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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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