Introduction to Elasticity/Principle of minimum complementary energy

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Complementary strain energy[edit | edit source]

The complementary strain energy density is given by

For linear elastic materials


Principle of minimum complementary energy[edit | edit source]

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[edit | edit source]

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.