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representative elementary volume of non-local continuum

WaiChing Sun's picture

Dear mechanicans,

        For non-local continuum, is there a proper approach to determine the size of the REV in experiment? For the classical linear elastic continuum, one can measure the homogeneized stress and strain (or local averaged stress/strain)  and compute a homogeneized elastic constitutive tensor. However, I am not sure how to do it for non-local continuum, since the constitutive response is now sensititve to the gradient term. Any comment/suggestion is appreciated. 


 WaiChing Sun



Reza Azizi's picture

Dear WaiChing

For non local homogenization, the Hill-Mandel condition has to be satisfied.
This condition just guarantees that the energy in micro scale is as same as the
one in macro scale. Note that a part of energy comes from the higher order
traction.  The result of this equivalency gives an equation to
homogenize the micro stress and higher order stress. This point is written in
the thesis of VARVARA KOUZNETSOVA in Technical University of Eindhoven (Number
ME97020). Let me know if you need more details.



WaiChing Sun's picture

Hi Reza,


     Thanks for your information. I will  check out the reference. 





WaiChing Sun's picture

Hi Reza,

   I have quickly read the papers from Dr. V.G. Kouznetsova's webpage. I think these papers provide the information I needed. According to my understanding, the first order homogenization scheme based on Hill-Mandel condition lead to a RVE with homogenized strain field, which does not vary within the RVE. Meanwhile, the second order homogenization scheme of Mindlin's type does allow linear variation in the macroscopic strain within the RVE and therefore allows one to determine the non-local terms in the constitutive model. 

   I do, however, have one additional question.  In the paper titled "Multi-scale computational homogenization: Trends and challenges" by Geers, Kouznetsova and Brekelmans, it is stated that even the second order homogenization scheme is not suitable for the completely softening analysis (at the end of section 3). Why is that? As far as I know, the softening is a necessary condition for strain localization in assoicative elasto-plastic materials. Therefore, does that mean the 2nd-order method is not suitable for the domain where shear band forms? 

Thanks for your kindly help.

 With Regards,



Pradeep Sharma's picture

A while back, I read a paper by Drugan and Willis on this topic. I think it is quite good. Here is the reference. I think there were some follow-up works too but I have not kept track.

Walter Drugan, John Willis, 1996, JMPS, A Micromechanics-Based Nonlocal Constitutive Equation and Estimates of Representative Volume Element Size for Elastic Composites, 44, 497


 As you mentioned this paper, I would like to draw your attention to my recent paper, and point out a fundamental mistake of the Drugan&Willis paper on RVE (in my opinion of course), and its claim that " the minimum RVE size is at most twice the reinforcement diameter for any
reinforcement concentration level".  The claim obviously conflicts with almost all other theoretical works and all experimental results (actually my paper shows that the minimum RVE size can be as large as containing millions of spheres for densely packed cases). The real problem roots in the misunderstanding of randomness of microstructure in a RVE. More discussion of the mistake and the real RVE size were given a recent paper -

Xu and X. Chen, “Stochastic
Homogenization of Random Multi-phase Composites and Size Quantification of
Representative Volume Element
”, Mechanics of Materials (2009) 41 (2)

Any critique of my viewpoint and/or discussion of the minimum RVE size are welcome, which should certinaly contribute to more understanding of this important issue.  

X. Frank Xu


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