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Stress permutations: Three-dimensional distinct element

analysis accounts for a common phenomenon

in brittle tectonics

Jyr-Ching Hu

Department of Geosciences, National Taiwan University, Taipei, Taiwan Jacques Angelier1

Observatoire Oce´anologique de Villefranche, Ge´osciences Azur, Villefranche-sur-Mer, France Received 8 June 2003; revised 16 May 2004; accepted 9 June 2004; published 16 September 2004.

[1] Using three-dimensional (3-D) distinct element modeling, we explored a variety of

simulations to characterize and interpret the stress permutations in brittle tectonics. Stress inversions of fault slip data or earthquake focal mechanisms often revealed such

permutations. The main aim of our study is to produce simple, mechanically consistent 3-D models that account for these switches between the principal stress axes s1/s2 or

s2/s3. Even with simple boundary conditions the stress changes induced by variations in

rheology are large enough to modify the local tectonic behavior and to produce permutations of principal stress axes. Rather than simple directional changes of stress axes, which exist but often remain limited, the relative variations in principal stress magnitudes are the major cause of permutations s1/s2 ands2/s3. In nature, permutations

being left apart, the orientations of axes often remain tightly clustered. In our experiments we adopted a ratio F = (s2  s3)/(s1 s3) of 0.5, which makes permutations difficult

(lowF favors s2/s3permutations, highF favors s1/s2permutations), and we explored a

variety of tectonic situations involving compression, extension, and strike slip. Our experiments indicate that the major causes of stress permutations are the heterogeneity of the brittle deformation (e.g., intact rock massifs between heavily faulted deformation zones) and the anisotropy of the mechanical properties that results from fracturing and faulting, which concur to modify the mechanical balance inside the analyzed volume and to produce stress permutations. Our models show that contrasts and anisotropy in rock properties favor stress permutations. Of major importance is the existence of relatively resistant zones at tips of the deformed zones, acting as channels where stress concentrates and switches occur. Because in nature such zones move in time and space, it is not surprising that stress permutations are so pervasive. INDEXTERMS: 1744 History of Geophysics: Tectonophysics; 8168 Tectonophysics: Stresses—general; 8020 Structural Geology: Mechanics; KEYWORDS: stress permutations, brittle tectonics, numerical modeling

Citation: Hu, J.-C., and J. Angelier (2004), Stress permutations: Three-dimensional distinct element analysis accounts for a common phenomenon in brittle tectonics, J. Geophys. Res., 109, B09403, doi:10.1029/2003JB002616.

1. Introduction

[2] Stress permutations are common in brittle tectonics at

local and regional scales [Angelier and Bergerat, 1983; Angelier et al., 1985; Larroque et al., 1987; Yeh et al., 1991; Hippolyte et al., 1992; Angelier, 1994; Bergerat et al., 2000; Angelier et al., 2000]. While reconstructing paleo-stress from fault slip data or present-day paleo-stress field from focal mechanisms, one often identifies several stress tensors recorded at a single site. Some tensors correspond to distinct

tectonic events, which may reflect either polyphase tecton-ism or block rotation for monophase tectontecton-ism. In such cases, the attitudes of the stress axes for the different stress states differ by a variety of angles, which have no particular reason to be close to 90.

[3] Other situations exist, so that all angles between the

stress axes for two or more contrasting stress states are right angles. In other words, for a given event accounted for by a main stress tensor, additional stress tensors are often recon-structed and display the same symmetry axes as the main tensor but principal values in a different algebraic order. That such associated stress regimes simultaneously occur is obvious in the case of present-day earthquakes, or can be demonstrated on the basis of geological evidence in the case of ancient brittle structures. These additional states of stress

1

Also at Institut Universitaire de France, Paris, France. Copyright 2004 by the American Geophysical Union. 0148-0227/04/2003JB002616$09.00

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do not reflect-dependent events with respect to the main stress state but often result from linked mechanisms that occurred during the tectonic event considered. Such changes occur in time as well as in space. They were described as permutations of stress axes [e.g., Angelier and Bergerat, 1983; Letouzey, 1986] because they can be regarded as simple switches between principal stresses. In this respect, permutations drastically differ from rotations of principal stress axes, which were also widely documented but will not discussed in this paper.

[4] In most permutation cases the intermediate principal

stresss2is replaced by the maximum compressional stress

s1or the minimum stresss3(permutationss1/s2ands2/s3,

respectively). There are many examples of such permuta-tions throughout the world. A simple example of wide-spread stress permutation in a compressional tectonic setting has been described in the Costal Range of eastern Taiwan [Barrier and Angelier, 1986]. The compressional stress axis s1determined from fault slip data inversion trends N120 –

130E on average and displays very shallow plunges, whereas the minimum stress axis s3 is either close to

vertical (a reverse faulting regime) or close to horizontal (a strike-slip faulting regime). The switches betweens2and

s3axes are expressed by close association between reverse

and strike-slip faults, which are all consistent with the same s1axis. This situation is illustrated in Figure 1. The results

were consistent with the behavior of the major Longitudinal Valley fault zone, dominated by NW – SE compression that induces reverse faulting with a minor left-lateral component. [5] Such permutations also occur in extensional settings.

Note that throughout this paper, ‘‘extension’’ refers to situations with a nearly horizontal minimum stress axis and does not imply actual tensional stress (which would be negative according to our convention). They result either in systems of conjugate normal faults with perpendicular trends (s2/s3 permutation) or in mixed conjugate sets of

normal and strike-slip faults (s1/s2permutation), as

sum-marized in Figure 2. In the first case, the vertical or nearly verticals1axis is in common, and the two main systems of

normal faults reveal perpendicular extensions. In the second case, the horizontal or nearly horizontals3axis is common,

and the nearly horizontal perpendicular axis iss1(strike-slip

type) ors2(normal type). A systematics1/s2permutation in

extensional tectonics was outlined for the Hoover Dam area, western United States [Angelier et al., 1985; Angelier, 1989], where crosscutting relationships revealed intricate sequences from normal dip slip to strike slip or from strike slip to normal dip slip. Widespread s2/s3

permuta-tions were revealed by fault slip analyses in Crete and the whole southern Aegean domain [e.g., Angelier et al., 1982; Angelier, 1994]. The origin of these permutations has not been clearly explained and may result from a variety of causes, which can be classified into two groups. The first group involves actual modifications in the stress field (induced by far-field changes, e.g., plate motion changes, or local conditions that affect the vertical stress, e.g., erosion, deglaciation, or burial). The second group involves little change in tectonic environmental conditions and is mainly under control of the ongoing brittle deformation response that induces stress changes (e.g., stress drop) and depends on the rheological properties of the affected rock masses. In this paper, we mainly consider the latter causes, and we aim to demonstrate that variations in rheology are sufficient to induce stress permutations.

[6] The ratio of stress differences,F = (s2 s3)/(s1 s3),

provides a convenient index to characterize the relationship between the principal stress magnitudes [Angelier, 1975, 1989]. This ratio F ranges from 0 (meaning that s2 = s3)

to 1 (meaning that s1 = s2). Whereas simple extension

generally corresponds to relatively high values of F (e.g., 0.5), multidirectional extension is characterized by low values that make s2/s3 stress permutation easier. In

com-pressional tectonics, changes between reverse and strike-slip faulting modes generally correspond to situations with low values ofF, down to about zero in the case of the Coastal Range of Taiwan mentioned above (Figure 1). Low values of F thus favor s2/s3permutations, regardless of the tectonic

setting. These considerations highlight the major role played by the ratio between principal stress differences: the impor-tance of reverse/strike-slip mixed modes of faulting increases asF decreases in a compressional stress regime. Conversely, where the tectonic regime is dominated by extension, a decrease in the ratioF results in more irregular trajectories ofs3and local permutations ofs2/s3, as in the case of the

southern Aegean domain mentioned above. Fault slip anal-yses in Crete effectively revealed low values ofF, consistent with frequent permutations betweens2ands3axes.

[7] Despite expert understanding of the stress

permuta-tion phenomenon in Earth sciences, little has been done to understand its physical origin. Many variations in stress state, such as the accommodation of deformation within complex inherited structural patterns, variations in lateral confining pressure, and overburden pressure, may produce stress permutations. The effects of the stress changes induced by faulting and controlled by fluid pressure varia-tions should also be invoked. They partly account for a complexity of stress patterns near major faults that could not be explained in terms of polyphase tectonics, as was shown Figure 1. Examples of stress permutations of two

conjugate fault systems in the Costal Range, eastern Taiwan. Principal stress axes: s1, s2, ands3. Stereograms

are in Schmidt’s projection, lower hemisphere (data from Barrier and Angelier [1986]).

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along an oceanic transform fault in northern Iceland [Angelier et al., 2000]. It is likely that the stress drop plays an important role in producing quick stress permutations during earthquake activity. Because a major aim of this paper is to demonstrate that even in the absence of elastic discharge (aseismic faulting) and fluid pressure changes, these effects are not involved in our modeling and will be analyzed in a future study. Rather than address short-term seismological phenomena, we examine the problem of stress permutations in terms of the effects that may affect major structures of the brittle crust during long tectonic periods.

[8] In this paper, our aim is to characterize the stress

permutations induced by changes in a stress field that resulted from the presence of preexisting structural grains and mechanical anisotropy of blocks. To address the prob-lem of stress field patterns around mechanical discontinu-ities with anisotropic block material, we followed an approach based on 3-D distinct element numerical model-ing. To clearly identify the sources of permutation, we analyze simple geometrical patterns in which large numbers of singularities concur to produce stress perturbations and permutations. For this reason, the relatively minor disconti-nuities inside fracture zones, which induce additional per-turbations in the deformation field, are not taken into account. We consequently restrict our study to simple cases, with few major discontinuities and simple patterns of blocks, so that mechanical constraints can be clearly defined and validated with geological data. We especially consider elongated zones of mechanical weakness with anisotropic properties, a common feature in structural geology. We thus combine a 3-D numerical simulation with knowledge of the

deformation pattern in compressional, extensional, and strike-slip tectonic settings in order to check the mechanical validity of the observed stress permutations. We finally show that although the stress permutation phenomenon certainly adds much to the structural complexity, it obeys relatively simple laws and can be accounted for by simple mechanisms that involve variations in rheology.

2. Numerical Modeling: Method

2.1. Principle of the Distinct Element Analysis

[9] The distinct element method is a numerical method in

geomechanics that enables one to simulate the mechanical response of systems composed of discrete blocks or particles [Cundall and Strack, 1979]. This method falls within the general classification of discontinuum analysis techniques. A discontinuous medium is distinguished from a continuous medium by the existence of interfaces or contact between the discrete bodies that constitute the system. The formulation and development of the distinct element method have been continuous improving for a period of more than 25 years, since the initial presentation by Cundall [1971]. Many authors [Cundall, 1971, 1988; Hart et al., 1988; Harper and Last, 1990] have published detailed descriptions of the numerical schemes, and a detailed description of the 3DEC computer code is beyond the scope of this paper. The distinct element method is based on a time marching integration scheme using central finite difference. It is an explicit method in which the equations of dynamics (e.g., Newton’s second law for rigid body) are integrated over time.

Figure 2. Stress permutations between stress axes and rift opening (modified after Angelier and Bergerat [1983] and Angelier [1994]). Case A is perpendicular extension ands2/s3permutation. Case B

is from normal extension to strike-slip regime and s1/s2 permutation. Mohr diagrams indicate the

magnitude of the vertical principal paleostress, sv, depending on the tectonic regime shear stress as

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[10] Within the distinct element method a rock mass

consists of a system of discrete blocks in mechanical inter-action at their boundaries. Blocks may behave as deformable elastic bodies. Block interactions are computed assuming finite stiffness at contact points along the interface between blocks. The case of deformable blocks is accounted for by considering a finite difference grid with mass points located at vertices of tetrahedra. The vertices of the finite tetrahedra zones located along the discontinuities are the contact points between blocks. The contact forces applied to rigid blocks or discrete particles are given by the elastic response at contact points, which are evaluated after each integration cycle over the time step. Opening or shear displacement along block interfaces occurs when the normal and the shear forces meet conditions for tensile and shear failure, respectively. The shear strength is defined through the Mohr-Coulomb failure criterion, which requires definition of the angle of sliding friction and cohesion as joint properties.

[11] The distinct element method enables one to take into

account the different rheological domains (considering different mechanical properties in each block, which means different values of density, Young’s modulus, and Poisson’s ratio) as well as the mechanical discontinuities (with a displacement law and a friction law), such as for preexisting faults. Thus using this method is particularly worthwhile for analyzing the mechanical behavior of prefractured geolog-ical domains. Several modeling experiments have already demonstrated the ability of this method to solve geological problems involving fault behavior [Dupin et al., 1993; Sassi et al., 1993; Homberg et al., 1997; Hu et al., 1997, 2001a, 2001b; Matsuda and Iwase, 2002; Pascal, 2002]. In the distinct element method the deformation of the medium is controlled by Newton’s second law (for details, see Appendix A). Deformation is continuous in the blocks and follows the elasticity theory. Sliding along discontinuities is allowed and controlled by a force-displacement law, and shear stresses along the discontinuities are limited according to a Mohr-Coulomb friction law. As a consequence, the required parameters to solve the calculation include (1) the density, the elastic constants of the medium, and (2) the coefficient of friction, the cohesion and the normal and shear stiffnesses of the discontinuities. For our models we will introduce the anisotropic constitutive law for the whole behavior of fractured zones; as a consequence, the mechanical properties of individual discontinuity are fixed for all models. The normal and shear stiffness is 220 GPa/m, and the coefficient of friction is 100 for numerical calcula-tion. The potential and accuracy of the distinct element method for quantitative modeling of the deformation have been shown using relatively simple models and case exam-ples [Itasca Consulting Group, 1998a, 1998b].

[12] The results of the numerical determination of

princi-pal stresses inside the model, which is crucial to characterize the stress patterns and the possible permutations, deserve attention. We first examined situations with similar values for the three principal stresses,s1 s3being close to zero,

indicating situations close to isotropic stress so that no stress attitude can be reliably determined. If there is very little difference between the numerically determined values of two principal stresses,s2ands3ors1ands2, the attitudes of

the axes cannot be determined unambiguously so that in terms of stress directions the degree of reliability of the stress

distribution model is low. We note that by definition the near-permutation situations that we are interested in are characterized by similar values for two principal stresses. The ratio of stress differences, F, as defined in section 1, provides a good indication. If this ratio is very close to 0 or 1, it reveals that two principal stresses have very close values (s2ands3for a value close to 0, ands1ands2for a value

close to 1), relative to their difference with the remaining principal stress value. The significance of the stress attitudes reconstructed in numerical modeling has been evaluated in light of these values, s1  s3 for detection of nearly

spherical stress ellipsoids ands1 s2ands2 s3(orF)

for stress ellipsoids that are nearly of revolution. 2.2. Constitutive Law and Anisotropy

[13] Many rocks exposed near the surface show

well-defined fabric elements in the form of bedding, stratification, layering, foliation, or jointing, and this means that their physical properties vary in different directions and are said to be inherently anisotropic. In elastic materials the ratio of linear strain to stress applied in the same direction is expressed by Young’s modulus. If a rock is anisotropic in its elastic properties, Young’s modulus is not a single value but is direction-dependent. In some directions a given stress will produce a greater strain than in other directions. Observed strains and structures manifesting such strains are not simply a function of the imposed stresses but also depend on rock anisotropy. The importance of anisotropy has been clearly shown in the analysis of overcoring tests conducted at the Underground Research Laboratory (URL) site located 100 km northeast of Winnipeg, Manitoba, Canada, in the Lac du Bonnet granite batholith [Martin and Simmons, 1993]. At the 240 level of the URL site the granite was found to be anisotropic because of pervasive microcracks and was found to be aligned with a major joint set [Amadei, 1996].

[14] Deformability experiment results on anisotropic rocks

are commonly analyzed in terms of the theory of elasticity for anisotropic media by assuming Hooke’s law. The stress state of three-dimensional stress at a point is described by a stress tensor consisting of six-dependent components (sx,sy,sz,

txy,txz,tyz), and the corresponding strain is described by an

infinitesimal strain tensor made up of six components (ex, ey,

ez, gxy, gxz, gyz). For a general linear elastic material each of

the six strains is a linear function of all the six stress components. The symmetrical tensor has 21-dependent com-ponents in the most general case [Jaeger and Cook, 1979; Ramsay and Lisle, 2000]. However, for most practical cases, anisotropic rock is often modeled as orthotropic or trans-versely isotropic media in a coordinate system attached to their apparent structure or directions of symmetry. In fact, a realistic approximation for many geological situations, such as that for a layered material affected by parallel bed-perpendicular fractures, is considered as an orthotropic case. Thus the orthotropic formulation has been used in the literature to characterize the deformability of rocks such as sandstones, schists, slates, gneisses, and granites. Nine-dependent elastic constants are needed to describe the deformability of the orthotropic materials, as depicted below. In our models we further simplify this situation by consider-ing transverse isotropy, which reduces the number of param-eters to five and still represents a realistic approximation for many fractures zones, as mentioned in section 2.2.2.

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2.2.1. Orthotropic Materials

[15] Many elastic materials are approximately

ortho-tropic, which means that their elastic properties have three mutually perpendicular planes of symmetry. For orthotropic materials with their planes of symmetry aligned parallel to the coordinate axes, Hooke’s law can be expressed as following the compliance tensor [Jaeger and Cook, 1979]:

ex ey ez gxy gyz gxz 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ¼ 1 E1 n21 E2 n31 E3 0 0 0 n21 E1 1 E2 n32 E3 0 0 0 n13 E1 n23 E2 1 E3 0 0 0 0 0 0 1 G12 0 0 0 0 0 0 1 G23 0 0 0 0 0 0 1 G13 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 sx sy sz txy tyz txz 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : ð1Þ In equation (1) the subscripts 1, 2, and 3 refer to measurements in the direction of x, y, and z axes, which are aligned with the principal directions of elasticity. E1, E2,

and E3 are Young’s moduli in the 1, 2, and 3 directions,

respectively. G12, G13, and G23 are shear moduli in planes

parallel to the 12, 13, and 23 planes, respectively. The variable nij (i, j = 1, 2, 3) represents Poisson’s ratio that

characterizes the normal strains in the symmetry directions j when a stress is applied in the symmetry directions i. 2.2.2. Transverse Isotropy

[16] Equation (1) can still apply if the rock is transversely

isotropic in one of the three 12, 13, or 23 planes. Thus the transverse isotropy is an example of orthotropic anisotropy involving a plane of isotropy where elastic properties have axial symmetry. The transverse isotropy formulation has been used to characterize the deformability of rocks such as schists, gneisses, phyllites, shales, and basalts. For such rocks the plane of transverse isotropy is assumed to be parallel to schistosity, bedding planes, and foliations. A simple illustration of this behavior is an elastic foliation parallel to the 1 – 3 plane as the plane of transverse isotropy demonstrated by equation (2). In that case, only five-dependent elastic constants are required to characterize this type of anisotropy: ex¼ 1 E1 sx n12 E1 sy n13 E1 sz; ey¼  n12 E1 sxþ 1 E2 sy n12 E1 sz; ez¼  n13 E1 sx n12 E1 syþ 1 E1 sz; ð2Þ gxy¼ 1 G12 txy; gyz¼ 1 G12 tyz; gxz¼ 2 1ð þ n13Þ E1 txz;

where E1is Young’s modulus for all directions within the

plane of transverse isotropy (foliation plane) and E2 is

Young’s modulus in the direction normal to the plane of transverse isotropy. There are two Poisson’s ratios: n13

refers to any direction within the plane of transverse isotropy, whereasn12quantifies the contraction

perpendic-ular to the plane of transverse isotropy as a proportion of the extension along the plane of transverse isotropy. In addition, G13is the shear modulus in the plane of transverse isotropy.

G12 = G23 is the shear modulus normal to the plane of

transverse isotropy. The modulus G12is often expressed in

the following equation based on laboratory testing of rock [Lekhnitskii, 1981] by assuming the 1 – 3 plane is the plane of isotropy:

G12¼

E1E2 E1ð1þ 2n12Þ þ E2

: ð3Þ

3. Common Features of 3-D Models

3.1. Basic Model Configuration and Boundary Conditions

[17] We have carried out a variety of 3-D numerical

experiments during a detailed investigation of stress permu-tations created with different boundary conditions and rheologies. All models, however, have common basic characteristics, which permit simple comparisons.

[18] First, the geometric configuration considered in our

3-D distinct element model is a rectangular block with fixed dimensions. The internal configuration and rheological properties of this block vary according to the experiment, as described in section 3.2. As Figure 3 shows, the studied block is always included in a much larger cube with isotropic properties (60 GPa for Young’s modulus and 0.25 for Poisson’s ratio). This geometric configuration precludes edge effects that otherwise would strongly affect the distribution of stress and deformation. The ratio between the sizes of the internal and external blocks is 10 (that is, the total volume is 1000 times the analyzed block), which greatly diminishes undesirable edge effects. The average mesh size is large inside the outer block, where only a few control points are necessary, and small inside the inner block where the stress deformation distribution is analyzed in detail (Figure 3a). On average, there are 1281 elements in the central block for a total of 17,219 elements. This equates to a network density of74 times greater in the analyzed block considering the large difference in volume.

[19] Second, all models were subject to identical

bound-ary conditions. The imposed boundbound-ary conditions involve typical triaxial stress with principal axes perpendicular to block faces, respectively (Figure 3b). The principal bound-ary stress values ares1= 30 MPa,s2= 20 MPa, ands3=

10 MPa. This means that theF value, that is, the ratio (s2

s3)/(s1 s3), is fixed as 0.5 for all numerical experiments.

In other words,s2= (s1+s3)/2. This value ofF has been

chosen because it makes permutations more difficult than smaller or larger values. Small and large values would result in easier s2/s3 and s1/s2 permutations, respectively. It

should be kept in mind, throughout this paper, that usual geological situations commonly involve large or small F values, which significantly favor stress permutations with respect to our experiments. As a consequence, as far as the far-field stress condition is concerned, our results

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provide minimum frequency estimates for the occurrence of permutations.

3.2. Model Structure

[20] The structure of the analyzed block was chosen

according to realistic, albeit simple, requirements. A com-mon situation involves alternating elongated zones with relatively weak and strong rheologies. This type of situation is imitated by adding two boundaries parallel to a block face, which bound an additional relatively resistant zone with isotropic properties inside the model (Figure 4). When the weaker lateral domains have anisotropic properties, two main types of situations are considered: assuming ortho-tropy with transverse isoortho-tropy as defined before, the E2axis

may be perpendicular (Figure 4a) or parallel (Figure 4b) to the interfaces between the resistant and weak domains. As stated before, E2is Young’s modulus in the direction normal

to the plane of transverse isotropy.

[21] In geological terms these situations resemble intact

rock zones inside heavily fractured ones. In the anisotropic case the pervasive fractures and faults are considered as having a large variety of attitudes around a common direction, the E2 axis. This results in stronger average

properties in the direction parallel to the axis (Figure 4), so that E2> E1. First, note that the ‘‘foliation’’ mentioned in

section 2.2 would be perpendicular to the fractures consid-ered now. Second, the assumption of transverse isotropy is a simplification because it would imply that the directions of the fractures in the plane perpendicular to the E2axis are

randomly distributed, which is generally not the case. This simplification, discussed in section 4.2, was adopted be-cause otherwise a large number of parameters would have precluded clear identification of the sources of stress permutations.

[22] The situation of Figure 4b, with resistant zone

parallel to E2, reflects a common case in structural

geology: a domain of relatively undeformed rock that trends parallel to the axis of a system of fractures and

faults. In contrast, the situation in Figure 4a enables us to investigate the case of an intact rock bridge left behind fractured zones and elongated perpendicular to their trend. These two cases are schematically illustrated in Figures 4c and 4d, with examples of normal faulting: relatively intact horsts exist between deformed grabens (Figure 4d) or faulted zones interrupted by relatively intact domains (Figure 4c).

[23] An important source of variety in the models deals

with the orientation of the far-field principal stresses, as compared with the rheological boundaries inside the model. In our study, we distinguished three types of situations, normal, strike slip, and reverse in type (that is, with vertical s1,s2, ands3, respectively). Considering the two

geomet-rical configurations discussed above, we analyzed the six typical patterns shown in Figure 5. Detailed boundary conditions and rheological properties for numerical experi-ments are illustrated in Table 1.

4. From Isotropic to Anisotropic Rheologies in Models

4.1. Isotropic Case

[24] Preliminary experiments were carried out with

isotropic properties in order to allow comparison with the experiments involving anisotropy. A very simple situation is depicted in Figure 6, with a weak zone between relatively Figure 3. Basic geometric configuration and boundary

condition of the 3-D distinct element experiment. (a) Three-dimensional block diagram showing the geometry and discretization of the 3-D distinct element model. (b) Schematic block diagram and boundary condition (triaxial stress with principal axes perpendicular to block faces). Principal stresses ares1= 30 MPa,s2= 20 MPa, and

s3 = 10 MPa, respectively. Detailed configuration of the

analyzed central block (in gray) is illustrated in Figure 4.

Figure 4. Structure of the analyzed block involving two types of alternating elongated zones corresponding to relatively weak and strong rheologies. (a) A relatively resistant zone with isotropic properties (in gray) located inside two weaker lateral domains with anisotropic proper-ties. E2axis is perpendicular to the interfaces between these

two domains (for detail, see text). (b) A similar definition of a resistant zone (in gray) and weak lateral domains. E2axis

is parallel to the interfaces between of these two domains (for detail, see text). (c) A geological resemble of the analyzed block in Figure 4a with an intact rock bridge left in between fractured zones and elongated perpendicular to their trend. (d) A geological analogue of the analyzed block in Figure 4b with the example of normal-faulting relatively intact horsts existing between deformed grabens.

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strong domains. Two experiments are illustrated, with identical boundary conditions (as described before) and rheological parameters (uniform Poisson’s ratio of 0.25 and Young’s moduli of 60 GPa for the strong domain and 6 GPa for the weak domain). The weak zone trends perpendicular to the minimum stress (extension) axis. The only difference between the two experiments is the relative width of the weak domain: one half of that of each strong domain in the first case (Figure 6a) and one fourth in the second case (Figure 6b).

[25] The results are illustrated in terms of both the

stereo-plots showing the attitudes of the principal stress axes calculated for each element of the ‘‘weak’’ zone and a horizontal cross section at midheight of the model. Not surprisingly, in both cases the cross sections show that the horizontal components of the extreme principal stresses s1

and s3 have more scattered directions, lower absolute

values, and less difference in amplitude in the weak domain, as compared with the ‘‘strong’’ domain.

[26] An interesting difference between the two

experi-ments of Figure 6 is the influence of the relative width of the weak domain. This difference is detectable in the cross section. It is best illustrated in the stereoplot, which refers to the entire analyzed volume. Whereas the s1

axes are relatively homogeneous, having nearly vertical orientations in both cases, the distributions of the s2 and

s3 axes are markedly different. In the first case (wide

weak domain, Figure 6a) the trends show an azimuthal dispersion so that they almost lie in two opposite quad-rants centered around the far-field s2 and s3 axes (N – S

and E – W, respectively, in Figure 6). In other words, the maximum deviation of the local stress axes s2 and s3 is

about ±45 relative to the corresponding far-field axis in the horizontal plane; the plunge variation is much smaller. In the second case (narrow weak domain, Figure 6b), there are two distinctive clusters of local axes, one being parallel to the far-field axis (N – S s2 or

E – W s3, depending on the stereoplot) and the other

Figure 5. Six different patterns of the far-field principal stresses boundary conditions corresponding to three types of tectonic settings: normal, strike slip, and reverse in type. Convergent couple arrows indicate thes1direction. Divergent couple arrows represent the s3direction. For clarity, s2direction is

not shown.

Table 1. Summary of Boundary Conditions and Rheological Properties for Numerical Experimentsa

Figure

Boundary Conditions and Coordinate Axes Versus Stress Axes

Principal Stress Boundary Conditions,

MPa Rheology, GPa

Axis of Anisotropy s1 s2 s3 s1 s2 s3 Isotropic Domain Anisotropic Domain

3 Y Z X 30 20 10 Z(s2) 6 Y Z X 30 20 10 E1= 60, E2= 6 Z(s2) 7b Y Z X 30 20 10 E = 60 E = 60, E0= 48 Z(s2) 7c E = 60, E0= 18 7d E = 60, E0= 6 9 Y Z X 30 20 10 E = 60 E = 60, E0= 12 Z(s2) 10 X Z Y 30 20 10 E = 60 E = 60, E0= 12 Z(s2) 11b X Y Z 30 20 10 E = 60 E = 60, E0= 6 Y(s2) 11c X Y Z 25 20 15 E = 60 E = 60, E0= 6 Y(s2) 11e Z Y X 30 20 10 E = 60 E = 60, E0= 6 Y(s2) 11f Z Y X 25 20 15 E = 60 E = 60, E0= 6 Y(s2) 12 Z Y X 30 20 10 E = 60 E = 12, E0= 60 X(s3) 13 X Z Y 30 20 10 E = 60 E = 60, E0= 12 Z(s2) 14 Y Z X 30 20 10 E = 60 E = 60, E0= 12 Z(s2) a

Examples of coordinates, stress axes, and block inclusion shown in Figure 3. In all models, X, Y, and Z axes indicate E – W, vertical, and N – S directions, respectively. E is Young’s modulus along the axis of anisotropy. E0isYoung’s modulus in the direction perpendicular to this axis.

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being perpendicular. Interestingly, the intermediate azi-muths (near NE – SW and NW – SE) are rather few, which indicates that the major phenomenon is not a simple azimuthal dispersion but a real permutation. Although the amplitude difference remains generally limited, the local occurrences of s2/s3 permutation are also observable in

the cross section in Figure 6b, showing that these permutations do not simply affect the tip zones of the weak domain but also its central portion.

[27] These preliminary experiments demonstrate that

pro-vided that sharp contrasts in rheology are present, stress permutation may develop even with isotropic materials. Two limitations, however, deserve consideration. First, it

was necessary to design a narrow weak zone (Figure 6b) to obtain significant amounts of permutations, which could not be reached with a relatively wide weak zone (Figure 6a). Second, many of the local stress permutations obtained in Figure 6b involve small local values of F, that is, little difference between s2 and s3 as compared with s1. This

observation indicates that with a far-field ratioF = 0.5 it is difficult to obtain drastic local differences between the principal values of permuted stresses.

4.2. Introducing Rock Anisotropy in 3-D Modeling [28] In the following experiments a central block is

considered as an intact isotropic block bounded by two Figure 6. Results of experiments with contrasting rheology with a weak zone (in gray) between

relatively strong domains. (bottom) Thick, horizontal cross sections of the model. Weak zone trends perpendicular to the extension axis. Poisson’s ratio is 0.25, and Young’s moduli is 60 GPa for the strong domain and 6 GPa for the weak domain. (a) Width of weak domain of one half of that of each strong domain. Thes1axes have relatively homogeneous, nearly vertical orientations. Distributions of the s2

ands3axes show an azimuthal dispersion about ±45 relative to the corresponding far-field axis in the

horizontal plane. (b) Width of weak domain of one fourth of that of each strong domain. Two distinctive clusters of local axes are observed, one being parallel to the far-field axis (N – Ss2or E – Ws3, depending

on the stereoplot) and the other perpendicular. Intermediate azimuths (near NE – SW and NW – SE) are rather few, indicating that the major phenomenon is not a simple azimuthal dispersion but a real permutation. Local occurrences of s2/s3permutation are also observable in the cross section, showing

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anisotropic blocks (Figure 4). In general, intact rocks are not strongly anisotropic as compared with faulted-fractured rocks. To illustrate the effect of anisotropy, we describe a simple experiment in Figure 7. Within the anisotropic domains the material is orthotropic with transverse isotropy, and the axis of maximum Young’s modulus trends parallel to both the s2 axis and the

elongated isotropic domain. The Poisson’s ratios parallel

and perpendicular to this axis are 0.25 and 0.15, respec-tively. The largest Young’s modulus is 60 GPa. For isotropic domain, Young’s modulus is 60 GPa. The Poisson’s ratio is 0.25. The thickness of central isotropic zone is 20 km. As Figure 7a shows, the boundary conditions are the same as for Figure 6.

[29] Let us denote Young’s modulus along the axis of

anisotropy by E and Young’s modulus in the direction perpendicular to this axis (which in this case coincides with thes2axis) by E0, with E > E0. To characterize the degree of

anisotropy and influence of the stress distribution in the model, we considered a variety of ratios E/E0. Three typical cases are illustrated in Figure 7 through presentation of a vertical cross section of the model in the anisotropic domain, parallel to the s2 axis (location in Figure 7a).

These three cases involve minor anisotropy (E/E0 = 1.25, Figure 7b), intermediate anisotropy (E/E0= 3.33, Figure 7c), and major anisotropy (E/E0= 10, Figure 7d).

[30] The cross sections show regular patterns of

projec-tion of principal axess1ands1in the situations of minor

and major anisotropy. However, these patterns markedly differ in that the greatest axis is vertical in the first case (Figure 7b), in agreement with the vertical s1 axis of the

far-field stress, whereas it is horizontal in the third case (Figure 7d). This contrast indicates that between these two stages a systematic stress permutation has occurred in the anisotropic domain of the model. The situation of inter-mediate anisotropy is characterized by widely scattered orientations of the principal axes projected in the cross section (Figure 7c), which denotes a transitory situation between the other two stages.

[31] To better illustrate this modeling experiment and the

influence of increasing anisotropy, it is worthwhile to examine the 3-D distribution of local stress axes for the entire analyzed block, which is done in Figure 8. In the stereoplots each individual plot reflects the result of calcu-lations and stress axis determination at each numerical element of the domain specified. It follows that some edges or corner effects, as those observable in the cross sections of Figure 7, are integrated in these stereoplots. In each case, we checked that these effects were not major and did not result in significant bias in the statistical evaluation of the stress axis distribution.

[32] First, with a limited anisotropy, no significant

perturbation of stress was noticed, and the results resem-bled those obtained in the isotropic case. Second, varia-tions in orientation are negligible, and permutavaria-tions are absent. Moreover, no significant difference occurs between the orientations of local principal stress axes in the anisotropic and isotropic domains (Figure 8a). Note that in Figure 8 the width of each anisotropic domain is twice that of the isotropic, stronger domain; as a consequence, the type of stress permutation evidenced in Figure 6b could not develop.

[33] The experiment shown in Figure 8b illustrates the

transition case, with significant rotations and permutations of local stress axes. In the isotropic portion of the model the stress axes suffer little perturbation and are generally parallel to the corresponding far field. However, a limited dispersion in azimuth, generally less than ±10, affects the axes s2 and s3. In contrast, in the anisotropic domains

the local stress axess1ands2show a ±90 dispersion in the

Figure 7. Distribution of principal stresses s1 and s2

along the anisotropic domain corresponding to the degree of anisotropy. (a) A model configuration similar to Figures 4b and 4d. Far-field boundary conditions ares1= 30 MPa,s2=

20 MPa, and s3= 10 MPa. The central block is an intact

isotropic block (in white) bounded by the two anisotropic blocks (in gray). Young’s modulus E is 60 GPa. E0 is variable in different models. Dashed rectangles indicate the location of the profile shown in Figures 7b, 7c, and 7d. (b) Model of E/E0= 1.25. No significant stress permutations occurred. (c) Model of E/E0 = 3.33. Limited stress permutations occurred in the anisotropic blocks. (d) Model of E/E0= 10. Significant stress permutations occurred in the anisotropic blocks.

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plane perpendicular to thes3axis, which remains stable. In

other words, a s1/s2 permutation phenomenon certainly

occurs, but intermediate attitudes of axes are numerous, which result in typical girdle distributions except for s3

(Figure 8b). This distribution is consistent with the 2-D distribution observed in the N – S trending vertical cross section of Figure 7c.

[34] The most interesting situation, in terms of stress

permutations, is that obtained with a strong anisotropy (Figure 8c). The distribution of the local stress axes in the

isotropic domain resembles the previous one, with orienta-tions that basically reflect the far-field stress. The distribu-tion of the local stress axes in the anisotropic domain is completely different, except for thes3axis. It is

character-ized by a complete permutation between thes1ands2axes

with respect to the isotropic domain. Because intermediate orientations of axes are absent, this phenomenon cannot be explained in terms of progressive rotations of axes, which would have been possible with girdle distributions as in Figure 8b.

Figure 8. Stereoplots of stress permutations corresponding to the degree of anisotropy. Boundary conditions, rheology, and model configuration are the same as in Figure 7. (a) Model of E/E0= 1.25. No stress permutations observed in the anisotropic blocks. (b) Model of E/E0 = 3.33. Limited s1/s2

permutation phenomenon occurs in the anisotropic blocks. Intermediate attitudes of axes are numerous, resulting in typical girdle distributions. (c) Model of E/E0= 10. Significant stress permutations occurred in the anisotropic blocks.

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[35] Amadei et al. [1987] analyzed 98 measurements of

elastic properties of different intact rock masses. They found that the ratio E/E0 varies between 1 and 4 for most intact transversely isotropic rocks. On the basis of the results of 200 sets of analysis of different rock types, Worotnicki [1993] obtained similar results. For those with the highest degree of anisotropy such as pelitic clay and pelitic mica rocks, Worotnicki [1993] also pointed out that the ratio E/E0 was found to be lower than 6 and in most cases below 4. However, the presence of fractures and faults markedly affects the anisotropy of the whole rock mass, so that large E/E0ratios, as in Figure 8c, are not uncommon in tectonized areas. We acknowledge that there is no assurance that densely distributed microcracks can be represented by the overall effective elastic constants of a medium having anisotropic properties. However, Hudson [1980, 1981] theoretically showed that at least a limited concentration of oriented microcracks might be represented by such an anisotropy. Consequently, we made the assumption that in each considered case, there is an anisotropic medium that reproduces the mechanical elastic response of the rock unit affected by fractures and faults.

[36] We conclude first that the stress changes induced by

variations in rheology are large enough to modify the local tectonic behavior and produce permutations of principal stress axes, despite the simplicity of far-field boundary conditions (Figures 6a, 8b, and 8c). As a second conclusion, the importance and clarity of the stress permutation

phe-nomenon increase with the degree of anisotropy (compare Figures 8a, 8b, and 8c).

5. Different Model Configurations

5.1. Extension and Compression With Grain-Parallel Anisotropy

[37] We have carried out 3-D numerical modeling for a

detailed investigation of the stress permutation phenome-non, with a variety of different situations as suggested in Figure 5. The influence of rheological contrasts has been evidenced in the experiment in Figure 8. To explore the influence of the regional tectonic regime, we now consider different far-field regimes, which are extensional (Figures 5a and 5d), strike slip (Figures 5b and 5e), and reverse (Figures 5c and 5f) in type. In the boundary conditions of these experiments the orientations of the principal stress axes differ (the vertical axis beings1,s2, and s3for these

three types, respectively), but the principal stress values remained the same (s1= 30 MPa,s2= 20 MPa, and s3=

10 MPa.). We also adopt different geometrical relationships between the isotropic, central portion of the analyzed block and the principal axes of the far-field stress: the interfaces between isotropic and anisotropic portions are perpendicular to the s3 axis in Figures 5a and 5b, to the s2 axis in Figures 5d and 5f, and to thes1axis in Figures 5c and 5e.

Some typical situations have been extracted from this variety of models and are discussed below.

[38] It is worth noting again that some of these models

refer to geological situations with elongated rheological domains that trend parallel to the structural grain responsi-ble for anisotropy (Figure 4b), as occurs for instance in a typical horst-and-graben pattern (Figure 4d). Some other models refer to relatively undeformed ‘‘bridges’’ that cut across the graben structure (Figure 4c), the less deformed blocks on both edges of the graben (Figure 4a) being represented in this case by the large block in which the analyzed block is included (Figure 3). These two types of situations are common in structural geology.

[39] Figure 9 illustrates the case of a simple

horst-and-graben system submitted to extensional stress, with a vertical s1 axis and a s3 axis that trend perpendicular to

the graben axis (Figure 9a). The distribution of isotropic and anisotropic domains is the same as for Figures 4b – 4d. This type of stress-structure relationship is very common (e.g., the Gulf of Suez), although in many cases the extension trends oblique to graben axes (e.g., the Rhinegraben). Not surprisingly, the results resemble those of Figure 8c, which involved the same geometrical configuration. Because the ratio E/E0was significantly less in Figure 9 than in Figure 8c (5 instead of 10), this analogy indicates that with the geometry adopted the permutation phenomenon is rather stable for values of E/E0 higher than 5. This brings confir-mation that the result obtained in Figure 8b, with girdle distributions of s1 ands2 axes in the anisotropic domain,

effectively characterized the transition from situations without permutation (Figure 8a) to situations with system-atic s1/s2 permutation in the anisotropic domain. In

Figure 9c the only evidence that the unstable situation of Figure 8b (E/E0 = 3.33) gets closer as the ratio E/E0 decreases from 10 to 5 is brought by the slight dispersion ofs1ands2axes in the vertical plane perpendicular tos3.

Figure 9. Model results of stress permutations of the horst-and-graben system. (a) Same boundary conditions and model configuration as Figure 7. Ratio E/E0 = 5. (b) Stereoplots of isotropic domain. Three principal stresses retain the same direction as applied to far-field boundary conditions. (c) Stereoplots of two anisotropic blocks. Significant stress permutations ofs1/s2occurred.

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[40] One should not infer, however, that permutations

occur for similar E/E0 ratios regardless of far-field stress orientation (relative to the rheological interfaces). The relative complexity of the permutation phenomenon is illustrated by the model in Figure 10, in which all param-eters are the same as in Figure 9, except for the switch between thes1ands3axes of the far-field stress (that is, the

interfaces are perpendicular tos1instead ofs3). This model

describes geological situations with parallel-trending ranges affected by reverse faults and subjected to compression with a verticals3axis and as1axis that trends perpendicular to

the structural grain (Figure 9a). Comparison between the distributions of local orientations of stress axes in Figures 9c and 10c reveals a typical s1/s2permutation in both these

cases, the invariant axis being s3(horizontal in Figure 9c

and vertical in Figure 10c). In contrast, the isotropic domain reveals markedly different situations. In the extensional case (Figure 9b) the distribution of local stress axes conforms to that of the far-field stress, with a minor additional disper-sion. In the compressional case (Figure 10b) this distribu-tion reveals partial permutadistribu-tion. The s1 axes concentrate

around E – W (according to the far-field stress) and N – S (indicating s1/s2 permutation), with many intermediate

trends. Most s3 axes are vertical (according to far-field

stress), but large scatter exists, forming a girdle in the E – W trending vertical plane. As a result, the distribution of thes2

axes is complex, with N – S, E – W, and vertical clusters and intermediate positions within two vertical planes, N – S and E – W (Figure 10b). This comparison between Figures 9 and 10 shows first that multiple stress permutations may occur even with extremely simple structural patterns and second that in a given situation the occurrence of permuta-tions cannot be inferred in a simple way from consideration of another situation.

5.2. Strike-Slip Regime WithS2-Parallel Anisotropy

[41] In the preceding experiments the axis of the

anisot-ropy chosen was horizontal and parallel to the structural grain of the model. Despite the simplifying assumption of transverse anisotropy mentioned in section 4.2, this fulfills a reasonable requirement in light of the geological observa-tion: the distribution of faults and fractures maintains a larger average Young’s modulus along the graben trend than perpendicular to it. For this reason, the axis of anisotropy and thes2axis are parallel in these experiments. Adapting

this geometry to the case of the strike-slip regime (horizontal s1 and s3 axes), a vertical axis of anisotropy may be

expected, partly in agreement with the vertical intersection of the strike-slip fault and related fractures with a variety of trends. The models of Figure 11 were built according to this anisotropy scheme. As before, two types of structural con-figurations were considered, with the isotropic/anisotropic interfaces perpendicular tos3(Figures 11a, 11b, and 11c) or

s1(Figures 11d, 11e, and 11f ).

[42] The results obtained with the same stress values and

Young’s moduli as before do not reveal any significant stress permutation in the anisotropic domains of the models, despite dispersion in the s1/s2 plane in the second case

(Figure 11e, compare with Figure 11b). In the isotropic domains the first model reveals local stress orientations that closely resemble those of the far-field stress, whereas in the second model the s1 and s2 axes are distributed

along a vertical N – S trending girdle (although most axes conform with the orientations of the far-field s1 and s2).

These experiments show that under comparable conditions regarding stress values and rheology, the strike-slip-type configuration is more ‘‘permutation resistant’’ than the normal-type and reverse-type configurations illustrated in Figures 9 and 10. To check the influence of the anisotropy ratio on the permutation occurrence, we performed experi-ments with increasing values of E/E0 (not shown in Figure 11). For instance, with the configuration of Figure 11a, adopting a value E/E0= 10 (instead of 5) simply resulted in a slight increase of the dispersion ofs1ands2

axes, which was not sufficient to induces1/s2permutation

in the anisotropic domain, and induced very little change in the isotropic domain. With the configuration of Figure 11d the same change in the anisotropy ratio resulted in large dispersion of thes1ands2axes in the planes1/s2inside the

anisotropic domain, but again the permutations stage was not reached. However, in the isotropic domain, increasing the ratio E/E0to 10 resulted in a large dispersion of local axes, and thes1/s2permutation was locally obtained, with a large

number of nearly verticals1axes. These results reflect the

general effect already observed with the normal-type and Figure 10. Model results of compression with

grain-parallel anisotropy. (a) Same boundary conditions and model configuration as Figure 7. Ratio E/E0 = 5. (b) Stereoplots of the isotropic domain with partial permutations ofs1/s2. Thes1axes concentrate around E –

W (according to the far-field stress) and N – S, with many intermediate trends. Mosts3axes are vertical (according to

far-field stress). There is a girdle in the E – W trending vertical plane. (c) Stereoplots of two anisotropic blocks. Significant stress permutations ofs1/s2occurred.

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Figure 11. Model results of strike-slip regimes with s2-parallel anisotropy. E is 60 GPa. E/E0= 5 for

models in Figures 11b and 11e, and E/E0= 10 for models in Figures 11c and 11f. (a) Structural configuration built with the isotropic/anisotropic interface strike perpendicular tos1. Far-field boundary conditions are

s1 = 30 MPa (E – W), s2 = 20 MPa (vertical), and s3 = 10 MPa (N – S). (b) Model with the same

configuration as Figure 11a with E/E0= 5. There are no stress permutations in both the anisotropic and isotropic domains. There is some dispersion in thes1/s2axes. (c) Model with configuration as Figure 11a

with E/E0= 10. There is a slight increase of the dispersion ofs1ands2axes, but it did not induce significant

s1/s2 permutation in the anisotropic domain and induced very little change in the isotropic domain.

(d) Structural configuration built with the isotropic/anisotropic interface strike perpendicular tos3. Far-field

boundary conditions ares1= 30 MPa along N – S direction,s2= 20 MPa along vertical direction, ands3=

10 MPa along E – W direction. (e) Model with configuration as Figure 11d with E/E0= 5. Somes1ands2

axes are distributed along a vertical, N – S trending girdle in the isotropic domain. In the anisotropic domains the local stress orientations resemble those of the far-field stress. (f) Model with the same configuration as Figure 11d with E/E0= 10. Dispersion of thes1ands2axes is in the planes1/s2inside the anisotropic

domains. In the isotropic domain, there was large dispersion of local axes, and thes1/s2permutation was

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reverse-type configuration; the permutations of stress axes are more common with high anisotropy ratios than low ones. [43] In our modeling, we also explored the effect of

variations in principal stress values. Increasing the F ratio simply favors s1/s2 permutation (because the difference

betweens1ands2decreases with respect to their differences

withs3); conversely, lowF ratios favor s2/s3permutations.

Such easily predictable effects are unwarranted for discus-sion. We thus concentrate, with the same configurations as in Figures 11a and 11d, on the effect of decreasing principal stress differences. We replaced the 5 MPa differencess1

s2ands2 s3by 10 MPa differences. Thus, in Figures 11c

and 11f the far-field principal stresses s1,s2, ands3have

values of 25, 20, and 15 MPa instead of 30, 20, and 10 (respectively). This change does not affect the ratio F but induces large variations in the distribution of local stress axes. Not only does dispersion increase in both the isotropic and anisotropic domains (Figures 11c and 11f), but also complicated local permutations of stress axes occur in the isotropic domain in the case of Figure 11f; these permuta-tions belong to both thes1/s2ands2/s3types. This shows,

as might also be expected, that decreasing differences between principal stresses results in increasing probability for stress permutation to occur. However, there is a signif-icant difference with respect to other experiments (such as in Figure 9c); this particular effect induces permutations that remain local and generally do not invade the entire domain. The same conclusion could be drawn from varia-tions inF ratios.

5.3. Strike-Slip Regime WithS3-Parallel Anisotropy

[44] Geological observation often shows that in regions

that experienced strike-slip tectonics a horizontal structural grain developed. This contradicts the reasoning that was presented to support the anisotropy pattern of Figures 11a and 11d, which has a larger Young’s modulus along the vertical s2 axis. The geological reasons explaining this

difference are numerous, such as that because of the horizontal layering that commonly prevails in the rock masses and diminishes the vertical resistance, the inherited patterns of normal faults or reverse faults, or the pervasive fissuring in the vertical plane parallel to the compression axis. In Figure 12 we simply present a model similar to that of Figure 11b but with a ratio E/E0= 0.2. This ratio lower than 1 means that the Young’s modulus is larger in the transverse plane (perpendicular to s3) than that along the

anisotropy axis (parallel tos3). This rheology mimics that

of a rock mass with pervasives3-perpendicular fissuring, a

common situation where tension fractures are numerous along with strike-slip faults.

[45] The model in Figure 12 shows stress permutations,

s1/s2in type, occurring in a systematic way in the

aniso-tropic domain. The results obtained in the isoaniso-tropic domain resemble the previous permutations (i.e., nonsystematic s1/s2permutations), although dispersion is larger and local

permutations are more numerous. The comparison between the results obtained in the anisotropic domains in Figures 11e and 12b illustrates the major role of the anisotropy pattern: with the same contrast in Young’s moduli the stress permu-tation is absent with the largest Young’s moduli along the vertical, s2-parallel axis, and systematically occurs

with the smallest Young’s moduli along the horizontal, s3-parallel axis (E/E0= 5 and E/E0= 0.2, with the definitions

adopted before regarding E and E0). 5.4. Across-Strike Bridge Cases

[46] In this section, we consider simple cases of

normal-type and reverse-normal-type stress regimes that affect a structural pattern characterized by a grain-parallel anisotropy axis, with a strike-perpendicular isotropic zone. This configuration reproduces common geological situations, with discontinu-ous systems of horsts and grabens subjected to normal-type stress (Figure 13a), or discontinuous systems of ranges and grabens subjected to reverse-type stress (Figure 14a). In both these cases the resistant, isotropic domain represents the relatively undeformed rock mass that separates the discontinuous graben segments. The boundary principal stresses are 30, 20, and 10 MPa, and the ratio E/E0equals 5. [47] The first experiment (Figure 13, compression)

reveals systematic s1/s2 permutation in the anisotropic

domain, whereas no permutation occurs in the isotropic domain where the local stress axes simply reproduce the far-field stress, with minor additional dispersion. This result Figure 12. Model results of strike-slip regimes with

s3-parallel anisotropy and with pervasives3-perpendicular

fissuring. (a) Same far-field boundary conditions as Figure 11d. Young’s modulus is 60 GPa in the isotropic domain (in gray). For anisotropic domains, Young’s modulus is larger in the transverse plane (E0 = 60 GPa, perpendicular to s3) than along the anisotropy axis (E =

12 GPa, parallel to s3). (b) Significant stress permutations

of s1/s2 in the anisotropic domains. For the isotropic

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shows that the stress regime inside the across-strike bridge of may drastically differ with respect to the stress regime that prevails inside the grabens (the anisotropic domain). There is no need to present the local state of stress in the graben-parallel isotropic domains (the ‘‘horsts’’) because Figure 10 already illustrated the stress permutations that may take place there (in Figure 13, the horsts that bound the grabens simply correspond to the isotropic envelope of the analyzed block, as shown in Figure 3).

[48] The case for the normal type is briefly illustrated in

Figure 14. The configuration is the same as in Figure 13, except for the stress regime, which is now compressional with a s1 axis perpendicular to the structural grain. As

before, the distribution of local stress axes in the isotropic domain (the across-strike bridge) reproduces the far-field stress, the additional dispersion remaining minor. In con-trast, systematic permutation occurs between the stress axes, s1ands2, in the anisotropic domains (the ‘‘grabens’’). In this respect, the stress permutation phenomena illustrated in Figures 13 and 14 are quite similar. It is interesting to note, in both these cases, that in each domain the orientations of stress axes show very little dispersion (not surprisingly, the angular dispersion is larger in the anisotropic medium but

remains small). Concerning the graben-parallel isotropic domains (the ‘‘ranges’’), the type of local stress that may prevail there has already been illustrated in Figure 9. Remember, however, that the comparison between the models of Figures 9 and 10 revealed large differences in stress distributions, as a function of the type of far-field stress.

[49] The local distribution of stress axes in the model in

Figure 14 is illustrated in two cross sections. The first cross section is horizontal (Figure 14c); in agreement with the stereoplots (Figure 14b), it shows high levels of homoge-neity in each domain. This cross section further indicates that the components of the principal stress values in the horizontal plane vary very little. The second cross section is vertical, perpendicular to thes3 axis, and thus shows that

the above conclusion remains valid in three dimensions.

6. Discussion and Conclusion

[50] Even with the simple configurations adopted

throughout the paper, the numerical modeling experiments successfully reproduced the stress permutation phenome-non, as observed in nature. This phenomephenome-non, however, involves a wide range of possible cases. The variety in stress permutation types especially depends on the orienta-tion of the far-field stress (e.g., normal, reverse, and strike slip in type), the principal stress values (e.g., the ratioF and the principal stress differences), the relative size of the different domains (e.g., Figure 6), and the type and degree of anisotropy (e.g., the ratio E/E0).

[51] The simple models shown above as examples

(Figures 7 – 14) illustrate a few typical cases from which other responses can be derived. As a major target, we considered the role of the anisotropy in Young’s modulus. We thus aimed at reproducing in a simple way the aniso-tropic distribution of mechanical properties in rock masses, as induced by layering, fissuring, and faulting. It is well known that major sources of stress variations are related to the presence of mechanical discontinuities in the crust and were not explicitly considered in this paper because our aim was to demonstrate that even in very simple cases, with a few homogeneous domains, stress permutations occur very often. One may easily infer that stress permutations are more frequent where mechanical discontinuities are present. A recent paper illustrated the importance of such phenom-ena in the case of an oceanic transform fault in Iceland, showing that a large variety of stress states can be attributed to variations in coupling along a major discontinuity [Angelier et al., 2000]. Such observations can be multiplied in various tectonic settings and deserve careful consider-ation in geodynamic reconstructions. To this respect, it is worth noting that stress deviations and stress permutations were often interpreted in terms of dependent tectonic events, although they may simply reflect variations during a single event.

[52] As mentioned above, most intact rocks show less

anisotropy than heavily fractured and faulted rock masses. The intact blocks in our models were considered to be isotropic and homogeneous. This is, however, an approximation. Several authors [e.g., Amadei et al., 1987; Worotnicki, 1993; Telesnick et al., 1995] have indicated that the ratio E/E0 approximately varies between 1 and 4 for Figure 13. Reverse type of across-strike bridge cases.

(a) (left) Geological configuration. Grain-parallel anisotropy zones with discontinuous systems of ranges and grabens with a strike-perpendicular isotropic zone submitted to reverse-type stress. Principal stress boundaries are 30 (E – W), 20 (N – S), and 10 MPa (vertical). For simplicity, only far-field maximum boundary principal stress is shown (couple convergent open arrows). (right) Model analogue. Isotropic domain, in gray, has a Young’s modulus of 60 GPa along all directions. Anisotropic domains have E/E0 = 5. (b) Stereoplots of results. Systematic s1/s2 permutation

occurs in the anisotropic domains, whereas no permutation occurs in the isotropic domain.

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most intact transversely isotropic rocks. For example, most of the quartzofeldspathic and basic/lithic rocks show the low to moderate degree of anisotropy with a maximum to minimum Young’s modulus ratio Emax/Eminto be less than

1.5; carbonate rocks show an intermediate degree of rock anisotropy with Emax/Emin not exceeding 1.7 [Worotnicki,

1993]. As for the highest degree of anisotropy such as pelitic clay and pelitic mica rocks, this ratio was found for most cases to be lower than 4. If the anisotropic properties really exist inside the intact rock due to foliation and stratification, the partial stress permutations or additional stress perturba-tion should be expected. On the basis of the results of tests of the degree of anisotropy (Figure 8) a ratio E/E0 of 3.33 seems to be a critical value for the delimitation of significant stress permutations inside the anisotropic blocks, as a result

of the preexisting fracturing in the deformation zone. In our models we assumed the intact rock mass bounded by two anisotropic blocks (Figure 4) to be isotropic, which was a simplification. Because the Emax/Emin ratio was found to

vary between 1 and 4 for the above mentioned rocks, more frequent and complex permutation patterns can be expected in reality, as compared with our modeling. This comment is important because it suggests that stress permutations may occur as a consequence of rock anisotropy even in the absence of dense fracturing or faulting.

[53] From a mechanical point of view, one should expect

rock anisotropy to be stress-dependent with decrease in anisotropy associated with increase of stress [Homand et al., 1993]. This phenomenon was observed and analyzed by ultrasonic tests on slates during triaxial compression tests Figure 14. Normal type of across-strike bridge cases. (a) Geological configuration. Grain-parallel

anisotropy zones with discontinuous systems of ranges and grabens with a strike-perpendicular isotropic zone submitted to normal-type stress. Principal stress boundaries are 30 (vertical), 20 (N – S), and 10 MPa (E – W). Isotropic domain has a Young’s modulus of 60 GPa. Anisotropic domains have E/E0= 5. Dashed lines indicate the locations of horizontal cross section of Figure 14c and vertical cross section of Figure 14d. (b) Stereoplots of results. Significant s1/s2permutation occurs in the anisotropic domains,

and no permutation occurs in the isotropic domain. (c) Distribution of stress axes by horizontal cross section in the model of Figure 14a. In the isotropic domain,s2(N – S) ands3(E – W) axes are shown. In

the anisotropic domains,s1(N – S) ands3(E – W) axes are shown. (d) Distribution of stress axes in vertical

cross section of the model. In isotropic domain, s1(vertical) and s2(N – S) axes are shown. In the

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Figure 2. Stress permutations between stress axes and rift opening (modified after Angelier and Bergerat [1983] and Angelier [1994])
Figure 4. Structure of the analyzed block involving two types of alternating elongated zones corresponding to relatively weak and strong rheologies
Table 1. Summary of Boundary Conditions and Rheological Properties for Numerical Experiments a
Figure 8. Stereoplots of stress permutations corresponding to the degree of anisotropy
+2

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