Koren B., Vuik K. (Editors) Advanced Computational Methods in Science and Engineering

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Stefan Luding

Remarkably, the shear stress intensity |

|/p ≈

is almost constant for practically

all averaging volumina with strain rates larger than some threshold value, i.e.,

with

≈ 0.02 s

−1

. Whether the threshold has a physical meaning or is only an

artefact due to the statistical ﬂuctuations in the average data has to be examined

further by much longer runs with better statistics.

From the constant shear stress intensity in the shear zone, one can determine the

Mohr-Coulomb-type friction angle of the equivalent macroscopic constitutive law,

see Fig. 16, as

≈ arcsin

. Interestingly, without friction,

is rather large, i.e.,

much larger than expected from a frictionless material, whereas it is astonishingly

small with friction (data not shown), i.e., smaller than the microscopic contact fric-

tion

= 0.4 used, see Ref. [40].

500 400 300 200 100 0

|τ|

0.01

0.02

0.04

0.06

0.08

0.16

µ p

0.4

0.3

0.2

0.1

500 400 300 200 100 0

|τ|/p

Fig. 16 (Left) Shear stress |

|and (Right) shear stress intensity |

|/p plotted against pressure. The

size of the points is proportional to the shear rate, and the dashed line (right panel) separates the

data from simulations without (Bottom) and with (Top) friction, see [40].

7.2.4 Discussion

In summary, the example of a ring shear cell simulation in 3D has shown, that even

without the more complicated details of fancy interaction laws, experiments can be

reproduced at least qualitatively. A more detailed study of quantitative agreement

has been performed in 2D [27], and is in progress for the 3D case.

A challenge for the future remains the micro-macro transition, for which a ﬁrst

result has been shown, i.e. the yield stress can be extracted from a single 3D DEM

simulation for various pressures and shear rates. Open remains an objective contin-

uum theory formulation of the shear band problem.

488

From Particles to Continuum Theory

8 Conclusion

The present study is a summary of the most important details about soft particle

molecular dynamics (MD), widely referred to as discrete element methods (DEM)

in engineering, and hard particle event driven (ED) simulations, together with an

attempt to link the two approaches in the dense limit where multi-particle contacts

become important.

As an example for a micro-macro transition, the stress tensor was deﬁned and

computed for dynamic and quasi-static systems. This led, for example, to a global

equation of state, valid for all attainable densities, and also to the partial stresses

due to normal and tangential (frictional) contacts. For the latter situation, the micro-

macro average is compared to the macroscopic stress (=force/area) measurement

(with reasonable agreement) and, at least in 3D, a yield stress function can be ex-

tracted from a single ring shear cell simulation.

In conclusion, discrete element methods have proven a helpful tool for the un-

derstanding of many granular systems, while MD is the standard tool for atomistic

and molecular systems. The methods presented in this paper can be applied to both

DEM and MD simulation results with the goal to obtain micro- and particle-based

constitutive relations for continuum theory.

The qualitative approach on DEM of the early years has now developed into the

attempt of a quantitative predictive modeling tool for the diverse modes of complex

behavior in granular media. To achieve this goal will be a research challenge for

the next decades, involving enhanced kinetic theories for dense collisional ﬂows

and elaborate constitutive models for quasi-static, dense systems with shear band

localisation.

In the future this tool will allow to impose a desired behavior by control or de-

sign, with particular application in mind as, e.g., modern sintered materials, reactors

involving catalysts, and many others.

Acknowledgements We acknowledge the ﬁnancial support of several funding institutions that

supported the reviewed research, and also the helpful discussions with, and contributions from the

many persons that contributed to these results.

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60. U. Langer, M. Discacciati, D. Keyes, O. Widlund, W. Zulehner (eds.), Domain Decomposition

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63. M. Bebendorf, Hierarchical Matrices. A Means to Efﬁciently Solve Elliptic Boundary Value

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