Strain Life Method (Low Cycle Fatigue)
Theory and Physics
What is the Strain-Life Method?
Professor, how is the strain-life method different from the S-N curve method?
The S-N curve method is stress-based for high-cycle fatigue ($N > 10^4$). The strain-life method is strain-based for low-cycle fatigue ($N < 10^4$). It is essential for problems with large plastic strain.
Coffin-Manson Equation
The first term is elastic strain (Basquin's equation), the second term is plastic strain (Coffin-Manson equation).
$\sigma_f'$: Fatigue strength coefficient, $b$: Fatigue strength exponent, $\varepsilon_f'$: Fatigue ductility coefficient, $c$: Fatigue ductility exponent.
It includes both elastic and plastic strain. In low-cycle (large strain) fatigue, the plastic term dominates.
The S-N method assumes stress is below yield. The strain-life method can directly handle plastic strain exceeding yield.
Summary
The Independent Discovery of Coffin-Manson
The Coffin-Manson law, which forms the basis of the strain-life method, was independently discovered in 1954 as part of nuclear-related research. Coffin studied thermal fatigue in turbine blades at Pratt & Whitney, while Manson studied steam turbines at NACK, a predecessor of NASA. That both arrived at the same equation Δεp/2=εf'(2Nf)^c is considered a curious coincidence in the history of science.
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 away" 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 is "left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly so acceleration can be ignored". 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", 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" is incorrect. 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 contents" (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 common pitfall here: getting the load direction wrong. Intending "tension" but it becomes "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 by 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 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 inhomogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the 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, considering only the 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 and creep, constitutive law extensions are needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system 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 Strain-Life
1. Elastoplastic FEM Analysis — Track hysteresis loops under repeated loading.
2. Obtain stabilized strain range $\Delta\varepsilon$
3. Calculate life $N_f$ using Coffin-Manson equation
Fatigue software (nCode, fe-safe) automatically calculates life from FEM stress/strain results using Coffin-Manson.
Summary
Separation of Elastic and Plastic Strain Ranges
In the strain-life method, the total strain range Δε is separated into the Coffin-Manson equation (plastic component) and Basquin's law (elastic component) to evaluate life. The practical transition life 2Nt (the point where both components are equal) is typically around 2000 cycles for A7075 aluminum and about 4000 cycles for SS400 steel. In low-cycle fatigue (Nf<10^4), the plastic component becomes dominant.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but lower 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 every iteration. Shows quadratic convergence within convergence radius but has high computational cost.
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 $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies total load not all at once but in small increments. The arc-length method (Riks method) can track 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: it's more efficient to open it at an estimated location and adjust forward/backward (iterative) than to search 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 a "flexible curve"—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
Strain-Life in Practice
Used for thermal fatigue in engine components, pressure cycles in pressure vessels, and low-cycle fatigue in earthquakes.
Practical Checklist
Low-Cycle Fatigue Assessment for Nuclear Power Plant Piping
Piping in nuclear power plants experiences large strain ranges during startup, shutdown, and earthquakes, making low-cycle fatigue a major design concern. ASME Code Sec.III NB-3200 uses material design fatigue curves (strain-life based) and stipulates that the allowable cumulative damage factor must be kept below 1.0. For Δε=0.5%, the allowable cycle count is about 100,000 cycles for 304SS.
Analogy: Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare 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 mesh quality is poor, the results will be a mess no matter how good the solver is.
Pitfalls Beginners Often Fall Into
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 can 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 really fully fixed?" "Is this load really uniformly distributed?"—Correctly modeling real-world constraint conditions is often the most critical step in the entire analysis.
Software Comparison
Tools
Differences in ε-N Method Material Constant Databases
The MATLAB Fatigue Toolbox compliant with SAE J1099 standard and nodeDB (formerly MatDB) contain literature values for the fatigue ductility coefficient εf of the same steel S45C that differ by about 8%. ANSYS nCode uses both ASME material DB and its own test DB, and its material selection UI mandates source disclosure, a point valued in reliability design.
The Three Most Important Questions for Selection
- "What are you solving?": Does it support the physical models/element types needed for the strain-life method (low-cycle fatigue)? For example, for fluids, the presence of LES support; for structures, the ability to handle contact/large deformation makes a difference.
- "Who will use it?": For beginner teams, tools with rich GUIs 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 will you expand?": Selection considering future expansion of analysis scale (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technologies
Advanced
Thermo-Mechanical Fatigue (TMF) and Strain-Life
Related Topics
なった
詳しく
報告