Deformation-Induced Change in the Geometry of a General Material Surface and Its Relation to the Gurtin–Murdoch Model
Bibliographic record
Abstract
Abstract Small deformation theory plays an important role in analyzing the mechanical behavior of various elastic materials since it often leads to simple referential analytic results. For some specific mechanical problems however (for example, those dealing with small-scale materials/structures with significant surface energies or soft solids containing gas/liquid inclusions with high initial pressure), in order to obtain sufficiently accurate solutions, the classical boundary conditions associated with small deformation theory often require modification to incorporate the influence of deformation on the geometry of the boundary. In this note, we provide first-order approximate expressions characterizing the change in the geometry (normal vector, curvature tensor, etc.) of a general surface during deformation. In particular, using these expressions we recover without difficulty, the stress boundary condition in the original Gurtin–Murdoch surface model for an (initially) spherical interface with constant interface tension. We believe that the expressions established here will find widespread application in the mechanical analysis of problems requiring an extremely high level of accuracy in the description of the corresponding boundary conditions. In addition, higher-order approximate expressions representing the change in the geometry of a general surface during deformation could be also obtained using the same procedure.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".