Elasticity/Energy methods example 2

Example 2

Given:

The potential energy functional for a membrane stretched over a simply connected region $𝒮$ of the $x_1-x_2$ plane can be expressed as

$\mathrm{\Pi }[w(x_1,x_2)]=\frac{1}{2}\underset{𝒮}{\int }\eta [(w_{,1}{)}^2+(w_{,2}{)}^2]dA-\underset{𝒮}{\int }pwdA$

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 $𝒮$.
2. The permissible boundary conditions at the boundary $\partial 𝒮$ of $𝒮$.

Solution

The principle of minimum potential energy requires that the functional $\mathrm{\Pi }$ be stationary for the actual displacement field $w(x_1,x_2)$. Taking the first variation of $\mathrm{\Pi }$, we get

$\delta \mathrm{\Pi }=\frac{\eta }{2}\underset{𝒮}{\int }[(2w_{,1})\delta w_{,1}+(2w_{,2})\delta w_{,2}]dA-\underset{𝒮}{\int }p\delta wdA$

or,

$\delta \mathrm{\Pi }=\eta \underset{𝒮}{\int }[w_{,1}\delta w_{,1}+w_{,2}\delta w_{,2}]dA-\underset{𝒮}{\int }p\delta wdA$

Now,

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

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 $\delta \mathrm{\Pi }$,

$\delta \mathrm{\Pi }=\eta \underset{𝒮}{\int }[(w_{,1}\delta w{)}_{,1}+(w_{,2}\delta w{)}_{,2}-(w_{,11}+w_{,22})\delta w]dA-\underset{𝒮}{\int }p\delta wdA$

or,

$\delta \mathrm{\Pi }=\eta \underset{𝒮}{\int }[(w_{,1}\delta w{)}_{,1}+(w_{,2}\delta w{)}_{,2}]dA-\eta \underset{𝒮}{\int }{\mathrm{\nabla }}^{2}w\delta wdA-\underset{𝒮}{\int }p\delta wdA$

Now, the Green-Riemann theorem states that

$\underset{𝒮}{\int }(Q_{,1}-P_{,2})dA={\oint }_{\partial 𝒮}(Pd{x}_1+Qd{x}_2)$

Therefore,

$\delta \mathrm{\Pi }=\eta {\oint }_{\partial 𝒮}[(w_{,1}\delta w)d{x}_2-(w_{,2}\delta w)d{x}_1]-\underset{𝒮}{\int }[\eta {\mathrm{\nabla }}^{2}w+p]\delta wdA$

or,

$\delta \mathrm{\Pi }=\eta {\oint }_{\partial 𝒮}[w_{,1}\frac{d{x}_2}{ds}-w_{,2}\frac{d{x}_1}{ds}]\delta wds-\underset{𝒮}{\int }[\eta {\mathrm{\nabla }}^{2}w+p]\delta wdA$

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

The potential energy function is rendered stationary if $\delta \mathrm{\Pi }=0$. Since $\delta w$ is arbitrary, the condition of stationarity is satisfied only if the governing differential equation for $w(x_1,x_2)$ on $𝒮$ is

$\eta {\mathrm{\nabla }}^{2}w+p=0\mathrm{\forall }(x_1,x_2)\in 𝒮$

The associated boundary conditions are

$w_{,1}\frac{d{x}_2}{ds}-w_{,2}\frac{d{x}_1}{ds}=0\mathrm{\forall }(x_1,x_2)\in \partial 𝒮$