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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">τ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">ZJ</mml:mi> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">τ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">ZJ</mml:mi> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">τ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">ZJ</mml:mi> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , by means of special tensor products and their symmetrisations. In some past works, the major symmetry of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">τ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">ZJ</mml:mi> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> has been missed because not all necessary symmetrisations were applied. The analogous tangent stiffness tensor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">ZJ</mml:mi> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , relating the Cauchy stress tensor σ to the Zaremba–Jaumann corotational derivative is also obtained, with both methods used for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">τ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">ZJ</mml:mi> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . 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 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) | 0.001 | 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".