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Simulation of localized compaction in Tuffeau de Maastricht based on evidence from X-ray tomography
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Abstract
Deformation experiments with concurrent X-ray microtomography are used to characterize the mechanical properties of a high-porosity carbonate rock from Central Europe (the Tuffeau de Maastricht). Evidence of deformation behavior and pressure-dependent strain localization for this rock are taken into account to define the parameters of an elastoplastic constitutive law with competing hardening and softening mechanisms. Full-field data regarding the compaction localization regime have been used to identify a compromise between compaction hardening and structure deterioration. It was shown that by placing emphasis on the destructuration behavior it was possible to reproduce features such as the elongation of the deformation plateau emerging during compaction band formation, as well as the pressure-dependent inclination of the ensuing compaction bands. Most notably, the comparison between measurements and simulations showed that the strategy here used to constrain the model parameters allows a satisfactory depiction of both the macroscopic deformations and the portion of sample volume across which the compaction zones develop and propagate. These results point out the major benefits of parameter calibration strategies accounting synergistically for global measurements and spatially-distributed deformations, especially in the presence of constitutive laws meant to replicate compaction localization.
... The incorporation of breakage and/or damage mechanics has been attempted to better understand inelasticity in porous sandstones [32][33][34]. The elastoplastic constitutive framework incorporating cap plasticity proposed by Nova et al [35] has also been proven to accurately reproduce the mechanical responses of a variety of geological solids, including stiff clays [36,37], sands [38], and porous rocks [39,40]. Recently, this model has been further enhanced through viscoplasticity and used to investigate localized creep in sandstone [41,42]. ...
... Although the use of Eq. (7) has allowed for the inception of strain localization to be captured mathematically in different classes of plastic solids [24][25][26]40], negative values of A (θ)) indicate impending loss of material strength, which, for static problems, converts the partial differential equations (PDEs) governing the deformation process from elliptic to hyperbolic, thus leading to ill-posedness. Numerical solutions of ill-posed BVPs involve deformation zones with vanishing thickness upon mesh refinement, which is not compatible with observation. ...
... The model incorporates both hardening and softening mechanisms able to replicate the destructuration processes in soft rocks and hard soils. Prior works have illustrated its ability to capture the deformation response of sands [38] and porous rocks [40], including applications involving the time-dependent behavior of sandstones [41,42] once enhancements are introduced. Both the yield function and plastic potential follow the expression proposed by Lagioia et al [56]: ...
Nonlocal implicit gradient enhancements are widely used to suppress mesh dependency in simulations involving strain localization. For example, nonlocality is often introduced through internal variables that account for possible material softening. This work, however, shows that this approach may become ineffective if the solid displays plastic non-normality (i.e., non-associated plastic flow). For this purpose, we consider an over-nonlocal formulation and mathematically inspect the conditions at which regularization is lost in the presence of plastic non-normality. Specifically, such loss of regularization is linked to the loss of uniqueness and/or existence of the incremental plastic response that is kinematically compatible with the development of a deformation band. By doing so, we find a lower limit for the admissibility of the parameters controlling the effectiveness of nonlocal implicit gradient regularization. Furthermore, we show that such a lower limit is regulated by a plastic modulus reflecting the loss of controllability of the constitutive response, and, hence, depends on the degree of plastic non-normality. We also derive a closed-form expression relating the thickness of the deformation band to both the controllability modulus and gradient regularization constants, which suggests that the thickness of the process zone may change in response to the prevailing plastic flow characteristics and evolve during active plastic deformation. The proposed nonlocal enhancement is applied to a non-associated elasto-plastic model for porous sedimentary rocks, which is capable of displaying both shear-dominated and compaction-dominated bands. Numerical simulations reveal that effective regularization can be enforced only when the over-nonlocal weighting coefficient is larger than the above-mentioned lower limit.
... 1,2 In the domain of mechanics, strain localization has been extensively studied both from an experimental 3-6 and a numerical standpoint. [7][8][9][10][11] From a mathematical standpoint, strain localization is typically addressed as a bifurcation problem, [12][13][14] that is, a condition at which nonhomogeneous solutions of the underlying nonlinear field equations become possible without violations of equilibrium or compatibility. At the local material point level, such circumstances typically involve the loss of uniqueness and/or existence of the incremental constitutive response. ...
... In this context, the so-called overstress approach 61 (also referred to as the overstress approach) can be regarded as one of the most widely used frameworks to reproduce the time-dependent response of a wide range of geomaterials [62][63][64][65][66] and regularize ill-posed boundary value problems associated with strain softening. 9,[67][68][69] Despite these advantages, the analytical identification of delayed localized deformation can be challenging in that the standard strain localization theory and controllability theory are not directly applicable to elasto-viscoplasticity. In this context, recent contributions by Pisanò and Di Prisco 70 provide an alternative methodology to identify strain acceleration in viscoplastic solids subjected to stationary external perturbations. ...
... 30 For the analyses conducted in this work, has been specialized to simple shear control conditions. In Equations (8) and (9), the time derivative of the plastic potential can be developed as follows: ...
This work presents a viscoplastic localization criterion to detect quasi-instantaneous (i.e., load-induced) and delayed (creep-induced) strain localization in rate-dependent solids. The study is based on the theory of controllability and a viscoplastic description of the mechanical response. Analytical precursors of unstable states are defined through systems of ordinary differential equations (OEDs). The use of the proposed criteria is illustrated at the material point level through a set of strain localization analyses simulating active strain localization of a porous rock. In addition, full-field finite element simulations of compression tests conducted under various pressures are reported to demonstrate the role of local unstable viscoplasticity in the spontaneous propagation of deformation bands under stationary boundary conditions. The study shows that the viscoplastic localization criterion maintains a negative sign as long as the behavior is unstable, that is, the rate of deformation is accelerating. The sign switch coincides with the transition to decelerating deformation. The analyses revealed that pulses of overstress always emerge in correspondence with the growth of unstable behavior, and the peak matches the transition to stable behavior. The local responses recovered from full-field analyses were consistent with those observed in analyses at material point level and the predictions of the presented theory.
... The selected materials for this study are Bleurswiller sandstone and Maastricht Tuffeau, both porous rocks known to develop compaction bands over a wide range of confinements (Fortin et al., 2005;Baxevanis et al., 2006). Recent simulations of these materials based on elastoplastic constitutive laws have shown that Bleurswiller sandstone can be effectively simulated with hardening behaviour (Shahin et al., 2019a;Shahin & Buscarnera, 2019), while the compaction localization response of the Maastricht Tuffeau is better captured by strain-softening behaviour (Shahin et al., 2019b). ...
... The banding properties of the two abovementioned rocks were recently quantified in the context of plasticity models focusing on strain localization (Marinelli & Buscarnera, 2015;Shahin et al., 2019b). The constitutive framework adopted for such analysis was rooted on the competition between the hardening and softening contributions reflected by multiple state variables, according to techniques extensively developed for structured soils and soft rocks (Gens & Nova, 1993;Nova et al., 2003). ...
... The adopted model uses the function proposed by Lagioia et al. (1996) to model the yield surface and plastic potential in the context of a non-associated plastic flow rule (details are provided in the appendix). In addition, it regulates the evolution (Shahin et al., 2019b) and Bleurswiller sandstone (Marinelli & Buscarnera, 2015 of the yield surface through two internal variables, Ps and Pm: ...
This study examines the role of boundary friction in promoting heterogeneous compaction in soft rock specimens loaded at high confining pressure outside the domain of compaction localisation. An elastoplastic constitutive model characterised by tunable hardening/softening behaviour is used to conduct the analyses. Finite-element simulations suggest that material instability is a non-necessary condition for the emergence of compaction fronts. Such fronts propagated as a result of a severe deviation in the local responses induced by frictional constraints. These findings suggest that boundary effects can bias the assessment of the extent of the compaction localisation domain. Experimental countermeasures and informed model calibration procedures are therefore necessary to minimise such bias and enable more accurate predictions of soft rock compaction.
... Figure 1a, b shows that the Mount Gambier limestone has a geometrically complex percolative backbone through the macropores (10-200 μm) supported by a solid matrix with a population of micro-pores sizes ranging from 0.1 to 5 μm. Similar results have been reported by Shahin et al. (2019) using Tuffeau de Maastricht limestone (53% porosity) and Ji et al. (2014) using Majella limestone (30% porosity). Two distinct pore throat clusters are evident from Fig. 1c, i.e., incremental intrusion volume divided by the overall intrusion volume showing the pore size distribution. ...
... Characteristics of stress-strain response Figure 3 summarizes the results of axial stress versus axial strain at 5 MPa confining pressure of two samples. Experiments with a similar high porosity (52%) carbonate, Tuffeau de Maastricht calcarenite, (Baxevanis et al. 2006;Shahin et al. 2019) reported that at 4 MPa confining pressure, a transition from shear bands to compaction bands occurs. At lower confining pressure (< 4 MPa), the specimens developed high-angle shear bands while above 4 MPa confining pressure, compaction bands are formed. ...
... A detailed description of the incremental compressive strain shown in the DIC images will be further discussed in this section Chen et al. Progress in Earth and Planetary Science (2020) strain curves after yielding showed relatively large and periodic stress drops which are similar to those reported for low strength and porous rocks (Han et al. 2013;Shahin et al. 2019). Such large stress drops are not usually observed in rocks with relatively lower porosity e.g., low porosity carbonate rock (Nicolas et al. 2016) and sandstones (Chen et al. 2017). ...
The formation of compaction bands in porous brittle rocks such as sandstones and carbonates has a significant impact on the localization mechanisms preceding earth and planetary surface instabilities such as earthquakes, landslides, and plate boundary faults. The micromechanics underpinning the dynamics of the formation of compaction bands and its effect on alteration of pore fluid pathways are not yet fully understood. The current study seeks to understand the mechanical properties of compaction in highly porous carbonate at micro- and macro-scale using time-lapse triaxial experiments in an X-ray transparent flow and deformation cell. Images were obtained with increasing axial strain levels using X-ray computed tomography allowing mapping of the evolution of internal structures. In addition to the X-ray analysis, digital image correlation (DIC) was used to quantify the evolution of strain and precisely identify the nucleation mechanism of compaction bands and its dynamics. The effect of friction on the boundary platens was shown to be minimal as evidenced by shear strain obtained from DIC analysis. This comprehensive analysis allowed assessment of the role of heterogeneity for the initiation of compaction bands. Local regions with high porosity provide the initial seeds for discrete compaction followed by the nucleation of traveling waves that lead to diffuse growth of the compaction zone. This interesting phenomenon is expected to be a fundamental mode of compressional deformation in porous brittle media where discrete, often periodic, deformation bands are observed on compaction.
... To study strain localization in porous rocks, an elastoplastic model incorporating competition between compaction-induced hardening and debonding-induced softening is used. This model was developed by Nova and co-workers [24,44] and has been selected by virtue of its ability to replicate mechanical responses of different classes of porous rocks [38,56], as well as to replicate a wide range of strain localization patterns. A cap-shaped yield surface and non-associated plastic potential are used in the model. ...
... Although this choice does not imply reference to real field settings (e.g., it will involve mean stress levels much higher than those found in typical borehole stability applications), it is justified by the availability of extensive databases of compaction localization behavior for this rock [5,7], thus enabling us to (i) use constitutive parameters that accurately capture the pressure dependence of the brittle-ductile transition in porous rocks; (ii) inspect inelastic deformation patterns in idealized simulated boreholes embedded within a formation that can display multiple forms of compaction localization depending on the in situ stress conditions. The model parameters for Berea sandstone calibrated by Marinelli and Buscarnera [38] are used ( Expansion in size of the elastic domain for cemented media softening response of some porous rocks has been illustrated by Marinelli and Buscarnera [38] and Shahin et al. [56]. [5,7]) and model simulations Fig. 3 Compaction localization characteristics of Berea sandstone predicted with the used constitutive model. ...
Boreholes excavated in porous rocks exhibit a variety of inelastic deformation patterns. This paper investigates how nucleation, propagation, and coalescence of compaction zones, including those resulting from compaction localization, affect the mechanics of excavated boreholes in high-porosity rock formations. For this purpose, a recently developed controllability framework able to differentiate among multiple modes of compaction banding is used to guide the interpretation of numerical simulations for boreholes excavated in porous rock. The goal is to explain the link between heterogeneous deformation patterns forming around the excavated zone and in situ stress conditions. To carry out the analyses, the theory is combined with a strain-hardening constitutive law calibrated against Berea sandstone laboratory evidence, thus capturing the stress-dependence of both homogeneous and heterogeneous compaction. The resulting simulations of idealized boreholes show that the inelastic compaction around the excavated zone depends dramatically on the far-field stress state, leading to transitions from isotropic-distributed plastic deformation modes under in-plane isotropic stress states, to dog-ear-, slot-, and/or butterfly-shaped inelasticity in the presence of strongly anisotropic in situ stress. It was found that each of these scenarios can be explained in terms of controllability indices computed at locations exhibiting inelastic response, thus establishing a link between the computed modes of global borehole deformation and multiple types of compaction localization processes (e.g., compactive shear bands, shear-enhanced compaction bands, and pure compaction bands).
... In these experiments, researchers acquire three-dimensional images of rock cores during triaxial compression. The images enable calculating the evolving local strain tensor throughout the rocks using digital volume correlation analysis (e.g., Charalampidou et al., 2011;Ji et al., 2015;McBeck et al., 2018;Shahin et al., 2019;Stamati et al., 2019;Heap et al., 2020;Mao et al., 2021;Baud et al., 2021). Using such time series of strain fields, machine learning analyses indicate that the intermediate values of the dilative strain can predict the timing of system-scale failure in these experiments (McBeck et al., 2020a). ...
... We use the code TomoWarp2 (Tudisco et al., 2017) to perform the digital volume correlation analysis. This analysis searches for similar patterns of voxels in pairs of tomograms, or other three-dimensional images, and then calculates the displacement vector that best maps one set of voxels to the other (e.g., Charalampidou et al., 2011;Ji et al., 2015;McBeck et al., 2018;Shahin et al., 2019;Stamati et al., 2019;Heap et al., 2020;Mao et al., 2021;Baud et al., 2021). In TomoWarp2, the node spacing determines the spatial resolution and the correlation window size determines the size of the volume used to identify similar patterns of voxels. ...
Deformation localization is a widely observed, but rarely quantified process in the crust. Recent observations suggest that the localization of seismicity and fracture networks can help identify the approach to catastrophic failure. Here, we quantify the localization processes of the volumetric and deviatoric strain components in twelve triaxial compression experiments imaged with X-ray tomography. We capture three-dimensional images of the rock cores during triaxial compressing toward failure, and then calculate the local strain components using digital volume correlation. The divergence and curl of the incremental displacement vector field provide the volumetric and deviatoric components of the strain field. We quantify localization using the proportion of the rock occupied by high magnitudes of the volumetric and deviatoric strains, and the Gini coefficient of these high magnitude strains, which measures the deviation from a uniform process. We find that the vast majority, but not all, of the experiments experience strain localization toward failure. The rocks typically experience their maximum degree of strain localization not immediately preceding failure, but on average at 90% of the failure stress. The volumetric strain tends to localize more than the deviatoric strain. These observations support using the localization of the volumetric strain, along with the deviatoric strain, to identify the evolution of the precursory phase preceding earthquakes.
... An accurate geomechanical analysis of compaction band systems at any locality requires extensive stages of material characterization to constrain the deformation response and strain localization modes embedded in the constitutive model [Shahin et al., 2019b]. Such measurements are not available for any of the outcrops know to have compaction bands. ...
... The strain-hardening elastoplastic model adopted in the current study aims to reproduce pre-and post-localization deformations, as well as represent pressure-dependent strain localization patterns. The model was developed in a series of contributions by Nova and coworkers [Gens and Nova, 1993, Lagioia and Nova, 1995, Nova et al., 2003, and has been selected by virtue of its proven success in capturing aspects of porous rocks mechanics in cases of time-independent [Buscarnera and Laverack, 2014, Das and Buscarnera, 2014, Marinelli and Buscarnera, 2015, Shahin et al., 2019b and time-dependent properties [Shahin et al., 2019a, Shahin andBuscarnera, 2019]. The structure of the model is inspired by classical critical state plasticity laws such as Cam Clay [Wood, 1990], in which a single yield surface evolves homothetically upon plastic deformation. ...
The Buckskin Gulch locality in Utah is a landmark example of compaction localization. The outcrop of this locality involves distinct stratigraphic heterogeneity and was exposed to complex loading history. It features multiple sets of deformation bands with different kinematics and orientation. Similar formations were seen in the Valley of Fire, Nevada, and the Orange quarry, France, among other localities. The formation of such complex structures, their propagation mechanisms, and frequency is affected by numerous local and ambient factors whose impacts are not yet fully understood. The simulation of the above‐mentioned localities is not feasible because of the limited amount of available information. This work, instead, investigates from a geomechanics standpoint how the interplay among material nonlinearity, outcrop stratigraphy, and loading history interconnects with specific spatiotemporal patterns of compaction band propagation. Our study shows that the system stratigraphy can be responsible for the emergence of coexisting compaction bands with different inclination and kinematics. Specifically, we show that stiffness contrasts induce nonlocal stress changes which may favor the initiation of secondary structures with different compaction localization characteristics. Furthermore, systems of inclined compaction bands induced by burial increase display secondary, noncontemporaneous sets of vertical compaction bands under the effects of postburial shortening. Our results indicate that stages of intermediate burial decrease prior to tectonic shortening can promote the formation of such complex systems. Despite the simplifications involved in our analyses, these findings show how geomechanics computations complement field observations and could provide a mechanics‐based validation of site‐specific reconstruction hypothesis.
... After the compaction bands propagate across the entire width of the sample, softening in the stress-strain response ensues, matching the experimental laboratory test data. This softening is followed by propagation of the zones of compaction towards the sample top and bottom edges as axial deformation increases coupled with hardening in the stress-strain response, a feature observed in other numerical modeling efforts [29,59,93] and in experimental studies [10]. The initiation of hardening indicates that compaction and grain crushing have ended, and the crushed grains have obtained their stable configurations. ...
We propose a novel continuum damage model for describing the propagation of compaction bands formed by grain crushing in porous sedimentary rock using the kernel-based approximation and discretization of the meshfree Lagrangian smoothed particle hydrodynamics (SPH) method. In the model, damage is assumed to be caused by grain crushing, and the effects of plasticity are incorporated through a critical state type model which depends on a degradation function designed to capture the abrupt onset of pore collapse following grain crushing. In doing so, we account for the two main forces understood to drive compaction band formation, brittle failure at the grain scale, and plastic dissipation. Through the smoothing length parameter, the SPH method possesses nonlocal properties intrinsic to the method, allowing for the development of a nonlocal integral form of the equivalent strain variable used to compute the damage in addition to a distinct gradient enhanced approach that when discretized using SPH is endowed with two characteristic length scales. We compare and contrast both formulations, as well as the roles of the different length scales. We evaluate our model by performing numerical experiments on Bentheim and Berea sandstone as well as on Tuffeau de Maastricht, in both notched and unnotched specimens, matching experimentally observed results such as a transition from high-angled bands to horizontal bands with increasing confining pressure, and similar compaction band styles to those visible in the field. We lastly consider the effects of material heterogeneity on samples of Tuffeau de Maastricht noting that our model produces precursory low-magnitude localization events which may develop into persistent compaction bands involving localized grain crushing, which is consistent with our understanding of compaction band formation.
... This section aims for replicating these mechanical characteristics through elasto-plastic constitutive modeling. The work is based on the constitutive functions proposed in a series of contributions by Nova and coworkers 47,48,49 , which resulted into a constitutive law able to capture phenomena typical of soft rocks and hard soils including debonding and destructuration 50,51,52,53,54 . This approach is featured by unique capabilities to reproduce versatile modes of instability propagation ( 55 ). ...
Sedimentary rocks are inherently anisotropic and prone to strain localization. While the influence of rock anisotropy on the brittle/dilative regime has been studied extensively, its influence on the ductile/compactive regime is much less explored. This paper discusses the anisotropic behavior of a high‐porosity carbonate rock from central Europe (the Maastricht Tuffeau). A set of triaxial tests with concurrent x‐ray tomography has been performed at different confining pressures. The anisotropic characteristics of this rock have been investigated by testing samples cored at different inclinations of the bedding, thus revealing non‐negligible effects of the coring direction on yielding and compaction behavior. Specifically, samples cored perpendicular to bedding display higher strength and longer stages of post‐yielding deformation before manifesting re‐hardening. Despite such alterations of the inelastic response, Digital Image Correlation has revealed that the strain localization mode is independent of the coring direction, thus being primarily affected by the confinement level. To capture the observed interaction between material anisotropy and compaction behavior at the continuum‐scale, an elastoplastic constitutive law has been proposed. For this purpose, a set of tensorial bases has been introduced to replicate how the oriented rock fabric modulates the yielding and plastic flow characteristics of the material. The analyses show that the impact of the coring direction on yield function and plastic flow rule is fundamentally different, thus requiring the use of distinct projection strategies (a strategy here defined heterotopic mapping). The performance of the model, studied through parametric analyses and by calibrating the experimental results, illustrates the improved capability of the proposed constitutive approach when applied to strongly anisotropic porous rocks.
... There are many remedies (i.e., regularization techniques) that can be used to regularize such ill-posed BVPs and obtain mesh objectivity. The most common regularization techniques proposed in the literature are the enhanced assumed strain finite element method, strong discontinuity finite element method (Borja, 2008), viscosity theory (Oka et al., 2002;Shahin et al., 2019), multiscale approach (Chen et al., 2011;Guo and Zhao, 2016), micro-polar theory (Tejchman, 2006), and nonlocal integral theory (Lu et al., 2009;Mánica et al., 2018). Among these regularization techniques, the nonlocal integral theory is a reliable technique; it is constructed using nonlocal internal variables by spatially averaging local internal variables (e.g., effective plastic strain, damage variable, etc.) over a finite volume (Huang et al., 2018b). ...
This work looks into the passive failure of tunnel head in strain-softening soils modeled using a nonlocal constitutive model. The Drucker–Prager yield criterion and plastic potential are used, and an over-nonlocal approach is utilized to regularize the ill-posed boundary value problems emerging from material softening and strain localization inelasticity. The nonlocal variable is incorporated through a Helmholtz-type partial different equation (PDE), instead of nonlocal integration. A nonlocal implicit gradient enhancement is proposed, and its numerical implementation is discussed. The proposed methodology is verified to produce mesh-independent strain-softening responses accompanied by localized deformation. The passive failure analysis of tunnel head is conducted to study key parameters influencing plastic strain accumulations and ground surface deformation. It is revealed that the strain-softening mechanism can promote local concentrated passive deformation and substantial ground surface movement near the tunnel head. Increasing the friction angle increases face supporting force, but only affects the inelastic deformation in a minimal way. The increase in the dilation angle widens the extent of horizontal soil deformation and vertical ground deformation, while a larger cover depth suppresses the passive failure response of shallow tunneling.
... Different regularization techniques have been applied to deal with this issue, including nonlocal/gradient (e.g., Chen & Schreyer, 1987;Needleman, 1988;Nguyen & Korsunsky, 2008;Pijaudier-Cabot and Bažant, 1987;Poh & Swaddiwudhipong, 2009;Vardoulakis, 1989) or viscous (Duvaut and Lions, 1972;Perzyna, 1966;Needleman, 1988;Simo et al., 1988;Wang et al., 1997;Loret and Prevost, 1990) elastoplastic models. Viscous regularization is the simplest, not expensive numerically, and widely used approach (e.g., Das et al., 2013;Carosio et al., 2000;Heeres et al., 2002;Shahin, et al., 2019;Duretz et al., 2019), which is also applied here. ...
It is known that rock fracture includes inelastic straining or damage that should localize at a certain loading stage and result in fracture initiation. The details of this process are not clear, and it is frequently omitted in the models by imposing the initial microcracks (seeds) with certain lengths and orientations. Here we investigate 2-D systems of three layers in finite-difference models. The layers subjected to the horizontal extension are separated by cohesive-frictional interfaces and have contrasted properties typical of sedimentary piles. Fractures are initiated in a more brittle central layer in the vicinity of the interfaces with the adjacent layers. It starts with the initially distributed inelastic straining, which then localizes into narrow bands. The damage within these bands is strongly accelerated, resulting in complete material failure locally. Short initial fractures corresponding to narrow bands of failed material are normal to the least local stress. They then propagate from the interfaces to the layer center with further extension. We carefully investigate the impact of different regularization procedures, the grid geometry, and structure on all stages of the fracture process and define the optimal conditions that can be applied for fracture modeling in different structural and loading configurations.
... Typically, the relatively high porosity is necessary to promote the shear-enhanced compaction (Brace, 1978), while the lower bound remains uncertain (Loaiza et al., 2012;Cilona et al., 2014) and other microstructural factors such as grain size, mineralogy and cementation are certainly functional (Holcomb et al., 2007;Wong and Baud, 2012). Numerical J o u r n a l P r e -p r o o f Journal Pre-proof controlled by the artificial weaknesses in continuum models (Das et al., 2013;Rattez et al., 2018;Shahin et al., 2019;Abdallah et al., 2020b). In the absence of weaknesses, the localization is prone to generate from the boundary constraints (Shahin and Buscarnera, 2020;Wu et al., 2020). ...
Practical attempts are made to develop a generalized framework that is capable of characterizing the diverse forms of deformation band. A distinct element model enabling grain fracture and pore collapse is presented, to quantify attributes such as porosity, grain size, mineralogy, cementation, and boundary constraint. The micro-cracking activity, grain fragmentation and energy components are tracked to inspect the tempo-spatial development of localization under contractional regimes. Typically, the porosity exerts the first-order control on the evolution of failure mode with confining pressure: the low-confinement response is always dilatant with the generation of axial split or shear band, and the high-confinement behaviors depend on porosity, where distributed cataclasis dominates at low porosity and collapsed pores often coalesce into a compaction band at high porosity. The nucleation and propagation of localization relate closely to the micro-cracking activity and energy budget, on which grain sorting, boundary constraint, mineralogy and cementation attributes have a direct impact. When the maximum grain size is not more than five times the minimum, the relative abundance of coarser grains facilitates the compaction localization, which is accompanied by a smaller magnitude in both fragmentation and energy release. The mineralogy and cementation generally affect the competition of tensile and shear cracking events on the grain interior and boundary, and have minor effect on the rupture morphology; the frictional boundary does little to the cracking rate while seriously impact the low-confinement performance. Numerical results suggest that the compaction band may originate from the intrinsic defect characterized by pore structure, and the other attributes are of secondary importance.
... where the permeability slightly reduces; initial yield occurs around 1.0% axial strain; the results are consistent with that of the previous experiments as shown in Fig. 4; (ii) Stage ii features the development of compaction bands when axial strain proceeds from 1.0 to 16.7%; the permeability curve experiences a significant drop due to the formation of compaction or shear-enhanced compaction bands; the step-like permeability reduction corresponds to a reoccurring drop of the differential stress. Similar observations were reported by Han et al. (2013) for highly porous weathered rocks and by Shahin et al. (2019) for Tuffeau de Maastricht limestone. The following two small-scale experiments were conducted in this stage (Fig. 7) where the overall porosity evolution of the small specimens is shown in Fig. 8. Considering that the carbonate structure features two clusters of pores: micro and macro (Fig. 1c), we observe in Fig Content courtesy of Springer Nature, terms of use apply. ...
Mechanical damage and resultant permeability evolution during compaction of highly porous reservoir rocks have strong implications on the extraction of mineral and energy resources. Laboratory Experiments can be performed to quantify this effect; however, the effect of size on these processes and the information they provide need to be evaluated before any conclusion can be drawn. As part of this study, conventional triaxial compression tests under different confining pressures were carried out on large samples (30 mm diameter and 60 mm length). These experiments were compared to the same setup for small samples with 12.7 mm diameter and 25.4 mm length which allowed monitoring of the pore structure changes through the use of an X-ray transparent triaxial cell at constant confining pressure. Both scales showed a similar mechanical response. The large-scale experiments were used to investigate the transition from brittle to ductile deformation, and the small-scale experiments allowed detailed investigation of the microstructural changes affecting the permeability evolution. The permeabilities of the specimens were continually measured during the triaxial loading at both scales. At defined increasing axial strain levels, the small sample was imaged using X-ray computed tomography (XRCT) and internal structural changes were mapped. A series of digital rock analysis techniques and Pore Network Modelling allowed accurate analysis of the evolution of the microstructure and its effect on permeability evolution using Pore Network Models. An XRCT-based, microstructurally enriched, continuum model successfully describes the permeability evolution measured during triaxial testing. Self-organized criticality of the propagating front of compaction was also shown by R2 values > 0.95 for a double Pareto fractal scaling law. Both approaches, as well as the macroscale experiments, confirmed a phase change in permeability at ~ 5% axial strain which provided a solid basis for microstructurally enriched assessment of the dynamic permeability.
... Field, laboratory, and numerical studies have shown that subtle differences in microstructural attributes, such as grain size distribution, can promote or inhibit the development of compaction bands in granular materials (Wang et al., 2008;Cheung et al., 2012). Compaction bands formed in laboratory deformation experiments typically have a thickness of 2-4 grains, are oriented sub-perpendicular to the maximum principal stress, initiate at one side of the sample and propagate to the other, are associated with intense grain crushing and pore collapse, and their growth is marked by an uptick in acoustic emission activity (e.g., Baud et al., 2004;Louis et al., 2006;Fortin et al., 2006;Townend et al., 2008;Charalampidou et al., 2011;Heap et al., 2015b;Baud et al., 2015;Huang et al., 2019;Shahin et al., 2019). Compaction bands have also been found in porous tuffs (Cavailhes and Rotevatn, 2018) and laboratory studies on volcanic rocks have, so far, observed compaction bands in porous basalt (Adelinet et al., 2013), porous trachyandesite (Loaiza et al., 2012), porous andesite (Heap et al., 2015a, and porous dacite . ...
Strain localisation structures, such as shear fractures and compaction bands, are of importance due to their influence on permeability and therefore outgassing, a factor thought to influence eruptive style. In this study, we aim to develop a better understanding of strain localisation in porous volcanic rocks using X-ray tomographic images of samples of porous andesite (porosity = 0.26) acquired before and after deformation in the brittle and ductile regimes. These 3D images have been first analysed to provide 3D images of the porosity structure within the undeformed andesite, which consists of a large, well-connected porosity backbone alongside many smaller pores that are either isolated or connected to the porosity backbone by thin microstructural elements (e.g., microcracks). Following deformation, porosity profiles of the samples show localised dilation (porosity increase) and compaction (porosity reduction) within the samples deformed in the brittle and ductile regimes, respectively. Digital volume correlation (DVC) of the images before and after triaxial deformation was used to quantify the tensor strain fields, and the incremental divergence (volumetric strain) and curl (used as an indicator of shear strain) of the displacement fields were calculated from the DVC. These fields show that strain localisation in the sample deformed in the brittle regime manifested as a ~ 1 mm-wide, dilatational shear fracture oriented at an angle of 40–45° to the maximum principal stress. Pre- and post-deformation permeability measurements show that permeability of the sample deformed in the brittle regime increased from 3.9 × 10⁻¹² to 4.9 × 10⁻¹² m², which is presumed to be related to the shear fracture. For the sample deformed in the ductile regime, strain localised into ~1 mm-thick, undulating compaction bands orientated sub-perpendicular to the maximum principal stress with little evidence of shear. Taken together, our data suggest that these bands formed during large stress drops seen in the mechanical data, within high-porosity zones within the sample, and within the large, well-connected porosity backbone. Pre- and post-deformation permeability measurements indicate that inelastic compaction decreased the permeability of the sample by a factor of ~3. The data of this study assist in the understanding of strain localisation in porous volcanic rocks, its influence on permeability (and therefore volcanic outgassing), and highlight an important role for DVC in studying strain localisation in volcanic materials.
... One exception is a recent experimental study on Tuffeau de Maastricht which indeed reported the transition from CB to SB in a specimen subjected to triaxial compression of axial strain up to 50% (Papazoglou, 2018). Shahin et al. (2019) has offered a numerical prediction of the evolution of compaction zones and the global responses up to the homogeneous re-hardening stage based on FEM simulations with an elastoplastic constitutive law. It remains tremendously challenging, both numerically and experimentally, to capture and characterize the transition of localization patterns in porous rocks to understand the underlying mechanisms. ...
We employ a hierarchical multiscale modeling approach to investigate the transitions of localized deformation patterns in high-porosity sandstone subjected to sustained shear to understand their underlying physics. The multiscale approach is based on hierarchical coupling between finite element method (FEM) with discrete element method (DEM) to offer cross-scale predictions for granular rocks without assuming phenomenological constitutive relations. Our simulations show that when a high-porosity sandstone specimen is subjected to continuous deviatoric loading, compaction bands may occur and evolve, featuring a steady movement of the compaction front (i.e. the boundary between the compaction band and the rest uncompacted zone). The specimen reaches a homogeneous state of reduced porosity when the compaction fronts traverse the entire specimen. A re-hardening response is initiated in the specimen under further shear, which is followed by a shearing dominating stage with the emergence of shear bands. The material responses inside the ultimate shear bands approach a “steady state” of constant porosity and stress ratio. Cross-scale analyses reveal that debonding and pore collapse are dominant mechanisms for the compaction stage of the specimen, and debonding and particle rotation dictate the physics for the shear banding stage. The transitions from compaction to shear banding occurs due to the degradation of the cohesive contact network and significant reduction in porosity. There are limited number of interparticle bonds remaining at the “steady state” under sustained shear, with a preferential direction perpendicular to the loading direction, leading to a higher steady void ratio than the critical state void ratio of non-cohesive sand.
... In addition, the numerical and theoretical models are also employed to investigate the deformation localization of rocks. [18][19][20][21][22][23] Previous studies indicate that deformation localization plays an important role in understanding the cracking process from microscales to macroscales, contributing to the precursor development of macrofailure. ...
We present deformation localization and cracking process of sandstone with two flaws of different geometric configurations in uniaxial compression. The full field strain and progressive cracking processes are quantitatively monitored using three‐dimensional digital image correlation and acoustic emission techniques. The results show that peak strength and elastic modulus show a first decrease and then increase trend with regard to ligament angle under the same flaw geometry, achieving the minimum at the ligament angle 60°. The tensile strain develops at low stress level and then increases significantly. However, the shear strain only tends to be obvious while approaching peak stress. With the ligament angle increasing, the crack coalescence mode transfers from indirect to direct coalescence. In addition, the complete cracking processes of sandstone can be divided into six stages, together with six crack coalescence modes.
... Microscale information linking underlying mechanisms to compaction bands that cannot be directly measured in experiments, such as the particle-scale failure process, the evolution of interparticle contact forces and contact network, is further extracted from the multiscale simulation to provide a fresh, multiscale perspective to advance the understanding of compaction bands. Notably, an elastoplastic constitutive model has been developed and calibrated based on the same set of experimental data before being implemented into FEM to capture the pressure-dependent inclination of deformation bands and their evolution with increased axial strain [47]. ...
We present a study on compaction band in a high-porosity limestone (Tuffeau de Maastricht) based on comparison and analysis of X-ray tomography observations and computational multiscale simulations. We employ a hierarchical multiscale approach coupling the finite element method (FEM) with the discrete element method (DEM) to simulate the formation of compaction bands in Tuffeau de Maastricht. A high-porosity RVE is prepared according to X-ray tomography observations of material microstructure, and its grain-scale parameters are calibrated by data from laboratory isotropic compression and triaxial compression tests. Triaxial compression tests are simulated by FEM as a boundary value problem to observe compaction bands. The generated RVE is embedded into each integration point of the FE mesh, receiving displacement gradient as DEM boundary conditions, and is solved accordingly to produce the required mechanical responses for FEM computation without assuming phenomenological constitutive relations. The simulated global mechanical responses of the triaxial tests are found to show qualitative agreement with the experimental data. The evolution of compaction band patterns in the simulation matches remarkably well with the experimental observations in terms of fields of porosity and incremental strains. Both show two compaction fronts initiating from the two ends of the specimen and propagating toward the middle. By virtue of the multiscale approach, useful microstructural information is further extracted from the simulations to offer cross-scale insights into compaction bands. The study confirms that significant debonding accompanied by collapse of macropores and grain rearrangements is the major microstructural mechanisms causing the formation of compaction bands in high-porosity Tuffeau de Maastricht.
Compaction bands are localized failure patterns that appear in highly porous rock material under the effect of relatively high confining pressure. Being affected mainly by volumetric compression, these bands appear to be almost perpendicular to the most compressive principal stress of a stress state at the so-called "cap" of the yield surface. In this study, we focus on the mechanism that leads to the onset of compaction bands by using a viscoplasticity model able to describe the post-localization response of these materials. The proposed constitutive framework is based on the overstress theory of Perzyna and the anisotropic clay plasticity model of Dafalias which provides not only the necessary "cap" of the yield surface, but introduces a rotational hardening mechanism, thus, accounting for the effect of fabric anisotropy. Following the analysis of Veveakis and Regenauer-Lieb we identify the compaction bands as "static" cnoidal wave formations in the medium that occur at a post-yield regime and we study the effect of rotational and isotropic hardening on their onset. Moreover, we determine a theoretical range of confining pressures in triaxial compression tests for the compaction bands to develop. Under the assumption of coaxiality between stress and anisotropy tensors, the results show that the isotropic hardening promotes compaction localization, whereas the rotational hardening has a slightly negative effect on the onset of compaction localization.
Deformation experiments on porous rocks conducted at sufficiently high confining pressure may result into localized compaction. While strain localization is expected to be triggered by material heterogeneities, evidence of compaction zones propagating from the specimen boundaries is often observed, thus indicating that sample-platen friction plays a relevant role on the compaction banding patterns. This paper aims to examine this hypothesis and quantify the relative role played by field heterogeneities and boundary effects by using a numerical replica of specimens for which compaction localization was tracked through X-ray micro-tomography. For this purpose, finite element simulations are performed, where spatially-distributed heterogeneities of the porosity field are replicated through a computerized-tomography to finite-element (CT-FE) mapping scheme. The concept of representative elementary volume is used to extract material state variables directly from the CT scans, eventually associating each integration point to a corresponding volume in the digital image. Full-field numerical analyses conducted on the virtual replicas show that frictionless specimens promote the onset of compaction localization in zones characterized by higher porosity (i.e., lower strength). By contrast, the presence of platen-specimen friction nearly eliminates the impact of material heterogeneities, such that compaction localization is forced to occur at the specimen boundaries. Furthermore, the analyses reveals the existence of an intermediate range of platen-specimen friction coefficient which favors the emergence of a transitional regime of strain localization at which material heterogeneity and boundary effects concurrently control the compaction banding patterns. Numerical simulations based on quasi-synthetic spatial fields have also revealed a marked dependence of the three zones on the degree of heterogeneity, showing that increasing values of boundary friction are necessary to offset the effect of material weaknesses in strongly heterogeneous samples.
Recent laboratory evidence shows that compaction creep in porous rocks may develop through stages of acceleration, especially if the material is susceptible to strain localization. This paper provides a mechanical interpretation of compaction creep based on viscoplasticity and nonlinear dynamics. For this purpose, a constitutive operator describing the evolution of compaction creep is defined to evaluate the spontaneous accumulation of pore collapse within an active compaction band. This strategy enables the determination of eigenvalues associated with the stability of the response which is able to differentiate decelerating from accelerating strain. This mathematical formalism was linked to a constitutive law able to simulate compaction localization. Material point simulations were then used to identify the region of the stress space where unstable compaction creep is expected, showing that accelerating strains correspond to pulses of inelastic strain rate. Such pulses were also found in full‐field numerical analyses of delayed compaction, revealing that they correspond to stages of inception and propagation of new bands across the volume of the simulated sample. These results illustrate the intimate relation between the spatial patterns of compaction and their temporal dynamics, showing that while homogeneous compaction develops with decaying rates of accumulation, localized compaction occurs through stages of accelerating deformation caused by the loss of strength taking place during the formation of a band. In addition, they provide a predictive modeling framework to simulate and explain the spatio‐temporal dynamics of compaction in porous sedimentary formations.
In this work, we study compaction localization in porous rocks under creep conditions. We show through full-field numerical simulations that: -------
1- Spatial heterogeneity allows rock specimens to display compaction localization if subjected to creep stresses below the nominal yielding stress (in homogeneous specimens).
------
2- The stronger the intensity of heterogeneity, the greater the rate of creep deformation.
---------
3- The greater the creep-traction the greater the rate of deformation.
Recent laboratory evidence shows that compaction creep in porous rocks may develop through stages of acceleration, especially if the material is susceptible to strain localization. This paper provides a mechanical interpretation of compaction creep based on viscoplasticity and nonlinear dynamics. For this purpose, a constitutive operator describing the evolution of compaction creep is defined to evaluate the spontaneous accumulation of pore collapse within an active compaction band. This strategy enables the determination of eigenvalues associated with the stability of the response which is able to differentiate decelerating from accelerating strain. This mathematical formalism was linked to a constitutive law able to simulate compaction localization. Material point simulations were then used to identify the region of the stress space where unstable compaction creep is expected, showing that accelerating strains correspond to pulses of inelastic strain rate. Such pulses were also found in full‐field numerical analyses of delayed compaction, revealing that they correspond to stages of inception and propagation of new bands across the volume of the simulated sample. These results illustrate the intimate relation between the spatial patterns of compaction and their temporal dynamics, showing that while homogeneous compaction develops with decaying rates of accumulation, localized compaction occurs through stages of accelerating deformation caused by the loss of strength taking place during the formation of a band. In addition, they provide a predictive modeling framework to simulate and explain the spatio‐temporal dynamics of compaction in porous sedimentary formations.
Given their high porosity, carbonates form important water and hydrocarbons reservoirs, and they are also suitable for other applications such as CO2 storage and nuclear waste disposal. However, localized compaction in carbonates affects the stress field and the hydromechanical properties of these rocks leading to inelastic deformation and failure with potential economic, environmental and social impacts. Previous field and experimental studies have shown that in porous carbonates, unlike sandstones, a variety of micromechanisms such as pore collapse, grain crushing, debonding, crystal plasticity and pressure solution can potentially lead to inelastic compaction. Due to the coexistence of such multiple inelastic processes and the interplay among them, the dominant micromechanism responsible for failure remains poorly understood.This doctoral thesis presents an experimental investigation into the deformation mechanisms governing the mechanical behavior and failure mode of high porosity carbonate rocks. To this end, Tuffeau de Maastricht, a bioclastic sedimentary limestone exhibiting up to 52% porosity, has been tested under dry conditions. This study focuses on how stress path, confining pressure and bedding orientation affect the onset and propagation of localized compaction. Three main experimental campaigns are conducted on cylindrical specimens of 11.5 mm diameter and 22 mm height to study the brittle-ductile transition: (i) isotropic compression, (ii) uniaxial compression, and (iii) triaxial compression tests at confining pressures ranging from 1 to 5 MPa. A systematic analysis of the anisotropic behavior of Tuffeau de Maastricht is conducted on samples cored perpendicular, oblique 45° and parallel to the bedding plane. High resolution x-ray computed tomography (CT) is used to obtain 3D images of the entire specimen under loading. The acquired images are processed and full-field measurements have been used to elucidate the mechanics of initiation and propagation of localized compaction. Porosity variations during loading are measured macroscopically as well as locally. The porosity measurements are performed over a REV, which is defined with the use of statistical tools. The systematic use of x-ray micro tomography combined with the use of advanced image analysis and Digital Image Correlation (DIC) provides a quantitative 3D information the strain field inside a sample and its evolution during a test.Two failure modes are identified, based on porosity measurements and DIC: compactive shear bands at low confining pressure, and compaction bands (perpendicular to the maximum compressive stress) at higher confinement. These bands develop at essentially constant deviator stress and propagate through the whole sample punctuated by episodic stress drops. Triaxial compression tests at much higher axial strain present three distinct phases: (1) an initial quasi-linear increase of deviator stress, followed by (2) a plateau and (3) a post-plateau hardening. The essential observation from these experiments if the occurrence of a debonding phase which converts the specimen from rock-like to sand-like. A second localization, typical of dense sand, eventually occurs for very axial strain. Additional experiments that are performed on artificially debonded specimens emphasize this destructuration phase during the plateau of deviator stress. The experimental results also highlight the strong anisotropy of the mechanical behavior of the studied material.
TomoWarp2 is a Python based code that allows full-field vector displacements to be measured between 2D or 3D image sets, based on a local approach of Digital Image Correlation. These displacements can be used to calculate the complete 2D or 3D strain tensor field to study, for example, heterogeneous deformation responses of materials during loading. The code is provided with a GUI, and can handle different input and output formats. Moreover, new features can be implemented with the flexible code architecture.
This paper presents an experimental and constitutive study of compaction banding in Tuffeau de Maastricht, a bioclastic sedimentary limestone exhibiting up to 52% of porosity. An elasto-plastic constitutive model is used to simulate the mechanical behavior of the limestone within the compaction localization regime. It is shown that the simulated macroscopic behavior is in good agreement with the experimental data. In addition, image processing tools have been used to perform full-field measurements elucidating the mechanics of initiation and propagation of localized compaction zones. These findings emphasize the complex nature of localized compaction in porous rocks and represent a preliminary step towards the integrated use of multi-scale testing and mechanical modeling for their characterization across scales.
In this work, the consequences of using several different Discrete Element granular assemblies for the representation of the microscale structure, in the framework of multiscale modelling, have been investigated. The adopted modelling approach couples, through computational homogenization, a macroscale continuum with microscale discrete simulations. Several granular assemblies were used depending on the location in the macroscale Finite Element mesh. The different assemblies were prepared independently as being representative of the same material but their geometrical differences imply slight differences in their response to mechanical loading. The role played by the micro-assemblies, with weaker macroscopic mechanical properties, on the initiation of strain localization in biaxial compression tests is demonstrated and illustrated by numerical modelling of different macroscale configurations. This article is protected by copyright. All rights reserved.
This paper presents a summary of recent work on cemented soils at Milan University of Technology (Politecnico). Oedometric and triaxial tests have been performed on lightly bonded soils of medium to very high porosity. Soils tested vary from a rather conventional silica sand-lime mixture to more unusual materials, including expanded clay aggregates, fragmented marine shells or stabilized metallurgical residues. A simple but powerful elasto-plastic bonded soil model is employed to select testing procedures and interpret the results obtained. Both experimental results and model simulations are employed here to illustrate and explore the onset of compaction bands, a new form of localization previously observed in rocks but whose appearance is here first signaled for bonded soils.
This doctoral thesis presents an experimental investigation into the mechanics of granular media.The novelty that this work brings is that the specimens of sand tested in this work are systematicallyand non-destructively imaged using x-ray tomography. Sample size is considerably reducedfrom standard (specimens measure approximately 22 mm height by 11 mm diameter), allowingentire specimens to be scanned at a sufficiently high resolution to identify all the grains (morethan fifty thousand) in each specimen.A campaign of triaxial compression tests has been run on a series of three different naturalsands with different grain shapes (Hostun sand, Ottawa sand and Caicos ooids – all prepared atrelatively dense initial states), and tested at 100 or 300 kPa cell pressure. In each test around 15x-ray scans are performed. In the 3D images resulting from the reconstruction of the x-ray scansperformed, grains are identified each state using a standard watershed algorithm. Starting fromthese discretised data, techniques are developed in order characterise grain-to-grain contacts,as well as to measure the kinematics of all the identified grains between imaged states. Grainkinematics are measured with two specifically-developed tools: “ID-Track” to track grains yieldingtheir displacements, and a discrete image correlation technique to measure grain rotations.Grain-scale measurements are reported in detail for one test, and are then compared to testsin different conditions, in order to highlight the micro-mechanisms responsible for the observedmacroscopic behaviour. This comparison highlights some important micro-scale mechanisms suchas the increasing rotational frustration of more angular grains when the sample’s deformation isconcentrated in a fully developed shear band; this is used to explain to some extent the highervalue of their residual stress for these materials. Signs of localised deformation are seen to occurwell before the peak in many samples, and complex patterns of rotating grains (which match alocal, grain-based measurement of strain) are noticed around the peak of each sample’s response.
This work presents results from a series of triaxial compression tests on two quartz sands (differing principally in grain shape), at confining pressures high enough to cause grain breakage during shearing. Tests are performed inside an X-ray scanner, which allows specimens to be imaged non-destructively as they deform. Observation of the acquired images clearly shows different mechanisms of deformation, including shearing, dilation, compaction and grain breakage. These mechanisms are investigated quantitatively through 3D measurements of local porosity, as well as strain (obtained by 3D Digital image correlation), which is analysed in terms of volumetric and shear components. These tools allow the transition between macroscopically dilative (typically of a dense sand at low mean stress) and compactive behaviour to be investigated. The analysis reveals that at the high end of the confining pressure range studied (100–7,000 kPa), the more rounded sand deforms with highly localised shear and volumetric strain—the porosity fields show a dilative band within which a compactive region (due to grain crushing) grows. The more angular material shows shear strain localisation; however, its faster transition to compactive behaviour (due to a higher propensity for individual grains to crush) translates to much more distributed compactive volumetric strain.
Grain crushing and pore collapse are the principal micromechanisms controlling the physics of compaction bands in porous rocks. Several constitutive models have been previously used to predict the formation and propagation of these bands. However, they do not account directly for the physical processes of grain crushing and pore collapse. The parameters of these previous models were mostly tuned to match the predictions of compaction localization; this was usually done without validating whether the assigned parameters agree with the full constitutive behavior of the material. In this study a micromechanics-based constitutive model capable of tracking the evolving grain size distribution due to grain crushing is formulated and used for a theoretical analysis of compaction band formation in porous rocks. Linkage of the internal variables to grain crushing enables us to capture both the material behavior and the evolving grain size distribution. On this basis, we show that the model correctly predicts the formation and orientation of compaction bands experimentally observed in typical high-porosity sandstones. Furthermore, the connections between the internal variables and their underlying micromechanisms allow us to illustrate the significance of the grain size distribution and pore collapse on the formation of compaction bands.
Conditions for shear and compaction localization are examined for stress
states on an elliptic yield cap in the space of Mises equivalent shear
stress ? and mean normal compressive stress σ. Localization is
predicted to occur when the slope of the hydrostatic stress versus
inelastic volume strain curve falls to a critical value
kcrit. When applied to the standard triaxial test, the
analysis reveals a transition from compaction localization to shear
localization as lateral confining stress σc decreases
below a particular value. This value also corresponds to the largest
value of kcrit, which is zero if normality is satisfied and
slightly positive if not, for either shear or compaction bands. For
σc larger than the transition value, compaction bands
are the only mode of localized deformation predicted, and
kcrit becomes increasingly negative with increasing
σc. For σc smaller than the transition
value, both shear bands and compaction bands are possible, but the value
of kcrit is larger for shear bands. These results are
consistent with experimental observations of compaction bands on
relatively flat portions of the stress versus strain curve,
corresponding to kcrit ≈ 0, and with their occurrence in a
limited range of σc. As σc is
decreased from the transition value, the predicted angle between the
normal to the shear band and the maximum compression direction increases
rapidly from 0° (for a compaction band) to 20°-30°. This
rapid increase provides a possible explanation for the infrequent
observation of very low angle shear bands.
The failure of rocks in the brittle regime is generally associated with the appearance of strain localization bands. For very porous rocks, three types of strain localization can be distinguished: extension bands, shear bands, and compaction bands. The first is associated with an extensional normal strain concentration inside the band; the second, with a shear strain concentration: and the third, with a compressive normal strain concentration. This paper shows the continuous transition between pure extension bands and pure compaction bands, via shear bands that evolve from dilating shear bands to compacting shear bands. By an extension to the analysis of Rudnicki and Rice [1975] (RR) on strain localization in pressure sensitive rocks, the prediction of the strain type inside bands at the onset of localization shows that inside shear bands, the shear strain can be associated with a volumetric dilatancy or compaction depending on the constitutive parameters of the material. The theoretical determination of the strain type is in accordance with recent observations of dilating and compacting shear bands in laboratory tests on porous sandstone specimens. A limit for the existence of a localized reduction of porosity within the band is expressed. A physical limit to the RR model is also proposed to insure continuity of the strain mechanism of localization with respect to the constitutive parameters.
Reexamination of the results of Rudnicki and Rice for shear localization
reveals that solutions for compaction bands are possible in a range of
parameters typical of porous rock. Compaction bands are narrow planar
zones of localized compressive deformation perpendicular to the maximum
compressive stress, which have been observed in high-porosity rocks in
the laboratory and field. Solutions for compaction bands, as an
alternative to homogenous deformation, are possible when the inelastic
volume deformation is compactive and is associated with stress states on
a yield surface "cap." The cap implies that the shear stress required
for further inelastic deformation decreases with increasing compressive
mean stress. While the expressions for the critical hardening modulus
for compaction and shear bands differ, in both cases, deviations from
normality promote band formation. Inelastic compaction deformation
associated with mean stress (suggested by Aydin and Johnson) promotes
localization by decreasing the magnitude of the critical hardening
modulus. Axisymmetric compression is the most favorable deviatoric
stress state for formation of compaction bands. Predictions for
compaction bands suggest that they could form on the "shelf" typically
observed in axisymmetric compression stress strain curves of porous rock
at high confining stress. Either shear or compaction bands may occur
depending on the stress path and confining stress. If the increase in
local density and decrease in grain size associated with compaction band
formation result in strengthening rather than weakening of the band
material, formation of a compaction band may not preclude later
formation of a shear band.
The objective of this paper is to describe the subsidence-depletion delay from field observations with the aim to explain its cause and constrain its modelling. The nonlinearity in the surface subsidence response to the reservoir depletion can be seen as a shift between the start of the depletion and the (delayed) start of the subsidence. We have analysed data of eight hydrocarbon fields to quantify the subsidence-depletion delay effect and the corresponding time-delay. The time-delays appeared to be in the range of 1.6 to 13 years. The fields were categorised according to their depth, age and rock type. To explain the field observations, the data was tested against four categories of mechanisms: pore pressure diffusion effects; overburden inertia; reservoir compaction behaviour; deformation of the over- under- and side-burden. The relative importance of these mechanisms was assessed from an analysis of the field data. Some mechanisms could be rejected based on theoretical grounds and others appeared to be irrelevant. It is concluded that pore pressure diffusion always contributes to a delay effect but that its effect is too low to explain the field observations. For the shallow reservoirs, natural over-compaction has the potential to cause the observed subsidence-depletion delay. In general, the other reservoir compaction related mechanisms (such as creep, an intrinsic rate effect and an elastic-plastic transition) could also be the main cause of the subsidence-depletion delay effect.
Results of axisymmetric compression tests on weak, porous Castlegate Sandstone (Cretaceous, Utah, USA), covering a range of dilational and compactional behaviors, are examined for localization behavior. Assuming isotropy, bulk and shear moduli evolve as increasing functions of mean stress and Mises equivalent shear stress respectively, and as decreasing functions of work-conjugate plastic strains. Acoustic emissions events located during testing show onset of localization and permit calculation of observed shear and low-angle compaction localization zones, or bands, as localization commences. Total strain measured experimentally partitions into: A) elastic strain with constant moduli, B) elastic strain due to stress dependence of moduli, C) elastic strain due to moduli degradation with increasing plastic strain, and D) plastic strain. The third term is the elastic-plastic coupling strain, and though often ignored, contributes significantly to pre-failure total strain for brittle and transitional tests. Constitutive parameters and localization predictions derived from experiments are compared to theoretical predictions. In the brittle regime, predictions of band angles (angle between band normal and maximum compression) demonstrate good agreement with observed shear band angles. Compaction localization was observed in the transitional regime in between shear localization and spatially pervasive compaction, over a small range of mean stresses. In contrast with predictions, detailed acoustic emissions analyses in this regime show low angle, compaction-dominated but shear-enhanced, localization.
Particles exist in great abundance in nature, such as in sands and clays, and they also constitute 75% of the materials used in industry (e.g., mineral ores, formulated pharmaceuticals, dyes, detergent powders). When a load is applied to a bulk assembly of soil particles, the response of a geomaterial at the bulk (macro) scale, originates from the changes that take place at the particle scale. If particle breakage occurs, the shape and size of the particles comprising the bulk are changed; this induces changes in the contact network through which applied loads are transmitted. As a result, changes at the micro-scale can significantly affect the mechanical behaviour of a geomaterial at a macro-scale. It is therefore unsurprising that the mechanisms leading to particle breakage are a subject of intense research interest in several fields, including geomechanics. In this thesis, particle breakage of two dry granular materials is studied, both experimentally and numerically. The response of the materials is investigated under different stress paths and in all the tests grain breakage occurs. High resolution x-ray computed micro-tomography (XCT) is used to obtain 3D images of entire specimens during high confinement triaxial compression tests and strain controlled oedometric compression tests. The acquired images are processed and measurements are made of the temporal and spatial evolution of breakage, local variations of porosity, volumetric and shear strain and grading. The evolution and spatial distribution of quantified breakage including the resulting particle size distribution for the whole specimen and for specific areas, are presented and further related to the localised shear and volumetric strains that developed in the specimens. In addition, the discrete element method (DEM) was used to provide further micro-mechanical insight of the underlying mechanisms leading to particle breakage. Classical DEM simulations, using a Hertz-Mindlin contact model and non-breakable spheres, was first deployed to study the initiation and likelihood of particle breakage under oedometric compression. Moreover, a bonded DEM model was used to create clumps that represent each particle and simulate breakage of particles under single particle compression. The DEM model parameters were obtained from results of single particle compression test and the models were validated against the quantitative 3D information of the micro-scale, acquired from the XCT analysis.
A three-dimensional nonlocal multiscale discrete-continuum model has been developed for modeling mechanical behavior of granular materials. In the proposed multiscale scheme, we establish an information-passing coupling between the discrete element method (DEM), which explicitly replicates granular motion of individual particles, and a finite element continuum model, which captures nonlocal overall response of the granular assemblies. The resulting multiscale discrete-continuum coupling method retains the simplicity and efficiency of a continuum-based finite element model while circumventing mesh pathology in the post-bifurcation regime by means of staggered nonlocal operator. We demonstrate that the multiscale coupling scheme is able to capture the plastic dilatancy and pressure-sensitive frictional responses commonly observed inside dilatant shear bands, without employing a phenomenological plasticity model at a macroscopic level. In addition, internal variables, such as plastic dilatancy and plastic flow direction, are now inferred directly from granular physics, without introducing unnecessary empirical relations and phenomenology. The simple shear and the biaxial compression tests are used to analyze the onset and evolution of shear bands in granular materials and sensitivity to mesh density. The robustness and accuracy of the proposed multiscale model are verified in comparisons with single-scale benchmark DEM simulations.
KEY WORDS: multiscale discrete-continuum model; staggered nonlocal operator; strain localization; granular materials; homogenization; shear band; anisotropy
This paper discusses the parameter calibration procedure for an elastoplastic constitutive model for high-porosity rocks. The model selected for the study is formulated in the frame of the critical state theory, which is here used in a form able to accommodate non-associated plastic flow and softening effects due to volumetric and deviatoric plastic strains. The goal of this study is to generate a set of model constants able to capture both the stress-strain response and the compaction localization characteristics (e.g., stress and inclination at the onset of the deformation bands). For this purpose, data about the compaction localization properties of four extensively characterized sandstones have been considered. In particular, the strain localization theory has been used as a calibration tool, using explicitly information about the pressure-dependence of the localization mechanisms observed in experiments. The model constants have been defined by matching the constitutive response upon hydrostatic compression, as well as the stresses at the transition from high-angle shear bands to pure compaction bands, and from compaction bands to homogeneous cataclastic flow. It is shown that such procedure generates a set of model constants able to capture satisfactorily both the rheological response upon triaxial compression and the salient features of the compaction localization process.
This paper discusses strain localisation in granular media by presenting experimental, full-field analysis of mechanical tests on sand, both at a continuum level, as well as at the grain scale. At the continuum level, the development of structures of localised strain can be studied. Even at this scale, the characteristic size of the phenomena observed is in the order of a few grains. In the second part of this paper, therefore, the development of shear bands within specimen of different sands is studied at the level of the individual grains, measuring grains kinematics with x-ray tomography. The link between grain angularity and grain rotation within shear bands is shown, allowing a grain-scale explanation of the difference in macroscopic residual stresses for materials with different grain shapes. Finally, rarely described precursors of localisation, emerging well before the stress peak are observed and commented.
The bifurcation problem by shear band localization of an initially homogeneous sample, is studied. Continuous bifurcation, as well as discontinuous bifurcation (in the Rice sense) are both studied. Constitutive equations used are piecewise incrementally linear ones, for which the state is smoothly varying. A fundamental theorem is established. This theorem gives us a general method which involves the solution of a finite number of equations. These equations are not more difficult to solve than those obtained from the continuous bifurcation classical problem. The general method applied in particular cases gives us through another route some previous classical results.
This paper presents a study of localized deformation processes in sand with grainscale resolution. Our approach combines state-of-the-art x-ray micro tomography imaging with 3D Volumetric Digital Image Correlation (3D V-DIC) techniques. While x-ray imaging and DIC have in the past been applied individually to study sand deformation, the combination of these two methods to study the kinematics of shear band formation at the scale of the grains is the first novel aspect of this work. Moreover, we have developed an original grain-scale V-DIC method that enables the characterization of the full kinematics (i.e. 3D displacements and rotations) of all the individual sand grains in a specimen. We present results obtained with both "continuum" and "discrete" DIC on Hostun sand, and a few preliminary results (continuum DIC only) recently obtained on ooid materials, which are characterized by spheroidal, layered grains.
[1] The paper presents a theoretical investigation of compaction banding based on a plasticity model for high-porosity rocks. The selected model is featured by a non-associated flow rule and two internal variables simulating the competition between softening and hardening in the brittle-ductile transition. The model is calibrated for two extensively studied rocks that have exhibited localized compaction under laboratory conditions. In particular, the model constants that control the compaction banding domain are defined by matching the stresses at which localized compaction was found in the experiments. The resulting parameters are used to simulate the stress-strain response for two loading modes (i.e., triaxial compression and radially-constrained deformation), thus exploring the role of stress paths and kinematic constraints on the evolution of the compaction banding potential. The analyses suggest that the loading paths able to mobilize the plastic resources of the rock can alter the potential for compaction banding through irreversible effects. A notable example is one-dimensional compression, for which the potential to generate compaction bands tends to vanish during the initial stages of inelastic loading. On the one hand, these predictions suggest that constraints to the mode of deformation can hinder the occurrence of compaction bands in the field. On the other hand, they suggest that the interplay between kinematic constraints and evolution of the compaction banding potential can be used for experimental characterization purposes. As a consequence, these results emphasize the importance of complementing stress-controlled experiments with other tests able to explore different stress paths, thus providing additional data for an accurate characterization of rheological properties and strain-localization characteristics.
The Mount Simon Sandstone (Mt. Simon), a basal Cambrian sandstone underlying much of Midwestern US, is a target for underground CO2 storage and waste injection which requires an assessment of geomechanical behavior. The range of depositional environments yields a heterogeneous formation with varying porosity, permeability, and mechanical properties. Experimental deformational behavior of three distinct Mt. Simon lithofacies was examined via axisymmetric compressional testing of core samples. Initial yielding was confirmed with acoustic emissions in many tests and failure envelopes were determined for each lithofacies. Evolution of elastic moduli with stress and plastic strain was determined by use of unload–reload cycles, which permit separation of total measured strains into elastic and plastic strains. The Upper Mt. Simon lithofacies yields at higher shear stresses compared to two “Lower” lithofacies, with little modulus degradation with plastic strain. Lower Mt. Simon lithofacies are weaker and deform plastically with modulus degradation. This range in constitutive response is quantified with an elasto-plasticity model. Based on these results, Mount Simon Sandstone would likely deform elastically during CO2 injection and storage, with large pore pressure increases (∼8–9 MPa above hydrostatic) predicted to initiate plastic yielding. Nonetheless, near-wellbore damage could result in weaker lithofacies during injection and/or brine extraction.
We analyze the propagation of compaction bands in high porosity sandstones using a constitutive model based on breakage mechanics theory. This analysis follows the work by Das et al. [2011] on the initiation of compaction bands employing the same theory. In both studies, the theory exploits the links between the stresses and strains, and the micromechanics of grain crushing and pore collapse, giving the derived constitutive models advantages over previous models. In the current post localization analysis, the bifurcation instability of the continuum model is suppressed by the use of a rate-dependent regularization. This allows us to perform a series of finite element analyses of drained triaxial tests on porous sandstone specimens. The obtained numerical results compare well with experimental counterparts, in terms of both the initiation and propagation of compaction bands, besides the macroscopic stress-strain responses. On this basis, a parametric study is carried out to explore the effects of loading rate, degree of structural imperfections, and confining pressure on the propagation of compaction bands.
Many of the earliest laboratory studies of the brittle-ductile
transition were on porous rocks, with a focus on the evolution of
failure mode from brittle faulting to cataclastic flow with increasing
pressure. Recent advances in this area are reviewed. Porosity has been
demonstrated to exert critical control on the brittle-ductile
transition, and its phenomenology has two common attributes. Under low
confinement, brittle faulting develops as a dilatant failure mode. Under
high confinement, delocalized cataclasis is accompanied by
shear-enhanced compaction and strain hardening. Plasticity models such
as the cap and critical state models have been developed to describe
such constitutive behaviors, and many aspects of the laboratory data on
porous rock have been shown to be in basic agreement. Bifurcation
analysis can be used in conjunction with a constitutive model to predict
the onset of strain localization, which is in qualitative agreement with
the laboratory data. However, recent studies have also underscored
certain complexities in the inelastic behavior and failure mode. In some
porous sandstones, compaction bands would develop as a localized failure
mode intermediate between the end members of brittle faulting and
cataclastic flow. In limestones (and selected sandstones) under
relatively high confinement, cataclastic flow is accompanied first by
shear-enhanced compaction which then evolves to dilatancy. Various
techniques have been employed to characterize the microstructure and
damage, which have elucidated the deformation mechanisms associated with
the brittle-ductile transition. These observations have revealed a
diversity of micromechanical processes, and fundamental differences were
observed especially between sandstone and limestone with regard to
inelastic compaction. Micromechanical models that have been formulated
to describe these processes include the pore-emanated and sliding wing
crack models in the brittle faulting regime, and the Hertzian fracture
and cataclastic pore collapse models in the cataclastic flow regime.
Numerical techniques based on the discrete element method have also been
employed to simulate these processes. Comparison of the model
predictions with laboratory and microstructural observations has
provided useful insights into the mechanics of brittle-ductile
transition in porous rock.
Bifurcation theory is often used to investigate the inception of a shear band in a homogeneously deforming body. The theory predicts conjugate shear bands that have the same likelihood of triggering. For structures loaded symmetrically the choice of which of the two conjugate shear bands will persist is arbitrary. In this paper we show that spatial density variation could be a determining factor for the selection of the persistent shear band in a symmetrically loaded localizing sand body. We combine experimental imaging on rectangular sand specimens loaded in plane strain compression with mesoscale finite element modeling on symmetrically loaded sand specimens to show that spatial heterogeneity in density does have a profound impact on the persistent shear band.
Field investigators have recently discovered thin, tabular zones of pure compressional deformation that they called compaction bands. These bands were found in association with shear bands and were postulated to be genetically related to them. At the laboratory scale, compaction bands have been noticed in association with boreholes and preexisting, artificial shear cracks subjected to compressive stress fields. Natural compaction bands are noticeable in outcrops because of their resistance to weathering; however, they may be more difficult to discern on freshly cut rock surfaces such as drill core or borehole walls. Because of the much reduced porosity in the compaction bands, these structures are potentially important as permeability barriers in reservoirs and aquifers in porous rocks. For bands associated with boreholes, the crushed material can be washed into the borehole contributing to sand production and possibly altering the stability of the borehole. This paper examines a theoretical framework that explains these features as a constitutive instability leading to localized compaction in a way completely analogous to shear strain localization. Conventional triaxial experiments on Castlegate sandstone resulted in compaction bands. In addition, thick deformation bands having normals at low angles to the maximum compression are present in some specimens.
The localized deformation is a ubiquitous phenomenon in geomaterials. It occurs over a vast range of size scales, from the microscale level of grains to faults extending over hundreds of kilometers. It occurs in a variety of forms: a concentration or coalescence of cracks; a distinct, planar frictional surface; a gouge zone of finely comminuted material; or simply a region of higher shear strain. In geomaterials, the severe shearing in regions of localized deformation may be accompanied by dilatancy (inelastic volume increase) or compaction (inelastic volume decrease) and by chemical alteration. Localization can even occur purely by compaction without any evident shear. If the material is fluid-saturated, as is frequently the case, inelastic volume changes can induce the flow of fluid or changes in pore pressure that affect the response. Localization occurs under a variety of conditions. Although most often associated with the formation of shear bands or faults under nominally brittle conditions, localization can also occur by cataclastic flow of rocks at higher mean stresses and by ductile shearing at temperatures and pressures typical of depths of 10 km to 15 km in the earth's crust.
Thin, tabular, low-porosity, low-permeability compaction bands form pervasive, subparallel, anastomosing arrays that extend over square kilometers of exposure in the Aztec Sandstone of southeastern Nevada, an exhumed analog for active aquifers and reservoirs. In order to examine the potential flow and transport effects of these band arrays at scales relevant to production and management, we performed a suite of simulations using an innovative discrete-feature modeling technique to capture the exact pattern of compaction bands mapped over some 150,000 m2 of contiguous outcrop. Significant impacts related to the presence of the bands and their dominant trend are apparent: the average pressure drop required to drive flow between wells exceeds that for band-free sandstone by a factor of three and is 10% to 50% higher across the bands versus along them; reservoir production efficiency varies up to 56% for a typical five-spot well array, depending on its orientation relative to the dominant band trend; and contaminant transport away from a point source within an aquifer tends to channel along the bands, regardless of the regional gradient direction. We conclude that accounting for the flow effects of similar compaction-band arrays would prove essential for optimal management of those sandstone aquifers and reservoirs in which they occur.
In any elasto-plastic constitutive model there are three main ingredients, namely a yield surface, a plastic potential surface and a hardening/ softening rule. In this paper a versatile mathematical expression is presented which can be used to describe the yield and plastic potential surfaces. The expression is defined completely by a maximum of only four parameters. These parameters can easily be obtained from observable soil behaviour in simple triaxial tests. A major advantage of the expression is that by suitable adjustment of the parameters a wide range of surface shapes can be achieved. For example, it is possible to reproduce the so called “bullet shape” typical of the plastic potential used in the original Cam clay model and the “tear shape” yield surfaces employed in the more recent models. In fact the expression is capable of accurately reproducing the shapes of many of the yield and plastic potential surfaces currently in use. The expression is also shown to be in good agreement with experimental data.
Experimental work on shear localization in porous sandstone led to the
observation of nonuniform compaction. By analogy with shear
localization, the process is referred to as compaction localization. To
gain insight into the process of compaction localization, acoustic
emission locations were used to define and track the thicknesses of
localized zones of compaction during axisymmetric compression
experiments. Zones of acoustic emission, demarcating the boundaries
between the uncompacted and compacted regions, developed and moved
parallel to the sample axis at velocities an order of magnitude higher
than the imposed specimen shortening rate. Thus tabular zones of
compaction were found to grow (thicken) in the direction of maximum
compressive stress. These structures may form due to tectonic stresses
or as a result of local stresses induced during production of fluids
from wells, resulting in barriers to fluid (oil, gas, water) movement in
sandstone reservoirs.
Triaxial compression experiments were conducted to investigate the inelastic and failure behavior of six sandstones with porosities ranging from 15% to 35%. A broad range of effective pressures was used so that the transition in failure mode from brittle faulting to cataclastic flow could be observed. In the brittle faulting regime, shear-induced dilation initiates in the prepeak stage at a stress level C' which increases with effective mean stress. Under elevated effective pressures, a sample fails by cataclastic flow. Strain hardening and shear-enhanced compaction initiates at a stress level C* which decreases with increasing effective mean stress. The critical stresses C' and C* were marked by surges in acoustic emission. In the stress space, C* maps out an approximately elliptical yield envelope, in accordance with the critical state and cap models. Using plasticity theory, the flow rule associated with this yield envelope was used to predict porosity changes which are comparable to experimental data. In the brittle faulting regime the associated flow rule predicts dilatancy to increase with decreasing effective pressure in qualitative agreement with the experimental observations. The data were also compared with prediction of a nonassociative model on the onset of shear localization. Experimental data suggest that a quantitative measure of brittleness is provided by the grain crushing pressure (which decreases with increasing porosity and grain size). Geologic data on tectonic faulting in siliciclastic formations (of different porosity and grain size) are consistent with the laboratory observations.
The engineering properties of naturally occurring sedimentary and residual deposits which are usually treated in geotechnical engineering as ‘soils’ are reviewed, and it is shown that usually they have characteristics due to bonded structure which are similar to those of porous weak rock. While this structure can arise from many causes, its effects follow a simple general pattern that involves stiff behaviour followed by yield. This yield can be described in a similar way to that occurring due to overconsolidation, although it is a separate phenomenon. The effects of structure are as important in determining engineering behaviour as are the effects of initial porosity and stress-history, which are the basic concepts of soil mechanics. As it can be described in a general way, it is concluded that structure and its effects should be treated as a further basic concept of equal importance.
L'article passe en revue les propriétés des dépôts sédimentaires et résiduels naturels qui sont nor-malement traités comme des sols en géotechnique et démontre que généralement ils possèdent des caractéristiques dues à la structure liée qui ressemblent à celles du rocher tendre poreux. Bien que cette structure puisse provenir de beaucoup de facteurs, ses effets suivent un cours général très simple comprenant un comportement raide suivi par l'ecoulement. Cet écoulement peut être comparé a celui résultant de la surconsolidation, bien qu'il représente un phénomène different. Dans la détermination du comportement dans la construction les effets de la structure ont une importance égale a celle des effets de la porosité initiate et de l'historique des contraintes, qui représented les conceptions classiques de la mécanique des sols. On tire la conclusion que la structure et ses effets devraient être traités comme une autre conception de base d'importance égale, puisqu'il est possible de les décrire d'une façon génerate.
Compaction bands, a form of localized deformation found in field and laboratory specimens of high porosity rock, consist of planar zones of pure compressional deformation that form perpendicular to maximum compression. Experimentalists report compaction bands and/or shear bands (angled to maximum compression) in high porosity sandstone during a transitional loading regime with multiple active deformation mechanisms. Conditions for localized deformation are determined using a two-yield surface constitutive model and bifurcation theory. The shear yield surface corresponds to a dilatant, frictional mechanism while the cap corresponds to a compactant mechanism. Unlike a single yield surface model, the two-yield surface model predicts both experimentally observed band types for reported values of key material parameters. Observed and predicted shear band angles generally agree. Theory suggests that shear band formation may coincide with activation of the shear yield surface by a previously active cap. If the bulk hardening modulus, k, equals zero (corresponding to localization on the peak or plateau of the mean stress-volume strain curve) compaction band conditions are more favorable than for small positive values of k.
The mathematical foundation of the general bounding surface constitutive formulation in plasticity is presented. Along these lines the concept of hypoplasticity is formally introduced, and it is shown that a particular class of hypoplastic formulations arises naturally from certain bounding surface models, with the distinguishing feature being the dependence of the elastoplastic moduli and/or the plastic strain rate direction on the stress rate direction. The general analytical perspective allows the better understanding and improvement of existing bounding surface plasticity and hypoplasticity models, which are briefly discussed, and suggests the proper way to construct new ones for future applications.
Field and laboratory investigators have observed thin, tabular zones of localized compressional deformation without shear in high porosity sandstone. These ‘compaction bands’ display greatly reduced porosity, and may affect the withdrawal of fluids from reservoirs. Studies addressing band formation as a type of strain localization predict the onset of the bands in a range of constitutive parameters roughly consistent with experiments, but are highly dependent on the constitutive relation used. In particular, the hardening modulus in shear and the slope of the yield surface in a plot of shear stress versus mean compressive stress are critical to localization predictions. Previous yield cap constitutive models employed a single deformation criterion, linking hydrostatic and shear response. In this work, we propose an elliptic yield cap model employing separate inelastic deformation parameters along each axis of the ellipse. The two deformation parameters allow the proposed surface to change in aspect ratio as it deforms, and allow a negative hardening modulus in shear without a negative hydrostatic modulus. Some cases with simplified modeling are shown for illustrative purposes, followed by a comparison with existing models. The proposed model displays similar strain behavior to the other models, but predicts localization under less restrictive conditions.
A viscoplastic consistency condition is incorporated into the traditional viscoplastic rate format with two objectives in mind: (i) to develop a tangential material operator for rate sensitive behavior similarly to rate independent elastoplasticity, and (ii) to make an analytical reference solution available for evaluating the accuracy of new and well-established computational strategies to integrate the viscoplastic evolution equations.The viscoplastic tangent operator provides the missing link between rate independent plasticity and rate dependent viscoplasticity. It also furnishes the acoustic tensor, which is required for localization analysis of discontinuous failure. Besides the formulation of the viscoplastic tangent operator, we present an analytical reference solution for perfect J2 viscoplasticity. This analytical result is subsequently used to determine the accuracy of simplifying assumptions behind the algebraic evaluation of the viscoplastic multiplier, and to quantify the error of the radial return mapping strategy for viscoplastic computations.
Compaction bands are narrow planar zones of localized purely compressive (without shear) deformation that form perpendicular to the most compressive principal stress. Such bands have been observed in high porosity rocks in the laboratory and in the field. Because compaction presumably decreases permeability, these bands can act as barriers to flow within reservoirs. Reexamination of the results of Rudnicki and Rice (J. Mech. Phys. Solids, 1975) for shear localization, with corrections by Perrin and Leblond (J. Appl. Mech., 1993), reveals that they admit solutions for compaction bands in a range of parameters that is representative of porous rock. Solutions for compaction bands are possible when the inelastic volume deformation is compactive and is associated with a “cap” on the yield surface. The expression for the critical hardening modulus (related to the slope of the shear stress vs. shear strain curve at constant mean stress) at which compaction bands are predicted to form differs from that for shear localization. For parameters representative of porous rock, axisymmetric compression is the most favorable deviatoric stress state for formation of compaction bands. Comparison of conditions for shear localization and compaction band formation suggests that either may occur depending on the stress path and magnitude of the confining stress.
The essential feature of the observed behaviour is the occurrence of a destructuration phase, which marks the transition from rock-like to soil-like behaviour. During this phase the state of stress remains constant, while strains increase steadily. Three phases can be distinguished: an initial elastic, a destructuration phase and a hardening or softening phase which ends on an ultimate state locus which is linear in the p′-q plane. The observed behaviour is more and more ductile for increasing confining pressures. In the softening regime the specimen is unstable. It is shown that by means of a mathematical model based on the theory of strain-hardening plasticity it is possible to describe mathematically the overall behaviour of the calcarenite in various types of triaxial compression test. -from Authors
One of the open questions is the performance of rate-independent versus rate-dependent constitutive formulations when failure is evaluated at the material and the finite-element levels. In the case of rate-independent descriptions, the underlying tangential material operator exhibits singularities and material branching at limit points of the response regime. In addition discontinuous bifurcation can take place in the form of localization concomitant with the formation of spatial discontinuities. In contrast, rate-dependent descriptions resort to an instantaneous elastic stiffness operator that remains normally positive definite, while degradation is introduced through the time history of inelastic eigenstrains. In fact when the inelastic process does not contribute to the instantaneous material operator one speaks of elastic-inelastic decoupling. As a consequence viscoplastic material descriptions are often advocated to retrofit loss of stability, loss of uniqueness, and loss of ellipticity of rate-independent, inviscid material descriptions. In this paper the failure predictions of viscoplastic Duvaut-Lions and viscoplastic Perzyna material formulations are analyzed and compared with the inviscid elastoplastic formulation. Our attention will be focused on the loss of material stability and on discontinuous bifurcation in the form of localization. The results on the material and on the finite-element level indicate that Duvaut-Lions regularization fails in the limit, when we consider viscoplastic processes with relaxation times approaching zero. In this case, there exists an algorithmic tangent operator for the Newton-Raphson solution of implicit time integration procedures that exhibits loss of stability, loss of uniqueness, and loss of ellipticity in the form of discontinuous bifurcation similar to rate-independent elastoplasticity. On the other hand, localization at the material level indicates that Perzyna viscoplasticity does suppress localization for the entire range of viscosities and thus provides stronger regularization than the Duvaut-Lions viscoplastic overstress model at the cost of excessive degradation when the viscosity approaches zero. These theoretical observations are confirmed with computational simulations of dynamic failure of a flexural member.
Alternative constitutive models are employed to examine localization conditions for high porosity sandstone. Two deformation types are considered: shear bands and compaction bands (planar zones of pure compressional deformation, perpendicular to maximum compression). The proposed two-yield surface model, motivated in part by observations of two microstructural damage mechanisms, consists of a shear yield surface (dilatant, frictional mechanism), combined with a cap (compactant mechanism). Results from this model are consistent with limited experimental observations: both band types are permitted for probable values of key material parameters. Conversely, the single-yield surface model, previously employed for sandstone, cannot predict the observed compaction bands.
At low mean stresses, porous geomaterials fail by shear localization, and at higher mean stresses, they undergo strain-hardening behavior. Cap plasticity models attempt to model this behavior using a pressure-dependent shear yield and/or shear limit-state envelope with a hardening or hardening/softening elliptical end cap to define pore collapse. While these traditional models describe compactive yield and ultimate shear failure, difficulties arise when the behavior involves a transition from compactive to dilatant deformation that occurs before the shear failure or limit-state shear stress is reached. In this work, a continuous surface cap plasticity model is used to predict compactive and dilatant pre-failure deformation. During loading the stress point can pass freely through the critical state point separating compactive from dilatant deformation. The predicted volumetric strain goes from compactive to dilatant without the use of a non-associated flow rule. The new model is stable in that Drucker's stability postulates are satisfied. The study has applications to several geosystems of current engineering interest (oil and gas reservoirs, nuclear waste repositories, buried targets, and depleted reservoirs for possible use for subsurface sequestration of greenhouse gases).
Material instability occurs when ellipticity is lost for symmetric constitutive equations. Prior to loss of ellipticity it is possible that the second-order work of Hill or Drucker becomes negative. There are implications in the literature that numerical solutions cease to be meaningful when a material strain softens and the second-order work is not positive. The instant that the second-order work is zero or negative simultaneously with the additional restriction that the strain increments satisfy compatibility is equivalent to the loss of the ellipticity criterion for symmetric constitutive relations. The loss of ellipticity criterion is the appropriate one for identifying when numerical solutions cease to show convergence and may also be a suitable criterion for identifying the instant at which material failure is initiated. An analytical development is provided for loss of ellipticity together with an explicit expression for the normal to the bifurcation plane. Numerical solutions are given for several sample problems. For all cases, the numerical solutions based on the finite element method conform to the theoretical expectations that unique numerical solutions exist prior to the point at which ellipticity is lost.