Elasticity/Principle of minimum complementary energy

Complementary strain energy

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The complementary strain energy density is given by

 

For linear elastic materials

 


Principle of minimum complementary energy

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Let   be the set of all statically admissible states of stress.

Let the complementary energy functional be

 

Then the Principle of Stationary Complementary Energy states that:

Among all stress fields   in  , the functional   is rendered stationary only by actual stress fields which satisfy compatibility and the displacement BCs.

The Principle of Minimum Complementary Energy states that:

For linear elastic materials, the complementary energy functional is rendered an absolute minimum by the actual stress field.

Note that the complementary energy corresponding to the actual stress field is the negative of the potential energy corresponding to the actual displacement field.

Proof

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Let   be a solution of a the mixed boundary value problem of linear elasticity.

Let  .

Define

 

Then   satisfies equilibrium and the traction BCs, i.e.,

 

Since

 

and

 

we have,

 

We can also show that

 

Therefore,

 

Now,

 

Hence,

 

Therefore,

 

From equations (1), (2), and (3), we have,

 

Since   on  , we have,

 

Hence, proved.