Theory of Metal Cutting

Tool Geometry for Single Point Tool

Tool Designation (ASA)

\[\alpha_b, - \alpha_s, - \gamma_e, - \gamma_s, - C_e, - C_s, - R\]

$\alpha_b$ : back rake angle

$\alpha_s$ : Side rake angle

$\gamma_e$ : End relief angle

$\gamma_s$ : Side relief angle

$C_e$ : End cutting edge angle

$C_s$ : Side cutting edge angle

R: Nose radius

Tool Designation (Orhtogonal Rake system)

\[i - \alpha - \gamma - \gamma_1 - Ce - \lambda - R\]

i : Inclination angle

$\alpha$ : Side rake

$\gamma$ : Side relief

$\gamma_1$ : End relief

Ce : End cutting edge angle

$\lambda$ : Approach angle

R : Nose radius

Orthogonal Cutting

$t_o$ : Uncut chip thickness

$t_c$ : Produced chip thickness

α : Rake angle

$\phi$ : Shear angle

V : Cutting velocity

Cutting Ratio:

It is defined as uncut chip thickness to produced chip thickness.

\[r = \frac{t_o}{t_c} = \frac{sin \phi}{cos(\phi - \alpha)}\]

\[tan(\phi) = \frac{r cos \alpha}{1 - r sin \alpha}\]

Shear strain in chip:

\[\gamma = cot(\phi) + tan(\phi - \alpha)\]

Velocity relation:

• The velocity of the tool relative to workpiece is called cutting velocity (V).
• The velocity of the chip relative to the work in shear plane is called shear velocity(Vs).
• The velocity of chip on the rake face of tool is called chip velocity(Vc). 

By Triangle Rule,

\[\boxed{\frac{V}{cos(\phi - \alpha)} = \frac{V_s}{cos \alpha} = \frac{V_c}{sin \phi}}\]

Forces in Orthogonal Cutting

Assumptions:

• 2-D cutting process and plane strain process
• $t_o << w$
• Infinite sharp cutting edge
• Continuous chip with no BUE (Built Up Edge)
• Uniform shear and normal stresses along shear plane and tool chip interface.

$F_t$ : Thrust force

$F_c$ : Cutting  force

R : Resultant force

$F_s$ : Shear force (in shear plane)

$N_s$ : Normal force on shear plane

F : Friction force at tool chip interface

N : Normal force at tool chip interface normal to rake face of tool

Average coefficient of friction (μ) is given by:

\[μ = tan( \beta) = \frac{F}{N}\]

Force Circle

You don't need to remember following equations just remember above 'Force Circle' and derive:

\[F = F_c sin \alpha +  F_t cos \alpha \]

\[N = F_c cos \alpha -  F_t sin \alpha \]

\[F_s = F_c cos \phi -  F_t sin \phi  = R cos(\phi + \beta - \alpha)\]

\[N_s= F_c sin \phi +  F_t cos \phi  = R sin (\phi + \beta - \alpha)\]

If τ is the ultimate shear stress of work material and As is shear plane area then shear force can be written as:

\[F_s = τ * A_s = \frac{τ w t_o}{sin \phi}\]

Cutting Force:

\[F_c = R cos(\beta - \alpha) = \frac{F_s cos(\beta - \alpha) }{cos(\phi + \beta - \alpha)} = \frac{τ w t_o cos(\beta - \alpha) }{sin \phi cos(\phi + \beta - \alpha)}\]

Merchant’s theory

Shear angle $\phi$ takes value such that least amount of energy is consumed or minimize work done.

\[ \frac{d F_c}{d \phi} = 0\]

\[\boxed{\phi = \frac{\pi}{4} - \frac{(\beta - \alpha)}{2}}\]

Above relationship is OK for plastics but does not hold well for metals.

Modified Merchant’s Relationship : \( 2 \phi + \beta - \alpha = C\)

C is property of work material but it tends to increase with cold work.

NOTE: Remember that the value of $\phi$ obtained from Merchant's theory is always higher that actual value than the exact value in case of metals, therefore the forces calculated are lower.

Energy Dissipation

 Total power consumption $P_c = F_c * V$

The major plastic deformation take places in shear plane, this is primary heat source (Ps). Frictional energy loss due to sliding motion between chip and tool rake face, called secondary heat source( $P_f = F * V_c$).
\[P_c  \approx  P_s + P_f\]
\[P_s = P_c - P_f = F_c*V - F*V_c\]

Mean Temperature rise of material passing through shear zone:
\[\theta_s = \frac{(1-\Gamma)P_s}{\rho c V w t_o}\]

$\Gamma$ : Fraction of shear zone heat which goes to workpiece

$\rho$: density of material

c : specific heat

Mean Temperature rise of the chip due to frictional zone:
\[\theta_f = \frac{P_f}{\rho c V w t_o}\]

The final temperature is is given $\theta = \theta_o + \theta_s + \theta_f$ where $\theta_o$ is initial temperature of workpiece.

Mechanism of Tool Wear

Loss of material from surface is called wear.

1. Adhesive Wear: Due to high pressure and temperature wear particles transfer from softer surface to harder body.
2. Abrasive Wear: If one of the surface contains very hard particles then these particles during the processes of sliding may dislodge material from the other surface by abrasine action.
3. Fatigue Wear: loss of material from a surface due to fracture under cyclic loading conditions.
4. Diffusion: Atoms in a metallic crystal move from a region of high concentration to that of low concentration.
5. Corrosive Wear: It is due to chemical reactions between the surface and environment (like water, oxygen etc.). 

Types of Tool Wear

• Crater Wear : Wear on the rake face of tool. Its main reason id diffusion along with abrasion.
• Flank Wear : Wear on flank face of tool due to work hardening. The main reason is adhesion and abrasion.

Types of tool wear

Taylor's Tool Life Equation

Tool life is mainly affected by the cutting, higher the speed smaller the life. Relationship between cutting speed and tool life is given by:

\[\boxed{V T^n = C}\]

where C is constant based on tool and work and cutting condition.

Extended Taylor's equation which includes feed (f) and depth of cut (d) is given by:

\[VT^n f^x d^y = C'\]