formation of systems of incompact bands parallel to the compression axis in the unconsolidated...

ISSN 1069�3513, Izvestiya, Physics of the Solid Earth, 2011, Vol. 47, No. 10, pp. 886–901. © Pleiades Publishing, Ltd., 2011.Original Russian Text © Sh.A. Mukhamediev, D.A. Ul’kin, 2011, published in Fizika Zemli, 2011, No. 10, pp. 32–47.

886

Formation of Systems of Incompact Bands Parallelto the Compression Axis in the Unconsolidated

Sedimentary Rocks: A ModelSh. A. Mukhamedieva and D. A. Ul’kinb

a Institute of Physics of the Earth, Russian Academy of Sciences, ul. Bol’shaya Gruzinskaya 10, Moscow, 123995 Russiab Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047 Russia

Received May 3, 2011

Abstract—Uniaxial compression of poorly lithified rocks leads to the formation of thin incompact layers (orbands, in the two�dimensional case) parallel to the compression axis, which are characterized by increasedporosity. The standard model of the formation of such bands, as well as deformation bands of other types,associates them with the narrow zones of localization of plastic deformations. In the case of decompaction,it is assumed that transverse tensile deformations are localized within the band, which cause the band todilate. Here, the formation of a band of localized deformations is treated as a loss�of�stability phenomenon.Based on observations, we propose a fundamentally different model of incompact bands formation, accord�ing to which the microdefects in sediment packing (pores) rather than the deformations are localized in thenarrow zones. The localization of pores, which are initially randomly distributed in the medium, occurs as aresult of their migration through the geomaterial. The migration and subsequent localization of pores aredriven by a common mechanism, namely, a trend of a system to lower its total energy (small variations in totalenergy are equal to the increment of free energy minus the work of external forces). Migration of a single porein a granular sedimentary rock is caused by the force f driving the defect. This force was introduced byJ. Eshelby (1951; 1970). An important feature of our model is that the formation of an incompact band heredoes not have a sense of a loss of stability. Quite the contrary, the formation of incompact bands is treated asa gradual process spread over time. In this context, the origination of incompact band systems directly followsfrom our model itself, without any a priori assumptions postulating the existence of such systems and withoutany special tuning of the model parameters. Moreover, based on the proposed model, we can predict theincompact bands to always occur in the form of systems rather than as individual structures. A single incom�pact band may only be formed when the force resisting the pore motion, fc, is absent.The calculations were conducted for the case of a plane strain in an infinite elastic medium loaded by uniaxialcompressive stress, p. At the initial state, a random spatial distribution of N round pores having a diameter awas specified in some bounded domain Ω. At each iteration for each pore the driving force on a defect, f, wascalculated, which is caused by the influence of all other pores. The position of the pore was varied along thedirection of the acting force f if | |f| > fc. The iterative process for the given initial conditions was terminatedwhen the criterion | |f| ≤ fc had been met for all pores. Our calculations showed that the migration of poresresults in the formation of a relatively regular structure composed of quasi�parallel linear elements extendedalong the axis of compression. We associate these formations with the systems of incompact bands. Severalseries of calculations were carried out. Each series was characterized by its own values of N, fc, and spatialaverage pore density, ρ. With fixed N, the pore density ρ was varied by changing the size of the initial domainΩ within which the pores are distributed. Each series of calculations included 960 case computations of theformation of an incompact band. The cases within the series differ by the initial random distribution of pores.For each series, the frequency histograms were compiled for distances h between the incompact bands. Ourresults show that the initial pore density ρ (excluding its critically small values) does not have any effect onthe shape of the histogram. In particular, this means that the typical distance hm between the incompact bandsdoes not depend on ρ. Quite the opposite, variation in the resisting force, fc, with the other conditions beingthe same, drastically changes the frequency distribution of distances h. As fc decreases, hm increases; the his�tograms become more diffused, and their maxima become lower.

DOI: 10.1134/S1069351311100089

INTRODUCTION

In sedimentary rocks there are macrostructureswhose formation is largely controlled by porosity.These structures are narrow, a few millimeters to fewcentimeters wide bands (or layers, in the three�dimen�

sional case), which, according to the classificationsadopted in the western literature, are referred to asdeformation bands (Schultz and Fossen, 2008). It canbe easily seen that the definition “deformation” is notneutral but carries a connotation of a definite interpre�

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 47 No. 10 2011

FORMATION OF SYSTEMS OF INCOMPACT BANDS 887

tation: the researcher is forced to take for granted thatthere are deformations of some or other type localizedin the layers. The kinematical classification distin�guishes five types of layers, depending on the style ofdeformation. These are (1) compaction layers,(2) shear layers, (3) dilation layers, (4) compactiveshear layers, and (5) dilatant shear layers (see, e.g., thereviews (Aydin, Borja, and Eichhubl, 2006; Fossen etal., 2007)). The first type is more frequent in sands andporous sandstones, and the third, in sands. Followingmany authors, we assume that, in the sense of kine�matics, the compaction is understood as a transversalshortening of a layer. Then, the above�cited nomen�clature of the deformation layers means that in the lay�ers pertaining to the first three types, simple (dyadic)deformation modes are realized, while in the layers ofother types, combined deformation modes occur. Theviews of many researchers as to the origin of a layer ofsome or other type are based on the cited classifica�tion, which is reflected in the models suggested.

Indeed, the theoretical models meant for explain�ing the nucleation of the deformation bands (e.g., Ols�son, 1999; Issen and Rudnicki, 2000; Borja and Aydin,2004; Borja, 2004) often follow the scheme based onthe known approach by Rudnicki and Rice (1975),which has already become conventional. In thisscheme, in particular, the mentioned approach is gen�eralized by taking into account the volume plasticdeformations under hydrostatic pressure. In thescheme, which will be further referred to as the RRscheme (after the names of Rudnicki and Rice whofirst proposed this approach), a monotonic single�parameter loading of uniformly deformed elastoplas�tic material is studied. The loading is usually describedin terms of plastic hardening modulus, H, whichmonotonically decreases with the increasing intensityof plastic deformations in the classical RR model.Then, the maximum possible critical value of the

hardening modulus is determined, at whichbifurcation first becomes possible. In the case consid�ered, this is the transition from the homogeneous elas�tic strain state to the essentially nonuniform strainwith the formation of a narrow layer in which plasticdeformations of some or other type are localized.Here, one calculates the orientation of the layer rela�tive to the principal stress axes and estimates the rheo�logical constraints on the model, which would allowthe layer with a given type of deformations to beformed. The RR scheme is treated by its followers as atool for recognition of the mode of rheological insta�bility. Strictly speaking, this is not the case because,when using the scheme, it has not been proved that thelocalization form of the loss of stability is the only formpossible (Ryzhak, 2002).

In the present work, we do not intend to subject theaforementioned classification of deformation layers tooverall criticism. Nevertheless, instead of the term“dilation layer (or band)” we will use the term

max,RRH H=

“incompact layer (or band)”. The grounds for doing sowill be clear from the further considerations. We notethat both separate deformation layers, as well as thoseclustered in relatively regular systems, occur. Theexception is probably the incompact layers: we failedto find any mention in the literature of the existence ofsystems of such layers. This question will be discussedbelow.

Our interests primarily lie in the incompact defor�mation bands. However, before proceeding to theanalysis of these structures, it seems reasonable tobriefly discuss their antipode, the compaction bands.The latter seem first to have been described in(Mollema and Antonellini, 1996). The compactionlayers, which are characterized by the fracturing of thematerial grains (cataclasis) and high resistance toweathering, due to low porosity and permeability arenatural barriers for fluid flows. The presence of com�paction layers significantly affects the patterns of fluidprocesses in sedimentary basins. The properties men�tioned above have played a significant role in theappearance of a variety of publications focused on theexperimental (Olsson, 1999; Kaproth, Cashman, andMarone, 2010; etc) and theoretical (Olsson, 1999;Issen and Rudnicki, 2000; Sternlof, Rudnicki, andPollard, 2005; Rudnicki and Sternlof, 2005; Gara�gash, 2006; Stefanov and Tjerselen, 2007; Chemenda,2009; etc) study of the formation and development ofcompaction bands. Mathematical modeling in the rel�atively early works was carried out in the context of theRR scheme discussed above (Olssen, 1999; Issen andRudnicki, 2000). In some later works, since (Sternlof,Rudnicki, and Pollard, 2005), a new idea has pre�vailed. According to this idea, the development of thecompaction band was associated with the propagationof the so�called anticrack (mode I crack). Whenstudying the anticrack propagation, the formalism ofbrittle fracture mechanics for the mode I crack isretained, although with opposite�sign stresses andcrack opening displacements. The interpenetration ofthe crack surfaces, which appears in the solution, istreated as an inelastic compaction of the material,which is accompanied by the reduction in porosity.Regardless of whether the RR scheme or anticrackconcept is used in the modeling, in either case thecompaction band is orthogonal to the maximum com�pressive stress. For completeness of the review on themodel approaches, we mention also the works(Ryzhak, 1999; 2003) in which, by using the speciallaw of plasticity, the author showed that compactioncan be localized not only within narrow layers but,generally, within any small volumetric domains ofarbitrary shape.

The compaction layers in natural conditions occuras both the single separate forms and the systems ofsuch forms. The system of compaction layers, whichA.F. Grachev and Sh.A. Mukhamediev encounteredwhen studying the jointing of sedimentary rocks in theVienna and Pannonian Basins in 2010, is shown in

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MUKHAMEDIEV, UL’KIN

Fig. 1. The formation of the systems of compactionbands was theoretically modeled in the scope of a loss�of�stability concept in (Garagash, 2006; Chemenda,2009). It is worth noting that the authors of these worksa priori intend to find a solution which would describethe nucleation of a system of bands (instead of a singleband, as it is done in the classical RR scheme). There�fore, the system is implicitly postulated to exist at thevery beginning of the analysis, and further on in thestudy, the parameters of the constitutive model and ofthe process of loading are fitted to ensure that this pos�tulated assumption is valid. An important implicationis that the possibility of the formation of a system ofbands is rather sensitive to the behavior of the plastichardening modulus, H. The system of bands can beformed if, starting from the moment of transition to

the plastic state, either the condition isimmediately met (Garagash, 2006), or H falls in theadmitted region by decreasing stepwise instead ofgradually, as is the case with the RR scheme (Che�menda, 2009).

Now, we proceed to considering incompact bandsand note that the latter were first identified not verylong ago (Du Bernard, Eichhubl, and Aydin, 2002)and, probably due to this reason, have not been asextensively studied as the compaction and shearbands. Du Bernard and co�authors (2002) point outthat the incompact bands, which they observed inpoorly consolidated Pleistocene sands in NorthernCalifornia, are about 1–2 mm wide, which is 5–10�fold larger than the grain diameter. The authors of thecited work associate these bands with the tensile zonesgenerated by the conjugate shear bands coexisting in

maxRRH H<

the same outcrop. The decompaction revealed bymicroanalysis is reflected in the fact that the relativevolume of the pore space within the band is 7–9%larger than the corresponding value in the host rock.Decompaction is not accompanied by shear and doesnot involve the destruction of the grains of the mate�rial. Du Bernard and his co�authors believe that theincompact bands are formed parallel to the majorcompression axis. Their notion of the differencebetween the incompact band and the opening�modecrack is illustrated by Fig. 2. A real incompact bandobserved in the natural outcrop is shown in Fig. 3a.

Experimental studies of the origination of incom�pact layers have started quite recently (Chemenda etal., 2011). In the experiment, a specimen from thematerial imitating a consolidated rock was first sub�jected to hydrostatic compression by stress P, andthen, at fixed confining pressure P, unloaded along thex axis of the specimen until a discontinuity occurred.When P did not exceed some fully determined limitingvalue, the discontinuity occurred on the cross sectionorthogonal to the x axis (and, therefore, to the axis ofmajor tension, in a deviatoric sense). Here, for small Pand tensile stresses along x, the surface of discontinu�ity was smooth, and the discontinuity was classified asan opening�mode crack. At higher P and either tensileor compressive loading along x, the discontinuity hadtypically a plumose surface, and the discontinuityitself was a thin (with a thickness of a few grains’ diam�eter) layer with increased porosity, i.e., an incompactlayer. The microphotograph of the cross section of thislayer along x (the incompact band) is presented inFig. 3b. The characteristic features of the incompactbands reproduced in the experiments and those

Fig. 1. A system of compaction layers in the bleached silica Pannonian sands (Pannonian Basin, Hungary). Left bottom: the dipcompass shown for the scale. (Photo by A.F. Grachev and Sh.A. Mukhamediev).



Inco

mpa

ct

ban

d

Yo

int

(а) (b)

Fig. 2. (a) The schematic of an incompact band (in the terminology of the present paper) and (b) a joint (according to (Du Ber�nard, Eichhubl, and Aydin, 2002)). In (Du Bernard, Eichhubl, and Aydin, 2002) the enhanced porosity in the band (a) is assumedto be passively formed due to the horizontal divergence of grains.

LG

V

B

5 µ

LG

V

Incompact band

1.5 mm(а)

(b)

Fig. 3. The appearance of the incompact band observed (a) in the Nubi Sandstone layer, Sinai (after (Fossen et al., 2007), mod�ified) and (b) in the experiment conducted in (Chemenda et al., 2011) (see text). For the band obtained in the experiment, Che�menda et al. (2011) introduce the following notations: V (voids) are zones of coalesced pores; LG (loose grain zones) are loosezones with the decompacted grains, and B are bridges of visually intact material.

observed in natural geological conditions (Fig. 3a) aregenerally similar. A striking detail is the nonuniformdistribution of the pore space throughout the band: thesegments almost void of grains alternate with those ofreduced and normal grain density.

1. MOTIVATION FOR THE PROBLEM STATEMENT AND THE ANALYSIS

OF THE ALTERNATIVE MODEL CONCEPTS

When analyzing the mechanisms and models of theformation of incompact layers, one should clearly

890



demarcate facts and their interpretations. The fact isthat in these layers the porosity is increased comparedto the ambient medium. Moreover, the porosity isnonuniformly distributed along the layer, whereas thelayer itself may be not quite straight (Fig. 4a). In otherwords, in the layer, localization of the defects—thepores—takes place. However, it is not clear from theobservations if the localization of deformations in thelayer occurs. The answer depends on the interpreta�tion of the observations.

The standard interpretation of the formation anddevelopment of incompact layers relies on the theoret�ical result obtained in the context of the generalizedRR scheme (Borja and Aydin, 2004; Borja, 2004),which has been described above. The incompact layer,which is modeled as a zone of the localization of elon�gation deformations perpendicular to the layer, is ori�ented orthogonally to the axis of maximal deviatorictension. Surprisingly, this result, considered to be anexclusive case that might be of some interest only froma purely theoretical viewpoint, had been obtained ear�lier (see, e.g., (Perrin and Leblond, 1993; Issen andRudnicki, 2000)) than the incompact layers were rec�ognized in nature and studied in the experiments.Thus, according to the generally adopted interpreta�tion, an incompact layer is formed as a result of therheological instability of elastoplastic geomaterial; inthe course of development, this layer dilates due toplastic deformations, which are uniform along thelayer (Fig. 4b). This interpretation raises many ques�tions. To what extent is the assumption of a structuralformation, which is knowingly nonuniform through�

out the layer, residing in a uniform stress�strain state,adequate? Is the perpendicular dilation of an incom�pact layer obligatory? Is the formation of regular sys�tems of incompact layers similar to the sets of compac�tion layers shown in Fig. 1 possible?

1.1. Choosing a Model

These questions call for searching for an alternativemodel to explain the increased concentration of thedefects (pores) in the incompact layer. This modelshould reveal the physical prerequisites for localiza�tion of the defects in a narrow layer and describe themechanisms of this localization. The requirements ofthe model raise the following question: to what level ofdetail should the model take into account the micro�mechanics of the localization process? For example,when analyzing granular media, the macro�, meso�,and microscale levels of description are usually distin�guished (Sun, Wang, and Hu, 2009). There is no uni�fied mathematical model which describes the varietyof effects observed in a granular medium on variousscales.

The microscale analysis is focused on studying sep�arate grains: their physical properties, the pattern oftheir contact interactions with the neighboring grains,etc. The numerical experiments on the microscalelevel are based on the so�called discrete elementmethod (DEM), which was first suggested in (Cundalland Strack, 1979). DEM considers a finite number ofdiscrete particles interacting by contact forces. Thetranslation and rotation of each particle are describedby Euler’s equations of motion. DEM�based mathe�matical modeling provides information on the particletrajectories and the forces acting on separate particles.Numerous results found through research conductedover more than 20 years using DEM and its modifica�tions are summarized in a comprehensive review byZhu et al. (2008). Nevertheless, due to the computa�tional overburden involved, it seems unreasonable tosolve the problem of localizing the defects in narrowlayers on a microscale level by the methods of mathe�matical modeling described above.

The mesoscale analysis is intended for studying theso�called force chains, that is, the filamentary struc�tures observed by photoelastic methods, along whichthe force interactions in granular media are translated(see, e.g., (Socolar, Schaeffer, and Claudin, 2002;Blumenfeld, 2004)). The existence of the force chainsmakes the distribution of stresses essentially nonuni�form on a mesoscale level. Due to the arching effect,the contact stresses in the ensembles of granules unaf�fected by chains are negligible. The existence of forcechains will be formally ignored in our model.

Thus, we set our choice on the macroscale level ofdetail in our considerations, that is, in the field ofresponsibility of phenomenological theories. Never�theless, the question of taking into account the micro�

p1p1

p2p2

p2p2

p1 p1vv

1 2 3

(а) (b)

Fig. 4. An incompact band aligned parallel to the maximalcompressive stress p2. (a) A simplified schematic structurebased on real field observations (Fig. 3); (b) theoreticalmodel based on the concept of a uniform plastic banddilating horizontally at rates ±v; 1, host rock; 2, pores;3, plastic state of material.



mechanics of defects in a phenomenological modelstill needs to be discussed.

As a rule, phenomenological theories rely least onthe micromechanisms of the formation and accumu�lation of damage. Numerous criteria for damage sug�gested for lithified rocks (see, e.g., the review (Edel�bro, 2003)) are, in fact, various versions of the strengththeory which is applied in simplified calculations offailure. This theory is poorly tailored for describingmacroscopic dislocations in a rock massif that residesin a nonuniform stress state (Kondaurov et al., 1987).The approach based on the application of phenome�nological criteria of failure does not meet the con�straints of the desired model because it not onlyignores the kinetics of damage accumulation but isfundamentally unable to elucidate the physical rea�sons of their localization. As for a granular medium,some features of its macroscale behavior can be repro�duced in terms of phenomenological plastic theoriesby taking into account the nonassociative plastic flow,the dilatancy, the internal friction, the Bauschingereffect, and other factors (Nikolaevsky, 1996) in theconstitutive equations. Based on these theories, it ispossible to recognize the localization of deformationsbut not the localization of defects.

The kinetics of damage accumulation can be takeninto account by postulating the kinetic equation interms of the continual (distributed) damage theory.Here, the defects are considered in a poorly phenom�enological context by introducing into the model of acontinuum a scalar or tensor parameter, which istreated as a measure of damage. In the energy�basedcounterpart of this theory (Kondaurov, 1986; Kondau�rov and Nikitin, 1990, etc.), it is possible to recognizethe onset of rheological instability; however, this the�ory, as well as other variants of the continual damagetheory, fails to describe the effect of damage localiza�tion. As the condition of damage localization is notinherent in the continual damage theory, this condi�tion has to be postulated, just as it is done in thestrength theory with the failure condition. In particu�lar, Bai et al. (2001) and Xu et al. (2004) assumed thecriterion of defects' localization in the form of certainconstraints imposed on the growth rate of the gradientof the scalar measure of damage. Roughly speaking,the domain where the measure of damage starts toexperience sharp spatial variations is associated withthe region of nucleation of a macrofracture.

1.2. Choosing the Mechanism of Pore Concentration

It follows from the points mentioned above that thedesired phenomenological model of pore concentra�tion must explicitly take into account the microme�chanics of the process. What is the mechanism respon�sible for the increased porosity in an incompact layer?Here, two scenarios are possible: the pores are formedin�situ or imported to the layer from the host rocks.

One modification of the first variant is well studiedin the theory of ductile fracture, which was primarilydeveloped for the mechanics of materials. In this the�ory, it is stiff grains rather than soft inclusions such asmicrocracks and micropores that are typically sup�posed to be the sources giving rise to the nucleation ofdefects and to the formation of macrodamages in het�erogeneous media. For example, in metal alloys,mainly under tension and at the temperature exceed�ing the point of brittle–ductile transition, decohesionof the matrix material from grains occurs, followed bysubsequent micro�void growth to coalescence in theconditions of mature plasticity. This process results inthe formation of ductile macrofractures with ratherrough fracture surfaces. Detailed reviews on this prob�lem, including the latest findings, are presented in(Benzegra and Leblond, 2010; Li et al., 2011). Despitethe attempts to tailor the ductile fracture theory to theconditions of sedimentary rocks (e.g., in (Eichhubland Aydin, 2003)), it still seems that the scheme ofductile fracture micromechanics is inapplicable to thecase in which we are interested. Indeed, in the uncon�solidated granular rocks, the grains interact with eachother but not with the host matrix. Another modifica�tion of the first variant is dilation of the layer with theinevitable expansion of the pore volume. Setting asidethe heterogeneity of the formed pore volume and thenonuniform pattern of the stress�strain state in thelayer, this scheme coincides with that shown in Fig. 4b,while it is that scheme to which we are trying to findthe alternative.

Thus, we come to considering the second variantassociated with the migration of pores relative to theparticles of the material and their concentration innarrow bands. The principle of pore migration will bethe basic concept of our model. We note that mecha�nisms of micropore migration through the material arestudied in some technical fields for the processes inwhich the mobility of pores is of primary importance.These include the powdered metal engineering appli�cations where the micropore migration accompaniesthe processes of extrusion and sintering (Kang, 2005).Particular attention should be paid to the followingprocess, which presents a challenging problem fornuclear power stations. In the working fuel elements(cartridges) of nuclear reactors, with the lapse of time,a central void emerges, which runs along the axis of afuel rod, which is initially macroscopically continu�ous. The agreement on the physical causes of this phe�nomenon has long been reached among scientists. Inthe initial state, the micropores (with a radius of about1 µm) in a cartridge are randomly distributed through�out the volume. During the operation of a reactor, in arod cooled from outside, a sharp contrast in tempera�ture arises between the hot central part of the rod andits relatively cold marginal parts. Driven by tempera�ture and pressure gradients, the micropores radiallymigrate to the center and eventually form the centralopening (see, e.g., (Gill, 2009; Hu and Henager,

892



2010)). The only debatable issue is the mechanisms ofpore migration. One mechanism suggests that migra�tion of a pore is controlled by evaporation of the sub�stance on the hot side of the pore and its condensationon the cold side.

As follows from the above, if we leave aside the lin�ear dimensions of pores and the particular mechanismdriving their motion, the principle of formation of anopening in fuel rods will be perfectly similar to theprinciple of formation of an incompact layer in themodel we intend to construct. In both cases, the for�mation of a void (in the former case) or incompact (inthe latter case) zone is caused by the migration ofpores relative to the particles of the material.

Summarizing the discussion, we come to the con�clusion that the behavior of the material in the rockvolume should be described in terms of the phenome�nological model, taking into account the pore migra�tion relative to the material. The constitutive equa�tions of linear elastic theory will be used as constitutiverelations for the matrix material in our model. If theproposed model of localization of defects proves work�able in the simplest case of linear elasticity, there willbe confidence that the model has a margin of stabilityfor the selection of the class of constitutive relationsamong a variety of candidates. In addition, in the caseconsidered, the linear elasticity of the matrix is not theworst choice, as it might have seemed at first glance.Of course, the migration of a pore relative to the parti�cles of a granulated medium is accompanied by resid�ual strains, which are associated with the fact thatsome grains become detached from being interlockedwith the neighboring grains. However, one should notsuppose that a relatively long�distance migration of apore is associated with comparable displacements ofgrains of the material. Large displacements of a poreare caused by successive displacements of grains by thedistances comparable with the grain diameter (seebelow). Therefore, the residual strains of the matrixare small, and ignoring them does not lead to a signif�icant error.

2. STATEMENT OF THE PROBLEM

In solving the problem of incompact bands forma�tion, we rely on the model suggested by Mukhamediev(1990) for describing the formation of a single lineardiscontinuity along the direction of maximum com�pression in an infinite elastic unconsolidated granularmedium. In this model, based on the concept of adriving force on a defect (Eshelby, 1951; 1970; 1975),the behavior of the defect (a round pore) in the stressfield produced by the principal compressive stresses

, applied at infinity wasconsidered in a 2D context. With a quite definite rela�tionship between the loads p1 and p2 and the efficientsurface energy, the defect bifurcates into two newdefects and these newly formed ones diverge from one

1 1 0T p∞

= − < 2 2 0T p∞

= − <

another along the axis of maximum compression. Theclosed (due to compression conditions) trace left bythe pores moving relative to the particles of the mate�rial was interpreted as a dislocation that develops intoa crack in the course of time. This model (Mukhame�diev and Nikitin, 2007) was applied to interpreting theformation of a set of thin (having the thickness of ahair) perfectly straight lines observed in one of the out�crops in Northwestern Caucasus and thoroughly stud�ied in (Beloussov et al., 2005). In Sections 2.1 and 2.2below, we quote some elements of the describedmodel, which are necessary for the solution of theproblem of incompact band formation.

2.1. Force of Interaction between Two Pores in Special Positions

We consider generally not simply connected elasticbody B bounded in the reference configuration κ by apiecewise�smooth surface S

κ, with the vector element

of the surface dsκ pointing outwards. The variation in

the actual configuration χ of body B (χ → χ + δχ) atfixed configuration κ means a small deformation ofbody B caused by the field of small continuous dis�placements of its material points x(κ). At the sametime, the variation in the reference configurationκ (κ→ κ + δκ) means a change in the set of the mate�rial points x(κ) that compose the body B (Mukhamed�iev, 1990). Let the elastic body B be quasi�staticallydeformed (χ → χ + δχ) and its reference configurationκ be simultaneously varied. The latter variation meansa change in the reference surface S

κ, which can be

described by specifying on Sκ a surface vector field

δx(κ). Then, with body forces neglected, the variationδWB of elastic energy WB of body B has the followingform:

(1)

Here, δA is the work done on body B, • denotes ascalar product, w

κ is the density of elastic energy WB

per unit volume in the reference configuration. Thesecond term in the right�hand part of Eq. (1) is the fluxof elastic energy due to the addition ( ) orremoval ( ) of the material particles onthe surface S

κ.

Our interest lies in the special case of elastic body Bthat contains the defect in the form of a pore possess�ing a traction�free sufficiently smooth surface S

κ in the

reference configuration κ. At quasi�static variation inthe actual equilibrium state of body B, let the defectshift in the reference configuration κ as a “rigid” bodyby a small vector δx(κ) = const, relative to the particlesof the material. Then, in the absence of body forces,the variation δEB in the total energy of the system,which is the sum of the variation δWB of elastic energyof the body and the variation of the potential energy ofthe loading device (which is equal to the work of exter�

( )

κ

.B

S

W A w dκ κκ

δ = δ + δ∫ x si

( ) 0dκκ

δ >x si

( ) 0dκκ

δ <x si



nal forces δA taken with the opposite sign), is foundfrom Eq. (1) in the following form:

(2)

We note that formula (2) in the cited form is alsovalid for an infinite body.

According to J. Eshelby (1970; 1975), vector f is aforce that drives (or acts on) the defect. Recall that theoriented surface element dsκ is directed towards thepore. The total energy decreases (δEB < 0) if the dis�placement vector of a pore, δx(κ) forms an acute anglewith the direction of the force f.

In the further calculations, we will confine our�selves to considering the case of a plane strain in a lin�early elastic body. In this relation, the symbol κ of thereference configuration will be omitted. For a roundtraction�free pore, the elastic energy density, w, on thecontour of the pore (per unit thickness across thedeformation plane x1, x2) is

(3)

where μ is shear modulus, ν is Poisson’s ratio, and Tθ

is the circumferential normal stress.We consider an unbounded plane and introduce the

Cartesian coordinates x1 and x2 with unit vectors e1 ande2. The position of a point is specified by the vectorx = x1e1 + x2e2. Let the plane experience the action ofremote positive compressive stresses p1 and p2 and con�tain two similar round pores of diameter a, centered atthe points xi and xj (Fig. 5). The unit vector ki,j specifiesthe orientation of a pair of points xi and xj relative tothe coordinate axes. In contrast to the model consid�ered in (Mukhamediev, 1990), here, we only analyzethe case when the pores are not simply separated butspaced significantly apart one from another, namely

(4)

We consider a special case of the orientation ofpoints xi and xj when ki,j

· e1 = 1. In this case, the pair ofpoints xi and xj are oriented along the x1 axis, and theabscissa of point xj is larger than that of point xi. Usingdefinition (2), we obtain the following expression forthe force f = fi,j acting on the pore with the center at xj:

(5)

where

(6)

and the central angle θ is counted counterclockwisefrom positive x1. When deriving Eq. (5), we employedthe symmetry of the problem about the horizontal linecoming through the points xi and xj, as well as the rela�

( ) , .B B

S

E W A w dκ κκ

δ = δ − δ = −δ = −∫x f f si

( )21 1 ,

4w Tθ= − ν

μ

, 0.4.i ji j

aε = <

−x x

2

2, 1

0

cos ,i j c T d

π

θ= θ θ∫f e

( )1 1 ,8

c a= − νμ

tion ds · e1= –cos θ dθ, which is valid on the boundary

of the pore. The force fi,j should be treated as a forceacting from the left�hand pore with the center at xi onthe right�hand pore with the center at xj. Obviously,fi,j = –fj,i, that is, the force acting on the pore with thecenter in xi has the opposite direction. In other words,depending on the magnitudes of the applied compres�sive stresses p1 and p2, the pores experience eithermutual attraction or mutual repulsion. In (Mukhame�diev, 1990), with reference to the results obtained in(Savin, 1968; Kosmodamiansky, 1975), it was shownthat, provided (4),

(7)

Now, let ki,j • e2 = 1; i.e., let the pores be distributedalong the vertical axis x2. Then, with the forces fi,j andfj,i understood as previously, one should interchangeindices 1 and 2 in formula (7), that is, replace e1

by →e2, p1

→ p2, and p2 → p1.

Further, we consider the case of uniaxial compres�sion along the vertical axis, i.e., p = p2

p1. The terms

of the order of smallness are omitted. Then, forthe vertical and horizontal layout of the pair of pointsxi and xj, we have, respectively,

(8)

( ) ( ) ( )( ), ,

3 41 , 1 2 1 2 ,2 2 .

i j j i

i j i jc p p p p

= −

= π ε + − + Ο ε

f f

e

�

( )4,i jΟ ε

3, 2 , , 2 ,

3, 1 , , 1 ,

1 2 *

1 * ,

i j i j j i i j

i j i j j i i j

f

f

= → = − = ε

= → = − = − ε

k e f f e

k e f f e

i

i

x2

p2

p1

x1

a

e2e1

xj

xi

ki, j ϕi, j

Fig. 5. The geometry of the problem: the coordinate frame,the applied stresses and the layout of two round pores witha diameter a and centers at the points xi,

xj.

894



where the notation f* is introduced for the character�istic force (per unit thickness across the strain plane x1,x2):

(9)

From Eq. (8), it follows that two pores locatedalong the compression axis x2 experience mutualrepulsion whereas the pores located perpendicular tothe compression axis experience mutual attraction.Notably, in the two considered variants of orientationsof pore pairs with equal distances between the porecenters, the magnitude of the repulsive force is twicethe magnitude of the attractive force.

2.2. Mutual Motion of Two Pores at Special Positions

The existence of a force of interaction betweenpores only means the potential capability of theirmutual motion, either convergence driven by attrac�tive forces or divergence in the case of repulsive forces.In order for this potential capability to be actualized inthe form of a real motion (which will evidently reducethe total energy of the system), it is necessary that amechanism exists that would implement this realmotion. For two pores at special positions, which wereconsidered in the previous section (2.1), such a mech�anism has been suggested in (Mukhamediev, 1997) inthe scope of a 2D model of a granular monodispersemedium. A dense lattice packing of round grains was

( ) 21* .

4

a pf

π − ν=

μ

considered, and a pore was regarded as a packingdefect, which is a missing grain (Fig. 6). Each grain ina granular medium is in contact with six neighbors.These neighbors form a cage for the mentioned grain.Under the quasi�static loading of a densely packedrock, a grain cannot leave its “cage”. The only excep�tion is grains adjacent to the pore. They are sur�rounded by five neighbors; and for these grains, thepore is an adjacent void space. If such a grain is able tosurmount the cohesion with its five neighbors, achance arises for this grain to leave the cage and toreplace the pore. If this occurs, the grain swaps placeswith the pore, and in this case, we deal with the motionof a pore.

However, as can be seen in Fig. 6, the pore is sur�rounded by five grains. Which of the five grains willtake the place of the pore; and, thus, in which direc�tion will the pore migrate? In order to select a singlevariant from the five possible, we apply the principle,according to which the migration of the defect shouldreduce the total energy of the system. Moreover, fromseveral probable variants, the one is implemented thatprovides the maximal reduction of the total energy.This principle is rigorously followed throughout ourpresent work. We illustrate this principle by a concreteexample and consider a pore with center xj (Fig. 6a,left). Formally, each of six grains surrounding the poremay take its place. In order to make a unique selec�

p

xi' xj'

p

xi xj

xi'

xj'

pp

xj

xi

pp

1 2

(а) (b)

Fig. 6. The mechanism of mutual drift of the pores initially centered at xi, xj and oriented (a) along and (b) across the axis of com�

pression p. are the locations of the pore centers after displacement. The particles of the material: 1, bonded with neighbors,2, unbonded.

' ',i jx x



tion, we use Eq. (2) and the first equation of (8). Then,we can write out the following condition:

(10)

where δxj is a potentially possible displacement of thejth pore (i.e., the pore with the center at xj). For eachof the six variants of pore migration, |δxj| = a; there�fore, the maximal reduction in δEB will occur atδxj = ae2. From this it follows that the jth pore will shiftupwards by one diameter, having swapped places withthe grain above. Similar reasoning suggests that the ithpore (Fig. 6a) will migrate downwards. Thus, therepulsion between the pores centered at xi,

xj, whichare oriented parallel to the compression axis, causesmutual divergence of the pores along this axis and, viceversa, if the ith and jth pores are perpendicular to thecompression axis, an attractive force will cause thepores to converge (Fig. 6b).

It has been already mentioned above that, for beingable to leave “the cage” and swap places with the adja�cent pore, the grain must overcome the force of adhe�sion with its five neighbors. By recasting this statementin terms of a driving force on a pore, we may state that,in order to enable the migration, this force shouldexceed some threshold force, fc. To put it another way,the necessary condition for pore migration is that

(11)

In (Mukhamediev, 1990), it was shown thatrequirement (11) in the case of a pore moving alongthe principal stress axis (which is considered here) canbe equivalently reformulated as the necessity to takeinto account the efficient surface energy along thenewly formed surface. In the present work, we will notneed this interpretation.

2.3. The Statement of the Problem for a Finite Set of Pores

The solution for the problem of the comigration ofN pores in an infinite elastic plane under the action ofcompressive stress p along the vertical axis x2 is dis�cussed below in Section 3. The migration processoccurs in such a way as to decrease the total energy ofthe system. In order to solve this problem, the conceptof a driving force on a defect must be expanded on a setof N pores.

In the case of N pores, the domain under study is(N + 1)�connected, and the variational equation (2) isevidently recast in the following form:

(12)

Here, δxj is a potentially possible displacement of thejth pore, where fj is a force acting on the jth pore from

3, 2 ,2 * ,B j i j j i jE fδ = −δ = −δ εx f x ei i

, .i j cf>f

1

.N

B j j

j

E=

δ = − δ∑ x fi

the other N – 1 pores. Further, we assume the forces tobe additive, that is

(13)

where, the same as previously, fi,j, i = 1, …, j – 1, j + 1,

…, N, is a force acting on the jth pore from the ith pore.The force fi,j, for which Eqs. (8) hold true in the case ofa special position of pores, should be generalized tothe case of arbitrary pore positions, when the directionbetween xi and xj makes angle ϕi,j with the positivedirection of the x axis (Fig. 5). By making use of thefact that the force fi,j is directed along the line connect�ing the points xi and xj, we assume fi,j in the followingform:

(14)

Expression (14) at ϕi,j = 0, π/2, π, and 3π/2 coin�cides with the particular cases of the expression for theforce acting on the pore which were considered in Sec�tion 2.1. We suppose that condition (11) of the pres�ence of a threshold force, which is resisting the poremigration, remains in effect. In the case of a finite setof pores, this condition should be taken in the follow�ing form

(15)

A question remaining to be discussed is whether itis possible that the potentially feasible displacement ofa pore under the action of a driving force will turn intoa real displacement. As shown in the studies of poros�ity in granular media (e.g., see (Aste and Weiare, 2000;Zamponi, 2008)), there are three characteristic valuesγ that determine the fraction of monodisperse spheri�cal granules per unit volume. Kepler’s hypothesis sug�gesting that the most dense packing corresponds to aregular lattice with γ ≈ 0.74 was proven long ago byK.F. Gauss. In the case of a loose random packing ofbeads, γ = γlp ≈ 0.55. A similar value of γ should beexpected at the initial stage of sedimentation of drygranular monodisperse deposits. Due to compactionof dry monodisperse granules, γ attains the value γ =γdp≈ 0.64. The value of γdp corresponds to the so�calledstate of dense random packing. In the experiments,the transition from loose to dense random packing isachieved by tapping or shaking the container withgranules. Impacts of this kind are believed to be moreefficient for the compaction processes than the mereapplication of static pressure (Ribière et al., 2005;Richard et al., 2005). In natural conditions, the role ofsuch impacts is likely played by microseisms.

It is worth noting that the described impacts are notincluded in an explicit form in the statement of ourproblem. However, if the body forces of inertia causedby such impacts are negligible, these impacts can betaken into account in the statement of the problem ina formal way, e.g., by introducing a certain nonmono�tonic dependence of the applied compressive stress pon parameter t that plays the role of time. Let thedependence p(t) achieve the stationary value p = p*with increasing t; this attainment occurs before the

1, 2, 1, 1, ,... ... ,j j j j j j j N j− +

= + + + + + +f f f f f f

( )3, , 1 , 2 ,* cos 2 sin .i j i j i j i jf= ε − ϕ + ϕf e e

.j cf>f

896



pores stop migrating (see the solution algorithm belowin Section 3). Then, the final results in the processeswith p = p(t) and p = p* = const will not differ signifi�cantly. The distinctions will show up only in the meth�ods of attaining the final result.

In the context of our study, the micromechanics ofthe motion of spherical grains, which is observed in thecompaction experiments (Ribière et al., 2005; Richardet al., 2005), is an issue of importance. Here, theimpacts mentioned above disturb the granularmedium from the equilibrium state for a short time,after which repacking of the grains takes place. Theexperiments show that in the case of weak impacts, thegrains normally do not leave their “cages”; i.e., theydo not lose their neighbors. Their average displace�ments in this case are as small as 0.4% of the graindiameter. However, in rare cases, the impacts cause adisplacement as large as almost half the grain’s diam�eter, and then the grain escapes from the cage. Accord�ing to the estimates reported in (Ribière et al., 2005),there were only 100 such jumps among 10 000 impactsacting on a container with 4096 spherical granules.Despite their infrequent occurrence, these eventsdetermine the so�called slow dynamics of a granularmedium and, in particular, the pattern of the transi�tion from a loose to a dense random packing. Thepresence of such jumps ensures faster transition fromγlp to the asymptotic values γ, which are in this casecloser to the limiting value γdp than in the absence ofslow dynamics.

The same as in the kinematics of two pores, whichhas been considered in Section 2.2, in the model witha finite set of pores, one act of slow dynamics (a jumpwith a grain leaving its cage) means a small displace�ment of one, e.g., the jth, pore to a new position andobtaining new adjacent grains. Here, the probabilityincreases for one of the grains that are now adjacent tothe pore to be the next to experience a jump, etc. Thus,step by step, the jth pore will slowly migrate throughthe geological material, thereby implementing itsenergetically favorable potential possibility of motioninto real motion. In the real motion, for the totalenergy of the system to be lowered, it is sufficient thatthe displacement vector of the jth pore makes an acuteangle with the force vector fj. In the algorithm of thesolution to the problem, this angle is assumed to bezero (see below).

It is worth noting that, when solving the problem,we do not intend to keep to the cited values of the poredensity. This is because (1) our aim is to study in prin�ciple the feasibility of implementing the suggestedmechanism for the formation of incompact zones and(2) the values of γlp and γdp relate to a particular 3D caseof spherical granules. For granules of other shapes(particularly, in the 2D case), these values can be quitedifferent. However, the most convincing argument infavor of the arbitrary choice of pore density is providedby the results on solving the problem (see Section 3.2),which show that the value of the initial pore density

hardly affects the key characteristics of the systems ofincompact bands.

3. THE ALGORITHM AND THE SOLUTION

3.1. The Algorithm of the Solution

We consider the problem of a comigration of Npores in an infinite elastic plane which is subjected tothe action of compressive stress p along the vertical axisx2. At the initial state, the pores are randomly scatteredin a rectangular domain Ω: 0 ≤ x1≤

L1 , 0 ≤ x2

≤ L2. Eachjth pore migrates across the plane, being driven byforce fj (Eq. 13), which is resultant of the forces fi,j, i=1,…, j – 1, j + 1, …, N, calculated according to Eq. (14).The displacement occurs if the resultant force exceedsthe threshold value fc (see Eq. (15)). In the numericalmodeling, the process of migration is represented interms of a sequence of iterations. At each iteration,each pore accomplishes a displacement that is codi�rected and proportional in magnitude to the resultantforce acting on the given pore. Computations continueas long as at least one pore is affected by a forceexceeding the threshold value.

Let us consider the selection of the parameters formodeling in finer detail. The model parametersinclude the pore diameter a, the characteristic force f*defined by Eq. (9), the threshold force fc, and the poredensity where N is the number of poresand L1, L2 are linear dimensions of the initial domainΩ. Two of the four parameters have independentdimensions; therefore, according to the dimensiontheory, the number of parameters can be reduced totwo. We select a and f* as parameters with independentdimensions. Then, the other quantities involved in theproblem are transformed in the following way:

(16)

Here, Fc is the dimensionless threshold force and Ris the dimensionless density of pore distribution. Fur�ther in the text, the tildes in the notations of otherdimensionless parameters are omitted. All the follow�ing calculations and estimates are presented in termsof dimensionless units.

In the case of the uniform distribution of N pores ina square box with side L, the average distance

between the pores can be presented in thefollowing form:

(17)

Then, from the dimensionless analogue of expres�sion (14), the estimate follows for the maximal, interms of angle ϕi,j, force of interaction between the

pores: The mean of this force is

estimated from Eq. (17) as Thus, if

( )1 2 ,N L Lρ =

* *21,2

1,2, , , , .j cj c

L fL F R a

a a f f= = = = = ρ

fxx f��

�

i j−x x

1 .i j L N R− ≈ =x x

3, max

2 .i j i j= −f x x3 2

, max2 .i j R≈f



the pore concentration is as low that thenonly insignificant local displacements of pores arepossible in the domain Ω. Due to this fact, whenstudying the formation of incompact structures, whichimplies substantial alteration of the initial distributionof pores, one should specify the initial concentration R

such that is at least severalfold larger than .

After they have been converted into dimensionlessunits, L1 and L2 are the ratios of the domain size to thepore diameter a. In order to be able to study informa�tive scenarios of pore migration, one should specifythe values L1 and L2 to be severalfold greater than thenumber of pores N. For example, the main calcula�tions in the present work were carried out for N =

100, This means that

Here, in order for the condition

to be fulfilled, one has to assume

3.2. Results

We have conducted several series of calculations forvarious values of and R, which were kept constantthroughout the given series. Each separate seriesincluded 960 solutions of the problem with differentinitial distributions of pores in the domain. In all the

calculations where the condition was satis�fied, the pores, which were randomly scattered at theinitial moment of time, eventually clustered in anordered structure with the pores concentrated (local�ized) in the systems of quasi�parallel vertical bands(Fig. 7).

3/22 ,cR F<

3/22R cF

2 31 210 , 10 .L L≥ ≥

( )4

1 2 10 .R N L L −

= ≤

3 22 cR F>610 .cF −

≤

cF

3 22 ,cR F>

Due to the huge number of calculated cases, visualanalysis of the obtained results is not practical; there�fore, an algorithm for the recognition of vertical bandswas suggested and implemented in the computer pro�gram. The results of its operation can be tuned usingtwo parameters: ε, the distance separating two bands,and Nmin, the minimal number of pores in a band. Thepores are sorted in the order of increasing x1 coordi�nates; the first pore is assumed to pertain to the firstband. Then, if the distance (along x1) between the cur�rent and the next pore exceeds ε, it is assumed that thelatter pore belongs to a different band. The parameterNmin is introduced in order to sort out single (not per�taining to a system) pores and small sets of pores whichare bands only in terms of the proposed algorithm.

Periodicity is one of the characteristic featuresexhibited by the natural localization systems: the ele�ments of the systems are not only parallel but arespaced by approximately similar distances apart fromeach other. Based on the obtained numerical results,we have studied the properties of the distances betweenthe bands in the localization systems. In each calcula�tion case, the sets of pores forming the vertical bandswere recognized by the procedure described above,and the coordinates x1(b) of these bands were calcu�lated. The coordinates x1(b) were taken as the meanvalue of the minimal and the maximal coordinates x1of the pores pertaining to a band. Then, if severalbands were recognized in one calculation case, the dis�tances between the neighboring bands were calculated.The differences of x1(b) coordinates of the neighboringbands were computed for this purpose.

Based on the obtained sets of interband distances,histograms were constructed for each series of calcula�

00

200 400 600 800 100012001400160018002000

10

20

30

40

50

60

70

80

90

100x2

x1

200–800

400 600 800 1000 1200 1400 1600 1800

–600

–400

–200

0

200

400

600

800

1000x2

x1

(а) (b)

Fig. 7. An example of calculations from a series with the dimensionless parameters N = 100, L1 = 2000, L2 = 100, Fc = 10–7.(a) The initial pore distribution in the domain Ω; (b) the resulting pore distribution after the completion of migration of all pores.Proportions between the lengths L1, L2 and the pore diameter a are not kept.

898



tions (Fig. 8). The horizontal axes of the histogramsare the distances between two neighboring elements ofa system, and the vertical axes show how many timesthe given distance has been encountered in the givenseries of computations.

Our results show that all histograms have a clearlydistinct peak, i.e., most of the bands are spaced apartfrom each other by approximately equal distances.The position of the peak only depends on the value of

; as Fc decreases, the interband distance increases,and the maxima in the histograms become lower(Fig. 8a). The value of has almost no effect on thehistogram (Fig. 8b). Only if R decreases down to a crit�ically small value (Fc/2)2/3, the histogram starts tobecome slightly distorted (see the curve marked by cir�cles in Fig. 8b).

We note that the initial pore density, R, affects theheight of the bands that compose the system. Thiseffect can be interpreted as a feature introduced bymodeling. In order to understand why this happens,we must recall that the force of interaction between thepores attracts them towards each other in the directionof x1 and pushes them away from each other in thedirection of x2 (see Eq. (8)). In our calculations, wespecify the pores to be initially located within a limiteddomain Ω; therefore, the extreme pores in the direc�tion of x1, being attracted by the other pores, drifttowards the center, while the extreme pores in thedirection of x2, being repulsed, move off the center.The calculations show that in the direction of x1 thesize of the region occupied by the pores changes insig�nificantly (Fig. 7). The increase in the pore concentra�

cF

R

tion neither changes the distances between the bandsnor, correspondingly, their positions. Thus, theincrease in the concentration results in the increase inthe number of pores in each band, which causes theband to lengthen in the direction of x2.

CONCLUSIONS

In the present work, we suggest a fundamentallynew model for the formation of regular macrostruc�tures under uniaxial compression of sedimentaryrocks. This model significantly differs from the previ�ous models in a series of key points.

Under the uniaxial compression of poorly lithifiedsediments (typically sands), thin layers characterizedby increased porosity are formed along the compres�sion axis. We call these layers incompact layers (in the2D case, incompact bands). We proposed the term“incompact band” instead of the term “dilationband,” common in the western literature, which relieson the widely adopted theoretical interpretation of theorigination and development of the observed bands.The standard model of the formation of these andother types of deformation bands associates them withnarrow zones of localization of plastic deformations.In the case considered, it is implied that the elongationdeformations are localized across the band, whichcause the band to dilate. In terms of this approach, theformation of a band of localized deformations istreated as a phenomenon of a loss of stability.

Based on the experimental facts, we suggest a dif�ferent model for the formation of incompact layers,

0 200 400 600 800 1000 1200 1400 1600 1800 2000

100

200

300

400

500

600

700

0 200 400 600 800 1000 1200 1400 1600 1800 2000

100

200

300

400

500

600

700n

h

(а) (b)

h

n

Fig. 8. The histograms of distribution frequency n of the distances h between the systems of pores. (a) with fixed N = 100, L1 =2000, L2 = 100 for different Fс: the curves for Fc = 10–7 , 5–7, and 2.5–7are marked by asterisks, squares, and circles, respectively;(b) with fixed Fс = 10–7, N = 100, L1 = 2000 for different R: the curves for L2 = 100, 400, and 1000 are marked by asterisks,squares, and circles, respectively.



according to which it is not deformations but micro�defects in the sediment packing (the pores) that arelocalized in the narrow bands. Localization of pores,which have been initially randomly distributedthroughout the volume, is the result of their migrationin the geomaterial. A common mechanism driving theprocesses of pore migration and subsequent localiza�tion of the pores is the property of a system to tend toa lower total energy (the small changes in which areequal to the increments of free energy, excluding thework done by external forces).

In granular sedimentary rocks, migration of a sin�gle pore is caused by a driving force on a defect, f,which was introduced by J. Eshelby (1951; 1970). Animportant feature of our model is that the formation ofincompact layers here is not interpreted as a phenom�enon of a loss of stability. On the contrary, the forma�tion of incompact layers in terms of our model is con�sidered as a gradual process spread over time. Its timespan is likely associated with the interval of transitionof granular sediments from the state of loose randompacking into the state of dense random packing. Innatural conditions, the process of pore migration canbe facilitated by microseisms.

The two�dimensional model addressed in thepresent work is a multiple�pore generalization of themodel suggested by Mukhamediev (1990) for studyingthe bifurcation of a single pore and the mutual motionof two pores in an elastic medium under compressiveloading. If a pair of round pores is oriented along thecompression axis p, a force of mutual repulsion willarise between the pores. If, on the contrary, the poresare oriented perpendicular to the axis of compression,they will experience mutual attraction. In the presentwork, it is assumed that in the case of an arbitrarily ori�ented pair of pores, the value of the force of interactionbetween them continuously changes with the anglebetween the line connecting the centers of the poresand the compression axis. In the 2D case consideredhere, for sufficiently remote pores, the magnitude ofthe force of their interaction changes inversely as thecubic distance between the centers of the pores. Theforce f acting on a separate pore is a resultant of theforces acting from all other pores and a cause for apotentially possible motion of the pore.

Calculations were carried out for the case of a planestrain of an infinite elastic medium loaded by uniaxialcompressive stress p. At the initial state, a random spa�tial distribution of N round pores with a diameter a isspecified in a certain limited rectangular domain Ω. Inour calculations, at each iteration the force f driving adefect is found for each pore. When the magnitude |f|of this force exceeds the force fc resisting the poremotion, which is caused by cementation of the parti�cles of the material, the pore starts to move. In thiscase, in the course of calculations, the position of thepore was varied in the direction of the force f. The processof iterations is terminated when the condition |f| ≤ fc issatisfied for all pores. The calculations showed that,

due to pore migration which lowers the total energy ofthe system, the pores cluster in relatively regular struc�tures composed of quasi�parallel linear elements elon�gated along the axis of compression. We associate suchformations with the systems of incompact bands.

We have carried out several series of calculations.Each of the series was characterized by its own valuesof N, fc, and average spatial pore density ρ. With a fixednumber of pores N, the pore density ρ was varied byvarying the size of the initial domain Ω. Each series ofcalculations involved 960 cases with different initialrandom distributions of pores. For each series, the fre�quency histograms for distance h between the incom�pact bands were constructed. The histograms exhibit aclearly distinct hm mode; i.e., the value of h that occursmost frequently. The hm can be interpreted as a typicaldistance between the bands in the system over thegiven series of calculations. The left�hand branch ofthe histogram at h < hm is steep, while the right�handbranch (h > hm) is flatter. Our results show that the hm

mode is only affected by the dimensionless parameter

where with μ stand�ing for shear modulus and ν for Poisson’s ratio. As Fc

decreases, the hm mode increases; the maximum cor�responding to this mode becomes lower, and the histo�gram itself becomes more diffuse. In fact, with fixedparameters of the elastic model and known compres�sive stress p, the frequency distribution of the distancesbetween the systems is controlled by the force resistingthe pore motion, fc. At the same time, the dimension�less initial pore density, R, except for its critically lowvalues, affects neither the hm mode nor the shape of thehistogram. Probably, the latter fact indirectly indicatesthat the model we suggest is valid not only for themedium composed of regular monodisperse granulesbut also for a wider class of granular media.

An important feature of the approach proposed inthe present paper is that, as distinct from the standardmodels for the formation of deformation bands due toinstability, the formation of the systems of incompactbands is a direct sequence of the model used, withoutany assumptions on the existence of such systems andany special selection of the model parameters. More�over, based on the suggested model, we can predict theincompact bands to always occur in the form of regularsystems, which is a characteristic feature of many nat�ural processes of localization. A single incompactband can only be formed if a force fc resisting the poremotion is absent.

The present work leaves aside the question con�cerning the possibility for the incompact bands todevelop into joints in the sedimentary rocks in thecourse of lithification. This question is the subject offurther research.

*,c cF f f= ( )2* 1 4 ,f a p= π − ν μ

900



ACKNOWLEDGMENTS

The work was partially supported by the RussianFoundation for Basic Research and the Department ofEarth Sciences of the Russian Academy of Sciences(under Program 6). We are grateful to S.F. Kurtasovwho many years ago proposed to conduct such a workand carried out rough calculations.

REFERENCESAste, T. and Weaire, D., The Pursuit of Perfect Packing, Bris�tol, Philadelphia: Institute of Physics Publishing, 2000.Aydin, A., Borja, R.I., and Eichhubl, P., Geological andMathematical Framework for Failure Modes in GranularRock, J. Struct. Geol, 2006, vol. 28, pp. 83–98.Bai, Y.L., Xia, M.F., Ke, F.J., and Li, H.L., StatisticalMicrodamage Mechanics and Damage Field Evolution,Theor. Appl. Fract. Mech, 2001, vol. 37, pp. 1–10.Belousov, T.P., Kurtasov, S.F., Mukhamediev, Sh.A., andTsel’movich, V.A., Localization Instability of Sediements,in Geofizicheskie issledovaniya (Geophys. Res.), Moscow:IFZ, 2005, vol. 3, pp. 123–126.Benzerga, A.A. and Leblond, J.�B., Ductile Fracture byVoid Growth To Coalescence, Adv. Appl. Mech, 2010,vol. 44, pp. 169–305.Blumenfeld, R., Stresses in Isostatic Granular Systems andEmergence of Force Chains, Phys. Rev. Lett., 2004, vol. 93,no. 10, p. 108301.Borja, R.I. and Aydin, A., Computational Modeling ofDeformation Bands in Granular Media: I. Geological andMathematical Framework, Comput. Methods Appl. Mech.Engrg, 2004, vol. 193, pp. 2667–2698.Borja, R.I., Computational Modeling of DeformationBands in Granular Media: II. Numerical Simulations,Comput. Methods Appl. Mech. Engrg, 2004, vol. 193,pp. 2699–2718.Chemenda, A.I., The Formation of Tabular Compaction�Band Arrays: Theoretical and Numerical Analysis, J. Mech.Phys. Solids, 2009, vol. 57, pp. 851–868.Chemenda, A.I., Nguyen, S.�H., Petit, J.�P., and Ambre,J., Experimental Evidences of Transition from Mode ICracking To Dilatancy Banding, Comptes Rendus Meca�nique, 2011, vol. 339, pp. 219–225.Cundall, P.A. and Strack, O.D.L., A Discrete NumericalModel for Granular Assemblies, Geotechnique, 1979,vol. 29, pp. 47–65.Du Bernard, X., Eichhubl, P., and Aydin, A., DilationBands: A New Form of Localized Failure in GranularMedia, Geophys. Rev. Lett., 2002, vol. 29, no. 24, p. 2176.doi: 10.1029/2002GL015966Edelbro, C., Rock Mass Strength: A Review, Luleå, Sweden:Luleå University of Technology, 2003.Eichhubl, P. and Aydin, A., Ductile Opening�Mode Frac�ture by Pore Growth and Coalescence During CombustionAlteration of Siliceous Mudstone, J. Struct. Geol, 2003,vol. 25, pp. 121–134.Eshelby, J.D., The Force on An Elastic Singularity, Phil.Trans. R. Soc. London, 1951, vol. A133, pp. 87–112.Eshelby, J.D., Energy Relations and the Energy�Momen�tum Tensor in Continuum Mechanics, in Inelastic Behaviorof Solids, Kanninen, M.F, Alder, W.F, Rosenfield, A.R, and

Joffee, R.I., Eds., New York: Mc Graw�Hill, 1970, pp. 77–115.Eshelby, J.D., The Elastic Energy�Momentum Tensor, J.Elasticity, 1975, vol. 5, pp. 321–335.Fossen, H., Schultz, R.A., Shipton, Z.K., and Mair, K.,Deformation Bands in Sandstone: A Review, J. Geol. Soc.London, 2007, vol. 164, pp. 755–769.Garagash, I.A., Conditions of the formation of regular sys�tems of shear and compaction bands, Rus. Geol. Geophys.,2006, vol. 47, no. 5, pp. 655–666.Gill, S.P.A., Pore Migration Under High Temperature andStress Gradients, Int. J. Heat Mass Transport, 2009, vol. 52,pp. 1123–1131.Hu, S.Y. and Henager, C.H., Jr, Phase�Field Simulation ofVoid Migration in a Temperature Gradient, Acta Materialia,2010, vol. 58, pp. 3230–3237.Issen, K.A. and Rudnicki, J.W., Conditions for Compac�tion Bands in Porous Rock, J. Geophys. Res., 2000, vol. 105,pp. 21529–21536.Kang, S.�J.L., Sintering: Densification, Grain Growth, andMicrostructure, Oxford: Butterworth�Heinemann, 2005.Kaproth, B.M., Cashman, S.M., and Marone, C., Defor�mation Band Formation and Strength Evolution in Unlith�ified Sand: The Role of Grain Breakage, J. Geophys. Res.,2010, vol. 115, B12103. doi:10.1029/2010JB007406Kondaurov, V.I., Energy Approach to the Problems of Con�tinual Failure, Izv. Akad. Nauk SSSR, Fiz. Zemli, 1986,no. 6, pp. 17–22.Kondaurov, V.I., Mukhamediev, Sh.A., Nikitin, L.V., andRyzhak, E.I., Mekhanika razrusheniya gornykh porod(Fracture Mechanics of Rocks), Moscow: Nauka, 1987.Kondaurov, V.I. and Nikitin, L.V., Teoreticheskie osnovyreologii geomaterialov (Rheology of Geomaterials: Introduc�tion to Theory), Moscow: Nauka, 1990.Kosmodamiansky, A.S., Ploskaya zadacha teorii uprugostidlya plastin s otverstiyami, vyrezami i vystupami (Plane Elas�tic Problem for Plates with Holes, Notches, and Asperities),Kiev: Vysshaya shkola, 1975.Li, H., Fu, M.W., Lu, J., and Yang, H., Ductile Fracture:Experiments and Computations, Int. J. Plast., 2011, vol. 27,pp. 147–180.Mollema, P.N. and Antonellini, M.A., Compaction Bands:A Structural Analog for Anti�Mode I Cracks in AeolianSandstone, Tectonophysics, 1996, vol. 267, pp. 209–228.Mukhamediev, Sh.A., Protsessy razrusheniya v litosfereZemli (Failure Processes in the Lithosphere of the Earth),Moscow: IFZ AN SSSR, 1990.Mukhamediev, Sh.A., Failure Processes and Stress State ofthe Earth’s Lithosphere, Monograph Doctoral (Phys.–Math.) Diss., Moscow: Inst. Phys. Earth, Rus. Acad. Sci.,1997.Mukhamediev, Sh.A. and Nikitin, L.V., Application of theMaterial Force Concept To Formation of Discontinuities inSoft Sedimentary Rocks, Arch. Appl. Mech, 2007, vol. 77,pp. 155–163.Nikolaevsky, V.N., Geomekhanika i flyuidodinamika (Geo�mechanics and Fluid Dynamics), Moscow: Nedra, 1996.Olsson, W.A., Theoretical and Experimental Investigationof Compaction Bands in Porous Rock, J. Geophys. Res.,1999, vol. 104, pp. 7219–7228.



Perrin, G. and Leblond, J.B., Rudnicki and Rice’s Analysisof Strain Localization Revisited, J. Appl. Mech., 1993,vol. 60, pp. 842–846.Ribie�re, P., Richard, P., Delannay, R., Bideau, D., Toiya,M., and Losert, W., Effect of Rare Events on Out�Of�Equi�librium Relaxation, Phys. Rev. Lett., 2005, vol. 95,p. 268001.Richard, P., Nicodemi, M., Delannay, R., Ribie�re, P., andBideau, D., Slow Relaxation and Compaction of GranularSystems, Nature Materials, 2005, vol. 4, pp. 121–128.Rudnicki, J.W. and Rice, J.R., Conditions for the Localiza�tion of Deformation in Pressure�Sensitive Dilatant Materi�als, J. Mech. Phys. Solids, 1975, vol. 23, pp. 371–394.Rudnicki, J.W. and Sternlof, K.R., Energy Release Modelof Compaction Band Propagation, Geophys. Rev. Lett.,2005, vol. 32, p. L16303. doi:10.1029/2005GL023602Ryzhak, E.I., A Case of Indispensable Localized Instabilityin Elastic�Plastic Solids, Int. J. Solids Struct., 1999, vol. 25,pp. 4669–4691.Ryzhak, E.I., Criteria of Stability and Patterns of Instabilityof Weakened Elastoplastic Bodies, Doctoral (Phys.�Math.)Diss., Moscow, 2002.Savin, G.N., Raspredelenie napryazhenii okolo otverstii(Stress Distribution around a Hole), Kiev: NaukovaDumka, 1968.

Schultz, R.A. and Fossen, H., Terminology for StructuralDiscontinuities, AAPG Bulletin, 2008, vol. 92, pp. 853–867.Socolar, J.E.S., Schaeffer, D.G., and Claudin, P., DirectedForce Chain Networks and Stress Response in Static Gran�ular Materials, Eur. Phys. J. E, 2002, vol. 7, pp. 353–370.Stefanov, Yu.P. and Tjerselen, M., Modeling of the Behaviorof Highly Porous Geomaterials at the Formation of Local�ized Compaction Bands, Fiz. Mezomekh, 2007, vol. 10,no. 1, pp. 93–106.Sternlof, K., Rudnicki, J.W., and Pollard, D.D., AnticrackInclusion Model for Compaction Bands in Sandstone, J.Geophys. Res., 2005, vol. 110, p. B11403.doi:10.1029/2005JB003764Sun, Q., Wang, G., and Hu, K., Some Open Problems inGranular Matter Mechanics: Review, Prog. Nat. Sci, 2009,vol. 19, pp. 523–529.Xu, X.H., Ma, S.P., Xia, M.F., Ke, F.J., and Bai, Y.L.,Damage Evaluation and Damage Localization of Rock,Theor. Appl. Fract. Mech, 2004, vol. 42, pp. 131–138.Zamponi, F., Packings Close and Loose, Nature, 2008,vol. 453, pp. 606–607.Zhu, H.P., Zhou, Z.Y., Yang, R.Y., and Yu, A.B., DiscreteParticle Simulation of Particulate Systems: A Review ofMajor Applications and Findings, Chem. Eng. Sci., 2008,vol. 63, pp. 5728–5770.

formation of systems of incompact bands parallel to the compression axis in the unconsolidated...

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