# Rotational elastic-brittle bonds BrittleBeamPrms vs local plastic Mohr-Coulomb condition

Good morning!

When working on application of ESyS to sandstones modeling, I encountered three problems.
Namely, they are as follows:
1) How to link local plastic Mohr-Coulomb condition to the moments and forces of rotational elastic-brittle bonds BrittleBeamPrms.
2) Does the model include Rankine tensile cutt-off condition.
3) Is it true that strength of rocks cannot be reproduced by the Mohr-Coulomb criterion at least in macro-scale (only for limited stress ranges it may be valid) and for that reason Hoek-Brown criterion was introduced some time ago.

I would be very grateful for any hints or suggestions how to figure out that issues.

Thank you and best wishes,
PK

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Dion Weatherley
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 Revision history for this message Dion Weatherley (d-weatherley) said on 2021-05-04: #1

Hi Piotr,

I'm not sure I understand your 3 concerns but I will have a go at making some comments:

1) How to link local plastic Mohr-Coulomb condition to the moments and forces of rotational elastic-brittle bonds BrittleBeamPrms.

BrittleBeamPrms assumes a Mohr-Coulomb failure criterion governing the maximal shear stress within a beam interaction, as a function of the normal (tensile/compressive) stress. The formulae for calculating the shear and normal stresses involves the forces and moments within the beam interaction, accordingly:

shearStress_ij = shear_Force_ij/A_ij + torsion_Moment_ij*R_ij/J_ij
normalStress = -1.*(normal_Force_ij/A_ij + bend_Moment_ij*R_ij/I_ij)

where A_ij is the cross-sectional area of the beam, R_ij is the radius of the beam, I_ij is the bending moment of inertia and J_ij is the twisting moment of inertial. These are just formulae from linear elastic beam theory and typically employed in bonded particle DEM models involving rotational degrees of freedom.

2) Does the model include Rankine tensile cutt-off condition.

There is an option to apply a tensile cut-off for BrittleBeam interactions. I have not yet found this to have any appreciable macroscopic influence on the mechanical properties of the bonded particle assembly though. In the interest of keeping the model simple, I tend not to employ such a cut-off at the scale of individual beam interactions.

3) Is it true that strength of rocks cannot be reproduced by the Mohr-Coulomb criterion at least in macro-scale [..] for that reason Hoek-Brown criterion was introduced some time ago.

Essentially, yes this is correct. When triaxial tests are conducted on core samples over a wide range of confining pressures, the failure envelope is observed to deviate from a linear Mohr-Coulomb failure envelope for higher confining pressures. The Hoek-Brown criterion is a purely empirical formula that mimics this observed, nonlinear failure envelope. To my knowledge, there is currently no fundamental basis for the Hoek-Brown failure criterion; nor a fundamental understanding of the observed nonlinearity of the failure envelope at high confining pressures.

I hope this helps a bit. When considering such issues, be mindful that the interaction laws (such as BrittleBeamPrms) apply at the micro-scale; the scale of individual interactions between DEM spheres. The macroscopic mechanical response of the bonded particle model is only partially determined by the choice of interaction law. The assembly (or network) of bonded interactions also has a considerable role in the macroscopic response. Different arrangements of DEM spheres, with differing size-distributions, will result in quite different macroscopic response. The Hoek-Brown failure criterion was devised to mimic the macroscopic response of rocks. It is not clear that such a criterion should be employed at the micro-scale, between two bonded DEM spheres. Indeed, there is also no guarantee that doing so would necessarily result in a nonlinear, macroscopic failure envelope. My personal investigations would suggest that will not be the case in general.

Overall your concerns are very legitimate and probing at the boundaries of knowledge regarding the mechanical response of heterogeneous brittle-elastic materials, as well as how best to model that using numerical methods such as the DEM.

Cheers,

Dion

 Revision history for this message Piotr Klejment (glaubiger) said on 2021-05-05: #2

Hello Dion,

Thank you very much for your valuable response!
This issues arose when I tried to compare DEM simulations with laboratory geologists working on the strength of rocks. Your answer is very brightening!

Thank you very much,
Piotr Klejment

 Revision history for this message Piotr Klejment (glaubiger) said on 2021-05-05: #3

Thanks Dion Weatherley, that solved my question.