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Re: some questions (MacroMicroElasticRelationship)


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)


Kn = E * S/D * (1 + alpha) / [ beta * (1 + v) + gama * (1 - alpha*v) ]
Ks = Kn * (1 - alpha*v) / (1 + v)

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 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). 

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.

Therefore it was very interesting to me to learn how it is possible to use 
Young and Poisson in the simple and attractive to calculations linear models 
of interaction.
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