Mathematical model and numerical simulation of slow deformation waves in the earth’s
crust structural elements
P. V. Makarov and A. Yu. Peryshkin
Citation: AIP Conference Proceedings 1783, 020146 (2016); doi: 10.1063/1.4966439 View online: http://dx.doi.org/10.1063/1.4966439
View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1783?ver=pdfcov Published by the AIP Publishing
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Mathematical Model and Numerical Simulation
of Slow Deformation Waves in the Earth’s Crust
Structural Elements
P. V. Makarov
a)and A. Yu. Peryshkin
b)National Research Tomsk State University, Tomsk, 634050 Russia Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634055 Russia
b) Corresponding author: [email protected]
Abstract. Numerical calculations of the formation and propagation of slow deformation waves in geological media are
performed. The velocities of such a tectonic movements usually lie within the range of 1–100 km/year and these movements are treated as slow deformation waves. The deformation autowaves are shown to make a considerable contribution into the formation of fracture foci. When two such autowaves collide, they behave similar to solitons, reflecting from each other as elastic particles. The deformation autowaves form at the boundaries of structural elements, e.g., blocks of a geomedium during their fast movements. An autowave in a geomedium is developed due to a local loss of stability, and the velocity of its motion is found to be proportional to the velocity of crush movement (motion velocity of the grip during the formation of a Lüders front).
AUTOWAVE- AND SOLITON-BASED CONCEPTS OF SLOW
DEFORMATION WAVES
Recently there has been an increased interest in slow deformation wave movement due to the fact that the researchers began to associate energy redistribution in block media and fracture activisation in block geomedia with slow deformation waves [1–4]. A hypothesis has been put forward that a slow deformation wave propagating along a fault acts as a trigger, starting a sequence of earthquakes along the fault. The current evolution of the concept of deformation waves in geomedia is associated with a synergetic viewpoint of their physical nature. Slow deformation fronts in materials and geomedia have begun to be treated as autowave processes controlled by the instability of a damaged medium under loading, its cooperative response and parametric excitation [1, 3–5]. The autowaves in small-size specimens of plastic material and the slow deformation autowaves in geomedia are cooperative deformation processes developing at different scale levels, which appear to be similar in their physical nature.
Currently, the researchers are examining a variety of mathematical models and discussing an idea that the slow deformation waves represent solitons. Relying on this hypothesis, a number of authors propose to simulate the dynamics of fault zones and concurrent deformation and seismic effects using both a classical Sine-Gordon equation and various versions of the perturbed Sine-Gordon equation [2, 3, 5–7].
Due to the fact that real properties of slow deformation fronts, including their relation to the concept of solitons, are unknown (autowave parameters cannot be measured, meanwhile their presence is indirectly validated by variations in the geophysical fields), the interpretation of autowaves as solitons relies purely on investigation of the generation mechanisms of slow wave perturbations. According to some authors [2, 3], these mechanisms represent translational and rotational movements of block systems in the fault zones and displacements within the fault under conditions of rigid fixation of the blocks on their surface, which leaves no doubt. On the other hand, in order to prove the relevance of Sin-Gordon’s equation to the description of generation mechanisms and special features of slow deformation front propagation, use is made of the micropolar elasticity theory, wherein the asymmetrical part
of the stress tensor is assumed to be proportional to the sine of the block rotation angle in the fault. We believe this approach to be incorrect.
The micropolar theory is based on a fundamentally necessary condition of the presence of an external uncompensated moment, i.e., internal motion resulting in an independent material rotation of the macroscopic element. There are no such moments in a geomedium, nor there is an external field capable of ordering the internal moments like a magnet needle does in the Earth’ s magnetic field.
Nevertheless, the hypothesis itself that slow deformation fronts possess solitons’ properties is extremely fruitful and seems to be true [1, 2]; it has provided considerably more understanding of, primarily, the conditions of slow deformation wave generation.
For instance, well known that slow deformation fronts in metals, depending on the level of the actual stress, can move at different velocities, stop, forming localized shear bands, and interect like solitons. The velocities of such a slow wave movements usually lie in the range of 1–100 km/year (10–4–10–5 m/s). We believe the slow deformation
fronts in small-size specimens to have the same physical nature as those in the geomedia due to the scale invariance and self-similarity of deformation and fracture processes. It is for this reason that we would expect the autowaves in geomedia to exhibit similar properties.
In what follows we are going to demonstrate that investigation of a geomedium, or any other solid, as a multi-scale, non-linear dynamic system allows us, within the mathematical framework of the mechanics of deformed solids, to provide a successful description of the effects of evolution of the stress-strain state, including the generation of slow deformation autowaves. It should be underlined that the proposed model can do without the concepts of the micropolar theory, which appear to be controversial and hard to reason when applied to geomedia.
THE MATHEMATICAL MODEL OF GENERATION OF AUTOWAVE
DEFORMATION FRONTS AND THE CALCULATION DATA
There is large number of spatial-temporal variations of deformation autowaves, which is due to a multi-scale fractal-block organization of solids and geomedia. According to the concept developed in works [8–10], all solids, including geomedia, represent complex multiscale non-linear systems. Their hierarchical multi-scale block organization of geomedia (and any other solids), within which the blocks contact each other through the faults of different scales, which represent less strong, damaged medium, provides the conditions where a hierarchy of autowaves of differing energy and range of action could be generated.
Such dynamic phenomena at the boundaries of structural elements as slip and rotation of blocks result in a local loss of stability of the energy-saturated geomedium and generation of deformation autowaves. Thus, slow motions are generated in a geomedium due to different dynamic impacts of natural and technogenic origin.
A mathematical representation of the above concepts involves the basic computational core (equations of the mechanics of deformed solids, Eq. (1)). This computational core also contains kinetic equations (Eqs. (2) and (3)) prescribing the velocities of accumulation of inelastic strains and/or damage. In conjunction with the method of cellular automata, these equations allow calculating a consistent cooperative response of a medium to its loading [9]. This procedure allows self-organization of individual acts of inelastic deformation to be simulated as localized plasticity fronts or damage waves:
e v , d d 1 1 div 0, , , , ; d d 3 ij ij i i ij i i ij ij ij ij ii j p E F q P S S P t t x t t (1) p t p t p t e p t t t p ef ef 1 3 ( ) 2 ( ), , , , ; 2 2 j i ij ij ij ij ij ij ij ii ij xj xi ij Sij (2) 1 2 1 2 p p p 2 p p 2 p p 2 2 2 2 ef 1 2 2 3 3 1 1 2 2 3 3 1 ef 2 1 ( ) ( ) ( ) , ( ) ( ) ( ) . 3 2S S S S S S (3)
The constitutive macroscopic equations (2) are written for velocities in the relaxational form and allow the kinetic equations to be included into consideration in order to prescribe the velocities of inelastic deformation as, e.g., it was done in [9] by setting the dislocation kinetics for p
ef.
In the present calculations, stress relaxation was taken to be instantaneous and was performed by reducing the stresses to the prescribed yield surface.
(a) (b) FIGURE 1. The scheme for σ–ε diagrams with Δσ и Δε thresholds in relaxational model [8, 9] (a);
trajectory of deformation fronts (Va = 23 ± 1 m/s, Vb = 24 ± 1 m/s) (b)
Earlier the proposed model was used to investigate the formation of autowaves in different media [10] and their propagation velocities as a function of loading. The presence of the stress relaxation after the yield drop is essential for the formation of a slow deformation wave front (Fig. 1a). Its speed, as in the experiments [11], is approximately 20 times higher than the loading rate (grip velocity at tension is 1 m/s, front velocity Va = 23 m/s and Vb = 24 m/s
(Fig. 1b)). (1) (300) (300) (2) (325) (340) (4) (360) (360) (6) (380) (400) (7) (400) (420) (9) (430) (430) (10) (445) (445) (a) (b) FIGURE 2. Spatial distribution of inelastic deformation fronts for different time sequences:
In this work, we focus on the features of interaction of deformation autowaves. Figure 2 illustrates this interaction both for the case of autowave generation in a ductile metal specimen under tensile loading and in a brittle rock under compression. When collided, these waves behave as solitons, interacting as elastic particles.
CONCLUDING REMARKS
A hypothesis has been put forward and validated that all disturbances of the stress-strain state observed in solids and geomedia, which propagate as slow deformation fronts, including plastic deformation fronts in small-size specimens, metals and rocks, the fronts in geomedia and faults of different scales, have a common physical nature. These wave disturbances have been treated in this work as autowave processes. Deformation autowaves develop due to the loss of stability and parametric perturbation of the loaded elastoplastic medium.
The formation of deformation autowaves in solid media under loading represents the processes of self-organization in solids as non-linear dynamic systems possessing a criticality property. They develop in a self-similar manner at different scales. This is the reason for a wide variety of slow autowave disturbances.
When interacting, the slow deformation autowave disturbances demonstrate the properties of solitons.
The mathematical model of the process of generation and dynamics of slow autowaves involves equations of the mechanics of deformed solids, positive and negative feedback, and kinetic equations prescribing the velocities of inelastic deformation and damage. These are equations of mixed type and their solutions contain both ordinary stress waves, propagating at the velocity of sound, and slow perturbation waves, behaving as solitons. The latter circumstance is especially important and requires an in-depth investigation. The fact that solutions to the equations of the mechanics of deformed solids, expressing the laws of conservation, could, given the availability of non-linear constitutive equations prescribing the kinetics of accumulation of inelastic deformation and/or damage, give rise to the formation of solitary waves is an essentially new result. The data obtained indicate that soliton-like disturbances formed in a non-linear medium are quite stable, they can stop forming regions of inelastic strain localization, which suggests that their properties are close to those of solitons satisfying the solutions to the Sine-Gordon equation.
ACKNOWLEDGMENTS
The investigation is carried out in the frames of program of fundamental research of State Academies of Sciences for 2013–2020 years.
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