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];

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

dirPrin[2];

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

dirPrin[3];

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

Elasticity