残留応力解析
Theory and Physics
Residual Stress
Professor, what is residual stress?
Stress that exists within a structure in the absence of external forces. Generated by welding, heat treatment, forming, and surface treatment. Significantly affects fatigue life, buckling load, and stress corrosion cracking.
Mechanism of Residual Stress Generation
Professor, what is residual stress?
Stress that exists within a structure in the absence of external forces. Generated by welding, heat treatment, forming, and surface treatment. Significantly affects fatigue life, buckling load, and stress corrosion cracking.
Summary
What is Residual Stress: Generation Mechanism
Residual stress is a self-equilibrating stress that remains within a material after external forces are removed. The three main causes are non-uniformity of plastic deformation, temperature gradients, and phase transformations. In welding, tensile residual stress is generated by rapid cooling and contraction around the molten pool. Measurements confirm that residual stress near the weld line in butt welding of mild steel can reach near the yield point (355 MPa). On the other hand, shot peening (particle impact processing) intentionally introduces compressive residual stress into the surface layer, extending fatigue life by 2 to 5 times. The wing span of a Boeing 737 is about 30m, and compressive residual stress is crucial for fatigue management of the lower wing surface.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Haven't you experienced your body being thrown forward when braking suddenly? 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 absolutely 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 is Hooke's law $F=kx$, the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more under the same force? 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" — 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 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 — intentionally absorbing vibration 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 inhomogeneity
- 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 inertial and damping forces, considers only equilibrium 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 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) | Unified as N in mm system, N in m system |
Numerical Methods and Implementation
Residual Stress FEM
Welding residual stress:
1. Welding Heat Source Model (Goldak, etc.) — Calculate temperature field with moving heat source
2. Thermo-Elasto-Plastic Analysis — Temperature → thermal expansion → plastic deformation → cooling → residual stress
Inherent Strain Method: Input pre-determined inherent strain into the weld zone. Obtain residual stress with a single-step elastic analysis. Computation time reduced to 1/100 ~ 1/1000.
Summary
Comparison of X-ray Diffraction and Neutron Diffraction Methods
Comparison of the two main residual stress measurement methods: X-ray diffraction (sin²ψ method) is limited to a depth of a few μm from the surface but the equipment is compact and suitable for on-site measurement, accuracy ±20 MPa. Neutron diffraction can non-destructively measure up to several cm inside a structure, and measurements using the neutron beam at JRR-3 (Tokai Village, Ibaraki, Japan Atomic Energy Agency) achieved accuracy of ±5 MPa. In 2020, the dedicated residual stress beamline ENGINEERING at J-PARC (Ibaraki) enabled 3D residual stress mapping of welded joints (resolution 0.5mm³).
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 significantly improved, 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 mode (zero-energy mode). 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. Achieves 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 $\epsilon = 10^{-3}$〜$10^{-6}$). 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 accuracy improves 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 method) is more efficient than searching 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 "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
Practical Checklist
Reduction by Post-Weld Heat Treatment (PWHT)
Post-Weld Heat Treatment (PWHT) is the most reliable method to relieve residual stress. JIS B 8285 (Post-Weld Heat Treatment Standard) specifies holding at 600~650°C for 1~4 hours for carbon steel. For BWR (Boiling Water Reactor) piping fittings (SUS304), residual tensile stress causes stress corrosion cracking (IGSCC), so since the 1980s, alternatives to PWHT omission such as hydrostatic testing (Mechanical Stress Improvement Process, MSIP) and Low Plasticity Burnishing (LPB) have been adopted. In the Westinghouse AP1000, PWHT is incorporated as a design standard for all welded locations.
Analogy of 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 (visualization in post-processing). Here's an important question — which step in cooking is most prone to failure? Actually, it's "prep work". If mesh quality is poor, the results will be a mess no matter how excellent 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 is 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 actually the most critical step in the entire analysis.
Software Comparison
Tools
Comparison of Dedicated Solvers for Welding Residual Stress Analysis
For welding residual stress analysis, there are three major players: SYSWELD (ESI Group), Simufact Welding (MSC), and ABAQUS with Goldak model. SYSWELD comes with a measured database of arc efficiency η=0.85~0.95 and is an automotive industry standard, while Simufact Welding is strong in shipbuilding (for MHI) due to integration with process simulation. Verification accuracy against ASME Sec.IX welding qualification tests shows SYSWELD has the highest performance with error within 7%.
The Three Most Important Questions for Selection
- "What to solve?": Does it support the physical models and element types needed for residual stress analysis? For example, presence of LES support for fluids, contact/large deformation capability for structures make 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 transmission cars (GUI) and manual transmission cars (script).
- "How far to 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 Technology
Advanced
Non-Destructive Measurement of Residual Stress by Neutron Diffraction
Neutron diffraction boasts the highest accuracy for non-destructive residual stress measurement. The BL19 beamline at J-PARC (Tokai Village, Ibaraki) can measure residual stress inside steel with ±5 MPa accuracy. It is reported that non-destructive evaluation of welded axle parts of JR Central N700S Shinkansen up to 30mm depth achieved 95% agreement with SYSWELD analysis values.
Troubleshooting
Troubles
Related Topics
なった
詳しく
報告