Faculty of Engineering and Applied Science
ENGR 5243G: Mechanics and Dynamics of Machine Tools
Instructor: Dr. Sayyed Ali Hosseini, P.Eng.
Assignment # 1
First Name: Ravi
Last Name: Malla Thakuri
Student ID: 100801348
Section: MEM
Date Assigned: Monday, February 7th, 2022
Date Due:
Friday, February 18th, 2022, 6:00 PM
Word file to online drop box
Late submissions will not be accepted.
ENGR 5243G: Mechanics and Dynamics of Machine Tools
Assignment #1
Question #1
In the filed of metal cutting, there have multiple attempts to predict the shear angle theoretically. There have
two fundamental approaches to predict the shear angle. The first one is based on maximum shear stress principle
that was proposed by Krystof. This approach predicts the shear angle as �c = n — (�a — �r). The method
proposed by Krystof was later approved by Lee and
4
Shaffer using slip-line filed model instead of maximum shear stress principle. Another approach in this context
was proposed by Merchant based on minimum energy principle. It expresses the
shear angle as �c = n — þa–αr. Write a 4-5 pages report and review the background of each method,
4
2
derive its equation showing how they yield the corresponding equation, discuss about their accuracy and
application. Also, discuss about how slip-line filed model resulted in the same answer as predicted by the
maximum shear stress principle.
Question #2
In order to predict the machining forces, Oxley introduced the parallel-sided shear zone model in 1963. This
model is also known as thick shear zone model. His model is mainly based on single shear-plane model, but it
also take into consideration the effects of temperature, strain-rate, and strain hardening on the material property.
Oxley’s machining theory is usually accompanied by Johnson-Cook constitutive material model to represent
the material flow stress or flow behavior under cutting conditions. Write a 4-5 pages report and review the
background of these models and how they can be used to predict the cutting forces during machining.
Question 1
Background review
Maximum Shear Stress Principle (Krystof 1939) Followed by Lee & Shaffer (1951)
Krystof assume that Shear occurs when the maximum shear stress is applied. The resultant force
produces an angle with the shear plane in this situation.
R=𝞍+β-α
Which in further progress suggest that the Angle between Maximum Shear Stress and Resultant tends
to be 45 degrees.
𝜙=
𝜋
− (𝛽 − 𝛼)
4
On continuation to Krystof, Lee & Shaffer prove the Krystof theory by emphasize the Theory of
Plasticity and they assume the developing of a slip-Line field for Stress Zone in which no deformation
would take place even if it is stressed to its yield point.
Assumptions:
 Machining Work piece is ideal rigid plastic material (Elastic Stress is ignored and deformation
beyond yield point consider at constant rate)
 Chip separate from parent material and deformation takes place at shear plane
 No Strain Hardening
A’
All deformations occur inside a stress field defined by rigid bodies, which passes cutting forces from
the shear plane to the chip, resulting in the triangle plastic zone ABC. There is no deformation in this
region, but the material is strained to its yield point, resulting in a maximum shear stress on the shear
plane.
The slip-lines show the two directions in which the maximum shear stress occurs. Because the highest
shear stress must occur along the shear plane, the shear plane AB has only one pair of slip-lines.
Furthermore, BC can be considered a free surface because no forces apply on the chip after it
passes through BC, and no stresses can be transmitted from BC.
Then as per rule ACB=π/4
Normal stresses will reach the boundary at angles β and CAB=β+π/4 if stresses operate uniformly at
the chip-tool contact. Maximum shear stresses are directed in the direction of normal stresses π/4;
hence ABC is π/4-β
Take Triangular Portion of A’AB
𝞍+CAB=90+α
Or, 𝞍+ β+π/4= π/2+α
Or, 𝞍= π/2-π/4+α- β
Or, 𝞍= π/4+α- β
𝜋
𝜙 = 4 − (𝛽 − 𝛼) which matched the theoretical equation using Maximum Shear Stress Principle.
Accuracy
On Graphical representation of 𝞍 and β-α, it is found that the Intercept is π/4 but the slope is -1.
Hence the accuracy level decease and θ factor is taken in to consideration with small Build-Up
edge.
Even this correlation had not been commonly accepted due to assumptions such as material
behavior as perfect plastic. This built-up edge is actually considerably larger in size, and metal work
hardening must also be addressed.
Minimum Energy Principle (M. Eugene Merchant 1941)
Assumption: Shearing occur at the angle where minimum consumption of energy take place and
Shear stress τ will be at maximum level and constant., Also independent of Shear Angle𝞍.To
determine the Shear Plane Angle(θ) theoretically, Merchant presume that deformation process is like
ideal plastic with no strain hardening.
A high cutting width-to-unreformed chip thickness ratio in order to meet the plane-strain-condition
requirement for the analysis.
b. The cutting instrument is sharp.
c. The machined chip is continuous and does not have a built-up edge.
d. The cutting velocity is maintained constant.
A
B
E
D
C
Fig. Merchant Diagram
Fs-Shear Force
Fn-Normal force to Shear force,
F-Friction Force
N- Normal Force to Friction Force,
Fc-Horizontal Cutting Force
Ft-Vertical Thrust Force
R-Resultant force
α- Rake Angle
β – Friction Angle
𝞍- Shear angle
AB-Shear Plane, b- Width of Chip
t- Thickness of Uncut Portion
Derivation:
Let take right angle triangle BDE [Cutting force and Thrust force]
By trigonometric,
Sin(β−α) = P/H=Ft/R
∴ Ft= R* Sin(β−α) ------1
Cos(β−α) = B/H=Fc/R
∴ Fc= R* Cos(β−α) ------2
Similarly,taking right angle triangle ABD [Shear force and Normal force]
Sin (𝞍 +β−α) = P/H=Fn/R
or, Fn= R* Sin (𝞍 +β−α))
Or, Fn= R Sin 𝞍 Cos (β−α) + R Cos 𝞍 Sin (β−α) ∵ Sin(A+B) = Sin A Cos B + Cos A Sin B
Or, Fn=Fc Sin 𝞍 + Ft Cos 𝞍 ∵ Referring to Equation 1 & 2 ---------3
Also,
Cos (𝞍 +β−α) = B/H=Fs/R
or, Fs= R* Cos (𝞍 +β−α) ----------4
Or, Fs= R Cos 𝞍 Cos (β−α) - R SIn 𝞍 Sin (β−α) ∵ Cos(A+B) = Cos A Cos B - Sin A Sin B
Or, Fs= Fc Cos 𝞍 – Ft Sin 𝞍 ∵ Referring to Equation 1 & 2 ---------5
Similarly,taking right angle triangle BCD [Friction force and Normal friction force]
Sin β = P/H=F/R
or, F= R* Sin β or,F= R* Sin (α +β−α)
Or, F= R Sin α Cos (β−α) + R Cos α Sin (β−α) ∵ Sin(A+B) = Sin A Cos B + Cos A Sin B
Or, F=Fc Sin α + Ft Cos α ∵ Referring to Equation 1 & 2 ---------6
Also,
Cos β= B/H=N/R
or, N= R* Cos β or, N= R* Cos (α +β−α)
Or, N= R Cos α Cos (β−α) - R SIn α Sin (β−α) ∵ Cos(A+B) = Cos A Cos B - Sin A Sin B
Or, Fs= Fc Cos α – Ft Sin α ∵ Referring to Equation 1 & 2 ---------7
As we know
Fc Sin 𝛂 + Ft Cos 𝛂
Coefficient of friction, µ= Tan β=F/N or µ= Tan β= Fc Cos 𝛂 – Ft Sin 𝛂 ∵ Referring to Equ 5 & 6---8
Dividing by Cos α in Denominator and Numerator of Equ---7
Fc (Sin 𝛂/ Cos 𝛂 )+ Ft ( Cos 𝛂/Cos 𝛂)
Fc Tan𝛂+ Ft
– Ft Tan 𝛂
µ= Tan β= Fc (Cos 𝛂/ Cos 𝛂) – Ft (Sin 𝛂/Cos 𝛂) = Fc
-------------9
Similarly,taking right angle triangle ABF [Shear force Plane and Thickness of Uncut]
From the Trigonometry ratio, Sin𝞍 =P/H=t/AB
AB=t/Sin𝞍
We have,
Shear Stress(τ)= Shear Force (Fs) / Shear plane Area (As) = Fs / (AB * b)
Or, Fs= τ*AB*b
or, Fs= τ*b* t/Sin𝞍 -----------------10
Likewise,
Normal Stress(σs) = Normal Force (Fn) / Area (As)
Fn= σs* A= σs* AB*b= σs*t*b/ Sin𝞍 -----------------11
Now Divide Equ 2 by 4
Fc
Fs
R∗ Cos(β−α)
Cos(β−α)
= R∗ Cos (ϕ +β−α) = Cos (ϕ +β−α)
Hence,
Fs Cos(β−α)
Fc= Cos (ϕ +β−α) ----------12
After putting the value of Fs from the equation 10
τ∗b∗ t Cos(β−α)
Fc= Sinϕ Cos (ϕ +β−α)
As per the assumption, Minimum Energy Principle applied
𝑑𝐹𝑐
=0
𝑑𝜙
Or,
τ∗b∗ t Cos(β−α)𝑑1
𝑑𝜙 Sinϕ Cos (ϕ +β−α)
= 0 --------13
Let Say y= Sin𝞍 Cos (𝞍+β-α)
Then,
𝑑𝑦
=Cos𝞍Cos(𝞍+β-α)
𝑑𝜙
+ (Sin𝞍* -Sin(𝞍+β-α)) = Cos𝞍Cos(𝞍+β-α) - Sin𝞍* Sin(𝞍+β-α)
=Cos(𝞍+𝞍+β-α) or, Cos(2𝞍+β-α) -------14
Now,
𝑑1
𝑑𝜙 Sinϕ Cos (ϕ +β−α)
𝑑1
𝑑𝑦
𝑑𝑦
y-1 𝑑𝜙
𝑑𝑦
−y-2 𝑑𝜙
=0
We can also arrange like this equation
𝑑𝑦
or, −1/y2 𝑑𝜙 or, - Cos(2𝞍+β-α)/ Sin 2 𝞍 Cos 2 (𝞍+β-α)
Finally, Putting this value in equation ----13
−τ ∗ b ∗ t Cos(β − α)Cos(2ϕ + β − α)
=0
Sin^2 ϕ Cos ^2 (ϕ + β − α)
π
or, Cos(2ϕ + β − α) = 0 = Cos ( 2 )
π
Or, (2ϕ + β − α) = ( 2 )
Finally,
π
β−α
(ϕ) = ( ) − (
)
4
2
Which implies that to increase the shear plane angle, Rake angle need to increase and inversely
reduce the friction angle or coefficient of friction.
Accuracy level:
In practice, the value of Shear angle reveals that the equation
π
β−α
(ϕ) = ( ) − (
)
4
2
is more on synthetic plastic cutting but poorly agreed on steel Machining with sintered carbide tools.
Later, he changed the first equation by including the machining constant "C," whose value is usually
C≤π/2 and is dependent on the work piece, which is unaffected by the cutting condition. Instead,
concentrate on grain size and microstructure.
2 Φ + β – α = C -------(Modified Equation)
Overall Application:
 Determine the power consumption-Motor selection (After getting force and velocity)
[Pc=Fc*Vc] Kw
 Structural design of Cutting Machine
 To obtain Maximum productivity
Question 2
Why This Model?
When we doing the static analysis of any structure for the stimulation, we are imposing the young
modulus, Poison ratio and density etc. But if the load is not static (Impact, Sudden load or explosion)
then inertia effect cannot be neglected and Couple parameter such as strain effect, Strain rate and
temperature is required for stimulation.
Back ground:
Following Ernst and Merchant theory and supported by Lee and Shaffer, the research continue and
radical change came when Oxley and Welsh, purpose the Sheared plane model where relation of
the strain Hardening effect, Strain rate and temperature is well established. All Shear plane model
before Oxley assume that there is no Strain Hardening, Friction coefficient between Chip and the tool
is constant. However, experimental data contradicts the theoretical value.
The impacts of yield stress fluctuating with strain, as well as strain rate and temperature, were taken
into account, and the equilibrium and flow were simplified. Oxley's work is credited as being a
pioneer in this field. It is believed that the shear zone thickness is around one tenth of the shear zone
length based on experimental results where plastic flow patterns are found. The strain rate and strain
at each point in the primary deformation zone can then be calculated; strain rates are calculated
by integrating strain rates with respect to time along the flow streamlines, and strains are calculated
by integrating strain rates with respect to time along the flow streamlines. In the secondary
deformation zone, same assumptions are applied to determine strain rates and strains. The Oxley
Shear plane model differs from those other shear plane models in that it does expand up to generate
a finite thickness. The primary shear zone is considered to be parallel-sided, while the secondary shear
zone is simplified to a rectangle of uniform thickness.
Below are some assumptions taken in to consider for the Oxley Model.
 Simple strain conditions exist.
 The tool edge is razor sharp.
 There are no velocity discontinuities in the shear zone, indicating that the work material
velocity varies along smooth streamlines to the chip velocity.
 Throughout the shear zone, the shear strain rate remains constant.
 Along shear plane, the shear stress is constant.
 At Shear plane, half of the whole shear strain should occur.
 The impact of temperature gradient is not taken into account.
 The influence of the strain rate gradient is ignored.
 The hydrostatic pressure has a linear distribution.
 The relationship between stress and strain is linear.
 The usual stress distribution at the chip-tool rake contact is uniform.
 Shear tension is constant along the chip-tool rake interface.
The shear strength in the chip material next to the tool-chip contact will be utilized to describe the
friction parameter since sticking prevails in the secondary shear zone.
To characterize material qualities as a function of strain rate and temperature, Oxley adopted the
velocity modified temperature concept.
Term as,
Which shows, It raises in response to rising temperatures and falls in response to rising strain rates.
For a given material, the variables v and e0 are constant. The flow stress is proportional to the strain
according to the power law
, where the strength coefficient and the strain
hardening factor are functions of velocity modified temperature.
In Oxley's theory, high order curve fitting equations for low carbon steels define the link between
velocity modified temperature and both specific flow stress and strain hardening index n.
Unfortunately, other materials do not have access to these proven linkages. To adapt Oxley's
machining theory to a larger range of materials, the Johnson-Cook constitutive material model,
which includes constants for the most typically machined materials, is used in this study to describe
the material flow stress or flow behavior under cutting circumstances.
Johnson-Cook Equation
Where,
Strain Effect
Strain Rate
Temperature Effect
A- Yield Stress of the material under reference condition
B- Strain Hardening Constant
ε- Strain
n-Strain Hardening coefficient
C-Strengthen coefficient of strain rate
έ- Strain rate
έ0 – Reference value of strain rate
T- Actual Temperature
T0- Reference or Room Temperature
Tm- Melting Temperature of Material
m-Thermal Softening Coefficient
How we calculate
Strain Effect
If we plot the curve, Engineering Stress Vs
Strain
A, B & n can be derived as shown in fig.
Strain Rate
Temperature Effect
As the temperature increase the flow stress
also decrease. Value of m can be calculated
through the Minimum to Maximum range.
Experimental and Simulation results by using Johnson -Cook equation
Korkmaz, M. E. (2019). Determination and Verification of Johnson–Cook Parameters for 430 Ferritic Steels via Different
Gage Lengths. Transactions of the Indian Institute of Metals, 72(10), 2663–2672. https://doi.org/10.1007/s12666019-01734-9