fi=(pi/4) - (1/2)(beta - alfa)- the basic relation from Merchant theory of metal cutting.
Conditions for the relationship above to be true:
- fi- shear angle which it is adjusting itself to result minimum work
- beta- friction angle which is independent of fi and is constant
- the edge of the tool is sharp
- the distribution of the shear force and normal stress is uniform, in the shear plane
- the part material is rigid and plastic
- the width of the generated chip is constant
After experimental work, the relationship above was found to be over estimated, so that Merchant modified it and the result was another relationship:
2*fi + beta - alfa=C, C being the machining constant. In order to C<PI/2, grain size and the micro structure has to remain constant after the material is manufactured.
The normal force Fn, which is acting in a direction perpendicular to the shear plane, is changing due to the deformations and frictions.The normal stress affects the shear stress, in the shear's direction.