Major symmetry of the induced tangent stiffness tensor for the Zaremba–Jaumann rate and Kirchhoff stress in hyperelasticity: Two different approaches
Bibliographic record
Abstract
We recall in this note that the induced tangent stiffness tensor H τ ZJ ( τ ) appearing in a hypoelastic formulation based on the Zaremba–Jaumann corotational derivative and the rate constitutive equation for the Kirchhoff stress tensor τ is minor and major symmetric if the Kirchhoff stress τ is derived from an elastic potential W ( F ) . This result is vaguely known in the literature. Here, we expose two different notational approaches which highlight the full symmetry of the tangent stiffness tensor H τ ZJ ( τ ) . The first approach is based on the direct use of the definition of each symmetry (minor and major), i.e., via contractions of the tensor with the deformation rate tensor D . The second approach aims at finding an absolute expression of the tensor H τ ZJ ( τ ) , by means of special tensor products and their symmetrisations. In some past works, the major symmetry of H τ ZJ ( τ ) has been missed because not all necessary symmetrisations were applied. The analogous tangent stiffness tensor H ZJ ( σ ) , relating the Cauchy stress tensor σ to the Zaremba–Jaumann corotational derivative is also obtained, with both methods used for H τ ZJ ( τ ) . The approach is exemplified for the isotropic Hencky energy. Corresponding stability checks of software packages are shortly discussed.
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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 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.011 | 0.003 |
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".