Complex Systems

Amorphous materials are ubiquitous in natural and engineered systems. Granular fault gouge in earthquakes faults, thin film lubricants, and bulk metallic glasses are seemingly disparate systems which are similar in that they possess an amorphous structure. Colloids, emulsions, window glass, dense polymers, and even biological tissues are other examples.

Although ruptures on earthquake faults, nanoscale friction measured using a Surface Force Apparatus, and deformation in bulk metallic glasses appear to be very different phenomena, they share a common feature: the region where deformation or slip occurs is populated by an amorphous material. Amorphous solids are comprised of particles (atoms, grains, bubbles, molecules) arranged so that the locations of their centers of mass are disordered; their structure is essentially indistinguishable from a liquid. However, these materials are ``jammed'' and exhibit a yield stress like a solid. Other examples of amorphous materials include colloids and emulsions, foams, glass-forming molecular liquids, traffic jams, and even living tissue.

While amorphous materials differ in important and significant ways, they exhibit common features which we explain using a single continuum formalism. One such feature is a mode of deformation or failure called shear banding or strain localization. Strain localization is the spontaneous development of coexisting flowing and stationary regions in a sheared material.

For engineering or design purposes, materials like films, foams, and rocks are currently described by phenomenological laws based on fits to data. While these laws are useful for predicting macroscopic behavior in most typical situations, they do not predict the onset of fracture, deterioration, and propagation of cascading failure, which can be sudden, unexpected, and extremely sensitive to the history of the material. Such fragilities are often intrinsically linked to feedback between systems at different scales, such as granular fault gouge and networks of faults spanning hundreds of miles. Read more about constitutive laws and friction

Amorphous materials often comprise or lubricate sheared material interfaces and require more complicated constitutive equations than simple fluids or crystalline solids. They flow like a fluid under large stresses, creep or remain stationary under smaller stresses, and have complex, history-dependent behavior. Bulk metallic glasses, granular materials, and bubble rafts are just some of the disordered materials that exhibit a yield stress.

We model amorphous solids with a set of partial differential equations that describe Shear Transformation Zones (STZs) (Falk and Langer, 1998) with an effective temperature.

Snapshot from simulation by M. Falk illustrating an STZ. Darker particles have undergone more plastic deformation.

One difficulty in describing non equilibrium systems is that the thermal temperature no longer completely characterizes probability distributions for the system's many degrees of freedom. For example, the thermal temperature specifies the velocity and position distributions of particles in an equilibrium gas, while the thermal temperature reveals very little about a glass-forming liquid well below the glass temperature or a dense colloidal suspension driven in simple shear. Physicists have long searched for an internal variable that can be used to characterize far from equilibrium systems.

Mehta and Edwards were perhaps the first to point out that although thermal temperature does not determine statistical distributions for macroscopic particles such as powders, the statistical properties of these systems could still be characterized by a small number of macroscopic state variables, such as free volume.

They defined the effective temperature as the intrinsic variable which is the derivative of the volume or configurational energy with respect to the configurational entropy. This definition is based on the intuition that for many systems thermal temperature is not sufficient to cause configurational rearrangements, but slow shearing or stirring causes the particles to ergodically explore configuration space.

Ono, et al. and Haxton and Liu have used this entropic definition as well as fluctuation-dissipation based definitions in simulated glassy materials. Their results are very encouraging; they suggest that a single effective temperature describes the configurational degrees of freedom in slowly sheared amorphous packings. In other words, the effective temperature is a internal state parameter, much like an order parameter, that specifies the disorder in configurational packings. By combining a heat equation that accounts for effective temperature with a model for particle rearrangements, we generate a model for deformation in amorphous materials.

Strain localization is the spontaneous development of coexisting flowing and stationary regions in a sheared material. Strain localization has been identified and studied experimentally in granular materials, bubble rafts, complex fluids, and bulk metallic glasses. Shear banding may play an important role in the failure modes of structural materials and earthquake faults. Localization is a precursor to fracture in bulk metallic glasses and has been cited as a mechanism for material weakening in granular fault gouge on faults.

Using the STZ constitutive model, we find small perturbations in the effective temperature can lead to localized regions of higher strain, or shear bands, in our numerically integrated solution, and show that the system is linearly unstable with repect to perturbations to a time-varying trajectory.

Localization in an STZ model for amorphous materials

We have also used STZs equations that exhibit localized shear regions to generate constitutive relations for interfaces between sheared materials (such as fault planes in earthquakes). Shear banding is shown to be a strong dynamic weakening mechanism which has important implications for friction laws in these systems.