MODELING OF THERMOMECHANICAL SHEAR INSTABILITY IN MACHINING

intensity of heat liberation of the continuous line heat source, cal/cm s intensity of heat liberation of the plane heat source, cal/cm 2 s suffix "max" indicates the maximum value,, he

Pergamon

PII: S00207403(97)00017-9

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

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 S H E A R INSTABILITY IN

M A C H I N I N G

Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK 74078-5016, U.S.A

(Received 2 May 1996; and in revised form 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

intensity of heat liberation of the continuous line heat source, cal/cm s

intensity of heat liberation of the plane heat source, cal/cm 2 s (suffix "max" indicates the maximum value,,) heat liberation rate of the stationary plane heat source (main shear band heat source), cal/cm 2 s

thermal diffusivity, cm2/s

thermal conductivity, cal/cm s °C

specific heat, cal/g °C

density, g/cm 3

temperature rise at any point M in the conducting medium, °C

distance between point M and the line heat source (a segment of the plane heat source), cm

cutting depth, cm

width of the main shear plane, cm

coordinates of point M in the moving coordinate system in X and Y directions

width of the moving plane heat source at the time of observation, t, cm

velocity of the moving plane heat source in the direction of main shear plane, m/s

velocity of chip segment moving along main shear plane, m/s

the equivalent sliding speed along the secondary shear plane, m/s

cutting speed, m/s

sliding speed of chip segment along the tool rake face, m/s

rake angle of the cutting tool, deg

shear angle of the main shear plane

shear angle of the secondary shear plane

time of observation or the time of that moment when the temperature rise is to be determined, s

time when the heat source segment (regarded as a line heat source) begins to work, s

maximum time of duration of working of first and second heat sources, s

time required for a chip segment to move along the main shear plane through a distance, l, s

shear stress of the material, kgf/cm 2

1273

1274 Z.B Hou and R Komanduri

a true stress of the material, kgf/cm ~

e true strain of the material

1 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 t o o l - c h i p 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), p o o r 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) F o r 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 1 Similar results are obtained with titanium alloys and nickel-base superalloys

Table 2 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 1275

Fig I al IVlicrograph of a shear localized chip formed in the machining of a titanium 6 AI-4V alloy [11]

Fig l(b) Micrograph of a continuous chip formed in the machining of an A1SI 4340 steel [11]

Modeling of thermomechanical shear instability in machining

Table 1 Effect of work material hardness on shear localization

Cutting speed at which individual segments work were completely isolated (m/rain) (ft/min)

Titanium 0.0160 0.1135 4.43 0.14 0.0258 0.1585 0.0099 0.0816 6AI-AV

Inconel 718 0.0290 0.2057 8.19 0.104 0.0340 0.2081 0.0247 0.2031 AISI 4340 0.0797 0.5652 7.83 0.1t 0.0925 0.5663 0.0686 0.5641 Steel

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 thermomechani- cal 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

2 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 0 (d0/de) where z, s, and 0 refer to the shear stress, shear strain, and temperature, respectively Accordingly, material will shear catastrophically when this ratio lies between 0 and 1; catastrophic shear will be imminent when the ratio is equals to 1 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 non- uniform 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 5 F o r 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 F o r 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

N a k a y a m a [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, N a k a y a m a 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, N a k a y a m a 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, shear- separated region at the top, as evidenced by the dimple structure and a sliding region Where the

individual segments slid on the tool rakeface after the shear separation between the segments (see Fig 3 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 N a k a y a m a [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 H S M of some difficult-to-machine materials is considered

3 C R I T E R I O N F O R T H E R M O M E C H A N I C A L 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]

4 T H E R M O M E C H A N I C A L P R O P E R T I E S O F T H E W O R K M A T E R I A L S

Based on the experimental data available in the literature on the strain-hardening and thermal- softening 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 3

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

5 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 2 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, 4 and 5 in Fig 2 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 condi- tions 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 5 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

a new localized shear zone forms again due to thermomechanical instability, the increasing porition

of line 4 [a hot, freshly sheared (nascent) surface] that lies on the rake face remains at rest Shearing between the segments along line 3 ceases when the next localized shear band forms along line 5 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 1281

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

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 thermomechanical shear 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]

6 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 A B [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

1286 Z B, Hou and R Komanduri

Fig 4(a)-(c) Schematic of the different stages of shear localization in machining showing various heat sources

responsible for the temperature rise in the shear band

main shear plane a' At this instant the length of the shear zone on the main shear plane of tile former

chip segment has its m a x i m u m value A'C or A D B C [Fig 4(c)]

To predict the conditions for the onset of shear localization quantitatively, the 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, a, where intense shear takes place between the segments [see Fig 4(a)]: this heat source is the predominant contributor to the temperature rise in the shear band especially at the higher cutting speeds; (2) the secondary shear band heat source b [see Figs 4(b) and (c)]: this is formed during the upsetting stage of cyclic chip formation; (3) the frictional heat source,

c [Figs 4(b) and (c)] between the segment already formed and the rake face of the cutting tool, and (4) frictional (intense shear) heat source d between the segment just being formed and the tool face during the upsetting stage of the following segment This heat source is the second major contributor

to the temperature rise in the shear band after the shear band heat source (i.e the first heat source)

In addition, all the four heat sources also affect the temperature on the new shear band of the following chip segment, i.e every new segment where shear localization begins to take place will occur at a temperature higher than the room temperature This is the preheating effect on the main shear plane

7 H E A T T R A N S F E R M O D E L I N G Carslaw and Jaeger [15, 16] developed a series of classical solutions for the temperature rise for both cases of stationary and moving heat sources They pointed out that based on the solution of an instantaneous point (or line) heat source which involves integration of it with respect to appropriate space and time variables most heat transfer problems of practical interest can be solved In this paper, the instantaneous, infinitely long line heat source solution is used as a basis for ~the above-mentioned four primary heat sources as well as the preheating effects of these heat sources Alternatively, the instantaneous line heat source solution is used to determine the temperature rise of any point, M(x, y), and at any time z after the initiation of the instantaneous line heat source This general equation has the following form [15, 16]:

be similar to sliding friction and the heat source is modelled as a moving plane heat source with variable intensity of heat liberation The fourth heat source, d, also a frictional heat source, is between the tool face and the segment that is undergoing upsetting It may be noted that this heat source coincides with part of the shear band heat source and contributes additional heat due to intense shear [Fig 4(c)] The fourth heat source is assumed as a moving heat source with variable width and variable heat generation This is because the initial contact between the segment being formed and the tool face is at the apex of the tool and is extremely small The contact length as well

as the length of the heat source increases as the upsetting of the following segment takes place (Fig 4) This gradually shifts the shear separation between the segments to the frictional shear

1288 Z.B Hou and R Komanduri

between the segment and the rake face of the tool The first and the fourth heat sources are the two heat sources which coincide with the shear band Similar approach can be taken for calculating the temperature rise due to each of the four preheating sources

Appendix A gives details of the thermal modeling for the calculation of the temperature rise in the shear band due to each of the heat sources, including the primary, preheating effects, and the image heat sources, which are summarized in the following

8.1 Primary heat sources

(see Fig 4 for details) During shear, the shear stress is the yield stress of the material ~., and the strain rate is a function of the cutting speed Both are assumed to be constant in this investigation Thus, the total heat generation rate, qo (cal/cm 2 sec) is constant The total width of the band heat source increases with increase in shear strain, but a portion of it that is in contact with the work remains unchanged The rate of heat flow into the work material decreases with increasing shear strain because the ratio of the contact part to the total band width decreases Thus, the intensity of heat flow into the work material is a variable

The two side surfaces along the width of the work material can be regarded as adiabatic boundaries Thus, no heat flow is assumed along the Z-direction This is equivalent to an infinitely long (along the Z-direction) band heat source problem So, the first heat source is assumed as an infinitely long stationary, continuous heat source with a variable intensity of heat liberation It is shown in Appendix A that the temperature rise, OM, at point M in the shear band caused by the first heat source is given by

are special functions, and u is a non-dimensional positive value, and p = r ~ / x / ~ t

undeformed part of the work material ahead of the tool face which begins simultaneously with the beginning of the localized shear in the main shear band, a (Fig 4) During deformation, this shear plane acts as the heat source which can be considered as a moving plane heat source with variable width moving along the direction A B [Fig 4(b)] As shown in Appendix A, the temperature rise, 0M,

at point M in the shear band caused by the second heat source is given by

2a and

f / ~ e x p ( - 0) - ~-~)

is a special function, and co is a non-dimensional, positive value

formed and the rake face of the tool is assumed as a moving plane heat source with variable intensity

of heat liberation As shown in Appendix A, the temperature rise in the shear band, Ou at point M,

Modeling of thermomechanical shear instabilily in machining 1289 caused by the moving frictional plane heat source is given by

X = x + W i cos(q~ - :~), Y = y - V , t - W i sin(q~ - ~), u = ~ and r { = X 2 -k- y2

The f o u r t h h e a t s o u r c e is the frictional heat source of the chip segment sliding on the tool face during upsetting of the material of the wedge shape part of the following chip segment This heat source is assumed as a moving heat source with a variable width and a variable heat generation This

is because the initial contact between the segment being formed and the tool face is at the apex of the tool and is extremely small The contact length as well as the heat generation of the heat source increases as the upsetting of the following segment takes place (Fig 4) This gradually shifts the shear separation between the segments to the frictional shear between the segment and the rake face of the tool As shown in Appendix A, the temperature rise, OM at any point M ( x , y) and at any time t caused

by this moving frictional plane heat source with variable width is given by

qpl Y V :~t- to~),,4, do) exp - ~o -

be considered The mean temperature rise on the main shear band caused by this image heat source

is designed as 011 In addition, there will be a preheating effect of this image heat source which should also be considered The mean temperature rise due to the preheating effect of this heating source is designated as 011, Other heat sources are relatively far away from their boundaries; hence, the effects of these image heat sources can be neglected

a temperature higher than the r o o m temperature This, in brief, is the preheating effect caused by the four heat sources

1290 Z.B Hou and R Komanduri

As shown in Appendix A, the temperature rise, 0M, at point M in the next shear band caused by

the second heat source is given by (see Appendix A for details)

q p l w Y V v ' to,)/4a do9 exp - co -

For the preheating effect of the fourth heat source, the temperature rise, Ou, at point M(x, y) in the

following shear band is given by (see Appendix A for details)

Modeling of thermomechanical shear instability in machining 1291

In Appendix B, one of the equations, namely, the temperature rise due to the first primary heat source, or, the shear band heat source is used as an example for the calculation of the temperature rise in the shear band at a given cutting speed for an AISI 4340 A similar approach can be used to calculate the temperature rise due to each of the heat sources at various cutting speeds, as well as for different values of n Different stages of the deformation of the main shear band are expressed by

a number, n, which can vary from 0 to 1 At the very beginning n = 0, and when the deformation reaches its maximum value (AI = l~), n = 1

9 DETERMINATION OF THE CUTTING SPEED FOR THE ONSET

OF SHEAR LOCALIZATION

F o r the determination of the cutting speed above which shear instability occurs, it is sufficient to determine the temperature of the shear band in the initial stages of chip segmentation For, if shear localization can take place at this stage, it is bound to occur at the following stages where the temperatures will be higher than this value Hence, this temperature is used for the calculation of the shear stress of the material in the shear band [using Eqn (2)] If this is lower than the original shear strength of the material, shear localization will be imminent Knowing this temperature, the shear stress in the shear band was estimated and compared with the strength of the work material at the preheating temperature If the shear stress in the shear band, a', is greater than or equal to the strength of the material, a at the preheating temperature, no shear localization takes place; instead, strain hardening occurs If the shear stress in the shear band, a' is equal to or less than the original strength ~r, then shear localization is imminent The model is found to predict 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]

10 RESULTS AND DISCUSSION 10.1 Calculation of temperature rise in the shear band

Appendix B gives an example calculation for the temperature rise in the shear band due to the first primary heat source (primalry shear band heat source) for the case of machining an AISI 4340 steel work material Similar approach can be taken to calculate the temperature rise in the shear band due

to each of the primary as well as preheating effects of the primary heat sources The cutting conditions used in this example are as follows: orthogonal cutting, cutting speed: 100 m/min, depth

of cut : 0.2 mm, width of cut: 1.5 mm, and rake angle ~: 10 °, friction angle between the chip segment formed and the total rake surface, fl:45 ° (/~ = 1), main shear angle q5:27.5 °, and secondary shear angle ~b': 41.25 ° [19] The thermal properties of the work material used are: thermal conductivity,

2 = 0.0797 cal/m s °C and thermal diffusivity, a = 0.0925 cmZ/s

Based on the thermal modeling of the shear localization, the contributions to the total temper- ature rise in the primary shear band, ~d, due to all the heat sources for an AISI 4340 steel at varJious cutting speeds from 1 to 500 m/min during the initial stages of the shear localization (i.e n = 0.05) are summarized in Table 3 Also given in this table are the contributions of the four primary heat sources, the contributions of the preheating effects of the primary heat sources, the shear stress in the shear band, a', and the shear strength of the bulk material, a, at the preheating temperature The actual temperature, T, can be obtained by adding 20°C to ~ff, which is the assumed room temperature

Figures 5(a) and (b) show the variation of contributions to the total temperature rise in the shear band due to various primary heat sources and preheating effects of the primary heat sources, respectively, with cutting speed during the initial stages of chip segmentation (i.e n = 0.05) and Fig 5(c) shows the variation of the temperature rise in the shear band (~0), sum of the primary heat sources ( ~ 0 primary), and sum of the preheating heat sources ( ~ 0 secondary) with cutting speed It can be seen from Table 3, the shear band temperature rise due to the first heat sources, 01, increases with cutting speed (i.e from about 19.5°C at 1 m/min to 220°C at 100 m/rain) and becomes the single most dominant heat source with increase in cutting speed However, at very low speeds, the contributions of the preheating sources can be significant For example, at 1 m/min, 01 is only 19.5°C, while the sum of the preheating effects Z0' is 76.9°C (nearly 4 times) Hence, its effect is very significant

in determining onset of shear localization, especially if it occurs at low speeds Even at 100 m/min, while 0 ~ is 220°C, the sum of the preheating effect, ~0' is 121.2°C (about 55%) is significant Also,

Modeling of thermomechanical shear instability in machining 1293

Cutting Speed, v m/min

Fig 5(a) Variation of the temperature rise in the shear band due to the various primary heat sources with cutting speed during the initial stages of chip segmentation (i.e n = 0.05) in machining of AISI 4340 steel

Cutting Speed, v m/rain

Fig 5(b) Variation of the temperature rise in the shear band due to the various preheating effects of the primary heat sources with cutting speed during the initial stages of chip segmentation (i.e n = 0.05) in

machining of AISI 4340 steel

Cutting Speed, v m/min

Fig 5(c) Variation of the total temperature rise in the shear band ~ 0 , sum of the temperature rise due to primary heat sources ~ 0 primary and sum of the temperature rise due to preheating heat sources ~tq preheating with cutting speed during the initial stages of chip segmentation (i.e n = 0.05) in machining of AISI 4340 steel

1294 Z.B Hou and R Komanduri

as in the case of titanium 6 A1-4 V alloy [5, 6], the preheating effects can be very significant and hence should be considered in the analysis

10.2 Prediction of the onset of shear instability in machinin9

Figure 6 shows the variation of shear stress in the shear band, a', at the shear band temperature and shear strength of the bulk material, a, at the preheating temperature, with cutting speed (Table 3) They were obtained using Eqn (2) (relation between shear stress, temperature, and strain) which is developed on the basis of material properties data for a and the thermal properties of AISI

4340 steel available in the literature (see Section 4) When a' is greater than or equals to a, no shear localization takes place, since strain hardening predominates This is the case at cutting speeds up to

52 m/min Above this speed, thermal softening predominates over strain hardening, i.e a' < a, hence shear localization is imminent This is the case for values of cutting speed higher than 52 m/rain The intersection of these two curves gives the cutting speed for the onset of shear localization Experi-

mental results of Komanduri et al [8] showed the onset of shear instability at about 60 m/min which

agrees reasonably well with the analytical results Also, once the transition from a continuous to

a shear-localized chip occurs, no further transition or reversal to a continuous chip was observed experimentally with further increase in speed up to 1000 m/min The analytical results presented here thus supports the experimental observation

11 C O N C L U S I O N S

1 Based on the analysis of cyclic chip formation in machining of some of the difficult-to-machine materials, such as hardened alloy steels, titanium alloys, and nickel-base superalloys a physical mechanism for shear localization in machining is presented

2 Based on the thermal model of the shear localized chip formation process, four primary heat sources and the preheating effects of these four heat sources were identified for the temperature rise

in the shear band In addition, the image heat source due to the first primary shear band heat source