Given:
The potential energy functional for a membrane stretched over a simply connected region
S
{\displaystyle {\mathcal {S}}}
x
1
−
x
2
{\displaystyle x_{1}-x_{2}}
Π
[
w
(
x
1
,
x
2
)
]
=
1
2
∫
S
η
[
(
w
,
1
)
2
+
(
w
,
2
)
2
]
d
A
−
∫
S
p
w
d
A
{\displaystyle \Pi [w(x_{1},x_{2})]={\frac {1}{2}}\int _{\mathcal {S}}\eta \left[(w_{,1})^{2}+(w_{,2})^{2}\right]~dA-\int _{\mathcal {S}}pw~dA}
where
w
(
x
1
,
x
2
)
{\displaystyle w(x_{1},x_{2})}
p
(
x
1
,
x
2
)
{\displaystyle p(x_{1},x_{2})}
η
{\displaystyle \eta }
Find:
The governing differential equation (Euler equation) for
w
(
x
1
,
x
2
)
{\displaystyle w(x_{1},x_{2})}
S
{\displaystyle {\mathcal {S}}}
The permissible boundary conditions at the boundary
∂
S
{\displaystyle \partial {\mathcal {S}}}
S
{\displaystyle {\mathcal {S}}}
The principle of minimum potential energy requires that the functional
Π
{\displaystyle \Pi }
w
(
x
1
,
x
2
)
{\displaystyle w(x_{1},x_{2})}
Π
{\displaystyle \Pi }
δ
Π
=
η
2
∫
S
[
(
2
w
,
1
)
δ
w
,
1
+
(
2
w
,
2
)
δ
w
,
2
]
d
A
−
∫
S
p
δ
w
d
A
{\displaystyle \delta \Pi ={\frac {\eta }{2}}\int _{\mathcal {S}}\left[(2w_{,1})\delta w_{,1}+(2w_{,2})\delta w_{,2}\right]~dA-\int _{\mathcal {S}}p~\delta w~dA}
or,
δ
Π
=
η
∫
S
[
w
,
1
δ
w
,
1
+
w
,
2
δ
w
,
2
]
d
A
−
∫
S
p
δ
w
d
A
{\displaystyle \delta \Pi =\eta \int _{\mathcal {S}}\left[w_{,1}~\delta w_{,1}+w_{,2}~\delta w_{,2}\right]~dA-\int _{\mathcal {S}}p~\delta w~dA}
Now,
(
w
,
1
δ
w
)
,
1
=
w
,
11
δ
w
+
w
,
1
δ
w
,
1
(
w
,
2
δ
w
)
,
2
=
w
,
22
δ
w
+
w
,
2
δ
w
,
2
{\displaystyle {\begin{aligned}(w_{,1}~\delta w)_{,1}&=w_{,11}~\delta w+w_{,1}~\delta w_{,1}\\(w_{,2}~\delta w)_{,2}&=w_{,22}~\delta w+w_{,2}~\delta w_{,2}\end{aligned}}}
Therefore,
w
,
1
δ
w
,
1
+
w
,
2
δ
w
,
2
=
(
w
,
1
δ
w
)
,
1
−
w
,
11
δ
w
+
(
w
,
2
δ
w
)
,
2
−
w
,
22
δ
w
{\displaystyle w_{,1}~\delta w_{,1}+w_{,2}~\delta w_{,2}=(w_{,1}~\delta w)_{,1}-w_{,11}~\delta w+(w_{,2}~\delta w)_{,2}-w_{,22}~\delta w}
Plugging into the expression for
δ
Π
{\displaystyle \delta \Pi }
δ
Π
=
η
∫
S
[
(
w
,
1
δ
w
)
,
1
+
(
w
,
2
δ
w
)
,
2
−
(
w
,
11
+
w
,
22
)
δ
w
]
d
A
−
∫
S
p
δ
w
d
A
{\displaystyle \delta \Pi =\eta \int _{\mathcal {S}}\left[(w_{,1}~\delta w)_{,1}+(w_{,2}~\delta w)_{,2}-(w_{,11}+w_{,22})~\delta w\right]~dA-\int _{\mathcal {S}}p~\delta w~dA}
or,
δ
Π
=
η
∫
S
[
(
w
,
1
δ
w
)
,
1
+
(
w
,
2
δ
w
)
,
2
]
d
A
−
η
∫
S
∇
2
w
δ
w
d
A
−
∫
S
p
δ
w
d
A
{\displaystyle \delta \Pi =\eta \int _{\mathcal {S}}\left[(w_{,1}~\delta w)_{,1}+(w_{,2}~\delta w)_{,2}\right]~dA-\eta \int _{\mathcal {S}}\nabla ^{2}{w}~\delta w~dA-\int _{\mathcal {S}}p~\delta w~dA}
Now, the Green-Riemann theorem states that
∫
S
(
Q
,
1
−
P
,
2
)
d
A
=
∮
∂
S
(
P
d
x
1
+
Q
d
x
2
)
{\displaystyle \int _{\mathcal {S}}(Q_{,1}-P_{,2})~dA=\oint _{\partial {\mathcal {S}}}(P~dx_{1}+Q~dx_{2})}
Therefore,
δ
Π
=
η
∮
∂
S
[
(
w
,
1
δ
w
)
d
x
2
−
(
w
,
2
δ
w
)
d
x
1
]
−
∫
S
[
η
∇
2
w
+
p
]
δ
w
d
A
{\displaystyle \delta \Pi =\eta \oint _{\partial {\mathcal {S}}}\left[(w_{,1}~\delta w)~dx_{2}-(w_{,2}~\delta w)~dx_{1}\right]-\int _{\mathcal {S}}\left[\eta \nabla ^{2}{w}+p\right]~\delta w~dA}
or,
δ
Π
=
η
∮
∂
S
[
w
,
1
d
x
2
d
s
−
w
,
2
d
x
1
d
s
]
δ
w
d
s
−
∫
S
[
η
∇
2
w
+
p
]
δ
w
d
A
{\displaystyle \delta \Pi =\eta \oint _{\partial {\mathcal {S}}}\left[w_{,1}~{\frac {dx_{2}}{ds}}-w_{,2}~{\frac {dx_{1}}{ds}}\right]\delta w~ds-\int _{\mathcal {S}}\left[\eta \nabla ^{2}{w}+p\right]~\delta w~dA}
where
s
{\displaystyle s}
∂
S
{\displaystyle \partial {\mathcal {S}}}
δ
Π
=
0
{\displaystyle \delta \Pi =0}
δ
w
{\displaystyle \delta w}
w
(
x
1
,
x
2
)
{\displaystyle w(x_{1},x_{2})}
S
{\displaystyle {\mathcal {S}}}
η
∇
2
w
+
p
=
0
∀
(
x
1
,
x
2
)
∈
S
{\displaystyle {\eta \nabla ^{2}{w}+p=0~~~~~\forall ~~(x_{1},x_{2})~\in ~{\mathcal {S}}}}
The associated boundary conditions are
w
,
1
d
x
2
d
s
−
w
,
2
d
x
1
d
s
=
0
∀
(
x
1
,
x
2
)
∈
∂
S
{\displaystyle {w_{,1}~{\frac {dx_{2}}{ds}}-w_{,2}~{\frac {dx_{1}}{ds}}=0~~~~~\forall ~~(x_{1},x_{2})~\in ~\partial {\mathcal {S}}}}
![{\displaystyle \Pi [w(x_{1},x_{2})]={\frac {1}{2}}\int _{\mathcal {S}}\eta \left[(w_{,1})^{2}+(w_{,2})^{2}\right]~dA-\int _{\mathcal {S}}pw~dA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b48a4f18640d82d54dcd9e7e4efa53e269bb39)
![{\displaystyle \delta \Pi ={\frac {\eta }{2}}\int _{\mathcal {S}}\left[(2w_{,1})\delta w_{,1}+(2w_{,2})\delta w_{,2}\right]~dA-\int _{\mathcal {S}}p~\delta w~dA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20166f5f850bc82d7d3b37b22630c2208f7a33a0)
![{\displaystyle \delta \Pi =\eta \int _{\mathcal {S}}\left[w_{,1}~\delta w_{,1}+w_{,2}~\delta w_{,2}\right]~dA-\int _{\mathcal {S}}p~\delta w~dA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c593dd542e5865edc577f74ef031854930b262)
![{\displaystyle \delta \Pi =\eta \int _{\mathcal {S}}\left[(w_{,1}~\delta w)_{,1}+(w_{,2}~\delta w)_{,2}-(w_{,11}+w_{,22})~\delta w\right]~dA-\int _{\mathcal {S}}p~\delta w~dA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc708e65504bdef961a62e8f011d40cb4b7fee2)
![{\displaystyle \delta \Pi =\eta \int _{\mathcal {S}}\left[(w_{,1}~\delta w)_{,1}+(w_{,2}~\delta w)_{,2}\right]~dA-\eta \int _{\mathcal {S}}\nabla ^{2}{w}~\delta w~dA-\int _{\mathcal {S}}p~\delta w~dA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c632564482eb5f0524181997b0bcb248f67df85b)
![{\displaystyle \delta \Pi =\eta \oint _{\partial {\mathcal {S}}}\left[(w_{,1}~\delta w)~dx_{2}-(w_{,2}~\delta w)~dx_{1}\right]-\int _{\mathcal {S}}\left[\eta \nabla ^{2}{w}+p\right]~\delta w~dA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7175e4a2c35688661f59e0c05786022a9570e09a)
![{\displaystyle \delta \Pi =\eta \oint _{\partial {\mathcal {S}}}\left[w_{,1}~{\frac {dx_{2}}{ds}}-w_{,2}~{\frac {dx_{1}}{ds}}\right]\delta w~ds-\int _{\mathcal {S}}\left[\eta \nabla ^{2}{w}+p\right]~\delta w~dA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2fde0cfe462c06c48bdf6dccbc2bdb49f677dbb)
