Equations of Granular Materials: Transmission of Stress

Abstract

Transmission of stress and statistics of force fluctuations in static granular arrays are fundamental, but unresolved problems in physics. Despite several theoretical attempts [1,2, 3] and a vast engineering literature [4,5] the connectivity of granular media is still poorly understood at a fundamental level. We propose a theory of stress transmission in disordered arrays of rigid cohesionless grains with perfect friction. A real granular aggregate (e.g. sand or soil) is a very complex object [5]. However, simple models are easier to comprehend, and extra complexities can always be incorporated subsequently. In our case the rigid grain paradigm provides a crucial starting point from which to appreciate the theoretical physics of the problem. We model the granular material as an assembly of discrete rigid particles whose interactions with their neighbours are localized at pointlike contacts. Therefore the description of the network of intergranalar contacts is essential for the understanding of force transmission in granular assemblies. Grain α exerts a force on grain β at a point ${{\vec{\mathcal{R}}}^{{\alpha \beta }}} = {{\vec{\mathcal{R}}}^{\alpha }} + {{\vec{r}}^{{\alpha \beta }}}$. The contact is a point in a plane whose normal is ${{\vec{n}}^{{\alpha \beta }}}$. The vector ${{\vec{R}}^{\alpha }}$ is defined by:

$${{\vec{R}}^{\alpha }} = \frac{{{{\sum }_{\beta }}{{{\vec{\mathcal{R}}}}^{{\alpha \beta }}}}}{z},$$

(1)

So that ${{\vec{R}}^{\alpha }}$ is the centroid of contacts, and hence

$$\begin{array}{*{20}{c}} {\sum\limits_{\beta } {{{{\vec{r}}}^{{\alpha \beta }}} = 0,} } & {{{{\vec{R}}}^{{\alpha \beta }}} = {{{\vec{r}}}^{{\alpha \beta }}} - {{{\vec{r}}}^{{\beta \alpha }}},} \\ \end{array}$$

(2)