Example 3

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Given:

Unit square   with displacement fields :

  1.  .
  2.  .
  3.  .

Sketch: Deformed configuration in   plane.

Solution

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The displacement  . Hence,  . In the reference configuration,   and  . Hence, in the   plane, the initial square is the same shape as the unit square in the   plane. We can use Maple to find out the values of   and   after the deformation  .

  with(linalg):</code>
  X := array(1..3): x := array(1..3): u = array(1..3):
  e1 := array(1..3,[1,0,0]): 
  e2 := array(1..3,[0,1,0]): e3 = array(1..3,[0,0,1]):
  ua := evalm(k*X[2]*e1 + k*X[1]*e2):
  ub := evalm(-k*X[2]*e1 + k*X[1]*e2);
  uc := evalm(k*X[1]^2*e2);
 
 
 
  xa := evalm(ua + X);
  xb := evalm(ub + X);
  xc := evalm(uc + X);</code>
 
 
 

Plots of the deformed body are shown below

 
Deformed shapes