Pavel V. Makarov

Model of formation of large cracks in brittle solids taking into account various space–time scales of crack accumulation during deformation is constructed. In the framework of the phase field method, a system of two coupled nonlinear parabolic equations which describes the deformation of a loaded medium during crack initiation is obtained on two spatial and time scales. The conditions at which a homogeneous distribution of cracks becomes instable are found. The development of the instability is accompanied by the formation of large cracks.

A new autosoliton view is developed for the seismic process. In physical terms, faults correspond to stationary autosolitons, and inter- and intrafault deformation disturbances are traveling autosolitons. Slow dynamics reveals itself only on large time scales because slow autosoliton disturbances, as a rule, have velocities 4–7 orders of magnitude lower than the sound velocity. It is shown that, in the loaded strong medium, slow autowave and autosoliton disturbances are generated by short dynamic actions (pulses) at interfaces. In real geomaterials, these are block boundaries and various-scale faults. Dynamic movements of structural elements cause the deformation autowaves and autosolitons to propagate from the interfaces into blocks and along faults. Velocities of such deformation autowaves and autosolitons are low and proportional to velocities of the related movements of structural elements in the geomedium. Propagating in structural elements that are in a certain stress-strain state, deformation autowaves and autosolitons can be taken as small disturbances of the existing fields of the stress-strain state. A mathematical model is represented for the geomaterial treated as a nonequilibrium randomly inhomogeneous medium. Special features of the generation and propagation of deformation autosolitons in such media are studied.

In the autosoliton view, the complete mathematical model of the seismic process taken as the deformation and fracture process of a loaded geomedium combines dynamic equations of solid mechanics and specific constitutive equations for geomedium rheology. These equations describe both the conventional stress-strain evolution due to the stress wave propagation with sound velocities, which are governed by special features of constitutive equations, and slow dynamics of the loaded strong medium. Numerical investigation is given to the generation of deformation autosolitons, front structure, and propagation of intra- and interfault deformation disturbances. Slow deformation disturbances in real geomedium elements are numerically modeled.

The paper describes numerical modeling of the generation and propagation of the fronts of moving deformation autosolitons in a loaded nonlinear strong medium. It presents solving a system of dynamic equations for solid mechanics, using an equation of state written in a relaxation form that takes into account both an overload of the solid medium and subsequent stress relaxation. The structure of a deformation autosoliton front is investigated in detail. It is shown that the front of a deformation autosoliton that is moving in an elastoplastic medium is a shear band (i.e. a narrow zone of intense shearing strain), which is oriented in the direction of maximum shear stress. Consecutive formation of such shear bands can be viewed as deformation autosoliton perturbations propagating along the axis of loading (compression or extension). A fine structure of a deformation autosoliton front is revealed. It is shown that slow autosoliton dynamics is an integral component of any deformation process, including the seismic process, in any solid medium. In contrast to fast autosoliton dynamics (when the velocities of stress waves are equal to the speed of sound), slow deformation autosoliton perturbations propagate at velocities 5–7 orders of magnitude lower than the velocities of sound. Considering the geomedium, it should be noted that slow dynamics plays a significant role in creating deformation patterns of the crust elements.

Doppler laser interferometry is used to measure the transient time between the slow quasi-stationary stage of damage accumulation in rock samples to the ultrafast catastrophic stage of failure as well as the duration of the autocatalytic stage of macroscopic fracture. Small rock samples are tested for compression and three-point bending, and the velocity of displacement of their lateral surfaces is measured up to macroscopic fracture. The surface velocity at the catastrophic stage proves to be three orders of magnitude higher than the average surface velocity at the quasi-stationary stage of damage accumulation. The transient time to catastrophic failure is estimated at 60–100 ms, and the duration of the ultrafast catastrophic failure stage is 15–20 ms for small marble samples. The transient stage is the process of self-organization of individual acts of fracture into the state of self-organized criticality. At this stage, the distribution of individual acts of fracture evolves into power-law distributions. A simple fracture model with power laws is proposed, which is in full agreement with the experimental data. The developed mathematical model is used to calculate fracture of small rock samples, reproducing uniaxial compression and three-point bending tests, as well as fracture in rock masses with mine openings. We also model the process of faulting and fracturing in the mountains of Central Altai, including the foreshock process, main event (the Chuya earthquake of September 27, 2003) and aftershock process. The calculated seismic process fully corresponds to the Gutenberg–Richter recurrence law, and the calculated aftershock process conforms to the Omori law.

A model of the large cracks formation in brittle solids is formulated taking into account different space-time scales of crack accumulation during deformation. Within the framework of the phase field method, a system of two coupled nonlinear parabolic equations is obtained that describe the deformation of a loaded medium during crack initiation on two spatial and temporal scales. The conditions are found under which the uniform distribution of cracks becomes unstable. The development of instability is accompanied by the formation of large cracks.

A variant is considered for the relaxation model of a loaded elastic-plastic medium with dislocation kinetics of plastic shearing. The model is formulated in rates and includes two independent strain rates: total strain rate, which corresponds to the rate of external action, and local rate of plastic response of the material, which represents the ability of the medium to generate strain-induced defects. This makes it possible to describe both local relaxation processes in the elastic-plastic medium and average relaxation of stresses in a loaded specimen. The model being developed amounts to microscopic ones. All model parameters are determined from independent experiments for the evolution of the dislocation continuum during loading of macroscopic specimens. The model provides an adequate description of the dynamic effects of the macroscopic response of materials depending on the strain rate: the upper and lower yield points (yield drop, yield plateau), subsequent strain hardening as well as features of cyclic and alternating loading, ideal and nonideal Bauschinger effect.

The paper considers an autosoliton model of slow deformation disturbances propogating in the elements of the Earth’s crust. An autosoliton model of the fault’s formation is also presented. The faults themselves are represented as static autosolitons. It is shown that running (moving) autosoliton perturbations are actively involved both in the formation of fault zones and in the formation of fracture foci by the means of an increase in the degree of inelastic strain and damage in the medium. The autosoliton concept developed by the authors for slow deformation processes in geomedia allows explaining from one point of view both the processes of faults formation, as stationary (pulsating) autosolitons, and the processes of fracture foci formation.