Linear Elasticity theory has been traditionally developed within a continuum framework. In this context, state variables ( i.e. stress, strain, or elastic energy ) are treated as smooth continuum fields with spatial fluctuations that tend to average out beyond some characteristic scale. The latter scale is associated with the representative volume element (RVE) which has a strong relevance on homogenization techniques and constitutive modeling commonly used in engineering contexts. The RVE size is often taken to be larger than relevant micro-structural lengths but negligible compared to the sample size.

However, the notion of RVE and implicit scale-separation hypothesis is a debated concept in the context of particulate matters. A granular packing, for example, forms "force chains" in response to stress with complex fractal patterns that are correlated over large scales (compared to the grain size). Several numerical and experimental observations report on the scale-free nature of these fluctuations which persist up to the physical system size. Therefore, the RVE representation is irrelevant in this framework as the stressed medium lacks any characteristic scale.

Figure 1) Compressed granular packings form fractal units known as force chains. At what scale fluctuations average out?

Despite being visually obvious, characterization of these collective patterns remains a formidable task. Common approaches based on the force correlation functions or topological characterizations of stress chains did not yield promising results. A novel technique which combines both notions was utilized by Karimi & Maloney (2011) and Karimi et al. (2017) based upon a local statistical metric measuring scale-dependent anisotropy within a globally isotropic medium. This analysis indicates long-range scaling features corresponding to the chosen metric (order parameter) along with a characteristic size that exhibits some forms of growth (if not divergence!) on approach to the rigidity transition. In a broader context, this may be viewed as a signature of the true continuous phenomenon with universal statistical features that are not specific to scales, microscopic constituents, or interactions. This multi-scale analysis confirms that the RVE-based approach is too simplistic and breaks down in the vicinity of the rigidity transition.

Mechanical response in sheared amorphous solids also shows anomalous features that may not be predicted from linear elasticity theory. Karimi & Maloney (2015) showed that the elastic response is governed by certain characteristic scales that tend to diverge near the rigidity point. They argued that only above some certain scales scales one could recover the theoretical solutions and, therefore, the elasticity theory fails upon the divergence of relevant length-scales. Such elastic anomalies may be partially treated within the context of higher order continuum mechanics and gradient theories that take into account micro-structural scales directly in the governing equations.

Figure 2) Sheared disordered solid (compressed vertically and expanded horizontally). To what extent the elasticity theory can predict the linear response?