ISSN 1029-9599, Physical Mesomechanics, 2013, Vol.OF 16, No. 3, pp. 207–226 © Pleiades Publishing, Ltd., 2013. FRACTURE MODEL BRITTLE AND QUASIBRITTLE MATERIALS AND Original Russian Text © P.V. Makarov, M.O. Eremin, 2013, published in Fizicheskaya Mezomekhanika, 2013, Vol. 16, No. 1, pp. 5–26.

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Fracture Model of Brittle and Quasibrittle Materials and Geomedia P. V. Makarov* and M. O. Eremin Institute of Strength Physics and Materials Sciences, Siberian Branch, Russian Academy of Sciences, Tomsk, 634021 Russia National Research Tomsk State University, Tomsk, 634050 Russia * [email protected] Received June 15, 2012

Abstract—A general model and a unified mathematical formalism are proposed for description of inelastic deformation and fracture of any solids, where brittle or plastic, as their evolution in effective force fields. Loaded solids and media are considered as nonlinear dynamic systems. One of the main tasks of the work is to show that if deformation and fracture of a strong medium is treated as its evolution in effective force fields, numerical solutions of equations of solid mechanics do demonstrate the fundamental properties of nonlinear dynamic systems—self-organized criticality and two-stage evolution (a rather slow quasistationary stage and a superfast catastrophic stage or a blowup mode). The model is tested by simulation of fracture of quasibrittle composites under axial compression. Calculations are also presented for the now developing tectonic flows and seismic processes in Central Asia, including the Baikal rift zone and the Altai-Sayany folded region. It is shown that numerical solutions of all examined problems of inelastic deformation and fracture demonstrate self-organized criticality of loaded media, including peculiarities of slow dynamics and spatial-temporal migration of deformation activity, and as well as two-stage fracture: comparatively slow quasi-equilibrium damage accumulation and superfast catastrophic fracture. The predicted seismic events obey the Gutenberg–Richter law. DOI: 10.1134/S1029959913030041 Keywords: quasibrittle fracture, nonlinear dynamic systems, self-organized criticality, quasistationary phase, blowup mode, migration of deformation activity.

1. INTRODUCTION. GENERALIZED PLASTICITY AND BRITTLE FRACTURE CONDITIONS The development of a fairly general methodology for simulating inelastic deformation and fracture of solids and media remains an urgent problem which is far from being resolved in solid mechanics. This methodology should be based on most general physical and mathematical ideas and provide description of inelastic deformation and macrofracture of any strong medium (be it brittle or plastic) in effective force fields. The most general ideas and methods, in our opinion, are the concept of a loaded strong medium as a multiscale nonlinear dynamic system [1–4] and the methodology of nonlinear dynamics [5–7] with which deformation processes can be studied as evolutionary processes. The presented paper concerns the development of a mathematical evolution theory of loaded solids and mePHYSICAL MESOMECHANICS

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dia [4, 8]. One of the chief tasks of the work is to show that numerical solutions of equations of solid mechanics do demonstrate the most significant fundamental property of nonlinear dynamic systems—self-organized criticality—on examples of numerically solved test problems of inelastic deformation and fracture of quasibrittle media. Another task is to show the capability of the developed approach to provide description of macrofracture as a superfast catastrophic stage of evolution of a loaded strong medium—the so-called blowup mode [9, 10]. Among the central problems of solid mechanics remains formulation of fracture conditions and criteria. This problem in the work is also solved reasoning from the mathematical evolution theory of loaded solids and media [4, 8]. A material is considered to be destroyed when its strength parameters during evolution degrade to zero. This viewpoint is inconsistent with conventional concepts of solid mechanics. According to conventional

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fracture mechanics, local fracture in a solid occurs when its ultimate load is reached. All experience of limit design with this approach demonstrates its acceptable efficiency and correctness in solving many practical problems. However, nothing can be said about fracture as such, much less about its prediction. The fracture time, even in a dynamic problem, is in no way related to the properties of a medium and is determined by the rate of increase in elastic stress which is completely determined by the rate of external loading. If a certain constant or variable load is applied to a solid, we can calculate only the corresponding stress-strain state and answer the question of whether an ultimate load is reached somewhere in the solid. In some important engineering problems, these data are useful and sufficient but give nothing about the formation of a fracture nucleus, its mechanisms and patterns. Of tremendous significance for fracture mechanics is the quasibrittle fracture concept formulated by G. Irwin and E. Orowan and the impetus that it gave to go from ideal material in the Griffiths concept to real materials, including plastic materials. It was found that in the vicinity of the surface of a macroscopically brittle fracture, there always exist appreciable inelastic strains of lower scales [11, 12], and this means that fracture and plastic deformation are multiscale processes. Solid mechanics, being based on conventional macroscale concepts and force criteria, was not ready for dealing with such multiscale problems. So the macroscale approach alone, though it did provide many valuable data on individual cracks (see, e.g., the reviews [11, 13]), failed to offer a solution for the fracture problem of a material, a specimen, or a construction as a multiscale nonlinear system. In the criteria proposed for transformation of material to its plastic state or fracture, the loading time was in no way taken into account and thus the possibility to predict the onset of fracture was fully excluded. However, much experimental experience did allow mechanical engineers to develop key ideas and formulate correct qualitative concepts as to peculiarities of inelastic deformation and subsequent fracture of plastic and brittle media. So even in the 50s of the last century, N.N. Davidenkov pointed to the necessity of considering not plastic and brittle solids but their plastic or brittle response to loading and proposed a scheme for the plastic-to-brittle state transition of loaded material. Later on, understanding this fundamental property allowed formulating the concept that the limiting state of material is impossible to describe by a single equation [12]. Thus fracture of material, depending on the type of its stress-strain state,

can be both brittle (by cleavage) due to normal stress and ductile (by shear) due to tangential stress. From this followed the fundamental conclusion that the form of a limiting surface and its properties are completely determined by three parameters of the stress-strain state: octahedral normal stress σoct , octahedral shear stress τoct , and stress state type µ σ (Lode–Nádai parameter) [12]. As early as the 1970s, there was established a fundamental fracture mechanism for any materials: complete fracture (whether fatigue or any other) is preceded by a more or less long preparatory period. So for silicate glass whose fracture was considered instantaneous, the crack velocity early in the process was found to be thousand times lower than that at its final stage [11], even though the fracture as a whole took a few milliseconds. It was also clear that the main thing is to design and manufacture constructions so that development of cracks (as they already exist) be stable and preferably predictable in a wide range of allowable loads, and superfast critical fracture be put off as far as possible. It was found that a crack, if stable, fails to develop when external load is constant and is less than a certain critical value [11]. Another fundamental regularity disclosed in classical fracture mechanics is that a small increment in external load dP involves the same small increment in the length or area of a crack provided it is stable. Consequently, a stable crack obeys the inequality dP dt > 0, and an unstable unlimitedly growing crack (in its supercritical state) obeys the inequality dP dt < 0 [11]. A crack starts growing when a certain critical load is reached [11, 12], and the fracture velocity at this supercritical stage increases many orders of magnitude. Hence, fracture should be studied as a process whose final evolution proceeds in a superfast catastrophic mode or the so-called blowup mode by terminology of nonlinear dynamics [9, 10] and only dynamics can solve the fracture problem and describe various possible scenarios by which a fracture nucleus is formed. Critical loads were estimated both from the energy criteria [14] and from the force criteria pioneered by G. Irwin. Of importance here is the equivalence shown between these two (energy and force) approaches [15, 16]. Clearly, following the fundamental works by A. Griffiths and G. Irwin, fracture mechanics focused first on propagation mechanisms of individual cracks. The main lines of research in this field can be formulated as follows: (i) strength of crack-containing solids and media; (ii) crack geometry and stress-strain states near cracks; and (iii) crack dynamics [11]. Solving problems of joint PHYSICAL MESOMECHANICS

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development of two or several cracks with the above classical approaches ran into almost insurmountable mathematical difficulties and limitations of the average macroscale approach. In those years and in the next decades, there was rapid progress in nonlinear dynamics, giving birth to fresh ideas. So a new concept on superfast catastrophic evolution of nonlinear systems (blowup modes) was formulated by S.P. Kurdyumov, et al. [9, 10] and types and peculiarities of these modes were studied both analytically and numerically. The basic equation for blowup modes was a nonlinear heat conduction equation. It was found that when the distribution of parameters in any nonlinear system is localized in space, the evolution of the system becomes localized in time and passes into a superfast catastrophic stage—the system evolves in a blowup mode [9, 10]. These ideas and associated qualitative results are also a key to understand the process of fracture [3, 4, 8]. In brittle or quasibrittle fracture (as well as in fracture of any materials and constructions, plastic metals, brittle concrete, rocks, geomedia, etc.), the preliminary stage of inelastic strain or damage accumulation is localized in definite regions. This preliminary quasistationary stage, due to self-organized criticality of a deformed solid as a nonlinear dynamic system, transforms sooner or later to a superfast catastrophic stage—to a blowup mode by S.P. Kurdyumov [3, 4, 8, 17, 18]. It is clear that any fracture prediction is in principle impossible without knowledge of peculiarities of these stages and conditions for fracture transition from stable to unstable superfast development. Of significance for the present study is that classical approaches of fracture mechanics concerned, in one way or another, the existence of a quasistationary prefracture stage associated with inelastic strain and/or damage accumulation on scales much smaller than the macroscale as well as the existence of a superfast catastrophic fracture stage (supercritical unstable stage). However, classical mechanics failed to offer satisfactory multiscale mathematical models of these fracture stages and fracture as a whole. Solving these problems and the problem of correct fracture prediction required radically new ideas and approaches, which were just those formulated in the framework of physical mesomechanics of materials [1, 2] and nonlinear dynamics [3–7]. The concept of physical mesomechanics according to which inelastic deformation and fracture develop in the entire hierarchy of scales—from scales of interatomic spacing to macroscales equal to characteristic specimen PHYSICAL MESOMECHANICS

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sizes—are now commonly accepted. It is also recognized that the key place in the scale hierarchy belongs to the so-called mesoscale which combines a group of mesoscopic scales representative of most significant structural elements responsible for leading deformation mechanisms. The basic ideas and principles of physical mesomechanics have been given rather complete coverage, see, e.g., [1, 2]. Physical mesomechanics treats a loaded solid or medium as a hierarchically organized nonlinear dynamic system which evolves by the laws of nonlinear dynamics. It is also shown that to provide correct qualitative results in simulation and, in many cases, a good quantitative agreement, there is no need to explicitly account for many scales, as this is redundant [4, 8]; it is quite sufficient to consider three scales: the macroscale (an average level), the mesoscale at which most significant structural elements are considered (e.g., for ceramic composites, these are a matrix and rigid hardening particles, including their interfaces), and the microscale which integrally includes scales lower than the mesoscale on which structural elements are explicitly allowed for. Microscale description, as a rule, involves certain kinetic equations to trace the contribution of microscopic scales to macroscale deformation and/or fracture of particles [4, 19]. Thus, fracture is a multiscale hierarchically organized process, and any loaded solid is a nonlinear dynamic system whose evolution under applied loads obeys the laws of nonlinear dynamics. The mathematical evolution theory of solids and media combines the basic principles of classical solid mechanics and the concepts, approaches, and ideas of physical mesomechanics of materials and nonlinear dynamics. At the heart of the mathematical evolution theory of solids and media lie equations of solid mechanics as fundamental equations of mathematical physics reflecting the most general natural laws of conservation of mass, momentum, moment of momentum, and energy [4]. At this phenomenological level, all the variety of physical mechanisms of inelastic (plastic) deformation and dilation, i.e., development of discontinuities of different scales and various physical nature, is integrally described by nonlinear loading response functions of a medium—constitutive evolution equations of the first and second groups. Constitutive evolution equations of the first group provide average macroscale description and specify relations between macroscale (average) parameters of a medium: its stress rate, inelastic strain rate, and discontinuity (damage) accumulation rate. These equa-

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tions should also include sources of inelastic strains and damages on which nonstationary dissipative structures may arise [4] and positive and negative parameter feedbacks that control the formation of various structures in a loaded medium [4, 20]. Another fundamental function of the feedbacks is in guiding the evolution of a deformed medium by specific scenarios. In particular, they play a decisive role in fracture transition from quasistationary to superfast catastrophic evolution. It is the competition of stabilizing negative feedbacks and destabilizing positive feedbacks in the entire scale hierarchy which is responsible for inelastic deformation and fracture as a sequence of progressive catastrophes separated by quasi-equilibrium stages of rather slow damage accumulation. Evolution equations of the second group are kinetic equations that specify inelastic (including plastic) strain accumulation rates or damage accumulation rates and establish relations between the processes on the macroscale and lower scales, i.e., on the microscale. In this context, the microscale is understood as a hierarchy of scales which are taken into account not explicitly but integrally through specifying appropriate accumulation kinetics of various discontinuities and damages for a medium. This group of equations also includes equations descriptive of variation in average macroscale strength characteristics of a medium or geomaterial. They involve strength degradation rates due to damage accumulation and strength recovery rates due to healing of damages, if this process takes place [4]. In the present work, the limiting surface is a surface which meets the generalization requirement for plasticity and brittle fracture [12] and takes into account the stress state type. This surface can be represented in the following general form: f (σoct , τoct , µσ , ci ) = 0. (1) Here ci are certain material characteristics which, in this case, reflect the resistance of a brittle material to tension and compression; µ σ is the Lode–Nádai parameter. Based on generalization of the Coulomb–Mohr hypothesis, D. Drucker and V. Prager formulated the yield criterion [21]: α (2) f = I1 ( σij ) + J 12 2 ( σij ) − Y 3 which partly meets condition (1), as there is no dependence on the stress state type, and takes into account internal friction α and dilatancy. According to the notion of plastic potential, which coincides with yield function (2) in the associated plastic flow rule, we have the following expression for inelastic strain rates ε ijp [21]:

α  sij ∂f ε ijp = λ = λ  δij + 1 2 (σij )  . (3) ∂σij 2J 2 3  Hence, in the Drucker–Prager model, the dilatancy ε ijp is a dependent quantity which is completely determined by the internal friction parameter α:  . I1 (ε ijp ) = ε iip = λα (4)

In our calculations, we use the modification of equations (2), (3) which was proposed by V.N. Nikolaevsky [22]. This modification is a thoroughly studied model [19, 23] in which dilatancy is independent of internal friction, and thus inelastic deformation of material obeys the non-associated rule [22]. The model, as shown [19, 22], well describes the behavior of soils and rocks, in particular sandstones. In the presented version of the model, inelastic deformation of a quasibrittle material is ensured by microscale damage accumulation and is described through reducing the stress to an instantaneous limiting surface. This procedure, as shown by S.S. Grigoryan in his brief annotation to Wilkins’s study [24], is equivalent to inelastic flow theory. Thus, small accumulated inelastic strain corresponds to prefracture of a quasibrittle material. The function of the damage parameter D of a medium is described by a kinetic equation and depends on the stress state type µ σ . The model includes a procedure for degradation of strength characteristics and elastic moduli of a medium as damages are accumulated in it. At the final superfast catastrophic prefracture stage, the degradation rate increases many orders of magnitude resulting in an avalanche decrease in strength characteristics in corresponding local regions and hence in a blowup mode, with Y → 0 in (2), i.e., the material is considered to be destroyed. The fact that macrofracture occurs in degraded material regions with decreased strength characteristics was understood long ago. So in considering mechanical theories of limiting states, the authors of [12] pointed out that in shear fracture, crack nucleation is preceded by plastic deformation due to tangential stresses (of the 2nd kind), the material is “loosened” (i.e., accumulates damages and defects on the microscales), and the cohesive forces in slip planes vanish. However, due regard to this fact in macroscale models was not given and criterion theories of limiting states consider that fracture occurs when maximum critical stress is reached in a material. Moreover, fracture was treated as an instantaneous event, whereas it is a long-term process of defect and damage accumulation in the entire scale hierarchy, including the whole stable quasistationary stage which not merely precedes PHYSICAL MESOMECHANICS

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the superfast catastrophic mode but paves the way to this mode and completely determines it. Thus, the stage of avalanche increase in inelastic strain rate and catastrophic strength degradation of material is just its fracture as such in the model considered. This approach removes the contradictions between description of inelastic deformation prior to fracture of material and corresponding strength calculations. Actually, the developed approach implies that a deformed medium even with greatly decreased strength characteristics remains consolidated on the macroscale. In formal average macroscale description, this view on deformation makes no distinction between plastic deformation and fracture in their conventional sense, e.g., between plastic flow of metals and quasibrittle deformation of rocks or geomedia. In both cases, the process under study is inelastic deformation during which a loaded medium progressively looses its strength due to damage accumulation. The term “inelastic deformation” implies that deformation remains to be a continuous process, and a medium remains to be a continuum only on the macroscale, while being involved in discrete events of inelastic deformation—micro- and mesoscale fractures accumulated by defects of different scales and various physical nature. This general approach allows one to distract from numerous specific mechanisms and to focus on general mechanisms of deformation in solids as nonlinear dynamic systems. Note that at α = Λ = 0 in constitutive relations (2), (7), and (8), we obtain the Prandtl–Reuss plastic flow model in which the plastic shear rate ε p can be described, for example, by the Orowan equation ε p = bN f L(σ) (with b, Nf, and L(σ) being, respectively, the Burgers vector magnitude, the density of mobile dislocations, and the average dislocation velocity) and appropriate dislocation kinetics [19]. Thus, by more specifically defining the loading response functions of a deformed material (constitutive equations of the second group), it is possible to consider various mechanisms responsible for inelastic deformation of brittle and plastic media. In all cases, the general phenomenology and the mathematical formalism on which to describe deformation remain the same. In the opinion of A.H. Cottrell [25], the theories that concern phenomenology of fracture should stem only from common properties inherent to all or to most of solids and should not refer to specific mechanisms (e.g., those in which dislocations participate) inherent to only some of them. These common properties of deformed solids as nonlinear dynamic systems are self-organized criticality, slow dynamics, and blowup mode of deformation at its final stage. PHYSICAL MESOMECHANICS

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2. MATHEMATICAL PROBLEM STATEMENT. MODEL OF QUASIBRITTLE MEDIUM According to the evolution theory of inelastic deformation and fracture [4], the total system of equations includes the fundamental laws of conservation of mass, momentum, and energy as well as the evolution equations of the first and second groups. The fundamental conservation laws are written as follows: for mass, dρ + ρdiv v = 0; dt for momentum, dL ∂σ ρ i = ijj + ρFi ; (5) dt ∂ x and for energy, d E 1 dεij = σij . dt ρ dt The evolution equations of the first group are written in relaxation form (6). This means that the increment in stress ∆σij = σ ij ∆t is proportional to the increment in total strain rate ε ijt , and the stress relaxation is proportional to the inelastic strain rate ε ijp . The procedure of reducing the stress to an instantaneous limiting surface means instantaneous stress relaxation at each time step to some dynamic equilibrium defined by the relaxation and by the degradation rate of strength and elastic parameters of a medium. At ε ijp > ε ijt in (6), we have ∆σij < 0; that is, we have relaxation and negative feedback which stabilizes deformation in dynamic equilibrium [4, 20] determined by current values of strength parameters. At ε ijp < ε ijt , ∆σij > 0 and the stresses increase thus increasing the generation rate of inelastic strains or damages. In this case, positive feedback resulting in a catastrophe—macroscopic fracture—is realized. A decisive role in the transition of a medium to superfast catastrophic evolution of its strength belongs to peculiarities of localized inelastic strain accumulation by the medium and associated degradation of its physical and mechanical parameters (this issue is considered below): σ ij = λ (θ t − θ p )δij + 2µ (ε ijt − ε ijp ),

σij = − Pδij + Sij , V DSij 1   P = − K , = 2µ  ε ij − θδ ij  , 3 V Dt   DSij   jk − S jk ω  ik , = Sij − Sik ω Dt 1  ∂L ∂L j  1  ∂L ∂L   ij =  ij − ij  , ε ijt =  ij + i  , ω 2  ∂x 2  ∂x ∂x  ∂x 

(6)

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where D Dt is the Jaumann corotational derivative, which takes into account rotation of elements of a deformed medium. The evolution equations of the second group serve to determine the inelastic strain rates in equations (6). In the general case, these are kinetic equations which specify inelastic strain rates and provide elastic stress relaxation in (6) [4]. In the present work, the inelastic strain rate components are defined according to plastic flow theory and instantaneous stress relaxation at each time step. The limiting stress surface is given in Mises–Schleicher form (2) and is a generalization of the Coulomb– Mohr yield criterion. The basis is the Drucker–Prager– Nikolaevsky model that uses the non-associated flow rule and allows description of dilation as a process independent of internal friction. In the non-associated inelastic flow rule, the plastic potential does not coincide with the yield function and its form for limiting surface (2) is as follows [19, 22, 23]: Λ  α  g (σij ) = J 2 + J1  2Y − J1  + const; (7) 3  3  the inelastic strain rate components are defined as [19, 22]: 2  ∂g  α   ε ijp = λ =  sij + Λ  Y − I1  δij  λ , (8) 3  3   ∂σij 

I1p = 2Λ ( I2p )1 2 ; and this allows us to determine the relation between the bulk and shear plastic strain components (the second equation in (8), where Λ means the dilatancy factor. However, the model is yet unrelated to the stress state type. An appropriate dependence will be written in specifying the damage accumulation rate. Fracture of material in the developed approach is described as avalanche degradation of its strength to zero when a crack develops at the superfast catastrophic stage of stress-strain state evolution. This stage is a final prefracture stage and the medium at Y → 0 remains consolidated. Hence, all equations of inelastic deformation (1)– (8) remain valid. There is also no need to introduce strength parameters for the limiting state in the model. In the concept developed in the work, the limiting state in a loaded medium is formed as inelastic strains—damages—are accumulated in the medium and we are to specify only the initial material strength Y0 . According to the classical kinetic fracture concept (by N.S. Zhurkov, A.V. Stepanov, R. Becker, Ya.I. Frenkel, et al.) [26–30], bringing an undamaged crystal to the state of local shear requires a work propor-

tional to the difference in free energy F between the perfect crystal and the crystal in its current state: A(σ) ~ V (σ02 − σ2 ) (2µ), where V is the volume and σ0 is the theoretical strength. Following R. Becker [30] who assumed that A(σ) ~ V (σ0 − σcur )2 (2µ )  (σcur is the current stress acting on the material volume), fracture mechanics began wide use of phenomenological expressions for the measure of damage accumulation D in the form: t (σ − σ )2 D = ∫ 0 2 cur dt , 0 ≤ D ≤ 1. σ* t* 0 The strength of a crystal and its structural state, cold working, and hardening were first related by E. Orowan who put that activation energy depends only on plastic (inelastic) strain and that ∆σ = σ − σ0 = hεp , where h is the strain hardening coefficient [31]: F (σ) ~ h2V ε 2p (2µ). Let us use this Orowan’s idea to define the function of the damage parameter of a loaded medium. Analysis and criticism of various approaches used to obtain dependences for the energy stored in inelastic deformation can be found, e.g., in [32]. The degradation function of a medium D = D (t , µ σ , ε) in its simplest variant can be represented as a dependence on the inelastic strain accumulated by the medium ε p and its stress state. This phenomenological expression is given below, (9), and corresponds to Orowan’s idea that the activation energy required for the material volume V to reach the state of local shear is proportion to the accumulated plastic (inelastic) strain, because εcur − ε0 = ε p : t [(ε − ε ) 2 + K1 (εcur − ε′0 ) 2 ]dt , D = ∫ cur 0 ε∗2t∗ t0 ε∗ = ε0∗ (1 + µ σ ) n , Y = Y0 (1 − D ), D ≤ 1, (9) S −S µ σ = 2 2 3 − 1, S1 − S3 where ε cur is the second invariant of the total strain tensor; ε0 , ε′0 are the initial elastic strain degrees at which damage accumulation begins in compressive and tensile material regions, respectively, and ε′0 << ε0 . So, damages in tension-shear regions (µ σ < 0) start being accumulated at stresses much lower than those in compression-shear regions (µ σ > 0). The damage accumulation rate in local regions with µ σ < 0 is also much higher than that in compression-shear regions with µ σ > 0. This process is controlled by the parameter ε* = ε* (µ σ ) in (9). Thus, the response of a medium to the stress state type (its current strength) develops in the course of its PHYSICAL MESOMECHANICS

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(b)

Fig. 1. Model structures of the composites with a hardening particle percentage of 15% (a) and 40% (b).

loading. Consequently, the strength parameters will degrade much faster in those regions (particles) of a medium where the Lode–Nádai parameter is decreased (µ σ < 0), i.e., the stress-strain state is governed predominantly by tension-shear. This response also depends on loading history. If any particle of a medium experiences first, e.g., tension-shear and then the stress-strain state changes for compression-shear, the further fracture develops, though much slowly, at decreased strength parameters according to kinetics (9). At µ σ ≤ 0, K1 = 1 and at µ σ > 0,  K1 = 0, ε0* is a model parameter, and t* is the characteristic time of the process. Any solid under loading sooner or later fails; macroscopic fracture, as a rule, is considered as being instantaneous, while stable inelastic deformation as a prefracture stage can be quite long. Below we show that the superfast blowup stage in quasibrittle media under constrained deformation, due to dilation, can also take a long time. These processes in the model are described by the dependence for the damage parameter D and by the degradation law. The results presented below were obtained in simulation of inelastic deformation and fracture of brittle and quasibrittle media by solving the system of equations (2), (5)–(9) in the two-dimensional statement with the use of the second-order accurate finite difference scheme described in detail in [24]. 3. SIMULATION RESULTS FOR INELASTIC DEFORMATION AND FRACTURE OF QUASIBRITTLE MEDIA Let us consider fracture of quasibrittle media under axial compression on the example of ceramic composPHYSICAL MESOMECHANICS

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ites with a ZrO2 matrix and Al2O3 (corundum) hardening particles of different percentage. Model structures of the composites are shown in Fig. 1. In loaded composites, there always arises a complex stress state, and on the other hand, ceramic composites being quasibrittle media are bound to experience macroscopic fracture as a typical quasibrittle medium, despite local fracture peculiarities dictated by their mesostructural properties. Thus, these materials provide a way to most fully test the model under study. Table 1 presents physical and mechanical properties of the composite components which are necessary for calculation by the model. The hardening particles fill the matrix quasi-uniformly. The mesovolumes in Fig. 1 can be considered as representative volumes; therefore, their loading response is equivalent to the average response of a macroparticle in the conventional sense of continuum mechanics, while the character of local fracture is governed by mesostructural peculiarities. Because the medium is anywhere continuous up to the point at which its strength becomes equal to zero (Y = 0 at D = 1), there is no need to specify any boundary conditions at the interfaces of inciTable 1. Physical and mechanical properties of the composite components Density, g/cm

3

ZrO2

Al2O3

5.7

3.984

1.433

3.46

0.6615

1.6

0.021

0.0374

Dilatancy factor

0.22

0.12

Internal friction coefficient α

0.62

0.6

Initial compression modulus, 105 MPa 5

Initial shear modulus, 10 MPa 5

Initial yield strength, 10 MPa

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(a)

1.0 0.5 0.0 –0.5 –1.0

1.0 0.8 0.6 0.4 0.2 0.0

1.0 0.5 0.0 –0.5 –1.0

1.0 0.8 0.6 0.4 0.2 0.0

1.0 0.5 0.0 –0.5 –1.0

1.0 0.8 0.6 0.4 0.2 0.0



(b)



(c)



t1

t2

t3

µσ

D

Fig. 2. Calculated patterns of inelastic strains (for three successive times ti ), stress-strain state (from the Lode–Nádai coefficient µ σ ), and damage distribution (function D) in the composite with a hardening particle percentage of 15% (a, c) and 40% (b) for ideal slip (a, b) and rigid fixing of the specimen (c) at the boundary of applied load.

pient cracks and structural elements, e.g., at the grain– matrix interfaces [19]. As already noted, fracture as such is considered as avalanche strength degradation in local material regions. In this sense, regions of localized inelastic deformation are not necessary coincident with developing cracks. Below we will see that in local tension regions, cracks can develop very rapidly being only partly coincident with regions of localized inelastic strains accumulated, e.g., in compression-shear regions in which damage accumulation as a whole has not yet turned to a superfast catastrophic mode. A peculiarity of inelastic deformation and fracture of quasibrittle media is that the slope of bands of localized inelastic deformation is determined by the parameters of internal friction and dilatancy, and the formation of future cracks in compression-shear regions can long be quasistationary, whereas inelastic strain accumulation in

regions dominated by tension occurs orders of magnitude faster. Degradation in these tension regions very rapidly reaches its supercritical stage, forming systems of vertical cracks in the axially compressed specimens (Figs. 2a and 2b). The loading conditions for these calculations fit ideal slip at the specimen faces. The overall prefracture state of the specimen shows up as systems of vertical mesocracks and bands of localized inelastic deformation in which the residual strength of the material can be considerable. The fine structure of such cracks in the axially compressed specimens represents, as a rule, a system of small vertical mesocracks (Fig. 2b, t3 ). As can be seen from the calculations, the most severe damaging of the specimens corresponds to tensionshear regions with µ σ < 0 and D close to 1 (Fig. 2). Thus, the proposed model gives a qualitatively true prefracture pattern in quasibrittle media. PHYSICAL MESOMECHANICS

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(b)

Fig. 3. Average σ–ε diagrams for two representative volumes of the specimens with a hardening particle percentage of 40% and 15% (curves 1 and 2, respectively) for ideal slip at the boundary (a) and rigid fixing (b).

If the faces of the axially compressed specimens are fixed, the fracture pattern changes somewhat (Fig. 2c). At the faces, the specimen is less fractured and fracture is concentrated in the region of maximum tension—at the specimen center.

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Appropriate average macroscopic σ–ε diagrams for the mesovolumes shown in Fig. 1 are presented in Fig. 3. With rigid fixing of the quasibrittle specimen at the loading boundary, inelastic deformation preceding catastrophic fracture (blowup mode) is noticeably higher, resulting in somewhat higher degradation of macroscopic strength and, hence, in lower average maximum stress on the σ–ε diagram compared to ideal slip. It is seen from the presented fracture patterns that increasing the hardening particle percentage changes the fracture mechanism. The composite with 40% of hardening particles undergoes multiple cracking. The great many hardening particles create a possibility for the formation of numerous local tensile stress regions, while impeding the propagation of fine cracks in local compression regions. The main crack develops along a preliminary formed percolation network of smaller cracks at the final deformation stage. In the composite with a hardening particle percentage of 15%, there are almost three times less barriers to crack growth. Cracks nucleate and develop facing hardly any resistance from the hardening particles, and the main crack passes throughout the specimen. The supercritical unstable stage of catastrophic macrofracture proceeds in the blowup mode in a mere

(a)

(b)

1.0 0.8 0.6 0.4 0.2 0.0

1.0 0.5 0.0 –0.5 

–1.0

t1

t2

t3

µσ

D

Fig. 4. Calculated displacement fields in the composite with a hardening particle percentage of 15% in weak friction at the boundary of applied load for three successive times ti (a) and corresponding distribution patterns of strain εi , Lode–Nádai parameter µσ , and damage parameter D (b). PHYSICAL MESOMECHANICS

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(b)

Fig. 5. Monitoring of the damage function D for specimen points belonging to adjacent systems of mesocracks in the composite with a hardening particle percentage of 15%: calculation for rigid fixing at the boundary of applied load (a) and weak friction (b).

few time steps. The fracture velocity at this stage is several thousand times higher than that at the prior quasistationary stage in the calculations. In all cases presented, fracture occurs in tensionshear regions (µ σ < 0, D → 1), as evidenced from the Lode–Nádai parameter and damage distributions (Fig. 2) which fully coincide with the pattern of localized inelastic deformation. The character of fracture and the formation of a main crack can be traced in displacement fields constructed with respect to a certain characteristic point, e.g., to a hardening particle (an open circle in Fig. 4) near the mesovolume centre. It can be seen from Fig. 4 that at the prefracture stage, when mesocracks arise in the composites, the displacement fields are highly inhomogeneous. Analysis of the displacement fields in time shows that in the course of loading, the stress-strain state in the region of the fixed grain changes from compression at the initial deformation stage to shear and formation of vortex structures, and eventually to tension in fragmentation of the composite. During the formation of the main crack, the displacement field reveals correlated motion of composite fragments in the direction normal to the main crack. 4. BLOWUP MODE. SELF-ORGANIZED CRITICALITY OF SOLIDS AS NONLINEAR DYNAMIC SYSTEMS At present, the fact that all deformed solids as nonlinear dynamic systems possess the property of self-organized criticality can be considered as proven [8, 33–37]. The fundamental property of such nonlinear systems is

that due to internal nonlinear properties, they tend to a critical state—to a blowup mode. In these systems, catastrophes of any scales are possible [38]. Loaded solids are the most vivid representatives of nonlinear systems. Statistics of stress fluctuation distributions in them gives distributions with heavy tails [39]. For geological media (as well as for any loaded solids [8, 37, 40]), these distributions reflect the Guttenberg–Richter and Omori laws. A necessary condition for the transition of a system from stable inelastic deformation to a superfast stage is the presence of positive feedback which speeds up fracture as an autocatalytic process. Actually, the nonlinear properties of a medium are such that diffuse quasi-uniform damage accumulation very rapidly gives way to localized damage accumulation in rather narrow regions— localized shear bands—and this decreases the strength in the regions. The decreased strength amplifies localization, and hence degradation, and so on. At the final stage, the rate of similar autocatalytic processes is higher even than an exponentially increasing rate [9]. Hence, it is basically required that positive feedback be present in formulation of constitutive equations so that models of a medium and appropriate numerical solutions demonstrate these fundamental evolutionary features of nonlinear dynamic systems—transition of fracture to a superfast catastrophic mode. The calculation presented below show that a loaded solid as a nonlinear dynamic system reveals self-similar properties on different scales, and mesocracks in its local regions and a main macrocrack develop following the same qualitative pattern. All mesocracks as well as fracture of the specimen as a whole develop in two stages: a slow quasistationary stage and a superfast catastrophic PHYSICAL MESOMECHANICS

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(b)

Fig. 6. Stress-strain diagrams of intermittent flow in aluminum 5454 alloy (a), stress distributions (PDF) at the stage of plastic flow for different materials ( —Al 5454, —03Cr18Ni10 steel, —armco iron) (b) [37].

stage. Figure 5 shows monitored functions of the damage parameter D for local specimen points which belong to neighbor systems of mesocracks in the expected zone of their mutual dynamic influence. It is seen from Fig. 5 that damage accumulation is first quasistationary and then it becomes superfast: a blowup mode begins and damaging in the composite particle reaches its maximum. Because the damage function was monitored in the expected zone of dynamic influence, the successive increase in D to its maximum at different points of the zone suggests that the deformation activity switches from one mesocrack to another. So when one point of the medium starts evolving in the blowup mode, its neighbor particle starts or continues evolving in the quasistationary mode, and then this particle also passes into the blowup mode; thus, the deformation activity switches to the next particle and so on. As can be seen from the diagrams descriptive of average behavior of the representative mesovolume as a whole (Fig. 3), macroscopic fracture also develops in the blowup mode. This stage of superfast evolution corresponds to the formation of a main crack in the specimen (Figs. 2, 4, t3). The fracture scenarios discussed above suggest that the loaded model medium displays the property of selforganized criticality which is in correlation of all dynamic processes in a solid and their span over the entire scale hierarchy up to the scale of the specimen as a whole. Thus, one of the most significant features of dynamic systems with self-organized criticality is that statistically independent mesoscales in them are impossible to define. It is this feature which leads to migration of deformation activity: stagnation of strain and damage localization in one specimen region and its activation in another, for example, like with migration of earthquakes in the Earth crust. PHYSICAL MESOMECHANICS

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Self-organized criticality is possible only in those nonlinear dynamic systems in which there is information exchange between all their elements (in our case, between all particles of a loaded medium). In solids, this data transfer is through stress waves propagating with velocity of sound. Clearly, deformation as such, including migration of their activity and fronts [8, 17, 18], represents slow dynamics of a solid as a nonlinear dynamic system. In real solids, including geomedia, self-organized criticality makes itself evident in peculiar statistics of stress distributions (probability density functions, PDFs) as a universal statistical self-similarity law of coherently evolving stress fluctuations in all scale hierarchy of a loaded solid [8]. This statistics of stress distributions reveals the so-called heavy tails in them and their obedience to the power law. In the Gaussian distribution, all events are random and independent, and in power distributions, they are correlated in the whole scale hierarchy. Classical examples of these distributions are the Guttenberg–Richter and Omori laws for seismic events. Consequently, numerical solutions simulative of deformation processes should also give power laws and appropriate statistics of stress distributions. Figure 6 shows an experimental intermittent flow diagram for aluminum 5454 alloy and stress distributions for different materials [37]. Figure 7 presents a calculated σ–ε diagram for elastoplastic flow, corresponding stress fluctuation distribution, and recurrence diagram for stress fluctuations. Thus, the calculations give the distribution that agrees with the experimental one and fit the property of self-organized criticality of deformation as a whole. Similar distributions were also obtained in calculation of fracture of quasibrittle specimens. In the calculations, fracture under constrained conditions was considered. Compressive stress was applied to the side boun-

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MAKAROV, EREMIN (a)

(b)

(c)

Fig. 7. Test calculation of the σ–ε diagram (a), corresponding stress distribution (b), and calculated recurrence diagram of stress fluctuation (c).

dary of a loaded specimen. The deformation constraint grossly changed the character of fracture at the supercritical stage (Fig. 8). The blowup mode collapsed and the stress decreased slowly (Fig. 8a) or even increased (Fig. 8b) at larger constraint. The duration of the supercritical stage shown in Fig. 8 is more than 103 times longer than that of the blowup mode in Fig. 5. Figure 9 shows a diagram for stress fluctuations about the average trend and stress distribution for the ascending branch in Fig. 8 (contoured by an ellipse). It is seen that the distribution is a typical heavy-tailed distribution corresponding to a power dependence. This character of the supercritical stage under constrained deformation owes to dilatancy peculiarities. The commenced catastrophic fracture stage (Fig. 8a) causes decompaction in the fracture zone and hence an increase in stress rather than its release. In the local region of a developing macrocrack, the stress-strain state changes from shear-tension to shear-compression. This process is shown in Fig. 10. The local deformation constraint, in this case, owes to both rigidly fixed specimen boundaries and hardening particles. In tension-shear zones of the band (µ σ < 0), the dilatancy increases, and in shear-compression zones (µ σ > 0), compaction takes place. Under severe con-

strained conditions (the lateral faces of the loaded specimen are fixed, Fig. 8b), the stress can even increase against small mesocracks formed in shear-compression. The foregoing results demonstrate that dilation under constrained deformation can considerably lengthen (thousand times) the blowup mode such that the latter becomes a comparatively slow process rather than being avalanche strength degradation. There is a very rare class of slow earthquakes—bradyseisms—the physical nature of which is still unclear. So, in 1968, a similar earthquake occurred in the Phlegraean Fields of Naples. A vertical displacement of amplitude up 7 m took 48 h [41]. One of the most probable mechanisms of such phenomena, in our viewpoint, can be “lengthened” fracture at the unstable supercritical stage under constrained deformation. If dilatancy is high and covers large volumes of fractured material, constrained deformation will radically change the stress-strain state and tension-shear will give way to compression-shear thus greatly decreasing the degradation rate of the medium and radically changing the pattern of its evolution. In answer to the naturally arising question of how far the phenomenon is possible in a real geomedium, it can be said that its occurrence requires many conditions ful(b)

(a)

Fig. 8. Fracture of the quasibrittle specimen at the unstable supercritical stage under constrained deformation. PHYSICAL MESOMECHANICS

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(b)

Fig. 9. Heavy-tailed distributions of stress fluctuations in the specimen at the supercritical deformation stage under constrained conditions: stress deviation from the average trend (a) and density function (b).

filled at a time. As shown in the experimental study by G.G. Kocharyan [42], the dilation developing in a real narrow fault zone is unable to greatly change the stressstrain state near it and hence the evolution scenario of the geological medium. For the effect of dilatancy on fracture to be strong, it is required that the fracture be extensive covering much space near a main plane of fault displacement. In other words, it is required that the fractured material volume be large rather than cover only a narrow zone of the main plane of fault displacement, which is quite obvious. Much influence on dilatancy is exerted by the granulometric composition of destroyed particles in a shear zone [42]. However, in zones of main fault planes, where most of displacements are typically localized, the particle size is very small ranging, as a µσ 0.8 0.4 0.0 –0.4 

–0.8 

Fig. 10. Change of the stress-strain state from shear-tension (µσ < 0) to shear-compression (µσ > 0) in the band of a developing crack under constrained deformation. The loaded specimen fragment corresponding to Fig. 2c is contoured by an ellipse. PHYSICAL MESOMECHANICS

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rule, from 30 to 200 µm [42], and the effect of dilatancy on the stress-strain state in the near-fault zone is thus bound to be marginal. Similar conclusions were also made by the authors of [43] in their numerical study of the constraint effect on dilation in block media. According to [43], the fracture developing in a block medium under equiaxial compression impedes involvement of higher scale dilation and deformation mechanisms thereby decreasing the main dilatancy parameters of the medium. Note that the results presented in Fig. 8 correspond to multiple fracture in the whole volume of the loaded specimen rather than only in the narrow zone of the main fault, and this is just the fact responsible for the radical change in the evolution scenario of the specimen as a whole. Figure 8 shows specimen-averaged macroscopic σ–ε diagrams that reflect damage accumulation and fracture in the specimen volume as a whole. As pointed out by the authors of [42], a necessary condition for strong influence of dilatancy on the formation of a fracture nucleus (to which we also add fracture in a rather slow quasistationary mode but already at the unstable supercritical stage) is a peculiar structure of developing main fault planes that interact through extensive fractured regions or “bridges”, i.e., fracture in the volume. It is to this case that the results presented in Fig. 8 correspond. The change in fracture scenario in local regions of a loaded quasibrittle medium is illustrated in Fig. 10. For solution of many applied problems, the conventional limit design approach is acceptable and there is no sense to consider the blowup mode. However, of fundamental significance is to understand that a medium fails not where the stress is high but where the medium is most damaged, its strength is low, and hence the stress is the least. This fact is of special significance for understanding the formation mechanisms of a fracture nuc-

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(b)

Fig. 11. Macroscale evolution of a loaded solid, including the quasistationary stage of inelastic deformation (II) and the superfast catastrophic stage of macroscopic fracture in the blowup mode (III).

leus, in particular, in earthquakes. Analysis of the stressstrain state in a geomedium in the context of limiting states leads to knowingly false conclusions and makes impossible any prediction theory based on this idea. The diagrams shown in Fig. 11 illustrate the basic idea which allowed joint description of inelastic deformation and macroscopic fracture of any media, whether brittle or plastic, in the framework of the general formalism. Evolution of inelastic deformation in time can be illustrated by the diagram shown in Fig. 11a. Here, region I corresponds to appreciable damage accumulation at the “elastic” stage almost without the quasistationary stage (brittle and quasibrittle fracture). Region II corresponds to the quasistationary stage of damage accumulation and plastic response of material. Region III corresponds to the superfast catastrophic mode. The macroscopic σ–ε diagram appropriate to these regions is presented in Fig. 11b (curve 2). This diagram reflects only the average macroscopic response of the medium to loading, and this response reflects dynamic equilibrium between loading of a solid and its average response to loading. The strength of the medium (Fig. 11b, curve 1) evolves individually in each particle, depending on individual evolution of the stress-strain state in each particle and associated damage accumulation. Macrofracture is just avalanche strength degradation in the multitude of particles in the region of localized macroscale deformation (the intersection point of curves 1 and 2). It is clear that stage II in Fig. 11b can be the least and correspond to brittle response of the material, for example, at a very high level of applied stress. In summarizing the forgoing results, it can be stated that the developed methodology allows correct description of inelastic deformation and fracture of solids in the framework of the unified formalism.

5. SIMULATION OF TECTONIC FLOWS AND SEISMIC PROCESSES IN GEOLOGICAL MEDIA The next series of calculations allowed numerical reproduction of both tectonic flows and attendant seismic processes. The approach proposed for evolution of the stress-strain state is applied to evolution of folded zones in Central Asia, in particular in the Baikal rift zone and Altai-Sayany folded region, as the result of Indo-Eurasian collision. The authors of [44] point out that the collision of the Indian and Eurasian plates governs the character of deformation processes in Central and Eastern Asia. Of fundamental importance in the simulation is that submeridional compression of the crust in this region gave rise to two types of neotectonic units—rigid domains (microplates, e.g., Tarim, Dzungarian depression, etc.) and compliant mylonitization zones with different crushing degrees (e.g., Great Altai) bordering the rigid domains [45]. Without going into details of solving this problem (this issue is the subject of a separate paper), we will briefly dwell on some of the results obtained. The continental structure was considered using the model of zone-block lithospheric structure of Central and Eastern Asia proposed by K.Zh. Seminsky [45]. This structure is shown in Fig. 12a and is a hierarchy of blocks subdivided into two types of neotectonic units: rigid domains (white color) and compliant zones (dark color). Figure 12b shows a map of the computational domain constructed on the basis of the Seminsky model with three groups of zone-block regions corresponding to different densities of seismogenic ruptures. The strength parameters of rigid blocks with respect to those of compliant zones are characterized by the parameter δ which varies from 1.1 to 1.8 for different groups of the zoneblock structure. Tectonic flows calculated with this rePHYSICAL MESOMECHANICS

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(a)

(b) N S

North American plate

Eurasian plate

Indian plate = > = >

0 500 1000 km

Fig. 12. Zone-block lithospheric structure in Central and Eastern Asia by K.Zh. Seminsky [45] (a) and map of the computational domain with three groups of zone-block regions and boundary elastic blocks (b).

fined zone-block model taking into account significant faults are presented in Fig. 13a. The calculated displacement field relative to Irkutsk demonstrates the current pattern of tectonic flows in Central Asia. There is a clearly defined rotation of the Amur plate and global shear with tension in the Baikal rift region. The obtained tectonic flow represents a typical rift zone and agrees well with observation data on displacements of crustal elements in the Baikal rift zone; the rift opens (Fig. 13a). Figure 14a shows the structure of the Chuya-Kuray zone with indication of epicenters of the mainshock and

aftershocks of the Chuya earthquake. Also shown in the figure is the spatial distribution of calculated seismic events broken into conditional classes by the formula K = logE, where K is the class of an event, E is the energy released in a seismic event. As shown in the discussion of self-organized criticality of loaded solids (Sect. 4), the stress fluctuations in deformation and fracture of a quasibrittle composite (Figs. 6–9) obey the Guttenberg–Richter law. In this sense, destruction of crustal elements, including their destruction in earthquakes as the final fracture stage or the

(a) N

(b)

S

Irkutsk

Fig. 13. Calculated displacement field relative to Irkutsk (a) and recurrence diagram of calculated seismic events for the entire computational domain (b). PHYSICAL MESOMECHANICS

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MAKAROV, EREMIN 87°30' k ula Aig

88°30'

ge rid

Ku ray rid ge Kuray valley

Sept. 27, 18-52 K = 16.6, Ms = 6.4 No rt h -

50°

Ch u

Oct. 01, 01-03 K = 16.4, Ms = 6.6

ya rid ge

Sept. 27, 11-33 K = 17.0, Ms = 7.3

Chuya valley

Sou th -Chu ya ri

87°30'

dg e

88°

(a)

(b)

K 17 16 13-14 12 10-11 8-9 Block-separate faults Surface ruptures

88°30'

Fig. 14. Zone-block lithospheric structure in the Chuya-Kuray zone with indication of the mainshock and aftershock epicenters in the Chuya earthquake in 2003 (a) and spatial distribution of inelastic strains (b); the Chagan-Uzun rigid block is marked by a circle.

blowup mode, is no different from fracture of a composite or destruction of a material in jerky flow. In the calculations, the energy of a seismic event was determined as the stress work on the corresponding strain increment  t at the final fracture stage. ∆ε = ε∆ Comparison of the spatial localization of calculated seismic events (Fig. 15a) and observed seismic events (Fig. 14a) shows adequacy of the structural model of the Chuya-Kuray zone and correctly chosen conditions of submeridional compression in the first approximation. The spatial distribution of calculated seismic events demonstrates activity of the fault (Fig. 14a) along which catastrophic motion took path in the Chuya earthquake in 2003. Noticeable concentration of calculated seismic events is also observed at the southeast boundary of the Chagan-Uzun block suggesting that in reality, activation of a seismic process along this boundary is potentially possible. Statistical analysis of the calculated seismic events for Central Asia (Fig. 13b) and Chuya-Kuray

zone (Fig. 15b) shows that they meet the Guttenberg– Richter law, and this is indicative of a power dependence of the number of predicted seismic events on released energy. Surely, the procedure of simulating seismic events is idealized and the calculations demonstrate only a fundamental possibility of numerical simulation of seismic processes as a catastrophic final fracture stage in a geomedium on a corresponding scale. By and large, the calculations of tectonic flows and seismic events in the model geomedium suggest that numerical solutions of equations of solid mechanics describe deformation as typical evolution of a nonlinear dynamic system with selforganized criticality. Thus, the developed approach and the model approximation provide correct qualitative and, in some cases, correct quantitative description of mechanical behavior of real geological media. Figure 18 compares the space-time localization of predicted seismic events and seismic events observed in the Chuya-Kuray zone [46].

(a)

(b)

K=8 9 10 11 12 13

Fig. 15. Spatial distribution of calculated seismic events by classes (a) and recurrence diagram of calculated seismic events (b). PHYSICAL MESOMECHANICS

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Estimated time K=7 8 9 10 11 12 13

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06.12

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4

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17.10

2

07.10

1 10

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30 50 70 Distance along the activation line in projection onto the Ox axis, km

–20 0 20 40 Distance to the epicenter along the activation line, km

–40 

Fig. 16. Space-time localization of calculated seismic events (a) and observed seismic events [46] (b).

According to the space-time localization of aftershocks in the Chuya earthquake [46], the seismic process migrated along the activation line from the focus of the mainshock to the North-Chuya ridge where two largest aftershocks occurred. Then the process decayed gradually, going to background seismicity with individual large events. The calculated pattern of seismic activity migration (Fig. 16a) agrees qualitatively with the observations (Fig. 16b). Of fundamental importance in the numerical simulation is that the geodynamic process is studied as an evolutionary phenomenon, and this allows simulating both slow dynamics and associated strain activity migration. The calculations reproduce the seismic process and the presence of slow dynamics in geological media. Thus, mathematical modeling allows us to reproduce the fundamental properties of real nonlinear dynamic systems such as self-organized criticality, slow dynamics, strain activity migration, and superfast blowup fracture at the final supercritical stage, and this evidences adequacy of the developed methodology for simulation of a loaded geomedium in the evolutionary approach [4]. Despite the qualitative similarity between the calculated tectonic flows and the observed displacements of crustal elements in Central Asia as well as between the calculated space-time localization of the seismic process and the observed seismicity, we consider that the calculations presented in the paper are test calculations. The PHYSICAL MESOMECHANICS

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issues related to correctness of the structural model, boundary conditions, model approximation of geomedium rheology, and fracture peculiarities require closer examination and further research. 6. CONCLUDING REMARKS Thus, we presented the general methodology and the unified mathematical formalism for describing inelastic deformation and fracture of any solids, whether brittle or plastic, as a process of their evolution in effective force fields. Loaded solids and media are described as nonlinear dynamic systems. One of the main tasks of the work was to show that if deformation and fracture of a strong medium is considered as its evolution in effective force fields, numerical solutions of equations of solid mechanics do demonstrate the fundamental properties of nonlinear dynamic systems—self-organized criticality and evolution in two stages: a rather slow quasistationary stage and a superfast catastrophic stage or the so-called blowup mode. At present, we can see tangible progress in the field of prediction of catastrophic events to which fracture as a whole and earthquakes in particular are referred. This progress is associated with the new concept of self-organized criticality of nonlinear dynamic systems [8, 33– 37]. The new concept comes into conflict with conventional concepts in which rare catastrophic phenomena are considered as independent random events whose fu-

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ture depends almost not at all on the past. Such an approach leads to the Gaussian statistics—to the normal Gaussian probability distribution of independent random events. Statistics of natural catastrophes (earthquakes, hurricanes, floods, technogenic accidents (failure of various constructions, explosions at production sites), and many other disasters (stock market collapse, information leakage, etc.), as a rule, obeys power laws [38, 39]. The power-law distribution is a fundamental property of most multiscale nonlinear dynamic systems, including deformed solids, and reflects the following major qualities of loaded media: (i) long-range space-time correlations in an evolving system which cover the entire scale hierarchy; statistically independent mesoscales in such a system are impossible to define; (ii) self-similarity of fracture due to self-similar defect and dame accumulation in the entire scale hierarchy; (iii) slow dynamics; and (iv) strain activity migration in the region of space-time-correlations. The systems that possess the foregoing properties were given the name systems with self-organized criticality [33]. A fundamental feature of these systems is that their internal properties drive them to a critical state. So if fracture is statistically correlated in the entire scale hierarchy up to the macroscale, the process inevitably reaches this macroscale. It is shown that numerical solutions of equations of solid mechanics demonstrate all the above listed characteristic features of nonlinear dynamic systems with selforganized criticality. This is evidenced by stress fluctuation statistics (calculated probability density functions) obtained for elastoplastic flow and quasibrittle fracture of a medium under constrained conditions. Similar heavy-tailed distributions, which correspond to power dependences, were also obtained in calculations of tectonic flows in geomedia on the examples of simulating the current evolution of the stress-strain state in Central and Eastern Asia and in the Altai-Sayany folded region. The calculated recurrence diagrams of seismic events obey the Guttenberg–Richter law. Apparently, simulation of a seismic process through calculation of deformation in a geomedium was performed for the first time. Another key point of the approach presented in the work is the idea to treat inelastic deformation of a medium as its prefracture on the macroscale, on the one hand, and as its growing fracture on the micro- and mesoscales, on the other hand. In this context, inelastic deformation, including plastic deformation, can be described as fracture which grows from scale to scale

through appropriate kinetic accumulation of defects and damages of various physical nature in the entire scale hierarchy. This was the idea that made possible the unified mathematical formalism for description of destruction of solids and media (i.e., compatible development of inelastic or plastic deformation and fracture in conventional terms). The foregoing ideas, though realized in the work on the examples of simplest idealized kinetics of damage accumulation, allowed good qualitative illustration of the fundamental evolutionary peculiarities of solids and media as nonlinear dynamic systems. Clearly, in each specific case, one should consider appropriate kinetics, depending on rheology, fracture peculiarities, and loading conditions, to properly reflect leading physical mechanisms of inelastic deformation and fracture. The examples of simulation show that fracture of a quasibrittle medium under constrained conditions reveals a strong influence of dilatancy on evolution of the medium at the supercritical stage. It is demonstrated that when fracture is not localized in the region of main shear but covers a considerable (all) volume of a loaded specimen, there can be a radically different evolution scenario. The blowup mode collapses and macrofracture passes through a series of much smaller catastrophes extended in time. In this case, the total time of macrofracture at the unstable supercritical stage is several orders of magnitude longer than the time of macrofracture in the blowup mode. These results allowed us to come up with the idea that the most probable mechanism of slow earthquakes—bradyseisms—can be the dilation mechanism occurring under specific conditions of constrained deformation. A necessary condition for dilatancy to exert a strong effect on the evolution scenario in a geomedium is multiple fracture covering considerable volumes of loaded material. The same conditions were pointed out by G.G. Kocharyan [42]. One of the keystones of nonlinear dynamic systems is a threshold of evolutionary processes. However in reality, it is almost impossible to define the threshold at which the evolution scenario of a system is radically changed, for example, the system transforms from its stable state into an unstable catastrophic state of evolution. In this connection, it has been said many times that prediction of evolution of nonlinear dynamic systems with self-organized criticality is in principle impossible, in particular, that we can not exactly predict an earthquake (we can only estimate its probability). As shown earlier [8], the situation is not so dramatic and there are other properties of nonlinear dynamic systems (peculiaPHYSICAL MESOMECHANICS

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rities of slow dynamics, formation of lull zones) to give optimism and hope of the possibility to predict transition of a system to a superfast catastrophic mode. The way to make the prediction possible is associated with a search for appropriate precursors and with calculation of threshold parameters or dimensionless criteria responsible for transition of a system to a blowup mode. The simplest idealized models of similar threshold criteria for solids and media are conventional limit design criteria (limits of elasticity, plasticity, strength, etc.). However, it should be kept in mind that they allow only engineering estimates to define the upper bound of permissible loads. The question as to the behavior of material destroyed in a band also remains open. Of importance for our work is that at D → 1, Y → 0, a material is considered to be destroyed on the macroscale: its bearing capacity is lost. In geological media and materials under constrained conditions, this is not the case: they will resist to compression and will not resist to tension. However in reality, there is always a combination of tension-shear (λ < 0) or compression-shear (λ > 0) and it is required to construct a model taking into account cohesion and shear strength of a destroyed medium under constrained conditions. This problem is the subject of further research. ACKNOWLEDGMENT The authors are thankful to A. Peryshkin for providing a series of calculations of tectonic flows in Central Asia. The work was performed under RFBR project No. 12-05-00503, integration project of SB RAS No. 90, fundamental research program of the Presidium of RAS No. 4.1, and basic project No. VII.64.1.8. REFERENCES 1. Physical Mesomechanics of Heterogeneous Media and Computer-Aided Design of Materials, Panin, V.E., Ed., Cambridge: Cambridge Interscience Publishing, 1998. 2. Panin, V.E., Korotaev, A.D., Makarov, P.V., and Kuznetsov, V.M., Physical Mesomechanics of Materials, Russ. Phys. J., 1998, no. 9, p. 856. 3. Makarov, P.V., Loaded Material as a Nonlinear Dynamical System. Simulation Problems, Phys. Mesomech., 2006, vol. 9, no. 1–2, pp. 37–52. 4. Makarov, P.V., Mathematical Theory of Evolution of Loaded Solids and Media, Phys. Mesomech., 2008, vol. 11, no. 5–6, pp. 213–227. 5. Nicolis, G. and Prigogine, I., Exploring Complexity: An Introduction, New York: W H Freeman, 1989. 6. Malinetskii, G.G. and Potapov, A.B., Modern Problems of Nonlinear Dynamics, Moscow: Editorial URSS, 2000. PHYSICAL MESOMECHANICS

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2013