Given:
A homogeneous stress field with components in the basis
given by
![{\displaystyle \left[{\boldsymbol {\sigma }}\right]={\begin{bmatrix}3&1&1\\1&0&2\\1&2&0\end{bmatrix}}{\text{(MPa)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d0a4455887ad2026fb7d4b952381e195c93ac7)
Find:
- The traction (
) acting on a surface with unit normal
.
- The normal traction (
) acting on a surface with unit normal
.
- The projected shear traction (
) acting on a surface with unit normal
.
- The principal stresses.
- The principal directions of stress.
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5433eed8871ca79ffe6dfd5848f8631fff57cdc)
e2 := linalg[matrix](3,1,[0,1,0]);
![{\displaystyle {\mathit {e2}}:={\begin{bmatrix}0\\1\\0\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6274dfc3d45a753b18179bbcbef9e51e3a447e24)
e3 := linalg[matrix](3,1,[0,0,1]);
![{\displaystyle {\mathit {e3}}:={\begin{bmatrix}0\\0\\1\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b669df61d1e36b69786702c9493604fbbd4480a8)
n := evalm((e2+e3)/sqrt(2));
![{\displaystyle n:={\begin{bmatrix}0\\{\frac {\sqrt {2}}{2}}\\[2ex]{\frac {\sqrt {2}}{2}}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bdb7c0ebdbd9e1522df84b363825a23e9964f80)
sigmaT := transpose(sigma);
![{\displaystyle {\mathit {sigmaT}}:={\begin{bmatrix}3&1&1\\1&0&2\\1&2&0\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac988725cd2db39a5a4068c8a29d71c721269b74)
t := evalm(sigmaT&*n);
![{\displaystyle \mathbf {t} :={\begin{bmatrix}{\sqrt {2}}\\{\sqrt {2}}\\{\sqrt {2}}\end{bmatrix}}~~~~{\text{Solution for Part 1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e20371bcc95d901c419eb6186a263b5031bd04a9)
tT := transpose(t);
![{\displaystyle {\mathit {tT}}:={\begin{bmatrix}{\sqrt {2}}&{\sqrt {2}}&{\sqrt {2}}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c31efa188dd307f239d63d4a731756f4930dd0dd)
N := evalm(tT&*n);
![{\displaystyle N:={\begin{bmatrix}2\end{bmatrix}}~~~~{\text{Solution for Part 2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4d6dceddf0ce1d85ad025413a6962a46e13f01b)
tdott := evalm(tT&*t);
![{\displaystyle {\mathit {tdott}}:={\begin{bmatrix}6\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84916541886f98c47b3e52a01b390a5e86cf8ae1)
S := sqrt(tdott[1,1] - N[1,1]^2);
![{\displaystyle S:={\sqrt {2}}~~~~{\text{Solution for Part 3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b745ac98552b2001c6a6b999db7089ef7ff7090f)
sigPrin := eigenvals(sigma);
![{\displaystyle {\mathit {sigPrin}}:=1,\,-2,\,4~~~~{\text{Solution for Part 4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d03e983498e6cc950810db9f69782373500e3eae)
dirPrin := eigenvects(sigma);
![{\displaystyle {\mathit {dirPrin}}:=[1,\,1,\,\{[-1,\,1,\,1]\}],\,[-2,\,1,\,\{[0,\,-1,\,1]\}],\,[4,\,1,\,\{[2,\,1,\,1]\}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6afd1c728e862ac4940ccec49907c09c7648ca73)
dirPrin[1];
![{\displaystyle [1,\,1,\,\{[-1,\,1,\,1]\}]~~~~{\text{Solution for Part 5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63b61aabf84daa98088731227649039c24b03f9e)
dirPrin[2];
![{\displaystyle [-2,\,1,\,\{[0,\,-1,\,1]\}]~~~~{\text{Solution for Part 5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce446beee5074b46867f469f3e910a12932863b2)
dirPrin[3];
![{\displaystyle [4,\,1,\,\{[2,\,1,\,1]\}]~~~~{\text{Solution for Part 5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1313c307c5105997b53bd6f3b25b6f7900d353c4)