Will the virtual work be valid in finite deformation?

The short answer is yes. From a mathematical perspective, "virtual work" is just a projection of the equations of motion on an appropriate space of functions. The name "virtual work" comes from the fact that these "test" functions may be interpreted as variations of the motion about the current configuration. In some fortunate cases, the linear elasticity being the most notable case, a variational (stationarity and even optimal) principle may be associated with the virtual work (or weak) form of the equations of motion. Beautiful books on the subject are:

1) Belytscho et al: Nonlinear finite elements for continua and structures.

2)Bonet and Wood: Nonlinear continuum mechanics for finite element analysis.

3)Ciarlet: Numerical analysis of the finite element method (or The finite element method for elliptic problems)

4)Simo and Hughes: Computational inelasticity.

5)Oden: Finite elements of nonlinear continua.

Hi, Weijie,

The answer is yes. I wrote a short note on the principle of virtual work in terms of nominal stress and deformation gradient tensor. Hopefully it helps. Please find the note here.

BTW, you can always write the principle of virtual work with other finite stress/deformation measures. For more information, please refer to Prof. Alan Bower's online book

Best,

Dear Weijie,

First of all, good that you raised this question.

I noticed your question some time yesterday or the day before, and thought, sure, this one needed to be promoted to the main page. ... Though, I initially actually wasn't sure what to think of it. ... As far as I can tell, there is a certain deep mathematical sense to your query.

There have been what I guess are really good replies. ... Good enough to allow me to think of trying out an ``answer''! [Even if only at iMechanica!!]

Let's take a rather extreme case.

Let's suppose, that the constitutive law isn't just nonlinear, it's totally random. [I'been thinking of such things just recently---not in the solid mechanics sense, but anyway... Say, springs and load-vs-deformation rather than stress-vs-strain.] ...

Say, the constitutive law (y = f(x)) begins from zero (y = 0 at x = 0), and then, goes haywire. Sort of like the special yield-point behavior in mild steels due to the ``environment'' by the C atoms---except that, here, the initial linear portion is absent. And, except for the fact that it must go through the origin.

So, it's a really random constitutive law.

If truly random, differentiability will be lost; cf. chaos theory.

But, integrability won't be. [And, hence, here, the mathematicians can now feel bold enough to enlighten the rest of us all.]

So long as the curve y = f(x) is continuous and finite in ``y,'' you can always integrate it to a finite value. (Here, I mean definite integrals.)

And, if you are like real mathematicians, you can also set the resulting value equal to that of some other curve of whatever form you like, and therefore assert that the two thereby become ``math-wise'' [sic] equal.

For instance, a horizontal line having the same area as the area under that random curve.

And, well, coming back to your question, the principle of virtual work applies to an integration. $\int \vec{F} \cdot dx$.

Guess that means that it does it. But don't know, really speaking.

... Would love to know if there are any arguments going counter to the above story.

Thanks, anyway, for a neat question.

Best,

--Ajit

[E&OE]

The name "virtual" is misleading. Virtual displacement or virtual velosity are just "test functions“ with which we can reformulate the strong form (governing PDEs) to weak form (variational) by integration by part. In linearized elasticity (small deformation in this case), fortunately, the test function has the form of linearized strain. There is another explaination for linearized elasticty that the weak form can come from the variation of total potential energy.