Draft:String vibration/Young's modulus

• Draft:String vibration/Young's modulus started by Guy vandegrift July 2021
• Talk:Draft:String vibration/Young's modulus

Formulas

• Tensor operators explained at https://en.wikipedia.org/wiki/Tensor_operator

Isotropic case

Bulk: $K=-V(dP/dV)$

Poisson's ratio Typically $0<\nu <.5$, where $ϵ=\mathrm{\Delta }L/L,$ and:

$\nu =-\frac{{ϵ}_{\text{transverse}}}{{ϵ}_{\text{axial}}}$

w:Young's modulus#Calculation: $E=\frac{P}{(\mathrm{\Delta }L/L)}$

shear

w:Shear modulus: $G=\frac{{\tau }_{xy}}{{\gamma }_{xy}}=\frac{F/A}{\mathrm{\Delta }x/l}=\frac{Fl}{A\mathrm{\Delta }x}$

where

${\tau }_{xy}=P$ = shear stress

${\gamma }_{xy}$ = shear strain. In engineering $:=\mathrm{\Delta }x/l=\tan \theta$, elsewhere $:=\theta$

$\mathrm{\Delta }x$ is the transverse displacement

$l$ is the initial length of the area.

• Shear from Young's

$G=\frac{E}{2(1+\nu )}$

• Shear from Bulk

$G=\frac{3K(1-2\nu )}{2(1+\nu )}$

• Bulk from Young's

$K=\frac{E}{3(1-2\nu ))}$

• Bulk from Shear

$K=\frac{2G(1+\nu )}{3(1-2\nu ))}$

• Young's from Shear

$E=2G(1+\nu )$

• Young's from Bulk

$E=3K(1-2\nu )$

w: Hooke's Law

• https://en.wikipedia.org/w/index.php?title=Infinitesimal_strain_theory&oldid=1029508502

General linear

w:Hooke's_law#Hooke's_law_for_continuous_media: Stiffness tensor Template:Math is represented by a matrix of 3 × 3 × 3 × 3 = 81 real numbers Template:Math. Hooke's law then says that

${\sigma }_{ij}=\sum _{k,l}{c}_{ijkl}{\epsilon }_{kl}$ and ${\epsilon }_{ij}=\sum _{k,l}{s}_{ijkl}{\sigma }_{kl}$

From w:Infinitesimal_strain_theory#Infinitesimal_strain_tensor:

$${\epsilon }_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i})={\partial }_i{u}_j={\partial }_j{u}_i$$

From w:Cauchy stress tensor:: $\frac{d{F}_i}{dS}={\sigma }_{ij}{n}_j$

Introduction to the Rotation Matrix

<math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> $\tilde{a}$

Isotropic

$\begin{array}{cc}{\epsilon }_{11}& =\frac{1}{E}({\sigma }_{11}-\nu ({\sigma }_{22}+{\sigma }_{33}))\\ {\epsilon }_{22}& =\frac{1}{E}({\sigma }_{22}-\nu ({\sigma }_{11}+{\sigma }_{33}))\\ {\epsilon }_{33}& =\frac{1}{E}({\sigma }_{33}-\nu ({\sigma }_{11}+{\sigma }_{22}))\\ {\epsilon }_{12}& =\frac{1}{2G}{\sigma }_{12};{\epsilon }_{13}=\frac{1}{2G}{\sigma }_{13};{\epsilon }_{23}=\frac{1}{2G}{\sigma }_{23}\end{array}$

From w:Hooke's Law. Also available at Rod Lakes' website

Wikipedia formulas

See continuummechanics.org/tractionvector

The tensor consists of nine components ${\sigma }_{ij}$ that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length w:direction vector n to the traction vector T⁽ⁿ⁾ across an imaginary surface perpendicular to n:

${𝐓}^{(𝐧)}=𝐧\cdot \mathit{\sigma }\text{or}{T}_j^{(n)}={\sigma }_{ij}{n}_i.$

where, $\mathit{\sigma }=$

$\left[\begin{array}{ccc}{\sigma }_{11}& {\sigma }_{12}& {\sigma }_{13}\\ {\sigma }_{21}& {\sigma }_{22}& {\sigma }_{23}\\ {\sigma }_{31}& {\sigma }_{32}& {\sigma }_{33}\end{array}\right]\equiv \left[\begin{array}{ccc}{\sigma }_{xx}& {\sigma }_{xy}& {\sigma }_{xz}\\ {\sigma }_{yx}& {\sigma }_{yy}& {\sigma }_{yz}\\ {\sigma }_{zx}& {\sigma }_{zy}& {\sigma }_{zz}\end{array}\right]\equiv \left[\begin{array}{ccc}{\sigma }_x& {\tau }_{xy}& {\tau }_{xz}\\ {\tau }_{yx}& {\sigma }_y& {\tau }_{yz}\\ {\tau }_{zx}& {\tau }_{zy}& {\sigma }_z\end{array}\right]$

The SI units of both stress tensor and stress vector are N/m², corresponding to the stress scalar. The unit vector is dimensionless.

Plane Stress & Strain

$\overrightarrow{u}$

Remember: f = kx → Stress = σ =c εTemplate:SpacesandTemplate:Spacesx = (1/k)x → Strain = ε = sσ

From Wikipedia articles on Hooke's law, Plane stress and Plane strain:

$\underset{\_}{\underset{\_}{\mathit{\sigma }}}=\left[\begin{array}{cc}{\sigma }_{xx}& {\sigma }_{xy}\\ {\sigma }_{yx}& {\sigma }_{yy}\end{array}\right]\equiv \left[\begin{array}{cc}{\sigma }_x& \tau \\ \tau & {\sigma }_y\end{array}\right]$

where the double underline indicates a second order tensor. Following the notation used by most Wikipedia articles, the plane strain tensor is:

$\underset{\_}{\underset{\_}{\mathit{\epsilon }}}=\left[\begin{array}{cc}{\epsilon }_{xx}& {\epsilon }_{xy}\\ {\epsilon }_{yx}& {\epsilon }_{yy}\end{array}\right]\equiv \left[\begin{array}{cc}{\epsilon }_x& \gamma /2\\ \gamma /2& {\epsilon }_y\end{array}\right]$

To sort this out, see , and also:

• w:Viscous stress tensor
• w:Linear_elasticity#(An)isotropic_(in)homogeneous_media

w:Infinitesimal strain theory#Geometric_derivation

• w:Deformation_(physics) versus w:Deformation_(engineering): These pages do NOT explain why γ/2 appears in the shear strain. We use the former but not the latter in this resource. This w:special:permalink/1032796196#Normal_and_shear_strain to w:Deformation_(physics) does explain it.

Infinitesimal strain theory

Footnotes

• Oklahoma University Multimedia Engineering Mechanics is part of https://www.ecourses.ou.edu/. My favorite discussion of this Gramoll's https://www.ecourses.ou.edu/cgi-bin/ebook.cgi?topic=me&chap_sec=01.4&page=theory: