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[(2 w_{,1})\delta w_{,1} + (2 w_{,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{align}

(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{align}$$ 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}  } $$