Example 2

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Given:

The potential energy functional for a membrane stretched over a simply connected region   of the   plane can be expressed as

 

where   is the deflection of the membrane,   is the prescribed transverse pressure distribution, and   is the membrane stiffness.

Find:

  1. The governing differential equation (Euler equation) for   on  .
  2. The permissible boundary conditions at the boundary   of  .

Solution

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The principle of minimum potential energy requires that the functional   be stationary for the actual displacement field  . Taking the first variation of  , we get

 

or,

 

Now,

 

Therefore,

 

Plugging into the expression for  ,

 

or,

 

Now, the Green-Riemann theorem states that

 

Therefore,

 

or,

 

where   is the arc length around  .


The potential energy function is rendered stationary if  . Since   is arbitrary, the condition of stationarity is satisfied only if the governing differential equation for   on   is

 

The associated boundary conditions are