Yade

CPMMat Stiffness formulation versus other cohesive laws

Asked by loiseaurare

Hi everybody,

I have been browsing through the description of different cohesive contact laws, and I am now trying to understand the differences between JCFpmMat, CPMMat and cohFrictMat.

My question is on the formulation on the contact normal stiffness Kn,

I read in the JcfPMMat and CPMMat description that this is defined by the function kn = 2*EcRaRB/(Ra+Rb)
Ec being the "Young modulus" parameter introduced in each of those contact laws, that actually does NOT define macroscopical youngs modulus. Ra and Rb being the radiuses of the two interacting particles.

In Dr Smilauer thesis, this formulation is introduced in a more general way, stating that kn in the algorithm is computed with the formula :

kn = k1k2/(K1 + K2) => kn = E1l1~*E2l2~/(E1l1~+E2l2~) . Assuming that E1 and E2 are the same between the two particles, this yields

kn = E(l1~*l2~)/(l1~+l2~)

It is then precised in the thesis that for 'The most used class computing interaction properties Ip2_FrictMat_FrictMat_FrictPhys uses ̃ l i~= 2ri. "

I think that Cohfrictmat and JcfPMat also use this formulation, am I correct ?

Then the concept of the equivalent cross section is introduced, as being another way to define the li~ length.

"Some formulations define an equivalent cross-section A eq , which in that case appears in the li~ term
as Ki = E i * l i~ = Ei Aeq/li . Such is the case for the concrete model (Ip2_CpmMat_CpmMat_CpmPhys)
described later, where A eq = min(r 1 , r 2 ) "

And that is where I get lost. I do not understand the dimension of this Aeq over li quantity. I think that in the way it is expressed here, it actually has no dimension, where it should have a m dimension ?

If anybody can give a hint, I will appreciate !
Thanks in advance,
Manon

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Robert Caulk (rcaulk) said : 		#1

Hello Manon,

>And that is where I get lost. I do not understand the dimension of this Aeq over li quantity. I think that in >the way it is expressed here, it actually has no dimension, where it should have a m dimension ?

You've found a typo in the DEM background/Václav's thesis. It should read "A_eq = pi * min(r1, r2)^2" since that is in fact what is used in the concrete model [1]. The unit of stiffness is force/displacement or in SI, N/m. So in this case, the units work out since E Pa * A_eq m^2/ l_i m -> Pa*m = N/m.

I am curious though why he chose to use min(r1, r2). This seems like it would skew the interaction stiffness distribution left resulting in a less stiff specimen. At the end of the day, we always calibrate the micro-properties to experimentally observed macro behavior. In this case, we'd just have to compensate for that skewed distribution with a different combination of micro-parameters. Idk maybe someone can shed light on that.

[1]https://github.com/yade/trunk/blob/dafe23a8e34ab581edc0425d28290fc5ba591ce8/pkg/dem/ConcretePM.cpp#L315

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loiseaurare (loiseaurare) said : 		#2

Hi Robert,

Thanks for the answer on the Aeq quantity, I suspected something like that but was unsure...

Well, you've asked my second question, I also wonder why in the formulation of this contact law he chose to use the minimum of the two. Maybe the CPM mat was meant to be used for materials with an homogeneous size distribution in the particles ? Or to model granular material at a scale where particles do not represent material particles, and thus might not need a specific size distribution. Then the difference of radius between two given spheres could be neglected ?

However, since in this formulation the li~ equals Aeq over li, I am right in assuming that the li distance corresponds to ri ?

Cheers,
Manon

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Jan Stránský (honzik) said : 		#3

Hello,

as Manon said, CPM was designed for particles with the same radius (because particles was just artificial discretization and not representing physical grains). So if min is good or not was not actually tested..

The main difference is that CPM assumes Aeq = pi*r*r, while Ip2_FrictMat_FrictMat_FrictPhys (if I remember correctly) assumes Aeq=r*r..

> However, since in this formulation the li~ equals Aeq over li, I am right in assuming that the li distance corresponds to ri ?

yes

cheers
Jan

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