I'm having some trouble understanding the proof of the Principle of Virtual Work for deformable bodies. I'll give below the proof that I've read, and, next, I'll remark what I'm not understanding.(adsbygoogle = window.adsbygoogle || []).push({});

The first thing to remember before going through the proof is that the virtual work done by a system of virtual forces in equilibrium as a rigid body undergoes a small, compatible displacement is zero.

PROOF:

Suppose that a deformable body is in static equilibrium under the external loads of a virtual Q-force system.

Since the body as a whole is in equilibrium, any particular particle can be isolated and will be in equilibrium under the internal virtual Q stresses developed by the external virtual Q forces.

Now suppose that the body is subjected to a small change in shape caused by some other source than the virtual Q-force system. Owing to this change in shape, any particle might be deformed as well as translated and rotated as a rigid particle. Hence, the boundaries of such a particle would move and hence do virtual work. Let the virtual work done by the Q stresses on the boundaries of the differential particle be designated by dWs. Part of this virtual work will be done because of the movements of the boundaries of the particle caused by the deformation of the particle itself; this part will be called dWd. The remaining part of dWs will be the virtual work done by the Q stresses during the remaining part of the displacement of the boundaries and will be equal to dWs-Dwd. However, this remaining is caused by the translation and rotation of the particle as a rigid body, and, as reminded above, the virtual work done in such a case is equal to zero. Hence

dWs=dWd.

If the virtual work done by the Q stresses on all particles of the body is now added, this equation becomes

Ws=Wd

To evaluate first Ws, we recognize that this term represents the total virtual work done by the virtual Q stresses on all the boundaries of all the particles. However, for every internal boundary of a particle there is an adjoining particle whose adjacent boundary is actual the same line on the body as whole, and therefore these adjacent boundaries are displaced exactly the same amount. Since the forces acting on the two adjacent internal boundaries are numerically equal but opposite in direction, the total virtual work done on the pair of adjoining internal boundaries is zero. Hence, since all internal boundaries occur in pairs, there is no net virtual work done by the forces on all the internal boundaries. Ws therefore consists only of the work done by the external Q forces on the external boundaries.

Wd, on the other hand, was obtained by integrating the virtual work associated with deformation of the element. This work includes the effects of all forces on the element, both stress resultants and external forces. However, when an element deforms, only the stress resultants perform any work. Thus, Wd represents the virtual work done by the stress resultants alone.

END OF PROOF

I understood the part concerning Ws.

However, I don't understand why "when an element deforms, only the stress resultants perform any work".

With respect to Wd, there's this also I don't understand "Wd represents the virtual work done by the stress resultants alone". But, according to the paragraph pertaining to Ws, this work should be zero, since all internal boundaries occur in pairs and the stresses on one side of an internal boundary of a particle undergoes the same displacement as the common side of a neighboring particle and such two sides are subjected to equal and opposite stresses, respectively. So, this result would contradict the result concerning Ws, where the work would be non zero.

Where's the flaw in my reasoning?

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# On the proof of the Principle of Virtual Work

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