Meaning of PoissonsRatio in BrittleBeamPrms

Hi Robert,

The rotational interactions described in Wang et al. (2006) and a few subsequent papers by Dr. Yucang Wang, are "generalised" 3D rotational bonded interactions, of which BrittleBeamPrms (...) is a particular simplification.

To be more specific, Dr. Wang's formulation coincides with the RotBondPrms interaction group which requires specification of eight model parameters, namely:
* normalK = elastic stiffness for compression/tension
* shearK = elastic stiffness for shear
* bendingK = elastic stiffness for bending
* torsionK = elastic stiffness for torsion
* normalBreakForce = maximum tensile/compressive force
* shearBreakForce = maximum shear force
* bendingBreakForce = maximum bending moment
* torsionBreakForce = maximum torsion moment

The first four model parameters determine the elastic properties of the interaction between two particles. The second four parameters determine when a bond will break via a generalised Mohr-Coloumb failure criterion described in Dr. Wang's papers.

Dr. Wang's formulation is quite "general" in the sense that one can explore quite a broad (eight-dimensional) parameter space for the relative impact of normal/shear/bending/torsion components of deformation between two particles. However, Dr. Wang's formulation is also largely only applicable for models comprised of spheres of equal size. For models comprised of spheres with a range of sizes (the usual case for ESyS-Particle simulations) the general formulation (based on elastic stiffnesses) is not scale-invariant. It is now quite well-known in the DEM community that interactions that use a constant elastic stiffness (e.g. normalK in Newtons/metre) to bind particles of differing radii, results in a macroscopic particle-size dependency of the elastic properties (e.g. Young's modulus) of the particle assembly. By contrast, a linear elastic material does not have such a size-dependency (elastic properties are scale-invariant).

It is quite easy to eliminate the particle-size dependency from DEM models, simply by assigning a constant bond Young's modulus between bonded particles, and then assigning the elastic stiffness (such as normalK) as a function of the constant bond modulus and geometrical properties that depend on the radii of the two bonded particles. More specifically, one must assume the bonded interaction between the two particles has a particular shape (i.e. an equilibrium length, L and a cross-sectional area, A). The linear elastic beam theory predicts that the elastic normal stiffness (Kn) is related to the bond modulus (E) by:

Kn = E A / L

similarly the shear, bending and torsion stiffnesses can be computed from linear elastic beam theory.

The BrittleBeamPrms interaction group is thus a simplification of Dr. Wang's rotational interaction model; one that is also scale-invariant. For BrittleBeamPrms, I have assumed the interaction between two particles is a cylindrical beam whose equilibrium length is:

L = R1 + R2

and cross-sectional area is:

A = pi (R1+R2)^2 / 4

i.e. the effective radius of the cylindrical beam is the arithmetic mean of the two particle radii (R1 and R2).

Applying these assumptions and linear elastic beam theory, permits one to define the four elastic stiffness parameters as functions of only two elastic properties - the bond Young's modulus and bond Poisson's ratio.

If one further assumes that these beam interactions break according to a simple Mohr-Coulomb failure criterion (governed by two model parameters: cohesive strength, C and internal friction angle, phi), the four breakage forces can be computed as functions of C and phi. Consequently, only four model parameters must be specified for BrittleBeamPrms.

So in answer to your question, BrittleBeamPrms assumes particles are connected by cylindrical beams and the relationship between prescribed elastic properties (youngsModulus and poissonsRatio) and Dr. Wang's four elastic stiffnesses, is given by linear elastic beam theory. BrittleBeamPrms is consequently less general than RotBondPrms but in practical terms, much easier to use, given:
A) the reduced number of model parameters requiring calibration, and
B) the scale-invariance of macroscopic elastic properties this simplification affords when using models with particles of variable radius.

I hope this helps.

Cheers,

Dion

Hi Dion,

Thanks for your answer. I take it, then, that the bond parameters used by the code to influence and control bonded particle motion are Kn, Ks, Kb and Kt, but that these are calculated from E (Young's modulus) and nu (Poisson's ratio), the parameters input into BrittleBeamPrms.; as well as the length and cross-sectional area of the bond. As you point out, an isotropic linear elastic solid has only two parameters, so in such a material the four K's would be functions of only two parameters.

Kn, then, is calculated by the code from EA/L, with A and L determined by particle sizes and E a parameter in BrittleBeamPrms.

What are the formulas used by the code to calculate Ks, Kb and Kt?

Hi Dion,

As a follow up to my message, elementary beam theory would give the following formulas:

Ks = GA/L = EA/2(1+nu)L

Kt =GA^2/2*pi*L = EA^2/4(1+nu)*pi*L

Kb = 3EA^2/2*pi*L

where E, G and nu are the Young's modulus, the shear modulus and Poisson's ratio.

Are these the formulas used in the code?

Thanks Dion Weatherley, that solved my question.

Thanks, Dion -- it all makes sense.