The incremental response of random aggregates of identical round particles

Abstract

This paper is concerned with a dense, randomly packed, granular material that consists of identical spheres or disks with elastic, frictional interactions, that is first isotropically compressed and subsequently loaded along an arbitrary stress path. An analytical relationship between the overall stress and strain increments is determined for the pre-failure regime. The purpose of the modelling is to understand how this relation depends upon the features of the packing and the particle interactions. From the outset it is recognised that the packing and interactive properties for these materials may vary substantially from grain to grain and the heterogeneity introduced in this manner is fully accounted for. Moment equilibrium equations are solved for each particle and force equilibrium equations are solved for each neighbourhood. Then, the heterogeneity of the aggregate is taken into account by introducing means and fluctuations in the description of the local deformations and the measures of the particles and interactions. The general development is illustrated with an example in two dimensions in which the packing and contact interactions are approximated by angular distributions and the heterogeneity is introduced by variations in these. For an isotropic medium with constant contact stiffnesses the theory provides predictions that compare well with results obtained from numerical simulations.

Estimates for the localised stiffness and heterogeneity of auxetic foams

“…The Poisson's ratio takes the same sign as the elements iijj Z and so it is seen that the Poisson's ratio is predicted to be positive for materials known to have a negative Poisson's ratio. Previous work in granular mechanics [14,18] shows how poor a mean strain estimate is at predicting the elastic moduli of heterogeneous materials. Therefore the absence of a negative Poisson's ratio is an expected result and the advanced modelling to better this has been referred to above.…”

“…4 Mean strain estimate 4.1 Model Jenkins and Koenders [14] introduce a modelling technique within a granular mechanics setting that can account for microstructure within a heterogeneous setting. It is founded on the requirements of equilibrium in the microstructure by summations of forces and force moments.…”

From Granular Matter to Generalized Continuum

“…To specify the parametersL n in (34) for some subset of m branch vectors l k , several authors advocate strict polynomial fits, with a maximal value of m, or else some other "best" fit of (34) to continuum kinematics [27,33,42]. Given the well-known pathology ("overfitting") of polynomial fits of random data, and in view of the paramount importance of energy, we take the position that a "best" fit should rather be based on minimization of an appropriate norm of stress-power fluctuations plus some norm of the variation implied by (34), in the spirit of the so-called "generalized additive models" (GAM) [49].…”

The auxetic properties of a network of bending beams

“…These nodal points, therefore, act as 'particles' and the beams themselves are interacting elements between the particles. The formalism set up in Jenkins and Koenders [31] can be employed.…”

“…This model is therefore similar to a granular material and the methods of granular mechanics for densely packed aggregates (see [31]) may be more or less directly employed. The question is, how applicable is that approach for cells that interact with one another not through contacts, but via some other mechanism.…”