Notch Fatigue (Notch Effect)
Theory and Physics
Notch Effect
Professor, how do we evaluate fatigue at notches?
Notches (holes, fillets, grooves) significantly reduce fatigue life due to stress concentration. The relationship between the theoretical stress concentration factor $K_t$ and the fatigue notch factor $K_f$ is important.
$q$ is the notch sensitivity (depends on material and notch radius, 0~1). Higher strength materials have $q \to 1$.
Notch Stress in FEM
FEM directly calculates notch stress including $K_t$. This stress is evaluated using an S-N curve (notch stress-based).
Neuber's Rule
Neuber's rule estimates elastoplastic local strain from elastic FEM stress:
$K_\sigma$ is the stress concentration factor, $K_\varepsilon$ is the strain concentration factor. Estimates local strain without elastoplastic FEM.
Summary
Neuber's Kt-Kf Problem
The ratio (sensitivity factor q) between the theoretical stress concentration factor Kt and the fatigue notch factor Kf varies with material strength and notch dimensions. For high-strength steel (1500MPa class), q≈1.0 (Kt and Kf are almost equal), but for mild steel, q≈0.6. Neuber (1936) explained this difference is due to the stress gradient at the notch root, and this forms the basis of current ISO/ASME standards for structural strength design.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried forward" is precisely the inertia 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 so acceleration is negligible". It absolutely cannot be omitted for impact loads 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. So a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber band. 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"—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), 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 common mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression"—sounds like a joke, but it actually happens when coordinate systems rotate 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. That's because vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibration energy for a smoother ride. What if damping were zero? Buildings would keep shaking 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 specified otherwise): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertia/damping forces, considers only balance between external and internal forces.
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity or creep, constitutive law extension is needed.
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. Note unit inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note 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 Notch Fatigue
Two approaches:
1. Direct Approach — Elastoplastic FEM → local strain → Coffin-Manson
2. Neuber Approach — Elastic FEM → Neuber's rule to estimate local strain → Coffin-Manson
Fatigue software (nCode, fe-safe) supports both approaches.
Summary
Practical Estimation Formula for Notch Fatigue Limit
Peterson's formula (Kf=1+q(Kt-1)) is widely used to estimate notch fatigue limits. q is a parameter representing the material's "gradient sensitivity", increasing with higher tensile strength. For tool steel SUJ2 (Rm=2200MPa), q=0.98, while for S45C (Rm=700MPa), q=0.75, resulting in Kf differences from 1.5 to 2.1 for a notch depth of 1mm and r=0.5mm.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with Midside Nodes)
Can represent curved deformation. Stress accuracy improves significantly but degrees of freedom increase ~2-3x. 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.
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. Quadratic convergence within convergence radius, but computationally expensive.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Lower cost per iteration but linear convergence speed.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$~$10^{-6}$). 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 limit 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: opening to an estimated page and adjusting forward/backward (iterative) is more efficient than searching sequentially from the first page (direct).
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 "flexible curves"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
Notch Fatigue in Practice
Essential for fatigue evaluation of bolt holes, fillets, keyways, and weld toes.
Practical Checklist
Crack Prevention Measures for Press Dies
Fatigue failure at notch areas (corner R) in press dies directly leads to production stoppage. In practice, ensure a minimum curvature radius r≥0.5mm, verify stress concentration at the notch root via FEM, then perform life evaluation using the Kf method. Denso Corporation standardized FEM + notch fatigue analysis in die design around 2015, improving die life by 1.5x compared to conventional methods.
Analogy: Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare CAD model), do prep work (mesh generation), apply heat (solver execution), and finally plate it (post-processing visualization). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work". If mesh quality is poor, results will be garbage no matter how excellent the solver is.
Common Pitfalls for Beginners
Are you checking mesh convergence? Do you think "calculation ran = results are 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 is far from reality. Verify that results stabilize across at least three mesh density levels—neglecting this leads to the dangerous assumption that "the computer's answer 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 constraints is often the most critical step in the entire analysis.
Software Comparison
Tools
Simulation Driven Design with OptiStruct
Altair OptiStruct has a Fatigue Quick Setup feature that automatically calculates Kf for notch fatigue evaluation. Through collaboration with HBM-Prenscia, it directly links with fe-safe, enabling fatigue evaluation of all components including notches in suspension arm shapes in a single flow. BMW shortened the design verification period for a new suspension by 3 months using this flow.
Three Most Important Questions for Selection
- "What to solve?": Does the required physical model/element type for notch fatigue (notch effect) have support? For example, presence of LES support for fluids, contact/large deformation capability for structures makes a difference.
- "Who will use it?": For beginner teams, tools with rich GUI are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between automatic (GUI) and manual (script) transmission cars.
- "How far to expand?": Selection considering future analysis scale expansion (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technology
Advanced Topics in Notch Fatigue
Taylor's Theory at the Notch Tip
Taylor's cylinder (critical distance) theory evaluates fatigue based on stress at a point distance L away from the notch tip. L is material-dependent, approximately 0.1mm for high-strength steel, 0.1~1mm for cast iron. More accurate than traditional evaluation using total stress concentration, and its concept is incorporated into ASTM E739.
Troubleshooting
Notch Fatigue Troubleshooting
Discrepancy Between FEM Stress Concentration and Actual Measured Life
When there's a large difference between Kt calculated by FEM and Kf obtained from tests, surface roughness effects are suspect. For turned surfaces (Ra=1.6μm), finishing
Related Topics
なった
詳しく
報告