Elasticity/Minimizing a functional

Minimizing a functional in 1-D

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In 1-D, the minimization problem can be stated as

Find   such that

 

is a minimum.

We have seen that the minimization problem can be reduced down to the solution of an Euler equation

 

with the associated boundary conditions

 

or,

 

Minimizing a Functional in 3-D

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In 3-D, the equivalent minimization problem can be stated as

Find   such that

 

is a minimum.

We would like to find the Euler equation for this problem and the associated boundary conditions required to minimize  .

Let us define all our quantities with respect to an orthonormal basis  .

Then,

 

and

 

Taking the first variation of  , we get

 

All the nine components of   are not independent. Why ?

The variation of the functional   needs to be expressed entirely in terms of  . We do this using the 3-D equivalent of integration by parts - the divergence theorem.

Thus,

 

Substituting in the expression for  , we have,

 

For   to be minimum, a necessary condition is that   for all variations  .

Therefore, the Euler equation for this problem is

 

and the associated boundary conditions are