Bruce Hobbs

Earthquakes in the ductile regime

Pseudotachylytes from a crustal scale shear zone in Central Australia have developed in a cyclical manner: once developed, an individual pseudotachylyte is deformed in a ductile manner, only to be overprinted at a later stage by a new generation of pseudotachylytes. Such cyclic generation and deformation of pseudotachylyte has been interpreted in the past as representing conditions at the brittleductile transition; a different interpretation, however, is presented here. It is proposed that psuedotachylytes and associated ultramylonites can develop entirely within the ductile regime as ductile instabilities. Such instabilities are different in nature to those previously discussed at length in the geophysical literature but are identical in principle with the instabilities that develop for velocity-weakening frictional behavior in spring-slider systems. At a given strain rate a critical temperature,Tc, is defined, at which the transient work hardening equals the product of stress relaxation due to a thermal fluctuation and the heat generated by shearing. A necessary condition for ductile instability at a given strain rate is that the temperature is belowTc; then the rate of change of stress with respect to strain is negative. An additional requirement is that this rate of change exceeds, in magnitude, the effective elastic stiffness of the loading system. Ductile instabilities are marginally possible at geological strain rates in quartzites but are possible at mid-crustal temperatures in other rock types. On the basis of these observations a new interpretation is presented for the base of the seismogenic zone in crustal regions.

Coulomb Constitutive Laws for Friction: Contrasts in Frictional Behavior for Distributed and Localized Shear

We describe slip-rate dependent friction laws based on the Coulomb failure criteria. Frictional rate dependence is attributed to a rate dependence of cohesionc and friction angle ϕ. We show that differences in the stress states developed during sliding result in different Coulomb friction laws for distributed shear within a thick gouge layer versus localized shear within a narrow shear band or between bare rock surfaces. For shear within gouge, shear strength is given by τ=c cosφ + σn sinφ, whereas for shear between bare rock surfaces the shear strength is τ=c cosφ + σn tanφ, where τ and σn are shear and normal stress, respectively. In the context of rate-dependent Coulomb friction laws, these differences mean that for a given material and rate dependence of the Coulomb parameters, pervasive shear may exhibit velocity strengthening frictional behavior while localized shear exhibits velocity weakening behavior. We derive from experimental data the slip-rate dependence and evolution ofc and ϕ for distributed and localized shear. The data show a positive rate dependence for distributed shear and a negative rate dependence for localized shear, indicating that the rate dependence ofc and ϕ are not the same for distributed and localized shear, even after accounting for differences in stress state. Our analysis is consistent with the well-known association of instability with shear localization in simulated fault gouge and the observation that bare rock surfaces exhibit predominantly velocity weakening frictional behavior whereas simulated fault gouge exhibits velocity strengthening followed by a transition to velocity weakening with increasing displacement. Natural faults also exhibit displacement dependent frictional behavior and thus the results may prove useful in understanding the seismic evolution of faulting.

From point defects to plate tectonic faults

Understanding and explaining emergent constitutive laws in the multi-scale evolution from point defects, dislocations and two-dimensional defects to plate tectonic scales is an arduous challenge in condensed matter physics. The Earth appears to be the only planet known to have developed stable plate tectonics as a means to get rid of its heat. The emergence of plate tectonics out of mantle convection appears to rely intrinsically on the capacity to form extremely weak faults in the top 100u2009km of the planet. These faults have a memory of at least several hundred millions of years, yet they appear to rely on the effects of water on line defects. This important phenomenon was first discovered in laboratory and dubbed “hydrolytic weakening”. At the large scale it explains cycles of co-located resurgence of plate generation and consumption (the Wilson cycle), but the exact physics underlying the process itself and the enormous spanning of scales still remains unclear. We present an attempt to use the multi-scale non-equilibrium thermodynamic energy evolution inside the deforming lithosphere to move phenomenological laws to laws derived from basic scaling quantities, develop self-consistent weakening laws at lithospheric scale and give a fully coupled deformation-weakening constitutive framework. At meso- to plate scale we encounter in a stepwise manner three basic domains governed by the diffusion/reaction time scales of grain growth, thermal diffusion and finally water mobility through point defects in the crystalline lattice. The latter process governs the planetary scale and controls the stability of its heat transfer mode.

On the thermodynamics of listric faults

We investigate a novel fully coupled thermal-mechanical numerical model of the crust in order to trace the physics of interaction of its brittle and ductile layers. In a unified approach these layers develop in a natural transition as a function of the state variables pressure, deviatoric stress, temperature and strain-rate. We find that the main storage of elastic energy lies in the domain where brittle and ductile strain-rates overlap so that shear zones are attracted to this zone of maximum energy dissipation. This dissipation appears as a local heat source (shear heating). The brittle-ductile transition zone evolves through extreme weakening by thermo-mechanical feedback. The physics of the weakening process relies on repeated breaching of a critical energy flux threshold for feedback within this sub-horizontal brittle-ductile flow layer, thus developing unstable slipping events at postand pre-seismic strain-rates. The width- and the temperature domain of the feedback layer is controlled by the activation enthalpy Q of the material. For olivine rheology (Q ∼ 500 kJ/mol) the layer can be extremely thin <500 m and adheres to the 875 K isotherm. For quartz (Q ∼ 135 kJ/mol) the width of the feedback layer fans out into multiple interacting ductile faults covering a temperature domain of 450–600 K. The weakening by thermal-mechanical feedback entirely controls the location and rejuvenation of upper crustal shear zones propagating from the detachment upwards in the form of listric faults. Within the detachment shear layer we identify an astonishing rich dynamics featuring distinct individual creep bursts. We argue that the rich ductile dynamics holds the key to earthquakes in the brittle field.

First Steps Towards Modeling a Multi-Scale Earth System

Recent advances in computational geodynamics are applied to explore the link between Earth’s heat, its chemistry and its mechanical behavior. Computational thermal-mechanical solutions are now allowing us to understand Earth patterns by solving the basic physics of heat transfer. This approach is currently used to solve basic convection patterns of terrestrial planets. Applying the same methodology to smaller scales delivers promising similarities between observed and predicted structures which are often the site of mineral deposits. The new approach involves a fully coupled solution to the energy, momentum and continuity equations of the system at all scales, allowing the prediction of fractures, shear zones and other typical geological patterns out of a randomly perturbed initial state. The results of this approach are linking a global geodynamic mechanical framework over regional-scale mineral deposits down to the underlying micro-scale processes. Ongoing work includes the challenge of incorporating chemistry into the formulation.

Mineral Reactions: Equilibrium and Non-Equilibrium Aspects

Mineral reactions are a fundamental component of the development of metamorphic rocks but the subject is commonly discussed without reference to deformation which ubiquitously accompanies the chemical processes. In this chapter we concentrate on the coupling between mineral reactions and deformation with particular reference to the controls that processes involved in mineral reactions exert on the development of metamorphic fabrics. Since we are concerned with the processes involved in chemical reactions and the coupling to deformation our emphasis is on systems not on equilibrium. We first discuss the conditions for coexisting minerals to be at chemical equilibrium during deformation and the controls on the position of the equilibrium phase boundary during deformation and diffusion. We then proceed to discuss the behaviour of chemical systems not at equilibrium and the roles of non-equilibrium stationary states; we consider the evolution of chemical systems to non-equilibrium stationary states driven by the Ross excess work instead of differences in the Gibbs energy which drive systems to an equilibrium stationary state. The fundamental importance of processes associated with the nucleation and growth of new mineral grains during deformation is emphasised particularly the isochoric replacement of old grains by new ones, the role of stress-assisted mass transfer (‘pressure solution’) and the significance of networked chemical reactions. Chemical dissipation is discussed in terms of the Prigogine principle of minimum entropy production.

Transport of Heat

This chapter is concerned with the transport of heat by conduction and thermal advection in deforming metamorphic rocks. We first consider the diffusion of heat governed by Fouriers Law of heat conduction and the heat conservation law. This enables the thermal diffusivity to be defined together with the characteristic time for heat conduction over a characteristic distance. Typical values of thermal conductivity are given along with the dependence on temperature and pressure. Solutions to the heat diffusion equation in terms of the complementary error function are considered. We then discuss internal heat production by radioactive decay of uranium, thorium and potassium and the influence such decay has on heat production for various tectonic models of metamorphic systems. We conclude with a discussion of thermal expansion and entropy production during thermal conduction.

Energy Flow – Thermodynamics

This chapter is concerned with the thermodynamics of systems not at equilibrium. Thermodynamics is the study of the flow of physical and chemical quantities (such as momentum, heat, fluid and chemical components) through or within a system driven by thermodynamic forces. These forces comprise gradients in deformation, the inverse of the temperature, hydraulic potential and chemical potential. Systems not at equilibrium are characterised by competition between thermodynamic forces that tend to drive the system away from equilibrium and thermodynamic flows that tend to return the system towards equilibrium. If the work done by the forces balances the dissipation from the flows the system is at a non-equilibrium stationary state. If there is no flow then the system is at equilibrium which is another stationary state a system can approach. Systems held away from equilibrium by the addition of mass and/or heat or by deformation are driven to one or more non-equilibrium stationary states that may or may not be stable. We develop the basic framework that enables the behaviour of deforming chemically reacting solids to be described in terms of microstructural evolution and entropy production.

Chapter 10 – Damage Evolution

This chapter is concerned with damage evolution, both brittle and ductile. Damage is a term that describes any process that results in degradation of strength or load-bearing capacity and is expressed both as localised and distributed microfractures, intra- and intergranular voids, chemical damage, such as stress corrosion, grain size reduction, ductile fracture and the formation of dislocation patterns. The degradation may be in yield stress, elastic moduli, flow stress or the coefficient of friction. The damage models of Lyakhovsky and Karrech are treated in some detail because they illustrate many of the constitutive and thermodynamic principles discussed in chapters 5, 6xa0Chapter 5xa0Chapter 6 and 7, particularly the use of the second law of thermodynamics in defining thermodynamically admissible rules for damage evolution. The difference between damage localisation criteria based on loss of convexity in the Helmholtz energy and on loss of ellipticity in the governing equations is also illustrated. The application of damage mechanics to strain localisation is emphasised. Damage is also closely related to criticality in deforming systems and we consider this concept in some detail with an emphasis on the development of avalanches and correlation lengths during damage evolution.

Chapter 8 – Brittle Flow

This chapter is concerned with brittle deformation, a process whereby macroscopic deformations are achieved by fracturing at the crystal structure scale. The classical theories of Griffith and Barenblatt are treated with an emphasis on the energy associated with fracture. We concentrate on two aspects of fracture mechanics, namely, those processes associated with fracture pattern formation and those associated with cataclasis (Breakage Mechanics), especially in shear zones. The fundamental principles involved in pattern formation during deformation are considered in a general case and then applied specifically to fracture pattern formation. These principles are two in number: first, the energy associated with deformation is minimised and second, the resulting deformation field must be compatible at both the local and macroscopic scales. These two principles mean that, for a non-convex energy function, no single homogeneous deformation can satisfy both requirements and in general some form of patterned structure develops distinguished by two or more deformation gradients. Fracture pattern formation is treated in terms of defects that enable the imposed deformation to be achieved. These defects are the classical elastic defects of Volterra and include dislocations (Mode 2 and 3 cracks), dilclinations (‘tension’ joints) and disclinations (Mode I cracks). We discuss the number of fracture systems required to accommodate a general imposed deformation. A kinematic view of fracture system development points to the eigenvectors of the velocity field being important in controlling the orientation of fracture systems. In a general macroscopically isochoric deformation, accommodation structures are important and involve the development of both diffuse and localised compaction zones, stylolites and mineral reactions.