# Plastic potentials/ Flow rules

I have a fundamental question regarding flow rules of finite plasticity models especially those used in soil mechanics. In most of the papers and books, I have seen the usage of an associated flow rule with the plastic potential similar to the yield surface. However, I am unable to understand the means of obtaining a non-associative flow rule. I am using Abaqus with cap plasticity model (modified Drucker Prager Cap model) to simulate powder compaction process. Abaqus uses a non-associative plastic flow rule definition in the shear yield space but doesn't provide any valid explanation for the equation of the flow potential used. The non-associated flow rule definition results in a lower dilation than that predicted by an associated flow rule.Is the flow rule equation something that is obtained arbitrarily and then fit using experiments or is it driven by plastic strain data obtained by compaction experiments?

If it is the latter case, then I am confused how does on obtained the plastic flow potential surface by plotting incremental plastic strains in the stress space. Wouldn’t it just give a qualitative estimate of the surface and one will have to come up with a proper surface by trial and error.

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## 1) if you work with soil,

1) if you work with soil, you will find out that not all material is obey assoc flow rule. Most does not dialate that much. For closer to reality, you need to use non-assoc flow rule.

2) material parameter can not come from "trial and error" period. For complicated problem, you probably won't get the same respond at all. However, you can propose an envolope that satisfy all physics requirements.

## Soil plasticity

You should take a look at Prof. Brannon's manual for the Sandia Geomodel where calibration issues are discussed. The physicality of a non-associated flow rule is still disputed, though it is a very useful abstraction for certain features of pressure-dependent plasticity.

Prof. Brannon's work can be found at http://www.mech.utah.edu/~brannon/public/GeoModel8.pdf

-- Biswajit

## Non-associative Flow Rule

Hi Biswajit,

Why does the physicality of non-associative plasticity disputed? Is it because of the violation of the max. dissipation theoerm? or the lack of normality? Would you mind to give us some reference about this topic?

Thanks,

WaiChing sun

## Non-associativity

I too am interested in the reasoning behind your statement that "The physicality of a non-associated flow rule is still disputed..." I know that such theories aren't as "nice" as associative theories and can "cause problems" from a thermodynamics standpoint, but I think there are physical justifications. In crystalline plasticity, consider the case of bcc metals whose plastic deformation is dominated by screw dislocations with "star-shaped" non-planar dislocation cores. In such materials, the "yield function" (stress state required to recombine the sessile dissociated dislocation into a full glissile one) depends on the resolved shear stress on several intersecting slip planes, but the "flow potential" is the usual one for crystal plasticity depending on which slip plane the recombined partials move in. In this case, the derivative of the yield function with respect to the stress is not the flow direction, and is therefore non-associative - see reference 1 for more details. This is just one example for metallic systems, but there are other physical reasons as well. The interested reader could refer to Section 3.1.2 "Non-associative flow and non-Schmid effects at various length scales" of the review article given in reference 2 below.

1. "Complex macroscopic flow arising from non-planar dislocation core structures," (2001) Bassani, Ito and Vitek, Mat Sci Eng A.

2. "Viscoplasticity of heterogeneous metallic materials," (2008), McDowell, D.L., Mat Sci Eng R.

## For granular materials,

For granular materials, usually you need to use non-associate plastic flow law, since many of them do not obey the associate flow rules. If you want to check the physical background, it will be better that you switch from the ABAQUS manual to the original paper about this model. Like this one,

Drucker and Prager (1953) Soil mechanics and plastic analysis or limit design. Quarterly of Applied Mathematics, 10:157-164.

If you look at the number of the parameters in the DPC model, e.g. slope and intercept of the shear surface, and the shape and size of the cap surface, it is difficult to determine all the parameters from simple test data. But you could find some correlations between these parameters...

## Thanks a lot for your

Thanks a lot for your comments. I will definitely look into the above mentioned references about the flow rules.

Tuhin

## Another way of looking at the non-associated problem

In order to understand what is going on here you have to look at the problem in a diffrent way. The easiest way is to start with a dissipation function and a dilatancy rule and then produce a yield function from them. I will explain how this can be done in a minute. The special property of granular materials (like soil ) is that the dissipation function is a function of the current stress and not just the strain increments. If it is not a function of the current stress then the procedure I will outline gives the associated flow rule.

When plastic deformation takes place the work work done equals the energy dissipated. If you guess values for the components of strain increment that obey the dilatancy rule then you have a surface in stress space. Pick other values of strain incremnt and you get another curve - pick enough and you get an envelope that represents the yield surface. Inside the yield surface the work done is too low to match the dissipation whatever the strain increment. This procedure can be done mathematically.

## another way to think of non-associative plasticity is friction

You can also see non-associativity easily in friction problems. Here, an associative rule would expect sliding to lift up with respect to the sliding plane.

Clearly, all plasticity coming from friction, like in geomaterials, is expected to show some effects of this.

Some references in the context of this start from Drucker 1954, to the Drucker medal lecture of Jim Barber in 2009, which Jim sent me and I can forward to you if you ask me !

Shakedown in elastic contact problems with Coulomb friction

International Journal of Solids and Structures,Volume 44, Issues 25-26,15 December 2007,Pages 8355-8365A. Klarbring, M. Ciavarella, J.R. Barber

Preview PDF (1095 K)

Shakedown of coupled two-dimensional discrete frictional systems

Journal of the Mechanics and Physics of Solids,Volume 56, Issue 12,December 2008,Pages 3433-3440Young Ju Ahn, Enrico Bertocchi, J.R. Barber

Preview Purchase PDF (275 K)

## Thanks a lot for your

Thanks a lot for your comments, overoll .