疲労亀裂伝播（Paris則）
Theory and Physics
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.).
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, meaning "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being pulled" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly enough that acceleration can be ignored". It cannot be omitted in impact loading or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That's Hooke's Law $F=kx$, the essence of the stiffness term. Now a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong". Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body force $f_b$ (gravity, etc.) and surface force $f_s$ (pressure, contact force, etc.). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A typical pitfall here: getting the load direction wrong. Intending "tension" but modeling "compression"—sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades away. That's because vibrational energy is converted to heat through air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibrational energy to improve ride comfort. What if damping were zero? Buildings would keep swaying forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, stress-strain relationship is linear
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions)
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considers only equilibrium between external and internal forces
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity or creep requires constitutive law extensions
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify load/elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation) |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
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.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Mid-side Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended: when stress evaluation is critical.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix each iteration. Provides quadratic convergence within convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies total load in small increments rather than all at once. The arc-length method (Riks method) can trace beyond extremum points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to estimate where to open it and adjust forward/backward (iterative method) than to search sequentially from the first page (direct method).
Relationship Between Mesh Order and Accuracy
1st order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judgment should be based on total cost-effectiveness.
Practical Guide
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.
Analogy for Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy the ingredients (prepare the CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work". If the mesh quality is poor, the results will be a mess no matter how good the solver is.
Common Pitfalls for Beginners
Are you checking mesh convergence? Do you think "the calculation ran = the result is correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer will be far from reality. Confirm that results stabilize across at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct".
Thinking About Boundary Conditions
Setting boundary conditions is like "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface truly fully fixed?" "Is this load truly uniformly distributed?"—Correctly modeling real-world constraint conditions is often the most critical step in the entire analysis.
Software 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.
The Three Most Important Questions for Selection
- "What are you solving?": Fatigue crack propagation (Paris Law
Related Topics
この記事の評価ご回答ありがとうございます！参考に
なったもっと
詳しく誤りを
報告