The Janssen effect and the Chini ordinary differential equation
Bibliographic record
Abstract
The Janssen effect describes the increase in pressure with depth in granular materials. However, the original Janssen formulation treats the density as constant. In this work, we examine parametric density models that depend on both pressure and moisture content and allow for temperature dependence to be easily included. We focus on physically-motivated empirical models that are built on power-law terms. These power-law terms correspond to thermodynamic equations of state, such as the ideal gas law, and polytropic processes. When coupled with the Janssen ordinary differential equation (ODE), these power-law models have the form of a Chini ODE. We study the properties of the Chini ODE for both constant and variable coefficients. We review Chini’s method for transforming these equations into separable forms and the conditions under which this approach fails. We derive a variety of novel expressions, including a critical value for the vertical-to-horizontal stress ratio which varies with depth.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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 source (direct Gemma or distilled Codex), 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".