Introduction to Elasticity/Energy methods example 2

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Example 2[edit | edit source]

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[edit | edit source]

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