The sort of factors that could affect chip radius are variations of friction along the chip/tool contact length and the roundness of the cutting edge, and also the work hardening behavio
Trang 1No better relationship has ever been found, for machining with plane-faced tools The reason for this is easy to understand Qualitatively, a curled chip may be regarded as shorter (more compressed) at its inner radius than at its outer radius Only rarely are chips
so tightly curled that (r/t) < 5; even then the variation in compression from the chip centre-line to its inner and outer radii is only ± 0.1, i.e t/(2r) Average chip equivalent strains
(equation 2.4(b)) are typically greater than 1 Thus, the modifications to flow associated with curvature are secondary relative to the magnitude of the flow itself The sort of factors that could affect chip radius are variations of friction along the chip/tool contact length and the roundness of the cutting edge, and also the work hardening behaviour and variations
of work hardening behaviour through the thickness of the chip (most chips are formed from surfaces which themselves have previously been strained by machining)
2.2.4 Shear plane angle prediction
The previous section gives data that show that chip thickness, and hence shear plane angle, depends on tool rake angle, friction and work hardening; and it records how forces and tool stresses can be estimated if shear plane angle, rake angle and friction angle are known In this section, early attempts, by Merchant (1945) and Lee and Shaffer (1951), to predict the shear plane angle are introduced Both attempted to relate shear plane angle to rake angle and friction angle, and ignored any effects of work hardening
Merchant suggested that chip thickness may take up a value to minimize the energy of cutting For a given cutting velocity, this is the same as minimizing the cutting force
(equa-tion (2.5(b)) with respect to f The well-known equa(equa-tion results:
Lee and Shaffer proposed a simple slip line field to describe the flow (see Appendix 1 and Chapter 6 for slip line field theory) For force equilibrium of the free chip, it requires that the pressure on the primary shear plane is constant along the length of the shear plane
and equal to k If (p/k) = 1 and Dk = 0 are substituted in equation (2.7), Lee and Shaffer’s
result is obtained:
f = p/4 – (l < a) or (f < a) = p/4 – l (2.10) Neither equation (2.9) nor (2.10) is supported by experiment Although they correctly
show a reducing f with increasing l and reducing a, each predicts a universal relation between f, l and a and this is not found in practice However, they stimulated much
exper-imental work from which later improvements grew
It is common practice to test the results of experiments against the predictions of
equa-tions (2.9) and (2.10) by plotting the results as a graph of f against (l – a) It is an
obvi-ous choice for testing equation (2.9); and equation (2.9) was the first of these to be derived
As far as equation (2.10) is concerned, an equally valid choice would be to plot (f – a) against l Different views of chip formation are formed, depending on which choice is taken The first choice may be regarded as the machine-centred view: (l – a) is the angle
between the resultant force on the tool and the direction of relative motion between the
work and tool The second choice gives a process-centred view: (f – a) is the complement
of the angle between the shear plane and the tool rake face Figures 2.13 and 2.14 present selected experimental results according to both views
The data in Figure 2.13 (from Shaw, 1984) were obtained by machining a free-cutting
Trang 2steel at a low cutting speed (0.025 m/min), with high speed steel tools with rake angles from 0˚ to 45˚ A range of cutting fluids were applied to create friction coefficients from 0.13 to 1.33 When the results are plotted as commonly practised (Figure 2.13(a)), data for each rake angle lie on a straight line, with a gradient close to 0.75, half way between the expectations of equations (2.9) and (2.10) When the process-centred view is taken (Figure
2.13(b)), an almost single relation is observed between the friction coefficient and (f – a).
Figure 2.14 collects data at higher, more practical, cutting speeds for turning a range of
ferrous, aluminium and copper alloys (Eggleston et al., 1959; Kobayashi and Thomsen,
1959) Both parts of the figure show each material to have its own characteristic behaviour Both show that annealed steel machines with a lower shear plane angle than the same steel cold-rolled Figure 2.14(b) marginally groups the data in a smaller area than does Figure 2.14(a) Certainly part b emphasizes the range of friction angles, common to all the mater-ials, from 25˚ to 40˚ (friction coefficient from 0.47 to 0.84) As this book is machining-process centred, the view of part b is preferred
Figure 2.15 gathers more data on this basis Figure 2.15(a) shows that free-cutting steels
Fig 2.13 φ–λ–α relationships for low speed turning of a free cutting steel with tools of different rake angle (0ºx, 16º+, 30ºo, 45º•), varying friction by selection of cutting fluid: (a) φ versus (λ–α) and (b) (φ–α) versus λ (after Shaw, 1984)
Fig 2.14 φ–λ–α relationships for normal production speed turning by high speed steel tools, with rake angles from 5º to 40º, of cold rolled (•) and annealed (o) free cutting steel, an aluminium alloy (+) and an α-brass (×): (a) φ versus (λ–α) and (b) (φ–α) versus λ (data from Eggleston et al., 1959)
Trang 3generally have lower friction coefficients (from 0.36 to 0.70) than non-free-cutting steels (from 0.47 to 1.00) when turned with high speed steel or cemented carbide tools (Childs, 1980a) Figure 2.15(b) extends the data to the machining of difficult materials such as austenitic stainless and high manganese steels, nickel-chromium and titanium alloys, by carbide and ceramic tools Friction angles remain in the same range as for other materials but larger differences between shear plane and rake angle are found Care must be taken
in interpreting this last observation Not only are lower rake angles used for the difficult to machine materials (from +10˚ to –5˚ for the data in the figure), biasing the data to larger
(f – a), but these materials also give serrated chips The data in Figure 2.15(b) are
aver-aged over the cycle of non-steady chip formation
2.2.5 Specific energies and material stress levels in machining
In the preceding sections, basic force and moment equilibrium considerations have been used, with experimental observations, to establish the mechanical conditions of continu-ous chip formation With the exception of the Merchant and Lee and Shaffer laws, predic-tion of chip shape has not been attempted Predictive mechanics is left to Chapters 6 and after In this section, by way of a summary, some final generalizations are made, concern-ing the energy used to form chips, and the level of contact stresses on the tool face
The work done per unit machined volume, the specific work, in metal cutting is Fc/(fd) The dimensionless specific work, may be defined as Fc/(kfd) Equation (2.11) takes
equa-tion (2.5b) and manipulates it to
—— = ———————— ≡ —— + tan(f + l < a) (2.11)
kfd sin f cos(f + l < a) tan f From Figures 2.13 to 2.15, the range of observed (f + l – a) is 25˚ to 55˚ (except for the nickel-chromium and titanium alloys); and the range of l is 20˚ to 45˚ With these
Fig 2.15 φ–λ–α relationships compared for (a) free-machining (o) and non-free machining (•) carbon and low alloy steels; and (b) austenitic stainless and high manganese steels (o), nickel-chromium heat resistant (•) and titanium alloys (+) turned by cemented carbide and ceramic tooling
Trang 4numbers, the non-dimensional specific work may be calculated for a range of rake angles.
Figure 2.16(a) gives, for rake angles from 0˚ to 30˚, bounds to the specific work for tan(f + l – a) from 0.5 to 1.5 and for l = 20˚ to 45˚ It summarizes the conflicts in designing a machining process for production For a high rake angle tool (a = 30˚), specific work is relatively low and insensitive to changes in f and l In such conditions an easily controlled
and high quality process could be expected; but only high speed steel tools are tough enough to survive such a slender edge geometry (at least in sharp-edged, plane rake face
form) At the other extreme (a = 0˚), cutting edges can withstand machining stresses, but
the specific work is high and extremely sensitive to small variations in friction or shear plane angle In practice, chamfered and grooved rake faces are developed to overcome these conflicts, but that is for a later chapter
Of the total specific work, some is expended on primary shear deformation and some
on rake face friction work The specific primary shear work, Up, is the product of shear
force kfd/sinf and velocity discontinuity on the plane (equation (2.3)) After ‘non-dimen-sionalizing’ with respect to kfd,
kfd sin f cos(f < a) which is the same as the shear strain g of equation (2.4a) The percentage of the primary
work to the total can be found from the ratio of equation (2.12) to (2.11) For the same ranges of numbers as used in Figure 2.16(a), the percentage ranges from more than 80%
when tan(f + l – a) = 0.5, through more than 60% when tan(f + l – a) = 1.0, to as little
as 50% when tan(f + l – a) = 1.5 The distribution of work between the primary shear
region and the rake face is important to considerations of temperature increases in machin-ing Temperature increases are the subject of Section 2.3
Finally, equations (2.5) can be used to determine the normal and friction forces on the tool face, and can be combined with equations (2.6) and (2.2) for the contact length between the chip and tool, in terms of the feed, to create expressions for the average normal and friction contact stresses on the tool:
Fig 2.16 Ranges of (a) dimensionless specific cutting force, (b) maximum normal contact stress and (c) maximum
fric-tion stress, for observed ranges of φ, λ, α (º) and m/n
Trang 5sn n 2cos2l tn n 2cos l sin l
(——)av.
= — —————— ; (——)av.
k m sin2(f + l < a) k m sin2(f + l < a)
In Section 2.2.3, the influence of m/n on contact stress distribution was considered,
lead-ing to Figure 2.12 The same considerations can be applied to derivlead-ing the peak contact stresses associated with the average stresses of equations (2.13) Figures 2.16(b) and (c) show ranges of peak normal and friction stress for the same data as given in Figure 2.16(a),
for the practically observed range of m/n from 1.3 to 3.5 Peak normal stress ranges from one to three times k Peak friction stress is calculated to be often greater than k This, of
course, is not physically realistic The loads in machining are so high, and the lubrication
so poor, that the classical law of friction – that friction stress is proportional to normal stress – breaks down near the cutting edge Section 2.4 gives alternative friction modelling, first widely disseminated by Shaw (1984)
It has already been mentioned that the focus of this introductory mechanics section is descriptive and not predictive However, the earliest predictive models for shear plane angle have been introduced – equations (2.9) and (2.10) In most cases, they give upper and lower bounds to the experimental observations It may be asked what is the need for better prediction? The answer has two parts First, as shown in Figure 2.16(a), the specific forces in machining (and hence related characteristics such as temperature rise and machined surface quality) are very sensitive to small variations in shear plane angle, for commonly used values of rake angle Secondly, the cutting edge is a sacrificial part in the machining process, with an economic life often between 5 and 20 minutes (see Chapter 1) Small variations in mechanical characteristics can lead to large variations in economic life
It is the economic pressure to use cutting edges at their limit that drives the study of machining to ever greater accuracy and detail
2.3 Thermal modelling
If all the primary shear work of equation (2.12) were converted to heat and all were
convected into the chip, it would cause a mean temperature rise DT1in the chip
rC sin f cos(f < a) rC where rC is the heat capacity of the chip material Table 2.2 gives some typical values of k/(rC) Given the magnitudes of shear strains, greater than 2, that can occur in machining
(Section 2.2), it is clear that significant temperature rises may occur in the chip This is without considering the additional heating due to friction between the chip and tool It is important to understand how much of the heat generated is convected into the chip and what are the additional temperature rises caused by friction with the tool
The purpose of this section is to identify by simple analysis and observations the main parameters that influence temperature rise and their approximate effects The outcome will
be an understanding of what must be included in more complicated numerical models (the subject of a later chapter) if they are also to be more accurate Thus, the simple view of chip formation, that the primary and secondary shear zones are planar, OA and OB of
Trang 6Figure 2.17(a), will be retained Convective heat transfer that controls the escape of heat from OA to the workpiece (Figure 2.17(b)) is the focus of Section 2.3.1 How friction heat
is divided between the chip and tool over OB (Figure 2.17(c)) and what temperature rise
is caused by friction is the subject of Section 2.3.2 The heat transfer theory necessary for all this is given in Appendix 2
2.3.1 Heating due to primary shear
The fraction of heat generated in primary shear, b, that flows into the work material is the main quantity calculated in this section When it is known, the fraction (1 – b) that is
carried into the chip can also be estimated The temperature rise in the chip depends on it
Table 2.2 Mechanical and physical property data for machining heating calculations
∆T1 [ °C] a 230–470 180–300 110–340 250–430 470–680
Ktoolb [W/m K] 20–50 80–120 100–500 80–120 50–120
a∆T1 for γ ≈ 2.5 and β = 0.85; b tool grades appropriate for work materials.
Fig 2.17 (a) Work, chip and tool divided into (b) work and (c) chip and tool regions, for the purposes of temperature
calculations
Trang 7Figure 2.17(b) shows a control volume AA′ fixed in the workpiece The movement of the
workpiece carries it both towards and past the shear plane with velocities u˘ z and u˘ x, as
shown u˘ ˘z = Uwork sinf and u˘ x = Uworkcosf When the control volume first reaches the
shear plane (as shown in the figure), it starts to be heated By the time the control volume
reaches the cutting edge (at O), some temperature profile along z is established, also as
shown in the figure The rate of escape of heat to the work (per unit depth of cut), by
convection, is then the integral over z of the product of the temperature rise, heat capacity
of the work and the velocity u˘ x:
∞
Qconvected to work= ∫u˘ x (T – To)rC dz (2.15a)
0
The temperature profile (T – T0) is given in Appendix 2.3.1: once a steady state
tempera-ture is reached along Oz
Qconvected to work = ∫—— q1e–u˘ z z/k dz ≡ ——— (2.15b)
z
where q1is the shear plane work rate per unit area and k is the thermal diffusivity of the work material The total shear plane heating rate per unit depth of cut is the product of q1 and the shear plane length, q1(f/sinf) The fraction b of heat that convects into the work is
the ratio of equation (2.15b) to this After considering that equation (2.15b) is a maximum estimate of heat into the work (the steady temperature distribution might not have been
reached), and also after substituting for values of u˘ x and u˘ z in terms of Uwork
k
Uworkf tan f
According to equation (2.16), the escape of heat to the work is controlled by the
ther-mal number [Uworkf tanf/k] This has the form of the Peclet number, familiar in heat
trans-fer theory (Appendix A2.3.2) The larger it is, the less heat escapes and the more is convected into the chip A more detailed, but still approximate, analysis has been made by Weiner (1955) Equation (2.16) agrees well with his work, provided the primary shear Peclet number is greater than 5 For lower values, equation (2.16), considered as an equal-ity, rapidly becomes poor
Figure 2.18(a) compares Weiner’s and equation (2.16)’s predictions with experimental and numerical modelling results collected by Tay and reported by Oxley (1989) Weiner’s
result is in fair agreement with observation b varies only weakly with [Uworkf tanf/k]: a change of two orders of magnitude, from 0.1 to 10, is required of the latter to change b from 0.9 to 0.1 There is evidence that as [Uworkf tanf/k] increases above 10, b becomes
limited between 0.1 and 0.2 This results from the finite width of the real shear plane The implication from Figure 2.18(a) is that numerical models of primary shear heating need
only include the finite thickness of the shear zone if [Uworkf tanf/k] > 10, and then only if (1 – b), the fraction of heat convected into the work, needs to be known to better than 10% Figure 2.18(b) takes the mean observed results in Figure 2.18(a) and, for f = 25˚, converts them to relations between Uworkand f that result in b = 0.15 and 0.3, for k = 3, 12
and 50 mm2/s These values of k are representative of heat resistant alloys (stainless steels,
Trang 8nickel and titanium alloys), carbon and low alloy steels, and copper and aluminium alloys respectively The speed and feed combinations that result coincide with the speed/feed ranges that are used in turning and milling for economic production (Chapter 1) In
turn-ing and millturn-ing practice, b≈ 0.15 is a reasonable approximation (actual variations with cutting conditions are considered in more detail in Chapter 3) A fraction of primary shear
heat (1 – b), or 0.85, then typically flows into the chip The DT1of Table 2.2 give primary
zone temperature rises when f ≈ 25˚ and b = 0.85 For carbon and low alloy steels, copper
and Ni-Cr alloys, these rises are less than half the melting temperature (in K): plastic flow stays within the bounds of cold working However, for aluminium and titanium alloys, temperatures can rise to more than half the melting temperature: microstructural changes can be caused by the heating Given that the primary shear acts on the workpiece, these simple considerations point to the possibility of workpiece thermal damage when machin-ing aluminium and titanium alloys, even with sharp tools
The suggested primary shear temperature rise in Table 2.2 of up to 680˚C for titanium alloys is severe even from the point of view of the edge of the cutting tool The further heating of the chip and tool due to friction is considered next
2.3.2 Heating due to friction
The size of the friction stress t between the chip and the tool has been discussed in Section 2.2.5 It gives rise to a friction heating rate per unit area of the chip/tool contact of qf=
tUchip Of this, some fraction a* will flow into the chip and the remaining fraction (1 – a*)
will flow into the tool The first question in considering the heating of the chip is what is
the value of a*?
The answer comes from recognizing that the contact area is common to the chip and the tool Its temperature should be the same whether calculated from the point of view of the flow of heat in the tool or from the flow of heat in the chip Exact calculations lead to the
conclusion that a* varies from point to point in the contact Indeed so does qf To cope with such detail is beyond the purpose of this section Here, an approximate analysis is
devel-oped to identify the physically important properties that control the average value of a*
Fig 2.18 (a) Theoretical (—, -) and observed (hatched region) dependence of β on [Uworkf tan φ/κ]; (b) iso-β lines ( β = 0.15 and 0.3) mapped onto a (Uwork, f ) plane for κ = 3, 12 and 50 mm 2 /s and φ = 25º
Trang 9and to calculate the average temperature rise in the contact It is supposed that qfand a* are constant over the contact, and that a* takes a value such that the average contact
temperature is the same whether calculated from heat flow in the tool or the chip Figure
2.17(c) shows the situation of qfand a* constant over the contact length l between the chip and tool The contact has a depth d (depth of cut) normal to the plane of the figure.
As far as the tool is concerned, there is heat flow into it over the rectangle fixed on its
surface, of length l and width d Appendix A2.2.5 considers the mean temperature rise over
a rectangular heat source fixed on the surface of a semi-infinite solid To the extent that the nose of the cutting tool in the machining case can be regarded as a quadrant of a semi-infi-nite solid, equation (A2.14) of Appendix 2 can be applied to give
(1 – a*)tavUchipl (T – T0)av.tool contact = sf——————— (2.17)
Ktool where T0is the ambient temperature, K is thermal conductivity and sfis a shape factor
depending on the contact area aspect ratio (d/l): for example, its value increases from 0.94
to 1.82 as d/l increases from 1 to 5.
As far as the chip is concerned, it moves past the heat source at the speed Uchip Its temperature rise is governed by the theory of a moving heat source This is summarized in
Appendix A2.3 When the Peclet number Uchipl/(4k) is greater than 1, heat conducts a
small distance into the chip compared with the chip thickness, in the time that an element
of the chip passes the heat source In this condition, equation (A2.17b) of Appendix 2 gives the average temperature rise due to friction heating Remembering that the chip has already been heated above ambient by the primary shear,
kg a*tavUchipl kwork 1 /2
(T – T0)av.chip contact= (1 – b) ———— + 0.75 ————— (——— ) (2.18)
(rC)work Kwork Uchipl Equating (2.17) to (2.18) leads, after minor rearrangement, to an expression for a*:
tav Uchipl kwork 1 /2 Kwork tav (rC)work a* —— ——— [0.75 (——— )+ sf——— ]= sf—— ———— Uchipl – (1 – b)
(2.19)
tavis related to k, l to f and Uchipto Uworkby functions of f, l, a and (m/n), as described previously, by combining equations (2.2), (2.3), (2.6) and (2.13) g is also a function of f and a After elimination of tav, l and Uchipin favour of k, f and Uwork, equation (2.19) leads to
0.75 Ktool n cos l cos(f – a)tanf 1/2 kwork 1 /2
a* [1 + —— —— sf Kwork (m— ———————— sin(f + l – a) ) ( ————— Uworkf tanf ) ]
(2.20a)
(1 – b) Ktool cos a cos(f + l – a)
sf[Uworkf tan f/kwork] Kwork sin l cosf
Trang 10The manipulation has introduced the thermal number [Uworkf tanf/kwork] b depends on this too (Figure 2.18(a)) If typical ranges of f, l, a and (m/n), from Figures 2.10, 2.14 and
2.15 are substituted into equation (2.20a), the approximate relationship is found
(0.45 ± 0.15) Ktool kwork 1 /2
a* [1 + —————— sf (——— Kwork )(————— Uworkf tan f) ]
(2.20b) (1.35 ± 0.5) (1 – b) Ktool
≈ 1 – ————— ———————— sf [Uworkf tan f/kwork](——— Kwork )
Figure 2.19(a) shows predicted values of a* when observed b values from Figure
2.18(a) and the mean value coefficients 0.45 and 1.35 are used in equation (2.20b) A
strong dependence on [Uworkf tanf/kwork] and the conductivity ratio K* = Ktool/Kworkis
seen, and a smaller but significant influence of the shape factor sf Predictions are only
shown for [Uworkf tanf/kwork] > 0.5: at lower values the assumption behind equation
(2.18), that Uchip1/(4k) is greater than 1, is invalid; and anyway friction heating becomes small and is not of interest As a matter of fact, the assumption starts to fail for [Uwork
f tanf/kwork] < 5 Figure 2.19(a) contains a small correction to allow for this, according to low speed moving heat source theory (see Appendix A2.3.2)
Figure 2.19(a) reinforces the critical importance of the relative conductivities of the tool
and work When the tool is a poorer conductor than the work (K* < 1), the main propor-tion of the fricpropor-tion heat flows into the chip As K* increases above 1, this is not always so Indeed, a strong possibility develops that a* < 0 When this occurs, not only does all the
friction heat flow into the tool, but so too does some of the heat generated in primary shear The physical result is that the chip cools down as it flows over the rake face and the hottest
part of the tool is the cutting edge When a* > 0, the chip heats up as it passes over the
tool: the hottest part of the tool is away from the cutting edge
Fig 2.19 Dependence of (a) α* and (b) friction heating mean contact temperature rise on [Uworkf tanφ/κwork], K* =
K /K from 0.1 to 10; s = 1 (—) and 2 (- -)