## Eq

where T —T(z, t) is the temperature of the workpiece, t is time, and R is the fraction of the generated heat that flows into the workpiece. The solution to this boundary value problem is (Ref 17):

where erfc is the complementary error function. On the surface:

Assuming that at a particular point on the surface the heat is applied beginning at x = -a, the time of heating for a point on the surface is t = (a - x)IV and the temperature at a point on the surface is:

The average surface temperature is:

3VJC k 1L

The form of this analysis is consistent with that of a similar analysis given by Ramanath and Shaw (Ref 18) for full wheel grinding.

Equations 2 and 5 can be written for the abrasive particle (or wheel) as well as the workpiece if R is replaced by 1 - R and the properties of the abrasive are used. Writing such an equation and setting the average temperature of the grain equal to the average temperature of the workpiece under conditions of thermal equilibrium gives (Ref 18):

where the subscripts A and W refer to the abrasive and the workpiece, respectively.

The concentrated contact between the abrasive and the workpiece is very similar to that occurring in a microhardness test. If we assume that the contact is a circular patch of radius a, the radius of contact can be obtained as:

where N is the normal grinding force and H is the Knoop hardness of the workpiece. (The workpiece generally has a smaller hardness than the abrasive.) The heat generated per unit area per unit time is:

where F is the tangential grinding force. Note that this model of the moving heat source does not require any knowledge of the fundamental deformation mechanism (such as brittle fracture, plastic deformation, or plowing) leading to heat generation. Substituting Eq 1, 6, 7, and 8 in Eq 5, the average surface temperature (T avg) of the workpiece and the particle in single-point grinding is obtained as:

and (Favg) is independenJ_of the thermal properties of the work material. Equation 9 or 10, along with Eq 7 and 8, can now be used to calculate if the normal and tangential grinding forces acting on an abrasive particle are known.

The normal and tangential forces on a single abrasive particle can be directly measured during single-point grinding or in single-grain fly cutting (Ref 1, 9, 19). When grinding is done with a full wheel, the statistical distribution of forces on individual grits can be estimated from a statistical analysis of grit sizes on the wheel surface. Such a calculation has recently been carried out for lapping and polishing (Ref 20). There has, however, been no direct measurement of forces on individual grits during full wheel grinding of metals and ceramics.

There are two important points to be noted in Eq 8 and 9. First, the work-surface (or particle-surface) temperature (Eq 9) is a function of the grinding specific energy (i.e., the energy required for a unit volume of material removal) and the heat partition coefficient, R. This implies that to achieve lower grinding temperatures, a lower specific energy and a smaller value of R are needed. The specific energy depends on the physics of deformation processes leading to material removal, but R is dependent on the thermal properties of the abrasive and the workpiece (Eq 6). The second point is that the main thermal property influencing R is not just the conductivity (k), but the product (k c) of the abrasive and workpiece materials, respectively. This fact has been highlighted by Shaw (Ref 3). Table 1 gives physical properties of various abrasive and work materials, including values of k c. From Table 1, it can be seen that both cubic boron nitride (CBN), and diamond have significantly higher values of k c than alumina (sapphire). Thus, Eq 6 and 9 imply that all other factors remaining the same, grinding temperatures would be lower with a system using CBN or diamond wheels than with one using aluminum oxide wheels. This is consistent with both direct and indirect experimental measurements of temperatures and its derived variables. Quantitative comparisons between measured and predicted temperatures (using Eq 9) are made in the next section.

 Property (YZ-110)(a) Si3N4 Sapphire Ni-Zn ferrite 1070 steel Diamond CBN E, GPa 210 300 390 191 203 1000 660
 v 0.24 0.26 0.23 0.2 0.26 0.2 0.15 k, W/m/°C 2.2 33 35 8.7 47 1000 300-600 c, J/gm/°C 0.63 0.72 0.95 0.71 0.432 0.525 0.51 , gm/cc 6.1 3.22 3.9 5.3 7.84 3.5 3.48 k c 8.45 76.5 129.7 32.7 159.2 1837.5 532-1065 HK, GPa 12 16.7 19.6 7.3 88 40-70

(a) YZ-110 is a tetragonal zirconia polycrystal manufactured by Norton Co.

(a) YZ-110 is a tetragonal zirconia polycrystal manufactured by Norton Co.

Measurement of Grinding Temperatures. The abrasive-tip and work-surface temperatures in grinding are highly localized spatially (~100 ^m or less in spot size) and decay rapidly with time (within microseconds). Measurement of these temperatures is therefore a challenging task.

Some of the early measurements of grinding temperatures were carried out with thermocouples embedded into the workpiece (Ref 9, 21). The relatively large time constant and poor spatial resolution of thermocouples enable only an estimate of the average temperature to be obtained. In the grinding of steels, such measurements have reported values of ~800 °C (1475 °F) for the work-surface temperature (Ref 9). This is in the neighborhood of the austenitizing temperature for many low-carbon steels.

Another technique for measuring grinding temperatures, which is much more sensitive both spatially and in time, involves monitoring and analysis of infrared radiation being emitted by the abrasive particle and/or the workpiece (Ref 1, 6, 22, 23). Using such a technique, the authors recently measured full-wheel, single-particle, and work-surface temperatures during the dry grinding of metals and ceramics (Ref 1, 23). For more details about the technique, see Ref 1; Table 2 and Fig. 3, 4, 5, and 6 show some of the results. In Table 2, the measured temperatures in single-particle (diamond) grinding of ceramics are compared with the analytical estimates (Eq 9) for this temperature at different wheel velocities. There is good agreement between the measured and predicted values. It must be noted here that the measured particle temperatures are high, ~600 to 1600 °C. When grinding is done with a full wheel, different abrasive particles on the wheel surface are exposed to different depths of cut due to their varying amounts of protrusion from the wheel surface. Therefore, each of these particles cuts out a different volume of material, and their tip temperatures are likely to differ. Indeed, this is seen to be the case in Fig. 3, 4, 5, and 6, where the particle temperatures are plotted as histograms. The mean of the particle-surface temperatures on the wheel (and consequently work-surface hot-spot temperatures) are found to vary between 592 °C for zirconia and 721 °C for 1070 steel.

 Material R Temperature, °C, at wheel velocity of: 25 m/sec 32 m/sec 37 m/sec E A E A E A
 Zirconia 0.064 1260±51 1320 1601±74 1494 Si3N4 0.17 1133±60 1110 1452±49 1255 Ni-Zn ferrite 0.12 570±30 537 620±35 607 690±30 653 Sapphire 0.21 920±65 1060±45 1270±80

(a) R, fraction of the generated heat that flows into the workpiece, as defined in Eq 6; E, experimentally measured value (including standard deviation; A, analytically measured value (depth of cut 10 /'m, table velocity 23.4 mm/s, diamond indenter tip radius 15 /Jm)

Fig. 3 Distribution of wheel temperatures in full wheel grinding of silicon nitride. Depth of cut, 12.5 /'m; table velocity, 23.4 mm/s; wheel velocity, 32 m/s; 220-grit
Fig. 4 Distribution of wheel temperatures in full wheel grinding of zirconia. Depth of cut, 12.5 /'m; table velocity, 23.4 mm/s; wheel velocity, 32 m/s; 220-grit
Fig. 5 Distribution of wheel temperatures in full wheel grinding of 1070 carbon steel. Depth of cut, 12.5 /'m; table velocity, 23.4 mm/s; wheel velocity, 32 m/s; 220-grit
Fig. 6 Distribution of wheel temperatures in full wheel grinding of ferrite. Depth of cut, 20 /Jm; table velocity, 23.4 mm/s; wheel velocity, 32 m/s; 320-grit

Sometimes, abrasive-particle temperatures are observed to reach the melting point of steel during the grinding of hardened steels. It is possible that under such conditions, there is actually melting of the steel at some regions along the wheel-work contact. Such melting could cause the spherical swarf particles shown in Fig. 7. Indeed, there is compelling evidence in support of this hypothesis (Ref 24). Many investigators (Ref 9, 25), however, have attributed the formation of such spherical particles only to melting during exothermic oxidation of the chips in air; this is, no doubt, a parallel mechanism. Infrared temperature measurements have also shown that the subsurface temperature in a workpiece decays rapidly with depth (Fig. 8). The high temperature gradients are the primary source of the tensile residual stresses, microcracking, and phase transformations sometimes observed on ground surfaces. Another observation from temperature measurements is that an abrasive grain on the wheel surface cools down rapidly after leaving the grinding zone and almost reaches room temperature in one revolution of the wheel before it begins its next cut (Ref 1).

Fig. 7 Scanning electron micrograph of grinding swarf from 52100 steel showing spherical particles

Fig. 8 Distribution of subsurface workpiece temperatures in full wheel grinding. Table velocity, 23.4 mm/s; wheel velocity, 32 m/s. • , silicon nitride (measured, 220-grit, 12.5 |jm depth of cut); •, ferrite (measured, 320-grit, 20 pm depth of cut); CD, 1070 steel (measured, 220-grit, 12.5 pm depth of cut). The solid lines are calculated values for the subsurface temperature obtained by matching the experimentally measured temperatures at the surface.

Fig. 8 Distribution of subsurface workpiece temperatures in full wheel grinding. Table velocity, 23.4 mm/s; wheel velocity, 32 m/s. • , silicon nitride (measured, 220-grit, 12.5 |jm depth of cut); •, ferrite (measured, 320-grit, 20 pm depth of cut); CD, 1070 steel (measured, 220-grit, 12.5 pm depth of cut). The solid lines are calculated values for the subsurface temperature obtained by matching the experimentally measured temperatures at the surface.

Coolant and Grinding Temperatures. Thus far, focus has been concentrated primarily on dry surface grinding. In practice, grinding operations on metals usually involve use of a coolant-lubricant fluid (Ref 4, 9). The purpose of this fluid is twofold: to provide lubrication and reduce frictional forces at the wheel-work interface and to cool the workpiece. No well-validated analysis of grinding temperatures takes into account heat transfer into the fluid, but the physical principles underlying such an analysis would not alter the broad nature of the temperature calculations given in Eq 9. It is generally accepted that the coolant has little influence on the peak abrasive tip temperature but does reduce the average bulk temperatures of the wheel and workpiece, thereby limiting thermally induced dimensional distortions. Although there is no experimental verification of the former hypothesis, the proper application of coolant is very beneficial in many grinding operations with metals due to its influence on the bulk temperature.

Surface Damage. The high surface temperatures and steep temperature gradients at the surface of the workpiece material during grinding are responsible for many forms of damage (Ref 26). During the grinding of ceramics, which are relatively brittle compared to metals, the high thermal stresses developed near the surface cause microcracking (Ref 27). Such microcracking is one of the main causes of strength degradation and strength anisotropy (with respect to grinding direction) commonly observed in ground ceramic components (Ref 28). Control of these thermal stresses is critical to the development of better grinding methods for ceramics. In electronic ceramic materials, the high temperatures and temperature gradients cause the near-surface electromagnetic properties to be changed, which affects the performance of these materials when they are fabricated into devices. Examples in this category are the ferrites (Ni-Zn, Mn-Zn, etc.) used in recording heads in magnetic storage systems. Grinding residual stresses or heating of the material during grinding to above the Curie temperature (typically between 800 and 900 °C, or 1470 to 1650 °F, for these materials) usually leads to the formation of magnetically damaged surface layers that have an adverse effect on recording head performance during the read/write process (Ref 8).

During the grinding of plain carbon or hardened alloy steels, one of the most common types of thermally induced damage is workpiece burn (Ref 29). The burn is characterized visually by bluish temper colors on the work surfaces, which are usually attributed to oxide-layer formation. The temper colors can be removed by sparkout at the end of the grinding cycle. However, this does not mean that the effects of burn are removed. Microhardness measurements on workpieces show that visible burn is accompanied by reaustenitization of the work surface during grinding. When hardened steels are ground without any burning, there is generally some softening due to tempering of the material close to the surface. This is seen in the "no burn" curve in Fig. 9 (Ref 29). Here, the material near the surface is seen to have a smaller microhardness value than the material in the bulk. However, under burn conditions in hardened steels, quite the opposite effect is seen at the surface; the "burn" curve in Fig. 9 shows that the surface layer has been hardened by grinding. This occurs as a consequence of reaustenitization of the work material near the surface during grinding, followed by the formation of hard, untempered martensite as a result of quenching. The martensitic layer can be identified upon etching of the surface as a white phase, usually occurring in patches (Ref 11, 12, 29). When grinding of soft steels is done, however, workpiece burn is not seen to cause any surface hardening. In such instances, burn is best detected by metallographic techniques (Ref 11, 12). The occurrence of workpiece burn in steels has been shown to be virtually coincident with the work-surface temperature exceeding the austenizing temperature (~800 °C, or 1470 °F, for low- to medium-carbon steels) during grinding. This observation has provided a basis for analytically deriving "burn limit" conditions for steels through calculations of grinding temperature from power or force measurements (Ref 29).

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