Elasticity/Energy methods example 2

Example 2

Given:

The potential energy functional for a membrane stretched over a simply connected region ${\mathcal {S}}$ of the $x_{1}-x_{2}$ plane can be expressed as

$\Pi [w(x_{1},x_{2})]={\frac {1}{2}}\int _{\mathcal {S}}\eta \left[(w_{,1})^{2}+(w_{,2})^{2}\right]~dA-\int _{\mathcal {S}}pw~dA$

where $w(x_{1},x_{2})$ is the deflection of the membrane, $p(x_{1},x_{2})$ is the prescribed transverse pressure distribution, and $\eta$ is the membrane stiffness.

Find:

1. The governing differential equation (Euler equation) for $w(x_{1},x_{2})$ on ${\mathcal {S}}$.
2. The permissible boundary conditions at the boundary $\partial {\mathcal {S}}$ of ${\mathcal {S}}$.

Solution

The principle of minimum potential energy requires that the functional $\Pi$ be stationary for the actual displacement field $w(x_{1},x_{2})$. Taking the first variation of $\Pi$, we get

$\delta \Pi ={\frac {\eta }{2}}\int _{\mathcal {S}}\left[(2w_{,1})\delta w_{,1}+(2w_{,2})\delta w_{,2}\right]~dA-\int _{\mathcal {S}}p~\delta w~dA$

or,

$\delta \Pi =\eta \int _{\mathcal {S}}\left[w_{,1}~\delta w_{,1}+w_{,2}~\delta w_{,2}\right]~dA-\int _{\mathcal {S}}p~\delta w~dA$

Now,

${\begin{aligned}(w_{,1}~\delta w)_{,1}&=w_{,11}~\delta w+w_{,1}~\delta w_{,1}\\(w_{,2}~\delta w)_{,2}&=w_{,22}~\delta w+w_{,2}~\delta w_{,2}\end{aligned}}$

Therefore,

$w_{,1}~\delta w_{,1}+w_{,2}~\delta w_{,2}=(w_{,1}~\delta w)_{,1}-w_{,11}~\delta w+(w_{,2}~\delta w)_{,2}-w_{,22}~\delta w$

Plugging into the expression for $\delta \Pi$,

$\delta \Pi =\eta \int _{\mathcal {S}}\left[(w_{,1}~\delta w)_{,1}+(w_{,2}~\delta w)_{,2}-(w_{,11}+w_{,22})~\delta w\right]~dA-\int _{\mathcal {S}}p~\delta w~dA$

or,

$\delta \Pi =\eta \int _{\mathcal {S}}\left[(w_{,1}~\delta w)_{,1}+(w_{,2}~\delta w)_{,2}\right]~dA-\eta \int _{\mathcal {S}}\nabla ^{2}{w}~\delta w~dA-\int _{\mathcal {S}}p~\delta w~dA$

Now, the Green-Riemann theorem states that

$\int _{\mathcal {S}}(Q_{,1}-P_{,2})~dA=\oint _{\partial {\mathcal {S}}}(P~dx_{1}+Q~dx_{2})$

Therefore,

$\delta \Pi =\eta \oint _{\partial {\mathcal {S}}}\left[(w_{,1}~\delta w)~dx_{2}-(w_{,2}~\delta w)~dx_{1}\right]-\int _{\mathcal {S}}\left[\eta \nabla ^{2}{w}+p\right]~\delta w~dA$

or,

$\delta \Pi =\eta \oint _{\partial {\mathcal {S}}}\left[w_{,1}~{\frac {dx_{2}}{ds}}-w_{,2}~{\frac {dx_{1}}{ds}}\right]\delta w~ds-\int _{\mathcal {S}}\left[\eta \nabla ^{2}{w}+p\right]~\delta w~dA$

where $s$ is the arc length around $\partial {\mathcal {S}}$.

The potential energy function is rendered stationary if $\delta \Pi =0$. Since $\delta w$ is arbitrary, the condition of stationarity is satisfied only if the governing differential equation for $w(x_{1},x_{2})$ on ${\mathcal {S}}$ is

${\eta \nabla ^{2}{w}+p=0~~~~~\forall ~~(x_{1},x_{2})~\in ~{\mathcal {S}}}$

The associated boundary conditions are

${w_{,1}~{\frac {dx_{2}}{ds}}-w_{,2}~{\frac {dx_{1}}{ds}}=0~~~~~\forall ~~(x_{1},x_{2})~\in ~\partial {\mathcal {S}}}$

Elasticity