EN224: Linear Elasticity
Division of Engineering
8.5 The Principle of Stationary and Minimum Complementary Energy
In our potential energy formulation, we considered a family of kinematically admissible displacement fields, and defined V(v) so that the elastostatic state minimized V.
We can devise an alternative variational method in which we consider a family of statically admissible stress fields, and define complementary energy so that the elastostatic stress field minimizes C.
Let us make this precise.
Let with
In addition, define the components of the compliance tensor such that
Now, define a statically admissible stress field which satisfies
Define the strain energy density associated with as
Define the complementary energy C as
We will proceed to show several important properties of C.
(1) C is stationary for
(2) If C is stationary for some , then
(3)
The Principle Of Stationary Complementary Energy
C is stationary if and only if with on .
Proof. We will show first that if with on then C is stationary.
Begin by computing the first variation of C. We could use the same procedure we followed in discussing the principle of stationary potential energy, but just for variety, we will this time use the formal machinery of the calculus of variations.
First, let us define a statically admissible variation in stress Let
where
Then, the first variation of C is defined as
Hence
Note that we may write
where we have used the symmetry of .
Now, substitute into the expression for and integrate by parts
where we have noted that on , while on .
This completes the first part of the proof.
Next, consider the converse. We wish to show that if C is stationary, then , with on .
If C is stationary, then
In particular, choose
where is a tensor function of position and is the permutation symbol. One may readily verify that
as required for a statically admissible variation. Hence
Now, recall that , so we may choose and on . Then, integrate by parts twice to see that
Since this holds for all , it follows that
This is the strain equation of compatibility (See Section 1.5) and implies (at least for a simply connected region) that there exists a single valued displacement with
Finally, we need to show that . Follow the procedure in the first part of the proof to see that
Then, choose
to obtain the required result.
The Principle of Minimum Complementary Energy
We will now go a step further, and show that C is a global minimum only for the compatible stress field. We could show this using the procedure we used to prove minimum potential energy, but just for variety we will do things slightly differently.
As before, let with
In addition, define the components of the compliance tensor such that
Define a statically admissible stress field which satisfies
Denote the strain energy density associated with as
and define
Now, consider
Noting that both and are equilibrium fields, we may write
Integrate by parts
The last two integrals evidently sum to zero if and . Expand the remaining terms in terms of strains
Finally, using the properties of the stiffness tensor, we see that
with equality if and only if .