Talk:String vibration/Young's modulus

From Wikiversity
Jump to navigation Jump to search

Guy's Draft[edit source]

See Draft:String vibration/Young's modulus

Outline[edit source]

Linear Young's modulus image.

2D image stretch rotation / goodshear badshear

I am also stepping on these two resources:

  1. 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 is strain and vector is stress. Poisson's ratio is , 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. 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.
  • 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).

pages that might prove these relations[edit source]

How many fourth order isotropic tensors are there? (three or only two???[edit source]

Footnotes[edit source]