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Zhigang Suo's picture

The attached notes are written for a course on plasticity.  I will update new posts on my twitter account:  

A metal undergoes plastic flow as atoms change neighbors.  A liquid undergoes viscous flow as molecules change neighbors. But the plastic flow of the metal differs from the viscous flow of the liquid in an important way.  When a metal undergoes plastic flow, the stress depends on the amount of deformation, but is insensitive to the rate of deformation.   By contrast, when a liquid undergoes viscous flow, the stress is insensitive to the amount of deformation, but depends on the rate of deformation.  In studying plasticity, it is useful to recall the law of viscosity.

PDF icon viscosity 2014 10 05.pdf3.16 MB


Dear Zhigang,

1. Excellent verbal description in the blog post here, and also very good notes.

2. Just a minor aside: You could highlight one point that is often missed or mis-understood by students. (In any case, at least I did, for quite some time in the past!)

In fluids, the pressure is not a part of the viscous, or dynamic stress tensor, but exists independent of it.

In a stationary fluid, pressure of course exists. However, since all the velocity components are zero, so are the velocity gradients, and thus, the viscous (or dynamic) stress tensor components are all zero.

Now, when it comes to a moving fluid, then it too, can experience this static force of a pressure, and on top of that, now the viscous stress components also arise due to the relative motion of a fluid parcel with respect to the surrounding fluid. (That is to say, the zeroth-order term and the first-order terms are both nonzero and significant in a moving fluid. And of course, these are separate, independent, terms. Hence the independence of pressure and viscous stress tensor.)

Now, if one at all wishes to apply the idea of "pressure" also in the context of solids, then it has to be seen as a part of the stress tensor itself---it would be the hydrostatic part. But there is no equivalent of the real pressure (as the zeroth-order term), because there is no basic time-dependence, in the definition.

With that said, let me share a difficulty pertaining to explaining all this part to students. They can get the dynamic stresses in the moving fluid pretty easily, so long as the shear components of the viscous stress tensor are concerned. (They can apply the duster analogy---a duster moving on a table experiences surface friction, etc.) But the fact that there also are similar but normal stress components in the viscous stress tensor (i.e., the fact that these arise through the velocity gradients), is something they find harder to accept.

... Would you have something to add here?

I may try to explain it by invoking the Lennard-Jones fluid, and that might help. I may also try to explain it (I don't know, may be I would) by restricting the interaction to the shear mode, by introducing a second differential element rotated through 45 degrees, and then getting the shear stress on that element transformed back to the original one. That, too, might help, but I am not sure---it's too indirect.

But, anyway, since you have put these matters of fluids vs. solids in such amazingly simple terms in your post above, it made me sit and wonder: how would you describe the normal viscous stress in fluids, i.e. the one in the "head-on" (or "tail-off") context?




Zhigang Suo's picture

Dear Ajit:  Thank you very much for this comment.  I will try out these notes in class in a few days, and your comment puts an issue in sharp focus.  I was thinking how much I will talk about this issue in class.  

In a longer version of the notes on the basic equations in fluid mechanics, posted a few days earlier, I wrote a paragraph with the heading “The dilation of a fluid is viscoelastic”.  There I estimated the relaxation time for water, and found it to be on the order of 10^-12 s.  I then wrote a paragraph as follows.

“For many substances, as the temperature drops, viscosity increases steeply, but the elastic modulus does not.  It is conceivable that viscoelastic dilation can be important.  It is also conceivable that when viscoelastic dilation is important, viscoelastic shear is also important.  To describe shear, viscoelasticity of the Kelvin type is clearly wrong, and we have to invoke viscoelasticity of the Maxwell type, or some form of hybrid.  The matter is complex.  We should deal with viscoelastic fluids in a separate formulation of the theory”

I then added a paragraph with the heading “partial thermodynamic equilibrium”.  The model assume that the fluid and the external forces are not in thermodynamic equilibrium with respect to shear, but are in thermodynamic equilibrium with respect to dilation.

After seeing your comments, I think I should bring this issue up in class, and see how students react.

Zhigang wrote "When a metal undergoes plastic flow, the stress depends on the amount of deformation, but is insensitive to the rate of deformation."  This is not strictly correct and has to be qualified.  See for example our paper "An extended mechanical threshold stress plasticity model: modeling 6061-T6 aluminum alloy" B Banerjee, A Bhawalkar - Journal of Mechanics of Materials and Structures, 2008. Link ->

-- Biswajit

Zhigang Suo's picture

Dear Biswajit:  Thank you so much for this timely comment.  Indeed, rate-independent plasticity and viscous flow are two idealized models.  They are commonly used to describe metals and liquids, for reasons noted in my original note.  The two idealized models also have pedagogical values: They form a clear contrast between the strain and the rate of deformation.  

But idealization is not the reality, as you have pointed out.  The deformation of a metal does depend on the rate of deformation.  The best known case is creep at high temperatures.  Even at room temperature, the rate-sensitivity may have profound effect on the behavior of the metal.  I will describe an example in class:

A.K. Ghosh, Tensile instability and necking in materials with strain hardening and strain-rate hardening, Acta Metallurgica 25, 1413-1434 (1977).

A.K. Ghosh, Influence of strain-hardening and strain-rate sensitivity on sheet-metal forming, Journal of Engineering Materials and Technology 99, 264-274 (1977).

Hutchinson, J.W., Neale, K.W. " Influence of Strain-Rate Sensitivity on Necking under Uniaxial Tension." Acta Metallurgica , 25, 839-846 (1977).

The experimental data are striking (Fig. 1).

Dear Zhigang, Figure 8 from our paper will give you a taste of the unusual behavior observed in aluminum alloys.

-- Biswajit

Zhigang Suo's picture

I have just posted the updated notes on viscosity.  I took the suggestion of Ajit, and highlighted the difference between the applied hydrostatic stress and the thermodynamic pressure.  In this version, this discussion happens early in the notes, in the simple comtext of hydrostatic state.  I have also added many other items to improve clearity.

As always, your input is valued, and will help me imrove the notes further.

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