Accurate modelling of the elastic behavior of a continuum with the Discrete Element Method

Abstract

The Discrete Element Method (DEM) has been used for modelling continua, like concrete or rocks. However, it requires a big calibration effort, even to capture just the linear elastic behavior of a continuum modelled via the classical force-displacement relationships at the contact interfaces between particles. In this work we propose a new way for computing the contact forces between discrete particles. The newly proposed forces take into account the surroundings of the contact, not just the contact itself. This brings in the missing terms that provide an accurate approximation to an elastic continuum, and avoids calibration of the DEM parameters for the purely linear elastic range.

Appendices

Appendix A. Shortcomings of the classic DEM when trying to reproduce the Young’s modulus

In order to emphasize the importance of the extra terms proposed in the constitutive expressions of Sects. 3.1 and 3.2, an example is shown to evidence some shortcomings of the classical DEM for cohesive materials:

Let us enforce a vertical strain of $\varepsilon _{v}=\frac{\delta _{v}}{L_{0v}}$ on two layers of circles, where $\delta _{v}$ is the relative vertical displacement between the layers and $L_{0v}$ is the initial vertical distance between the layers (distance between centers).

1.1 Case 1

In a Cartesian arrangement (Fig. 13), a two-layer sample formed by circles can be taken as a representative cell (Fig. 14). Let us impose a descending displacement $\delta _{v}$ to the upper layer, while keeping the lower layer fixed. $L_{0v}$ is then equal to 2R, so the vertical strain has a value of $\varepsilon =\frac{\delta _{v}}{L_{0v}}=\frac{\delta _{v}}{2R}$.

Fig. 13

Two layers of a Cartesian packing of particles

Fig. 14

Representative cell of the Cartesian packing for vertical force analysis

In the standard DEM, the vertical force between these two circles is $F_{v}=k_{n}\delta _{v}$, where $k_{n}$ is a fixed, calibrated value, or is obtained by $k_{n}=\frac{EA}{2R}=\frac{E2R}{2R}=E$. Let us choose the latter, where the contact area A has been assumed to be 2R for this case, which is a value that ensures that the sum of all areas between both layers is equal to the whole section, without gaps or overlaps.

1.2 Case 2

In a structured dense packing of circles (Fig. 15), a sample of one upper circle and two halves of lower circles can be taken as a representative cell (Fig. 16). In order to impose the same vertical strain, let us impose a descending displacement $\delta _{v}^{*}=\delta _{v}\cos \theta$ and $L_{0v}^{*}=2R\cos \theta$, where $\theta$ is the angle that the inclined bonds form with the vertical. With this configuration, the vertical strain is $\varepsilon ^{*}=\frac{\delta _{v}^{*}}{L_{0v}^{*}}=\frac{\delta _{v}\cos \theta }{2R\cos \theta }=\frac{\delta _{v}}{2R}=\varepsilon$.

Fig. 15

Two layers of a dense, structured packing of particles

Fig. 16

Representative cell of the dense packing for vertical force analysis

In the classical DEM, the vertical force acting on the upper circle is:

$$\begin{aligned} F_{v}^{*}=2\left( F_{n}^{*}\cos \theta +F_{t}^{*}\sin \theta \right) \end{aligned}$$

(36)

where $F_{n}^{*}$ and $F_{t}^{*}$ are the normal and tangential forces between the upper circle and one of the lower ones.

Using the concepts of the classical DEM, these two forces can be written as a stiffness constant multiplying a relative displacement, i.e.

$$\begin{aligned} F_{v}^{*}=2\left( k_{n}\delta _{v,n}^{*}\cos \theta +k_{t} \delta _{v,t}^{*}\sin \theta \right) \end{aligned}$$

(37)

where $\delta _{v,n}^{*}$ is the relative normal displacement at the contact point and $\delta _{v,t}^{*}$ is its tangential counterpart. In Eq. (37) $k_{n}$ is a fixed calibrated value, or it is obtained by $k_{n}=\frac{EA^{*}}{2R}$. Also in Eq. (37) $k_{t}$ is usually taken as a fraction of $k_{n}$, but it can also be estimated as $k_{t}=\frac{GA^{*}}{2R}$. No matter which option is chosen, in general $k_{n}\ne k_{t}$. For further developments we have chosen the second choice for defining $k_{t}$. Then, if we substitute $\delta _{v,n}^{*}$ and $\delta _{v,t}^{*}$ by $\delta _{v}^{*}\cos \theta$ and $\delta _{v}^{*}\sin \theta$, the expression of the total vertical force on the upper circle is:

$$\begin{aligned} F_{v}^{*}=2\left( \frac{EA^{*}}{2R}\delta _{v}^{*}\cos ^{2} \theta +\frac{GA^{*}}{2R}\delta _{v}^{*}\sin ^{2}\theta \right) \end{aligned}$$

(38)

Taking into account that $A^{*}=\frac{A}{2}\frac{1}{\cos \theta }=\frac{R}{\cos \theta }$ and $\delta _{v}^{*}=\delta _{v}\cos \theta$ we finally obtain:

$$\begin{aligned} F_{v}^{*}=E\delta _{v}\cos ^{2}\theta +G\delta _{v}\sin ^{2}\theta \end{aligned}$$

(39)

1.3 Comparison

In both cases, the total horizontal contact area is 2R (in Case 2, the two contact areas must be projected to the horizontal direction to recover this value). Having the same vertical strain in both cases, the vertical stress should be equal as well, understanding the vertical stress as the vertical force divided by the total area. Since the area is the same, the vertical forces must be equal. However, $F_{v}^{*}\ne F_{v}$ in general. They can only be equal if we assume that $G=E$. In other words, $k_{t}$ must be equal to $k_{n}$ in order to recover the same vertical stiffness.

1.4 Conclusion

From the above exercise we conclude that the micro parameters used in the standard DEM ($k_{n}$ and $k_{t}$) yield different stiffness values for a sample depending on the position of the particles or the direction of the bonds. This means that a random packing of spheres modelled with the standard DEM is extremely heterogeneous in terms of internal stiffness. It also means that a calibration obtained for one sample is not necessarily useful for other samples, as most probably the orientation of the bonds will be different.

Appendix B. Dynamics of the DEM and mass adjustment

When the cohesive DEM is used to model a continuum, any dynamic response is directly linked to the mass of the particle (a circle in 2D, and a sphere in 3D). However, the voids between particles are not typically considered. The computed sample is too porous, less dense than the real one, and the dynamic waves travel faster than expected.

In order to get a better approximation to the actual mass associated to each particle, the volume of the voids should be distributed among the neighbour particles. Instead of doing this, the volume of the particle can be computed by the ‘representative volume’ expressed in Eq. (12). Multiplying this volume by the bulk density of the material yields a mass for the particle which allows a better capture of any dynamic wave in the modeled continuum.