Merchant Model of Chip Formation

The goal is to describe work material deformation in the chip formation. With a description of deformation and work material mechanical properties (at the deformation zone conditions) the cutting forces and other process mechanics quantities can be obtained.

Model Development Plan:
The chip formation model we will use is the Merchant model of chip formation. The procedure that will be followed in developing this mechanics of chip formation model is
a) - describing the goal of the development,
b) - specifying what processes will be modeled and what quantities predicted,
c) - presenting a physical model of the process,
d) - abstracting the physical model to a mechanical model,
e) - developing detailed, mathematical descriptions of the process variables,
f) - applying a physical principle which governs the physical process and which can be applied to the mathematical model,
e) - producing the final mathematical model.

We will not be able to experimentally test the model.

a) The goal is to calculate / estimate / predict the cutting force.
The rationale for this effort,
- it is probably easier to calculate cutting force than running machining tests in which force is measured,
- the concept that the cutting force will depend on work material properties and the amount of deformation imposed to form the chip,
- the belief that material strength data is available

b) The lathe turning process will be considered since it includes all the important aspects of machining processes and has a simpler geometry or configuration than other processes. Cutting force is calculated since it is the largest of the forces acting, and combined with the speeds in the process accounts for the greatest portion of the power required. With material property data available, workpiece deformation will have to be calcualted in order to calculate cutting force.

c) There are two common ways to observe the chip formation process with the intent of describing the physical situation. One is the use of high speed video to observe the chip formation zone during machining. The other is the rapid stopping of the process, polishing and etching the chip formation zone and looking at the deformation patterns indicated in the workpiece structure.The same result can be had by rapidly removing the cutter from contact with the work.

What is observed is,
- chips form due to very intense shear deformation
- the chip formation zone or shear zone is narrow and extends from the tool edge to the free surface of the workpiece

d) The mechanical model, or the abstraction of the actual process, arises after a number of steps.
- concentrating on the chip formation zone, lathe turning looks similar to the turning of the end of a cylinder,
- the feed speed is so small compared to the cutting speed that we ignore it and the diameter of the workpiece is large enough that it's curvature is unimportant and so the process can be viewed as cutting the edge off of a plate,
- since there is little change in deformation pattern across the plate width as the chip is formed, we need only to consider a two-dimensional representation of the process,
- the shear zone is so narrow that we can approximate it as a shear plane
- the shear plane is at inclination or shear angle, ø to the cutting speed direction
- in the 2-d model the only tool geometry is the cutter rake angle, a and clearance angle, c4.

Our idealized chip formation model is then - chip formation by shear on a shear plane in which a layer of material (the uncut chip thickness, t) is removed from the workpiece forming a chip of thickness tc and producing the finished surface.

e) The outputs of the process or the dependent variables of the process model that are of interest are the shear angle ø and the cutting force Fc.

f) To predict ø a physical principle that describes how the chip formation process occurs is needed and is - mechanical processes occur in such a way that the energy needed for the process is minimum. That is, chip formation occurs in such a way that the energy input to the process in minimum.

The power required for cutting is given by

P = Fc V
The cutting speed is constant so changes in power or energy input to the chip formation process depend only on Fc. The minimum energy input condition is
dFc / dø = 0
and the appropriate second derivative condition.

This last equation is the process behavior description.

The cutting force, Fc, can be put in terms of work material strength, specifically material shear strength, by considering the forces acting in the chip formation zone.

The resultant force, R can be resolved into several different force systems. With interest in the cutting force

Fc = R cos( ß - a )
To put R in terms of the material shear strength, tau,
the shear force, Fs and shear plane area As can be used
R = Fs / cos( ø + ß - a ) = ( tau )( As ) / cos( ø + ß - a )
Fc = { ( tau )( As ) cos( ß - a ) } / { sin( ø ) cos( ø + ß - a )

Taking the derivative with the reasonable assumption of constant work material shear strength and the tool-chip coefficient of friction, i.e., not dependent on the shear angle, gives

cos ø cos( ø + ß - a ) - sinø sin( ø + ß - a ) = 0
tan ø tan( ø + ß - a ) = 1
tan( ø + ß - a ) = cot ø = tan( 90o - ø)

ø = 45° + a/2 - ß/2

Back to the Chip Formation Exercise

© 2001 by Barney E. Klamecki. All rights reserved.