Introduction to Elasticity/Stress example 2

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[edit] 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.

[edit] 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}
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