Generalised homogenisation procedures for granular materials

Pasternak, Elena and Muhlhaus, Hans-Bernd (2005) Generalised homogenisation procedures for granular materials. Journal of Engineering Mathematics, 52 1: 199-229. doi:10.1007/s10665-004-3950-z

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Author Pasternak, Elena
Muhlhaus, Hans-Bernd
Title Generalised homogenisation procedures for granular materials
Journal name Journal of Engineering Mathematics   Check publisher's open access policy
ISSN 0022-0833
Publication date 2005-07
Sub-type Article (original research)
DOI 10.1007/s10665-004-3950-z
Open Access Status File (Author Post-print)
Volume 52
Issue 1
Start page 199
End page 229
Total pages 31
Place of publication Dordrecht, Netherlands
Publisher Springer Netherlands
Collection year 2005
Language eng
Subject 290699 Chemical Engineering not elsewhere classified
290501 Mechanical Engineering
290704 Geomechanics
240500 Classical Physics
290800 Civil Engineering
260206 Earthquake Seismology
780104 Earth sciences
Abstract Engineering materials are generally non-homogeneous, yet standard continuum descriptions of such materials are admissible, provided that the size of the non-homogeneities is much smaller than the characteristic length of the deformation pattern. If this is not the case, either the individual non-homogeneities have to be described explicitly or the range of applicability of the continuum concept is extended by including additional variables or degrees of freedom. In the paper the discrete nature of granular materials is modelled in the simplest possible way by means of finite-difference equations. The difference equations may be homogenised in two ways: the simplest approach is to replace the finite differences by the corresponding Taylor expansions. This leads to a Cosserat continuum theory. A more sophisticated strategy is to homogenise the equations by means of a discrete Fourier transformation. The result is a Kunin-type non-local theory. In the following these theories are analysed by considering a model consisting of independent periodic 1D chains of solid spheres connected by shear translational and rotational springs. It is found that the Cosserat theory offers a healthy balance between accuracy and simplicity. Kunin's integral homogenisation theory leads to a non-local Cosserat continuum description that yields an exact solution, but does not offer any real simplification in the solution of the model equations as compared to the original discrete system. The rotational degree of freedom affects the phenomenology of wave propagation considerably. When the rotation is suppressed, only one type of wave, viz. a shear wave, exists. When the restriction on particle rotation is relaxed, the velocity of this wave decreases and another, high velocity wave arises.
Keyword Cosserat continuum
non-local continuum
rotational degrees of freedom
wave propagation
Q-Index Code C1
Institutional Status UQ
Additional Notes Citation: E. Pasternak and H.-B. Mühlhaus (2005) Generalised homogenisation procedures for granular materials, Journal of Engineering Mathematics 52 (1): 199 - 229. DOI: 10.1007/s10665-004-3950-z Copyrighr 2005 Springer-Verlag. All rights reserved. The original publication is available at

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Created: Tue, 21 Mar 2006, 10:00:00 EST by Hans-Bernd Muhlhaus on behalf of Earth Systems Science Computational Centre