The purpose of theory of grinding is to establish relationship between the radial feed, force on individual grits of grinding wheel, velocity of grinding wheel, velocity of work and their diameters. Fig. 20.5 shows a magnified portion of the grinding wheel and work in contact with each other.

It will be noted that when an abrasive gain starts to enter or penetrates the material, such as at A, the depth of cut is zero, it increases gradually as the wheel and the work revolve, and becomes minimum somewhere, along the arc of contact of the wheel and the work.

Since the wheel usually rotates much faster than the work, the point of maximum cut depth is almost at the point where the wheel leaves the work. The maximum depth is known as the gain depth of cut (represented by letter t).

Let diameters of work and grinding wheel bed and D, and their surface velocities be v and V respectively. Let T be the time taken by a grain on grinding wheel to move from A to B. So arc AB = V x T.

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During this time, a point on wheel at A will be able to move only upto C as shown in Fig. 20.5. Now arc AC = v x T. Obviously ACB shown by shaded area becomes the chip with its maximum thickness of CD.

By regulation of grain depth of cut, grinding wheels can be made to act softer or harder, either by increasing or decreasing the grain depth of cut. CD can also be varied by varying work speed or radial feed.

AC being a very small arc could be treated as straight line.

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∴ CD = AC sin (α + β) = v x T sin (α + β)

(α and β are the angles subtended by the arc of contact at the centre of wheel and work.)

Since there is not a single grit which is doing the cutting action, so if there be N number of grits per unit length of the wheel circumference (N can be measured by rolling wheel upon smoked glass and counting the marks left under a microscope) then the maximum chip thickness per grit or grain depth of cut-

From the equation (1), it is obvious that grain depth of cut varies directly as work speed, inversely as wheel speed and directly as sin (α + β).

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From the above, following facts are arrived at, relating to wheel action during cutting. These are derived with the assumption that there is only one variable and other factors remain constant. In actual practice, these must be tempered with other factors to produce results that will be satisfactory.

(as the radial feed (f) is very small in comparison to D and d, f^{2} can be omitted)

It is obvious from above equation that decrease in average chip thickness ‘t’ is possible by increase in wheel speed V. The decrease in chip thickness leads to better surface finish, tighter geometrical tolerances due to lower grinding forces, surface integrity and lower stresses in the component.

Thus all these advantages are possible with increase in speed of grinding wheel and thus there is tendency to attain as high grinding speed as possible in precision grinding applications.

Now force on individual grits of grinding wheel is proportional to the area of the chip formed, which is proportional to the square of grain depth of cut;

From equation (3), very important conclusions about behaviour of grinding wheel can be arrived at.

Obviously the grits will break from the wheel if the force exceeds the bond strength; so from equation (3), increasing of work speed is more effective in breaking the grits than increasing the radial feed.

For soft wheel, V should be high, and for hard wheels v should be high. Also if D and d are nearly equal as in internal grinding, then [(Z) + d)/Dd] is also small and, therefore, soft wheels are required. In external grinding where [(D +d)/Dd] is very large, F will be more and, therefore, hard wheels are required to counteract high force per grit. Similarly from equations (1), (2) and (3), very important conclusions can be derived.

For higher productivity, the material removal rate should be high. For this purpose, the abrasives should be able to withstand higher grinding forces, stay sharper for a longer period of time and fracture to expose new cutting edges.

**Chip/Dimensions in Surface Grinding: **

Undeformed chip length l in surface grinding l = √Dd

**Unformed chip thickness t, **

C = number of cutting points per unit area of periphery of the wheel and is estimated in the range of 0.1 to 10 per mm^{2}

r = ratio of chip width to average undeformed chip thickness. It has approximate value of between 10 and 20.