Elasticity - A Reminder
============================
> Assuming everyone has taken ME035043, we will simply recap on some basic concepts. In case you need a more in-depth reminder, I recommend [Applied Mechanics of solids by A.F. Bower](https://technion.primo.exlibrisgroup.com/permalink/972TEC_INST/17d0lpa/alma9926527489703971) and [Elasticity: theory and applications by A.S. Saada](https://technion.primo.exlibrisgroup.com/permalink/972TEC_INST/q1jq5o/alma9926694182203971) 
> Consider the following not as a lecture notes but more as a  quiz for yourslef. If you are not familiar with any of the terms or equations - you need to go to the recommended sources.
## Strain
Given a displacemnts vector $u$ the infinitsimal strain is defined as :

$$ { \epsilon_{ij} = \frac{1}{2} \left( \frac{\partial{u_j}}{\partial{x_i}} + \frac{\partial{u_i}}{\partial{x_j}} \right ) \quad ; \quad \pmb{\epsilon} = \frac{1}{2} \left( \pmb{u} \nabla + (\pmb{u} \nabla)^T \right )}
$$

$$ 
 \epsilon_{ij} = \Large {\begin{bmatrix} 
\frac{\partial{u_1}}{\partial{x_1}}& 
\frac{1}{2} \left( \frac{\partial{u_1}}{\partial{x_2}} + \frac{\partial{u_2}}{\partial{x_1}} \right ) &
 \frac{1}{2} \left( \frac{\partial{u_1}}{\partial{x_3}} + \frac{\partial{u_3}}{\partial{x_1}} \right ) \\
\frac{1}{2} \left( \frac{\partial{u_2}}{\partial{x_1}} + \frac{\partial{u_1}}{\partial{x_2}} \right )& 
\frac{\partial{u_2}}{\partial{x_2}}  &
 \frac{1}{2} \left( \frac{\partial{u_2}}{\partial{x_3}} + \frac{\partial{u_3}}{\partial{x_2}} \right ) \\
\frac{1}{2} \left( \frac{\partial{u_3}}{\partial{x_1}} + \frac{\partial{u_1}}{\partial{x_3}} \right )& 
\frac{1}{2} \left( \frac{\partial{u_3}}{\partial{x_2}} + \frac{\partial{u_2}}{\partial{x_3}} \right ) &
\frac{\partial{u_3}}{\partial{x_3}} \\
\end{bmatrix} }
$$

```{note}
In many FE codes the shear strains output is given as "Engineering shear strain" such that 

$$ \gamma_{ij}=2\epsilon_{ij} \quad for \quad i\ne j $$
```

The strain tensor can be divided to deviatric $\pmb(e)$ and volumetric $\epsilon_{kk}$ parts such that

$$
 J \approx 1+ \epsilon_{kk} \quad \text{\&} \quad \pmb{e} = \pmb{\epsilon} - \frac{1}{3}\pmb{I}\text{trace}(\pmb{\epsilon})
$$

The eigenvalues of $\pmb{\epsilon}$ and corresponding eigenvectors will define the **principal** values $(e_i)$ and directions $(\pmb{n}^i)$ of the strain tensor.
## Rotations

## Stress

```{image} ../LEFM/stress_cube.png
:alt: stress cube
:width: 400px
:align: center
```

For (isotropic) linear elasticity, we can correlate the strains with the stresses using :

$$
 \sigma_{ij} = \frac{E}{1+\nu}\left[ \epsilon_{ij} + \frac{\nu}{1-2\nu}\epsilon_{kk}\delta_{ij} \right] - \frac{E\alpha\Delta T}{1-2\nu}\delta_{ij}
$$
or, fiven the stress tensor, we can obtain the strains following:

$$
 \epsilon_{ij} = \frac{1+\nu}{E} \sigma_{ij} - \frac{\nu}{E}\sigma_{kk}\delta_{ij}+\alpha \Delta T\delta_{ij}$$

Here, $E$ is Young's modulus, $\nu$ is Poisson's ratio, $\alpha$ the thermal expansion coefficient and $\Delta T$ stands for temperature increase (decrease)

The shear modulus $G$ is defined as :

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

## Strain Energy density
The strain energy density $[Joule/m^3]$ is defined as :

$$W = \int_{0}^{\epsilon} \sigma d\epsilon $$ 
Under linear elasticity we obtain 

$$W_{elastic}=\frac{1}{2}E\epsilon^2 = \frac{\sigma^2}{2E}$$

## Plane stress
Thin solids, (i.e. one dimension is significantly smaller than the other two) loaded in-plane,  can be approximated using 2D plane stress assumptions. under this assumption :

$$ \sigma_{33} = \tau_{23} = \tau_{31} = \epsilon_{23} =\epsilon_{31} = 0$$

## Plane strain
A solid body, whose deformation in one direction are severely restricted (consider the mid-sction of a thick body) can be assumed to be in a plane strain condition such that:

$$ \epsilon_{23}=\epsilon_{13}=\epsilon_{33}=0 \quad \& \quad \sigma_{33} \approx \nu \left( \sigma_{11} + \sigma_{22} \right) $$

## Solution to elasticity problems (static)
Given an elastic body with applied tractions $t_j^{ap}$ and displacements $u_i^{ap}$ we seek to find a solution which will satisfy: 

$$ 
\begin{align*}
& \epsilon_{ij} = \frac{1}{2} \left( \frac{\partial{u_j}}{\partial{x_i}} + \frac{\partial{u_i}}{\partial{x_j}} \right ) \\
& \sigma_{ij} = \frac{E}{1+\nu}\left[ \sigma_{ij} + \frac{\nu}{1-2\nu} \right] - \frac{E\alpha\Delta T}{1-2\nu}\delta_{ij} \\
& \frac{\partial \sigma_{ij}}{\partial x_i} + F_j =0 \quad (\text{equilibrium}) 
\end{align*}
$$ 

as well as the B.C:

$$
u_i = u_i^{ap} \quad \& \quad \sigma_{ij} n_j=t_i^{ap} 
$$
on the portions of the boundary where they are defined