Dear RG members:
Screenshot from:
Vonsovskii, S. V. and Svirskii, M. S. 1961. About the spin of phonons. Soviet Physics of the Solid State. 3:2160. In Russian.
Do you know any article/book which uses explicit expressions containing Anisotropic Elastic Langragians L( Cij )?
For instance: cubic, hexagonal, or tetragonal ones? Other symmetries are also welcome.
L can be written in the 4th index range - ijkl or Voig - ij notations, but the math expressions must contain explicitly the elastic stiffness components Cij or compliances Sij (if S-1 C ~ 1) according to the point symmetry group.
For example: in the isotropy case: 2, in the cubic case 3, in the hexagonal case 6 and so on.
A lagrangian is defined as L(Cij) = K( ρ v2 ) - U( Cij ), therefore the potential term U(Cij) does have to include an expression invariant to the point group symmetry considered.
I did an intensive search on the web, so far only two papers with two expressions (isotropic and cubic cases) both papers from the '60s.
Thank you all so much for the interest.
Highly Research specific to Solid Mechanics with Symmetry theory -
https://www.google.com/search?sxsrf=ACYBGNRi-VwgQ25e0PrQkuBJY2p-w31RDQ:1580458136131&q=Lagrangian+expression+for+elastic+anisotropic+bodies+with+symmetry+point+grs&tbm=isch&source=univ&sa=X&ved=2ahUKEwiJt6Susa3nAhU963MBHUvpD3cQsAR6BAgKEAE
Thank yo so much for the link Dear Prof. Jaydip Datta. I will---check and replay--- next week. My Best Regards.
Indeed Prof. Jaydip Datta
the work entitled: Lie point symmetries and conservation laws in microstretch and micromorphic elasticity
- November 2006
- International Journal of Engineering Science 44(20):1571-1582
- DOI:
- 10.1016/j.ijengsci.2006.08.015
- 📷Markus Lazar
- 📷Charalampos Anastassiadis
Research Interest12.0Citations📷📷📷 gives some insights, but only for the isotropic case (2 elastic constants). You see, symmetries are a very powerful tool in low symmetry systems, they are the only way to guess the math expressions due to high number of terms involved.
Dear Prof Pedro L. Contreras E. ,
Actually whatever being a Chemical people , symmetry theo is very important and
it is coupled with elastic solid state . Although you are expert in this specialised
research .
Indeed Dear Prof. Jaydip Datta. I totally agree with you. Regards.
The following class notes by Prof. K. Sun helps to answer this thread for the case of an isotropy crystal:
http://www-personal.umich.edu/~sunkai/teaching/Winter_2013/chapter4.pdf
Dear ResearchGate thread participants; P. Contreras hopes that you all are in good health.
Internet connectivity was finally powered off in Pedro's town by chavista regime, last Friday May 23rd at midnight.
Pedro apologize's, he won't be able to coach this thread for some time, however feel free to carry on with the discussion and stay tuned.
God bless & protect you all. Thanks so much.
Dear ResearchGate thread participants; P. Contreras hopes that you all are in good health.
Internet connectivity was finally powered off in Pedro's town by chavista regime, last Friday May 23rd at midnight.
Pedro apologize's, he won't be able to coach this thread for some time, however feel free to carry on with the discussion and stay tuned.
God bless & protect you all. Thanks so much.
Dear thread participants. I hope you all are in good health.
I apologise for being off-line. We are facing major issues including any kind of online communication & general supplies.
My God be with all.
Dear all, the following interesting paper is related to this thread.
Euler-Lagrange Elasticity: Differential Equations for Elasticity without Stress or Strain, by H. H. Hardy, Journal of Applied Mathematics and Physics, 1, 7, 2013.
Article Euler-Lagrange Elasticity: Differential Equations for Elasti...
Although all the work seems to be for isotropy elastic bodies it is worth to mention the following paper:
Article In memory of Ronald S. Rivlin
This paper goes further in order to propose the original idea of this thread:
Article The role of transverse anisotropic elastic waves and Ehrenfe...
The answer to this thread is the following:
- The quantization of the transverse elastic field was carried out by two Soviet physicists: Professors S. Vonsovskii and M. Svirskii in 1961.
They demonstrated in 1961 that the spin of transversal phonon waves is equal to 1.
Vonsovskii, S. V. and Svirskii, M. S. 1961. About the spin of phonons. Soviet Physics of the Solid State. 3:2160. In Russian.
- In 1962, Prof. A. T. Levine noticed their paper:
Levine, AT. 1962. A note concerning the spin of the phonon. Nuovo Cimento. 26:190-193.
In 2019, Prof. V. G. Yarzhemsky kindly found for me, the lost work in a Library in Moscow.
In 2020, I pointed out that proof was made in 1961 by the two Soviet physicists in a CJPAS paper.
Article The role of transverse anisotropic elastic waves and Ehrenfe...
I have públished a preprint with the answer to this thread. The Lagrangian for D4 point group symmetry.
Preprint The role of transverse anisotropic elastic waves and Ehrenfe...
The following research paper is related to this thread:
Article Quantized intrinsically localized modes of a 3D lattice
Best Regards.
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