# Introduction to Elasticity/Kinematics example 3

## Example 3

Given:

Unit square $(X_{1},X_{2})\in [0,1]$ with displacement fields :

1. $\mathbf {u} =\kappa X_{2}{\widehat {\mathbf {e} }}_{1}+\kappa X_{1}{\widehat {\mathbf {e} }}_{2}$ .
2. $\mathbf {u} =-\kappa X_{2}{\widehat {\mathbf {e} }}_{1}+\kappa X_{1}{\widehat {\mathbf {e} }}_{2}$ .
3. $\mathbf {u} =\kappa X_{1}^{2}{\widehat {\mathbf {e} }}_{2}$ .

Sketch: Deformed configuration in $x_{1},x_{2}$ plane.

### Solution

The displacement $\mathbf {u} =\mathbf {x} -\mathbf {X}$ . Hence, $\mathbf {x} =\mathbf {u} +\mathbf {X}$ . In the reference configuration, $\mathbf {u} =0$ and $\mathbf {x} =\mathbf {X}$ . Hence, in the $(x_{1},x_{2})$ plane, the initial square is the same shape as the unit square in the $(X_{1},X_{2})$ plane. We can use Maple to find out the values of $x_{1}$ and $x_{2}$ after the deformation $\mathbf {u}$ .

  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*e1 + k*X*e2):
ub := evalm(-k*X*e1 + k*X*e2);
uc := evalm(k*X^2*e2);

${\mathit {ua}}:=\left[k{X_{2}},k{X_{1}},0\right]$ ${\mathit {ub}}:=\left[-k{X_{2}},k{X_{1}},0\right]$ ${\mathit {uc}}:=\left[0,k{X_{1}}^{2},0\right]$ xa := evalm(ua + X);
xb := evalm(ub + X);
xc := evalm(uc + X);</code>

${\mathit {xa}}:=\left[k{X_{2}}+{X_{1}},k{X_{1}}+{X_{2}},{X_{3}}\right]$ ${\mathit {xb}}:=\left[-k{X_{2}}+{X_{1}},k{X_{1}}+{X_{2}},{X_{3}}\right]$ ${\mathit {xc}}:=\left[{X_{1}},k{X_{1}}^{2}+{X_{2}},{X_{3}}\right]$ Plots of the deformed body are shown below