Elasticity/Stress example 2

Example 2

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

[σ] = [\begin{matrix} 3 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0 \end{matrix}] (MPa)

Find:

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

σ : = [\begin{matrix} 3 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0 \end{matrix}]

e2 := linalg[matrix](3,1,[0,1,0]);

𝑒 2 : = [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}]

e3 := linalg[matrix](3,1,[0,0,1]);

𝑒 3 : = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]

n := evalm((e2+e3)/sqrt(2));

n : = [\begin{matrix} 0 \\ \frac{\sqrt{2}}{2} \\ [2 e x] \frac{\sqrt{2}}{2} \end{matrix}]

sigmaT := transpose(sigma);

𝑠 𝑖 𝑔 𝑚 𝑎 𝑇 : = [\begin{matrix} 3 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0 \end{matrix}]

t := evalm(sigmaT&*n);

𝐭 : = [\begin{matrix} \sqrt{2} \\ \sqrt{2} \\ \sqrt{2} \end{matrix}] Solution for Part 1

tT := transpose(t);

𝑡 𝑇 : = [\begin{matrix} \sqrt{2} & \sqrt{2} & \sqrt{2} \end{matrix}]

N := evalm(tT&*n);

N : = [\begin{matrix} 2 \end{matrix}] Solution for Part 2

tdott := evalm(tT&*t);

𝑡 𝑑 𝑜 𝑡 𝑡 : = [\begin{matrix} 6 \end{matrix}]

S := sqrt(tdott[1,1] - N[1,1]^2);

S : = \sqrt{2} Solution for Part 3

sigPrin := eigenvals(sigma);

𝑠 𝑖 𝑔 𝑃 𝑟 𝑖 𝑛 : = 1, - 2, 4 Solution for Part 4

dirPrin := eigenvects(sigma);

𝑑 𝑖 𝑟 𝑃 𝑟 𝑖 𝑛 : = [1, 1, {[- 1, 1, 1]}], [- 2, 1, {[0, - 1, 1]}], [4, 1, {[2, 1, 1]}]

dirPrin[1];

[1, 1, {[- 1, 1, 1]}] Solution for Part 5

dirPrin[2];

[- 2, 1, {[0, - 1, 1]}] Solution for Part 5

dirPrin[3];

[4, 1, {[2, 1, 1]}] Solution for Part 5