This paper concerns the mathematical modelling of the deformation of granular materials. It showcases modern applications of undergraduate-level vector calculus in the context of evolving vector fields--specifically, the study of the kinematics (displacements, rotations and velocities) of an assembly of "grains" as they engage in self-organised pattern formation.
Granular materials form a class of materials encompassing everything from soil and natural grains to pharmaceutical pills and chemical powders. They are ubiquitous in nature and commercially vital in many industries such as chemical, agricultural, cosmetics, food manufacturing, pharmaceutical and mining industries (see figure 1). Considered as the ultimate paradigm of complex systems, granular materials exhibit emergent behaviour that has eluded scientists for centuries. Today, there is still no universally-accepted constitutive model to predict how granular materials will behave under load. Consequently, systems and processes involving granulates rarely reach 60% of their design capacity, whereas processes involving fluids operate on average at 96% design capacity (Duran, 2000; Oda & Iwashita, 2005). Thus, even a fractional advance in our understanding of how granular media behave can have a profound economic and social impact.
Numerous models of granular materials are constructed within the framework of classical continuum theory (Oda & Iwashita, 2005). This theory is based on the assumption that the body is continuous and comprises material points that bear only translational degrees of freedom. By contrast, a granular material is a discrete assembly of solid particles, each of which has translational as well as rotational degrees of freedom. Classical continuum mechanics asserts that a material should undergo homogeneous or affine deformation in response to homogeneous boundary forces, that is, the change in shape of the material should be uniform throughout the whole material. Therefore, in an affine deformation the components of the strain tensor are independent of position (ie. are constants or functions of time only). But experiments have shown that a granular material subjected to uniform boundary forces exhibits significant nonaffine deformation, ie. local deviations from affine deformation (see, for example, Tordesillas et al (2009) and references therein). Non-affine deformation becomes particularly important during self-organised pattern formation, a common phenomenon that arises almost from the onset of loading, especially in densely packed granular systems (LTB Group, 2004; Australasian journal of Engineering Education, Vol 15 No 2 Behringer, n. d.; Duran, 2000; Oda & Iwashita, 2005; Tordesillas et al, 2008; 2009; Tordesillas & Arber, 2005; Majmudar & Behringer, 2005; Rechenmacher, 2006; Oda et al, 2004; Kuhn & Bagi, 2002; Mueth et al, 2000; Tordesillas, 2007).
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By far the most striking example of emergent pattern formation in a deforming granular medium resides in the manner by which the material transmits force via a dual complex force network, as shown in figure 2. The first, known as the strong network, comprises force chains: quasi-linear particle chains through which above average contact forces are transmitted (LTB Group, 2004; Behringer, n. d.; Majmudar & Behringer, 2005). The second, known as the weak network, comprises the remaining particles bearing relatively small forces: these surround and provide lateral stability to the force chains. Force chains may span only a few grains in length, or they may extend for hundreds of grain diameters. Physical experiments using techniques of photo-elasticity have shown that force chains align themselves in the direction where the material feels the greatest compression (Oda & Iwashita, 2005; Tordesillas et al, 2009). In essence then force chains are emergent columnar structures that are subject to axial compression. …