Given:
Unit square
(
X
1
,
X
2
)
∈
[
0
,
1
]
{\displaystyle (X_{1},X_{2})\in [0,1]}
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}}
The displacement
u
=
x
−
X
{\displaystyle \mathbf {u} =\mathbf {x} -\mathbf {X} }
x
=
u
+
X
{\displaystyle \mathbf {x} =\mathbf {u} +\mathbf {X} }
u
=
0
{\displaystyle \mathbf {u} =0}
x
=
X
{\displaystyle \mathbf {x} =\mathbf {X} }
(
x
1
,
x
2
)
{\displaystyle (x_{1},x_{2})}
(
X
1
,
X
2
)
{\displaystyle (X_{1},X_{2})}
x
1
{\displaystyle x_{1}}
x
2
{\displaystyle x_{2}}
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
![{\displaystyle (X_{1},X_{2})\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfbc8add749a9b09c6f1c21d7af5d5139adcd67)
![{\displaystyle {\mathit {ua}}:=\left[k{X_{2}},k{X_{1}},0\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f989d5cbf94effe54d6957aede173cd41cc737b)
![{\displaystyle {\mathit {ub}}:=\left[-k{X_{2}},k{X_{1}},0\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa93466a29aa804d3c364a3e2a611872418471f2)
![{\displaystyle {\mathit {uc}}:=\left[0,k{X_{1}}^{2},0\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58fdde51ced9b0fca671a2639ec236a26d9720db)
![{\displaystyle {\mathit {xa}}:=\left[k{X_{2}}+{X_{1}},k{X_{1}}+{X_{2}},{X_{3}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee8f7990dab88551d80011e9571138bf1905a4c1)
![{\displaystyle {\mathit {xb}}:=\left[-k{X_{2}}+{X_{1}},k{X_{1}}+{X_{2}},{X_{3}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19514435f2a0daaf997569c5ebc64021ca7a8d8c)
![{\displaystyle {\mathit {xc}}:=\left[{X_{1}},k{X_{1}}^{2}+{X_{2}},{X_{3}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29f48acadb6efeaad6a8eea3eb051c39b05128f3)
