Fatigue Crack Propagation (Paris Law)
Fatigue Crack Propagation (Paris Law): Theoretical Foundations
Paris Law
Professor, how do we predict fatigue crack propagation?
Paris Law (1963) describes the fatigue crack growth rate in terms of the stress intensity factor range:
$$ \frac{da}{dN} = C(\Delta K)^m $$
Professor, how do we predict fatigue crack propagation?
Paris Law (1963) describes the fatigue crack growth rate in terms of the stress intensity factor range:
$da/dN$: Crack growth per cycle, $\Delta K = K_{max} - K_{min}$: Stress intensity factor range, $C, m$: Paris constants.
The larger $\Delta K$ is, the faster the crack grows. It's a straight line on a log-log graph.
Typical values for steel: $C \approx 10^{-12}$ (m/cycle, MPa$\sqrt{m}$ units), $m \approx 3$. $m$ indicates the material's sensitivity to crack growth.
Three Stages of Fatigue Crack Growth
1. Region I — $\Delta K < \Delta K_{th}$ (below threshold). Crack does not propagate
2. Region II — Region where Paris Law holds. Stable propagation
3. Region III — $K_{max} \to K_{IC}$. Transition to rapid fracture
Remaining Life Calculation
Integrate from initial crack $a_0$ to critical crack $a_c$ ($K = K_{IC}$):
Summary
Paris Law and NASA Funding
The fundamental law for fatigue crack growth rate "da/dN = C(ΔK)^m" was published by Paris and Gomez in 1961. Initially, it was rejected multiple times by major academic journals, but after NASA recognized its applicability to the structural integrity of commercial aviation and provided funding, it became widely adopted. Today, it forms the basis for crack evaluation standards worldwide (ASTM E647, BS 7910, etc.).
Computational Methods for Fatigue Crack Propagation (Paris Law)
FEM for Crack Propagation
1. Calculate SIF $\Delta K(a)$ via FEM for each crack length — Extend crack stepwise
2. Calculate $da/dN$ using Paris Law
3. Determine cumulative cycle count $N$
Dedicated Tools
Summary
Utilizing SIF Handbooks for ΔK Calculation
Applying Paris Law requires calculating the stress intensity factor range ΔK = Δσ√(πa)·F. The shape factor F is obtained from analytical solutions (F=1 for an infinite plate) or from handbooks (Stress Intensity Factor Handbook). In practice, the semi-elliptical surface crack (with Q-factor correction) is the most frequently used, and FEM-based SIF calculations are used to verify its accuracy.
Fatigue Crack Propagation (Paris Law) in Practice
Crack Propagation in Practice
Aircraft damage tolerance design (FAR 25.571), pressure vessel API 579 FFS assessment, crack growth evaluation for nuclear reactors.
Practical Checklist
Transition from FAA Safe-Life to Damage Tolerance
Damage tolerance design became mandatory for aircraft with the 1974 revision of US FAR 25.571. This was prompted by the 1969 F-111 wing spar defect accident. Today, all commercial aircraft are required to perform fatigue crack propagation life analysis, and the procedure for conservatively evaluating remaining life using Paris Law and setting inspection intervals is standardized.
Fatigue Crack Propagation (Paris Law): Software & Solver Comparison
Crack Propagation Tools
NASGRO Software and NASA's Legacy
NASGRO is a crack growth analysis software jointly developed by NASA, SwRI (Southwest Research Institute), and ESA. The commercial version is widely used in the US aerospace industry as an FAA-certified software. The NASGRO equation extends the Paris model to express R-ratio dependence, threshold ΔKth, and fracture toughness Kc in a single formula, and contains data for over 5000 materials. Pratt & Whitney and GE use it for engine component certification analysis.
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