Integro-Differential Equation for a Finite Crack in a Strip with Surface Effects
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Bibliographic record
Abstract
We present a mathematical investigation of a nonstandard boundary value problem for Laplace's equation in an infinite strip containing a finite crack. This unusual problem arises when surface elasticity is incorporated in the description of deformation of the crack faces. We use the Fourier transform to reduce the original boundary value problem to an integral equation that is further shown to reduce to a vector Riemann–Hilbert problem. The latter is solved analytically. The solution consists of two parts. The first is found explicitly, and the second is in a series form. The series converge exponentially, and the series coefficients are obtained by solving an infinite system. It is shown that the rate of convergence of an approximate solution to the exact one is exponential. We illustrate our solution with a particular example which, among other interesting phenomena, reveals the effect of the surface mechanics on the shape of the crack face and shows that the stress is bounded at the crack tip.
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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 it