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The most commonly used procedure for prediction of the behaviour of laterally loaded piles is the P–y curve formulation, which gives a simple but efficient framework to predict the response of the pile. This framework is limited to a single direction of loading, while there are several situations in which a pile is subjected to lateral loads with varying direction, as for example in the case of wind or wave loads. Here an extended framework for P–y curve modelling is presented, in which several springs are considered around the pile perimeter at each depth. The advantage of this framework is that it remains as simple and practical as the original P–y curve method and does not need any further information or parameters. A procedure is proposed for the extension of a given unidirectional model to the corresponding multi-directional one. The effects of multi-directional loading are discussed based on the simulation results. With a change in loading direction, misalignment between load direction and total displacement occurs. In addition, this quite simple model enables deduction of the profile of irreversible soil displacements around the pile at various depths.

International audience; An accurate monitoring of axial and radial thermal strains of specimens of the Callovo–Oxfordian claystone, heated under constant confining stress conditions close to that prevailing in the Bure Underground Research Laboratory at 490 m depth, evidenced a particular behavioural feature. The effect of the transverse isotropy of the claystone on thermal strains was not only confirmed, but it was also shown that the thermo-elasto-plastic overall volumetric response was mainly due to the axial response, in which some influence of the adsorbed water is suspected. Conversely, the reversible radial response observed along a temperature cycle could be modelled through a simple homogenisation method that confirmed that it was governed by the thermo-elastic expansion/contraction response of the constitutive minerals. Hence, the simultaneous occurrence of elastic thermal radial strains and elasto-plastic thermal axial strains was evidenced.

Field monitoring programs (e.g., convergence measurements and stress measurements in the support system) play an important role in following the response of the ground and of the support s...

International audience; Tunnel excavation in squeezing ground exhibits large time-dependent and often anisotropic deformation. Within the context of the Fréjus road tunnel and its safety gallery excavated under the Alps between France and Italy, an interesting configuration of two parallel tunnels under squeezing ground conditions is observed. The special feature of this case study lies in the fact that both tunnels have been excavated in similar geotechnical conditions but with different excavation techniques. The road tunnel was excavated with conventional drill and blast methods in the 70s, whereas the safety gallery was excavated between 2009 and 2016 with a single-shield tunnel boring machine (TBM). This paper presents monitoring data processing and numerical simulations of both tunnels with the aim of studying the influence of the excavation method on the time-dependent tunnel response. A calibration of a visco-elasto-plastic anisotropic constitutive model based on the back-analysis of convergence measurements retrieved during the excavation of the Fréjus road tunnel is carried out. The identified ground behavior can be extrapolated to the parallel zones of the safety gallery. In particular, we are interested in the prediction of the stress state in the segmental lining of the gallery during its excavation and the comparison with in situ measurements. It is shown that the time-dependent behavior of the ground is affected by the excavation technique. Finally, an attempt to predict the long-term response of both tunnels is proposed.

In this paper, the Creation of Residual Strain during the hydration of cement paste is studied by performing oedometric experiments on class G cement pastes during the six first days of hydration. Various conditions of temperature (between 7 and 30 °C) and pressure (between 0.3 and 45 MPa) are explored. It is found that after a hydration degree of about 18%, mechanical loadings can induce residual strain. This state is reached at a time called critical time for Creation of Residual Strain (CRS). It corresponds to the maximum axial strain rate in an oedometric test or to the maximum rate of wave-velocity evolution versus time recorded in a Ultrasonic Cement Analyzer (UCA) test. The Boundary Nucleation and Growth (BNG) model is used to estimate the critical time for CRS. Within the range of studied temperatures and pressures, the predictive capacity of the BNG model for estimation of the critical time for CRS is demonstrated.

International audience; Pore pressure evolution induced by tunnel excavation in low permeability anisotropic ground is investigated by means of a fully coupled hydro-mechanical finite element simulation. The goal is to explore the main trends of the pore pressure evolution on the basis of a poroelastic analysis, taking into account: on the one hand, the influence of excavation and face advance; on the other hand, the inherent anisotropy of the material. It is shown that overpressures can be induced before the passage of the face, depending on: the consolidation time of the rock, the rate of excavation and the degree of anisotropy of both the material and the initial stress state. Finally, a discussion about pore pressure evolution at tunnel wall is presented. An analysis is performed in order to propose a pore pressure release rate at the tunnel wall for 2D models.

International audience; A squeezing Carboniferous formation was met at a depth of 300 m during the excavation of the Saint-Martin-la-Porte access gallery (SMP2) in France within the Lyon-Turin railway link project. Large, time-dependent and anisotropic deformation was observed around the tunnel wall during and after excavation, and difficulties related to tunneling in squeezing ground have been encountered. An anisotropic visco-elastic plastic constitutive law (Tran- Manh et al. 2015) has been proposed in order to model the ground behavior. This model was validated by field auscultation carried out in SMP2. As a part of the base tunnel, a new survey gallery (SMP4) began to be excavated in the recent years across the same squeezing rock formation at a depth of about 600 m. A yield control support system was adopted in the zones of large deformation, which contains highly deformable concrete elements to stabilize the high convergence (Bonini and Barla 2012). In the present work, the studies of SMP2 are extended to SMP4 by considering the constitutive model developed for SMP2. The innovative excavation and support method is taken into account. Numerical modeling is performed using FLAC3D to analyze the tunnel response. A good agreement can be obtained between the field measurements and the numerical results.

International audience; Identifying the mechanisms that control the formation of compaction bands is of high interest in reservoir mechanics since these structures may drastically affect the performance of geosystems operations. Considering the difficulty to identify compaction bands in carbonate samples tested in the laboratory, the Digital Volume Correlation technique is applied here and proves to be a relevant method. X‐Ray Computed Tomography (XRCT) images of Saint‐Maximin limestone centimetric samples are recorded before and after several triaxial loading stages and deformation maps are built. A new postprocessing method based on the analysis of the kinematics throughout the observed localization bands is proposed to identify their type. Compaction bands are identified at relatively high‐confining pressures, while shear bands are observed at lower confinements. The brittle‐ductile transitional regime reveals the formation of compactive shear bands, while a diffuse compaction is observed under hydrostatic loading. The effect of porosity heterogeneities on strain localization is explored by computing 3D porosity maps from calibrated XRCT images. Compaction bands are found to dominantly lay inside high‐porosity zones, while shear bands can cross both high‐ and low‐porosity zones. The band orientation is found to be controlled primarily by the confining pressure. Moreover, the porosity heterogeneity strongly affects the volumetric behavior inside the deformation bands, with a dilatant behavior identified in low‐porosity zones in contrast to a compactive behavior observed in high‐porosity zones. Finally, Scanning Electron Microscopy observations reveal that calcite grain crushing is dominant in high‐porosity zones, while intergranular cement cracking occurs in denser zones.

International audience; Strain localization zones in the form of shear bands or compaction bands in geomaterials are observed across scales from sub-millimetric (grain size) to kilometric scale (geological structures). Triggering and evolution of such narrow zones of localized deformation depend on many factors. The mechanical behavior of geomaterials is central for the formation of such zones. However, thermal, pore-pressure and chemical effects play a crucial role in shear and compaction banding. Moreover, the inherent heterogeneous microstructure of geomaterials plays a significant role during strain localization. As for faults, compaction bands significantly influence the stress field and fluid transport. In this paper, we shall review some recent advances in experimental testing and numerical modelling on strain localization in geomaterials. The effect of grain crushing as observed in deformation bands under high confinement can be introduced by combining Breakage models and higher order continuum theories leading to a new framework of constitutive models with evolving microstructure. A major difficulty of these models is to establish reliable methods for the calibration of the higher order parameters (such as the internal length) in laboratory experiments. An example of a direct calibration of these parameters based on Digital Image Correlation of images provided by X-Ray tomography is proposed for the study of compaction banding in a carbonate rock.

In the framework of a deep geological radioactive waste disposal in France, the hydromechanical properties of the designated host rock, the Callovo-Oxfordian claystone (COx), are investigated in laboratory tests. Experiments presented in this study are carried out to determine several coefficients required within a transversely isotropic material model. They include isotropic compression tests, pore pressure tests, and deviatoric loading tests parallel and perpendicular to the bedding plane. We emphasize the adapted experimental devices and testing procedures, necessary to detect small strains under high pressures, on a material, which is sensitive to water and has a very low permeability. In particular, we discovered a significant decrease of elastic stiffness with decreasing effective stress, which was observed to be reversible. In both isotropic and deviatoric tests, a notable anisotropic strain response was found. The Young modulus parallel to bedding was about 1.8 times higher than the one perpendicular to the bedding plane. A notably low Poisson ratio perpendicular to the bedding plane with values between 0.1 and 0.2 was evidenced. While the anisotropy of the back-calculated Biot coefficient was found to be low, a significant anisotropy of the Skempton coefficient was computed. The performed experiments provide an overdetermined set of material parameters at different stress levels. Using all determined parameters in a least square error regression scheme, seven independent elastic coefficients and their effective stress dependency are characterized. Parameters measured under isotropic loading are well represented by this set of coefficients, while the poroelastic framework with isotropic stress dependency is not sufficient to describe laboratory findings from triaxial loading., accepted manuscript

Antigorite is a key constituent of subducted slabs, and its dehydration is thought to be responsible for the generation of intermediate-depth earthquakes. The mechanical behaviour of antigorite at elevated pressure and temperature remains difficult to constrain experimentally: intracrystalline slip systems are hard to activate under typical laboratory timescales and microstructures do not always provide unambiguous evidence for dislocation creep. Here, we present recent laboratory data showing that antigorite might deform due to intracrystalline frictional slip and delamination, at least in the low temperature regime (

The use of the classical ConVergence ConFinement (CV-CF) method which is a simplified tunnel design tool is very extended nowadays. A major limitation of this plane-strain approach is encountered when it is applied to the study of tunnels with a stiff lining near the tunnel face. Under this configuration, the pressure exerted by the ground over the lining can be underestimated. The reason stems from the fact that this method is not able to simulate the tridimensional arch effect taking place between the ground and the lining. To account for this ground-support interaction, some authors have proposed to enhance the classical CV-CF method by resorting the so-called implicit methods. However, these approaches still show some limits for very stiff lining. In this paper, the applicability domain of the existing CV-CF methods when applied to the design of full-face excavated circular tunnels with a stiff support system is discussed. The results of the ground-lining equilibrium state obtained with the plane-strain approaches are compared with the equilibrium state obtained with an axisymmetric numerical model which can properly capture the arch effect taking place at the vicinity of the tunnel face. A sensibility analysis with a focus on the support lay distance has been carried out. Finally, a simple chart that clarifies which CV-CF method is more adapted to each tunnel configuration is proposed.

International audience; Continuous monitoring around drifts excavated in the Underground Research Laboratory (URL) of The French National Radioactive Waste Management Agency (Andra) in Callovo Oxfordian claystone revealed the development of a fractured zone (extensional and shear fractures) induced by the excavation. Moreover, the field monitoring during the excavation showed an anisotropic pore pressure field around the drifts with zones of marked overpressure. For drifts parallel to the horizontal major principal stress, even though the initial stress state in the drift section is quasi-isotropic, the pore pressure evolution and the mechanical response are anisotropic. These observations suggest that the intrinsic anisotropy of the material plays a key role in the response of rock formation. A poroelastic analysis of the pore pressure change induced by drift excavation is performed. The principal objective is to simulate the main trends of the pore pressure evolution with a simple model taking into account the elastic anisotropy of the material. It is shown that depending upon the permeability and the degree of anisotropy of the rock, strong overpressures can be induced by the excavation. Furthermore, it is observed that the distribution of overpressures around drifts depends also on the drifts orientation. Finally, an analysis of the onset of failure shows the key role of the hydro-mechanical coupling on the extension of the failed zone around the drifts.

Curing temperature has a significant influence on cement paste microstructure and the properties of its principal hydrate C-S-H. In this paper, the effect of the hydration temperature in the range of 7 °C to 90 °C on the microstructure of a class G oil-well cement is studied. This is done by combining various experimental methods, including X-ray diffraction associated with the Rietveld analysis, thermo-gravimetric analysis, mercury intrusion porosimetry and porosity evaluation by drying. The experimental results show an increase of the capillary porosity and a decrease of the gel porosity by increasing the hydration temperature. This is attributed to a decrease of the C-S-H intrinsic porosity and a corresponding increase of the C-S-H density for higher curing temperatures. The experimental results are used in a simple analysis method to evaluate the density of C-S-H, as well as its C/S ratio and H/S ratio in dry and saturated conditions. The evaluated C-S-H density varies from 1.88 g/cm 3 at 7 °C to 2.10 g/cm 3 at 90 °C. The results also show a decrease of molar C/S ratio with increasing hydration temperature from 1.93 at 7 °C to 1.71 at 90 °C and of the H/S ratio from 5.1 at 7 °C to 2.66 at 90 °C.

Energy linings are receiving great interest due to their potential to transform a tunnel into a smart energy system for enhancing thermal comfort of buildings and metro stations, as well as to absorb waste heat from the tunnel environment. Yet, besides their thermal performance, the impacts of equipping segments with a net of pipes are still challenging in the framework of energy tunnels structural design. Indeed, the thermal regime of the concrete members is altered due to heat carrier fluid circulation. Hence, the need to shed some light on the possible limit states attainment arises. In this paper, field monitoring of stresses and strains are presented for the first time in relation to a full scale monitored prototype of Enertun energy tunnel lining installed in Turin, Italy. Experimental data are compared to the results of a coupled thermo-mechanical numerical model, showing to be in good agreement. Criticalities from the ultimate limit state do not emerge, although a deeper analysis of serviceability limit states attainment should be performed.

International audience; 22 23 The ConVergence-ConFinement (CV-CF) method is widely used in conventional tunneling at 24 a preliminary stage of the design. In this method, the rock-support interaction is simplified by 25 means of a two-dimensional plane-strain assumption. However, when the ground exhibits large 26 deformation and/or when the support is very stiff and installed close to the tunnel face, the 27 results obtained with the CV-CF method may significantly differ from those obtained using 3D 28 numerical computations. The strong interaction taking place between the rigid lining and the 29 rock mass is not considered in the most common use of the CV-CF method. Some 30 improvements of the CV-CF method as the so-called implicit methods have been developed in 31 order to better account for this interaction. 32 In this paper, the applicability of the CV-CF methods is discussed for full face excavation 33 tunneling with a stiff support system. An in-depth comparison between plane-strain closed form 34 solutions and numerical results which properly accounts for the 3D effects at the vicinity of the 35 tunnel face is carried out. The range of application of the different approaches of the CV-CF 36 method is discussed. Finally, some simple empirical formula which can be used in preliminary 37 design for a large range of ground conditions are proposed. 38 39

The effects of temperature on the water transport properties of intact and damaged Opalinus (OPA) clay are investigated by using a recently developed hollow cylinder triaxial cell [Monfared (Int J Rock Mech Min Sci 48:637–649, 2011b)] that allows full saturation and drainage conditions in low-permeability clays and shales. The volumetric response of saturated OPA clay sample during a drained heating test shows an irreversible contraction after a temperature threshold. The permeability tests which are performed before and after the heating test show that the induced irreversible sample contraction by thermal loading reduces the permeability of OPA clay sample. In order to study the effect of temperature on the permeability of a damaged sample of OPA clay, the permeability tests are performed on a saturated sample previously sheared by a standard drained triaxial loading. The test results show no significant effect of shear-type damage on the permeability of the sample at 25 °C and 80 °C. The experimental results presented in this paper show the crucial role of the thermally induced strains on sample permeability. Thermo-elastic dilation leads to a slight increase of the permeability, whereas thermoplastic contraction leads to a reduction.

Velocity stepping experiments have been performed on a simulated calcite gouge using an annular shear apparatus to investigate the effect of dissolution on the frictional properties of a carbonate fault. The tested material was put in contact with hydrochloric acid at different concentration in order to dissolve the grains. Particle size analysis shows that the small grains tend to disappear due to the chemical reaction, whereas the large grains are not much affected. The dissolution process mainly induces a decrease of the fractal number of the grain size distribution. The study of the rate-and-state parameters of the gouge, which enables to characterize the ability of the fault to generate earthquakes, shows a weakening due to the dissolution and the confining stress. This weakening implies that faults can become seismogenic after the injection in a carbonate reservoir of an acid fluid. This effect is explained by the removal of small particles, countering the increase of rugosity of the grains. Based on the experimental results, a constitutive law is suggested, in order to capture the influence of dissolution and confining stress on the frictional properties of fault. It enables to determine in which conditions of maturity and degradation of a fault and at which depth, a seismic slip can be triggered by anthropogenic activities like acid/CO2 injection.

to appear; International audience; Strain localization zones in the form of shear bands or compaction bands in geomaterials are observed across scales from sub-millimetric (grain size) to kilometric scale (geological structures). Triggering and evolution of such narrow zones of localized deformation depends on many factors. The mechanical behavior of geomaterials is central for the formation of such zones. However, thermal, pore-pressure and chemical effects play a crucial role in shear and compaction banding. Temperature increase and activation of chemical reactions such as mineral dehydration, carbonate decomposition, as well as dissolution/precipitation control the triggering and the evolution of localized deformation zones. Moreover, the inherent heterogeneous microstructure of geomaterials plays a significant role during strain localization. The purpose of this paper is to provide a review of recent research regarding the effects of temperature, pore-pressure, chemical reactions and microstructure on strain localization in geomaterials. Examples have been taken in relation with seismic slip and with compaction banding. Strain localization is treated as an instability from a homogeneous deformation state. Different types of instabilities may (co-) exist depending on different multi-physical couplings and micro-mechanisms. Finally, a comparison of rate dependent Cauchy continuum and rate independent generalized continua (Cosserat continuum) is made, which leads to an analog expression for the critical perturbation wave length, which scales the thickness of the localized zone and defines the region that non-homogeneous deformations are possible.

We have examined microstructures, mineralogical composition, geochemical alteration, and texture of four selected fault rock samples from the Deep Geodynamical Laboratory (DGLab) Gulf of Corinth project using optical microscopy, cathodoluminescence microscopy (CL), scanning electron microscopy (SEM), transmission electron microscopy (TEM), and synchrotron X-ray diffraction measurements. The fault core is composed of red and gray clayey gouge material and surrounded by a damage zone of brecciated limestones. Pressure solution features, calcite veins and calcite clasts in the breccia and gouge material attest the presence of paleo-fluids and fluid-driven mass transfer during deformation. Differences in CL-colors between the matrix and calcite vein cement and inside the vein cement suggest repeated infiltration of fluids with different composition from various sources (formation water and meteoric water). Twin lamellae densities estimated in calcite veins are used as paleo-piezometer. The deduced differential stress is ∼140 ± 70 MPa for the older vein generation and appears to be higher than stress for the youngest veins (45 ± 23 MPa). In spite of the relatively small clay content in both samples, newly formed clay minerals have been observed in gray as well as red clayey gouge material. Differences between gray and red clay gouge material are found in fault rock composition, porosity and clay fabric. The proportion of chlorite in the red gouge is significantly less than that in the gray gouge whereas the initial porosity is significantly higher than in the gray gouge material. The detection of a well-oriented clay fabric in red clay gouge samples is unique in comparison to other major fault zones.

Summary The macroscopic behavior of materials is affected by their inner micro-structure. Elementary considerations based on the arrangement, and the physical and mechanical features of the micro-structure may lead to the formulation of elastoplastic constitutive laws, involving hardening/softening mechanisms and non-associative properties. In order to model the non-linear behavior of micro-structured materials, the classical theory of time-independent multisurface plasticity is herein extended to Cosserat continua. The account for plastic relative strains and curvatures is made by means of a robust quadratic-convergent projection algorithm, specifically formulated for non-associative and hardening/softening plasticity. Some important limitations of the classical implementation of the algorithm for multisurface plasticity prevent its application for any plastic surfaces and loading conditions. These limitations are addressed in this paper, and a robust solution strategy based on the singular value decomposition technique is proposed. The projection algorithm is then implemented into a finite element formulation for Cosserat continua. A specific finite element is considered, developed for micropolar plates. The element is validated through illustrative examples and applications, showing able performance. Copyright © 2016 John Wiley & Sons, Ltd.

Time-dependent response of deep tunnels is studied considering the progressive degradation of the mechanical properties of the rock mass. The constitutive model is based on a rock-aging law for the uniaxial strength of the rock and for the Young’s modulus. A semi-analytical solution is developed for the stresses and displacements around a deep circular tunnel taking into account the face advance. The evolution of the plastic and damage zones over time is determined. Numerical examples are presented for the case of Saint-Martin-La-Porte access adit in France of the Lyon–Turin Base Tunnel. The computed results which are compared with the field data in terms of the convergence of tunnel wall and of the displacements inside the rock mass monitored by multi-point extensometers show the efficiency of the approach to simulate the time-dependent deformation of a tunnel excavated in squeezing ground. Simple relationships are proposed to evaluate the parameters of the constitutive model directly from those of the empirical convergence law presented in previous work.

Les couplages thermo-hydro-chimio-mécaniques jouent un rôle central dans les grands glissements gravitaires et sismiques. À partir du travail original de Pierre Habib sur la vaporisation du fluide interstitiel comme mécanisme pouvant expliquer les vitesses extrêmes atteintes durant le glissement d’une masse rocheuse, nous donnons un éclairage sur les développements récents sur la physique des tremblements de terre.

La méthode convergence-confinement permet de simuler le creusement d’un tunnel avec la mise en place d’un soutènement par un modèle de déformation plane. La connaissance de la convergence du massif au moment où le soutènement devient actif est une démarche essentielle pour l’application de la méthode. Celle-ci nécessite la détermination du profil longitudinal de déplacement (LDP), qui représente le déplacement radial à la paroi du tunnel en fonction de la distance au front. Dans cet article, nous rappelons les contributions importantes de Pierre Habib et de ses collègues du Laboratoire de mécanique des solides (LMS) dans le développement de la méthode, en particulier sur la prise en compte de l’influence de la pose d’un soutènement rigide près du front sur la courbe LDP.

Analysing the response of cement sheath in oil, gas or geothermal wells submitted to mechanical loadings needs an appropriate constitutive law for the cement paste in wells conditions. The knowledge of the actual initial state of stress is a key point which requires a simulation of cement paste behaviour from placement to hardened state. A chemo-poro-elastoplastic model for macroscopic shrinkage and stress-strain behaviour of cement paste at the early age is developed in this paper. A modified Cam-Clay type yield surface and an associate plastic flow rule are used. The hydration is considered as a hardening mechanism of the yield surface, which also evolves with the development of plastic strains. The model shows a good capacity in capturing the basic mechanisms of both macroscopic shrinkage and stress-strain behaviour of an oilwell class G cement paste hydrated under various conditions of temperature and pressure. A strong coupling is found between the macroscopic shrinkage and the stress-strain response.

International audience; In this study, we analyzed the influence of the excavation size and the sequence of excavation on the convergence evolution in drifts excavated in Callovo-Oxfordian claystone. The drifts, which are part of the Underground Research Laboratory of the French National Radioactive Waste Management Agency (ANDRA), had diameters ranging between 0.7 m and 6.3 m. The measurements showed an anisotropic convergence with the maximum and minimum values in horizontal and vertical directions. Using a semi-empirical law, convergence evolution was fitted independently for each drift. The results show that the values for the parameters describing the time-dependent properties of the system rock-support depended upon the time elapsed for the support installation and the rate of excavation. For drifts following the same sequence of excavation, these parameters did not depend upon the size of the excavation. Finally, the analysis of a drift cross section enlargement showed that the convergence evolution was more isotropic and that the values for the parameters describing the time-dependent properties were lower, suggesting that overexcavation could limit the delayed response of the ground.

International audience; A numerical model for thermo-hydro-mechanical strong couplings in an elasto-plastic Cosserat continuum is developed to explore the influence of frictional heating and thermal pore fluid pressurization on the strain localization phenomenon. This model allows specifically to study the complete stress–strain response of a rock specimen, as well as the size of the strain localization zone for a rock taking into account its microstructure. The numerical implementation in a finite element code is presented, matching adequately analytical solutions or results from other simulations found in the literature. Two different applications of the numerical model are also presented to highlight its capabilities. The first one is a biaxial test on a saturated weak sandstone, for which the influence on the stress–strain response of the characteristic size of the microstructure and of thermal pressurization is investigated. The second one is the rapid shearing of a mature fault zone in the brittle part of the lithosphere. In this example, the evolution of the thickness of the localized zone and the influence of the permeability change on the stress–strain response are studied.

In this paper we study the phenomenon of localization of deformation in fault gouges during seismic slip. This process is of key importance to understand frictional heating and energy budget during an earthquake. A infinite layer of fault gouge is modeled as a Cosserat continuum taking into account Thermo-Hydro-Mechanical (THM) couplings. The theoretical aspects of the problem are presented in the companion paper (Rattez et al., 2017a), together with a linear stability analysis to determine the conditions of localization and estimate the shear band thickness. In this Part II of the study, we investigate the post-bifurcation evolution of the system by integrating numerically the full system of non-linear equations using the method of Finite Elements. The problem is formulated in the framework of Cosserat theory. It enables to introduce information about the microstructure of the material in the constitutive equations and to regularize the mathematical problem in the post-localization regime. We emphasize the influence of the size of the microstructure and of the softening law on the material response and the strain localization process. The weakening effect of pore fluid thermal pressurization induced by shear heating is examined and quantified. It enhances the weakening process and contributes to the narrowing of shear band thickness. Moreover, due to THM couplings an apparent rate-dependency is observed, even for rate-independent material behavior. Finally, comparisons show that when the perturbed field of shear deformation dominates, the estimation of the shear band thickness obtained from linear stability analysis differs from the one obtained from the finite element computations, demonstrating the importance of post-localization numerical simulations.

The problem of a single tunnel in anisotropic ground has been largely studied in the literature but the interaction of two tunnels is mostly addressed for the isotropic case. Simple superposition of the solutions of two non-interacting tunnels may give reasonable results if the distance between the tunnels is large enough, but fails when the tunnels are excavated close to each other. Based on complex variable theory, closed-form solutions for the stresses and displacements around two deep circular tunnels in a homogeneous, transversely isotropic and linear elastic ground with non-isotropic far-field stress conditions are developed. The results obtained with the proposed analytical solution are compared with those obtained with the numerical code Flac3D. The effects of the orientation of the isotropic plane and of the distance between two tunnels on the stress and displacements fields are discussed.