Example 3
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Given:
Unit square ( X 1 , X 2 ) ∈ [ 0 , 1 ] {\displaystyle (X_{1},X_{2})\in [0,1]} with displacement fields :
u = κ X 2 e ^ 1 + κ X 1 e ^ 2 {\displaystyle \mathbf {u} =\kappa X_{2}{\widehat {\mathbf {e} }}_{1}+\kappa X_{1}{\widehat {\mathbf {e} }}_{2}} .
u = − κ X 2 e ^ 1 + κ X 1 e ^ 2 {\displaystyle \mathbf {u} =-\kappa X_{2}{\widehat {\mathbf {e} }}_{1}+\kappa X_{1}{\widehat {\mathbf {e} }}_{2}} .
u = κ X 1 2 e ^ 2 {\displaystyle \mathbf {u} =\kappa X_{1}^{2}{\widehat {\mathbf {e} }}_{2}} . Sketch:
Deformed configuration in x 1 , x 2 {\displaystyle x_{1},x_{2}} plane.
Solution
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The displacement u = x − X {\displaystyle \mathbf {u} =\mathbf {x} -\mathbf {X} } . Hence, x = u + X {\displaystyle \mathbf {x} =\mathbf {u} +\mathbf {X} } . In the reference configuration, u = 0 {\displaystyle \mathbf {u} =0} and x = X {\displaystyle \mathbf {x} =\mathbf {X} } . Hence, in the ( x 1 , x 2 ) {\displaystyle (x_{1},x_{2})} plane, the initial square is the same shape as the unit square in the ( X 1 , X 2 ) {\displaystyle (X_{1},X_{2})} plane. We can use Maple to find out the values of x 1 {\displaystyle x_{1}} and x 2 {\displaystyle x_{2}} after the deformation u {\displaystyle \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[2]*e1 + k*X[1]*e2):
ub := evalm(-k*X[2]*e1 + k*X[1]*e2);
uc := evalm(k*X[1]^2*e2);
u a := [ k X 2 , k X 1 , 0 ] {\displaystyle {\mathit {ua}}:=\left[k{X_{2}},k{X_{1}},0\right]}
u b := [ − k X 2 , k X 1 , 0 ] {\displaystyle {\mathit {ub}}:=\left[-k{X_{2}},k{X_{1}},0\right]}
u c := [ 0 , k X 1 2 , 0 ] {\displaystyle {\mathit {uc}}:=\left[0,k{X_{1}}^{2},0\right]} xa := evalm(ua + X);
xb := evalm(ub + X);
xc := evalm(uc + X);</code>
x a := [ k X 2 + X 1 , k X 1 + X 2 , X 3 ] {\displaystyle {\mathit {xa}}:=\left[k{X_{2}}+{X_{1}},k{X_{1}}+{X_{2}},{X_{3}}\right]}
x b := [ − k X 2 + X 1 , k X 1 + X 2 , X 3 ] {\displaystyle {\mathit {xb}}:=\left[-k{X_{2}}+{X_{1}},k{X_{1}}+{X_{2}},{X_{3}}\right]}
x c := [ X 1 , k X 1 2 + X 2 , X 3 ] {\displaystyle {\mathit {xc}}:=\left[{X_{1}},k{X_{1}}^{2}+{X_{2}},{X_{3}}\right]} Plots of the deformed body are shown below
Deformed shapes