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[Honza Stránský <_honzik@xxxxxxxxxx>]

Hi Nasibeh,

imagine two bars (or rods) a and b in serie. Rod a has cross section area A, length La and material stiffness ka, rod b is anlogous.
Let's define rod extension dL, strain e, stress s and apply force F.

F <--- o ---rod a--- o ----rod b--- o ---> F

e = dL/L
s = k.e = F / A
kL = k.A / L is rod stifness

because rods are in serie, Fa = Fb = F -> sa = sb = F/A = s 

dLa = s.La / ka = F / kaL
dLb = s.Lb / kb = F / kbL

dL = dLa + dLb = s.(La / ka + Lb / kb) = F.(1 / kaL + 1 / kbL)

from now, there are two ways how to define equivalent stiffness:

a) equivalent rod stifness
kL = F / dL = F / [ F.(1 / kaL + 1 / kbL) ] = kaL.kbL / (kaL + kbL)

b) equivalent material stifness
k = s.L / dL = s.(La + Lb) / [s.(La / ka + Lb / kb)] =
   = (La + Lb) / (La / ka + Lb / kb)
under assumption La = Lb
k = 2.ka.kb / (ka + kb)

Is this explanation at least a little understandable and useful? :-)

an example: let's have two springs (rods) made of steel with Young's modulus Ea=10. the rod stiffness of each spring is kLa = EaA / La = 10*2/5 = 4. If You put them into a serie and apply force F, the extension of such system would be twice as big as the extension of one spring. The rod stifness of the system is halved: kL = kLa/2 = 2. On the other hand, the value of the material stiffness E remains the same, E... You can try to apply aforementioned formulas :-)

In Yade, (I suppose :-) the stiffness is considered as the material stiffness, and therefore the value of the contact stiffness is computed using harmonic mean..

Best wishes

Honza

______________________________________________________________
> Od: "Nasibeh Moradi" <nasibeh.moradi@xxxxxxxxx>
> Komu: yade-users@xxxxxxxxxxxxxxxxxxx
> Datum: 19.04.2010 10:54
> Předmět: [Yade-users] harmonic average of stiffness
>
>hi,
>
>in contact law, harmonic average is used  for computing kn and ks in contact
>between 2 body, why harmonic average?
>in some papers, the springs are in serie, then k=ka*kb/(ka+kb),
>but in harmonic average k=2*ka*kb/(ka+kb)
>
>Nasibeh Moradi
>
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