# Manufacturing Processes - MFRG 315 - 2.3 Cutting Forces

2.3 Forces in Orthogonal Cutting

A dynamometer capable of measuring forces in 3 directions (only 2 are needed in orthogonal cutting) can be used to measure cutting forces. In the diagram below the reactions to these forces, which are acting on the tool, are shown:

Fc acting in the direction of cutting

Ft thrust force needed to keep the tool in the workpiece (direction perpendicular to the workpiece surface). At high positive rake angles the thrust force is negative and the tool is pulled into the workpiece.

The resultant force on the tool, Pr, is made up of a normal force, Pn acting perpendicular to the tool face and F the friction force acting along the face.

Pn = Fc cos(alpha) - Ft sin(alpha)

F = Fc sin(alpha) + Ft cos(alpha)

where alpha is the rake angle.

Knowledge of the shear angle, phi, will allow the deformed chip thickness to be estimated.

2.4 Cutting Theories

Experimentally it has been found that the shear angle and hence the cutting ratio, depend upon the workpiece and tool material and the cutting conditions. Several attempts have been made to establish a theoretical law which predicts the shear angle.

Ernst and Merchant developed the first reasonable theory. This assumed that the shear angle would take up a value to make the work done in cutting a minimum.

For given cutting conditions, the work done is proportional to Fc, an expression has been developed for Fc in terms of the shear angle and then a value for the shear angle found for which Fc is a minimum.

After some algebra this gives:

where tau is the mean angle of friction between the chip and the tool and

tau = tan- 1 (F/Pn)

This was later developed and other researchers have developed other expressions.

Lee and Shaffer realised that the above equation could not apply where tau = 45o and the rake angle, alpha = 0 as these values give a shear angle, phi = 0. They considered that such conditions of high friction and low rake angle are just the conditions that lead to the formation of a built up edge. To meet this a second solution was presented where a built up edge is present on the tool face.

Experimental results show that no simple unique relationship can agree with all experimental results.

It is difficult to predict forces because the relevant flow stress is not easy to predict.

Large shear strains (1 - 5 natural strain) across a narrow zone in a very short period of time mean that strain rates are very high, several 1000 s-1

2.5 Approximate Calculations - use experimentally determined constants.

2.5.1 Specific Cutting Pressure The cutting force, Fc, divided by the cross section area of the undeformed chip gives the nominal cutting stress or the specific cutting pressure, pc

NB pc is not a true stress even though it has dimensions of stress.

2.5.2 Specific Cutting Energy The energy consumed in removing a unit volume of material is called the specific cutting energy, E1. Energy (or work) is force Fc x distance, L, over which the force acts. The volume of material removed is V = h w L, so the specific cutting energy can be written as:

Such calculations are required to determine the size of the drive motor.

2.5.3 Material Removal Factor, K1 is the reciprocal of the specific cutting energy and indicates the volume of material which can be removed in unit time with a drive of unit power.

These 'material constants' can not be simply used in calculations as they are not true constants but depend upon process parameters such as undeformed chip thickness, rake angle and cutting speed.

As well as energy for shearing in the primary shear zone, flank friction (between the tool flank and the newly formed surface) and friction at the cutting edge (together often referred to as the ploughing force, P) absorb energy. The energy dissipated here is almost independent of the undeformed chip thickness. When h is small a greater proportion of the energy used is absorbed here. Hence the energy required to remove a unit volume of material increases with decreasing undeformed chip thickness.

2.5.4 Adjusted Specific Cutting Energy For the above reason 'material constants', E1 are published for agreed conditions eg: h = href = 1 mm and the adjusted specific cutting energy, E, for any other h can be found from the empirical law:

where the constant 'a' ranges between 0.2 and 0.4 and may be taken as 0.3 for most materials.

For an undeformed chip thickness below 0.1 mm, the energy requirement increases even more rapidly.

2.5.5 Power Required for Cutting

The power needed by a machine tool can be estimated if the rate of material removal, Vt and m/c tool efficiency (typically in the region of 0.7 to 0.8) are known.

Power, W = E Vt/efficiency.

The table below indicates values of 'Unit Power' for for some metals, it should be noted that values vary somewhat in different sources:

 Material Hardness, BHN Unit Power, W.s/mm3 Steel 1020 150 1.5 Steel 1040 200 1.8 Steel 1330 260 2.5 Stainless steels 135 - 275 1.4 Cast irons 110 - 190 0.8 Cast irons 190 - 320 1.6 Titanium 250 - 375 3 Super alloys (Ni and Co) 200 - 360 3 to 9 Aluminium alloys 30 - 150 0.6 Magnesium alloys 40 - 90 0.3 Copper 40 2 Leaded brass 75 0.7 Zinc alloys 0.3

2.5.6 Machinability

Machinability is not a precisely defined term, it is an attempt to account for several factors: tool life, power required for cutting, surface finish obtained, cost of removing material. The most important factor in most situations is usually tool life and machinability ratings are frequently based on this.

For many components the strength of the part is not as important as economical machining. Consequently for these applications materials are often selected for ease of machining - good tool life.

Factors that affect machinability are:
Hardness and ductility. Increasing hardness makes penetration by the tool more difficult, decreasing machinability. Generally lower ductility, which promotes discontinuous chips, is beneficial to machinability.
Because the wear of the cutting tool is heavily dependent upon the temperature it reaches, the material properties that govern this are critical:
Low workpiece: thermal conductivity, density and specific heat all give higher mean tool temperatures.
Increasing the the workpiece total cutting energy per unit volume increases the tool temperature.
So for similar tool wear rates, titanium can only be cut at about one quarter the speed of steel, which in turn can only be cut at about one tenth of the speed of aluminium.

Additions of elements such as lead, phosphorus, sulphur and tellurium to steel improves its machinability, but there are some disadvantages.

David J Grieve, 17th December 2004.