Deformation in amorphous materials

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. These materials exhibit lots of interesting behaviors, which we study using theoretical and computational techniques:

• Shear bands
• Glassy dynamics and STZ theory
• Looking for structural signatures of STZs
• Friction and earthquakes

Shear bands

Strain localization, or shear banding, 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.

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. 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.

Paper: Localization in an STZ model for amorphous materials

Recently, we have shown that an STZ model with a rate-dependent effective temperature predicts differet types of localization behavior as a function of two important parameters: applied strain rate and quench rate for initializing the sample. For quickly strained or slowly quenched systems, thin diffusion limited shear bands are predicted. In contrast, slowly sheared or quickly quneched samples undergo homogeneous deformation, and in between there is a regime of thick shear bands where the length scale is determined by the steady state density of STZs.

Paper: Rate dependent shear bands in an STZ model for amorphous solids

Glassy Dynamics

Thomas Haxton and Andrea Liu have shown that an "effective temperature" measured using a fluctuation-dissipation relation correlates with flow and stress in a simulated glass [Phys. Rev. Lett. 99, 195701 (2007)]. The extensive Haxton and Liu (HL) data sharply test the basic assumptions of STZ theory, especially the central role played by the effective disorder temperature as a dynamical state variable. We find that the theory survives these tests, and that the HL data provide important and interesting constraints on some of its specific ingredients. Our most surprising conclusion is that, when driven at various constant shear rates in the low-temperature glassy state, the HL system exhibits a classic glass transition, including super-Arrhenius behavior, as a function of the effective temperature.

Paper: Steady-state, effective-temperature dynamics in a glassy material

As explained in the reference above, the effective temperature STZ theory provides a mechanism for aging in glassy materials: in thermal systems the effective temperature is weakly coupled to the thermal bath and in the absence of applied strain the effective temperature approaches the bath temperature. Joerg Rottler has shown that two variations of STZ theory, one which includes an effective temperature and another which includes an aging timescale qualitatively explain simulation data for aging. In collaboration with Joerg, I am exploring whether the effective temperature STZ model quantitatively matches aging dynamics.

Looking for STZs

Localized regions of deformation have been seen in many experiments and simulations of amorphous solids, including simulated lennard jones glasses and experimental dense foams and colloids. Michael Falk has introduced a metric, D2min, which is an excellent way of identifying these regions after the zone has deformed. However, there is not yet a way to identify an STZ by its structure before the zone deforms. In collaboration with Andrea Liu at UPenn, I am using a combination of dissipative particle dynamics and normal mode analysis to identify STZs before they accomodate strain and find statistical descriptions of their properties.

Friction and earthquakes

We use STZ equations that exhibit localized shear regions to generate constitutive relations for interfaces between sheared materials (such as fault planes in earthquakes). We have recently shown that shear bands are a strong strain-rate weakening mechanism, and when they form they greatly reduce the shear stress supported by the fault. This new friction law, which permits spontaneous shear bands, changes the rupture dynamics on simulated earthquake faults. This is a novel mechanism for fault weakening at high velocities.

Paper: Shear strain localization in elastodynamic rupture simulations

Biological Tissues

The mechanical properties of embryonic tissues likely play an important role in cell movements and pattern formation during embryogenesis. In the Schoetz lab at Princeton, the physical properties of developing zebrafish embryos and cell aggregates using a battery of rheological and genetic techniques. In collaboration with Eva-Maria and her lab, I am developing mechanical models to explain and predict features of these experiments.

• Dynamic cell interaction models
• Surface tension in tissues

Cell-cell interation model

It is clear that biological tissues are not simple liquids. In fact, the structure and rheology of cell packings share striking similarities with foams and emulsions, although active processes generate significant differences. I have developed a new model for tissue dynamics and cell migration which is an extension of Durian's "bubble model" for wet foams [ D. Durian PRE 55 1996]. With Eva-Maria Schoetz, I am using experimental data to constrain model parameters and make predictions about cell sorting experiments. In collaboration with Mikko Haataja, I am working to determine the implications of this model for large scale flows in developing tissues.

Surface tension in tissues

Biological tissues share many properties with liquids, including a reproducible surface tension. Different types of tissues have different surface tensions, which can be used to predict cell sorting behavior in aggregates. We develop a "broken bond" model for surface tension in ordered 2D and 3D cell aggregates, and generalize this model to random aggregates of varying size. The model accounts for adhesive contacts, bulk elasticity, and cortical elasticity, and suggests that measured surface energies result from a combination of packing topology and these forces.

Transition to turbulence in fluid flowing past a compliant boundary

In the fluid/compliant wall system, we are interested in determining how the coupling of elastic equations for the wall with the Navier-Stokes equations for the fluid might hasten or delay transition to turbulence in channel flows. We perform an input/output analysis of the system. This is a generalized stability analysis that captures highly amplified modes in addition to those which are linearly unstable.

Notes on fluid/compliant wall problem

Spurious Eigenvalue problem

Spurious eigenvalues are unphysical, numerically-computed eigenvalues with large positive real parts that often occur in hydrodynamic stability problems. We propose that these unphysical eigenvalues are possible in the analysis of any set of differential-algebraic equations when the number of independent variables is analytically reduced using the algebraic constraints and the system is subsequently approximated using finite difference or spectral collocation methods.

An alternative approach to analyzing differential-algebraic equations is the descriptor framework, posed as a generalized eigenvalue problem, which explicitly retains the algebraic constraints during the numerical computation of eigenvalues. We reformulate two common hydrodynamic stability problems using descriptor notation and show that this method of computation avoids the spurious eigenvalues generated by other methods. The descriptor formulation is a simple, robust framework for eliminating spurious eigenvalues that occur in hydrodynamic stability analysis and in the study of other dynamical systems with algebraic constraints. Additionally, this formulation reduces the order of the numerically approximated differential operators and accommodates complex boundary conditions, such as a fluid interacting with a flexible wall.

Paper: Eliminating spurious eigenvalues

Highly Optimized Tolerance

Optimization-based models for complex systems that exhibit robust yet fragile behavior

I have also studied systems that generate distributions with heavy tails or power law behavior. In the past, these power laws have been explained by several different mechanisms including preferrential growth, self organized criticality (SOC), highly optimized tolerance (HOT), and allometry (scaling with biological mass). I am interested in how the different mechanisms generate power laws with different exponents and cutoffs. For example, different HOT models predict different exponents for distributions from similar systems. We determined that these different HOT models describe two separate regimes -- one where there are a lot of resources that can be used to prevent large, cascading events, and one where there are few resources available.

Paper: Highly Optimized Tolerance in dense and sparse resource regimes

Mixtures

I have also studied mixture models. We would like to determine what happens when several data sets which have a power-law distribution (with large and small-scale cutoffs) are mixed together indescriminately.

Notes on mixtures