# Introduction to Elasticity/Stress example 2

## Example 2

Given: A homogeneous stress field with components in the basis ${\displaystyle (\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3})\,}$ given by

${\displaystyle \left[{\boldsymbol {\sigma }}\right]={\begin{bmatrix}3&1&1\\1&0&2\\1&2&0\end{bmatrix}}{\text{(MPa)}}}$

Find:

1. The traction (${\displaystyle \mathbf {t} \,}$) acting on a surface with unit normal ${\displaystyle {\widehat {\mathbf {n} }}=({\widehat {\mathbf {e} }}_{2}+{\widehat {\mathbf {e} }}_{3})/{\sqrt {2}}}$.
2. The normal traction (${\displaystyle \mathbf {t} _{n}\,}$) acting on a surface with unit normal ${\displaystyle {\widehat {\mathbf {n} }}=({\widehat {\mathbf {e} }}_{2}+{\widehat {\mathbf {e} }}_{3})/{\sqrt {2}}}$.
3. The projected shear traction (${\displaystyle \mathbf {t} _{s}}$) acting on a surface with unit normal ${\displaystyle {\widehat {\mathbf {n} }}=({\widehat {\mathbf {e} }}_{2}+{\widehat {\mathbf {e} }}_{3})/{\sqrt {2}}}$.
4. The principal stresses.
5. The principal directions of stress.

### Solution

Here's how you can solve this problem using Maple.
 with(linalg): 

 sigma := linalg[matrix](3,3,[3,1,1,1,0,2,1,2,0]); 

${\displaystyle \sigma :={\begin{bmatrix}3&1&1\\1&0&2\\1&2&0\end{bmatrix}}}$

 e2 := linalg[matrix](3,1,[0,1,0]); 

${\displaystyle {\mathit {e2}}:={\begin{bmatrix}0\\1\\0\end{bmatrix}}}$

 e3 := linalg[matrix](3,1,[0,0,1]); 

${\displaystyle {\mathit {e3}}:={\begin{bmatrix}0\\0\\1\end{bmatrix}}}$

 n := evalm((e2+e3)/sqrt(2)); 

${\displaystyle n:={\begin{bmatrix}0\\{\frac {\sqrt {2}}{2}}\\[2ex]{\frac {\sqrt {2}}{2}}\end{bmatrix}}}$

 sigmaT := transpose(sigma); 

${\displaystyle {\mathit {sigmaT}}:={\begin{bmatrix}3&1&1\\1&0&2\\1&2&0\end{bmatrix}}}$

 t := evalm(sigmaT&*n); 

${\displaystyle \mathbf {t} :={\begin{bmatrix}{\sqrt {2}}\\{\sqrt {2}}\\{\sqrt {2}}\end{bmatrix}}~~~~{\text{Solution for Part 1}}}$

 tT := transpose(t); 

${\displaystyle {\mathit {tT}}:={\begin{bmatrix}{\sqrt {2}}&{\sqrt {2}}&{\sqrt {2}}\end{bmatrix}}}$

 N := evalm(tT&*n); 

${\displaystyle N:={\begin{bmatrix}2\end{bmatrix}}~~~~{\text{Solution for Part 2}}}$

 tdott := evalm(tT&*t); 

${\displaystyle {\mathit {tdott}}:={\begin{bmatrix}6\end{bmatrix}}}$

 S := sqrt(tdott[1,1] - N[1,1]^2); 

${\displaystyle S:={\sqrt {2}}~~~~{\text{Solution for Part 3}}}$

 sigPrin := eigenvals(sigma); 

${\displaystyle {\mathit {sigPrin}}:=1,\,-2,\,4~~~~{\text{Solution for Part 4}}}$

 dirPrin := eigenvects(sigma); 

${\displaystyle {\mathit {dirPrin}}:=[1,\,1,\,\{[-1,\,1,\,1]\}],\,[-2,\,1,\,\{[0,\,-1,\,1]\}],\,[4,\,1,\,\{[2,\,1,\,1]\}]}$

 dirPrin[1]; 

${\displaystyle [1,\,1,\,\{[-1,\,1,\,1]\}]~~~~{\text{Solution for Part 5}}}$

 dirPrin[2]; 

${\displaystyle [-2,\,1,\,\{[0,\,-1,\,1]\}]~~~~{\text{Solution for Part 5}}}$

 dirPrin[3]; 

${\displaystyle [4,\,1,\,\{[2,\,1,\,1]\}]~~~~{\text{Solution for Part 5}}}$