Draft:String vibration/Young's modulus


 * started by Guy vandegrift July 2021
 * Talk:

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 $$\epsilon=\Delta L/L,$$ and:


 * $$\nu = -\frac{\epsilon_\text{transverse}}{\epsilon_\text{axial}}$$

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

Shear modulus: $$G = \frac {\tau_{xy}} {\gamma_{xy}} = \frac{F/A}{\Delta x/l} = \frac{F l}{A \Delta x} $$

where
 * $$\tau_{xy} = P \,$$ = shear stress
 * $$\gamma_{xy}$$ = shear strain. In engineering $$:=\Delta x/l = \tan \theta $$, elsewhere $$ := \theta$$
 * $$\Delta x$$ 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
Hooke%27s_law : Stiffness tensor $c$ is represented by a matrix of 3 × 3 × 3 × 3 = 81 real numbers $c_{ijkl}$. Hooke's law then says that


 * $$\sigma_{ij} = \sum_{k,l}c_{ijkl} \varepsilon_{kl}$$ and $$\varepsilon_{ij} =   \sum_{k,l} s_{ijkl} \sigma_{kl}$$

From Infinitesimal_strain_theory :

$$ \varepsilon_{ij} = \frac{1}{2} \left(u_{i,j}+u_{j,i}\right) = \partial_iu_j=\partial_ju_i$$

From Cauchy stress tensor: : $$\frac{dF_i}{dS}=\sigma_{ij}n_j $$

Introduction to the Rotation Matrix
$$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$\tilde a$$

Isotropic

 * $$ \begin{align}

\varepsilon_{11} & = \frac{1}{E}\big(\sigma_{11} - \nu(\sigma_{22}+\sigma_{33}) \big) \\ \varepsilon_{22} & = \frac{1}{E}\big(\sigma_{22} - \nu(\sigma_{11}+\sigma_{33}) \big) \\ \varepsilon_{33} & = \frac{1}{E}\big(\sigma_{33} - \nu(\sigma_{11}+\sigma_{22}) \big) \\ \varepsilon_{12} & = \frac{1}{2G}\sigma_{12} \,;\qquad \varepsilon_{13} = \frac{1}{2G}\sigma_{13} \,;\qquad \varepsilon_{23} = \frac{1}{2G}\sigma_{23} \end{align}$$ From Hooke's Law. Also available at Rod Lakes' website

Wikipedia formulas


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 direction vector n to the traction vector T(n) across an imaginary surface perpendicular to n:


 * $$\mathbf{T}^{(\mathbf n)} = \mathbf n \cdot\boldsymbol{\sigma}\quad \text{or} \quad T_{j}^{(n)}= \sigma_{ij}n_i.$$

where, $$\boldsymbol{\sigma} =$$

$$ \left[{\begin{matrix} \sigma _{11} & \sigma _{12} & \sigma _{13} \\ \sigma _{21} & \sigma _{22} & \sigma _{23} \\ \sigma _{31} & \sigma _{32} & \sigma _{33} \\ \end{matrix}}\right]

\equiv \left[{\begin{matrix} \sigma _{xx} & \sigma _{xy} & \sigma _{xz} \\ \sigma _{yx} & \sigma _{yy} & \sigma _{yz} \\ \sigma _{zx} & \sigma _{zy} & \sigma _{zz} \\ \end{matrix}}\right] \equiv \left[{\begin{matrix} \sigma _x & \tau _{xy} & \tau _{xz} \\ \tau _{yx} & \sigma _y & \tau _{yz} \\ \tau _{zx} & \tau _{zy} & \sigma _z \\ \end{matrix}}\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
$$\vec u$$
 * Remember: f = kx → Stress = &sigma; =c &epsilon;   and    x = (1/k)x → Strain = &epsilon; = s&sigma;

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


 * $$\underline{\underline{\boldsymbol{\sigma}}} = \begin{bmatrix}

\sigma_{xx} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{bmatrix} \equiv \begin{bmatrix} \sigma_{x} & \tau \\ \tau & \sigma_{y} \end{bmatrix}$$

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


 * $$\underline{\underline{\boldsymbol{\varepsilon}}} = \begin{bmatrix}

\varepsilon_{xx} & \varepsilon_{xy} \\ \varepsilon_{yx} & \varepsilon_{yy} \end{bmatrix} \equiv \begin{bmatrix} \varepsilon_{x} & \gamma/2 \\ \gamma/2 & \varepsilon_{y}\end{bmatrix}$$

To sort this out, see , and also:
 * Viscous stress tensor
 * Linear_elasticity

Infinitesimal strain theory

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

Infinitesimal strain theory