Stress Wave Propagation Analysis
Stress Wave Propagation Analysis: Theoretical Foundations
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.
Computational Methods for Stress Wave Propagation Analysis
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 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
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.
Stress Wave Propagation Analysis in Practice
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.