# Introduction to Elasticity/Stress example 2

## Example 2

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

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

1. The traction ($\mathbf {t} \,$ ) acting on a surface with unit normal ${\widehat {\mathbf {n} }}=({\widehat {\mathbf {e} }}_{2}+{\widehat {\mathbf {e} }}_{3})/{\sqrt {2}}$ .
2. The normal traction ($\mathbf {t} _{n}\,$ ) acting on a surface with unit normal ${\widehat {\mathbf {n} }}=({\widehat {\mathbf {e} }}_{2}+{\widehat {\mathbf {e} }}_{3})/{\sqrt {2}}$ .
3. The projected shear traction ($\mathbf {t} _{s}$ ) acting on a surface with unit normal ${\widehat {\mathbf {n} }}=({\widehat {\mathbf {e} }}_{2}+{\widehat {\mathbf {e} }}_{3})/{\sqrt {2}}$ .
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]); 

$\sigma :={\begin{bmatrix}3&1&1\\1&0&2\\1&2&0\end{bmatrix}}$ e2 := linalg[matrix](3,1,[0,1,0]); 

${\mathit {e2}}:={\begin{bmatrix}0\\1\\0\end{bmatrix}}$ e3 := linalg[matrix](3,1,[0,0,1]); 

${\mathit {e3}}:={\begin{bmatrix}0\\0\\1\end{bmatrix}}$ n := evalm((e2+e3)/sqrt(2)); 

$n:={\begin{bmatrix}0\\{\frac {\sqrt {2}}{2}}\\[2ex]{\frac {\sqrt {2}}{2}}\end{bmatrix}}$ sigmaT := transpose(sigma); 

${\mathit {sigmaT}}:={\begin{bmatrix}3&1&1\\1&0&2\\1&2&0\end{bmatrix}}$ t := evalm(sigmaT&*n); 

$\mathbf {t} :={\begin{bmatrix}{\sqrt {2}}\\{\sqrt {2}}\\{\sqrt {2}}\end{bmatrix}}~~~~{\text{Solution for Part 1}}$ tT := transpose(t); 

${\mathit {tT}}:={\begin{bmatrix}{\sqrt {2}}&{\sqrt {2}}&{\sqrt {2}}\end{bmatrix}}$ N := evalm(tT&*n); 

$N:={\begin{bmatrix}2\end{bmatrix}}~~~~{\text{Solution for Part 2}}$ tdott := evalm(tT&*t); 

${\mathit {tdott}}:={\begin{bmatrix}6\end{bmatrix}}$ S := sqrt(tdott[1,1] - N[1,1]^2); 

$S:={\sqrt {2}}~~~~{\text{Solution for Part 3}}$ sigPrin := eigenvals(sigma); 

${\mathit {sigPrin}}:=1,\,-2,\,4~~~~{\text{Solution for Part 4}}$ dirPrin := eigenvects(sigma); 

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

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

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

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