R th  = OT – OU = r – (r 2 – t 2 /4) ½  ∴R th ≅ t 2/ 8r Case I When medium values of the turned feedrate are utilised – as is the situation in ‘Case  II’ (Fig. 166a), then both the tool nose (r) and the partial eect of the ‘End-cutting angle’ (C e ) must be considered, when estimating the deeper ‘R th ’ value. Assuming that ‘t’ is equal to ‘JL’ (Fig. 166a and b), then the following geo- metrical conditions are met: t = JK+HE+NL (Fig. 166a) ∴ t = [r 2 – (r – R th  ) 2 ] ½  + r sinC e + r(R th /r – 1 + cosC e )cotC e Case II NB  is equation is valid, just as long as the ‘t’ length lies between positions: ‘SE’ and ‘FR’. If even larger values of ‘t’ are utilised (i.e. higher feed- rates), then the three geometrically curved and linear portion’s of the cutting insert’s prole, will aect resul- tant turned surface topography. Namely, this surface topography aer machining, will now be comprised of the sum of the portions of the: end-cutting angle ‘C e ’; nose radius ‘r’; together with the side-cutting angle ‘C s ’ , as follows: R th  = AB – AG + r (Figs 166a and b) � R th = t tan C s +cot C e − r cos( −C e �−C s �) sin(−C e �+C s �) + r Case III e equations given in ‘Cases:  I, II and III’ , can be utilised to produce a ‘non-dimensional’ graphical plot for various cutting insert approach angles (i.e. see Shaw, 1984 – p 513) enabling the estimated value of ‘R th ’ to be found – for any combination of: ‘t’; ‘r’; ‘C e ’; plus ‘C s ’. Historically, it has been well-known that there exists a ‘minimum undeformed chip thickness’ 57 and if a value 57 ‘Minimum undeformed chip thickness’* was proposed some years ago by Sokolowski (1955), where it was suggested that a ‘limiting value’ occurred for a chip thickness, below which a tool simply rubs. *In the experiments undertaken at the time by Sokolowski, using a very sharp and honed single-point turning tool with a cutting edge nose radius of 12 µm at a cutting speed of 210 m min –1 , the smallest cut that could be taken had a depth of 4 µm. For additional information on succeeding research work on this sbject, also see Footnote 58. is smaller than this actual chip-thickness, it is not pos- sible to form a successful chip, as a result, only rubbing will occur – this being a combination of many inter- related factors. Applying this ‘minimum undeformed chip thickness’ concept to the insert’s secondary cutting edge, it was found that a small triangular portion of workpiece material which should have been removed, is normally le ‘in-situ’ (Fig. 167). is fraction of the workpiece le behind, has been the subject of intensive interest by previous machinability researchers over a number of years and is known as the ‘Spanzipfel’ 58 . In Fig. 167b, the raised portion (i.e. Spanzipfel) occurs at increments equivalent to that of the feed rev –1 and is il- lustrated in Fig. 167a. An expression has been derived to take account of this Spanzipfel on the theoretical surface texture, as follows: R′ th  = t 2 /8r + t m /2 (1+ rt m /2) NB  In this equation, the 2 nd term represents the con- tribution of the Spanzipfel. Both the theoretical values for the surface texture and actual ones from the trials are in close agreement, which was not expected, as the Spanzipfel will be sub- ject to some plastic deformation – as it comes into di- rect contact with the tool’s clearance face. In both end- and face-milling operations (Fig. 168a), the machining process is one of interrupted cutting, tending to impart an isotropic surface topog- raphy to the milled surface (Fig. 168b). If in Fig. 168b (i.e. top diagram) the excess stock material is removed by face-milling, the resultant surface exhibits quite a complex surface topography. is milled surface complexity results from the cutter’s trailing edge, as it moves over the previously cut surface at the periodic and pre-selected feedrate. is periodic surface topog- 58 ‘Spanzipfel’ , was initially investigated and analysed by Bram- mertz (1960). He was particularly interested in the Spanzip- fel’s aect of the resulting machined surface topography/tex- ture. Later work by Pahlitzsch and Semmler (i.e. between 1960 to 1962), looked at the ‘minimum undeformed chip thickness’ and the Spanzipfel’s inuence when ne-turning AISI 1045 steel workpieces with specially-sharpened ceramic tooling. Here, they found that machining with this much more abra- sive-resistant ceramic tooling, the ‘minimum undeformed chip thickness’ height could be dramatically-reduced to just 1 µm. Cutting speed utilised in these tests was 400 m min –1 and the machinability trials used an in-cut time of just 6 seconds – in order to maintain a sharp cutting edge on the tools. Machinability and Surface Integrity  Figure 167. The aect of the ‘minimum undeformed chip thickness’ on the machine topography, producing the so- called ‘Spanzipfel eect’ .  Chapter  raphy will not occur, if either a ‘slight’ milling spindle camber is utilised (i.e. see Fig. 86), or some form of ‘torque-controlled machining’ (TCM) system is em- ployed, that usually incorporates some form of ‘adap- tive control constraint’ (ACC) engagement of the feed- ing-system – additional information on the subject of TCM will be given in the nal chapter. e major rea- son for TCM feeding variability, is because the torque is monitored and as the stock height varies along the workpiece’s length, the torque is either lessened by slowing the feedrate, or increased, thereby maintain- ing relatively constant cutting forces on the tooling. is form of ‘adaptive control’ (ACC) by constraining the cutter’s feed, will impart a variable surface topog- raphy to the milled surface. In the previous milling scenario where there was no necessity for an TCM requirement, the milling cutter’s rotation in combination with the feed for a given ‘cut-o’ slightly changes the milled surface to- pography. It introduces some variability to the respec- tive cusp heights along the workpiece’s milled surface (Fig. 168b). Here, the periodic nature of the surface topography is regular (i.e. a constant ‘Sm’), but its pe- riodicity marginally changes according to whether the surface is measured at the centre of the cut, or oset across the face-milled surface, which has an some ef- fect on the relative cusp heights. Conversely, across the milled surface at arbitrary positions denoted in these examples as: ‘X-X’ and ‘Y-Y’ (Fig. 168b), the topog- raphy uctuates somewhat at a predetermined and quantiable interval, depending upon where the sur- face trace was taken. Hence, any milled surfaces with a non-directional, or indenable Lay – as is the case for most re-cutting eects introduced by either end-, or face-milling operations, should not simply have an ar- bitrary Ra quoted on the engineering drawing, as this has been shown (Fig. 168b) to be somewhat meaning- less. Milled surfaces having these latter characteristics, requiring the need to indicate the maximum tolerable Ra value for a given Lay direction – in a similar fash- ion to that of an anisotropic machined surface topog- raphy. Returning to the longitudinal turned surface to- pography once more. If consideration is given to the ‘idealised’ turning surface (Fig. 169a and b), then for a constant tool nose insert geometry and undeformed chip thickness, as the feed rev –1 is increased, the sur- face texture will be degraded. e residual cusps that periodically occur on the turned surface aer the tool’s passage along the part, are a product of two previ- ously described inter-related phenomena (Fig. 154), namely: the ‘moving-step eect’; in conjunction with the ‘emerging diameter’. is relationship diminishes the notable height of the turned cusps with a reduced feed rev –1 , while it increases with larger feed rev –1 – this aspect being depicted in Figs. 169a and b, respectively. If a proportionally larger feed rev –1 is selected, this in- creases the residual inuence of the tool nose contact region on the surface – as formerly mentioned when discussing Fig. 166. As a result of higher feeds the RSm increases, which inevitably heightens the cusps, promoting a larger recorded value in Ra, in associa- tion with greater angles for ∆q (Fig. 169b). Of course, the opposite is true in the case of reduced feeds (Fig. 169a). Explicitly, as a smaller tool nose contact region occurs – with reduced feed rev –1 (i.e ‘Case I’ in Fig. 166a), this has the eect of producing a smaller cusp height (R th ) and its accompanying Ra, giving a more shallow value of ∆q; due to the partial curvature of the tool nose radius tending toward zero as it approaches tangency with that of the workpiece’s axis (Fig. 169a). e dominant factor here is the feedrate, as it has an enormous inuence on the resultant turned cusps, af- fecting their: height; prole shape; periodicity; in asso- ciation with the pre-selected tooling geometry; these factors radically inuencing both the measurement and magnitude of the machined surface topography, which in turn, aects the surface texture parameters. If just the Ra parameter had been specied, it could not adequately describe the nature and condition of the surface topography in any consequential manner. Assuming that standardised cutting conditions are selected for a turning operation: workpiece com- position; rotational speed; feedrate, undeformed chip thickness, with only the tool nose geometry change, then the resulting surface topography will be markedly dierent. is divergence in machined topography is illustrated to good eect in Figs. 169c and d, where turned ferrous P/M compacts are depicted. Here, two extremes of cutting insert nose radii are utilised: Fig. 169c the nose radius was 0.8 mm; whereas in Fig. 169d a button-style insert (φ12 mm) having the equivalent of 6 mm nose radius was used. e turning insert with the 0.8 mm nose radius produced visually-apparent regularly-spaced cusps and despite the fact that a new turning insert was employed, there is evident signs of tears, laps and burrs present around the turned pe- riphery. In contrast in Fig. 169d, the turned surface topography appears appreciably smooother in prole, although even here, the surface topography is marred by similar tears, etc., which might be a cause for its potential rejection. is smoother surface was due to Machinability and Surface Integrity  Figure 168. In face- and end-milling operations, due to the ‘re-cutting eect’ of the trailing insert cutting edges, they impart an ‘almost’ isotropic milled surface topography to the part .  Chapter  Figure 169. How the feedrate inuences the machined cusp/surface roughness value ‘Sm’ and its aect on the wavi- ness parameter ‘Δq’, plus surface topography of actual longi- . tudinal CNC turned P/M ferrous compacts – cut with diering nose radii. [Courtesy of Joel (UK) Ltd.] Machinability and Surface Integrity  the considerably larger eective tool nose radius, act- ing like a ‘wiper-blade’ blending-out and obliterating the surface’s cusps. is technique of utilising a large tool nose geometry has traditionally been used by pre- cision turners to improve the overall surface nish. .. Manufacturing Process Envelopes e principal features of manufacturing process en- velopes and indeed, for many amplitude distribution curves is that they can be approximated by the so- called ‘beta-function’ – ‘β’ (Fig. 170a). Here, the func- tion has two parameters that are independent of one another enabling them to be used as a means of sur- face characterisation. e notation ‘a’ is the allocated weighting for the prole ordinates measured from the lowest valley and above, with notation ‘b’ being given to weighting the prole from the highest peak down. Hence, peaks and valleys have accordingly dierent weights. One of the problems that has arisen from util- ising this technique for a topographical prole, which has somewhat discredited them for certain applica- tions, is how and in what manner can one determine ‘a’ and ‘b’. e ‘beta-function’ is normally dened within a set range of: 0→1, being expressed in the following man- ner: β(a, b) =  �  z a− (a − z) b− dz . If by changing the range of the ‘beta-function’ equa- tion above, from: 0→1 to Rp + Rv, or indeed with that of Rt, then substituting σ (i.e. the standard deviation of the distribution) with Rq, the beta-function param- eters ‘a’ and ‘b’ become: a = Rv (Rv Rp – Rq 2 /Rt Rq 2 ) b = Rp (Rv Rp – Rq 2 /Rt Rq 2 ). e fact that any dominant peak, or valley within the assessment length is only raised to a unit power, in- fers additional stability over the ‘skewness/kurtosis ap- proach’. e problem with this ‘beta-function’ method is in accurately determining ‘sound’ results from the Rv and Rp, which conrms the diculty that obtain- ing information from peak/valley measurement and then deriving valid information is fraught with com- plications. In Figs. 170: ‘ai’ it is symmetrical; with ‘aii’ being asymmetrical; for their respective ‘beta-func- tions’ , these relationships are based upon a class of sta - tistically-derived ‘Pearson distributions’ 59 . In the sym- metrical case (Fig. 170ai) the skewness equates to zero; conversely, for an asymmetrical series of results (Fig. 170aii), skewness can be either positively, or negatively skewed (i.e see Fig. 164bii). Nevertheless, even allow- ing for these limitations, an example of the groups of manufacturing process envelopes for a range produc- tion processes is illustrated in Fig. 170b. Here, the pro- duction processes can be simplistically classied and grouped into either a ‘bearing’ , or ‘locking’ surface topography. e ‘bearing-/locking-groupings’ indicate that certain production processes can achieve specic functional surfaces for particular industrial applica- tions. ese ‘groupings’ (Fig. 170b), also indicate that the general classications are less distinct than might otherwise be supposed, as certain processes can be provide either a ‘locking’ , or a ‘bearing’ surface condi - tion – termed ‘intermediate groupings’. Typical of such an ‘intermediate group’ are the P/M drilled compacts. One reason for this is that any P/M ‘secondary ma- chining’ oen utilises twist drills, which may produce 59 ‘Pearson product moment correlation coecient’* – to give it its full title, is a statistical association utilised when a ‘rela- tionship’ exists between several quantities and it is a measure of the extent of this aliation, thus producing its ‘correlation coecient’ which can then be utilised. *For example, having a sample of pairs of observations ‘x’ and ‘y’ , the value ‘r’ of this ‘Pearson coecient’ , is given by the ‘generalised formula’ , below: r = Σ(x – x ) (y – y) /√ Σ(x – x) 2 Σ(y – y) 2 (Bajpai, et al., 1979): e calculated ‘Pearson coecient’ should lie either close to –1, or +1. If the calculated value is close to –1 then the ‘straight-line trend’ is in a downward direction, conversely, if the calculated value is close to +1 then the ‘straight-line trend’ direction is upward. If the value is close to zero (0), then no correlation exists, so the pairing of the data is disparate and cannot be utilised. With calculated data that is reasonably close to either –1, or +1, a ‘regression line’ (i.e. using linear regression) can be calculated based upon the general straight- line formula: Y = a + bX   Where a ‘regression line of Y on X’ for the two constants ‘a’ and ‘b’ respectively are: b = nΣxy – (Σx)(Σy)/n(Σx 2 ) – (Σx) 2   a = Σy – bΣx/n Where a ‘regression line of X on Y’ for the two constants ‘a’ and ‘b’ respectively are: b = nΣxy – (Σx)(Σy)/n(Σy 2 ) – (Σy) 2   a = Σx – bΣy/n NB e ‘regression line’ being the equivalent of the ‘least squares line’ , allows data on each axes to be compared – with some degree of condence.(Wild, et al.,1995)  Chapter  Figure 170. The ‘beta-function’ and typical ‘manufacturing process envelopes’. Machinability and Surface Integrity  a ‘saw-toothed prole’ to the hole’s surface, along with pores in the compact that are open to the ‘free-surface’ of the hole. e hole topography may have this ‘saw- toothed eect’ present, it being a combination of the drill’s partial lip and margin occurring at the feed rev –1 periodicity, formed by the drill spiralling-down and around the hole’s periphery. Hence, the drill’s passage creates a positive skewness via drilled ‘saw-toothed cusps’ , while the pores can introduce negative skew - ness – creating a potential ‘intermediate group’ to the manufacturing process envelope groupings. .. Ternary Manufacturing Envelopes (TME’s) In machining operations the dominant factor that in- uences surface topography has been shown to be the tool’s feedrate. In Fig. 171, the feedrate, in conjunc- tion with the principal factors such as surface texture (Ra) and roundness (i.e least squares circle – LSC), are utilised to dene the limits for these ‘Ternary manu- facturing envelopes’ (TME’s). By using such diverse factors as: surface texture, roundness and processing parameters (feedrate), for the major axes on the ter- nary graph, enables the surface to be characterised in a unique manner. Such TME’s dier quite consider- ably from the more usual and restricted ‘manufactur- ing process envelopes’ alluded to in the previous section – the skewness and kurtosis axes of the manufacturing envelopes, might otherwise mask crucial information. e ‘TME approach’ gives a psuedo three-dimensional representation on its ternary axes, which can be ex- ploited to illustrate how the inuence of changing a parameter – such as feedrate – modies the relation- ship of the associated surface texture and roundness values for the nal machined result. As an example of the eectiveness of this TME ap- proach to the complex problem of machining data analysis, Fig. 171 has been drawn from an actual machinability trial. If one observes this TME graph closely for a pre-selected range of turning and boring processes, indicated in Fig. 171, with specic reference here, to turning operations – by way of illustrating the TME’s expediency. e TME shows how – for the turning operations – at low feedrate (0.10 mm rev –1 ) the surface texture is closely conned to a relatively small spread of values – nominally around 0.5-1.5 µm Ra, whereas its associated roundness lies between approximately 5 and 50 µm LSC. As the feedrate in- creased in an arithmetic progression to 0.25 mm rev –1 , the range of the surface texture bandwidth propor- tionally expanded to 1.5 at approximately 5-6.5 µm Ra, with a corresponding roundness ranging from 8 to 48 µm LSC, giving a proportional bandwidth of 1.6. As the feedrate was raised even higher, to 0.40 mm rev –1 , it was not surprising to note that this also produced increases in both the surface texture and its propor- tional bandwidth, with similar values with respect to its roundness. ese ‘machinability and metrology trends’ allow examination of both the bandwidth vari- ability and the aect of dierent feedrates on other disparate factors – such as its machined roundness. Similar trends occurred for the boring operation, but here only two feedrates were employed, by applica- tion of this analysis technique via the ‘TME-approach’ to a concise machinability trial, complex analysis of the TME is possible. e pseudo three-dimensional graph, oers perhaps an unusual insight into the mul- tifaceted inter-relationships that exist aer workpiece machining. e TME shows that simply examining one metrological parameter in isolation to those that could aect it, may mask vitally important relation- ships and trends that would otherwise remain unseen. By careful selection of the parameters for the respec- tive axes, perhaps based upon the feedrate (i.e. here, normally situated along the X-axis), allows an appre- ciation of the whole surface at any instant along the three graph’s axes. 7.6 Machining Temperatures Ever since Taylor in 1907, recognised that elevated tool and workpiece temperatures in metal cutting played a crucial role in inuencing tool edge wear rates, the subject has been one of intensive study. Moreover, that the tool/chip interface temperature has a control- ling inuence on the rate of crater wear and the fact that tool life can be drastically curtailed by these in- duced machining temperatures, as such, the topic has received considerable research attention. Here, space will only allow a brief resumé of this complex temper- ature-induced machining problem. During metal cutting in particular, there are sev- eral temperature eects that need to be considered. In Fig. 51, an orthogonal single-point cutting operation is schematically illustrated, indicating the distribution of heat sources within the three deformation zones. In  Chapter  Figure 171. ‘Ternary manufacturing envelopes’ for the production processes of turning and boring, axes: feedrate, roundness and surface texture . Machinability and Surface Integrity  particular, the heat generated in the main ‘body’ within the cutting region via both the primary and secondary zones is a result here, of the workpiece’s plastic defor- mation. Still more intensive heat is generated at the tool/chip interface – along the rake face, with the ma- jority of heat being swept away with the chips, while the remainder of heat is either conducted through the tool, or conducted/convected into the workpiece. As- suming that no coolant application is present in the machining operation, then any heat loss to the ambi- ent air becomes insignicant. An equation has been developed that governs the temperature distribution – via its isothermal gradients – in machining (Fig. 172), this being an ‘energy-based equation’ as follows: ρC∂T∂t  V  T  k  T  ˙ q   (Source: Tay, et al., 1993). erefore, in steady-state machining operations, the transient term will disappear and at the region of the tool (i.e. insert) only the conduction term remains. ∴ Rate of heat generation (˙q) = σ ˙ε Where: ‘σ’ was obtained from an emprical function of: ‘ε’ , ‘˙ε’; plus ‘T’. NB  is rate of heat generation only exists within the primary and secondary deformation zones. By way of example of how the temperature genera- tion/distribution occurs in orthogonal cutting, in the more-easily understood ‘Boothroyd machining model’ (i.e. being in a slightly modied form – by the author), this workpiece material is in a state of ‘continuous motion’ during cutting (Fig. 172). If specic points are selected to show how temperatures occur as they pass along/through these deformation zones then, the points: ‘X, Y and Z’ can be considered for special observation. So, as the workpiece material enters the cutting region at point ‘X’ , it begins to move toward the cutting insert. It approaches and passes through the primary deformation zone where it is heated-up until it leaves this zone, it is then swept-away by the formed chip. Equally point ‘Y’ , passes through both the primary and secondary deformation zones (i.e see Fig. 51 for these deformation zones) and contin- ues to heat-up until it leaves the secondary deforma- tion zone. In both of the above cases, these points (i.e. namely: X and Y) are cooled as heat is conducted into the chip’s body (as it exit’s the cut), where it eventually achieves a uniform temperature right the way through. Prior to this occurring, the maximum temperature occurs along the cutting insert’s rake face, some dis- tance from the actual cutting edge (i.e see Fig. 172). Conversely point ‘Z’ , which remains attached to the workpiece, is heated by conduction from the primary deformation zone and some heat is also conducted from the secondary deformation zone into the body of the cutting insert, while the tertiary deformation zone will also impart some heat into the machined surface of the workpiece. Many thermal and thermographical techniques have been developed over the years to obtain accurate isothermal temperatures within the: cutting zones: tool/insert interface plus rake face vicinity; together with the machined surface region of the workpiece. Moreover, ‘indirect methods’ have been utilised to obtain similar thermal historical data from within these dynamic and harsh environments, but only one of these techniques will be mentioned in the next sec- tion. .. Finite Element Method (FEM) e popular approach today, to obtaining ‘simulated’ thermal data is by employing the ‘Finite element method’ (FEM), to calculate temperature distributions in the vicinity of the cutting regions (Fig. 173). Typical of this approach and worth mentioning in some de- tail, was that conducted and described by Tay (1993), where he experimentally-obtained information re- garding: velocity, strain and strain-rate distributions, by utilising a printed-grid and quick-stop technique. e rate of heat generation within the primary defor- mation zone was determined from the equation: (˙q) = σ ˙ε. From the deformed grid pattern (Fig. 173a), the ac- tual dimensions of the triangular deformation zone, as well as the velocity distribution along the tool/chip interface can be established and analysed. By this FEM technique, it is possible to determine the shear-strain rate within the secondary deformation zone at the tool/chip interface: (˙γ int ) has been found to be approximately constant and equal to: V c/ δt 2 . e shear strain-rate within the  Chapter  . min –1 and the machinability trials used an in-cut time of just 6 seconds – in order to maintain a sharp cutting edge on the tools. Machinability and Surface Integrity  Figure 167 . The. envelopes’ for the production processes of turning and boring, axes: feedrate, roundness and surface texture . Machinability and Surface Integrity  particular, the heat generated in the main ‘body’. the surface topography is marred by similar tears, etc., which might be a cause for its potential rejection. is smoother surface was due to Machinability and Surface Integrity  Figure 168 .