Key Formulas & Equations
Engineering Strain: ε = ΔL/L₀
True Stress: σt = F/Ai
True Strain: εt = ln(Li/L₀)
Shear Modulus: G = τ/γ
Bulk Modulus: K = -V(dP/dV)
Poisson's Ratio: ν = -εlateral/εaxial
Unit Cell Density: ρ = nA/(VcNA)
Packing Factor: PF = Vatoms/Vcell
FCC PF = 0.74; BCC PF = 0.68; SC PF = 0.52
Fick's Second Law: ∂C/∂t = D(∂²C/∂x²)
Diffusion Coefficient: D = D₀e(-Q/RT)
Diffusion Distance: x ≈ √(Dt)
Dislocation Density: ρd = L/V
Critical Crack Size: ac = (KIC/Yσ)²/π
Fracture Toughness: KIC = Yσ√(πa)
Core Concepts
Crystal Structures
Arrangement of atoms in solids: BCC, FCC, HCP. Crystal defects (vacancies, dislocations, grain boundaries) strongly influence mechanical properties.
Phase Diagrams
Maps showing stable phases of a material system as a function of temperature and composition. Essential for understanding heat treatment and alloy design.
Mechanical Properties
Response of materials to applied forces: strength, hardness, ductility, toughness, and fatigue resistance. Determined by microstructure and processing.
Strengthening Mechanisms
Methods to improve material strength: solid solution strengthening, grain boundary strengthening, precipitation hardening, and work hardening.
Corrosion
Deterioration of materials due to chemical or electrochemical reaction with environment. Types include uniform, pitting, galvanic, and stress corrosion cracking.
Polymer Science
Study of macromolecules and their properties. Includes polymerization mechanisms, molecular weight, crystallinity, and viscoelastic behavior.
Composite Materials
Materials made from two or more constituent materials with different properties. Fiber-reinforced composites combine high strength with low weight.
Heat Treatment
Controlled heating and cooling to alter microstructure and properties. Processes include annealing, quenching, tempering, and surface hardening.
Worked Examples
Example 1: Stress-Strain Calculation
Problem: A steel rod (E = 200 GPa, A₀ = 0.001 m²) is loaded with 20 kN. Calculate stress and strain.
Solution:
σ = F/A₀ = 20,000/0.001 = 20 MPa
ε = σ/E = 20×10⁶/200×10⁹ = 0.0001 = 0.01%
Example 2: Diffusion Distance
Problem: Carbon diffuses in iron with D = 2×10⁻¹¹ m²/s at 900°C. Calculate diffusion distance after 1 hour.
Solution:
x ≈ √(Dt) = √(2×10⁻¹¹ × 3600) = √(7.2×10⁻⁸) = 0.268 mm
Example 3: Fracture Toughness
Problem: A material has KIC = 30 MPa√m and is subjected to stress of 100 MPa. Calculate critical crack size (Y = 1).
Solution:
ac = (KIC/σ)²/π = (30/100)²/π = 0.09/π = 0.0286 m = 28.6 mm
Practice Problems
Problem 1
Calculate Young's modulus for a material with stress = 150 MPa and strain = 0.0015.
Answer: E = 150×10⁶/0.0015 = 100 GPa
Problem 2
A copper wire (ν = 0.33) experiences axial strain of 0.002. Calculate lateral strain.
Answer: εlateral = -0.33 × 0.002 = -0.00066
Problem 3
Calculate the density of FCC iron (a = 0.365 nm, atomic weight = 55.85 g/mol).
Answer: ρ = 4 × 55.85/(0.365×10⁻⁹)³ × 6.022×10²³ = 7870 kg/m³
Problem 4
Calculate shear modulus if τ = 30 MPa and γ = 0.0015.
Answer: G = 30×10⁶/0.0015 = 20 GPa
Problem 5
A crack of 2 mm exists in a material with KIC = 25 MPa√m. Calculate critical stress (Y = 1).
Answer: σ = KIC/(Y√(πa)) = 25/√(π × 0.002) = 315 MPa
Problem 6
Calculate diffusion coefficient at 800°C if D₀ = 2×10⁻⁵ m²/s and Q = 142 kJ/mol.
Answer: D = 2×10⁻⁵ × exp(-142000/(8.314 × 1073)) = 1.2×10⁻¹¹ m²/s
Problem 7
Calculate true stress if load = 10 kN and instantaneous area = 0.0008 m².
Answer: σt = 10,000/0.0008 = 12.5 MPa