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Re: some questions (MacroMicroElasticRelationship)


Hello Sega

I'm not the author of those relations, but I'll do my best to clarify a 
little bit (is Frederic Donze around to give better explanations?).

"/therefore, in my opinion, the contact stiffness (kn, ks) and the 
viscous damping coefficients (cn,cs) should be calculated proceeding from   
the coefficient of restitution and duration of impact."/

There is no such damping coefficient in current contact laws. There is 
only a numerical damping, called "non-viscous damping" in Cundall's 
papers, which applies at the level of resultant forces and moments on 
each particle (see damping classes).

> /I have found in the Annual Report 2006 of Discrete Element Group for Hazard 
> Mitigation, page E5
> (http://geo.hmg.inpg.fr/frederic/Discrete_Element_Group_FVD.html) expressions 
> for kn, ks: 
> if 
> E = D/S * Kn * ( beta + gamma * Ks/Kn ) / (alpha + Ks/Kn)
> v = (1 - Ks/Kn) / (alpha + Ks/Kn)/
Is it clear for you that in the equations above *E and v are elastic 
parameters of a sphere packing*, measured in a simulated test? I guess 
alpha, beta, gamma are ad-hoc coefficients that fit the curves obtained 
after a parametric study of E/kn =f(ks/kn) and v=f(ks/kn) in a simulated 
triaxial test (otherwise, i don't see how 3 coefficients can be computed 
from 2 equations!). Note that these relations probably depend on the 
compacity of the packing and particle size distribution.

> /hence
> Kn = E * S/D * (1 + alpha) / [ beta * (1 + v) + gama * (1 - alpha*v) ]
> Ks = Kn * (1 - alpha*v) / (1 + v)/
Here, kn and ks are constants stiffness coefficients that WILL give you 
given values of macroscopic elastic moduli Em and vm. Hence, the model 
is not based on Eg, vg of the grains, it is based on Em, vm that you 
want to obtain at the scale of the packing. This is in fact a common 
situation when you want to simulate soil or concrete.

> where alpha, beta, gamma is the parameters will be identified; E, v is Young's 
> Modulus and Poisson ratio.
> However, I on former do not understand as they have turned out and how can be 
> used in the linear contact model 
> Fn = kn * xn, 
> Ft = kt * xt 
> where xn - depth penetration, xt - relative tangential displacement.
I'm not sure I understand the problem. If constants kn and ks are 
defined, then you can compute the forces based on 
positions/displacement, there is nothing more here (note that second 
equation is in fact used in an incremental vectorial form : dFt = kt * dxt).

> I'm interested to that I write the PhD thesis about modelling the granulated 
> materials in which there is a review of various models of interaction: linear 
> and nonlinear viscoelastic models (Cundall*Strack, Kuwabara&Kono), 
> Hertz theory, elastoplastic models (Walton&Braun, Thronton), linear and 
> nonlinear tangential interaction (Mindlin&Derisevich, Walton&Braun). 
The model of interaction in ElasticContactLaw in Yade is linear elastic 
in normal direction, and linear elasto-plastic in tangential direction, 
exactly the same as in most Cundall's papers. The relations with alpha, 
beta, gamma is just a trick to choose the values of kn and ks.
> In the linear models factors of elasticity and dissipation are deduced from 
> the decision of the differential equation of pair interaction and  
> __empirically__ defined parameters, such as coefficient of restitution and 
> duration of pair impact. In the nonlinear models constructed on the basis of 
> the theory of elasticity Hertz, the given parameters define on the basis of 
> __clearly interpreted physical parameters__ of a body: the Young's Modulus 
> and Poisson's ratio.
Again, to be sure it is clear : there is no dissipation at contact, and 
there is nothing like a restitution coefficient or a priori duration of 
impact in Yade, appart from the ones you can observe in the results of a 
simulation. There is just kn, ks, and numerically damped Newton's laws.

I hope it helps.


Chareyre Bruno
Maitre de conference

Institut National Polytechnique de Grenoble
Laboratoire 3S (Soils Solids Structures) - bureau I08
BP 53 - 38041, Grenoble cedex 9 - France
Tél : 33 4 76 82 52 76
Fax : 33 4 76 82 70 00

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