Constitutive Laws

Rate and State Laws

rocks Over the last decade there has been a growing appreciation of the importance of incorporating history dependence in the description of materials far from equilibrium in a manner which also is amenable to modeling at coarser resolutions. An excellent example of this approach is being developed in the context of the earthquake problem. This is a field in which multiscale modeling still involves major uncertainties at all levels, since the unknown model ingredients span scales ranging from atomic to tectonic. On the other hand, recent results in theory, computer simulation, and laboratory experiments along with the increasing resolution of seismic data are leading to new opportunities to use constraints imposed by physical mechanisms and models to reduce the uncertainties associated with seismic observations.

One major contribution involves the development of phenomenological rate and state constitutive relations. The rate and state approach was introduced by Dieterich, Ruina, and Rice in the rock mechanics community to describe the friction associated with quasistatic deformation of rough, dry surfaces in the lab. More recently this approach has been generalized to lubricated surfaces by Carlson and Batista, and to fracture by Langer and Falk.

stz A rate and state law represents a reduction of some underlying microscopic materials process into a macroscopic force law described in terms of thermodynamic-like variables characterizing the system. Rate refers to the fact that the force law depends on the instantaneous rate of deformation, and state refers to the fact that the force law depends on the internal state of the system, which incorporates the history dependence. To date, the state variable(s) have been hand crafted, based on physical mechanisms which in some cases have been deduced from experiments or molecular dynamics simulations. In the case of Carlson's work on friction in boundary lubrication, the state variable was associated with the degree of melting in the lubricant. In Falk and Langer's work on fracture the state variable was associated with the density of Shear Transformation Zones (STZ's), defined to be local regions in the material which undergo reorientations in packing as a result of applied stress. The degree of molecular order and the density of STZ's depend on the preparation of the material, described in terms of some macroscopic external driving mechanisms. More recently, an effective temperature has been identified as an important internal state variable in these amorphous materials, and strain localization has been shown to affect the macroscopic constitutive friction law.

Rate and state laws are inspired by microscopic physics, and allow study of the implications of the microscopic phenomena at larger scales. However, this work is just beginning, and a great deal remains to be done at every scale of this problem. The microscopic models are highly simplified, and the choice of appropriate state variables remains something of an art. This represents a natural opportunity for application of the systems oriented mathematical techniques like model reduction which emerge from engineering for modeling the behavior of systems in response to their environments, and presents opportunities to link our work on seismology with the other aspects of complex systems theory which form the second major research focus area in my group.

worm More rigorous derivations of rate and state laws represent a future direction for developing a materials application of the HOT mechanism, describing the robust yet fragile nature of complex systems. In HOT complexity arises when a system with internal degrees of freedom develops structure in response to the environment and the history. While the complexity of materials under stress does not involve deliberate design or evolution by natural selection which is associated with the applications of HOT to man made networks and biological systems, the evolution of state variables in response to materials preparation in friction and fracture naturally lead to structured sensitivities, in the same spirit as other systems described by HOT. Compared to more complex phenomena in e.g. biology, detailed studies of applications of HOT in materials science intrinsically involve fewer design degrees of freedom. This is both an advantage and a necessity in the initial stages of the development of robust mathematical tools for multiscale modeling of complex systems. Techniques developed here will ultimately be extended to other, more complicated scenarios.

atoms The study of stress, deformation, and slip in granular materials is a good example, where history dependence leads to structured internal states of the system through the development of sharply localized force chains supporting the weight and applied stress. We have been developing rate and state descriptions for granular and amorphous materials based on the STZ theory. The response to applied stress depends on how the system is perturbed relative to its past history, though this aspect has not yet been investigated in detail. The robust yet fragile nature of complexity, arises, e.g., when granular material subject to shear deformation in one direction is abruptly subject to shear along a different axis. This disrupts the orientation of the force chain network, leading to a sharp change in the macroscopic friction characteristics. Such phenomena have direct implications for a wide range of geophysical phenomena, and special relevance to seismicity, landslides, and erosion which arise from (typically human) causes not associated with the natural evolution of geophysical systems.

Specific friction studies

Friction is an old and a modern problem. It was already identified by egyptians who pulled large stones on wetted sand; it is still a major concern in all industrial processes. The first quantitative studies were performed by Leonardo da Vinci, followed two centuries later by Amontons, Euler and Coulomb. They studied friction between pieces of wood or metal. Modern studies focus either on friction between rocks, or between atomic scale lubricated interfaces.

Friction and complexity. Our interest for friction has two major thrusts.
Friction forces plays an essential role in earthquake dynamics: they are a source of complexity.
Friction forces emerge from the microscopic collective dynamics: they originate from complexity.
Our work concerns both the origin of frictional force from microscopic complexity, and the origin of macroscopic complexity resulting from non-linear friction between interfaces.

Rate-and-state laws for boundary lubrication. Stick-slip is a problem of choice to study how atomic-scale collective dynamics can lead to an instability of macroscopic motion. Motivated by phenomenological approaches to rock friction, we have introduced rate-and-state laws to model experiments on boundary lubrication. A state variable is introduced, to account for the internal state of an atomic-scale layer of material shear between two plates. Simple mean-field like dynamical equations present a Hopf bifurcation, that account for the transition from steady sliding to stick-slip motion when the driving velocity is lowered.

Rheological chaos and the transition to stick-slip.

In the STZ framework, it is possible to perform a careful study of the type of macroscopic dynamics as a function of experimental parameters: the driving velocity, and the stiffness of the experimental apparatus. In this parameter space, a zone of positive Lyapunov exponent indicates the existence of rheological chaos as a secondary instability develops after the Hopf bifurcation.