A numerical model for the nonlinear interaction of elastic waves with cracks
Bibliographic record
Abstract
It is reasonably well accepted that cracks play a significant role in the nonlinear interactions of elastic waves, but the precise mechanism of why and how this works is less clear. Here, we simulate wave propagation to understand these mechanisms. Following existing techniques, we formulate the stress in terms of its linear and nonlinear contributions. The linear stress is the generalized Hooke’s law involving only the fourth-rank elastic stiffness tensor. The nonlinear stress comes from the product of the fourth- and sixth-rank tensors, and the spatial derivatives of the displacement vector. In a nonlinear isotropic medium, we show that the speeds of P- and S-waves generated by a time-harmonic source-function are not constant over time. In an anisotropic medium, P-wave speed is commonly estimated using effective medium theory. In the linear slip theory, we represent a crack by a displacement discontinuity embedded in an isotropic background. In a cracked medium, the estimated wave speeds show nonlinear behaviors similar to the ones estimated using the direct nonlinear approach. When the particle motion is parallel to the normal of the crack, the variation in P-wave speed is large, indicating that the crack is clapping (opening and closing).
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".