Volumetric-Distortional Decomposition of Deformation and Elasticity Tensor
Bibliographic record
Abstract
The deformation gradient admits a multiplicative decomposition into a purely volumetric component and a purely distortional component. For a hyperelastic material, based on this decomposition, the elastic strain energy potential, the stress, and the elasticity tensor can be expressed in general as a function of both the volumetric deformation and the distortional deformation. However, the volumetric—distortional decomposition of deformation has often been employed in a fully decoupled form of the elastic strain energy potential, which is expressed as the sum of a term depending solely on the volumetric deformation and a term depending solely on the distortional deformation. This work has three main objectives. First, to derive the elasticity tensor in the general (non-decoupled) case, in its material, spatial, and linear forms; this is achieved by extensive use of fourth-order tensor algebra, and in particular of the properties of the so-called spherical operator, which is largely used, but very seldom given the dignity of being assigned a symbol and a name, in the literature. Second, to show that a fully decoupled potential gives rise to an elasticity tensor which may be inconsistent with its linearized counterpart, as some components of the linear elasticity tensor in general do not have a corresponding term in nonlinear decoupled elasticity tensor. Third, to obtain the conditions under which a purely hydrostatic stress causes a purely volumetric deformation, by means of the developed theory; the results show that this condition is satisfied if and only if the elastic potential is fully decoupled. While the whole approach is completely independent of the material symmetry, the cases of isotropy and transverse isotropy are shown as an example.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
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".