Theory of Metal Cutting and Tool Geometry Notes

Fundamental Concepts of Metal Cutting

The Wedge Form: Nearly all cutting tools utilized in metal cutting operations (including turning, drilling, shaping, slotting, and planning) are fundamentally based on the shape of a wedge. A wedge is defined as an object having inclined planes in the shape of a triangular prism.
Wedge Orientation: The cutting edge must be oriented at specific angles relative to the work surface, determined by the operation's nature:
- Parting: The wedge is set at a right angle to the work surface to ensure the driving or cutting force acts in the direction of parting.
- Chipping: The wedge is set at an inclined angle to the work surface to facilitate chip separation.
Machining Definition: Machining is essentially a chip-forming metal cutting process. Excess material is removed in the form of chips to ensure the workpiece meets the desired shape, size, and dimensional accuracy. All metal cutting operations involve a special wedge-shaped element known as a cutting tool.

Anatomy of a Single Point Cutting Tool

Tool Point: The specific portion of the tool that takes part in the cutting process.
Single Point Cutting Tool: A tool containing one tool point (e.g., turning tools, shaping tools).
Multi-Point Cutting Tool: A tool containing more than one tool point (e.g., drill bits, milling cutters, saws).
Tool Components:
- Rake Face (Face): The top face or surface of the tool point.
- Flank: The surface of the tool located below and adjacent to the cutting edge.
- Cutting Edge: The line/edge formed where the face and flank meet. It is categorized as:
  - Principal/Major Cutting Edge: Performs the major portion of cutting; located nearer to the cutting zone.
  - Auxiliary/Minor Cutting Edge: Performs a minor portion of cutting; located further from the cutting zone.
- Nose: The point where the face and flanks meet. It is often rounded with a specific radius of curvature.
- Shank: The main body of the tool, which is secured by the tool holder.
- Base: The bearing surface of the tool when held in a tool holder.
- Heel: The point where the base and the flank meet.
Tool Handedness:
- Right Hand Tool: Moves from the right-hand side (RHS) to the left-hand side (LHS).
- Left Hand Tool: Moves from the left-hand side (LHS) to the right-hand side (RHS).

Tool Signature and Nomenclature Systems

Definition: Tool signature (also called tool designation or nomenclature) is a fixed sequence used to describe the geometry of the cutting part of a tool, consisting of six angles and the nose radius.
Major Systems:
- ORS: Orthogonal Rake System.
- ASA/ANSI: American Standards Association or American National Standards Institute.
- DIN: German System.
- NRS/ISO: Normal Rake System (International Standards Organisation).
- MRS/British: Maximum Normal Rake System.
Standard Abbreviation Glossaries:
- ISO: International Standards Organisation.
- SI: System International.
- IS: Indian Standard.
- BIS: Bureau of Indian Standard.

Orthogonal Rake System (ORS) Geometry

Tool Signature Sequence: ̀λ - ̀γ - ̀α - ̀α_a - ̀ϕ_a - ̀ϕ - r\,mm
- $λ$ : Inclination angle (positive, negative, or zero).
- $γ$ : Orthogonal rake angle (positive, negative, or zero).
- $α$ : Principal clearance (relief) angle.
- $α_a$ : Auxiliary clearance (relief) angle.
- $ϕ_a$ : Auxiliary plan approach angle (End cutting edge angle).
- $ϕ$ : Plan approach angle (Principal cutting edge angle).
- $r$ : Nose radius in millimeters.
Note: All angles are in degrees, and nose radius is in millimeters.
Geometric Planes in ORS:
- Tool Reference Plane / Principal Plane ( $\Pi_R$ ): A horizontal plane lying perpendicular to the cutting velocity vector.
- Cutting Plane ( $\Pi_C$ ): Tangential to the principal cutting edge and perpendicular to $\Pi_R$ .
- Orthogonal Plane ( $\Pi_O$ ): Perpendicular to $\Pi_R$ and $\Pi_C$ ; contains the cutting velocity vector.
- Auxiliary Cutting Plane ( $\Pi_C'$ ): Tangential to the auxiliary cutting edge and perpendicular to $\Pi_R$ .
- Auxiliary Orthogonal Plane ( $\Pi_O'$ ): Perpendicular to $\Pi_R$ and $\Pi_C'$ .
ORS Angle Definitions:
- Inclination angle ( $\lambda$ ): Angle of inclination of the face from $\Pi_R$ , measured in $\Pi_C$ .
- Orthogonal rake angle ( $\gamma$ ): Angle of inclination of the face from $\Pi_R$ , measured in $\Pi_O$ .
- Principal clearance angle ( $\alpha$ ): Angle of inclination of the principal flank from $\Pi_C$ , measured in $\Pi_O$ .
- Auxiliary clearance angle ( $\alpha_a$ ): Angle of inclination of the auxiliary flank from $\Pi_C'$ , measured in $\Pi_O'$
- Auxiliary plan approach angle ( $\phi_a$ ): Angle of inclination of the auxiliary cutting edge from the direction of conventional longitudinal feed, measured in $\Pi_R$ .
- Plan approach angle ( $\phi$ ): Angle of inclination of the principal cutting edge from the direction of conventional longitudinal feed, measured in $\Pi_R$ .

ASA (American Standards Association) System Geometry

Tool Signature Sequence: $\gamma_y - γ_x - α_y - α_x - ϕ_e - ϕ_s - r\,mm$
- $γ_y$ : Back rake angle (top rake angle).
- $γ_x$ : Side rake angle.
- $α_y$ : Front clearance angle (end relief angle).
- $α_x$ : Side clearance angle (side relief angle).
- $ϕ_e$ : End cutting edge angle ( $\phi_a$ in ORS).
- $ϕ_s$ : Side cutting edge angle.
- $r$ : Nose radius in millimeters.
Geometric Planes in ASA:
- Tool Reference Plane / Principal Plane ( $\Pi_R$ ): Horizontal plane perpendicular to the cutting velocity vector.
- Machine Longitudinal Plane ( $\Pi_X$ ): Perpendicular to the horizontal plane; contains the direction of conventional longitudinal feed ( $X$ ).
- Machine Transverse Plane ( $\Pi_Y$ ): Perpendicular to both $\Pi_R$ and $\Pi_X$ ; contains the direction of conventional cross feed ( $Y$ ).
ASA Angle Definitions:
- Back rake ( $\gamma_y$ ): Inclination of the face from $\Pi_R$ , measured in $\Pi_Y$ .
- Side rake ( $\gamma_x$ ): Inclination of the face from $\Pi_R$ , measured in $\Pi_X$ .
- Front clearance ( $\alpha_y$ ): Inclination of the principal flank from $\Pi_X$ , measured in $\Pi_Y$ .
- Side clearance ( $\alpha_x$ ): Inclination of the principal flank from $\Pi_Y$ , measured in $\Pi_X$ .

The Mechanism of Chip Formation

Deformation Process: Chip formation is governed by plastic deformation followed by shearing. As the tool contacts metal, pressure causes compression near the tool tip, leading to severe plastic deformation in the Primary Shear Zone.
Shear and Flow: The deformed metal fails via shear and flows over the rake face. Adhesion between the rake face and chip causes sticking, leading to a Secondary Shear Zone deformation.
Chip Curl: The curvature the chip takes after lifting away from the tool face.
Shear Zone Properties: The primary shear zone is extremely narrow (approximately $0.025\,mm$ ). At high cutting speeds, this width decreases to $1-10\,μm$ . Under normal speeds, it is treated as a Shear Plane where maximum shear force acts.
Piispanen’s Card Analogy: This model visualizes metal as thin lamellae that move over the tool face successively, representing chip formation as a process of successive slip through shear.
Built-Up Edge (BUE):
- Formation: At high speeds, temperature and adhesion cause plastically deformed material to stick to the cutting edge, forming a lump.
- Evolution: It grows until the pressure from adjacent flowing material causes it to break.
- Effects: While BUE protects the tool rake face from heat and friction wear (increasing tool life), it results in poor surface finish because it interferes with the finished surface.
- Prevention: Can be avoided using suitable cutting fluids.

Heat Generation in Metal Cutting

Heat Sources:
1. Primary Heat Source: Located in the primary shear zone.
2. Secondary Heat Source: Located in the secondary shear zone, caused by the sliding motion of the chip on the rake face.
3. Tertiary Heat Source: Located in the tertiary shear zone, caused by the rubbing action of the tool flank surface against the job.
General Heat Distribution:
- Chip: Carries away $80\%$ of total heat.
- Tool: Retains $15\%$ ( $10\%$ for carbide tools).
- Workpiece: Retains $5\%$ ( $10\%$ for carbide tools).

Classification of Chips

Continuous Chips without BUE: Formed with ductile work material, small uncut chip thickness, high cutting speeds, large rake angles, and suitable cutting fluid.
Continuous Chips with BUE: Formed with strong adhesion between chip/tool face, low rake angles, and large uncut thickness.
Discontinuous (Segmented) Chips:
- Formed with brittle work material (not necessarily hard), large uncut thickness/depth of cut, low cutting speed, and small rake angles.
- Can also occur in ductile materials with low speeds and no lubricant due to excessive friction causing intermittent rupture instead of continuous shearing.
- Brittle materials fail by maximum normal stress, rather than maximum shear stress.

Cutting Parameters and Forces

Kinematic Parameters:
- Cutting Speed ( $V_C$ ): Velocity of the tool through the workpiece, expressed in $m/min$ .
- Feed ( $f$ ): Tool advancement per revolution, stroke, or unit time in the longitudinal ( $X$ ) direction. Expressed in $mm/min$ or $mm/rev$ .
- Depth of Cut ( $d$ or $t$ ): The normal distance between unmachined and machined surfaces.
Resultant Cutting Force ( $R$ or $F_{xyz}$ ): Resolved into three components:
- Tangent Force ( $F_z$ ): Main cutting force in the direction of cutting velocity ( $70-80\%$ of total force).
- Thrust Force ( $F_{xy}$ ): Acting in the horizontal plane, further resolved into:
  1. Feed Force ( $F_x$ ): Axial thrust acting in the feed direction.
  2. Radial Force ( $F_y$ ): Radial thrust perpendicular to $F_x$ .
- Calculation: $R = \sqrt{F_x^2 + F_y^2 + F_z^2}$ .

Orthogonal vs. Oblique Cutting

Orthogonal Cutting Process:
- Principal cutting edge is perpendicular to the cutting velocity vector.
- Chip flow angle ( $\psi$ ) is zero; chip flow is normal to the principal cutting edge. Stabler’s rule: $\psi = C\lambda$ ( $C ≈ 0.9$ to $0.98$ ; thus $\psi ≈ λ$ ).
- Resultant horizontal force $P_{xy}$ is perpendicular to the principal cutting edge (requires $\lambda = 0$ ).
- First Kind: 0 < ϕ < 90^{\circ}, $\lambda = 0$ . $F_x = F_{xy} \sin(ϕ)$ and $F_y = F_{xy} \cos(ϕ)$ .
- Second Kind (2-D System): Obtained by choosing $\lambda$ and $\phi$ so that either $F_x$ or $F_y$ is zero.
Non-Free/Restricted Cutting: A process where both principal and auxiliary cutting edges are active.
Free Orthogonal Cutting: Occurs when the cutting edge length exceeds the width of cut.
Oblique Cutting: Common workshop cutting; involves a 3-D force system where $\lambda \neq 0$ and chip flow angle $\psi$ has a specific value.

Mathematical Observations and Formulas

Chip Thickness Ratio ( $r$ ): Ratio of uncut thickness ( $a_1$ ) to cut chip thickness ( $a_2$ ). $r = \frac{a_1}{a_2} = \frac{l_2}{l_1} = \frac{\sin(β)}{\cos(β - γ)}$
- Note: For continuous chips without BUE, a_2 > a_1, hence r < 1.
Chip Reduction Coefficient ( $k$ ): k = \frac{1}{r} = \frac{a_2}{a_1} > 1
Relationship for Shear Angle ( $\beta$ ): $\tan(β) = \frac{r \cos(γ)}{1 - r \sin(γ)}$
Geometrical Relationships:
- Depth of cut: $d = t = b \sin(ϕ)$
- Uncut thickness: $a_1 = f \sin(ϕ)$
- Area: $f \times d = a_1 \times b$
- Shear plane area ( $A_s$ ): $A_s = \frac{A_1}{\sin(β)}$
Shear Strain ( $ε$ ): $\text{Shear strain} = \cot(β) + \tan(β - γ)$
Velocity Relationships ( $V_C$ , $V_f$ , $V_s$ ):
- $V_C$ = Cutting velocity; $V_f$ = Chip flow velocity; $V_s$ = Shear velocity.
- $V_f = \frac{V_C \sin(β)}{\cos(β - γ)} = r \times V_C$
Volume Conservation: $V_C \times a_1 \times b_1 = V_f \times a_2 \times b_2$
ORS to ASA Transformation Matrices: $\begin{pmatrix} \tan(γ_x) \\ \tan(γ_y) \end{pmatrix} = \begin{pmatrix} \sin(ϕ) & \cos(ϕ) \\ \cos(ϕ) & -\sin(ϕ) \end{pmatrix} \begin{pmatrix} \tan(γ) \\ \tan(λ) \end{pmatrix}$ $\begin{pmatrix} \cot(α_x) \\ \cot(α_y) \end{pmatrix} = \begin{pmatrix} \sin(ϕ) & \cos(ϕ) \\ \cos(ϕ) & -\sin(ϕ) \end{pmatrix} \begin{pmatrix} \cot(α) \\ \tan(λ) \end{pmatrix}$