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Contact Model and Contact Stiffness


 Dear All,
I am losing myself a little to understand the contact model and contact stiffness used in YADE.Is it Linear Contact Model at present? I am trying to explain my question more details as follows,first explanation then my question.
01) Belheine.N et.al "Numerical simulation of drained triaxial test using 3D discrete element modeling"Computers and Geotechnics,2008
Kn=(kn(A)*kn(B))/(kn(A)+kn(B)))/r   page number 03 (Article in press)
02) Kozicki.J et.al."A new open-source software developed for numerical simulations using discrete modeling methods"
kn=r*(Kn(A)*Kn(B))/(Kn(A)+Kn(B)))  page number 04
03) PFC3D,Theory and Background,Page Number 2-2,Article 2.1.1
kn, normal secant stiffness=Kn,normal tangent stiffness
>>My question: what is the significance of "r" average radius,what are the difference among reference 01,02,03.
Now inside of YADE (12.rc1)
04) pkg>dem>engine>engineunit>SimpleElasticRelationships.cpp
          //Real Eab      = 2*Ea*Eb/(Ea+Eb);
          //Real Vab      = 2*Va*Vb/(Va+Vb); 

            Real Dinit      = Da+Db;                        
          //Real Sinit    = Mathr::PI * std::pow( std::min(Da,Db) , 2);

             Real Kn = 2*Ea*Da*Eb*Db/(Ea*Da+Eb*Db);//harmonic average of two stiffnesses                        
             Real Ks = 2*Ea*Da*Va*Eb*Db*Vb/(Ea*Da*Va+Eb*Db*Va);//harmonic average of two stiffnesses with ks=V*kn for each sphere

  //    Real alpha      = sdecContactModel->alpha;
 //     Real beta       = sdecContactModel->beta;
//      Real gamma      = sdecContactModel->gamma;
 //      Real Kn         = abs((Eab*Sinit/Dinit)*( (1+alpha)/(beta*(1+Vab) + gamma*(1-alpha*Dab) ) ));
//      Real Ks         = abs(Kn*(1-alpha*Vab)/(1+Vab));
>>In 04 and 05 the equation is different,can anybody explain a little the difference and significance of 04 and 05.
Thanks in advance.
Mohammad Nurul Islam

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