Mechanical Metallurgy Syllabus

April 02, 2025

Mechanical Metallurgy

Complete Guide for GATE Metallurgy (MT) - Section 6 (TestUrSelf)

Table of Contents

• 6.1 Stress and Strain Analysis
  • Stress and Strain Tensors
  • Mohr's Circle
  • Elasticity and Stiffness
  • Yield Criteria
• 6.2 Plastic Deformation
  • Slip and Twinning
• 6.3 Dislocation Theory
  • Dislocation Types
  • Dislocation Dynamics
• 6.4 Strengthening Mechanisms
• 6.5 Fracture Behavior
  • Griffith Theory
  • Fracture Mechanics
• 6.6 Fatigue
• 6.7 High Temperature Deformation
• TestUrSelf's Course

6.1 Stress and Strain Analysis

Stress and Strain Tensors

Stress Tensor (σ_ij)

σ_xx

τ_xy

τ_xz

τ_yx

σ_yy

τ_yz

τ_zx

τ_zy

σ_zz

Strain Tensor (ε_ij)

ε_xx

γ_xy/2

γ_xz/2

γ_yx/2

ε_yy

γ_yz/2

γ_zx/2

γ_zy/2

ε_zz

Mohr's Circle Representation

Mohr's Circle for 2D stress state

Center: C = (σ_x + σ_y)/2

Radius: R = √[((σ_x - σ_y)/2)² + τ_xy²]

Principal Stresses: σ_1,2 = C ± R

Elasticity, Stiffness and Compliance Tensors

Generalized Hooke's Law

σ_ij = C_ijklε_kl

Where C_ijkl is the stiffness tensor (4th rank, 81 components)

For Isotropic Materials

C₁₁₁₁ = C₂₂₂₂ = C₃₃₃₃ = λ + 2μ

C₁₁₂₂ = C₁₁₃₃ = C₂₂₃₃ = λ

C₁₂₁₂ = C₁₃₁₃ = C₂₃₂₃ = μ

Where λ and μ are Lamé constants:

λ = Eν/[(1+ν)(1-2ν)]

μ = G = E/[2(1+ν)]

Yield Criteria

Von Mises Criterion

(σ₁ - σ₂)² + (σ₂ - σ₃)² + (σ₃ - σ₁)² = 2σ_y²

Tresca Criterion

max(|σ₁ - σ₂|, |σ₂ - σ₃|, |σ₃ - σ₁|) = σ_y

Comparison of Von Mises and Tresca yield criteria

6.2 Plastic Deformation

Slip and Twinning

Slip Systems

Crystal Structure	Slip Plane	Slip Direction	# Slip Systems
FCC	{111}	<110>	12
BCC	{110}, {112}, {123}	<111>	48
HCP	(0001)	<11-20>	3

Schmid's Law

τ_CRSS = σcosφcosλ

Where φ is angle between stress axis and slip plane normal, λ is angle between stress axis and slip direction

Twinning

K₁ = {10-12} ⟨10-1-1⟩ for HCP metals

Twinning shear (γ) for common metals:

γ = s/2 where s is twinning shear magnitude

6.3 Dislocation Theory

Dislocation Types

Edge Dislocation

Edge dislocation with extra half-plane

b ⊥ dislocation line

Screw Dislocation

Screw dislocation with spiral ramp

b ∥ dislocation line

Mixed Dislocation

b = b_edge + b_screw

Dislocation Dynamics

Stress Fields

Edge dislocation:

σ_xx = -[Gb/(2π(1-ν))][y(3x²+y²)/(x²+y²)²]

σ_yy = [Gb/(2π(1-ν))][y(x²-y²)/(x²+y²)²]

Screw dislocation:

σ_xz = -[Gb/(2π)][y/(x²+y²)]

σ_yz = [Gb/(2π)][x/(x²+y²)]

Dislocation Energy

E = [Gb²/(4πK)]ln(R/r₀)

Where K = 1 for screw, K = 1-ν for edge dislocations

Frank-Read Source

Critical stress: τ_c = Gb/L

Where L is segment length between pinning points

6.4 Strengthening Mechanisms

Work/Strain Hardening

σ = σ₀ + Kεⁿ

Where n is strain hardening exponent (0.1-0.5 for metals)

Grain Boundary Strengthening (Hall-Petch)

σ_y = σ₀ + k_yd^-1/2

Solid Solution Strengthening

Δτ = Gb√(cε)

Where c is solute concentration, ε is size misfit parameter

Precipitation Strengthening

Δτ = Gb/L (Orowan bypassing)

Δτ = γ_APBb/r (Ordered precipitates)

6.5 Fracture Behavior

Griffith Theory

σ_f = √(2Eγ_s/πa)

Where γ_s is surface energy, a is crack length

Modified for Plasticity

σ_f = √(E(2γ_s + γ_p)/πa)

Where γ_p is plastic work per unit area

Linear Elastic Fracture Mechanics

Stress Intensity Factor

K_I = Yσ√(πa)

Where Y is geometry factor (~1 for center crack)

Fracture Toughness

K_Ic = √(EG_c)

Where G_c is critical strain energy release rate

Ductile-to-Brittle Transition

T_c = T₀ + (1/β)ln(K̇/K̇₀)

Where β is material constant, K̇ is loading rate

6.6 Fatigue

S-N Curve

σ_a = σ'_f(2N_f)^b

Where σ'_f is fatigue strength coefficient, b is exponent

Crack Growth (Paris Law)

da/dN = C(ΔK)^m

Where C and m are material constants

Endurance Limit

σ_e ≈ 0.5σ_u (for steels)

6.7 High Temperature Deformation

Creep Stages

ε = ε₀ + βt^1/3 + kt (Primary + Secondary)

Norton's Law

ε̇ = Aσⁿexp(-Q/RT)

Where n is stress exponent, Q is activation energy

Larson-Miller Parameter

P = T(C + log t_r)

Where C ≈ 20 for many metals

TestUrSelf's Course Features

Mechanical Metallurgy Mastery Course

Course Highlights

• 600+ practice problems with detailed solutions
• Interactive dislocation dynamics simulations
• 12 full-length mock tests with performance analysis
• Microstructure interpretation workshops
• Fracture surface analysis exercises

Special Modules

Module	Topics	Problems
1	Stress-Strain Analysis	120
2	Dislocation Theory	150
3	Fracture Mechanics	130
4	Fatigue & Creep	100
5	Numerical Problem Solving	100