## Grinding

Figure 1 shows a schematic of a surface grinding process that will serve as the framework for the discussion of the thermal aspects of grinding. In this operation, the workpiece is moved back and forth across a grinding wheel that is rotated at surface speeds typically exceeding 30 mIs.

The Moving Heat Source. In grinding, significant heat is generated at the localized contacts between the abrasive particle and the workpiece material. Moving heat source theory, with the heat generation being taken as the product of a frictional force (idealized to act tangentially at the particle-workpiece contact) and the relative sliding velocity between the abrasive and the workpiece, is applied to calculate the temperature field produced. The moving heat source model does not address the mechanism of heat generation, which is thought to be primarily due to plastic deformation of the workpiece very near the surface (Ref 13). However, this phrase near the surface begs the question of defining the workpiece-particle interface. This question is especially difficult in grinding, where material is being constantly removed from the workpiece.

The temperature field produced by a heat source moving over the surface of a semi-infinite solid (here, the workpiece) has been analyzed extensively (Ref 14, 15, 16, 17). One of the earliest analyses of such problems is that due to Blok (Ref 14); this was in the context of calculating sliding contact temperatures. He calculated the maximum surface temperature produced by frictional heating when a single asperity (attached to a semi-infinite solid) was slid against the surface of another semi-infinite solid. Blok's main contribution was in the estimation of the partition of the frictional heat between the two bodies, which was based on the assumption that the maximum temperature in the contact region was the same in either of the two bodies. Subsequently, Jaeger (Ref 15) provided a detailed analytical solution of the temperature field in a semi-infinite solid due to an infinitely wide strip heat source, having finite length in the direction of motion, moving over it. The calculations of Blok and Jaeger showed that the maximum surface temperature produced by the moving heat source in the solid was reached at the trailing edge of the contact region. Shaw (Ref 3) combined the work of Blok and Jaeger and applied it to estimate wheel and work temperatures in dry surface grinding. In this analysis the frictional heat generated at the wheel-work interface was applied as a moving heat source to the surface of a semi-infinite solid. Partition of the heat into the wheel and workpiece was carried out based on Blok's procedure. The mean work-surface temperatures thus calculated were found to be in good agreement with measured surface temperatures in dry grinding. The application of this analysis to calculate the maximum temperature of the abrasive particle and the workpiece during grinding is described below. More details can be found in Ref 1.

Calculation of Abrasive Grain Temperatures. Figure 2 shows a schematic of the moving heat source model used in the following calculations. The moving heat source model approximates the workpiece as a semi-infinite solid whose boundary is insulated away from the heat source. The heat source has length 2a in the sliding direction and is of infinite extent perpendicular to the sliding direction. It is moving across the solid with a velocity V and is assumed to generate heat at a rate of q per unit area per unit time. This solution of the temperature distribution for this problem involves modified Bessel functions. If the Peclet number

where a= k/( c) is the thermal diffusivity of the solid, k is the thermal conductivity, is the mass density, and c is the specific heat, then the flow of heat parallel to the surface can be neglected with respect to the velocity of the moving heat source. This approximation is appropriate for single-point (particle) grinding experiments where the abrasive particle is moving with a velocity of ยป 30 m/s.

The differential equation and the boundary conditions governing the one-dimensional transient temperature distribution are:

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