Introduction to Elasticity/Kinematics example 3

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Example 3[edit | edit source]

Given:

Unit square with displacement fields :

  1. .
  2. .
  3. .

Sketch: Deformed configuration in plane.

Solution[edit | edit source]

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