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Will the virtual work be valid in finite deformation?

Weijie Liu's picture

"We see from the above that the virtual work statement is precisely the weak form of equilibrium equations and is valid for non-linear as well as linear stress-strain (or stress-strain rate) relations." --- written by O.C. Zienkiewicz etc. at Page 71 of <The Finite Element Method: Its Basis and Fundamentals  Sixth edition>


Here is my question: Will the virtual work still be valid for the case of non-linear strain-displacement relations (finite deformation)?

Because during the calculations of weak form in the book, the linear strain-displacement relation is assumed.


If someone gives me a note, a key word or an explanation, I'll be appreciate that.


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.

Zheng Jia's picture

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


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.





Weijie Liu's picture

Dear All,

I am grateful for your comments.

If I understand right, Zienkiewicz used the virtual displacement principle to make an example for case of small strain; and correspondingly the virtual velocity principle will be used for case of finite strain expressed in Cauchy Stress.


Bonne journée.

Zheng Jia's picture

Virtual displacement and Virtual velocity principle are actually the same thing. Both can  be applied to cases with small/finite deformation.



Weijie Liu's picture

Hi Zheng,

I do agree with you that both the two principles belong to virtual power principal. Clearly, the differences are the two kinds of test functions.

But in my opinion, the virtual velocity principle works well for both small and finite deformation; not the virtual displacement principle.

I suppose that we always have the equations in rate form in finite deformation instead of total strain,

as you can find that the rate form is adopted in Prof. Alan Bower's online book which you provided above.


Zheng Jia's picture

Hi, Weijie,

I like your thinking process. Please see Zhigang's Lecture on elasticity with finite deformation and search the key word: virtual work. The virtual work principle there is derived based on virtual displacement for finite deformation. The link is given below:

Virtual velocity and virtual displacement both works for small and finite deformation. If you express energy variation by the variation of work, you need to use virtual displacement; if you express energy variation by variation of power, you need virtual velocity. The reason that Dr. Alan Bower chose virtual velocity is he would like to prove the virtual work principle in terms of power. It is not because virtual velocity principle is the only right way for finite deformation.

I guess you have a strong FEM background and in that case I would suggest you to stick with virtual velocity since 1) it's consistent with FEM framework and thus convenient to use and 2) you trust it:). BTW, Dr. Alan Bower worked a lot on FEM coding. I assume that's why he preferres virtual velocity in his textbook. Just my 2cents.

Best regards,

Junwei Xing's picture

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. 

Junwei Xing's picture

In adition, for linearized elasticity as I learned when I was an undergraduate the teachers always said the virtual displacement must be samll. But now I do not think so. Test function can be abitrary provided it belongs to a certain functional space. 

Weijie Liu's picture

Test function can be abitrary only under the constraint of kinematic admissible (boundary conditions).

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