多軸疲労
Theory and Physics
What is Multiaxial Fatigue?
Professor, what is multiaxial fatigue?
Standard S-N methods assume uniaxial stress, but in real structures, stresses in multiple directions fluctuate simultaneously. This is multiaxial fatigue.
Multiaxial Fatigue Criteria
| Criterion | Parameter | Characteristics |
|---|---|---|
| von Mises Equivalent Stress | $\sigma_{vm}$ | Simplest. For proportional loading only. |
| Critical Plane Method | Plane with maximum damage | Applicable to non-proportional loading |
| Fatemi-Socie | $\gamma_{max}(1+k\sigma_{n,max}/\sigma_y)$ | Shear strain based |
| Smith-Watson-Topper | $\sigma_{max} \cdot \Delta\varepsilon/2$ | Normal stress based |
| Dang Van | Mesoscopic stress | High-cycle multiaxial fatigue |
Is the Critical Plane Method the most common?
For non-proportional loading (multiaxial stresses with phase shifts), the Critical Plane Method is essential. The von Mises equivalent can be non-conservative.
Summary
The Problem von Mises Couldn't Solve
Directly applying uniaxial fatigue theory to a multiaxial stress field can lead to dangerously optimistic predictions. Evaluating with von Mises equivalent stress can predict a fatigue limit up to 40% higher for biaxial stress with a 180° phase difference. The Brown-Miller criterion (1973) solved this by evaluating the combination of maximum shear strain and maximum principal strain, achieving accuracy within 15% of experimental data.
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 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 is negligible". It 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" is incorrect. Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (e.g., pressure, contact forces). 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 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 ending up with "compression"—it 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 inhomogeneities.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the 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 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 or 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, ensure loads and elastic modulus are also unified 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 deformations). |
| 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 Multiaxial Fatigue
1. Calculate time histories of all stress components via FEM ($\sigma_x(t), \sigma_y(t), \tau_{xy}(t)$)
2. Search for the critical plane using fatigue software (scan all directions)
3. Perform rainflow counting and damage calculation on the critical plane
nCode, fe-safe support the Critical Plane Method for multiaxial fatigue.
Summary
Calculation Procedure for the Critical Plane Method
In the Critical Plane Method for multiaxial fatigue, the crack initiation parameter is calculated for all directions, and the plane giving the maximum value (the critical plane) is identified. In implementation, it's common to calculate for 18 directions in 10° increments from 0° to 180°. Implementing this as an FEM post-processing step takes a few seconds for a solid model with 10,000 elements, but its versatility is high, applicable to both proportional and non-proportional loading.
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 the tangent stiffness matrix every iteration. Achieves quadratic convergence within the convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the 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 the full 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: 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 judgment should be based on total cost-effectiveness.
Practical Guide
Multiaxial Fatigue in Practice
Components where multiaxial stress dominates, such as crankshafts, vehicle axles, nozzle connections on pressure vessels, etc.
Practical Checklist
Multiaxial Evaluation of a Turbofan Engine Disk
Aircraft engine compressor disks are a typical field of multiaxial fatigue where centrifugal force (tension) and thermal stress (compression) act simultaneously. For the CFM56 engine disk, life predicted by multiaxial fatigue criteria was 30% shorter than uniaxial evaluation, forming the basis for early overhaul scheduling.
Analogy of the 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—in cooking, which step is most prone to failure? Actually, it's the "prep work". If mesh quality is poor, no matter how good the solver is, the results will be a mess.
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
Capability of FEMFAT Multiaxial Module
FEMFAT by Austrian company ECS implements the Critical Plane Method, Integral Method, and Equivalent Stress Method in its Multiaxial module, and is used by AB company for certification analysis of engine mount brackets. Direct import from FEM is possible, supporting ABAQUS/ANSYS/Nastran. Full-direction critical plane search for a 10,000-element model completes within 3 minutes.
The Three Most Important Questions for Selection
- "What are you solving?": Does it support the physical models and element types needed for multiaxial fatigue? For example, for fluids, the presence of LES support; for structures, the ability to handle contact and large deformations 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 Topics in Multiaxial Fatigue
Relationship Between Non-Proportional Hardening and Fatigue Life
Under multiaxial stress with phase differences, materials undergo additional hardening (non-proportional hardening), leading to 20-40% shorter life than uniaxial fatigue predictions. For stainless steel SUS304, hardening is maximum at a 90° phase difference, and fatigue life can be less than half even at the same equivalent stress amplitude. The Itoh-Katakoke model developed in the 2000s can quantitatively handle this phenomenon.
Troubleshooting
Multiaxial Fatigue Trou
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