithm and the velocity field contour map of the region surrounding the cutting tool was obtained. 4.1. Positive rake angle
3. Experimental
Fig. 4 and electronic supplementary material, movie M1, shows velocity field contour maps for the case of d = 3.0 mm and α = 30° cutting tool. Velocity field maps captured at 9 different time instances are presented in the figure. A quiver plot was superimposed on the contour maps to identify the direction of material traverse. The horizontal and vertical axis in the velocity field map was normalized with respect to the depth of cut - d. From the figure, two distinct regions can be identified in the velocity field. The first region predominantly occurs below the horizontal level of the cutting tool tip and exhibits a constant velocity in the horizontal direction that is equal to the cutting velocity. The second region that can be clearly differentiated is the chip, which is removed from the material and has velocity higher than the cutting velocity. In addition to in situ imaging of the deformation field, cutting forces were also measured to understand the nature of the loading on the cutting tool. Fig. 5 summarizes the force data during the machining process, measured in the direction of cutting velocity. From the figure, the cutting force oscillates as a function of time, this ranging between 50 N and 180 N throughout the process. To facilitate mapping of these two data sets, time instances corresponding to the velocity field contour maps shown in Fig. 4 are marked as points on the force-time trace in Fig. 5. Using this data, the physics of the cutting process is presented in the ensuing. Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.ijrmms.2018.03.003. The cutting mechanism for the model brittle solid can be classified into three distinct stages. In the first stage, the cutting tool engages with the material as shown in Fig. 4a. When the tool engages with the workpiece, correspondingly, the cutting force reaches a peak value of 164 N. This peak value of the cutting force, marked in Fig. 5 as point ‘a′, showing a local maxima. With progress in the cutting, a second stage in the cutting mechanism can be identified wherein a crack initiates from the tool cutting edge and propagates towards the free surface of the material, resulting in formation of a chip (i.e. removal of a chip from the bulk of the material). This is shown in Fig. 4b where a chip separates from the bulk of the material at a higher velocity than the cutting velocity. This chip moves along the rake face of the cutting tool upon disengaging from the bulk of the material. At this stage, the cutting force drops significantly to 31 N and reaches a local minima, also highlighted in Fig. 5 as point ‘b′. Thirdly, the tool scratches the work material surface until it re-engages as shown in Fig. 4c. At this stage the cutting force increases gradually to 87 N as can be seen from point ‘c′ in
Plane strain orthogonal cutting experiments on model soft rock, gypsum,31–34 were carried out in this study. Samples of 135 mm × 25 mm × 15 mm were prepared by hydration of calcium sulphate hemihydrate (> 97% purity, Sigma Aldrich). Initial water content of 70% by weight of calcium sulphate hemihydrate powder was used to cast samples which resulted in final porosity of 52%. Detailed description about model material and sample preparation technique has been given in.35 The resulting porosity of the as cast solid was measured through mercury intrusion porosimetry (52%). For the given porosity, compression and tension tests have been performed as per ASTM C47299 and ASTM D2936-08 respectively. Fig. 1 shows mean compressive and tensile stress vs strain plot along with standard deviation (surrounding grey region), for three tests each of gypsum samples tested under uniaxial tension conditions. The peak compressive strength was found to be 8.19 MPa with Young's modulus of 0.49 GPa while the peak tensile strength and Young's modulus was significantly lower at 1.42 MPa and 0.025 GPa respectively. The experiments presented here have been performed in orthogonal cutting configuration as shown in Fig. 2, wherein the tool is placed stationary while the work piece was moved through a cutting die. This steel cutting die was designed with rake angles of 30°, 60° and −30° and a relief angle of 5°. 2D plane strain conditions were ensured by pressing a soft tempered glass on the workpiece so as to eliminate out of plane deformation. The experiments were performed at two different depths of cut (d: 3.0 mm and 1.5 mm). The width of cut was 15 mm. A constant cutting speed of 1.67 × 10−3 mm/s was used. The cutting force was measured at a frequency of 10 Hz. The images of the region
Fig. 2. Schematic diagram showing orthogonal cutting. 125
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Fig. 3. Sample images of cutting experiments with (a) −30°, (b) 30° and (c) 60° rake angle cutting tool.
in Fig. 7. With further progress of cutting, some material gets removed from the ‘dead zone’ as well, resulting in the reduction of the size of the ‘dead zone’. Figs. 6f and 6g show that the gradual reduction in the size of the ‘dead zone’. The corresponding points marked in Fig. 7 as point ‘f′ and point ‘g′ show that the size of the dead zone reduces to 1.5 ‘d′ and ‘d′ respectively. With further progress of cutting, the material again starts depositing at the tool cutting edge as shown in Fig. 6h, resulting in the increase of the size of ‘dead zone’ to 2 times ‘d′ (point ‘g′ in Fig. 7). This cycle of increase and decrease in the size of the ‘dead zone’ is repeated with the progress of cutting. The two cycles showing increase and decrease of the size of the ‘dead zone’ is shown in Fig. 6d - 6f and Fig. 6g - 6i. Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.ijrmms.2018.03.003. The size of the dead zone can be elucidated through maps showing velocity discontinuity. Velocity discontinuity maps were obtained by spatial differentiation of the velocity field maps, as given by Eq. (4.2).
Fig. 5. These three distinct stages occur cyclically throughout the cutting process. Further, two cycles of cutting are shown in Fig. 4d-4f and Fig. 4g-4i. The corresponding time instances marked in Fig. 5 exhibit the corresponding cyclic rise and drop in the cutting force values. At the initiation of the cutting, (i.e. the formation of the first chip), we find that the characteristics of the chip are significantly different from the chips formed beyond the first cycle. The chips formed in the first cycle were usually larger than the rest of the chips. Therefore, we refer to the first cycle as a transient stage and is highlighted in Fig. 4a - 4c. 4.2. Negative rake angle Fig. 6 presents the velocity field contour maps at nine time instances for experiments performed with cutting tool of α = −30° and d = 1.5 mm. These velocity field maps can also be seen in the electronic supplementary material, movie M2. In this velocity field map, a third stage can be identified, in addition to the two identified in Fig. 4 for a d = 3.0 mm and an α = 30° cutting tool. This third stage occurs in the vicinity of the cutting tool edge, where the velocity of the work material becomes zero. This is indicative of material in this region forming a ‘dead zone’, which adheres to the tool as the cutting progresses. The normalized size of the ‘dead zone’ in the horizontal direction was calculated at each time instant and plotted in Fig. 7, along with corresponding labeled points for the velocity field maps of Fig. 6. From the figure, the size of this ‘dead zone’ in the horizontal direction varies with time and was always found to be between 1 and 2 times d. The variation of the size of the ‘dead zone’ occurred cyclically, with the progress of the cutting process. At the start of the cutting process, a transient phase is present wherein the size of the ‘dead zone’ increased up to approximately 1.2 times d (e.g., point ‘a′). This transient formation of the dead zone is also clearly shown in Figs. 6a - 6c. As the cutting progresses, the size of ‘dead zone’ increased up to 2 times d as can be seen from point ‘e′
̇ = εxx
̇ = εeff
∂u , ∂x
̇ = εyy
∂v , ∂y
γ ̇ = 2ε ̇xy =
∂v ∂u + , ∂x ∂y
4 1 3 2 2 ⎛ [(εxx ̇ − εyy ̇ )2 + εxx ̇ + ε yy ̇ ] + γ ̇2 ⎞ 9 ⎝2 4 ⎠
(4.1)
(4.2)
Fig. 8 shows the velocity discontinuity maps obtained by spatial differentiation of the velocity field maps (as presented in Fig. 6). The regions showing high values of velocity discontinuity in the order of 5.0 × 10−3 s−1 indicate a drastic jump in the velocity values through a small spatial region. The region of high velocity discontinuity magnitudes surrounding cutting edge of the tool forms the boundary of the ‘dead zone’. Fig. 8a - 8c show a transient stage where initially the boundary of the ‘dead zone’ is diffuse (Fig. 8a), this boundary becomes sharper as the ‘dead zone’ develops (Fig. 8c). Fig. 8d - 8i show that the 126
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Fig. 4. Velocity field contour map showing different stages of cutting with a 30° rake angle tool.
4.3. Cutting force Fig. 9 shows the change in cutting force as a function of the length of traverse during the machining operation (cutting length) plot for three values of α = [−30°, 30°, 60°] and two values of d [1.5 mm, 3 mm]. Figure shows cyclic rise and drop in the force values for these different rake angles. For the case of d = 1.5 mm and α = −30° (Fig. 9a), the maximum cutting force is around 250 N and minimum is about 80 N. A reduction or drop in the cutting force corresponds to initiation of crack in the solid and the gain in the cutting force corresponds to the re-engaging of the cutting tool after the removal of the chip. The distance between two peak values of cutting forces is related to the length of the chip removed from the material. For the case shown in Fig. 9a, the distance between two peaks is around 6 times d. This distance for any particular α and d is not constant due to the porosity and heterogeneity of the material. However, average distance between two peak values of the cutting force increases with increase in depth of cut and with decrease in rake angle.
Fig. 5. Force vs Normalized Cutting Length plot for a 30° rake angle tool.
‘dead zone’ is triangular in shape with one of the edge coinciding with the rake face of the tool and one vertex at the tool cutting edge. The edge opposite the tool tip forms an approximately 45° angle from the vertical direction. The normalized size of the base of the triangle is here taken to represent the characteristic size of the ‘dead zone’. The high velocity discontinuity region adjacent to the rake face is the boundary of the chip climbing up the rake face of the tool.
5. Discussion A series of 2-D orthogonal cutting experiments were performed to understand the mechanics of cutting in a model porous brittle solid as a 127
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Fig. 6. Velocity field contour map showing different stages of cutting with a −30° rake angle tool.
angle, the orthogonal cutting process involved removal of chip from the material predominantly through fracture. In this case, fracture initiates from the tool cutting edge and propagates towards the free surface of the material. The cutting tool scratches the work surface until it reengages with the workpiece and the removal of a subsequent chip occurs. Correspondingly, the cutting force also reaches local minima when the chip is removed and local maxima when the cutting tool re-engages with the material. This cyclic process is depicted in Figs. 4 and 5. For the case of negative rake angle, the cutting process involves crushed rock ahead of the face of the cutting tool. This mode of failure has been described as ductile mode of failure by Huang et al.22 It can also be observed from the present experiments that with an increase in the rake angle, the effect of the depth of cut on cutting force diminishes. This is also clear in Fig. 10 where the cutting force averaged over the cutting length is summarized as a function of the rake angle and depth of cut. The average cutting force for α = −30° is 154.1 N and 355.8 N for d = 1.5 mm and d = 3.0 mm, respectively. As α increased to α = 60°, the average cutting force reduces to 53.3 N and 54.8 N for d = 1.5 mm and
Fig. 7. Normalized size of ‘dead zone’ vs Normalized Cutting Length plot for a −30° rake angle tool.
function of tool rake angle and depth of cut. From our experiments, we observe that the cutting mechanism changes from a ductile to brittle type response as a function of rake angle. For the case of a positive rake 128
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Fig. 8. Contour map showing velocity discontinuity during different stages of cutting with a −30° rake angle tool.
be observed that the horizontal spacing between the points in the bottom three rows, which corresponds to the region of the material which lies below the horizontal level of the cutting tool, remains unchanged. This highlights the fact that there is no deformation in the material below the horizontal level of the cutting tool, unlike observations made of surface deformation in metals.7 However, the horizontal distance between the marker points in the row designated B reduces as the material moves towards to the cutting tool, evident in Fig. 11d - 11f. This reduction in the marker distance is an indicator of localized compression and corresponding densification in the material as the solid moves towards the cutting tool. In this region, the velocity tends towards zero in the vicinity of the cutting tool, indicating the formation of a ‘dead zone’. The first row of markers (row A) of Fig. 11 also shows the reduction in the horizontal distance as the material moves towards the cutting tool, however the trajectory is significantly different. This triangular region between the points taking curved path in the first row and the points in the second row has been marked as ‘dead zone’ in Fig. 11f. The existence of a dead zone is also suggested
d = 3.0 mm, respectively. 5.1. Evolution of dead zone The present experiments also reveal significant differences in the mechanism of cutting in case of a negative α versus a positive α tool. Apart from the cyclic process of chip formation-scratching, development of a ‘dead zone’ also occurs at the tool cutting edge, as shown in Fig. 11f. Fig. 11a shows an image captured during orthogonal cutting experiment with α = −30° cutting tool and d = 1.5 mm. 115 points (markers) are established in the image in 5 horizontal rows. The spacing between each row is 50 pixels (0.5 mm) and between each point the spacing is 10 pixels (0.1 mm). With the progress of cutting, the material traverse throughout the field is tracked by the marker points using the local velocity of the material obtained from the experimental measurements. Fig. 11 shows six of these time instances captured after every 0.6 mm traverse of the material. These grid deformation maps can also be seen in the electronic supplementary material, movie M3. It can 129
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Fig. 9. Cutting force vs. Normalized length of traverse during cutting for (a) α = −30°, d = 1.5 mm, (b) α = 30°, d = 1.5 mm, (c) α = 60°, d = 1.5 mm, (d) α = −30°, d = 3.0 mm, (e) α = 30°, d = 3.0 mm, (f) α = 60°, d = 3.0 mm.
depends primarily on the amount of strain imposed by a given α of the cutting tool. In the current experiments the size of the ‘dead zone’ remains constant for a particular α = −30° cutting tool (Fig. 12). Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.ijrmms.2018.03.003. 5.2. Effect of α and d on crack length The mechanics of cutting in porous brittle solids is dominated by initiation and propagation of fracture. The crack initiates from the tip of the cutting tool and propagates towards the free surface of the material. The removal of the chip by propagation of cracks occurs in cyclic manner as explained above. Variations in the length of the crack in each cycle can be linked to the inhomogeneous distribution of the pores in the material, which serve as nodes of crack initiation and facilitate crack propagation.23 The extent of the crack normalized with respect to the depth of cut is shown in Fig. 13 by mapping the crack path from the PIV. Each panel in Fig. 13 shows the crack path map for multiple trials of the orthogonal cutting experiment for a particular rake angle and depth of cut. In order to obtain a crack path map, an image corresponding to the time instant when the crack propagates to the material surface is considered. A corresponding binary image is obtained by highlighting the crack in black while the rest of the solid work piece is rendered in white. A series of maps corresponding to the time instant when the crack propagates to the material surface for similar depth of cut and rake angle were then overlapped to obtain crack path maps for specific conditions, as shown in Fig. 13. Slight variation in the length of the crack is observed in each case which can be attributed to the stochastic distribution of pores in the material. The crack path for the case of α = 60° is presented in Figs. 13c and 13f. The crack path for this case is initially parallel to the material surface and curves towards the free surface of the material. In contrast, the path of the crack for the case of lower α = [300] sharply tends towards the surface of the solid. A stark difference in the extent of crack in the horizontal direction can be seen when α changes from positive to negative. The crack extent in the horizontal direction changes from around 3 times of d to 7 times of d with the change in α from 30° to −30°, respectively. This signifies that the geometry of the problem has predominant effect on the extent of the
Fig. 10. Average cutting force as a function of α and d.
in16 as a region of crushed material which adheres to the cutting tool and which is similar to the ‘built-up edge (BUE)’ in metal cutting. The normalized size of this ‘dead zone’ in the horizontal direction is calculated with the help of velocity discontinuity maps as explained in Section 4(b). The size of the ‘dead zone’ reduces when the chip is removed from the material because a portion of the ‘dead zone’ is removed along with the chip. The size of the ‘dead zone’ again increases when the cutting tool starts re-engaging the work material and material accumulates near the tool cutting edge. The cyclic character of the ‘dead zone’ size follows variation between 1 and 2 times d for d = 1.5 mm and 0.5–1.0 times d for d = 3.0 mm. We conjecture that for a cutting tool of a given rake angle and friction condition, the size of the ‘dead zone’ remains independent of d. This is based on direct observations of deformation zone size in surface generation processes. For example, Madhavan et al.37 compared the strain imposed in work materials through post process examination of textural lines in metals for the case of wedge indentation and orthogonal machining. They further propose that cutting is mechanistically equivalent to an asymmetric wedge indentation. Additionally, Yadav et al.35 performed wedge indentation experiments on porous brittle solids and showed that the extent of deformation zone depends on the amount of strain applied to the material and it increases with the increase of wedge angle. In the case of the present study, the extent of the ‘dead zone’ 130
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Fig. 11. Particle traverse with progress of cutting for α = −30° and d = 1.5 mm A quantitative assessment of the amount of densification in the ‘dead zone’ is been made by calculating the volume change in the ‘dead zone’. The volume change map has been obtained by the method described in Yadav et al.35 Fig. 12 shows a typical volume change map for α = −30° and d = 1.5 mm. A threshold has been applied to highlight the regions having volume change above 50%. The volume change map shows the rock in the dead zone marked in Fig. 11 is crushed and the voids are reduced leading to the densification of the material in the dead zone.
6. Conclusions The cutting mechanism for porous brittle solids is presented here through a series of orthogonal cutting experiments on a model porous brittle solid for a series of boundary conditions of both positive and negative tool rake angles. In the case of machining with a tool of positive rake angle, the cutting mechanism is dominated by fracture and the initiation of the crack occurs from the cutting edge of the tool. There is a cyclic increase and decrease in the cutting force which corresponds to the engaging of the tool in the material and crack propagation, respectively. The velocity vectors in the velocity field suggest that the material removal occurs due to shear as proposed by Nishimatsu.16 The cutting force for the case of positive rake angle tool was estimated using Nishimatsu's16 model. The theoretical and experimental values of the cutting force were comparable at lower values of α and d whereas for the case of higher values of α and d, a significant deviation of the theoretical estimate from the experimentally obtained cutting force values was observed. In case of cutting with a negative rake angle, a triangular shaped dead zone formed at the cutting edge of the tool and the size of this dead zone changed cyclically with the progress of cutting. The cutting force was also found to be more sensitive to changes in depth of cut for negative rake angle tools. Finally, the length of the crack in this process was found to increase with increase of depth of cut and with reduction in rake angle.
Fig. 12. Volume change map for the ‘dead zone’ observed in cutting with α = −30° and d = 1.5 mm.
crack. However, no significant difference in the crack length in the horizontal direction is observed for α = 30° and 60° cutting tools. For a particular α, the normalized crack length remains almost constant with the change in d, suggesting that the absolute length of the crack changes in the same order of magnitude as d. Lastly, the changes in porosity and friction condition of the material will also significantly affect the cutting force required for severe plastic deformation of the solid. Yadav et al.38 observed indentation force to increase with a decrease in the porosity of similar porous brittle solid. The overall cutting mechanism remained by and large unchanged with change in material porosity. However, the extent of the ‘dead zone’ and extent of the crack path are functions of the material porosity.
Funding statement This work was supported in part by DST grant no. SR-CE-0057-2010 (Indian Institute of Science) and NSF grant nos. CMM I1130852/ 1254818 (The Georgia Institute of Technology).
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Fig. 13. Crack path map for (a) α = −30°, d = 1.5 mm, (b) α = 30°, d = 1.5 mm, (c) α = 60°, d = 1.5 mm, (d) α = −30°, d = 3.0 mm, (e) α = 30°, d = 3.0 mm, (f) α = 60°, d = 3.0 mm.
References
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