Hi Reza,

To me the issue is not so-much associativity vs non-associativity as is rate independent vs rate dependent. One can always formulate yield criteria for shear deformation (conceptually in the same way as you formulate them for tensile conditions) and use an associative flow rule. However you still have to deal with the use of the plastic multiplier, which to me introduces a pretty high degree of ad-hocness. Instead, formulating a true flow rule that depends on the slip rates is a better avenue in my opinion.

I will add that according to my moderate experience the rotation tensor appearing in simple shear can lead to considerable difference in behaviour between simple and pure shear. That is exactly the case when two flow rules lead to noticeable differences. You can find my paper " Continuum models in dynamics of granular media. Review", where some remarks on the influence of rotation tensor are given (unfortunately in Russian).

Dear Reza,

A truism first: which model you use, depends on your material. Pressure dependence does not necessarily requires a non-associated flow rule (think of a porous metal with a matrix obeying the normality rule); furthermore, even when a material is not damaged but pressure sensitive, its flow rule may be quite close to a normality rule (in fact, all cubic lattices enjoy a particular form of normality). In a bit of a friendly contrast with Jaime's  opinion, I would say that there are differences between associate and non-associate models: the predicted deformations follow different rules !  And the plastic multiplier is a direct consequence of the elasticity of the material - no... "ad-hocness" (the way its magnitude is calculated - consistency condition - may be subject to debate, but this is rather an exotic topic).

The first question is then: does your material require a non-associate flow rule ? If yes, then answering  your question reduces to choosing particular constitutive elements (yield surface and flow potential, as simple as possible at first) and then to a straightforward derivation of the stress-strain relationship (in rate form). This should be fun  - good luck and ...have fun ! 

Dear Stefan

Looking at behavior of concrete materials and pressure sensitive constitutive behaviors in form of Drucker Prager and not Gurson, we usually are using a non-associate flow rule. However, there are some famous damage-plasticity models like Duvaut-Lion (used in LS-DYNA), using associated flow rule, which is not the perfect case in my opinion, but is has been used by many people. I want to take a look the non-trivial differences between this type of material models and Lee-Fenves type of damage-plasticity models under different loading scenarios in terms of dilatancy behavior, localization properties and also plastic work density before, during and after peak by doing a “numerical experiment”. This will help me answer the first question: why we have to choose non-associate flow rule?

Dear Reza,

It seems to me that you already have a sound research program !

I've never worked on the modeling of concrete, so my advice comes by analogy with what I've encountered in metal plasticity.

The devil lurks in the details. That is, associated or not, if a model satisfies some basic requirements (positive dissipation, fair approximation of the flow direction, etc),

then the broad features of the deformation and even failure of the target material will be captured. In general, this is satisfactory enough for practitioners, i.e. industry,

and perhaps that's why a well-established model like Duvaut-Lions continues to be a preferred work-horse. If fine details and accuracy are important, then you have a motivation for further investigation. However:

1) Simple comparison between numerical predictions may not be sufficient for discerning which model is most appropriate (what if you added more parameters on a model ?);

2) An experimental data set that cannot be described accurately by a certain class of models (no matter how many parameters they are endowed with) may give a good hint about the structure of a better model;

3) Ultimately, the answer lies in the physics/morphology of your material;

homogenization arguments could shed light on the problem.

Good luck !

Thanks for the comments. I am not sure if I could understand the homogenization part.

Well it is a case study on applications of Bromwich theorem in symmetric vs non-symmetric structural stiffness matrices to predict the localization in constitutive level, and associate vs non-associate flow rule seems good example of this induced non-symmetry. I hope I present different aspects my results in upcoming ASME and/or FRAMCOS.

Dear Reza,

Sometimes it is possible to infer the structure (that is, the constitutive elements and the relationships between them) of the overall response of a representative volume element (rve), from that of its constituents, by employing some averaging scheme. Concrete is a composite material, and hence its macro-response (what is modeled at macro-scale by phenomenological models, such as the ones you mentioned), at least its ("algebraic/analytic") structure, should be somehow related to that of its constituents.

Best