The Principle of Virtual Work

Explanation of Principle of Virtual Work :-

The principle of virtual work has nothing to do with linear elasticity, but it is convenient to start thinking about energy methods by reviewing a principle that you’ve probably seen before.

Let be a state of stress satisfying

Furthermore, let be a displacement field on and define

Then

• This is known as the Virtual Work equation.

Proof: It has been suggested that every result in solid mechanics is obtained by applying the divergence theorem. So, observe that

Hence

• Where we have used the equilibrium equations (including the symmetry of stress) to obtain the last line.

The virtual work equation is not particularly interesting. The converse is much more exciting:

Let be a symmetric stress field satisfying

for all displacement and strain fields with

Then

Proof: For sufficiently smooth , we may write

Then, use the divergence theorem to express the virtual work equation as

Recall that we required the virtual work equation to be satisfied for all displacement fields, so in particular, we may choose

with

Then

Next, suppose that the displacement field satisfies

Then

• This completes our proof.

Let be a region, and suppose thathas a jump discontinuity across some surface B.

We now define an equilibrium state of stress to be one that satisfies the virtual work equation for all displacement fields. It follows that

Hence