[1] Field observations of mature crustal faults suggest that slip in individual events occurs primarily within a thin shear zone, <1–5 mm, within a finely granulated, ultracataclastic fault core. Relevant weakening processes in large crustal events are therefore suggested to be thermal, and to involve the following: (1) thermal pressurization of pore fluid within and adjacent to the deforming fault core, which reduces the effective normal stress and hence also the shear strength for a given friction coefficient and (2) flash heating at highly stressed frictional microcontacts during rapid slip, which reduces the friction coefficient. (Macroscopic melting, or possibly gel formation in silica-rich lithologies, may become important too at large enough slip.) Theoretical modeling of mechanisms 1 and 2 is constrained with lab-determined hydrologic and poroelastic properties of fault core materials and lab friction studies at high slip rates. Predictions are that strength drop should often be nearly complete at large slip and that the onset of melting should be precluded over much (and, for small enough slip, all) of the seismogenic zone. A testable prediction is of the shear fracture energies that would be implied if actual earthquake ruptures were controlled by those thermal mechanisms. Seismic data have been compiled on the fracture energy of crustal events, including its variation with slip in an event. It is plausibly described by theoretical predictions based on the above mechanisms, within a considerable range of uncertainty of parameter choices, thus allowing the possibility that such thermal weakening prevails in the Earth.

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Heating and weakening of faults during earthquake slip

A microstructure-dependent Timoshenko beam model is developed using a variational formulation. It is based on a modified couple stress theory and Hamilton's principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Timoshenko beam theory. Moreover, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, which differ from existing Timoshenko beam models. The newly developed non-classical beam model recovers the classical Timoshenko beam model when the material length scale parameter and Poisson's ratio are both set to be zero. In addition, the current Timoshenko beam model reduces to a microstructure-dependent Bernoulli–Euler beam model when the normality assumption is reinstated, which also incorporates the Poisson effect and can be further reduced to the classical Bernoulli–Euler beam model. To illustrate the new Timoshenko beam model, the static bending and free vibration problems of a simply supported beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. Also, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that by the classical model, with the difference between them being significantly large only for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally. Finally, the Poisson effect on the beam deflection, rotation and natural frequency is found to be significant, which is especially true when the classical Timoshenko beam model is used. This indicates that the assumption of Poisson's effect being negligible, which is commonly used in existing beam theories, is inadequate and should be individually verified or simply abandoned in order to obtain more accurate and reliable results.

A microstructure-dependent Timoshenko beam model based on a modified couple stress theory

A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering

A new model for the bending of a Bernoulli–Euler beam is developed using a modified couple stress theory. A variational formulation based on the principle of minimum total potential energy is employed. The new model contains an internal material length scale parameter and can capture the size effect, unlike the classical Bernoulli–Euler beam model. The former reduces to the latter in the absence of the material length scale parameter. As a direct application of the new model, a cantilever beam problem is solved. It is found that the bending rigidity of the cantilever beam predicted by the newly developed model is larger than that predicted by the classical beam model. The difference between the deflections predicted by the two models is very significant when the beam thickness is small, but is diminishing with the increase of the beam thickness. A comparison shows that the predicted size effect agrees fairly well with that observed experimentally.

https://iopscience.iop.org/article/10.1088/0960-1317/16/11/015/pdf

Bernoulli–Euler beam model based on a modified couple stress theory

Reservoir Geomechanics: Frontmatter

Introduction. Basic concepts from continuum mechanics. Incremental continuum mechanics. Buckling of layered elastic media. Mechanics of water-saturated granular materials. Plasticity theory for granular materials. Bifurcation analysis of element tests. Shear band bifurcation in granular media. Cosserat continuum model for granular materials. Second grade plasticity theory for geomaterials. Stability of undrained deformations. Index.

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Bifurcation Analysis in Geomechanics

Effective stress law for the permeability of a limestone

[1] During an earthquake, the heat generated by fault friction may be large enough to activate the devolatilization of minerals forming the fault rocks. In this paper, we model the mechanical effects of calcite thermal decomposition on the slip behavior of a fault zone during an earthquake. To do so, we introduce the coupled effects of calcite volume loss, heat consumption, and CO2 production in the theoretical analysis of shear heating and thermal pressurization of pore fluids. We consider a rapidly deforming shear band consisting of a fluid-saturated carbonate rock. The equations that govern the evolution of pore pressure and temperature inside the band and the mass of emitted CO2 are deduced from the mass and energy balance of the multiphase-saturated medium and from the kinetics of the chemical decomposition of calcite. Numerical simulation of seismic slip at depths of 5 to 8 km show that decarbonation has two critical consequences on fault slip. First, the endothermic reaction of calcite decomposition limits the coseismic temperature increase to less than ∼800°C (corresponding to the initiation of the chemical reaction) inside the shear band. Second, the rapid emission of CO2 by decarbonation significantly increases the slip-weakening effect of thermal pressurization. The pore pressure reaches a maximum and then decreases due to the reduction of solid volume, causing a restrengthening of shear stress. Our theoretical study shows, on the example of decarbonation, that the thermal decomposition of minerals is an important slip-weakening process and that a large part of the frictional heat of earthquakes may go into endothermic devolatilization reactions.

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Jean Sulem

Papers

Bifurcation Analysis in Geomechanics

Effective stress law for the permeability of a limestone

Thermal decomposition of carbonates in fault zones: Slip-weakening and temperature-limiting effects

An analytical solution for time-dependent displacements in a circular tunnel

Effect of the hydration temperature on the microstructure of Class G cement: C-S-H composition and density