Talk:Physics/A/String vibration/Young's modulus

Guy's Draft

Linear Young's modulus image.

2D image stretch rotation / goodshear badshear

I am also stepping on these two resources:

https://www.ecourses.ou.edu/cgi-bin/ebook.cgi?topic=me&chap_sec=01.4&page=theory cc-by-nc-nd Simple exposition of simplfications due to symmetries that are not defined. Scalar E is young's modulus and scalar G is the shear modulus. Vector $\epsilon$ is strain and vector $\sigma$ is stress. Poisson's ratio is $\nu$ , which is a function of E and G. No mention of the symmetries required for this simplification. Introduces tau and gamma for off diagonal elements of stress/strain.

https://serc.carleton.edu/NAGTWorkshops/mineralogy/mineral_physics/tensors.html Explains the symmetrical nature of strain due to removal of simple rotations. Uses Einstein notation. Stress is symmetric to prevent rotational forces. Strain is defined so that rigid rotations are subtracted out. Due to this symmetry, stress and strain each have 6 independent terms. The matrix connecting them has 36 independent terms (instead of the 81 expected from a tensor relating two 9 term items).

w:Poisson's ratio The Poisson's ratio of a stable, isotropic, linear elastic material must be between −1.0 and +0.5 because of the requirement for Young's modulus, the shear modulus and bulk modulus to have positive values.

https://polymerdatabase.com/polymer%20physics/Moduli.html For all isotropic polymeric materials where Hook's law is valid, simple relationships exist between the elastic constants such as Young's modulus E, shear modulus G, bulk modulus B, and Poisson's ratio ν) As long as two of them are known, all others can be predicted. See also P. H. Mott and C. M. Roland, Physical Review B 80, 132104 (2009).

http://silver.neep.wisc.edu/~lakes/PoissonIntro.html seems to prove using algebra EXCELLENT WEBSITE with internal links. Apparently no proof. But they do define the " elementary isotropic form for Hooke's law." (as what I will probably introduce)
https://aip.scitation.org/doi/10.1063/1.1713863 citation paper unavailable
http://web.mit.edu/16.20/homepage/3_Constitutive/Constitutive_files/module_3_with_solutions.pdf lots of tensors. Might offer proof?
https://www.hkdivedi.com/2017/01/derive-relation-between-youngs-modulus.html Claims to derive, but I think it is only the young's modulus versus bulk modulus. Not shear modulus.
Maybe w:Hooke's_law#Isotropic_materials derives the relation beteen the bulk, young's and shear moduli hidden in a {{cot}}/{{cob}} template.

https://farside.ph.utexas.edu/teaching/336L/Fluid/node252.html Most general fourth order isotropic tensor seems to have three parameters...not the two I expected.???

w:Hooke's_law#Isotropic_materials argues that one more contraint reduces this to two free parameters. The seemed to have added symmetric to the list of conditions.