Mechanics of Cosserat Generalized Continuum and Modelling in Structural Geology Mecânica do Contínuo Generalizado de Cosserat e Modelagem em Geologia Estrutural

The conception and interpretation of models in structural geology, specially the numerical ones, are eminently based on the principles of classic continuum mechanics, where the intrinsic formulation has high degree of symmetry in its very essence. However, the geological structures carry in all scales a remarkable level of asymmetry. On the other hand, the mechanics of Cosserat generalized continuum, by incorporating characteristic lengths of the matter structure in its constitutive and strength laws, leads by itself to the asymmetry and the heterogeneity of the kinematic and dynamic fields responsible for rock fabric. Based on the intrinsic formulation of the Cosserat continuum, the present work emphasizes conceptual, analytical and numerical models that suggest the expansion in the interpretation of the genesis and development of geological structures, in particular those found in fault zones.


Introduction
Models in structural geology, being conceptual or coming from rock testing laboratory, from analogue experiments or, even more, from analytical and numerical methods, use the concepts of continuum mechanics in their design and particularly along the interpretation of their results. In particular, the recognition of the role of numerical modelling in structural geology has increased tremendously in recent decades mainly because of the possibility to test different concepts and physical conditions related to the development of geological structures in rocks. However, most of these models are based on the classic continuum mechanics.
In the classic continuum mechanics the stress and strain fields and the material physical properties are collapsed to points having three degrees of freedom for displacements in 3D. On the other hand, the mechanics of Cosserat generalized continuum, pioneered by the Cosserat brothers (Cosserat & Cosserat, 1909), incorporates finite lengths in microscale and, in addition to the three degrees of freedom for displacements, it allows three independent degrees of freedom for rotations. As this length with such properties is directly inserted in the mechanical formulation of a Cosserat continuum, asymmetries are generated for the stress and strain tensor fields, opening possibilities to improve the description of the mechanical behaviour of the materials, specially rocks in their different geological environments.
The present work strives to strengthen the potential of the application of the mechanics of Cosserat generalized continuum in structural geology. Through the presentation of its basic formulation, it will be highlighted how simple models based on the mechanics of Cosserat generalized continuum may suggest the genesis and development of some geological structures, specifically those related to fault zones.

Background on the Mechanics of Cosserat Generalized Continuum
This section presents the basic formulation of the mechanics of Cosserat generalized continuum, regarding the main points addressed in this work. Further details on the mechanics of the generalized continuum media and their variants can be found in the classic work of Germain (1973) and also in Maugin (2017). Specifically, a comprehensive formulation on the mechanics of Cosserat generalized continuum is showed in Figueiredo et al. (2004).
Fundamentally, the mechanics of Cosserat generalized continuum, also called micropolar continuum mechanics, is a particularization of the mechanics of generalized continuum. In a Cosser-

Mechanics of Cosserat Generalized Continuum and Modelling in Structural Geology
Anderson Moraes; Rodrigo P. de Figueiredo & Eurípedes do A. Vargas Jr at continuum, it is conceived the existence of rigid particles with finite lengths in microscale that suffers displacements and rigid rotations but do not deform (i.e., they do not change shape or volume), and these rotations are independent of the macroscale behaviour. In kinematic terms, being u' i the microscale displacements (relative to the local coordinate system x' i ), u i the macroscale displacements (relative to the global coordinate system x i ) and w c ij the Cosserat micro-rotation tensor, we can write: Cosserat micro-rotation vector (dual-vector of w c ij ) and k i as the macroscopic gradient of the micro-rotations vector (dual-vector of k ijk ), from Equation 1 to Equation 4, we can write in 2D: (1) The indexes i and j range from 1 to 3 in 3D (1 to 2 in 2D). The first index refers to the unit normal vector acting on the face of the elementary unit of the continuum and the second index refers to the direction that que component acts. The relative displacement gradient tensor g ij is defined as: (2) Therefore, w c ij = (∂ j u' i -∂ i u' j )/2, where ∂ i represents the gradients in the i directions. One can divide g ij into its symmetric part, symbolized by indexes between ( ), and anti-symmetric part, symbolized by indexes between [ ], in the form: (3) where ɛ ij is the strain tensor in macroscale and w ij is the macroscale rotation tensor. Equation 3 is of fundamental importance for the mechanical implications of a Cosserat continuum. It is seen that the symmetric part of g ij coincides with the macrostrains, since there are no microscale strain, and the anti-symmetric part of g ij contains the rigid rotation difference between the macroscale and the microscale. As a result, it is seen that g ij is an asymmetric tensor. Additionally, the macroscopic gradient of the micro-rotations tensor k ijk (unit of inverse of length) is defined as: representing the curvatures along the continuous medium. Note that k ijk is the gradient of w c ij , and thus it is also an anti-symmetric tensor. Defining w c i as the (5) In terms of stresses, as shown in Figueiredo et al. (2004), there are stress fields energetically related to the kinematic fields presented above. In this sense, the relative stress tensor τ ij is directly related to the anti-symmetric tensor w c ij , so that it is also an anti-symmetric tensor. Since s ij is the stress tensor of the classic continuum, the Cosserat stress tensor s c ij is defined as: As a consequence, being the sum of a symmetric tensor with an anti-symmetric tensor, s c ij is an asymmetric tensor. As a result, as shown in Figure 1 in 2D , the equilibrium conditions required for the Cosserat continuum arise from the emergence of moment (or couple) stresses m i (unit of stress times length) applied to the arm dx' j orthogonal to m i in the elementary unit of the continuum. Thus, the moment stresses vectors m 1 and m 2 induce local momentum forces that balance the forces relative to the shear stresses s c 12 and s c 21 of the Cosserat stress tensor. In this way, the equilibrium equations for a Cosserat continuum in 2D are: (7) where ρ is the density, X i are volumetric forces, Θ is the volumetric moment and s a is the anti-symmetric shear stress given by:

Mechanics of Cosserat Generalized Continuum and Modelling in Structural Geology
Anderson Moraes; Rodrigo P. de Figueiredo & Eurípedes do A. Vargas Jr representing at some extent a measure of the degree of asymmetry of s c ij . In fact, by inspection of the last expression in Equation 7, it is clear that with the emergence of non-zero moment stresses gradients the shear components s c ij and s c ji of the Cosserat stress tensor are no longer necessarily equal, resulting, in general, in an asymmetric Cosserat stress tensor. physical meaning. As suggested by Mindlin (1963) it is written: Figure 1 Representation in 2D of the cause of Cosserat stress tensor asymmetry. Moment stress vectors m 1 and m 2 induce local moment forces that equilibrate forces related to the different shear stresses s c 12 and s c 21 of the Cosserat stress tensor.
The kinematic and dynamic characteristics of the mechanics of Cosserat generalized continuum presented above make it possible to delineate constitutive relationships and flow criteria that are particularly important in structural geology. An isotropic elastic Cosserat continuum, as presented by Figueiredo et al. (2004) by slightly modifying the formulation of Teodorescu (1975), can be expressed by: (9) where δ ij is the Kronecker delta (values of 1 for i = j and 0 for i ≠ j), λ and G are parameters analogous to the Lamé's parameters, G r is the rotational modulus and B is the flexural modulus. The rotational modulus controls how much the microstructure influences the distribution of the macroscale stresses, being the parameter that couples the expressions of Equation 9. The relationship between the shear modulus G and the flexural modulus B has an interesting (10) where l is the called characteristic length. The characteristic length may be regarded as the length of the microstructure of the medium. In this sense, the presence of characteristic lengths, a single one when is considered an isotropic medium, is the seminal difference between the Cosserat and the classic continuum mechanics. Equation 10 shows that for l approaching zero, and therefore B also approaching zero, from the last expression of Equation 9, it can be seen that the moment stresses tend to vanish.
There are many proposals for working with elastoplasticity in a Cosserat continuum (e.g., Lippmann, 1969;Besdo, 1985;Bogardanova-Bontcheva & Lippmann, 1975;Vardoulakis & Sulem, 1995). The one implemented in the numerical models presented below is a modification of Bogdanova-Bontcheva & Lippmann (1975) proposed by Figueiredo (1999). Specifically, flow criteria f and plastic potential functions g are formulated for the instauration of mechanical instability for shear, for the moment stresses and for the rotation of the microstructure with a characteristic length l. Thus, analogously to the Mohr-Coulomb criterion, one can write: (11) where J 2 is the second invariant of the symmetric part of the Cosserat deviatoric stress tensor, I 1 e is the first invariant of the Cosserat effective stress tensor, s 0 is the cohesion, φ is the internal friction angle and ψ is the angle of dilatancy. The flow criterion involving the moment stresses is given by: (12) where J 2 m is the second invariant of the symmetric part of the deviatoric moment stresses tensor. In its turn, the flow criterion for the rotation of the microstructure, originally proposed by Figueiredo (1999), is written as:

Mechanics of Cosserat Generalized Continuum and Modelling in Structural Geology
Anderson Moraes; Rodrigo P. de Figueiredo & Eurípedes do A. Vargas Jr where s N ec is the Cosserat effective normal stress acting in the plane of rupture and φ r is the rotational friction angle, to be defined below.

Modelling Geologic Materials as Cosserat Generalized Continuum Media
The mechanics of Cosserat generalized continuum allows broadening the interpretation and implications of structural models for rocks. In fact, its core characteristics as formulation with characteristic lengths of the medium, the possibility of microstructure rotations as an independent field from macroscale and the establishment of tensor asymmetry can cast another light in structural geology modelling. In this sense, it will be suggested some simple but illustrative examples based upon conceptual, analytical and numerical arguments.
As shown in Figueiredo et al. (2004), the properties of Mohr diagrams in 2D classic continuum can be extended to a Cosserat continuum. In their parametric form, the equations for the Cosserat normal s c N and Cosserat shear s c C stresses in a plane can be written as a function of the angle θ between the maximum principal stress and the plane as: pointed by De Paor (1994), one can suggest that the asymmetric structuring patterns in shear zones may perhaps be better compatibilized by using the mechanics of Cosserat generalized continuum. In fact, as synthesized in Simpson & De Paor (1993), shear zones bear strain non-coaxial fields, vorticity complexes fields and non-orthogonal strain rates eigenvectors for defining the apophyses, elements not yet satisfactorily explained by the classic continuum mechanics. In this sense, it could be possible to extend expressions relating stresses and strains in shear zones in a more cause-consequence fashion.
(13) (14) Note that the anti-symmetric shear stress, given by Equation 8, appears in the expression for the Cosserat shear stress. As such, Equation 14 implies, as shown in Figure 2, that the shear stresses of the classic continuum are "displaced" in the s c N x s c C space along the shear stress axis by a amount s a . An immediate consequence is that the principal stresses (i.e. those acting on planes where s c C vanishes) should not be orthogonal as in the classic continuum and, depending on the value of s a , could even no longer exist. Taking these stress and strain equivalent frameworks, as at some extent already  Figure 2 suggests how the asymmetry of the Cosserat stress tensor may favour the emergence of fault zones with a single orientation and not as conjugated structures. In this case, there would be a combination of rotation of the microstructure and sliding as the two envelopes are touched by the Cosserat stress state. Indeed, in nature fault zones appear less frequently as conjugate and symmetric structures. An inspection of several seismic sections and analogue physical models, for example as shown in Jackson & Hudec (2017), suggests that the most expressive faults have rough parallel geometry and hardly appear as sistematic conjugated faults.

Mechanics of Cosserat Generalized Continuum and Modelling in Structural Geology
Anderson Moraes; Rodrigo P. de Figueiredo & Eurípedes do A. Vargas Jr The spatial arrangement of the microstructures in rocks, coming from either the geometry of single mineral grains or of their associations due to the stress loading conditions forming chains of mineral grains, may determine the rock strength. Similar to what was performed by Nascimento & Teixeira (1971), Figure 3 represents an allegory illustrating the impact of the structure geometry on the study of the mechanical stability in any material. Ultimately it is the classic slope plane-block problem in which the conservation of linear and angular moments for the block is analyzed by regarding the weight, normal and friction forces. Considering a the height and b the base of a block with mass m and g the acceleration due to gravity, for the case in Figure 3a it is known that in the limit for block sliding (slope α): inner microstructures in rocks. Note that the base and height ratios for the blocks are inverse in Figs. 4a and 4b. Thus, though the shear strength criteria is the same for both the rotation criteria are different.  Figure 4a and clockwise rotation of the blocks and later shearing in the contacts among the blocks in Figure 4b. It is very suggestive that the rotation patterns and relative shearing of the microstructures in the rock could explain, for example, the sinthetic Y, R and P and the antithetic R' and X structures of the classic Riedel-Tchalenko experiment, favoring one to other in the progress of deformation. Regarding the specific conditions in Figure 4, we could postulate the onset of Y structures in Figure 4a and, due to the clockwise rotation of the blocks, of R' or even of X structures (depending on the amount of rotation) and of Y structures in Figure  4b. Of course, a paramount of scenarios may arise depending on the stresses conditions and the spatial arrangement of the microstructure in rocks. (15) where m e is the static friction coefficient and φ e is the static friction angle. In this case, the base of the block has a length greater than its height, so that it can slide without rolling when the plane reaches the angle α corresponding to the sliding condition. The resulting torque is zero as the torque due to the weight force is balanced by the torque due to the contact forces (normal and friction forces). On the other hand, in Figure 3b the base of the block has a length smaller than its height. Assuming that φ e is high enough to prevent sliping, the torque due to the weight calculated relatively to the lower right corner of the block for the toppling limit condition (slope β) is given by: (15) where φ r is the rotational friction angle. In words, the occurrence or not of block rotation is dramatically dependent on the relationship between its geometry and the load distribution. When b > a we have β > α and φ r > φ e , what would cause the block to slip. Otherwise, when b < a we have β < α and φ r < φ e , what could lead the block to rotate.
Although this allegory is quite naive at a first glance, its mechanical context can be heuristically extend to what occurs to rocks. Figure 4 shows the loading condition in a set of blocks representing the Figure 3 Impact of block geometry in the analysis of its stability in the classic slope plane-block system problem. Different aspect ratios between block base b and height a determine if the block slides or topples. Considering Equation 16, in (a) as b > a and φ r > φ the block could slide without toppling; in (b) as b < a and φ r < φ block could topples before sliding.

Mechanics of Cosserat Generalized Continuum and Modelling in Structural Geology
Anderson Moraes; Rodrigo P. de Figueiredo & Eurípedes do A. Vargas Jr From what was pointed out, it is suggestive that the relation between the spatial arrangement of the rock structure and the normal and shear stresses that it can support depends not only on the analysis of the slip conditions but also on the analysis of the microstructure rotation conditions. Regarding the internal friction angle of rocks as the fundamental concept related to their strength, although without intending to offer a physical interpretation for it (as it is an empirical parameter), we can do some digressions. The basic one is that the internal friction angle could be weighted either by the static friction angle and by the rotational friction angle depending on the spatial distribution of the rock structure in relation to the stresses. In this context, the formulation of the mechanics of Cosserat generalized continuum should be very important in assessing the role of microstructure rotation in what is classically called internal friction coefficient for rocks.
Pristine rock fabric can readily be expressed by the characteristic lengths in the formulation of the mechanics of Cosserat generalized continuum. In order to assess the role of the characteristic length in the inner structure of fault zones, some numerical models were conducted by using the computer system TECTOS, a finite-element based system developed by Petrobras and the Catholic University of Rio de Janeiro (Moraes et al., 2002). Figure 5 provides the geometry, the mesh and the boundary conditions for the models. They have a non-associated elastoplastic rheology with an internal friction angle of 30° and a dilatancy angle of 12°. A rotational modulus G r of 25 GPa was used as well as isotropic characteristic lengths l of 0.01 m, 0.1 m, 1 m, 2.5 m and 10 m. Further properties as well the boundary conditions are suitable for the conditions at a depth of 5 km in the crust. A progressive dextral shear stress from 10 to 90 MPa was prescribed at the top and the bottom of the models. Figure 6 shows the results for rupture and the anti-symmetric shear stress at the latter stage for the models. A model for the classic continuum is also shown for comparison. Here, the concept of anti-symmetric shear stress rigorously does not apply, being basically zero. Comparing the failure patterns in the models, it is seen that the model carried by the classic continuum mechanics shows a generalized rupture in the fault zone. In contrast, the models governed by the mechanics of Cosserat generalized continuum show a strong banded pattern for the strain localization, with intercalation of sub-parallel strands with and without rupture that are in general proportional to the characteristic lengths. These results suggest a consistent spatial complexity in the onset of faulting not captured by the the classic continuum mechanics model. By observing the anti-symmetric shear stresses, negative values for clockwise rotations, the role of the characteristic length in structuring a fault zone becomes clear. The complexity in terms of microscale rotation fields, with heterogeneous clockwise and counter-clockwise distribution, is larger for smaller characteristic lengths and less pronouced for larger characteristic lengths. As previously stated, fault structural fabric, specially Riedel-Tchalenko structures with their sinthetic or antithetic shearing patterns, could be a consequence of the heterogeneous moment stresses field along the fault zone. In a broad sense, the results of the numerical models suggest that models using the mechanics of Cosserat generalized continuum bear a greater appeal to discuss the inner organization of fault zones in terms of rupture pattern and generation of their asymmetric fabric.

Mechanics of Cosserat Generalized Continuum and Modelling in Structural Geology
Anderson Moraes; Rodrigo P. de Figueiredo & Eurípedes do A. Vargas Jr In this way, we postulate that the discussion on the spatial arrangement of the fabric in fault zones is rarely extended as most of the works in fault zones modelling usually makes use of the classic continuum mechanics. For example, many authors in the structural geology literature endeavour to explain the very common antithetic features relying only on classic continuum mechanics. Ramsay (1967), page 283, points out the need to have a "disequilibrium component" in the stress tensor as "the disequilibrium components cause the material to undergo a rotation in space". In fact, this component need not to exist as the relative stress tensor t ij , the anti-symmetric part of the Cosserat stress tensor, leads to anti-symmetric shear stresses and related rotations. Mandl (1999) recognizes the need for the emergence of antithetic features when studying many structural and geodynamic problems (e.g. antithetic faults, bookshelf mechanism, San Andreas Fault Paradox). However, the author follows a complex path for the explanation of these geodynamic contexts by making several transformations in the space s e N x s C together with the Mohr-Coulomb criterion. Nevertheless, by using the mechanics of Cosserat generalized continuum, many antithetic features may easily emerge. Ahlgren (2001) places R' structures as "enigmatic features" and considers that "the antithetic shear sense of these bands ... is difficult to interpret mechanically". Once again, the presence of the anti-symmetric shear stresses could provide an adequate interpretation.

Concluding Remarks
It is suggestive that asymmetric geological structures are generated by asymmetric physical fields. According to what was discussed, we ponder that the mechanics of Cosserat generalized continuum allow a better comprehension when building and interpreting conceptual and, in special, numerical models in structural geology. In a broad sense, in comparison to the classic continuum mechanics, the mechanics of Cosserat generalized continuum deals more widely with heterogeneous and anisotropic media, that is, the seminal characteristics present in rocks. In this sense, we argue that by using the Cosserat continuum mechanics it would be possible to better define the observation scales to be considered in defining how heterogeneous is the deformation responsible for the structural symmetry/asymmetry in rocks.
Considering future perspectives, it would be of fundamental importance to develop more refined models based on the intrinsic formulation of a Cosserat's continuum and to extend its relationship with the experimental base and the observations in nature. In particular, it would be very useful to characterize the role of the characteristic lengths either in isotropic or in anisotropic rocks and their physical meaning and relevance in studying any particular set of geological structures.

Ackowledgements
Anderson Moraes is grateful to Petrobras for allowing to publish this work. We thank two anonymous reviewers who greatly helped to improve the manuscript.