On Spencer’s displacement function approach for problems in second-order elasticity theory
Bibliographic record
Abstract
The paper describes the displacement function approach first proposed by AJM Spencer for the formulation and solution of problems in second-order elasticity theory. The displacement function approach for the second-order problem results in a single inhomogeneous partial differential equation of the form [Formula: see text], where [Formula: see text] is Stokes’ operator and [Formula: see text] depends only on the first-order or the classical elasticity solution. The second-order isotropic stress [Formula: see text] is governed by an inhomogeneous partial differential equation of the form [Formula: see text], where [Formula: see text] is Laplace’s operator and [Formula: see text] depends only on the first-order or classical elasticity solution. The introduction of the displacement function enables the evaluation of the second-order displacement field purely through its derivatives and avoids the introduction of arbitrary rigid body terms normally associated with formulations where the strains need to be integrated. In principle, the displacement function approach can be systematically applied to examine higher-order effects, but such formulations entail considerable algebraic manipulations, which can be facilitated through the use of computer-aided symbolic mathematical operations. The paper describes the advances that have been made in the application of Spencer’s fundamental contribution and applies it to the solution of Kelvin’s concentrated force, Love’s doublet, and Boussinesq’s problems in second-order elasticity theory.
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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".