Mechanical conservation and balance laws
Bibliographic record
Abstract
In the previous chapter, we derived kinematic fields to describe the possible deformed configurations of a continuous medium. These fields on their own cannot predict the configuration a body will adopt as a result of a given applied loading. To do so requires a generalization of the laws of mechanics (originally developed for collections of particles) to a continuous medium, together with an application of the laws of thermodynamics. The result is a set of universal conservation and balance laws that apply to all bodies: conservation of mass; balance of linear and angular momentum; thermal equilibrium (zeroth law of thermodynamics); conservation of energy (first law of thermodynamics); second law of thermodynamics. These equations introduce four new important quantities to continuum mechanics. The concept of stress makes its appearance in the derivation of the momentum balance equations. Temperature, internal energy and entropy star in the zeroth, first and second laws, respectively. In this chapter we focus on the mechanical conservation laws (mass and momentum) leaving the thermodynamic laws to the next chapter. Conservation of mass A basic principle of classical mechanics is that mass is a fixed quantity that cannot be formed or destroyed, but only deformed by applied loads. Thus, the total amount of mass in a closed system is conserved. For a system of particles this is a trivial statement that requires no further clarification.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.012 | 0.005 |
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".