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Re: some questions (MacroMicroElasticRelationship)
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:
> 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.
> 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.
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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