The Merchant equation

• One of the important relationships in metal cutting was

derived by Eugene Merchant. Its derivation was based on the
assumption of orthogonal cutting, but its general validity
extends to three-dimensional machining operations.
• Merchant started with the definition of shear stress derived
by combining the following equations.
 Cont.

• Shear stress

• Merchant reasoned that, angle Ф angle at which shear stress is

just equal to the shear strength of the work material, and so
shear deformation occurs at this angle. In effect, the work
material will select a shear plane angle that minimizes energy.

 How shear plane angle is determined?

 Cont.

• This angle can be determined by taking the derivative of the

shear stress with respect to Ф and setting the derivative to
zero. Solving for Ф , we get the relationship named after
Merchant:
 Example

In an orthogonal cutting operation, the tool has a rake angle =

15°, the shear plane angle of 25.4°. Determine:
(a) the friction angle and
(b) the coefficient of friction.
 Cont.

the friction angle

β = 2 (45) + 10 - 2 (25.4) = 49.2°

The coefficient of friction

μ= tan 49.2 = 1.16
 From merchant equation
 The shear plane angle can be increased by
(1) increasing the rake angle and
(2) decreasing the friction angle (and coefficient of friction)
between the tool and the chip.
 Rake angle can be increased by proper tool design
 friction angle can be reduced by using a lubricant cutting fluid.

 What is the importance of increasing the shear plane angle?

• higher shear plane angle results in a smaller shear plane area. Since
the shear strength is applied across this area, the shear force
required to form the chip will decrease when the shear plane area
is reduced. A greater shear plane angle results in lower cutting
energy, lower power requirements, and lower cutting
temperature.

• Effect of shear plane angle : (a) higher Ф with a resulting lower

shear plane area;(b) smaller Ф with a corresponding larger shear
plane area.
 Power and energy relationship in machining
• A machining operation requires power. The product of cutting
force and speed gives the power (energy per unit time)
required to perform a machining operation:
Pc = Fcv
Where Pc = cutting power, N-m/s or W (ft-lb/min); Fc = cutting
force, N (lb); and v = cutting speed, m/s (ft/min)
• In U.S. customary units, power is traditionally expressed as
horse power

Where HPc = cutting horse power, hp.

 Gross power

• The gross power required to operate the machine tool is

greater than the power delivered to the cutting process
because of mechanical losses in the motor and drive train in
the machine. These losses can be accounted for by the
mechanical efficiency of the machine tool:

Where Pg = gross power of the machine tool motor, W; HPg = gross

horse power; and E = mechanical efficiency of the machine tool.
Typical values of E for machine tools are around 90%.
 Unit power
• It is often useful to convert power into power per unit volume
rate of metal cut. This is called the unit power, Pu (or unit
horse power, Hpu ), defined:
 Cont.

• Unit power also known as the specific energy U is:

The units for specific energy are typically N-m/mm3(in-lb/in3).

However, the last expression in Eq. (21.21) suggests that the units
might be reduced to N/mm2 (lb/in2)
 Cutting temperatures

 Cutting temperatures are important because high

temperatures
(1) reduce tool life,
(2) produce hot chips that pose safety hazards to the
machine operator, and
(3) can cause in accuracies in work part dimensions due to
thermal expansion of the work material.
 Cont.

• The following equation can be used to predict the increase in

temperature at the tool–chip interface during machining:
 Tool life

 The high forces and temperatures during machining create a

very harsh environment for the tool.
– If cutting force becomes too high, the tool fractures.
– If cutting temperature becomes too high, the tool material
softens and fails.
– If neither of these conditions causes the tool to fail,
continual wear of the cutting edge ultimately leads to
failure.
 There are three possible modes by which a cutting tool can fail
in machining:
1. Fracture failure. This mode of failure occurs when the cutting
force at the tool point becomes excessive, causing it to fail
suddenly by brittle fracture.
2. Temperature failure. This failure occurs when the cutting
temperature is too high for the tool material, causing the material
at the tool point to soften, which leads to plastic deformation and
loss of the sharp edge.
3.Gradual wear. Gradual wearing of the cutting edge causes loss of
tool shape, reduction in cutting efficiency, an acceleration of
wearing as the tool becomes heavily worn, and finally tool failure.
 Tool wear
 two main types of tool wear
1. Crater wear:
– consists of a cavity in the rake face of the tool that forms
and grows from the action of the chip sliding against the
surface.
– High stresses and temperatures characterize the tool–chip
contact interface, contributing to the wearing action.
– The crater can be measured either by its depth or its area.
2. Flank wear:
– occurs on the flank, or relief face, of the tool.
– It results from rubbing between the newly generated work
surface and the flank face adjacent to the cutting edge.
– Flank wear is measured by the width of the wear band
 The mechanisms that cause wear at the tool-chip and tool-work
interfaces in machining:
 Abrasion:
– a mechanical wearing action caused by hard particles in the
work material gouging and removing small portions of the tool.
– The abrasive action occurs in both flank wear and crater wear;
it is a significant cause of flank wear.
 Adhesion:
– When two metals are forced into contact under high pressure
and temperature, adhesion or welding occur between them.
– These conditions are present between the chip and the rake
face of the tool. As the chip flows across the tool, small
particles of the tool are broken away from the surface,
resulting in gradual wear of the surface.
 Diffusion:
– occurs at the tool–chip boundary, causing the tool surface
to become depleted of the atoms responsible for its
hardness. As this process continues, the tool surface
becomes more susceptible to abrasion and adhesion.
– Diffusion is believed to be a principal mechanism of crater
wear.
 Chemical reactions;
– The high temperatures and clean surfaces at the tool–chip
interface in machining at high speeds can result in chemical
reactions, in particular, oxidation, on the rake face of the
tool.
– The oxidized layer, being softer than the parent tool
material, is sheared away, exposing new material to sustain
the reaction process.
 Plastic deformation:
– The cutting forces acting on the cutting edge at high
temperature cause the edge to deform plastically, making
it more vulnerable to abrasion of the tool surface.
– Plastic deformation contributes mainly to flank wear.

Note: Most of these tool-wear mechanisms are accelerated at

higher cutting speeds and temperatures. Diffusion and chemical
reaction are especially sensitive to elevated temperature.
 The general relationship of tool wear versus
cutting time
• break-in period, in which the sharp cutting edge wears rapidly
at the beginning of its use.
• steady-state wear, wear that occurs at a fairly uniform rate.
• failure region, cutting temperatures are higher, and the
general efficiency of the machining process is reduced.
Tool wear as a function of cutting time
 Factors Affecting the Life of a Cutting Tool
• Type of material being cut
• Microstructure of material
• Hardness of material
• Type of surface on metal (smooth or scaly)
• Material of cutting tool
• Profile of cutting tool
• Type of machining operation being performed
• Speed, feed, and depth of cut
 Economic Advantages to Using Cutting Fluids
• Reduction of tool costs
– Reduce tool wear, tools last longer
• Increased speed of production
– Reduce heat and friction so higher cutting speeds
• Reduction of labor costs
– Tools last longer and require less regrinding, less
downtime, reducing cost per part
• Reduction of power costs
– Friction reduced so less power required by machining
 Characteristics of a Good Cutting Fluid
1. Good cooling capacity
2. Good lubricating qualities
3. Resistance to rancidity
4. Relatively low viscosity
5. Stability (long life)
6. Rust resistance
7. Nontoxic
8. Transparent
9. Nonflammable