応力波伝播解析
Theory and Physics
What is Stress Wave Propagation?
Professor, what is a "stress wave"?
When an impact is applied to a structure, stress (deformation) propagates through the material as a wave. This wave is the stress wave. It propagates at the speed of sound $c = \sqrt{E/\rho}$.
Types of Elastic Waves
| Wave Type | Velocity | Characteristics |
|---|---|---|
| Longitudinal Wave (P-wave) | $c_L = \sqrt{(\lambda+2\mu)/\rho}$ | Propagates in compression/tension direction |
| Transverse Wave (S-wave) | $c_S = \sqrt{\mu/\rho}$ | Propagates in shear direction. $c_S < c_L$ |
| Rayleigh Wave | $\approx 0.9 c_S$ | Propagates on the surface. Earthquake S-wave |
| Lamb Wave | Dispersive | Propagates in thin plates. Depends on plate thickness and frequency |
The longitudinal wave speed in steel is $c_L \approx 5900$ m/s, right?
It reaches 1 meter away in $0.17$ ms. In impact analysis, the wave propagation time determines the structural response time.
Stress Wave Analysis with FEM
Mesh requirements for accurately tracking stress waves:
$n_{ppw}$ is the number of elements per wavelength. A guideline is 20 for linear elements, 10 for quadratic elements.
20 elements per wavelength! That requires a very fine mesh to track high-frequency waves.
That's why stress wave propagation is a strong suit of explicit FEM. Since $\Delta t$ automatically becomes small to match wave propagation, wave sampling is naturally ensured.
Application Examples
Summary
Key Points:
- Stress waves propagate at the speed of sound — $c = \sqrt{E/\rho}$
- P-wave (longitudinal) > S-wave (transverse) > Rayleigh wave (surface) — In order of speed
- Mesh requirements are strict — 10~20 elements per wavelength
- Explicit FEM is optimal — $\Delta t$ automatically adapts to wave tracking
- Hopkinson bar, ultrasound, seismic waves — Main applications
Did you know there are 3 types of elastic waves?
Elastic waves propagating in solids are of three types: longitudinal waves (P-waves: compression/expansion, sound speed 5000m/s@steel), transverse waves (S-waves: shear, approx. 3000m/s@steel), and surface waves (Rayleigh waves: propagate near the surface, approx. 2800m/s@steel). The basic principle of seismology is to calculate the epicenter distance using the velocity difference between P-waves and S-waves. The sound wave from the 1883 Krakatoa eruption was calculated to have circled the Earth 3.5 times based on atmospheric propagation waves. Industrially, ultrasonic flaw detection tests use P-waves and S-waves to detect internal defects in materials.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward during sudden braking? 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, which is the assumption that "acceleration can be ignored because the force is applied slowly". It absolutely cannot be omitted in 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 is Hooke's law $F=kx$, and it's the essence of the stiffness term. So here's 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 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 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 typical mistake here: getting the load direction wrong. Intending "tension" but it becomes "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 the vibration energy is converted into heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is important.
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 (tensor definition required separately for anisotropic materials)
- Quasi-static assumption (for static analysis): Ignores inertial/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 and creep requires constitutive law extension
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 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: approx. 210 GPa, Aluminum: approx. 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 to N in mm system, N in m system |
Numerical Methods and Implementation
LS-DYNA
```
*CONTROL_TIMESTEP
0.0, 0.6 $ Safety factor 0.6 (for wave propagation. Smaller than the usual 0.9)
*CONTROL_TERMINATION
0.001 $ 1 ms (approximately the wave round-trip time)
```
Why lower the safety factor to 0.6?
For stress wave propagation, the safety factor for the CFL condition is reduced to minimize numerical dispersion (the phenomenon where wave speed depends on mesh size). With 0.9, numerical dispersion can become significant.
Numerical Dispersion Problem
When propagating waves with FEM, shorter wavelengths become slower (numerical dispersion). Physically, all frequencies should propagate at the same speed, but FEM discretization causes speed to depend on wavelength.
Countermeasures:
- Make mesh sufficiently fine (20+ elements per wavelength)
- Use quadratic elements (half the number of elements for same accuracy as linear elements)
- Spectral Element Method (Higher-order elements)
Spectral Element Method
Spectral Element Method (Spectral Element Method) is an element method using high-order GLL (Gauss-Lobatto-Legendre) points. It has significantly smaller numerical dispersion than standard FEM. It is the standard for seismic wave propagation simulation (e.g., SPECFEM3D).
Summary
Spectral Element Method (Spectral Element Method) is an element method using high-order GLL (Gauss-Lobatto-Legendre) points. It has significantly smaller numerical dispersion than standard FEM. It is the standard for seismic wave propagation simulation (e.g., SPECFEM3D).
High-density mesh is required for wave analysis with the finite element method
In FEM analysis of elastic waves, a minimum of 8~10 elements per wavelength is required to ensure accuracy (rule of thumb). For example, to accurately capture 100kHz ultrasound in steel (wavelength 50mm), elements of 5mm or less are needed. Applying this to a 1m test specimen results in millions to tens of millions of elements, causing computational costs to explode. Therefore, in practice, a graded mesh that refines the area of interest and coarsens distant areas is combined with PML (Perfectly Matched Layer) boundary conditions that absorb reflections at the boundaries.
Linear Elements (1st-order elements)
Linear interpolation between nodes. Computational cost is low, but stress accuracy is low. Beware of shear locking (mitigated with 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 important.
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. There are h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix every iteration. Has quadratic convergence within the convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix with initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (generally $\epsilon = 10^{-3}$〜$10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Instead of applying the full load at once, it is applied in small increments. The Arc-length method (Riks method) can track 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"—the initial answer is rough, but accuracy improves 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 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
Simulation of Hopkinson Bar Tests
SHPB (Split Hopkinson Pressure Bar) tests obtain high strain rate ($10^2 \sim 10^4$ /s) properties of materials. FEM simulates stress wave propagation through the incident bar → specimen → transmission bar, aiding in the interpretation of test data.
Simulation of Ultrasonic NDT
FEM simulation of wave propagation in ultrasonic flaw detection (UT). Estimates crack size and location from the pattern of reflected waves from cracks. Combination of FEM and inverse problem analysis.
Practical Checklist
What are "absorbing boundary conditions"?
Artificial boundary conditions that prevent waves from reflecting back at the model boundaries. Lysmer-Kuhlemeyer (viscous boundary) and PML (Perfectly Matched Layer) are representative. Without them, infinite space cannot be represented.
FEM utilized for ultrasonic inspection of Shinkansen rail weld joints
JR East has utilized FEM wave propagation analysis to optimize ultrasonic flaw detection systems for rail weld joints. They analyzed mode conversion and scattering patterns of 0.5~5MHz ultrasound propagating through rail cross-sections using Abaqus/Explicit, optimizing probe placement and angle. This significantly improved the detection rate of defects near the weld line, which were difficult to detect with conventional manual inspection, in the 2010s.
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