Amorphous solids are present in many engineering contexts. Examples include metallic glasses, granular solids, concrete, or even brittle crust of earth at geological scales. It has been known for many years now that the macroscopic failure obeys a scale-invariant universal pattern known as “shear band localization”. However basic questions remain unanswered; what is the validity range of Coulomb’s law in characterizing localization bands in frictional materials or why certain materials form bands plastically but others go through brittle fracture.

Figure 1) Shear banding localization a) Plastic deformation in a granular medium (Desrues et al. 2002) b) Brittle fracture in a metallic glass (Hofmann et al. 2008) c) Geological fault at Blue Anchor Bay, Somerset, England.

Some of the fundamental questions my research aims to answer are: i) what is the nature of spatio-temporal strain patterns that emerge prior to the ultimate failure? ii) how does this pre-yielding transient dynamics relate to the permanent shear banding? iii) is it feasible to use the former as a failure precursor in order to forecast the latter? Such issues have practical applications ranging from damage progression in brittle materials or micro-plasticity in granular media to fore-shock occurrence prior to the main earthquake rupture.

My research generally addresses these problems from a multi-scale point of view. Using particle-based modeling, the microscopic fluctuation patterns can be quantified within a statistical mechanics framework. Coarse-grained meso-scale view of the deformation involves elementary volumes that go through intense localized shear but have long-ranged non-local consequences. At continuum scales, macro failure may be viewed as a form of mechanical instability and bifurcation problem associated with spatial localization patterns. Integrating these different notions of failure remains central to my work.

Figure 2) Multiscale nature of the shear banding problem: from particle level to continuum scales

Emergence of the macroscopic failure is not abrupt but is commonly preceded by critical fluctuation patterns that have been recently described in the context of the so-called yielding transition. The microscopic basis of this viewpoint is the appearance of recurring plastic bursts that are quite localized in space but have long-range elastic-type consequences. In this framework, local isolated events are initially activated by external deformation (in the absence of thermal fluctuations), but then further instability may be triggered and propagates due to non-local interactions. This co-operative dynamics leads to the formation of scale-free clusters near the failure transition which then may be viewed as a signature of the true second-order phenomenon with unique scaling properties associated with it.

Figure 3) Amorphous plasticity is characterized by interactions between localized shear zones that incur non-local shear perturbations. The perturbed stress closely resembles that of an edge dislocation.

An important part of my work was to quantify the collective and intermittent features that precede longer-lived large scale shearing bands. I was able to observe the elasticity-based polar features that characterize the structure of correlations between plastic bursts using discrete element methods (DEM) Karimi et al. (2019). I made use of the long-range elasticity in a mesoscopic finite elements (FEM) based model of interacting blocks with the Coulomb friction law Karimi & Barrat (2018). The model led to the development of fully formed localization bands that followed the accumulation of transient mini bursts. The shear band inclination as well as the precursors’ patterns were compatible with the experimental observations.

Figure 4) Pure shear setup in the context of the mesoscopic elasto-plastic model: Left) Finite Elements-based discretization in a bi-periodic domain Right) Activity map featured in the movie evolves toward multiple localization bands as the loading continues (the imposed strain is also indicated). The color map is based on the height of each active site.

This universal deformation mechanism in flowing soft solids is despite their diversity in terms of scales, microscopic constituents, or interactions. The presence of frictional interactions in granular solids, however, alters the dynamics of flow by nucleating micro shear cracks that continually coalesce to build up system-spanning fracture-like formations on approach to failure. This is, in fact, quite similar to the progressive damage mechanism that is common in pressure-sensitive materials and often results in catastrophic failure events. My analysis in Karimi et al. (2019) showed that the plastic-to-brittle transition is uniquely controlled by the degree of frictional resistance which is in essence analogous to the role of heterogeneities that separate the abrupt and smooth yielding regimes in glassy structures.