Elasticity/Stress example 2

Example 2

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Given: A homogeneous stress field with components in the basis   given by

 

Find:

  1. The traction ( ) acting on a surface with unit normal  .
  2. The normal traction ( ) acting on a surface with unit normal  .
  3. The projected shear traction ( ) acting on a surface with unit normal  .
  4. The principal stresses.
  5. The principal directions of stress.

Solution

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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]);

 


e2 := linalg[matrix](3,1,[0,1,0]);

 


e3 := linalg[matrix](3,1,[0,0,1]);

 


n := evalm((e2+e3)/sqrt(2));

 


sigmaT := transpose(sigma);

 


t := evalm(sigmaT&*n);

 


tT := transpose(t);

 


N := evalm(tT&*n);

 


tdott := evalm(tT&*t);

 


S := sqrt(tdott[1,1] - N[1,1]^2);

 


sigPrin := eigenvals(sigma);

 


dirPrin := eigenvects(sigma);

 


dirPrin[1];

 


dirPrin[2];

 


dirPrin[3];