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

Guy's Draft
See Draft:String vibration/Young's modulus

Outline
Linear Young's modulus image.

2D image stretch rotation / goodshear badshear

links

 * Hooke's law main wikipedia resource

I am also stepping on these two resources:


 * 1) 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.


 * 1) 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).


 * 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).


 * https://www.mydatabook.org/solid-mechanics/convert-elastic-modulus-constants-shear-youngs-bulk/

pages that might prove these relations

 * 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 Hooke%27s_law derives the relation beteen the bulk, young's and shear moduli hidden in a / template.

How many fourth order isotropic tensors are there? (three or only two???

 * 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.???


 * Hooke%27s_law argues that one more contraint reduces this to two free parameters. The seemed to have added symmetric to the list of conditions.


 * https://users.soe.ucsc.edu/~avg/Papers/tensors-vg08-long.pdf pdf on 4th order tensors that was really complicated. 3d color visualization was too much for me.
 * https://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-chapter3.pdf Looked interesting, but no mention of the symmetric requirement.
 * Two stackexcanges https://math.stackexchange.com/questions/3589647/general-form-of-an-isotropic-fourth-rank-tensor gives three parameters, but also combines two of them into symmetric and antisymmetric parts. See also https://mathematica.stackexchange.com/questions/63937/symmetrizedarray-of-stiffness-compliance-tensor/120099#120099
 * http://mmc.rmee.upc.edu/documents/Tensor_Analysis/tensors.pdf again three parameters
 * https://www.weizmann.ac.il/chembiophys/bouchbinder/sites/chemphys.bouchbinder/files/uploads/Courses/2019/TA2-IndexGymnastics.pdf They seem to state that there are 3 independent tensors...but then they get to the Navier Stokes equation with what seems like only two???
 * https://www.researchgate.net/publication/225783114_A_Concise_Proof_of_the_Representation_Theorem_for_Fourth-Order_Isotropic_Tensors I downloaded this as isotropic.pdf