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Cheatsheet: Fatigue & Failure Theories
1. Static Failure Theories
1.1 Maximum Normal Stress Theory (Rankine)
| Parameter | Description |
|---|---|
| Criterion | Failure occurs when maximum principal stress equals yield strength: σ₁ ≥ Sy or σ₃ ≤ -Sy |
| Material Type | Brittle materials |
| Safety Factor | n = Sy/σmax |
1.2 Maximum Shear Stress Theory (Tresca)
| Parameter | Description |
|---|---|
| Criterion | Failure when τmax = Sy/2; for principal stresses: (σ₁ - σ₃)/2 ≥ Sy/2 |
| Material Type | Ductile materials |
| Safety Factor | n = Sy/(σ₁ - σ₃) |
| Biaxial Stress | σ₁ - σ₂ = Sy when σ₁σ₂ <> |
1.3 Distortion Energy Theory (von Mises)
| Parameter | Description |
|---|---|
| Criterion | Failure when distortion energy reaches critical value |
| Formula (3D) | σ' = √[(σ₁ - σ₂)² + (σ₂ - σ₃)² + (σ₃ - σ₁)²]/√2 ≥ Sy |
| Formula (2D) | σ' = √(σ₁² - σ₁σ₂ + σ₂²) ≥ Sy |
| Formula (σ, τ) | σ' = √(σ² + 3τ²) ≥ Sy |
| Material Type | Ductile materials; most accurate theory |
| Safety Factor | n = Sy/σ' |
1.4 Maximum Normal Strain Theory (St. Venant)
| Parameter | Description |
|---|---|
| Criterion | Failure when ε₁ ≥ Sy/E or ε₃ ≤ -Sy/E |
| Formula | ε₁ = (σ₁ - ν(σ₂ + σ₃))/E |
| Application | Limited use; less accurate than other theories |
1.5 Coulomb-Mohr Theory
| Parameter | Description |
|---|---|
| Criterion | σ₁/Sut - σ₃/Suc ≥ 1 for brittle materials |
| Material Type | Brittle materials with different tensile/compressive strengths |
| Variables | Sut = ultimate tensile strength; Suc = ultimate compressive strength |
1.6 Modified Mohr Theory
- Combines maximum normal stress and Coulomb-Mohr for brittle materials
- When σ₁ > 0, σ₃ < 0:="" σ₁/sut="" -="" σ₃/suc="" ≥="">
- When both tensile: σ₁ ≥ Sut
- When both compressive: |σ₃| ≥ Suc
2. Fatigue Fundamentals
2.1 Fatigue Terminology
| Term | Definition |
|---|---|
| Fatigue | Progressive structural damage under cyclic loading below static strength |
| Stress Amplitude | σa = (σmax - σmin)/2 |
| Mean Stress | σm = (σmax + σmin)/2 |
| Stress Range | Δσ = σmax - σmin = 2σa |
| Stress Ratio | R = σmin/σmax |
| Amplitude Ratio | A = σa/σm |
| Endurance Limit | Se: stress level below which infinite life is expected (ferrous metals) |
| Fatigue Strength | Sf: stress level for N cycles to failure (non-ferrous metals) |
2.2 S-N Curve (Stress-Life)
- Plots stress amplitude (S) vs. cycles to failure (N) on log-log scale
- Three regions: low-cycle fatigue (N < 10³),="" high-cycle="" fatigue="" (10³="">< n="">< 10⁶),="" infinite="" life="" (n=""> 10⁶)
- Steel endurance limit Se' ≈ 0.5Sut for Sut ≤ 200 ksi (1400 MPa)
- Se' = 100 ksi (700 MPa) for Sut > 200 ksi
- For N = 10³ cycles: Sf = 0.9Sut
2.3 Endurance Limit Modification
| Factor | Formula/Value |
|---|---|
| Modified Endurance | Se = ka·kb·kc·kd·ke·kf·Se' |
| ka (Surface) | ka = aSutᵇ; Ground: a=1.58, b=-0.085; Machined: a=4.51, b=-0.265; Hot-rolled: a=57.7, b=-0.718; As-forged: a=272, b=-0.995 |
| kb (Size) | kb = (d/0.3)⁻⁰·¹⁰⁷ for 0.11 ≤ d ≤ 2 in; kb = 0.91d⁻⁰·¹⁵⁷ for 2 < d="" ≤="" 10=""> |
| kc (Loading) | Bending: 1.0; Axial: 0.85; Torsion: 0.59 |
| kd (Temperature) | kd = 1.0 for T < 450°c;="" varies=""> |
| ke (Reliability) | 50%: 1.000; 90%: 0.897; 95%: 0.868; 99%: 0.814; 99.9%: 0.753 |
| kf (Miscellaneous) | Accounts for residual stress, corrosion, etc.; default = 1.0 |
2.4 Stress Concentration in Fatigue
| Parameter | Description |
|---|---|
| Kt | Theoretical stress concentration factor (geometric) |
| Kf | Fatigue stress concentration factor: Kf = 1 + q(Kt - 1) |
| q | Notch sensitivity: 0 ≤ q ≤ 1; q = 0 (no sensitivity), q = 1 (full sensitivity) |
| Neuber Equation | q = 1/(1 + √(a/r)) where a = Neuber constant, r = notch radius |
3. Fatigue Life Prediction
3.1 Mean Stress Effects
3.1.1 Goodman Relation
| Parameter | Formula |
|---|---|
| Linear Goodman | σa/Se + σm/Sut = 1/n |
| Safety Factor | n = 1/(σa/Se + σm/Sut) |
| Characteristics | Conservative; straight line; most common for design |
3.1.2 Gerber Relation
| Parameter | Formula |
|---|---|
| Gerber Parabola | σa/Se + (σm/Sut)² = 1/n |
| Characteristics | Less conservative; parabolic; better fit for experimental data |
3.1.3 Soderberg Relation
| Parameter | Formula |
|---|---|
| Soderberg | σa/Se + σm/Sy = 1/n |
| Characteristics | Very conservative; uses yield strength; prevents yielding |
3.1.4 Modified Goodman (ASME Elliptic)
- σa/Se + (σm/Sy)² = 1/n² for σm > 0
- Prevents yielding while accounting for mean stress
3.2 Basquin Equation (High-Cycle Fatigue)
| Parameter | Formula |
|---|---|
| Basquin Relation | σf = σf'(2N)ᵇ |
| σf' | Fatigue strength coefficient ≈ Sut for steels |
| b | Fatigue strength exponent ≈ -1/6 for steels |
| N | Number of cycles to failure |
3.3 Strain-Life (Low-Cycle Fatigue)
3.3.1 Coffin-Manson Relation
| Parameter | Formula |
|---|---|
| Total Strain | Δε/2 = (σf'/E)(2N)ᵇ + εf'(2N)ᶜ |
| Elastic Component | Δεe/2 = (σf'/E)(2N)ᵇ |
| Plastic Component | Δεp/2 = εf'(2N)ᶜ |
| εf' | Fatigue ductility coefficient ≈ εf for steels |
| c | Fatigue ductility exponent ≈ -0.6 for steels |
3.4 Cumulative Damage
3.4.1 Miner's Rule (Palmgren-Miner)
| Parameter | Description |
|---|---|
| Formula | Σ(ni/Ni) = 1 at failure, where ni = applied cycles at stress level i, Ni = cycles to failure at stress level i |
| Safety Factor | n = 1/Σ(ni/Ni) |
| Assumption | Linear damage accumulation; order-independent |
| Limitation | Does not account for load sequence effects |
4. Fracture Mechanics
4.1 Stress Intensity Factor
| Parameter | Description |
|---|---|
| K | Stress intensity factor: K = Yσ√(πa) where Y = geometry factor, a = crack length |
| Mode I | KI: opening mode (tensile) |
| Mode II | KII: sliding mode (in-plane shear) |
| Mode III | KIII: tearing mode (out-of-plane shear) |
| KIc | Fracture toughness; critical stress intensity for crack propagation |
4.2 Crack Growth Relations
4.2.1 Paris Law
| Parameter | Formula |
|---|---|
| Paris Equation | da/dN = C(ΔK)ᵐ |
| ΔK | ΔK = Kmax - Kmin = YΔσ√(πa) |
| C, m | Material constants; m ≈ 2-4 for metals |
| Integration | Nf = ∫(da/C(ΔK)ᵐ) from ai to af |
4.3 Fracture Criteria
| Condition | Description |
|---|---|
| Plane Stress | Thin sections; KIc is stress-state dependent |
| Plane Strain | Thick sections; B ≥ 2.5(KIc/Sy)²; minimum toughness |
| Failure Criterion | K ≥ KIc or σ√(πa) ≥ KIc/Y |
4.4 Energy Release Rate
| Parameter | Formula |
|---|---|
| G | Energy release rate: G = K²/E' where E' = E (plane stress), E' = E/(1-ν²) (plane strain) |
| Critical Value | Gc = KIc²/E' |
5. Design Considerations
5.1 Safety Factor Methods
| Method | Application |
|---|---|
| Static Loading | n = Sy/σ' (von Mises) or n = Sy/(σ₁ - σ₃) (Tresca) |
| Fully Reversed | n = Se/σa where σm = 0 |
| Fluctuating Stress | n = 1/(σa/Se + σm/Sut) (Goodman) |
| Combined Loading | Calculate equivalent stresses σa,eq and σm,eq then apply mean stress relation |
5.2 Infinite Life Design
- Target: N > 10⁶ cycles for steel, N > 10⁸ for aluminum
- Keep σa < se="" after="" all="">
- Account for mean stress using Goodman or Gerber
- Apply stress concentration factor Kf
5.3 Finite Life Design
- Use Basquin equation when 10³ < n=""><>
- Calculate Sf at required cycles: Sf = a(N)ᵇ
- Determine a from: a = (fSut)²/Se where f = 0.9 for steel
- b = -(1/3)log(fSut/Se)
5.4 Variable Amplitude Loading
- Apply Miner's rule for multiple stress levels
- Calculate equivalent stress for spectrum loading
- Use rainflow counting for complex load histories
- Consider load sequence effects for critical applications
5.5 Failure Mode Selection
| Material Type | Recommended Theory |
|---|---|
| Ductile (Sy/Sut <> | von Mises (best), Tresca (conservative) |
| Brittle (elongation <> | Modified Mohr, Coulomb-Mohr |
| Cyclic Loading | Fatigue theories with mean stress correction |
| Cracked Components | Fracture mechanics (LEFM) |
6. Key Formulas Summary
6.1 Static Failure
- von Mises: σ' = √(σ² + 3τ²) for combined stress
- Principal stresses: σ1,2 = (σx + σy)/2 ± √[((σx - σy)/2)² + τxy²]
- Maximum shear: τmax = (σ₁ - σ₃)/2
6.2 Fatigue Analysis
- Modified endurance: Se = ka·kb·kc·kd·ke·Se'
- Goodman: σa/Se + σm/Sut = 1/n
- Effective stress: σa,eff = Kf·σa; σm,eff = Kf·σm (conservative) or σm,eff = σm (common)
- Finite life: Sf = a(N)ᵇ where b = -(1/3)log(0.9Sut/Se)
6.3 Fracture Mechanics
- Stress intensity: K = Yσ√(πa)
- Paris law: da/dN = C(ΔK)ᵐ
- Failure criterion: K ≥ KIc
6.4 Combined Loading
- Alternating equivalent: σa,eq = √(σa² + 3τa²)
- Mean equivalent: σm,eq = √(σm² + 3τm²)
- Apply mean stress relation with equivalent stresses
7. Important Constants and Values
7.1 Material Properties (Typical)
| Material | Sut (ksi) |
|---|---|
| Low Carbon Steel | 60-80 |
| Medium Carbon Steel | 80-120 |
| High Carbon Steel | 100-150 |
| Aluminum Alloys | 30-90 |
| Titanium Alloys | 120-180 |
7.2 Standard Reliability Factors (ke)
- 50%: 1.000
- 90%: 0.897
- 95%: 0.868
- 99%: 0.814
- 99.9%: 0.753
7.3 Standard Cycle Counts
- Low-cycle fatigue: N < 10³="">
- High-cycle fatigue: 10³ < n="">< 10⁶="">
- Infinite life (steel): N > 10⁶ cycles
- Infinite life (aluminum): N > 10⁸ cycles
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