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Updated: 2 hours 6 min ago

mean stress vs. thermodynamic pressure

7 hours 56 min ago

In reply to Basic Equations in Fluid Mechanics

See a comment by Ajit on the difference between the applied stress and the thermodynamic pressure.

Recent experiments showed

7 hours 59 min ago

In reply to Representative Volume Element Calculations under Constant Stress Triaxiality, Lode Parameter, and Shear Ratio

Recent experiments showed that the Lode parameter, which distinguishes between axisymmetric and shear dominated stress states, has a profound effect on material ductility, especially at low stress triaxiality [2, 3]. Consequently, the theoretical framework for void growth and coalescence is currently being revisited, which often involve performing representative volume element (RVE) calculations. The present study investigates an RVE composed of a cubic unit cell containing a spherical void at its center. The void cell is subjected to a triaxial stress state with ∑11/∑22=ρ11, ∑33/∑22=ρ33, plus an additional shear stress component ∑12/∑22=ρ12. In the coordinate axes aligned with the edges of the cubic void cell, xi (i = 1, 2, 3), the non-dimensional stress ratios ρ11, ρ33, and ρ12 can be fully characterized by 3 parameters: the stress triaxiality, T, Lode parameter, L, and shear ratio, S. The aim of this paper is to provide an effective method to keep T, L, and S values constant in the entire course of the loading. The effectiveness of the proposed method is validated through several examples covering a wide range of T, L, and S values; the calculations are performed by using the general purpose finite element software ABAQUS.

Below, I attached the final version of the paper that I have submit to the International Journal of Solids and Structures. You may reach the online version of the paper from:

In a fluid does mean stress equal thermodynamic pressure?

8 hours 52 sec ago

In reply to Re. Viscosity: A minor aside

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.

Re. Viscosity: A minor aside

12 hours 40 min ago

In reply to Viscosity

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?




Hi! Have you solved this

Sun, 2014-09-21 11:46

In reply to Error in Ansys

Hi! Have you solved this "fatal error" problem? I have the same difficulties with my modell (reinforced semi preacast slab in case of erection).

--> nonlinear concrete (Miso)  Ex=30000N/mm2,  first point= 9/Ex, 9   [N/mm2].   So this doesnt cause this exit fatal error.

Any suggestions? (its my master thesis)

Hi all! Have you solved this

Sun, 2014-09-21 11:46

In reply to I am currently trying

Hi all! Have you solved this "fatal error" problem? I have the same difficulties with my modell (reinforced semi preacast slab in case of erection).

--> nonlinear concrete (Miso)  Ex=30000N/mm2,  first point= 9/Ex, 9   [N/mm2].   So this doesnt cause this exit fatal error.

Any suggestions? (its my master thesis)

Hi! Have you solved this

Sun, 2014-09-21 11:35

In reply to it's always greater than Ex

Hi! Have you solved this "fatal error" problem? I have the same difficulties with my modell (reinforced semi preacast slab in case of erection).

--> nonlinear concrete (Miso)  Ex=30000N/mm2,  first point= 9/Ex, 9   [N/mm2].   So this doesnt cause this exit fatal error.

Any suggestions? (its my master thesis)

Recording your lectures in class including discussion

Sun, 2014-09-21 11:09

In reply to YouTube mechanics vs. small classes

Hello Sir,

I wanted to say recording your lectures in class including discussion with students. 

Like this 'Lecture Series on Classical Mechanics' got more than 1 million views.

Classical Physics by Prof. V. Balakrishnan



YouTube mechanics vs. small classes

Sun, 2014-09-21 08:56

In reply to Free Video Lectures of Your Course !

Dear Dibakar:  Thank you so much for your kind words, and for the suggestion.  Our classes are rather small, typically around 10-20 people in the room.  The small classes allow conversations.  Students can ask me questions, and I also ask students questions.  My questions are usually directed to individual students.  The purpose is to engage the class, alert students possible difficulties, and make them picture a possible path of making the discovery.  This method of teaching may be not suitable for videos.  

Also, small classes allow me to experiment with new materials, such as a ten-minute description of deformation and fracture in lithium-ion batteries in a course on plasticity.  It also let me learn some old materials and try them out in class.  

But your suggestion about videos is really appealing.  I’ll find an effective way to do it at some point.

Suggested textbook

Sun, 2014-09-21 05:17

In reply to Plasticity

I would also suggest

Introduction to Computational Plasticity, by Fionn Dunne and Nik Petrinic

Did u got it?

Sun, 2014-09-21 03:16

In reply to Pressure load definition in ABAQUS when the surface is constantly changing

Did you got it? This is a micropipette aspiration type problem. I am having the same problem.. plz mail if possible.


Free Video Lectures of Your Course !

Sat, 2014-09-20 23:27

In reply to Plasticity

Dear Prof. Suo,

I have been following you for last one decade. You are a role model for many young mechanicians like me. 

If you record all your lectures and upload in YouTube for free, that will help many students around the world.  



Can anyone specify good books on Dynamics of structures

Sat, 2014-09-20 06:14

In reply to Lecture notes of interest to mechanicians


  Can anyone specify good books on Dynamics of structures and hydrodynamics?


Smartphone Thanks.  

Reminiscences of Ted

Fri, 2014-09-19 19:46

In reply to regarding Ted Belytschko

I consider myself very fortunate to have known Ted.  My Ph.D. advisor at N'western was Brian Moran, but I closely interacted with Ted, first as a graduate student and later on as a post-doc.  Brian gave me the latitude and support to explore new research avenues, and Ted kept me and others motivated to do research. The joint group meetings on Friday afternoons fostered learning and permitted many of us to interact on research.  We pick-up cues from our environment and what we observe, and Ted's style, presence, and feedback greatly enhanced our collective graduate experience.  His simple and direct writing style, eye for what's important, and how to conduct `methods development research' has had a direct impact on how many of us have shaped our own research.

Ted led by example -- when it comes to being committed to the academic profession, and to emphasize that there is no substitute for perseverance, dedication and hard-work.  I cherish the many memories of our interactions during my days at N'western, and later on whenever I have met him at conferences. He will be missed.

Too many attempts ...

Thu, 2014-09-18 16:17

In reply to [SOLVED] 3D crack growth modelling in Abaqus by XFEM

The default parameter of attempts per increment ist 5. You can increase the number in CAE by clicking on:

Model ==> Steps ==> "Step-1" (or how you called the step after the initial step with the loads) in the model tree on the left with double-click

==> the menu bar on the top changes ==> click on "Other" ==> "General Solution Controls" ==> "Edit" ==> "Step-1"(or the other name)

==> "Specify" ==> "Time Incrementation" => "more" (the first of the three) ==> parameter "IA" ==> increase this parameter

Ted Belytschko

Thu, 2014-09-18 16:01

In reply to About Ted Belytschko

It was with great sadness that I received the news on Monday. I had the chance to benefit from Ted's teaching and to attend the inspiring weekly group meetings on Fridays at Northwestern during my PhD with Brian. Ted taught us advanced finite elements and I think that his charisma and personality were a motivator for many to follow a research career in Computational Mechanics.


A two minute corridor discussion with Ted could be more revealing than a week of reading. His incisive brain and the passion he put into work were truly inspiring. Being a PhD student in the TAM lab, I did not yet comprehend the outstanding level abnegation that Ted showed for his students and close collaborators. Almost 15 years later, I am most admirative of Ted's caring style. He would visit most of his students every morning directly at their desk and support them in their work, propose new ideas, share his recent (possibly nightly) reading of a book. Realising the scarcity of academic's time, I realise now what these daily visits meant and I am most admirative.


My own students heard me many times use Ted as the archetype of a caring mentor.


Farewell, Ted, we miss you.


Hello Zheng,

Thu, 2014-09-18 10:58

In reply to convergence difficulty

Hello Zheng,

I have also convergence problems with using XFEM in Abaqus that are quite similar to the ones of the questioner. Can you send your offered PDF for improving the convergence in XFEM?

My email is:





It is maybe a bit late, but I

Thu, 2014-09-18 10:51


It is maybe a bit late, but I learned using XFEM in Abaqus by Matthew Pais in "".

There are a lot of examples with tutorials (step-by-step in CAE), the cea-files and the inputs, that you can use for learning it. He works with 6.10.

I think, you already have learned it within the 2 years :-)

Looking forward to read it.

Thu, 2014-09-18 10:22

In reply to A future review article in Advances in Applied Mechanics

Hi Stéphane,

Thanks for pointing out the article to me. I would love to take a look when the book is available.

BTW, I wonder if you have encounter similar inf-sup condition issue in isogeometric models especiually when fomrulating something that leads to a block system (e.g. incompressible elasticity with displacement and hydroastatic pressure as nodal solution)? Of course the basis functions are different between the conventional FEM and isogeometric Galerkin. Any important literature we should be aware of?

Best Regards,



About Ted Belytschko

Thu, 2014-09-18 04:20

In reply to Ted Belytschko passed away

As one of Ted’s PhD students, I feel devastated by this loss. Many of the comments and stories written here resonate with my own experience with Ted. He was an extraordinary researcher, teacher and mentor. It is difficult for me to express how genuinely these words are felt. 

Ted’s creativity, intuition or sense of humor were truly unique. He addressed complex and important problems with simple good ideas, going straight to the point. When developing his ideas--as in the group meetings--, or communicating them, he was able to combine contagious excitement with skepticism about his own work. This created a fantastic atmosphere to develop research. Despite being an extraordinarily accomplished researcher, he did not miss the opportunity to express his admiration for other people’s work. 

Ted was full of life, and he will be sorely missed.



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