MODELING OF THERMOMECHANICAL SHEAR INSTABILITY IN MACHINING

Pergamon Int J Mech Sci Vol 39, No 11, pp 12791314, 1997 ,~ 1997 Elsevier Science l_td Printed in Great Britain All rights reserved 0020-7403/97 $17.00 + 0.00 PII: S00207403(97)00017-9 M O D E L I N G O F T H E R M O M E C H A N I C A L SHEAR INSTABILITY IN MACHINING ZHEN BING HOU and RANGA KOMANDURI Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK 74078-5016, U.S.A (Received May 1996; and in revisedform 21 October 1996) Abstract The modeling of thermomechanical shear instability in the machining of some difficult-to-machine materials leading to shear localization is presented Shear instability was observed experimentally in high-speed machining (HSM) of some of the diffficult-to-machine materials, such as hardened alloy steels (e.g AISI 4340 steel), titanium alloys (e.g Ti-6AI-4V), and nickel-base superalloys (e.g Incone1718) yielding cyclic chips Based on an analysis of cyclic chip formation in machining, possible sources of heat (including preheating effects) contributing toward the temperature rise in the shear band are identified They include the four primary heat sources, the four preheating effects of the primary heat sources, the image heat source due to the primary shear band heat source, and the preheating effect of this heat source The temperature rise in the shear band due to each of the heat sources is calculated using Jaeger's classical model for stationary and moving heat sources Based on this temperature, Recht's classical model of catastrophic shear instability in metals under dynamic plastic conditions developed in 1964 is extended by predicting analytically the conditions for the onset of shear localization This is done by comparing the strength of material in the shear band, a' under the combined effects of thermal softening and strain hardening with that of the material in the vicinity of the shear band, a where the material has undergone small strains (i.e up to yield point) and at the temperature caused by the preheating heat sources Thus, a is nearly the original strength of the work material If a' < a, thermal softening predominates at the shear band and shear localization will be imminent The cutting speed for the onset of shear localization can be predicted based on thermomechanical shear instability model presented here High-speed machining results reported in the literature for an AISI 4340 steel agree reasonably well with the analytical values developed in this investigation © 1997 Elsevier Science Ltd Keywords: high-speed machining, shear instability, machining NOTATION heat liberated per unit length of the heat source, cal/cm q, intensity of heat liberation of the continuous line heat source, cal/cm s qpl intensity of heat liberation of the plane heat source, cal/cm s (suffix "max" indicates the maximum value,,) qo heat liberation rate of the stationary plane heat source (main shear band heat source), cal/cm s a thermal diffusivity, cm2/s thermal conductivity, cal/cm s °C C specific heat, cal/g °C P density, g/cm OM temperature rise at any point M in the conducting medium, °C r i distance between point M and the line heat source (a segment of the plane heat source), cm cutting depth, cm ao width of the main shear plane, cm X , Y coordinates of point M in the moving coordinate system in X and Y directions W width of the moving plane heat source at the time of observation, t, cm V velocity of the moving plane heat source in the direction of main shear plane, m/s v velocity of chip segment moving along main shear plane, m/s v~ the equivalent sliding speed along the secondary shear plane, m/s Vo cutting speed, m/s v sliding speed of chip segment along the tool rake face, m/s OZ rake angle of the cutting tool, deg ¢ shear angle of the main shear plane ¢, shear angle of the secondary shear plane t o r z time of observation or the time of that moment when the temperature rise is to be determined, s toi time when the heat source segment (regarded as a line heat source) begins to work, s t h maximum time of duration of working of first and second heat sources, s t o time required for a chip segment to move along the main shear plane through a distance, l, s Ty shear stress of the material, kgf/cm QI 1273 1274 Z.B Hou and R Komanduri a true stress of the material, kgf/cm~ e true strain of the material INTRODUCTION In the machining of some of the difficult-to-machine materials, such as hardened alloy steels, titanium alloys, and nickel-base superalloys, instabilities in the cutting process occur as the cutting speed is varied [1-4] These instabilities are a result of thermomechanical response of the work material under the conditions of cutting The consequence is localized shear and cyclic chip formation Shear localization results in cyclic variation of forces (both cutting and thrust) of significant magnitude and consequent vibration or chatter in the metal cutting process This occurs even with a very stiff machine tool-cutting tool-workpiece system [ 1] If, however, the stiffness of the system is low, self-excited vibrations may result, especially when the frequency of force excitation coincides with one of the natural frequencies of the machine tool system Also, high tool-chip interface temperatures (close to the melting temperature of the work material) are predicted :for this case especially at higher cutting speeds Consequently, an understanding of the process, the ,criteria for shear instability, and the conditions leading to shear localization are important considerations in our quest for improving productivity, tool life, and part quality as well as the overall efficiency of the cutting operation Shear localized chips [Fig l(a)] are likely to form in the machining of materials with a limited slip system (e.g hcp crystalline structure), poor thermal properties, and high hardnes.s, such as hardened alloy steels, titanium alloys, and nickel-base superalloys In contrast, continuous chips [Fig l(b)] are likely to form in the machining of materials with extensive slip system (e.g fcc/bcc crystalline structure), good thermal properties, and low hardness, such as conventional aluminum alloys (e.g A1 6061-T6) and low carbon steels (e.g AISI 1018 steel) While there :may be a transition from a continuous chip to a shear-localized chip with increase in cutting speed for some materials (e.g AISI 4340 steel and nickel-base superalloys), this type of chip persists with further increase in speed (with no additional transitions either into other chip forms or reversal to a continuous chip) at least up to 30,488 m/min (or 100,000ft/min) For example, for an AISI 4340 steel (325 BHN) at low speeds (below 15.25m/min) the chips generated are discontinuous and change to continuous in the speed range of 31-61 m/min At cutting speeds higher than this, the deformation of the chip is inhomogeneous on a gross level with two regions, one narrow band where deformation is very high (i.e between the segments) and the other where deformation is relatively low (i.e within the segments) Examination of the polished and etched chip midsections of an AISI 4340 steel (325 BHN) showed a continuous chip at 38 ml/min, a transition to a shear-localized chip at 122 m/min, to a fully developed shear-localized chip at 244 m/rain, and to a shear localized chip where the segments are almost completely detached at 975 m/min [8] The transition speed at which the chip form changes from a continuous to a shear-localized was found to be different for different work materials For example, it is only a few m/min or less in the case of titanium alloys [5, 6], about 61 m/min in the case of nickel-base superalloys [7], and to begin above 61 m/min and complete at about 244 m/min in the case of AISI 4340 steel (325 BHN) [18] The speed at which catastrophic shear completely develops and the cutting speed at which individual segments are completely isolated are found to decrease with increase in the hardness of an AISI 4340 steel as shown in Table Similar results are obtained with titanium alloys and nickel-base superalloys Table gives a thermomechanical properties of interest for some of the difficult-to-machine materials (titanium 6AI-4V and Inconel 718) along with two steels [a low carbon steels (AISI 1018 steel) and an alloy steel (AISI 4340 Steel)], and two aluminum alloys (AI 6061-T6 and A1 2024) for comparison Also given in the table is a comparison of the groups of thermal properties, namely, thermal diffusivity (2~pc), and the product of thermal properties (also termed thermal contact coefficient), (2pc) as well as the ratio of these values considering the values for a low carbon steel (AISI 1018 steel) as unity It can be seen that the thermal diffusivity of titanium 6AI-4V is ortly 16% that of AISI 1018 steel, and of Inconel 718 and AISI 4340 steel are 21 and 56% that of AISI 1018 steel, respectively In contrast, the thermal diffusivity of A1 2024 and A1 6061-T6 are 400 and 444% that of AISI 1018 steel, respectively Similarly, thermal contact coefficient, namely, ().pc) for titanium 6AI-4V is only 8% that of AISI 1018 steel, and of Inconel 718 and AISI 4340 steel are 20 and 56% Modeling of thermomechanical shear instability' in machining Fig I al IVlicrograph of a shear localized chip formed in the machining of a titanium AI-4V alloy [11] Fig l(b) Micrograph of a continuous chip formed in the machining of an A1SI 4340 steel [11] 1275 Modeling of thermomechanical shear instability in machining 1277 Table Effect of work material hardness on shear localization Hardnes of AISI 4340 steel material (BHN) Cutting speed at with catastrophic shear completely developed (m/min) (ft/min) Cutting speed at which individual segments work were completely isolated (m/rain) (ft/min) 520 325 215 61 (200) 244 (800) 488 (1600) 305 (1000) 976 (3200) 1952 (6400) Table Density of thermal properties of some work materials Thermal conductivity Material Titanium 6AI-AV Inconel 718 AISI 4340 Steel AISI 1018 Steel A1 2024 AI 6061-T6 Thermal diffusivity Density ) (2 of AISI p (cal/cm s ~C) 1018 steel = 1) (g/cm 3) Specific heat c, (cal/g'~C) Product of ().pc) ct (cmZ/s) (2 of AISI 1018 steel = 1) (cal/g °C) 2pc (2pc of AISI 1018 steel = 1) 0.0160 0.1135 4.43 0.14 0.0258 0.1585 0.0099 0.0816 0.0290 0.0797 0.2057 0.5652 8.19 7.83 0.104 0.1t 0.0340 0.0925 0.2081 0.5663 0.0247 0.0686 0.2031 0.5641 0.141 7.85 0.1099 0,165 0.1216 0.3917 0,4228 2.778 3.00 2.78 2.71 0.2109 0.2131 0.666 0.733 4.0388 4.4418 0.2286 0.2442 1.88 2.00 that of AISI 1018 steel, respectively In contrast, these values are 188 and 200% for A1 2024 and A1 6061-T6, respectively Consequently, much of the heat generated in machining of these materials which have good thermal properties or have extensive slip systems will be dissipated without concentrating in a narrow band Consequently, only continuous chips yield with these materials Loewen and Shaw [9] found the product of (2pc), namely, thermal contact coefficient, for a continuous chip formation process to be proportional to the inverse square root of the chip-tool interface temperature Of course, the actual temperature depends on other factors, such as specific energy, rake angle, cutting velocity, and cut depth In the following, Recht's criterion for catastrophic shear instability in metals under dynamiic plastic conditions [10] is briefly presented first This criterion is then extended for thermomechanical shear instability in machining Relevant thermomechanical properties of the work material (AISI 4340 steel) used in the analysis are given This is followed by a phenomenological description of the mechanism of shear localization process as well as the thermal modeling of shear localization in the machining of some of the difficult-to-machine materials that yield shear localized chips Various sources of heat (primary, preheating effects, as well as image sources) contributing towards the temperature rise in the shear band are identified The temperature rise in the shear band was calculated using Jaeger's classical method for stationary and moving heat sources starting with an instantaneous line heat source These equations were applied for the case of machining of an AISI 4340 steel to determine the effect of cutting speed on the temperature rise due to each of the heat sources The cutting speed for the onset of shear localization is predicted based on thermomechanical analysis This is done by comparing the resulting strength of the work material caused by both thermal softening and strain-hardening effects in the shear band at its working temperature and relevant strain with that of the original work material at the preheating temperature and yield point strain Experimental results of high-speed machining of an AISI 4340 steel reported in the literature [8] were found to agree reasonably well with the analytical values The notation for the paramete:rs used in this investigation is given at the beginning of this paper and the parameters generally are not defined again However, additional parameters where applicable are defined at the end of some of the equations 1278 Z.B Hou and R Komanduri CRITERIA FOR SHEAR INSTABILITY Recht in 1964 [10] developed a classical model for catastrophic shear instability in metals under dynamic plastic conditions Accordingly, catastrophic shear occurs at a plastically deforming region within a material when the slope of the true stress-true strain curve becomes zero, i.e the local rate of change of temperature has negative effect on strength (thermal softening) which is equal to or greater than the positive effect of strain hardening He formulated a simple criterion for catastrophic slip in the primary shear zone based on the thermophysical response of the work material under the conditions of cutting as ~< - - ~ / 0 (d0/de) ~< 1.0 (1) where z, s, and refer to the shear stress, shear strain, and temperature, respectively Accordingly, material will shear catastrophically when this ratio lies between and 1; catastrophic shear will be imminent when the ratio is equals to No catastrophic slip occurs when this ratio is greater than 1, in which case the material will strain-harden more than it will thermal-soften Assuming an approximate value of the temperature generated in the shear band, Recht estimated the values of thermomechanical properties and calculated the shear strength In this paper, this model was further developed by analytically predicting the conditions for the onset of shear localization Samiatin and Rao [11] developed another model for shear localization which incorporates a heat transfer analysis, and materials properties, such as the strain-hardening rate, the temperature dependence of the flow stress, and the strain rate sensitivity on the flow stress to establish the tendency towards localized flow Using the data available in the literature, they found the nonuniform flow in metal cutting is imminent when the ratio of the normalized flow softening rate to the strain rate sensitivity is equal to or greater than For an AISI 4340 steel (325 BHN), this speed was estimated to be about 60 m/min which agreed reasonably with the experimental results reported earlier [8] In addition to the thermoplastic instability (strain hardening versus thermal softening) leading to shear localization, there can be other mechanisms where an actual reduction in the shear strength of the work material in the shear band can take place without the thermal-softening effect For example, the generation of microcracks in the shear band and a reduction in the actual area undergoing stress Walker and Shaw [12] proposed this for material deformation at large strains and Komanduri and Brown [13] proposed this as a possible cause for chip segmentation in machining This concept seems to be valid particularly in the case of cyclic chips generated in the machining of titanium alloys at very low speeds At low cutting speeds the heat generated in the shear band could diffuse on either side with the result thermal softening would be somewhat difficult Instead, the nominal shear strength will be lowered by the presence of microcracks which will reduce the actual area undergoing shear Nakayama [14] investigated the formation of "saw-toothed chip" in metal cutting and proposed several plausible mechanisms of its formation They include (a) strain hardening, (b) hardening by heat treatment, (c) surface embrittlement by liquid, and (d) stress state, namely, the difference in the state of stress in the middle of the chip which is plane strain versus plane stress at the edges In describing the mechanism of saw-toothed chips, Nakayama invokes initiation of a crack at the surface arbitrarily, growth of this crack, and finally stoppage of crack growth as responsible for this type of chip formation In some respects the mechanism is similar to the one proposed in Ref [12] Also, during the growth of crack, Nakayama proposes slip to occur concentratively on a single plane as the mechanism of crack extension Thus, shear failure is implied similar to the model proposed here Further, no evidence of the crack formation was presented Also, this mechanism cannot explain why in some materials, such as in hardened alloys steel (e.g AISI 4340 steel) or nickel-base superalloys, the chip formation changes from a continuous to a shear-localized as the cutting speed is increased The thermomechanical shear instability proposed here, in contrast, can offer a plausible explanation In this paper, we propose the shear-mode failure between the chip segments as the mechanism responsible for the separation of segments, as evidenced by the dimple structure in the shear-localized region We will also show (based on the SEM examination of the surfaces of the chip segments that underwent shear localization) that there are two distinct regions, namely, shearseparated region at the top, as evidenced by the dimple structure and a sliding region Where the Modeling of thermomechanical shear instability in machining 1279 individual segments slid on the tool rakeface after the shear separation between the segments (see Fig in Section 5) It may thus be noted that the thermomechanical shear instability mechani,;m presented in this paper is distinctly different from the one proposed by Nakayama [14] Other mechanisms proposed for shear instability include structural transformation, as in the reversion of martensite to austenite in some steels [3] In this paper, only the first mechanism, namely, thermomechanical shear instability (namely, thermal softening versus strain hardening) in HSM of some difficult-to-machine materials is considered C R I T E R I O N FOR THERMOMECHANICAL S H E A R I N S T A B I L I T Y IN M A C H I N I N G Based on the analysis of cyclic chip formation in machining, possible sources of heat in the cutting region (including preheating effects by these heat sources) contributing towards the temperature rise in the shear band were identified Using Jaeger's classical method for stationary and moving heat sources as a basis, the temperature rise in the shear band due to various primary heat sources as well as preheating effects of these heat sources is calculated [15, 16] Knowing this temperature and a certain value of shear strain, the shear stress in the shear band, a' was estimated and compared with the strength of the work material, cr at the preheating temperature under very small strains (yield point strain) A thermo-mechanical model was developed wherein if the shear stress in the shear band, or' is greater than or equals to the strength of the material, cr at the preheating temperature, no shear localization takes place; instead strain hardening predominates If the shear stress in the shear band, or' is equal to or less than the original strength or, then shear shear localization is imminent The model proposed predicts the onset of shear instability (i.e cutting speed above which shear localization takes place) reasonably well with the experimental results reported in the literature [8] T H E R M O M E C H A N I C A L PROPERTIES OF THE WORK MATERIALS Based on the experimental data available in the literature on the strain-hardening and thermalsoftening characteristics of AISI 4340 steel [17], the following equation (based on best fit) was developed for true stress, or, in terms of true strain, e and temperature, T: a = (432.6572 - 0.3533T)e m'1213+6"4435× lo ~rl (2) The following values were used for the properties of an AISI 4340 steel Thermal diffusivity, (2/pc): 0.0925 cm2/s; specific heat, c: 0.11 cal/g °C ; density, p: 7.83 g/cm Only temperature and strain effects are considered here as the strain rate effects could not be considered due to non-availability of materials properties data in the open literature Of course, to a certain extent strain rate and temperature effects oppose each other and a balance may result fi)r some materials under certain conditions Similarly, where possible, thermal properties of the work material at different temperatures can be obtained and this in turn can be used in the analysis P H Y S I C A L M O D E L O F S H E A R L O C A L I Z A T I O N IN M A C H I N I N G Figure is a schematic of the shear-localized chip formation process showing various surfaces that participate in the process [6] It is based on extensive machining studies conducted at various cutting speeds from an extremely low speed (0.015 m/min or 0.50 in/min) which involves in situ machining inside an SEM to a moderately high cutting speeds using conventional lathes (up 1:o 2400 m/min) and high-speed photography, to very high cutting speeds (up to 30,488 m/min or 100,000 ft/min) using ballistic machining tests [-18] As the work material approaches the tool, it experiences a stress state which changes with time which is a cyclic, asymmetric process The chip segment enclosed in 1, 3, and in Fig is being upset (plastically deformed) by the advancing tool Appropriate stress, strain and temperature fields are thus being set up in the work material As the material begins to shear along line 5, these fields develop conditions which lead to thermoplastic instability, as governed by the thermomechanical response of the work material under the conditions of cutting The strain in the bulk of the segment due to upsetting, however, is rather small as can be seen by the very little deformation of the grains within the segment in Fig l(a) A very thin band between the segments accepts the burden of further strain, thus localizing shear The chip segment then moves up the ramp formed by the work material on the workpiece side of As the tool moves into the ramp, a new segment begins to form Its upper surface, represented by line 5, becomes 1280 Z, B, Hou and R Komanduri Ib/~- t,, Cn~p I., , j / Work Material Undeformed sudacea Part of the catastrophically shear failed sudsce separated from the following segment due to Intense sheer intense shear band formed due to catastrophic shear during the upsetting stage of the segment being formed 4, Intensely sheared surface of a segment in contact with the tool end subsequent slid on the tool face S, Intense localized deformation in the primary shear zone Machined sudace Fig Schematic of the shear-localized chip formation process showing various stages in cycle of chip segmentation [6] the surface through which the tool upsets the material ahead As upsetting progresses, this surface becomes that identified by lines and 4, the latter of which is pressed against the tool face Until a new localized shear zone forms again due to thermomechanical instability, the increasing porition of line [a hot, freshly sheared (nascent) surface] that lies on the rake face remains at rest Shearing between the segments along line ceases when the next localized shear band forms along line Once upsetting in the segment and shear between the segments have ceased, the chip segment moves up the tool face The sliding of the chip segment on the tool face is, therefore, characterized by a stick-slip motion Considering the pressure, temperature and heat transfer conditions at a chip-tool interface, sliding resistence would be much severe for shear localized chip than for a steady state continuous chip Thus, during shear localization, the region between the segments first undergo shear separation between the segments This will be followed by partial sliding between the chip segment and the tool face Figure 3(a) is a photomacrograph of isolated chip segments of an AIS14340 steel formed at high cutting speed Figure 3(b) is a micrograph of isolated chip segments of an AISI 4340 steel formed at high cutting speed Figure 3(b) is a micrograph at higher magnification of one of the chip segments showing two distinct regions: (1) shear-separated region at the top, as evidenced by the dimple structure and (2) a sliding region where the individual segments slid on the tool face after the shear separation between the segments Figures 3(b) and (d) show these two features clearly at still higher magnification It can thus be seen that this process is clearly a shear separation process and different from the fracture process proposed by Nakayama [14] Modeling of thermomechanical shear instability in machining Fig 3(a) Photomacrograph of isolated segments of an AISI 4340 steel obtained in shear localized chip formation process at 975 m/min [6] Fig 3(b) Micrograph at higher magnification of an isolated segment showing two distinct regions, namely, shear separated region at the top (as evidenced by the dimple structure) and a sliding region where the individual segments slid on the tool face after the shear separation between the segments 1281 Modeling of thermomechanical shear instability in machining 1283 25 L~ !ii Fig 3(c) Micrograph at higher magnification of Fig 3(b) showing details of the dimple structure at the top Fig 3(d) Micrograph at higher magnification of Fig 3(b) showing details of the sliding tracks on the underside of the chip segment at it slid past the tool face after the shear separation from the previous segment Modeling of thermomechanicalshear instability in machining 1:285 In summary, it may be pointed out that in the shear-localized chip formation process there are two stages involved One stage involves plastic instability and strain localization in a narrow band in the primary shear zone leading to a shear separation along a surface which originates from the tool tip almost parallel to the cutting velocity vector and gradually curves concavely upwards until it meets the free surface The other stage involves gradual buildup of the segment with negligible deformation by the upsetting of the wedge-shaped work material ahead of the advancing tool Initial contact between the segment being formed and the tool face is at the apex of the tool and is of extremely short duration The contact increases as the upsetting process progresses There is almost no relative motion between the bottom surface of the chip segment being formed and the rake face of the tool until almost to the end of the upsetting stage The gradual bulging of the chip segment slowly pushes the previously formed chip segment Also, the contact between the bulging segment and the tool face gradually increases (i.e between the segment being formed and the one before it) shifting gradually, beginning from close to the work surface to the tool face as flattering progresses As upsetting of the segment being formed progresses, the buildup of stresses in the primary zone causes intense shear between this segment and the one before it The highly intense concentrated shear bands (white etched bands) that are observed between the segments (in the micrographs of a longitudinal midsection of a shear-localized chip) at approximately 45 ° to the direction of cutting are actually formed between the segment already formed and the one just forming (i.e regions 2-4 in Fig 2) This is to be expected, as righdy pointed out by one of the reviewers, as the shear stress at the free surface will be maximum at about 45 ° to the principal stress direction This phenomenon repeats as cutting proceeds However, with increase in cutting speed, the intense shear takes place so rapidly that the contact area between any two segments gradually decreases With further increase in speed, a stage will be reached when the individual segments of the chip are actually separated Micrographs illustrating these features are given in Ref [8] THERMAL MODELING OF SHEAR LOCALIZATION IN MACHINING It may be noted that the nature of chip formation yielding shear localized chip is far different from that with a continuous chip In the case of a continuous chip, strain hardening always predominales over thermal softening Once shear takes place along the main shear plane, the stress required for further deformation is higher than before, so the weakest plane will be shifted to the next plane Thus, shear will also be shifted to the next plane This leads to a uniformly distributed deformation in the chips on a macroscale But in the case of chip formation with shear localization, thermal softening predominates over strain hardening Once shear takes place along the main shear plane, the strength there becomes lower than before So, the main shear plane is still the weakest plane and hence shear continues on the same plane In other words, shear localizes in a narrow band This results in an inhomogeneous deformation in the chips on a macroscale Figures 4(a)-(c) show schematically various stages of the shear-localized chip formation process Figure 4(a) shows the initial stage where chip segment I has just formed and under the pressure exerted by the tool face on the weakest plane a, shear S1 commences Thus, the main shear band is formed This highly intense, narrow shear band is designated as ABCD Note that segment II which is ahead of the shear band (i.e the segment to be deformed) undergoes very little plastic deformation Figure 4(b) shows an intermediate stage The cutting tool has moved a distance AA' The width of the shear zone has increased from AB [Fig 4(a)] to A'C ['Fig 4(b)] Also, the shear zone has rotated due to upsetting (or plastic indentation) of the segment ahead of the tool, and the deformation of segment II takes place by the movement of the cutting tool This deformation is caused by the shear $2 in the weakest plane b of that part of the chip segment which has its own shear angle 4~' and moves forward together with the cutting tool tip As the shear separation between the segments continues, frictional shear takes place between part of the segment being formed and the tool face This is represented by the heat source d As shown schematically in Fig 4, the initial contact between the segment being formed and the tool face is at the apex of the tool and is extremely small Figure 4(c) shows the final stage where the chip segment I has sheared along the main shear plane to Jits maximum extent and the weakest plane in segment II has reached its extreme position The heat source d can also be seen to increase during this stage After that, the weakest plane will shift to a' as shown in the figure Thus, the next chip segment is formed It will again begin to shear along the new 1300 Z B H o u and R K o m a n d u r i Fig A3(c), the weakest plane will shift to DC, as this new chip segment is being formed, another cycle of localized shear begins on the new main shear plane DC The secondary shear in this chip segment stops but another secondary shear begins in the next wedge-shaped part of the chip segment The heat transfer model for the second heat source is shown in Fig A5(a) The heat source begins to work at point (or A) at which time the width of the heat source is zero At time t, the heat source has moved a distance V t from the origin O to B The width of the plane heat source, W, at time t (see Fig A5 for details) is equal to V t sin ~b W = sin q~; ' If we consider a small segment d W~ of that heat source, this segment can be considered as a moving line heat source with its heat liberation intensity qj given by ql = qpl d W i where qp~ is the heat liberation intensity of the second heat source (secondary shear plane heat source), cal/cm s M(x, y) is a point in the conduction medium where the temperature rise is of interest and t is the time of interest At time t, the heat source has moved from to B a distance Vt Consider a small segment dWi of the moving plane heat source for analysis At this instant, the heat source B C has its width W This segment can be considered as a moving line heat source which begins to work only when the plane heat source has moved to Bz It means this segment begins to work at C~ at time t0~ s after the beginning of work of the plane heat source as a whole Prior to it, this segment was not generated: OBi toi = - V sin(180 - ~b') OBi = Wi sin Wz sin ~b' toi ~ - - V sin ~b" The time period from toi to t can be considered as a combination of m a n y small time intervals as d% If we consider a small time interval, dzz at the instant after the heat source dWi segment began to work for r~ s, the total heat liberated by the segment dWi [as shown in Fig A5(b)] in this time interval is given by qt d~i = qpl d W i dri cal/cm and which can be considered as liberated instantaneously The time from the instant of instantaneous liberation of this heat zi to the time of observation t is given by z M(x,y) " Y D •~ - B' v.E B~ I q q / d~ \ -= x'[i time 'Nt Fig A5 (a) Heat transfer model for the second primary heat source (b) Time relationships for the heat transfer model of the second primary heat source Modeling of thermomechanical shear instability in machining 1301 The temperature rise, dOu, at any point M(x, y) and at time t (refer to Fig A5) caused by this instantaneous heat source (line heat source) is given by dOu q~,~dWi dr, [(y y')2+(x x')2] cp(4zcaz) exp 4at (A4) where x' and y' are the coordinates of the heat source segment dW~, which is considered as an instantaneous line heat source, at time ~ x' = Wi sin(4~' - 4~) x' y' = OD + Vzi = ~ sin(4~' - ~) tan~ + Vri + Vrl = Wi Integrating Eqn (A4) from zi = to t - to~ and from W~ = to W, we get the temperature rise caused by the total moving plane heat source at any point M(x, y) and any time t, thus: 0q- P ' M ~ ]wi~w=o dWi ¢xP ( - ~-~ar) do~V( }/4ad¢~o ~ exp )( _f o)o _ (AS) where sin ~b W = Vt sin ~b" X = x - Wi sin(O' - ~b), Y = y - (Vt - Wi cos(~b' - ~b)) r i = x / X + Y2, (u2/46o)) is u = r y / a , t is the time of observation, f ~ d-o-Je x p ( - ~ o - 69 a special function, ~o is a non- dimensional, positive value A.3 Heat transfer modeling for the third primary heat source Figure A6 shows the relationships between the forces acting on the main shear plane and the rake face of the cutting tool It can be shown that R f~ cos(~ +/~ - ~) R=R' Fc = R' sin fl = Fs sin fl cos(q~ +/~ - ~)' At the beginning of shear, the area which bears the load on the main shear plane is given by lsb (where b is the width of cut), and F~ has its m a x i m u m value, i.e Fsmax= r~.lsb kgf Let the chip segment shearing along the main shear plane move up a distance l, The time needed is to, where to = lJV, and the area which bears the load on that plane becomes zero Thus, F~ = (refer to Fig A6) 907$+a v\/ p (1 V/ vc = v s i n $ /cos(t- a) vs =vsin(eOCC)lcos(#-a) Fig A6 (a) and (b) Relationships between the various forces acting on the main shear plane and on the rake face of the cutting tool 1302 Z B Hou and R Komanduri F s ~ _ to "* =, Time Fig A7 Variation of the shear force on the main shear plane of the chip segment with time for the third primary heat source The shear force at any time z~ after the initiation of shear is given by (refer Fig A7) t o - - Ti F~ = F~ m,, - to kgf Thus, the frictional force between the chip segment and the tool face, Fc, is given by /:'~ = F~ to - ~i max - - to sin fl cos(~ +/~ ~) - kgf The total heat liberated by the frictional plane heat source is given by qo = (F~ V, x 2.342)/100 cal/s The intensity of heat liberation of the friction plane heat source (heat liberated per unit area per unit time) is given by qo qpl = contact area Also, contact area = Wb where W is the width of the frictional contact area between the chip segment and the rake face of the cutting tool (see Fig A8) W =ao sin(4¢ - ~b) s m ,~ c o s ( ~ ' - ~)" Thus, qpl=i~smaxtO Zl to sinfl Vc ×2.342 cos(~b+ fl - ct) 100 Wb Relative to the main shear plane, the third heat source is a moving plane heat source with variable intensity of heat liberation and acts at an angle of inclination of (90 - 4> + ~) The velocity of motion of the heat source along the main shear plane, V~, is given by COS 0~ v ~ = Vo s m ( ~ - ~)" The heat transfer model for the third heat source, namely, the frictional moving plane heat source is shown in Fig A8 Consider a small segment of the plane heat source dWi at a distance Wi from the origin It can be regarded as a moving line heat source with intensity of heat liberation q~: q~ = qpl dWi to - q sin fl Vc ql = F~ max - - - X 2.342 d Wi to cos(~b+ fl - ~t) 100 Wb when zi = (at beginning of the work of the heat source), the plane heat source has its maximum intensity of heat liberation sin fl Vc qpl = Fsmax cos($ + fl ct) 100 W~ x 2.342 The line heat source dW~ also has its maximum value qlm~* = F~ m~x sin fl Vc l - - X 2.342 dWi cos(q5 +/3 - ~) 100 Wb Modeling of thermomechanical shear instability in machining Y 1303 l Y at moment t M "ti (x,y) ql max to ~ ~ Fig A8 Heat transfer model for the third primary heat source (the frictional moving plane heat source) Therefore, ql to ri q l m a x - = to Consider a small time interval drl at m o m e n t % At this instant, the moving line heat source has moved a distance vs rl along the ),-direction Thus, its location at time r~ is given by (xg, yi + v~r~)where y~ = W~ cos(90 - ~b + :t) The total heat liberated by the line heat source in this time interval is given by (see Fig A8) ql drl = qpl dWi dri cal/cm which can be considered as liberated instantaneously So, the temperature rise, d u , at any point instantaneous line beat source is given by dOM qp, dWidz, cp(4naz~ ( exp M(x, y) caused by this ( x - x,)2 + ( Y - y i - V~r,)2) - 4ar (A6) or dOM cp(47zar) exp - qpJm=*dWl drl f (x - xi) + (y - )'i -exp / cp(4nar) \ 4az 4at V,zi)2"~ ri qPlmaX -~odWi dzi cp(4nar) t exp (x _ xi)2 ~ (y~~ yi - V~ri)2I 4az " (A7) 1304 Z.B Hou and R Komanduri Using the moving coordinate system, and integrating it from z / = to t (t is the time of observation), the temperature rise at point M caused by the moving line beat source is given by 0~t qp,~,axdWi f_ ~ exp~ Y V ~ ~(1 t 2a ] [\ - ~ ) fl v2'"4a~ e x p ( c o - - - ~ - ) +~J0 dogexp -to ~ (A8) u2)t ~co (A9) For the total moving friction plane heat source, _ qplmax ;o j dco {( ,).v,,,,o exp[z YVsX - j, ~o + ~ 4a o ~ exp - co f~,/4~ dco exp ( - co where sin(¢' - 4) W = ao sin cos(4' - ct)' Wi = ~ W, X = x + W~cos(¢ - el), Y = y - V~t - W~ sin(4 - ~ ) , ry, u= 2a and r2=X 2+Y2, ;oOoxo(_ co \ co is a special function and co is a non-dimensional, positive value A.4 Heat transfer modeling for the fourth primary heat source The fourth heat source is the frictional heat source d of that part of the chip segment shearing on the tool rake face during upsetting of the material of the wedge shape part of the following unformed chip segment [AD in Fig 4(c)] Figure A9(a) is a schematic illustrating the upsetting process of an inclined wedge-shaped part of the segment as the tool advances into it During this process, the shear angle, 4', remains constant The tool tip begins contact with the wedge-shaped part at point A As the tool moves to A', part of the work material AA'B undergoing deformation changes its shape to CA'B The deformation process is composed of (1) shear $2 along the shear plane, element by element (i.e along A1Bt, A:,B2 ) with a velocity V,2 and (2) shear along the tool rake face with a velocity, V, The shear velocity, V,2; shear velocity along the tool face, V~;and the cutting speed, V0, have the following relationships [5] as shown in Fig A9(b): sin 4' Vc = Vo sin(90 - 4' + ~) sin(90 ~) V~2 = Vo sin(90 ~' + ~) - (a - sin 4' V0 ~ cm/s ~) cos~t Vo c ~s(4' - ~) cm/s A A2 A1 A (b Fig A9 Schematic illustrating the upsetting process of an inclined wedge-shaped part of the segment as tile tool advances into the work material Modeling of thermomechanical shear instability in machining 1305 a C B ~A' V at moment xi • A ~3 at moment ~1 Fig A10 Schematic illustrating the forces acting on the secondary shear plane and on the rake face of the tool, respectively At time 3~, when the tool tip reaches A', the forces acting on the secondary shear plane and on the rake face of the tool are shown in Fig A10 Consider the section A'BC as a free body For static equilibrium, the resultant force R~ acting on A'B and R'~acting on A'C should be equal, opposite, and collinear, i.e R~ = R', Hence, the angle of inclination of R~ (or R'~)with respect to the horizontal (fl-~) should be same (where ,6 is the frictional angle) During shear, F~ is given by F~ = zr A'B b R~ kgf F,, 3y A'B b cos(~' + ~ - ~) cos(¢' + p - ~) kgf The frictional force on the rake surface F , is given by Fc~ = R'~ sin fl = R~ sin fl - 3~ A'B b cos(4,' + ~ - ~) sin fl kgf The total heat generated in this frictional area at that instant is given by qo = F,.s (Vffl00) x 2.342 cal/s With respect to the main shear plane (plane AB), this heat source is a moving plane heat source with variable width When 31 = 0, the plane heat source is located at point A and has its width equal to zero At instant % the location and width of the heat source are shown as A'C in Fig A10 At different times (3~, 32, 33, ), the locations and widths are different as shown in Fig AI0 The velocity of this moving heat source, V, along the main shear plane A B is given by (refer to A AA'D) sin(90 - ~) AD V=Vo~-~7= V0sin(90_~+~) cos~ Vocos(~_a~ The intensity of heat liberation qp~ is qpl q0 contact area qo A'C b cal/cm s zrA'Bb ,sinfl~×2.342×l - cos(~' ~ fl ~- ~/ ,t~ A'C b Vc sin fl A'B = ry ] - ~ x 2.343 x c o s ( # + fl ~) cal/cm s where A'B A'C AA' sin ~b sin(90 - / + ct) sin(~b' - ~b) AA' sin q~' sin ~bcos(4~' - ~) sin 4¢ sin(# - ~b) (refer to the triangles A'BA and A'CA in Fig A9) 1306 Z B H o u and R K o m a n d u r i machining surface xI , / M(x,y) W2 x Ai ~ Y 90- mainyear plane o Ct (al ql d't i / t c - ~ t (b) Fig A l l Heat transfer model for the fourth primary heat source This analysis is to determine the mean temperature rise in the main shear band due to the fourth heat source For convenience in the analysis, let the origin of the coordinate system to coincide with the tool tip and the y-axis to coincide with the main shear plane as shown in Fig A l l It may be noted that it is actually the same as Fig A10, but rotated 180 ° and associate a coordinate system on it as mentioned above It is required to determine the temperature rise at any point M(x, y) caused by this moving plane heat source at any instant t At time t, the heat source has moved to the position AtCt as shown in Fig A l l m ODt = Vt The width W of the heat source at that instant is ArCt: W = W + W~ = Vt W sin(¢' - ¢) sin ~b W : Vt sin(¢' - ~b) lit - (refer to AODtC,) cos(¢' - ~) sin(90 - ¢' + :¢) - - sin ¢ sin(90 - ~) V[ cos (refer to AODtAt) Consider a small segment dWi which can be considered as a moving line heat source This line heat source begins to work only when the plane heat source has moved to AoCo (see Fig A10) Prior to this, this segment dWi was not produced The time, to~, this line heat source begins to work is given by tOi ODo V Referring to the triangle, ODoAo, we have m ODo sin(90 - ct) AoDo sin ~ Wi-W1 sin ¢ Modeling of thermomechanical shear instability in machining 1307 Therefore, COS ~X ODo = ( W i W l ) sin ( W i - W cos~ t0i V sin tk" The intensity of heat liberation of the line heat source is given by ql = qpl dW~ cal/cm s In this case qt is constant as s h o w n in Fig AI l(b) Consider a small time interval dz~ at any instant r~ after the initiation of the w o r k by this line heat source The total heat liberated, Q~, in this small time interval is given by QI = qt dri = qp, d W i dri cal/cm This heat can be considered as liberated instantaneously So the effect on temperature rise, dO, at any point M ( x , y) in the conduction medium caused by this instantaneous line heat source is given by qp, d W i dr, ( (x - xi)2 + (y - y ) c p ( u a r ) exp 4at dO where xi = ( W i - W l ) sin(90 - 4~ + ~) = (Wi - Wl) cos(q~ - ~) y~ = Vtol + Vr~ - ( W i - W O cos(90 - 4) + ~) = Vto~ + Vrl - ( W i - W I ) sin(4~ - ~) =Vtol + V ( t - toi - r) - (Wi - WL) sin(4~ - :t) = Vto~ + V z - Vto~ - V z - (W~ - W~) sin(4~ - =) = V t - ( W i - W1) sin(~b - =) - Vz y - yi = y - V t + (W~ - Wx)sin(~b - ~) + V r = Y + V r Y = y - V t + ( W i - Wa) sin(~b - ~) X = x - (W~ - W~) cos(q~ - ~) X and Y are the coordinates of point M in the moving coordinate system at the time of observation, t: qpj d W~ dr i { X + (Y4az+ V r ) ) c p ~ exp ~ dO (A l O) The temperature rise, 0~, at point M caused by the moving line heat source from the instant t0i to t is given by qpl dWi f -'0, dri e x p ( O'M = cp(4na-"-'~) :,,=o r (All) X + (Y4ar+ V r ) ) At the instant, t0~, r~ = 0, and at the instant, t, r~ = t - to~ F o r r~ = t and when r~ = t - t0~, r = 0: - t01 - r Therefore, dr~ = - dr when zl = O, r = t - tol dr e x p ( X + ( y + v r ) ) r 4at q p l d W i f c p ( n a ) J,=o -~-exp~ '~-a)f: 'O exp( Tar-)exp(- 4af Let 2z 4a =09, V 4a r=~ ico, dz 4a V2 -~- = V -5 dco 4aco r2 r2 V 4at a 4aco 4a dr=~-idco dco co (rV2~21 = \'-~a ] u2 4-"-~= 4o~ where r = X + y2, u = r V / a , when T = O, co = O, and w h e n z = t - toi, co = ( V / a ) (t - to3 1308 Z.B H o u and R K o m a n d u r i Then, 0~t -_ _ ~qpl -d W i exp ~ :|o -(o - e x p / - \- o9 (A12) The temperature rise, 0M, at any point M(x, y) and at any time t caused by the fourth heat source, namely, the moving frictional plane heat source with variable width is given by 0M=4 ~ dW, exp ~-a f exp )0 -~o-~-~o (A13) where w = vt [sin(~b' S ~ sin 4~-] Legs(4,' - ~) + ~-ff~J' sin(4¢ w, = v t cos((V X=x (W~-WOcos(4~-~), ril/ u= 2a and - q~) Y=y-Vt+(W~-WOsin($-~) r ~ = X + Y2 A.5 Preheating effect for the temperature rise in the main shear band caused by the four primary heat sources As mentioned in Section A.2, when the second heat source moves to its extreme position as shown in Fig A3(c), the weakest plane will shift to position DC, then the new chip segment is formed and the shear will be localized on the new main shear band It means these heat sources during the preceding localized shear cease to work from that instant and commence their cooling period It can be shown (for values of q~ = 27.5 c, ~ = 10°, qS' = 41.25 °) the time of that instant, th is given by th = 0.3722t0 (to = -~) It is clear that all the four heat sources have influence on the new main shear band of the following chip segment Temperature rise at any point on the main shear band is not only caused by current four heat sources at and near the main shear band, but also by similar heat sources at and near the preceding main shear band In other words, it is also caused by the preheating effect of these heat sources at or near the main shear plane of the preceding chip segment This, in brief, is the preheating effect of the primary heat sources on the following segment Let the observation time with respect to the begining of the four heat sources of current shear be t' (t' = nto) Then the observation time with respect to the beginning moment of similar heat sources of the preceding segment is given by t = t' + 0.3722t0 = (n + 0.3722)to All the heat sources considered for the shear localization operate simultaneously However, in this investigation the contributions of these heat sources on the temperature rise in the shear band are calculated separately and their combined effect obtained by s u m m i n g up the individual contributions As the temperature rise is a function of time, a separate time reference is necessary In this paper, we consider the initial time when the shear on the shear plane just commences as zero We take the time taken for the tool tip to move from A' to ,4l so that the segment formed moves along the main shear band to its m a x i m u m (i.e point A' to B) as to (Fig 4) It is necessary to calculate the temperature rise caused by the heat sources at any instant, 7, between zero and to, i.e., t = nto, where n is a real number between and "n" is the percentage of theoretically possible m a x i m u m shear in the main shear plane We take a small but reasonable value o f n to analyze the criterion for shear localization, say n = 0.05 The relevant true strain e is given by e, = In(0.05 + 1) = 0.0488 If the shear localization occurs under these conditions, it will definitely occur at higher values of n, when the temperature rise will be higher and thermal-softening effects predominate Figure AI2 shows the time relationships between the current and preceding localized shear processes It shows the times for the beginning of the preheating effects of the preceding heat sources on the current localized shear, which stop to work at time th Here, th = nhto The value ofnh in this case, as discussed in the paper, is 0.3722 The time of preheating effect t is (nh + n)to For example, if we consider the observation time as 0.25 to, for the calculation of the temperature rise on the current shear band caused by four heat sources of the current, we should take (0.25 + 0.3722)to as the observation time for the calculation of the preheating effect caused by heat sources of preceding shear ~rne for ~ beginning of current localized shear I I I •2 time for the beginning of preceding localized shear • i ! I lime for the end of preceding localized shear th = (nh - t o ) t' = n • to / J n [ i [ , I 1.0 t = (nh+n) • to Fig AI2 Time relationships between the previous segment and the current segment during shear localization in machining Modeling of thermomechanical shear instability in machining 1309 \-I "t t "to "to (a) qpl'dWi l (b) Fig A13 (a) Variation of the heat intensity of the first primary heat source (plane heat source) with time (b) Variation of the heat intensity of the second primary heat source (plane heat source) with time For the calculation of the preheating effect caused by those four heat sources of the preceding shear, it may be noted that these heat sources stopped their action at time nhto (or 0.3722to), which is earlier than (nh + n)to or (n + 0.3722)to So their preheating effects are proceeded at their cooling period Hence, Eqns (A3), (A5), (A9) and (A13) should be moditied accordingly Figures A 13(a) and (b) show the modified time relationships for the heat transfer model of first and second heat sources for the purpose of analyzing their preheating effects during their cooling period Here, to = Ijv~, th = nhto, and t = 4- n)to At time th, the heat sources of the preceding localized shear stopped to work Thus, all the integrations with respect to r~ for the solutions in their intermediate stages of derivations [Eqns (A1), (A4), (A7) and (A10)] should have their integration linfits r~ = to ~ = th (or z~ = to z~ = th to~ for the second heat source) By integrating Eqn (AI) from z~ = to r~ = 0.3722to and from y~ = to Is, an equation for the calculation of the temperature rise, 0M, due to the preheating effect caused by.first heat source of preceding shear can be obtained: (nh 0,,,I ~q° ~ot J,,:o'" a'(p,, p2) dyi (AI4) 16naAto Jr~=o where ¢p2 du fY(pl, p2)= | - - e x p ( - u Jpl U pl = ri v~, ), Z'(pl, p ) = f r2 du - - ~ e x p ( - u 2) Jpl U- ri p2 4a4Td~S_ th) with t being the time of observation and th the duration of working of first heat source See also the Notation section given at the beginning of the paper for the definition of various parameters Similarly, by integrating Eqn (A4) from r~ = to zi = th - tol and from Wi = to W, an equation for the calculation of the temperature rise, 0M, due to the preheating effect caused by second heat source of preceding shear can be obtained: qplf W 0~ = ~ (YV) w,=o dl4:i exp f;2(t-t°i)/4ad°) - ~-a 2(,_,,),,~ ~ ( exp u2) - o) - ~ where X = x - W, sin(~b' - d~) Y = y - (Vt - Wi cos(~b' - th)), ri=x/X 2+ y2 riV u = -a with t being the time of observation and th the duration of working of second heat source (A15) 1310 Z.B Hou and R Komanduri Similarly, the equations for calculating the temperature rise due to the preheating effect caused by third and fourth heat sources of preceding shear are shown as follows For third heat source: _qp~m~, fS~=0 d W , e x p ( - Y V ~ ) t ' ) f v~','4" -~exp(-co ~aj{(1 to/Jv~,t_th,,4~ ~-~) M-4r~2 - - dto exp (A16) where s i n ( # - 4~) W = a o s i n $ c o s ( ~ b , _ c 0, X = x + W~ c o s ( $ - Wi=0~W, r,V~ Y = y - V,t - Wi sin(~b - c0, u = - - 2a and ~) r { = X + y2 For the fourth heat source: qplmaxfW 0M=~ wi=o dWiexp ( Yl~( V2s( U4a-dos, (U 2) -exp -co2a / Jr2, ,h),*, (A17) ~o where WI = O ~ W, W = Wx + W2, X=x-(Wi-W0cos(~b-ct), riV u = ~a APPENDIX sin(th' - ~b) WI = Vth - Y=yand - cos(# - ~)' sin q~ Wz = Vtn - - cos Vt+(Wi-WOsin(~b-ct) r~ = S + y2 B: E X A M P L E CALCULATION B.1 Temperature rise in the shear band due to primary shear band heat source All the heat sources are used for the calculation of shear band temperature rise Depending on the cutting speed used, the influence of some of these heat sources will be more prominent than others Some can be neglected at higher speeds but become more significant at very low cutting speeds First, the temperature rise at any point and at any instant of tirae due to each of the heat sources can be obtained individually The temperature rise at 16 points evenly distributed along the shear band caused by the above-mentioned heat sources are calculated individually as a basis for the temperature rise in the shear band The mean of these values is taken as the temperature rise in the shear band A similar approach is taken for calculating the temperature rise due to preheating effects of each of the four sources The mean temperature rise in the shear band caused by each of the four primary heat sources and the four preheating effects of the primary heat sources are designated as 01, 02, 03, 04, and 0~, 0~,, 0~, and 0~, respectively In fact, the effect of the image heat source, 01 i, for the first heat source as well as preheating effect of this image heat source, 0'11, should also be considered Figure B is a flow chart for the calculation of the temperature rise in the shear band, 0M, considering the first primary heat source as an example To improve the accuracy, the shear band heat source is divided into a large number of parts (say, 100) and each part is considered as a continuous, stationary line heat source and then integrated along the whole length to obtain the total temperature rise Also, for convenience in the computation, 0~a is divided into two parts 0Mr and 0M2 Each of it is calculated separately and then combined Note that in the integration of the special functions fl(p) and X(P) for the first primary heat source (stationary) although the upper limits of integration is oo, the functions converge rapidly for values of p greater than 1.5 (see Fig B2) So, for all practical purposes, a value higher than this (say, 3) would be equivalent to its value at ~ It is, therefore, necessary to know in advance the nature of variation of these functions with respect to p, to determine the integration limits This can be done first by plotting their variation with respect to p and determining the cut-off value For other heat sot~rces (moving), we encounter a f(to) function with integration limits of zero to oo However, for numerical integration, we need to arrive at finite values of the limits Based on the analysis of the function,f(~o) it is possible to arrive at the upper limit of about (or say a maximum of 5-10 for sufficient accuracy) [see Fig B3(a)] The lower limit cannot be zero for the computation of this function Figure B3(b) shows the variation of the function, f(co) with m showing the function to converge very close to zero when co approaches zero Hence, a very small value, say 0.0001 can be taken without losing accuracy Modeling of thermomechanical shear instability in machining Define Input data: I I OM - OM~+ eM2 J 'Cy,;L, a, Vc, ao, o., I~,,0, n, x, y, and lol (tolerance of numerical integrationof sl~c~ func~ons) i I, Calculate:Vo, V=, to, t, qo II Numerical integration of the special 1311 I function :~f',(p) @~ yi=o i Calculate 101 different values of Yifrom to Is with increment For each value of p calculate ~(p) lower limit is p upper limit is 31 ta(pI = fp; exp (- : ) d it For each Yi,calculate: r, p iii Go sub to calculate D.(p) iv 101 values of fZ(p) are obtained conduct numerical integration IlL Calculate~,tl J, Numerical integration of the special function J'~=tsr2xlp) d~ i For each value of p calcutate X(P) yi=o i Calculate 101 different values ot Yi from o to Is with increment Is ii Calculate r2° X(P) Ay~= i~ it for each Yicalculate: r, r2 and p iii Go sub to calculate r2x{p) [ iv 101 valuesof r2"x(p) ara obtained; conduct numerical integra~on Calculateem Fig BI Flow chart for the calculation of the temperature rise, 0M, in the shear band due to first primary heat source For this case, the primary shear band coincides with the first primary heat source, which is a stationary heat source It should, however, be noted that all the other heat sources are moving heat sources and the shear band not coincide with the heat source The temperature rise in the shear band due to the first primary heat source, 0M, at point M is given by Eqn (4); thus OM_ ~q02 ( ')::' ~0 ',=o flip) dyl + ~ , =qoo :l, r2x(p) dyi" (Bl) In the following, an example calculation of the temperature rise in the shear band due to first heat source is given The temperature rise due to other heat sources and at other cutting conditions can be calculated similarly (i) Work material AISI 4340 steel (it) Thermal properties of the work material: thermal conductivity, ) = 0.0797 cal/cmsCC; thermal diffusivity, a = 0.0925 cm2/s (iii) Cutting conditions: orthogonal cutting; depth of cut, a0 = 0.02 cm; cutting speed, vc = 100 m / m i n (166.67 era/s); rake angle, ~ = 10°; coefficient of friction between the chip and the tool face,/z = 1.0 (friction angle/3 = 45°) (iv) Shear angle of the main shear plane 4) = 27.5L 1312 Z B H o u and R K o m a n d u r i 10 I I ? z iiiiiiiiii!! 0 P 0.5 1.5 2.5 U Fig B2 Variation of the function flu) with u showing rapid convergence of the function for values of u greater than 1.5 u-;,:5i i i i i i i ; ~ 0.4 0.8 1.2 1.6 o) f(o~) = 1/2 * e x p ( - c o - u / c o ) 16 u,,.5~ 14 12 10 ~ i i ! i iiii!ii 0.01 0.02 0.03 0.04 0.05 (9 Fig B3 (a) Variation of the functionf(~) with eo for various values of function u showing rapid convergence of the function for values of co greater than and (b) variation of the function,f (co) with w showing the function to converge very close to zero when o9 approaches zero Modeling of thermomechanical shear instability in machining 1313 (v) Length of shear plane l~ is given by 1, = ao ~ = 0.0433 cm sin q~ (vi) Sliding speed of chip segment along main shear plane v~ is 9iven by cos~ Vs=V,.-cos(~-~) cos 10 166.67 x - - = cos 17.5 172.10 cm/s (vii) to is defined as to I~ 0.0433 Vs 172.1 0.0002516 s (viii) t is given as t = nto = 0.05 x 0.0002516 = 1.288 x 10 -5 s Vs (ix) qo = r r ' i ~6 x 2.342 cal/cm2s where zr is the yield strength of the work material AISI 4340 Due to the preheating effect, the temperature at the main shear plane at the beginning of the shear is assumed initially to be in the range 150"-200 ° This, however, can be calculated and used as an input The tensile yield strength of AIS14340 steel, ,Ty, in this temperature range varies from 230 to 210 ksi (16,171-14,765 kgf/cm2) So, an average value of 220 ksi (15,468 kgf/cm 2) is taken as o~ for the calculation of qo Also, % = (0.5)a r Thus, 110,000 r r = 110 ksi = 0.0703 ~= 7734.1 kgf/cm Therefore, qo = 7734.1 x 172.1 100 x 2.342 = 31172.9 cal/cm2 s ~(p) and 7.(P) are two special functions whose argument p is given by r l / x ~ t where rl is the distance between the point M(x, y) and the elementary segment of the shear plane heat source which is regarded as a line heat source and has its coordinates and y~ along the x- and y-axis, respectively Hence, rl = x / ( x - - 0)2 + (y - yi)2 Therefore, ~ / x ~ + (y _ yi)2 P x~ 4at (xi) The mean temperature rise is determined as the average of the temperature rises of 16 points located along the length of main shear plane, i.e Mo(0, 0), Mx(0, Ay), M2(0, 2Ay), M3(0 , 3Ay), M4(0, 4Ay) M15(0, 15Ay) Here, Is Ay = - 15 0.0433 15 0.00298 The coordinates of the 16 points are as given in Table A1 Table A1 Coordinates of the 16 points M Mo M1 M2 M3 M4 - Mj3 M14 Mr5 x 0 0 - 0 3' 0.00289 0.00577 0.00866 0.01155 0.03753 0.03753 0.04330 1314 Z B Hou and R K o m a n d u r i Table A2 Temperature rise at points Mo, M1, Mz, and their mean values M Mo Mt M2 M3 M4 M5 M6 - MI3 M14 M15 0, 117.4 233.8 234.8 234.9 234.9 234.9 255.8 - 234.8 234.8 117.4 Based on this-data and using Eqn (1), the temperature rise at points Mo, M1, M3, , M15 and their mean value are obtained as given in Table A2 Here, 01 =z-7 Y~(0o + 01 + 0z + + 015) = 220°C 1o Thus, the mean temperature rise in the shear band due to first heat source, 0~ = 220°C (see Table of the main text for the value of 01 at 100 m/rain) The temperature rise in the shear band caused by other heat sources (both primary and preheating effect of the primary heat sources) can similarly be obtained by using the relevant equations presented in the paper It m a y be noted that these equations can also be used for other work materials (using appropriate thermomechanical properties), other cutting parameters, and other values of n [...]... and velocity of shear in the secondary shear plane, V,2 can be obtained using the well-known Merchant's equations for orthogonal machining [5] or from the Modeling of thermomechanical shear instability in machining 12'99 ®b (b) B2 D B t B1 ao C A C2 C1 C (c) Fig A3 (a)-(c) Schematics showing various stages (initial, intermediate, and final) of upsetting of a wedgeshaped part of a newly forming chip segment... definition of various parameters Additional parameters are defined at the end of equations when necessary Modeling of thermomechanical shear instability in machining 1297 C B/~ tool \ A O (a) ® A ~,A D ! D (c) (d) t to 2Io t Fig A1 (a)-(c) Schematic showing the deformation of the shear band (primary shear zone) at various stages (initial, intermediate, and final) of shear localization in machining, ... sources of preceding shear ~rne for ~ beginning of current localized shear I I I •2 time for the beginning of preceding localized shear • 4 i ! I 5 lime for the end of preceding localized shear th = (nh - t o ) t' = n • to / J 6 n [ i 8 [ , 9 I 1.0 t = (nh+n) • to Fig AI2 Time relationships between the previous segment and the current segment during shear localization in machining Modeling of thermomechanical. .. time, to~, this line heat source begins to work is given by tOi ODo V Referring to the triangle, ODoAo, we have m ODo sin(90 - ct) AoDo sin ~ Wi-W1 sin ¢ Modeling of thermomechanical shear instability in machining 1307 Therefore, COS ~X ODo = ( W i W l ) sin ( W i - W 0 cos~ t0i V sin tk" The intensity of heat liberation of the line heat source is given by ql = qpl dW~ cal/cm s In this case qt is... 60 70 v I 80 90 m/min I 100 110 120 Fig 6 Variation of shear stress in the shear band, tr', at the shear band temperature and shear strength of the bulk material at the preheating temperature, a with cutting speed in machining of AISI 4340 steel the contributions of each of the heat source varies with increase in cutting speed Hence, in the calculation of the temperature rise in the shear band at a given... mechanism of chip formation when machining titanium alloys Wear, 1981, 69, 179-188 6 Komanduri, R., Some clarifications of the mechanics of chip formation when machining titanium alloys Wear, 1982, 76, 15-34 7 Komanduri, R and Schroeder, T A., On shear instability in machining a nickel-iron base superalloy Transactions of American Society of Mechanical Engineers Journal of Engineeringfor Industry,... image heat sources in shear localized chip formation process are given A.1 Heat transfer modeling of the first heat source Figure 4 (of the main text) shows schematically various stages (initial, intermediate, and final) of shear localization in machining The primary shear band heat source a in the primary shear zone in the first primary heat source, where the shear strain, 7(CC/h) increases gradually... temperature rise in the shear band during cutting has to be determined Based on an analysis of the cyclic chip formation, the temperhture rise in the shear band is identified as due to the following four heat Modeling of thermomechanical shear instability in machining 1287 sources as well as the preheating effects of these four heat sources The four primary heat sources are (1) the shear band heat source,... A2 A1 A (b Fig A9 Schematic illustrating the upsetting process of an inclined wedge-shaped part of the segment as tile tool advances into the work material Modeling of thermomechanical shear instability in machining 1305 a C B ~A' V at moment xi • A ~3 at moment ~1 Fig A10 Schematic illustrating the forces acting on the secondary shear plane and on the rake face of the tool, respectively At time 3~,... catastrophic shear instability in high-speed machining of an AISI 4340 steel Transactions of American Society of Mechanical Engineers Journal of Engineeringfor Industry, 1982, 104, 121-131 9 Loewen, E and Shaw, M C., On the analysis of cutting tool temperatures Transactions of American Society of Mechanical Engineers, 1954, 76, 217 10 Recht, R F., Catastrophic thermoplastic shear Transactions of American ... the machining of a titanium AI-4V alloy [11] Fig l(b) Micrograph of a continuous chip formed in the machining of an A1SI 4340 steel [11] 1275 Modeling of thermomechanical shear instability in machining. .. shear localization in machining Modeling of thermomechanical shear instability in machining 1309 -I "t t "to "to (a) qpl'dWi l (b) Fig A13 (a) Variation of the heat intensity of the first primary... zero Modeling of thermomechanical shear instability in machining 1313 (v) Length of shear plane l~ is given by 1, = ao ~ = 0.0433 cm sin q~ (vi) Sliding speed of chip segment along main shear