Mechanics of Materials
Study of material behavior under various loading conditions, including stress, strain, and failure analysis.
Key Equations
Stress & Strain
σ = F/A
ε = ΔL/L₀
σ = E × ε
Bending
σ = My/I
τ = VQ/(It)
Applications
- • Structural Design
- • Machine Components
- • Pressure Vessels
- • Bridge Engineering
- • Aerospace Structures
- • Material Selection
Key Concepts
- • Normal and Shear Stress
- • Elastic Deformation
- • Failure Theories
- • Fatigue Analysis
- • Buckling Stability
- • Material Properties
Stress and Strain Fundamentals
Stress is the internal resistance of a material to deformation, defined as force per unit area. Strain is the measure of deformation representing the displacement of particles in the body relative to their reference position.
Normal Stress
σ = F/A
Where:
σ = normal stress (Pa or N/m²)
F = applied force normal to the surface (N)
A = cross-sectional area (m²)
Shear Stress
τ = V/A
Where:
τ = shear stress (Pa)
V = shear force (N)
A = cross-sectional area (m²)
Normal Strain
ε = ΔL/L₀
Where:
ε = normal strain (dimensionless)
ΔL = change in length (m)
L₀ = original length (m)
Shear Strain
γ = tan(θ) ≈ θ (for small angles)
Where:
γ = shear strain (radians)
θ = angle of deformation (radians)
Hooke's Law and Material Properties
Hooke's Law states that the strain in a solid is proportional to the applied stress within the elastic limit of that material. This fundamental relationship defines the elastic behavior of materials.
Linear Elastic Relationship
σ = E × ε
Where:
σ = normal stress (Pa)
E = Young's modulus or modulus of elasticity (Pa)
ε = normal strain (dimensionless)
Shear Modulus
τ = G × γ
Where:
τ = shear stress (Pa)
G = shear modulus (Pa)
γ = shear strain (radians)
Poisson's Ratio
ν = -ε_lateral/ε_axial
Where:
ν = Poisson's ratio (dimensionless)
ε_lateral = lateral strain
ε_axial = axial strain
Relationship Between Elastic Constants
G = E/[2(1 + ν)]
This relationship connects the three fundamental elastic constants: E, G, and ν.
Typical Material Properties
| Material | E (GPa) | G (GPa) | ν |
|---|---|---|---|
| Steel | 200 | 80 | 0.30 |
| Aluminum | 70 | 26 | 0.33 |
| Concrete | 30 | 12 | 0.20 |
| Copper | 110 | 40 | 0.35 |
Axial Loading and Deformation
Axial loading occurs when forces are applied along the longitudinal axis of a member, causing either tension or compression. Understanding axial deformation is crucial for structural design.
Axial Deformation
δ = PL/(AE)
Where:
δ = axial deformation (m)
P = applied axial force (N)
L = length of member (m)
A = cross-sectional area (m²)
E = Young's modulus (Pa)
Variable Cross-Section
δ = ∫[P(x)/(A(x)E(x))]dx
For members with varying cross-section, force, or material properties along the length.
Thermal Deformation
δ_T = αLΔT
Where:
δ_T = thermal deformation (m)
α = coefficient of thermal expansion (1/°C)
L = length (m)
ΔT = temperature change (°C)
Combined Loading
δ_total = δ_mechanical + δ_thermal
Total deformation includes both mechanical and thermal effects.
Torsion of Circular Shafts
Torsion occurs when a shaft is subjected to twisting moments or torques. Circular shafts are commonly used in mechanical systems to transmit power and rotational motion.
Torsion Formula
τ/r = T/J = Gθ/L
Where:
τ = shear stress at radius r (Pa)
r = radial distance from center (m)
T = applied torque (N·m)
J = polar moment of inertia (m⁴)
G = shear modulus (Pa)
θ = angle of twist (radians)
L = length of shaft (m)
Maximum Shear Stress
τ_max = Tr/J = T/(J/r) = T/Z_p
Where:
τ_max = maximum shear stress (Pa)
r = outer radius (m)
Z_p = polar section modulus (m³)
Angle of Twist
θ = TL/(GJ)
For a shaft of constant cross-section and material properties.
Polar Moment of Inertia
Solid circular: J = πd⁴/32
Hollow circular: J = π(d₀⁴ - d_i⁴)/32
Where d₀ = outer diameter, d_i = inner diameter
Bending of Beams
Bending occurs when beams are subjected to transverse loads, causing internal moments and stresses. Understanding beam behavior is fundamental to structural design.
Flexure Formula
σ/y = M/I = E/ρ
Where:
σ = bending stress (Pa)
y = distance from neutral axis (m)
M = bending moment (N·m)
I = moment of inertia (m⁴)
E = Young's modulus (Pa)
ρ = radius of curvature (m)
Maximum Bending Stress
σ_max = Mc/I = M/S
Where:
c = distance to extreme fiber (m)
S = section modulus = I/c (m³)
Shear Stress in Beams
τ = VQ/(It)
Where:
τ = shear stress (Pa)
V = shear force (N)
Q = first moment of area (m³)
I = moment of inertia (m⁴)
t = thickness at the point (m)
Beam Deflection
EI(d²y/dx²) = M(x)
EI(d⁴y/dx⁴) = w(x)
Where:
y = deflection (m)
w(x) = distributed load (N/m)
Failure Theories
Failure theories predict when materials will fail under complex stress states. These theories are essential for safe design under multiaxial loading conditions.
Maximum Normal Stress Theory (Rankine)
σ₁ ≤ σ_yield or σ₃ ≥ -σ_yield
Failure occurs when the maximum principal stress reaches the yield strength. Suitable for brittle materials.
Maximum Shear Stress Theory (Tresca)
τ_max = (σ₁ - σ₃)/2 ≤ σ_yield/2
Failure occurs when the maximum shear stress reaches half the yield strength. Conservative for ductile materials.
Von Mises Yield Criterion
σᵥₘ = √[(σ₁-σ₂)² + (σ₂-σ₃)² + (σ₃-σ₁)²]/√2 ≤ σ_yield
Most accurate theory for ductile materials under multiaxial stress states. Based on distortion energy.
Mohr-Coulomb Theory
τ = c + σ tan(φ)
Where:
c = cohesion
φ = angle of internal friction
Used for materials with different tensile and compressive strengths.
Buckling and Stability
Buckling is a sudden failure mode that occurs in slender compression members when the applied load exceeds the critical buckling load. Understanding stability is crucial for column design.
Euler's Buckling Formula
P_cr = π²EI/(KL)²
Where:
P_cr = critical buckling load (N)
E = Young's modulus (Pa)
I = minimum moment of inertia (m⁴)
K = effective length factor
L = actual length (m)
Effective Length Factors
Pin-pin: K = 1.0
Fixed-free: K = 2.0
Fixed-pin: K = 0.7
Fixed-fixed: K = 0.5
Slenderness Ratio
λ = KL/r
Where:
λ = slenderness ratio
r = radius of gyration = √(I/A)
Critical Stress
σ_cr = P_cr/A = π²E/(KL/r)²
Euler's formula is valid only when σ_cr ≤ σ_proportional_limit