No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Coupled phase-field and plasticity modeling of geological materials: From brittle fracture to ductile flow
Abstract
The failure behavior of geological materials depends heavily on confining pressure and strain rate. Under a relatively low confining pressure, these materials tend to fail by brittle, localized fracture, but as the confining pressure increases, they show a growing propensity for ductile, diffuse failure accompanying plastic flow. Furthermore, the rate of deformation often exerts control on the brittleness. Here we develop a theoretical and computational modeling framework that encapsulates this variety of failure modes and their brittle-ductile transition. The framework couples a pressure-sensitive plasticity model with a phase-field approach to fracture which can simulate complex fracture propagation without tracking its geometry. We derive a phase-field formulation for fracture in elastic-plastic materials as a balance law of microforce, in a new way that honors the dissipative nature of the fracturing processes. For physically meaningful and numerically robust incorporation of plasticity into the phase-field model, we introduce several new ideas including the use of phase-field effective stress for plasticity, and the dilative/compactive split and rate-dependent storage of plastic work. We construct a particular class of the framework by employing a Drucker–Prager plasticity model with a compression cap, and demonstrate that the proposed framework can capture brittle fracture, ductile flow, and their transition due to confining pressure and strain rate.
... It was then extended into exponential and hyperbolic forms (Alejano and Bobet 2012). Then, to account for pore collapse under volumetric compression, different cap models were added to the Drucker-Prager yield surface (Lubarda et al. 1996;Choo and Sun 2018;Liu et al. 2021). Finally, compressive failure and pressure dependence of elastic properties were captured in the Modified Cam-Clay model, originally developed for saturated soils (Borja et al. 1997;Charlez 1997). ...
... White et al. (2017a) developed a compositional reservoir simulator coupled with a hardening Drucker-Prager model for predicting wellbore failure and modeling field-scale CO 2 storage. Choo and Sun (2018) coupled a phase field approach with a Drucker-Prager model to capture the mechanical transition between brittle failure and ductile flow for geomaterials subject to loading and reservoir conditions typically encountered in subsurface energy production and CO 2 sequestration. To allow yield under hydrostatic stress, a compression cap model was incorporated into the original Drucker-Prager model. ...
... When loading rock samples beyond the yield strength, stress-strain curves move either up or down, indicating irreversible changes in rock strength. Multiple hardening laws (Charlez 1997;Armero 1999;Liu and Chen 2017;Choo and Sun 2018), or softening laws (Roshan and Fahad 2012;Lv et al. 2019;Guan et al. 2022) enrich elasto-plastic models to represent this strain strengthening or weakening behavior. These laws correlate changes in shape and position of yield surfaces to the plastic strain and to several other variables, such as the cohesion, friction angle, or dilatancy coefficient (Armero 1999;Conil et al. 2004;Choo and Sun 2018;Liu et al. 2021;Lu et al. 2022). ...
In geological CO2 storage operations, wellbore deformations and leakage pathways formations can occur around injection and abandoned wells subjected to high rates and long-term CO2 injection. To guide engineering design and prevent CO2 leakage risks, a full understanding of the underlying physics and robust numerical models is necessary to evaluate the response of underground formations in the near wellbore region and in the reservoir. In this study, a multi-scale and multi-physics open-source simulator (GEOS) is used to simulate multiphase flow and poromechanical deformations over time in three dimensions. The governing equations for mechanical deformations of the rock body and multiphase compositional fluid flow within the rock matrix are solved with a fully coupled finite element and finite volume approach. The Drucker–Prager model with friction hardening is applied to simulate elastoplastic deformation and a multiphase fluid model with power-law correlations for relative permeability is used to model the migration of CO2 plume, which are coupled with numerical implicit scheme. Simulation results are verified against multiple analytical solutions for multiphase flow and wellbore problems, thus demonstrating the accuracy of this advanced simulator. In two engineering applications, we highlight the impact of elastoplastic deformation and coupled modeling for assessing induced displacements and stress perturbations, which are more pronounced in the near wellbore regions. This work focuses on short-term processes in the vicinity of injection wells where stress evolutions, rock deformations and multiphase compositional flow and transport are simulated jointly to ensure wellbore stability and prevent damage. This fully coupled geomechanical model can simulate multiphase flow and any associated poromechanical effects within the CO2 storage site and in the surrounding formations. Such a large-scale, long-term, multi-physics simulation model is useful in many ways: it can guide operational decisions for CO2 injection, assess the containment potential and risks of a site, and analyze the wellbore stability and integrity during and after CO2 injection.
Highlights.We introduce a fully coupled finite element/finite volume approach to simulate multiphase fluid flow and the associated rock deformations.
This approach highlights how the coupling between rock deformations and multiphase fluid flow impacts short-term mechanical responses in the vicinity of the injection wells.
The results of this numerical model are successfully verified against reference analytical solutions for multiphase flow and wellbore problems.
We have tested the approach using both poroelastic and poroplastic deformations on an engineering problem, demonstrating the important effects of plasticity in CO2 injection scenario.
This work contributes to better operational decisions for designing CO2 injection operations by assessing the containment potential of a site, and by analyzing the wellbore integrity during and after CO2 injection.
... Implementing and evaluating nonlinear material models is a challenging yet essential task for achieving reliable predictions in many applications. For example, advanced models may be needed to represent the complex plastic behavior of metals, soils, and polymers, or to account for the softening of materials as damage and microcracks evolve [1][2][3]. Typically, modelers use a thermodynamic framework to derive these B Shahed Rezaei s.rezaei@access-technology.de 1 ACCESS e.V., Intzestr. 5, 52072 Aachen, Germany 2 Institute for Structural Analysis, Technische Universität Dresden, Georg-Schumann-Str. ...
... Typically, modelers use a thermodynamic framework to derive these B Shahed Rezaei s.rezaei@access-technology.de 1 ACCESS e.V., Intzestr. 5, 52072 Aachen, Germany 2 Institute for Structural Analysis, Technische Universität Dresden, Georg-Schumann-Str. 7, 01187 Dresden, Germany 3 Institute of Applied Mechanics, RWTH Aachen University, Mies-van-der-Rohe-Str. 1, 52074 Aachen, Germany models, which helps them to set up all the necessary equations and evolution laws for the internal state of the materials. ...
... In this algorithm, Δλ d = Δt λ d , where Δt is the pseudo time step. When the damage is active (φ d > 0), the damage residuals (r (1) d and r (2) d ) are linearized using the Newton-Raphson method to solve for the unknowns (d k+1 and ξ k+1 d ). The matrix K d = ∂ r d /∂U d includes the derivatives of the damage residual vector r d with respect to the unknowns ...
We applied physics-informed neural networks to solve the constitutive relations for nonlinear, path-dependent material behavior. As a result, the trained network not only satisfies all thermodynamic constraints but also instantly provides information about the current material state (i.e., free energy, stress, and the evolution of internal variables) under any given loading scenario without requiring initial data. One advantage of this work is that it bypasses the repetitive Newton iterations needed to solve nonlinear equations in complex material models. Furthermore, after training, the proposed approach requires significantly less effort in terms of implementation and computing time compared to the traditional methods. The trained model can be directly used in any finite element package (or other numerical methods) as a user-defined material model. We tested this methodology on rate-independent processes such as the classical von Mises plasticity model with a nonlinear hardening law, as well as local damage models for interface cracking behavior with a nonlinear softening law. In order to demonstrate the applicability of the methodology in handling complex path dependency in a three-dimensional (3D) scenario, we tested the approach using the equations governing a damage model for a three-dimensional interface model. Such models are frequently employed for intergranular fracture at grain boundaries. However, challenges remain in the proper definition of collocation points and in integrating several non-equality constraints that become active or non-active simultaneously. As long as we are in the training regime, we have observed a perfect agreement between the results obtained through the proposed methodology and those obtained using the classical approach. Finally, we compare this new approach against available standard methods and discuss the potential and remaining challenges for future developments.
... This means that shear fractures will initiate and propagate even when all principal strains of the material are compressive. In contrast to this fact, the compressive-shear fractures cannot be predicted by the current PFMs for rock-like solids [21,51,52], where the negative strains and the compressive part of elastic energy are assumed not to contribute to the evolution of phase field. More specifically, the first attempt of the PFM for mixed-mode crack propagation in rock-like materials was proposed by Zhang et al. [21]. ...
... This method was established by introducing the critical energy release rate of mode II and two historic energy references in the model of Miehe et al. [26]. Choo and Sun [51] coupled a pressure-sensitive plasticity model with a phase field approach to capture brittle fracture, ductile flow, and their transition in rocks. Bryant and Sun [52] proposed a kinematic-consistent phase field approach to model mixed-mode fractures in anisotropic rocks. ...
... Bryant and Sun [52] proposed a kinematic-consistent phase field approach to model mixed-mode fractures in anisotropic rocks. On the other hand, the contributions in Zhang et al. [21], Choo and Sun [51], Bryant and Sun [52] also show that the PFMs developed so far do not account for the influence of cohesion and internal friction angle on fracture propagation and the load-displacement curve. This is a missing aspect since it is generally accepted that a rock-like material will have a higher compressive strength if its cohesion or internal friction angle increases. ...
Compressive-shear fracture is commonly observed in rock-like materials. However, this fracture type cannot be captured by current phase field models (PFMs), which have been proven an effective tool for modeling fracture initiation, propagation, coalescence, and branching in solids. The existing PFMs also cannot describe the influence of cohesion and internal friction angle on load-displacement curve during compression tests. Therefore, to develop a new phase field model that can simulate well compressive-shear fractures in rock-like materials, we construct a new driving force in the evolution equation of phase field. Strain spectral decomposition is applied and only the compressive part of the strain is used in the new driving force with consideration of the influence of cohesion and internal friction angle. For ease of implementation, a hybrid formulation is established for the phase field modeling. Then, we test the brittle compressive-shear fractures in uniaxial compression tests on intact rock-like specimens as well as those with a single or two parallel inclined flaws. All numerical results are in good agreement with the experimental observation, validating the feasibility and practicability of the proposed PFM for simulating brittle compressive-shear fractures.
... More recently, only a few researches have been reported on adopting the variational phase-field fracture model to study ductile fracture in porous media [63][64][65][66]. However, the principle of maximum plastic dissipation in the variational formulation of plasticity indicates associative flow and associative hardening, geomaterials such as rocks and soils usually exhibit non-associative flow, care must be taken when using the variational description of fracture in geomaterials [84]. Therefore, it is necessary to develop new explicit phase-field fracture models derived based on the microforce balance law for efficiently solving complex large deformation fracture problems in the poro-elastoplastic media involving non-associative plasticity. ...
... where D represents the energy dissipation, and Ψ denotes the stored energy density per unit volume. Referring to the previous works [71,84], Ψ and its time differentiation are given as ...
... It can be seen that Eq. (19) represents the constitutive relationship for the elastoplastic large deformation analysis of the saturated porous media. In addition, the analytical expressions of ξ and π dis can be obtained by taking a series of algebraic operations on Eq. (21) as [71,84] ...
... The evolution of damage due to void growth at the microscale, described by Gurson-Tvergaard-Needelman (GTN) model (Gurson, 1977;Tvergaard and Needleman, 1984), or other porous plasticity models, has been extensively treated in several contributions (Miehe et al., 2016b;Aldakheel et al., 2018;Dittmann et al., 2020;Krüger et al., 2020;Azinpour et al., 2021;Dittmann et al., 2021;Tao et al., 2022;Chen et al., 2022). The simulation of crack propagation in geological materials of different types has been considered in , Choo and Sun (2018), Kienle et al. (2019), You et al. (2021), Ulloa et al. (2022) and Hu et al. (2022). The combination of phasefield description of damage with multisurface plasticity models has been investigated in Fang et al. (2019a). ...
... The phase-field approach has also been successfully applied to the modeling of cyclic plasticity problems and to low-cycle fatigue problems in the presence of plasticity, see, e.g., Seiler et al. (2020), Aygün et al. (2021), Seleš et al. (2021), Ulloa et al. (2021), Hasan and Baxevanis (2021), Song et al. (2022) and Tsakmakis and Vormwald (2022). Finally, while numerous contributions have been made within the context of small-strain kinematics, there has been an increasing interest in recent years towards the development of formulations within the large-strain plasticity regime (Aldakheel et al., 2014;Miehe et al., 2015Miehe et al., , 2016cMiehe et al., ,a,b, 2017McAuliffe and Waisman, 2015;Borden et al., 2016;Aldakheel, 2017;Shanthraj et al., 2017;Choo and Sun, 2018;Aldakheel et al., 2018;Dittmann et al., 2018;Chu et al., 2019;Kienle et al., 2019;Brepols et al., 2020;Krüger et al., 2020;Shishvan et al., 2021;Dittmann et al., 2021;Eldahshan et al., 2021b;Hu et al., 2021;Proserpio et al., 2021;Talamini et al., 2021;Felder et al., 2022;Han et al., 2022a;Abrari Vajari et al., 2022;Hu et al., 2022;Huber and Asle Zaeem, 2023;Han et al., 2022b). ...
... In contrast, in the nominal stress approach, the nominal stress decreases when the damage-softening branch of the response is entered, while the current yield stress does not decrease (weak plasticity-damage coupling), so that the yield condition cannot be satisfied anymore and the plastic strain evolution stops. In the latter case, the damage evolution becomes purely brittle and a non-physical elastic unloading is observed, as noted by several authors (see, e.g., Borden et al., 2016;Choo and Sun, 2018;Huang and Gao, 2019;Marengo and Perego, 2023). Other models (see, e.g., Alessi et al., 2014Alessi et al., , 2018bUlloa et al., 2016), consider different definitions for the degradation ( ) of stresses and static internal variables (see Section 2.4, Eq. (29), where ( ) ≠ 1 is used for the degradation of the current yield limit). ...
... This characteristic makes phase-field models effective in predicting complex crack propagation in heterogeneous materials like rocks, soils, and concretes, cf. [51][52][53][54][55][56]. However, the phase-field model demands a finely discretized domain as a fundamental requirement. ...
... In the elastic-plastic case, return mapping for plastic correction is necessary. For this purpose, Equations (47) and (51) are solved while satisfying the yield criteria, i.e., Equations (45) and (50), at the discrete time step t n+1 , see Figure 16 can be derived using return mapping, see [92,94]. This follows ...
A phenomenological material model has been developed to facilitate the efficient numerical analysis of fiber-reinforced high-performance concrete (HPC). The formulation integrates an elasto-plastic phase-field model for simulating fractures within the HPC matrix, along with a superimposed one-dimensional elasto-plasticity model that represents the behavior of the embedded fibers. The Drucker–Prager plasticity and one-dimensional von-Mises plasticity formulations are incorporated to describe the nonlinear material behavior of both the HPC matrix and the fibers, respectively. Specific steps are undertaken during the development and calibration of the phenomenological material model. In the initial step, an experimental and numerical analysis of the pullout test of steel fibers embedded in an HPC matrix is conducted. This process is used to calibrate the micro-mechanical model based on the elasto-plastic phase-field formulation for fracture. In the subsequent step, virtual experiments based on an ellipsoidal unit cell, also with the resolution of fibers (used as a representative volume element, RVE), are simulated to analyze the impact of fiber–matrix interactions and their physical properties on the effective material behavior of fiber-reinforced HPC. In the final step, macroscopic boundary value problems (BVPs) based on a cuboid are simulated on a single scale using the developed phenomenological material model. The resulting macroscopic stress–strain characteristics obtained from both types of simulations, namely simulations of virtual experiments and macroscopic BVPs, are compared. This comparison is utilized for the calibration of material parameters to obtain a regularized solution and to assess the effectiveness of the presented phenomenological material model.
... Several studies have developed phase-field hydraulic fracture models in poroelastic media based on linear elasticity (Mauthe and Miehe 2017;Heider and Markert 2017;Zhou et al. 2018), similar to the model developed herein. In higher magnitudes of the confining pressure, rock's (Choo and Sun 2018), and deformations become significant when the stiffness decreases; therefore, employing the kinematics of large deformations (finite strains) would be essential (Miehe and Mauthe 2016). The tensor � = Ψ eff ( )∕ is so-called the effective Cauchy stress tensor is a part of that only applies on the solid skeleton structure. ...
... The hydromechanical response of rocks under hydraulic fracturing operations can vary depending on the temperature (Kumari et al. 2018;Zhou et al. 2018;Cheng et al. 2021) and cyclic loadings (Zhao et al. 2018). Hence, developing the poroelastic material to poro-viscoelastic (Shen et al. 2019;Song et al. 2021), considering thermo-hydromechanics (Li et al. 2016), and capturing ductile behaviour of rocks (Choo and Sun 2018) can be counted as the potential areas of research to expand this current study. The versatility of the phase-field method in the finite-element implementation makes it easier for developing the presented HF model to capture the rupture characteristics of the rock under realistic environmental effects. ...
The interaction between a propagating hydraulic fracture (HF) and a pre-existing natural fracture (NF) embedded in saturated poroelastic rock formations is studied numerically in 2D plane–strain configurations. In this study, the phase-field method is further developed to be employed for modelling the HF propagation and the evolution of tensile and shear failure in geo-materials as gradient-type diffusive damaged zones. The shear slippage and dilation mechanisms inside the cemented NF are modelled using a Mohr–Coulomb–Griffith failure criterion that fitted in the framework of the phase-field fracture using appropriate energy functionals. The most important factors controlling the HF–NF interaction outcome are the approaching angle, differential in-situ stress, and hydro-mechanical characteristics of the NF. It is found out that higher tensile and shear strengths of the cemented NF are in favour of the crossing outcome when the differential stress is high enough to mobilise the resisting shear stresses against the slippage. Small hydraulic aperture (low hydraulic conductivity) for the NF is also in favour of the crossing outcome which helps to restrict the pressurised region local to the HF tip, lowering the possibility of shear slippage in the NF and the HF’s diversion. It is also concluded that the injection rate and the viscosity of fracturing fluid are operative factors to be adjusted for increasing the chance of crossing, a critical element for successful operation of hydraulic fracturing for effective use of subsurface energy resources.
... Over the past few decades, the phase-field model for fractures has been identified as a promising alternative to sharp interface approaches to model fracture propagation in rocks and rock-like materials. [13][14][15][16] The fracture, instead of being explicitly represented as an interface, is approximated by a diffuse variable (i.e., damage). The main advantage of this diffuse crack representation is the simplicity of representing complex geometries. ...
... As a consequence, the regularization length becomes a material-specific parameter that should be calibrated based on the tensile and compressive strengths of a given material. 14,32 This length dependency issue has been addressed in the recent works on phase-field models for quasi-brittle materials, 15,29,30 which are inspired by a gradient damage model introduced by Lorentz and Godard. 33 Unfortunately, these models are still derived through an energy minimization process, and as such damage nucleation is governed by a threshold that is energetic. ...
Many geo‐engineering applications, for example, enhanced geothermal systems, rely on hydraulic fracturing to enhance the permeability of natural formations and allow for sufficient fluid circulation. Over the past few decades, the phase‐field method has grown in popularity as a valid approach to modeling hydraulic fracturing because of the ease of handling complex fracture propagation geometries. However, existing phase‐field methods cannot appropriately capture nucleation of hydraulic fractures because their formulations are solely energy‐based and do not explicitly take into account the strength of the material. Thus, in this work, we propose a novel phase‐field formulation for hydraulic fracturing with the main goal of modeling fracture nucleation in porous media, for example, rocks. Built on the variational formulation of previous phase‐field methods, the proposed model incorporates the material strength envelope for hydraulic fracture nucleation through two important steps: (i) an external driving force term, included in the damage evolution equation, that accounts for the material strength; (ii) a properly designed damage function that defines the fluid pressure contribution on the crack driving force. The comparison of numerical results for two‐dimensional test cases with existing analytical solutions demonstrates that the proposed phase‐field model can accurately model both nucleation and propagation of hydraulic fractures. Additionally, we present the simulation of hydraulic fracturing in a three‐dimensional domain with various stress conditions to demonstrate the applicability of the method to realistic scenarios.
... It has been applied to different kinds of materials and problems, such as multi-physics coupling, 15 finite deformation, 16 and plastic damage coupling. 17,18 However, in most previous studies, homogeneous rock materials have been considered. The COx claystone is characterized by a multiscale heterogeneity. ...
... By substituting − for Eqs. (18), the evolution criteria for two crack fields are now expressed as: ...
This study is devoted to numerical modeling of cracking process induced by temperature change in saturated porous rocks in the context of geological disposal of radioactive waste. Effects of material anisotropy and heterogeneity are taken into account. The macroscopic elastic properties are determined from two steps of homogenization by considering pores and mineral inclusions at two different scales. An extended phase-field model is proposed to describe the initiation and propagation of localized cracks. Two damage variables are introduced to conveniently represent both tensile and shear cracks. New damage evolution criteria are defined by incorporating the pore pressure effect. Three application examples are presented. By assuming a random distribution of pores and inclusions, the efficiency of the proposed model for capturing the progressive cracking process is first verified in a triaxial compression test. The thermal cracking process in an anisotropic and heterogeneous sample is then investigated. The respective influences of elastic anisotropy and spatial variability of pores and inclusions are outlined. Finally, the proposed model is applied to a series of real laboratory thermal cracking tests. Both hydromechanical responses and cracking evolution patterns are investigated. Numerical results are compared with experimental measurements. The main mechanisms involved in the thermal cracking process are highlighted.
... Continuum-based approaches, in which the material domain of interest is treated as a homogeneous continuum body or a multiphase continuum mixture if the materials are fluid-infiltrated, are widely used, but are inherently incapable of dealing with material heterogeneity and they fail to a large extent to simulate appropriately damage localization and the resulting size-effect. In these approaches, the fracture surfaces are usually represented by various methods, including but not limited to, embedded discontinuities (Mohammadnejad and Khoei, 2013;Salimzadeh and Khalili, 2015;Jin and Zoback, 2017), phase field models (Mikelic et al, 2015;Miehe et al, 2015;Choo and Sun, 2018), or smeared crack bands (Chau et al, 2016;Li et al, 2017a). ...
A three-dimensional Multiphysics Lattice Discrete Particle Model (M-LDPM) framework is formulated to investigate the fracture permeability behavior of shale. The framework features a dual lattice system mimicking the mesostructure of the material and simulates coupled mechanical and flow behavior. The mechanical lattice model simulates the granular internal structure of shale and describes heterogeneous deformation by means of discrete compatibility and equilibrium equations. The network of flow lattice elements constitutes a dual graph of the mechanical lattice system. A discrete formulation of mass balance for the flow elements is formulated to model fluid flow along cracks. The overall computational framework is implemented with a mixed explicit-implicit integration scheme and a staggered coupling method that makes use of the dual lattice topology enabling the seamless two-way coupling of the mechanical and flow behaviors. The proposed model is used for the computational analysis of shale fracture permeability behavior by simulating triaxial direct shear tests on Marcellus shale specimens under various confining pressures. The simulated mechanical response is calibrated against the experimental data, and the predicted permeability values are also compared with the experimental measurements. Furthermore, the paper presents the scaling analysis of both the mechanical response and permeability measurements based on simulations performed on geometrically similar specimens with increasing size. The simulated stress-strain curves show a significant size effect in the post-peak due to the presence of localized fractures. The scaling analysis of permeability measurements enables prediction of permeability for large specimens by extrapolating the numerical results of small ones.
... With this class of models, it is easy to describe the transition from diffuse damage to localized cracks and to deal with three-dimensional multiple cracked problems. A high number of specific models have been so far developed, in particular those for rock-like materials (Choo and Sun, 2018;Fang et al., 2019;Wang et al., 2023;Yu et al., 2023) and multi-physical coupling problems (Miehe et al., 2015;Wang et al., 2022;Yu et al., 2024). ...
This work is devoted to numerical analysis of thermo-hydromechanical problem and cracking process in saturated porous media in the context of deep geological disposal of radioactive waste. The fundamental background of thermo-poro-elastoplasticity theory is first summarized. The emphasis is put on the effect of pore fluid pressure on plastic deformation. A micromechanics-based elastoplastic model is then presented for a class of clayey rocks considered as host rock. Based on linear and nonlinear homogenization techniques, the proposed model is able to systematically account for the influences of porosity
and mineral composition on macroscopic elastic properties and plastic yield strength. The initial anisotropy and time-dependent deformation are also taken into account. The induced cracking process is
described by using a non-local damage model. A specific hybrid formulation is proposed, able to
conveniently capture tensile, shear and mixed cracks. In particular, the influences of pore pressure and
confining stress on the shear cracking mechanism are taken into account. The proposed model is applied
to investigating thermo-hydromechanical responses and induced damage evolution in laboratory tests at
the sample scale. In the last part, an in situ heating experiment is analyzed by using the proposed model.
Numerical results are compared with experimental data and field measurements in terms of temperature
variation, pore fluid pressure change and induced damaged zone.
... Several experimental and numerical studies have indicated a transition in the fracture mode of geomaterials under tension as confining compression increases [19,77]. This phenomenon was experimentally investigated by Ramsey et al. [78] and further modeled numerically by Choo et al. [79] and Ulloa et al. [33] using a dog-bone-shaped sandstone specimen under monotonic loading. In this study, we subject the specimen to cyclic loading to investigate its fatigue fracture behavior. ...
... In all examples, we apply the proposed method with varied element sizes (h) and phase-field regularization lengths (L). The numerical results are produced using an in-house finite element code used in our previous work (e.g., Choo (2019), Choo and Borja (2015), Choo and Sun (2018a)), which is built on the deal.II library (Arndt et al., 2021). In all cases, we use quadrilateral elements with linear shape functions, neglect body force, and assume plane-strain conditions. ...
The phase-field method has become popular for the numerical modeling of fluid-filled fractures, thanks to its ability to represent complex fracture geometry without algorithms. However, the algorithm-free representation of fracture geometry poses a significant challenge in calculating the crack opening (aperture) of phase-field fracture, which governs the fracture permeability and hence the overall hydromechanical behavior. Although several approaches have been devised to compute the crack opening of phase-field fracture, they require a sophisticated algorithm for post-processing the phase-field values or an additional parameter sensitive to the element size and alignment. Here, we develop a novel method for calculating the crack opening of fluid-filled phase-field fracture, which enables one to obtain the crack opening without additional algorithms or parameters. We transform the displacement-jump-based kinematics of a fracture into a continuous strain-based version, insert it into a force balance equation on the fracture, and apply the phase-field approximation. Through this procedure, we obtain a simple equation for the crack opening, which can be calculated with quantities at individual material points. We verify the proposed method with analytical and numerical solutions obtained based on discrete representations of fractures, demonstrating its capability to calculate the crack opening regardless of the element size or alignment.
... The study of elasto-plastic constitutive models for granular materials is critical in civil engineering, especially geotechnical engineering, due to their complex and nonlinear behaviors under different loading conditions [1,2]. These models help predict the stress-strain relationship and the transition from elastic to plastic deformation, which is critical for the design of foundations, retaining structures, and other civil infrastructure [3,4,5]. ...
Multilayer perceptron (MLP) networks are predominantly used to develop data-driven constitutive models for granular materials. They offer a compelling alternative to traditional physics-based constitutive models in predicting nonlinear responses of these materials, e.g., elasto-plasticity, under various loading conditions. To attain the necessary accuracy, MLPs often need to be sufficiently deep or wide, owing to the curse of dimensionality inherent in these problems. To overcome this limitation, we present an elasto-plasticity informed Chebyshev-based Kolmogorov-Arnold network (EPi-cKAN) in this study. This architecture leverages the benefits of KANs and augmented Chebyshev polynomials, as well as integrates physical principles within both the network structure and the loss function. The primary objective of EPi-cKAN is to provide an accurate and generalizable function approximation for non-linear stress-strain relationships, using fewer parameters compared to standard MLPs. To evaluate the efficiency, accuracy, and generalization capabilities of EPi-cKAN in modeling complex elasto-plastic behavior, we initially compare its performance with other cKAN-based models, which include purely data-driven parallel and serial architectures. Furthermore, to differentiate EPi-cKAN's distinct performance, we also compare it against purely data-driven and physics-informed MLP-based methods. Lastly, we test EPi-cKAN's ability to predict blind strain-controlled paths that extend beyond the training data distribution to gauge its generalization and predictive capabilities. Our findings indicate that, even with limited data and fewer parameters compared to other approaches, EPi-cKAN provides superior accuracy in predicting stress components and demonstrates better generalization when used to predict sand elasto-plastic behavior under blind triaxial axisymmetric strain-controlled loading paths.
... Nevertheless, the standard phase-field model of fracture and its variants do not incorporate several distinct features of geologic fractures: frictional contact, pressure-dependence, quasibrittleness, mode-mixity, and roughness, among others. Therefore, although these phase-field models have been employed in many geomechanical applications (e.g., Lee et al., 2016;Zhang et al., 2017;Choo and Sun, 2018a;Ha et al., 2018;Zhou et al., 2018), there is a need for new phase-field models tailored to geologic fractures. ...
Geologic fractures such as joints, faults, and slip surfaces govern the stability and performance of many subsurface systems in the built environment. As such, a variety of approaches have been developed for computational modeling of geologic fractures. Yet none of them lends itself to a straightforward utilization with the classical finite element method widely used in practice. Over the past decade, phase-field modeling has become a popular approach for simulating fracture, because it can be implemented simply with the standard finite element method without any surface-tracking algorithms. However, the standard phase-field formulations do not incorporate several critical features of geologic fractures, including frictional contact, pressure-dependence, quasi-brittleness, mode-mixity, and their combined impacts on cracking. This article provides a brief report of a novel phase-field model that incorporates these features of geologic fractures in a well-verified and validated manner. Remarkably, the phase-field model allows one to simulate the combination of cohesive tensile fracture and frictional shear fracture without any algorithms for surface tracking and contact constraints. It is also demonstrated how phase-field modeling enables us to gain insights into geologic fractures that are challenging to investigate experimentally.
... Hu et al. [27][28][29] derived a phase-field fracture model based on the microscopic force balance principle to simulate the dynamic elastic-plastic fracture failure behavior in geomaterials. Since the PFM holds many attractive advantages attracting researchers, a series of numerical methods have been proposed to solve various fracture problems, such as quasi-static fracture [30][31][32][33], dynamic brittle fracture [34][35][36][37] and ductile fracture [38][39][40][41][42]. However, simulating fracture behaviors with large deformation is still a challenging problem due to the poor numerical convergence and stability caused by strong nonlinearities [43,44]. ...
An adaptive phase-field total Lagrangian material point method (APTLMPM) is proposed in this paper for effectively simulating the dynamic fracture of two-dimensional soft materials with finite deformation. In this method, the governing equations for the fracture of soft materials are derived by integrating the phase-field fracture model with the total Lagrangian material point method (TLMPM), and corresponding discrete equations are then formulated with explicit time integration. To address the significant computational issue in terms of memory and processing time, an adaptive technique for dynamically splitting particles and background grids in the phase-field TLMPM is proposed, based on the phase-field values of the particles. To further maintain continuity of the physical field throughout the computational process and consider the characteristics of the field update, an information remapping strategy is developed. Several representative numerical examples are presented to demonstrate the accuracy and efficiency of the proposed APTLMPM by comparing the simulation results with experimental data and those as obtained with other numerical methods.
... Owing to its compact and coherent framework, this method shows significant potential for addressing more complex cracking challenges, such as those induced by THM coupling. In domain of rock mechanics, various specific models have been developed for addressing cracking in brittle-ductile behavior (Fang et al., 2019;Choo and Sun, 2018;Wang et al., 2023). The phase-field method has also been applied to cracking modeling with thermo-hydromechanical coupling (Miehe et al., 2015;Wang et al., 2022;Yu et al., 2023a). ...
This study conducts a numerical analysis of long-term thermo-hydromechanical (THM) processes, with a particular focus on deep geological disposal of radioactive waste. It emphasizes the modeling of damage zones resulting from excavation activities, as well as changes in pore pressure and temperature. The numerical model presented in this paper delineates the fundamental relations of thermo-poro-elasticity. It also introduces a double phase field model specifically formulated for rock materials under compressive stress, taking into account THM coupling and time-dependent behavior. The proposed model has been implemented in a finite element code and applied to numerical analysis of short and long terms responses around a horizontal borehole in the context of the French High-Level Waste (HLW) disposal concept. In particular, displacements, temperature changes, pore fluid pressure variations as well as evolution of induced damage zones are presented and analyzed for different periods including excavation, operation and post-closure. It is found that the occurrence of damage zones is clearly related to the spacing distance between two neighboring disposal boreholes. The time-dependent deformation of host rock plays an significant role in the long-term responses.
... The phase-field method has gained prominence recently due to its advantages of not relying on explicit fracture criteria and its ability to accurately trace complex fracture surfaces [37]. This method has found extensive applications in simulating dynamic damage propagation [38][39][40][41][42][43][44][45], ductile fracture in metals [46][47][48][49][50][51], and fatigue crack propagation [19,[52][53][54][55][56][57]. ...
... A review of different approaches can be found in Alessi et al. (2018a) and Marengo and Perego (2023a). In the current work, the attention is focused on the effective stress approach (see, e.g., Ulloa et al. (2016), Miehe et al. (2017, Choo and Sun (2018), Huang and Gao (2019), Wambacq et al. (2021) and Marengo and Perego (2023b)), combined with an AT1 (Ambrosio and Tortorelli, 1990) phase-field dissipation model (see, e.g., Alessi et al. (2018b), Samaniego et al. (2021), Hu et al. (2021), Talamini et al. (2021), Wambacq et al. (2021) and Marengo and Perego (2023b)). The effective stress is defined as the true stress acting on the continuous part of the damaged material. ...
... With the approximation of sharp fractures by a regularized smear crack density field, this method is able to easily deal with the nucleation and propagation of cracks. In this framework, various specific models have been developed for addressing cracking in plastic materials [30][31][32], under dynamic loading etc. The phase-field method has also been applied to cracking modeling with thermo-hydromechanical coupling [33][34][35]. ...
This paper addresses numerical analysis of thermo-hydromechanical processes in the context of deep geological disposal of radioactive waste. The emphasis is put on modeling of damaged zones induced by excavation, pore pressure and temperature changes. The theoretical background of thermo-poroelasticity for saturated porous media is first recalled. The framework for modeling initiation and evolution of induced cracks is then presented by using a variational approach of phase-field method. A specific model with two crack phase fields and considering thermo-hydromechanical interaction is proposed. A particular attention is paid on the description of shear cracks. The proposed model is implemented in the standard finite element method. An example of application is finally presented on the analysis of thermo-hydromechanical responses and cracked zones evolution around a typical disposal repository in the context of French concept for high level waste disposal.
... In this paper, we focus on the phase-field model which originates from the variational fracture model [27,28] and can also be treated as a gradient damage model [20,21,29,30]. Its applications span cohesive fracture [31,32], ductile fracture [33][34][35][36][37][38], dynamic fracture [39][40][41], multi-physically induced fracture [42][43][44][45][46], and so on. To enhance the multiscale analysis of fracture, part of the authors proposed a micromechanics-enhanced phase-field model [16,47] that extends the well-developed micromechanical damage models to be nonlocal, thus resolving the first two problems mentioned above. ...
By incorporating two different fracture mechanisms and salient unilateral effects in rock materials, we propose a thermomechanical phase-field model to capture thermally induced fracture and shear heating in the process of rock failure. The heat conduction equation is derived, from which the plastic dissipation is treated as a heat source. We then ascertain the effect of the non-associated plastic flow on frictional dissipation and show how it improves the predictive capability of the proposed model. Taking advantage of the multiscale analysis, we propose a phase-field-dependent thermal conductivity with considering the unilateral effect of fracture. After proposing a robust algorithm for solving involved three-field coupling and damage-plasticity coupling problems, we present three numerical examples to illustrate the abilities of our proposed model in capturing various thermo-mechanically coupled behaviors.
... However, our purpose in this work is to model shock in which the material is subjected to an extreme environment with significantly higher mean pressure (and temperature) under shock, of which microstructural evolution under shock profoundly different than those obtained from the triaxial compression test used to calibrated the cap plasticity models, often under the normal temperature range. Due to the fact that significant increase in temperature as well as the increase of loading rate often reduce the strength but increase the ductility of the materials ( [10,48]), this extreme environment makes the location of the yield surface identified near the hydrostatic axis in the normal temperature range questionable. As such, we did not use cap-plasticity to replicate the plastic volumetric responses near the hydrostatic axis. ...
Shock waves in geological materials are characterized by a sudden release of rapidly expanding gas, liquid, and solid particles. These shock waves may occur due to explosive volcanic eruptions or be artificially triggered. In fact, underground explosions have often been used as an engineering solution for large-scale excavation, stimulating oil and gas recovery, creating cavities for underground waste storage, and even extinguishing gas field fires. As such, hydrocodes capable of simulating the rapid and significant deformation under extreme conditions can be a valuable tool for ensuring the safety of the explosions. Nevertheless, as most of the hydrocodes are often formulated in an Eulerian grid, this setting makes it non-trivial to track the deformation configuration of the materials without a level set. The objective of this paper is to propose the use of the material point method equipped with appropriate equation of state (EOS) models as a hydrocode suitable to simulate underground explosions of transverse isotropic geomaterials. To capture the anisotropic effect of the common layered soil deposits, we introduce a new MPM hydrocode where an anisotropic version of the Mie-Gruneisen equation of state is coupled with a frictional Drucker-Prager plasticity model to replicate the high-strain-rate constitutive responses of soil. By leveraging the Lagrangian nature of material points to capture the historical dependence and the Eulerian calculation of internal force, the resultant model is capable of simulating the rapid evolution of geometry of the soil as well as the high-strain-rate soil mechanics of anisotropic materials.
... Still, the overall ductile fracture behavior remains noticeable in our simulations (see Section 4). The generalization of our current model to incorporate plastic stored energy into the crack propagation driving force could be straightforward [Choo and Sun, 2018]. Combining the total internal energy input rate (9) with the total available free energy (12) and applying the Clausius-Duhem inequality and the Coleman-Noll argument, the following constitutive relations could be reached: ...
We present a phase-field fracture model for a stress resultant geometrically exact shell in finite deformation regime where the configuration manifold evolves according to deformation and fracture. The Reissner-Mindlin shell problem is first solved via the finite element method, where the independent unknown fields are the displacement and director. The phase-field ductile fracture model is then coupled with the verified geometrically exact shell by enriching the three-field elasto-plastic free energy with a regularized crack surface energy. To capture the geometrical nonlinearity due to the large plastic deformation exhibited in the processing zone, the corresponding finite-strain stress resultant elasto-plasticity model is coupled with the phase field model. This coupling between the stress resultant elasto-plasticity model and the phase-field fracture model enables us to predict the different amounts of energy dissipation attributed to plastic deformation and fracture and hence simulate the crack propagation in the ductile regime properly. We introduce a mixed finite element model with displacement, director, and phase-field order parameters as the nodal degree of freedoms formulated on the mid-surface. An energy-based arc-length method is generalized to track the equilibrium path and mitigate instabilities arising from material nonlinearity. Four numerical examples are presented to validate the implementation of the model and demonstrate its capability to simulate ductile fracture in shell structures. These examples include a plane-stress tension/shear fracture model, the Muscat-Fenech and Atkins plate, axial tension of a notched cylindrical shell, and ductile fracture of a simply supported plate under uniform pressure.
... Furthermore, the method can be readily extended to address multiple cracks in three-dimensional problems. Most recently, a phase field model considering irreversible thermodynamics was proposed to describe the ductile cracking in plastic material in [42]. The micro-structure based elasto-plastic phase field model is developed by [43,44]. ...
This study is part of numerical simulations performed on an in-situ heating test conducted by the French National Radioactive Waste Management Agency (Andra) at the Meuse/Haute-Marne Underground Research Laboratory (URL) to study the hydromechanical behavior of the host Callovo-Oxfordian COx claystone in quasi real conditions through the international research project DECOVALEX. In this study, we present a numerical analysis of damage and cracking behavior due to gas effects on hydromechanical processes in unsaturated claystone, while considering the distribution of material heterogeneity. The proposed model is implemented in a finite element code designed to solve hydromechanical coupling problems under unsaturated conditions. The nucleation and propagation of cracks are described using an extended phase-field method, which takes into account the effects of gas and liquid pressure on the evolution of the phase-field. In particular, a macroscopic
elastic model is determined using two steps of homogenization, which considers the effects of porosity and mineral inclusions. The spatial variability of these factors is modeled using the Weibull distribution function. Thus, the nucleation of cracks is directly influenced by the spatial distribution of material heterogeneity. The proposed model is applied to 3D benchmarks of gas injection in the context of radioactive waste disposal. The
process, in which the overpressure-induced damage zone affects the behavior of the two-phase flow during gas injection, is well reproduced.
... In geotechnical structures, rocks are mostly under complex compressive stresses and cracks show a composite extensional pattern of tension-compression-shear. Therefore, to describe the damage fracture behavior of rock materials, relevant preliminary studies have been conducted on the damage fracture phase field model of rock materials (Bryant & Sun 2018Fei & Choo, 2020a, 2020bZhang et al., 2017;. Choo and Sun (2018) proposed a coupling model of damage phase field and plastic deformation based on the Drucker-Prager plastic model with compression cap, considering the influence of confining pressure and strain rate on the elastic-brittle and plastic flow of rock materials. Based on the Palmer-Rice fracture theory, Fei and Choo (2020a) proposed a phase field model of frictional shear fracture of geological materials by considering the frictional slip of geological shear fractures and deducing the threshold of fracture shear energy. ...
Damage and fracture are the most extensive failure modes of rock materials, which may easily induce disaster and instability of engineering structures. This study developed a nonlocal damage fracture phase field model for rocks considering the heterogeneity of rocks. The modified phase field model introduced the heterogeneity of fracture parameters and modified the governing equations. Meanwhile, the free energy was constructed by the elastic strain energy sphere‐bias decomposition and the plastic strain energy. As for the numerical implementation, the three layers finite elements method structure was used in the frame of the finite element method. The ability of the modified phase field model has been illustrated by reproducing the experiment results of rock samples with pre‐existing cracks under compression.
... presents the post-failure images and schematics of the different failure modes observed in the mudstone specimens tested saturated conditions at 0, 50, and 130 MPa. Four failure modes can be identified: axial splitting fracture, shear fracture, shear bands, and plastic flow(Basu et al. 2013;Choo and Sun 2018). All tested mudstone lithofacies exhibited axial splitting fractures at 0 MPa confining pressure. ...
The influence of water on rock deformation and failure behavior is a critical variable to consider when developing unconventional shale reservoirs. To better understand this effect, we conducted a series of triaxial compression tests on four mudstone lithofacies, namely siliceous calcareous mudstones, siliceous mudstones, calcareous mudstones, and carbonate-rich mudstones from the Naparima Hill Formation, under water-saturated conditions at confining pressures up to 130 MPa. Our results showed that the mudstones displayed brittle, brittle–ductile transition, and ductile failure behaviors as the confining pressure increased. Mudstones with low strength, high porosity, and high silica and clay contents exhibited a strong water-weakening effect, leading to a significant reduction in failure strength. Conversely, high strength, low porosity, and carbonate-rich mudstones showed minor water-weakening effects. We also developed a failure behavior model to predict the brittle, brittle–ductile transition, and ductile zones in the mudstones. The model showed that the transition from brittle to ductile behavior occurs over a wide range of confining pressures, and it occurs at a lower pressure threshold in saturated compared to dry conditions. Additionally, the model suggests that the presence of water results in a wider transition zone between brittle and ductile behavior, as well as a larger ductile zone. Overall, our findings suggest that the water-weakening effect can significantly decrease the depth at which the brittle–ductile transition occurs in mudstones, emphasizing the importance of considering water as a critical variable in rock deformation and failure behavior.
... Extensions such as use of a phase field to represent ductile fracture in an otherwise elastic-plastic continuum have become widespread. 36,37 Enriched theories for structural transformations in nonlinear materials incorporating concepts and tools of Finsler geometry were set forth in 2016, [38][39][40] whereby connections with PFM, micropolar, micromorphic, 41 and other gradient-based models were demonstrated. Use of a Riemannian rather than Finslerian metric enabled reduction of the equilibrium equations derived for certain classes of energy potentials of the Finsler geometric theory 39,42 to those of the PFM. ...
... The starting point in these studies is the definition of a scalar-valued and space-time-dependent phase-field variable in the range [0, 1], which allows for differentiation between the cracked and the intact regions of the domain. With the PFM, the sharp edges of the crack are approximated by diffusive ones, whereas the width of this region is controlled via an internal length scale, see, e.g., [6,12,16,17,21,49,53,58,62,63,70,73,74] among others. In this context, it is worth mentioning that although the PFM is intensively applied for fracture modeling, it is also widely used in the simulation of many other engineering applications, as phase-change materials, see, e.g., [4,[94][95][96]104], for references. ...
This research aims to extend the isothermal continuum mechanical modeling framework of hydraulic fracturing in porous materials to account for the non-isothermal processes. Whereas the theory of porous media is used for the macroscopic material description, the phase-field method is utilized for modeling the crack initiation and propagation. We proceed in this study from a two-phase porous material consisting of thermomechanically interacting pore fluid and solid matrix. The heat exchange between the fluid in the crack and the surrounding porous environment through the diffusive fracture edges is carefully studied, and new formulations here are proposed. Besides, temperature-dependent solid and fluid material parameters are taken into account, which is of particular importance in connection with fluid viscosity and its effect on post-cracking pressure behavior. This continuum mechanical treatment results in strongly coupled partial differential equations of the mass, the momentum, and the energy balance of the thermally non-equilibrated constituents. Using the finite element method, two-dimensional initial-boundary-value problems are presented to show, on the one hand, the stability and robustness of the applied numerical algorithm in solving the emerged strongly coupled problem in the convection-dominated heat transport state. On the other hand, they show the capability of the modeling scheme in predicting important instances related to hydraulic fracturing and the role of the temperature field in this process. Additionally, they show the importance of using stabilization techniques, such as adding an artificial thermo-diffusivity term, to mitigate temperature fluctuations at high flow velocity.
... Of all the existing computational models, phase field modeling (PFM) has proven to be efficient in capturing crack nucleation and tracking crack paths [37,38]. PFM is based on the theoretical aspects of Griffith's fracture criteria and the early works of Bourdin [39], Francfort and Marigo [40]. ...
Elastomers and composites made thereof have wide applications, e.g., in automobile, aerospace, and civil engineering. Predicting fracture in such materials is crucial for efficient design and optimum utilization. These materials are oftentimes hyperelastic and anisotropic in nature and in general subjected to mixed mode loading rather than merely pure modes. Soft biological tissues can also be considered anisotropic hyperelastic materials. Computational modeling helps in studying the role of different sources influencing mixed-mode fracture. A unifying thermodynamically consistent anisotropic phase field formulation for modeling the mixed-mode fracture of hyperelastic soft materials like elastomers, elastomeric fiber-reinforced composites, and soft biological tissues at finite strains is proposed and formulated. To model the mechanical response of anisotropic hyperelastic materials subjected to mixed-mode loading, a coupled Neo-Hookean model with orthotropic anisotropy is adopted considering volumetric-deviatoric and a tension-compression decomposition. For modeling the complex crack initiation and propagation, a phase field method based on a power law criterion is adopted by considering a single order parameter as the damage variable. This model is suitable for capturing the overall response of soft fiber-reinforced elastomeric composites as well as soft biological tissues. The proposed model is validated by conducting fracture tests on (a) silicone elastomers, (b) unidirectional fiber-reinforced elastomeric composites, (c) natural rubber reinforced with black carbon, and (d) brain tissue reinforced with axons. The results obtained are compared with experimental and numerical investigations from literature.
This work aims to provide a computational model that can describe the complex behaviour of refractory industrial components under working conditions. Special attention is given to the asymmetric tension-compression behaviour and its evolution in the full range of working temperatures. The model accounts for inelastic flow in compression and brittle fracture behaviour in tension by leveraging the continuum-mechanics theory of plasticity and phase-field fracture damage. The model is implemented in the Finite Element open-source platform FEniCS and is used to analyze the fracture phenomenon in the refractory plate used in ladle slide gate systems to control the liquid steel flow from the ladle to the tundish.
The phase-field method has become popular for the numerical modeling of fluid-filled fractures, thanks to its ability to represent complex fracture geometry without algorithms. However, the algorithm-free representation of fracture geometry poses a significant challenge in calculating the crack opening (aperture) of phase-field fracture, which governs the fracture permeability and hence the overall hydromechanical behavior. Although several approaches have been devised to compute the crack opening of phase-field fracture, they require a sophisticated algorithm for post-processing the phase-field values or an additional parameter sensitive to the element size and alignment. Here, we develop a novel method for calculating the crack opening of fluidfilled phase-field fracture, which enables one to obtain the crack opening without additional algorithms or parameters. We transform the displacement-jump-based kinematics of a fracture into a continuous strain-based version, insert it into a force balance equation on the fracture, and apply the phase-field approximation. Through this procedure, we obtain a simple equation for the crack opening which can be calculated with quantities at individual material points. We verify the proposed method with analytical and numerical solutions obtained based on discrete representations of fractures, demonstrating its capability to calculate the crack opening regardless of the element size or alignment.
The diffuse damage and localized cracking of quasi-brittle materials (i.e., rocks and concrete) under compression can be delineated by a matrix-microcrack system, wherein a solid matrix phase is weakened by a large number of randomly oriented and distributed microcracks, and the macroscopic cracking is formed by a progressive evolution of microcracks. Several homogenization-based multiscale models have been proposed to describe this matrix-microcrack system, but most of them are based on a linear friction law on the microcrack surface, rendering a linear strength criterion. In this paper, we propose a new quadratic friction law within the local multiscale friction-damage (LMFD) model to capture the plastic distortion due to frictional sliding along the rough microcrack surface. Following that, a macroscopic Ottosen-type nonlinear strength criterion is rationally derived with up-scaling friction-damage coupling analysis. An enhanced semi-implicit return mapping (ESRM) algorithm with a substepping scheme is then developed to integrate the complex nonlinear constitutive model. The performance of LMFD model is evaluated compared to a wide range of experimental data on plain concretes, and the robustness of ESRM algorithm is assessed through a series of numerical tests. Subsequently, to effectively describe the localized cracking process, a regularization scheme is proposed by combining the phase-field model with the established LMFD model, and the discretization independent crack localization is numerically verified.
An adaptive phase-field total Lagrangian material point method (APTLMPM) is proposed in this paper for effectively simulating the dynamic fracture of soft materials with finite deformation. In this method, the governing equations for the fracture of soft materials are derived by integrating the phase-field fracture model with the total Lagrangian material point method (TLMPM), and corresponding discrete equations are then formulated with explicit time integration. To address the significant computational issue in terms of memory and processing time, an adaptive technique for dynamically splitting particles and background grids in the phase-field TLMPM is proposed, based on the phase-field values of the particles. To further maintain continuity of the physical field throughout the computational process and consider the characteristics of the field update, an information remapping strategy is developed. Several representative numerical examples are presented to demonstrate the accuracy and efficiency of the proposed APTLMPM by comparing the simulation results with experimental data and those as obtained with other numerical methods.
Crack initiation, growth, and coalescence in flawed rocks have been extensively studied for 2D (planar, penetrating) flaws under uniaxial/biaxial compression. However, little is known as to the mechanisms and processes of cracking from 3D flaws under true triaxial compression, where the intermediate principal stress (σ2) is distinguished from the major and minor principal stresses. In this work, we systematically investigate the effects of σ2 on the 3D cracking behavior of rock specimens with preexisting flaws, through the use of mechanistic simulations of mixed-mode fracture in rocks. We explore how two characteristics of σ2, namely, (i) its orientation with respect to the flaw and (ii) its magnitude, affect two aspects of the cracking behavior, namely, (i) the cracking pattern and (ii) the peak stress. Results show that the orientation of σ2 exerts more control over the cracking pattern than the flaw inclination angle. The peak stress becomes highest when σ1 is parallel to the flaw, whereas it becomes lowest when σ2 is parallel to the flaw. Also, the effects of σ2 magnitude are more significant when σ2 becomes more oblique to the flaw plane. On the basis of our observations, we propose mechanisms underlying the cracking behavior of 3D flawed rocks under true triaxial compression.
Highlights
The effects of the intermediate principal stress (σ2) on the 3D cracking behavior of flawed rocks under true triaxial compression are systematically investigated.
The orientation of σ2 exerts more control over the cracking pattern than the flaw inclination angle.
The effects of σ2 magnitude become more significant when σ2 are more oblique to the flaw plane.
Mechanisms underlying the cracking behavior of 3D flawed rocks under true triaxial compression are proposed based on the observations made.
This paper presents an overview of the theories and computer implementation aspects of phase field models (PFM) of fracture. The advantage of PFM over discontinuous approaches to fracture is that PFM can elegantly simulate complicated fracture processes including fracture initiation, propagation, coalescence, and branching by using only a scalar field, the phase field. In addition, fracture is a natural outcome of the simulation and obtained through the solution of an additional differential equation related to the phase field. No extra fracture criteria are needed and an explicit representation of a crack surface as well as complex track crack procedures are avoided in PFM for fracture, which in turn dramatically facilitates the implementation. The PFM is thermodynamically consistent and can be easily extended to multi-physics problem by 'changing' the energy functional accordingly. Besides an overview of different PFMs, we also present comparative numerical benchmark examples to show the capability of PFMs.
This study is part of numerical simulations performed on an in-situ heating test conducted by the French National Radioactive Waste Management Agency (Andra) at the Meuse/Haute-Marne Underground Research Laboratory (URL) to study the thermo-hydromechanical behavior of the host Callovo-Oxfordian COx claystone in quasi real conditions, through the international research project DECOVALEX. We present a numerical study of damage and cracking process in saturated claystone subjected to thermo-hydromechanical coupling by considering material heterogeneity distribution. For this purpose, a macroscopic elastic model is first determined by using two steps of homogenization by taking into account the effects of porosity and mineral inclusions. This model is implemented into a finite element code devoted to solving thermo-hydromechanical coupling problems. The nucleation and propagation of cracks are described by using an extended phase-field method, considering the effects of temperature and fluid pressure on the evolution of phase-field. The proposed model is applied to the numerical analysis of cracking process due to excavation and heating around a group of boreholes (CRQ). The numerical results of the 3D simulation are compared with in-situ measurements of temperature and pore pressure distribution. The excavation damage zone and heating fracture is reproduced and analysed according to the structure of the heating position and the heterogeneity of the rock.
In this study, earth blocks (EB) and compressed earth blocks (CEB) are fabricated and investigated along with the development of a mathematical model of compressive and tensile loads. The investigated specimens are manufactured from a mixture of soil, ground recycled concrete (GRC) powder and water in weight fractions of 4:1:1. EBs are molded and CEBs are obtained by quasi-static compression in wet state. Material samples extracted from the blocks are tested in dry state for compressive strength and three-point flexural strength according to ASTM standards. The results indicate that CEB exhibits 130% higher compressive load capacity and 63% higher tensile strength compared to EB. Furthermore, the proposed and calibrated mathematical model is able to adequately describe the strength and damage behavior of both materials. Finally, microstructural/micromechanical interrelationships with the modeled material response are established based on a characterization of postmortem CEB samples using EDS elemental analysis and SEM micrographs techniques.
We present an extended finite element framework to numerically study competing hydraulic fracture propagation. The framework is capable of modeling fully coupled hydraulic fracturing processes including fracture propagation, elastoplastic bulk deformation and fluid flow inside both fractures and the wellbore. In particular, the framework incorporates the classical orifice equation to capture fluid pressure loss across perforation clusters linking the wellbore with fractures. Dynamic fluid partitioning among fractures during propagation is solved together with other coupled factors, such as wellbore pressure loss (Δpw), perforation pressure loss (Δp), interaction stress (σint) and fracture propagation. By numerical examples, we study the effects of perforation pressure loss and wellbore pressure loss on competing fracture propagation under plane-strain conditions. Two dimensionless parameters Γ=σint/Δp and Λ=Δpw/Δp are used to describe the transition from uniform fracture propagation to preferential fracture propagation. The numerical examples demonstrate the dimensionless parameter Γ also works in the elastoplastic media.
We propose a mechanical and computational model to describe the coupled problem of poromechanics and cracking in variably saturated porous media. A classical poromechanical formulation is adopted and coupled with a phase-field formulation for the fracture problem. The latter has the advantage of being able to reproduce arbitrarily complex crack paths without introducing discontinuities on a fixed mesh. The obtained simulation results show good qualitative agreement with desiccation experiments on soils from the literature.
Phase-field formulations have recently emerged as promising tools to model brittle fracture. Based on the variational approach to fracture, these models aim at overcoming some of the computational challenges found in simulating complex fracture patterns and their evolution due to external or internal loads. Since most applications and validation exercises thus far have been restricted to academic benchmarks, the evaluation of phase-field fracture models against experimental results an practical engineering scenarios remains fragmented.
Here we introduce a straightforward phase-field approach to simulate fluid and mechanically-driven fractures based on energy minimization and thermodynamical principles. We apply our methodology to several laboratory experiments of brittle fracture, and to fracturing processes in two full-scale concrete dams, taking into account the hydraulic forces inside the fractures. We conclude that phase-field models represent a promising computational tool that may be applied to realistic engineering scenarios.
We performed an extensive suite of true triaxial experiments in two porous sandstones, Bentheim (porosity ≈ 24%) and Coconino (17%). The experiments were conducted using a novel loading path, which maintains constant Lode angle (Θ) throughout the test. This path enabled the examination of the effects of Lode angle and mean stress on failure. Our tests covered σ3 magnitudes between 0 and 150 MPa, and of Θ at -30° (axisymmetric extension), -16°, 0°, +11°, +21°, and +30° (axisymmetric compression). Test results revealed the respective contribution of each of the two stress invariants to failure stress, the failure-plane angle, and failure mode. In both sandstones, the shear stress required for failure increases with mean stress but decreases with Θ when shear failure mode dominates. However, the dependence of failure stress on mean stress and Θ is reversed when the compactive failure mode is in control. The compactive failure mode was evident in Bentheim sandstone when compaction bands were observed under high mean stress. The Coconino sandstone did not reach the compactive failure regime within the maximum confinement applied. The failure-plane angle monotonically decreases with increasing mean stress and Θ. For Coconino sandstone, failure-plane angle varies between 80° and 50° for σoct,f between 50 and 450 MPa whereas it drops to 0° as σoct,f, the mean stress at failure, approaches 250 MPa in Bentheim sandstone. We employed the bifurcation theory to relate the stress conditions at failure to the development of failure-planes. The theory is in qualitative agreement with the experimental data.
The phase-field model has been attracting considerable attention due to its capability of capturing complex crack propagations without mesh dependence. However, its validation studies have primarily focused on the ability to predict reasonable, sharply defined crack paths. Very limited works have so far been contributed to estimate its accuracy in predicting force responses, which is majorly attributed to the difficulty in the determination of the length scale. Indeed, accurate crack path simulation can be achieved by setting the length scale to be sufficiently small, whereas a very small length scale may lead to unrealistic force-displacement responses and overestimate critical structural loads. This paper aims to provide a critical numerical investigation of the accuracy of phase-field modelling of brittle fracture with special emphasis on a possible formula for the length scale estimation. Phase-field simulations of a number of classical fracture experiments for brittle fracture in concretes are performed with simulated results compared with experimental data qualitatively and quantitatively to achieve this goal. Furthermore, discussions are conducted with the aim to provide guidelines for the application of the phase-field model.
Hydromechanical interactions between fluid flow and deformation in porous geo- materials give rise to a wide range of societally important problems such as land- slides, ground subsidence, and injection-induced earthquakes. Many geomaterials in these problems possess two-scale porous structures due to fractures, particle ag- gregation, or other reasons. However, coupled hydromechanical processes in these multiscale porous materials, such as ground deformation caused by preferential flow, are beyond the modeling capabilities of classical frameworks.
This thesis develops theoretical and computational frameworks for fully cou- pled hydromechanical modeling of geomaterials with two-scale porous structures. Adopting the concept of double porosity, we treat these materials as a multiscale continuum in which two pore regions of different scales interact within the same continuum. Three major developments are presented.
First, we build a mathematical framework for thermodynamically consistent modeling of unsaturated porous media with double porosity. Conservation laws are formulated incorporating an effective stress tensor that is energy-conjugate to the rate of deformation tensor of the solid matrix. Based on energy-conjugate pairs identified in the first law of thermodynamics, we develop a constitutive framework for hydrological and mechanical processes coupled at two scales.
Second, we introduce a novel constitutive framework for elastoplastic mate- rials with evolving internal structures. By partitioning the thermodynamically consistent effective stress into two individual, single-scale effective stresses, this framework uniquely distinguishes proportional volume changes in the two pore regions under finite deformations. This framework accommodates the impact of pore pressure difference between the two scales on the solid deformation, which was predicted by thermodynamic principles. We show that the proposed frame- work not only improves the prediction of deformation of two-scale geomaterials, but also simulates secondary compression effects due to delayed pressure dissipa- tion in the less permeable pore region.
Third, we develop a finite element framework that enables the use of compu- tationally efficient equal-order elements for solving coupled fluid flow and defor- mation problems in double-porosity media. At the core of the finite element for- mulation is a new method that stabilizes twofold saddle point problems arising in the undrained condition. The stabilized finite elements allow for equal-order linear interpolations of three primary variables—the displacement field and two pore pressure variables—throughout the entire range of drainage conditions.
Rock response to confining pressure and strain rate can change dramatically from very brittle to ductile. Capturing this transition is crucial for a correct prediction of rock mass failure due to blasting, explosion or drilling in mining. In this work, a new constitutive model that accounts for the effects of both confining pressure and strain rate on the nominal strength and post peak behaviour is proposed for dry intact rocks and other similar geological materials. The key features of the proposed constitutive model are the employment of a single loading function that evolves from initial yielding to ultimate failure during damaging and the rate-dependent enhancement so that the strain rate effects can be faithfully described at different confining pressures. The model can adequately capture both the brittle and ductile responses as well as the brittle-ductile transition as a result of both strain rate and confining pressure.
An Arlequin poromechanics model is introduced to simulate the hydro-mechanical coupling effects of fluid-infiltrated porous media across different spatial scales within a concurrent computational framework. A two-field poromechanics problem is first recast as the twofold saddle point of an incremental energy functional. We then introduce Lagrange multipliers and compatibility energy functionals to enforce the weak compatibility of hydromechanical responses in the overlapped domain. To examine the numerical stability of this hydro-mechanical Arlequin model, we derive a necessary condition for stability, the twofold inf–sup condition for multi-field problems, and establish a modified inf–sup test formulated in the product space of the solution field. We verify the implementation of the Arlequin poromechanics model through benchmark problems covering the entire range of drainage conditions. Through these numerical examples, we demonstrate the performance, robustness, and numerical stability of the Arlequin poromechanics model.
Hydraulic fracturing is a big issue in the exploitation of oil and gas resources as well as in the production of heat in deep geothermal energy plants. Investigating hydraulic fracturing processes numerically by means of a finite-element analysis, one has to address the porous solid and its pore content within a fully coupled computational approach. For this purpose, the present article combines the well-established Theory of Porous Media with elements of fracture mechanics, especially, with the phase-field approach to fracture, which has proven as a successful tool for the computation of fracturing processes in the field of standard solid mechanics.
Although hydraulic fracturing is widely applied in practice, this procedure has not yet been investigated adequately by means of a full theoretical and computational framework on the basis of a multicomponent medium tackling a porous solid skeleton and its pore content with their mutual interaction of deformation and fracture, and fluid-driven processes both in the solid bulk and cracking domains. Addressing these features, the article concentrates on a permeable elastic solid skeleton, where the fracturing process is governed by brittle fracture driven either by a prescribed fluid pressure or by a prescribed fluid influx. Two- and three-dimensional numerical examples computed by use of the coupled solver PANDAS exhibit the possibilities of this approach.
We conducted an extensive suite of true triaxial experiments in two porous sandstones, Bentheim (porosity ≈ 24%) and Coconino (17.5%). Our experiments demonstrate that failure of both sandstones is not only a function of σ3, but also of σ2. For a given σ3, σ1 at failure (σ1,peak) increases as σ2 is raised above σ3 between tests. σ1,peak reaches a peak as σ2 is about halfway between σ3 and σ1 and then gradually decreases such that when σ2 ≈ σ1,peak, it approaches its initial magnitude when σ2 = σ3. For a constant σ3, failure-plane angle increases with σ2 by a maximum of less than 10° as σ2 rises from σ2 = σ3 to σ2 = σ1,peak. The effect of σ2 on both failure level and failure-plane angle is stronger in the lower porosity Coconino sandstone than in the Bentheim sandstone. The σ2-dependence of failure mode in the Bentheim is different than Coconino over the same σ3 range. Both sandstones failed dilatantly at low σ3 magnitudes. However, at high σ3 (100-120 MPa), Bentheim sandstone developed shear-enhanced compaction bands, followed by pure compaction bands at σ3 = 150 MPa. Compaction bands were not observed in the Coconino. Microscopic observations via SEM reveals that tensile microcracking is dominant when shear banding occurs (under low σ3), while pervasive grain crushing and pore collapse inside compaction bands are observed at high σ3.
Continuum porous media theories, extended by a diffusive phase-field modeling (PFM) approach, introduce a convenient and efficient tool to the simulation of hydraulic fracture in fluid-saturated heterogeneous materials. In this, hydraulic- or tension-induced fracture occurs in the solid phase. This leads to permanent local changes in the permeability, the volume fractions of the constituents as well as the interstitial-fluid flow. In this work, the mechanical behaviors of the multi-field, multi-phase problem of saturated porous media, such as the pore-fluid flow and the solid-skeleton deformation, are described using the macroscopic Theory of Porous Media (TPM). To account for crack nucleation and propagation in the sense of brittle fracture, the energy-minimization-based PFM procedure is applied, which approximates the sharp edges of the crack by a diffusive transition zone using an auxiliary phase-field variable. Furthermore, the PFM can be implemented in usual continuum finite element packages, allowing for a robust solution of initial-boundary-value problems (IBVP). For the purpose of validation and comparison, simulations of a two-dimensional IBVP of hydraulic fracture are introduced at the end of this research paper.
This paper deals with the hydromechanical modelling of the initiation of failure in soils with particular reference to landslides. To this end, localized and diffused failure modes are simulated with a finite element model for coupled elasto-plastic variably saturated porous geomaterials, in which the material point instability is detected with the second-order work criterion based on Hill’s sufficient condition of stability. Three different expressions of the criterion are presented, in which the second-order work is expressed in terms of generalized effective stress, of total stress and thirdly by taking into account the hydraulic energy contribution for partially saturated materials. The above-mentioned computational framework has been applied to study two initial boundary value problems: shear failure of a plane strain compression test of globally undrained water-saturated dense sand (where cavitation occurs at strain localization) and isochoric grain matter, and the onset of a flowslide from southern Italy due to rainfall (Sarno-Quindici events, May 5–6 1998). It is shown that the second-order work criterion applied at the material point level detects the local material instability and gives a good spatial indication of the extent of the potentially unstable domains in both the localized and diffused failure mechanisms of the cases analyzed, is able to capture the instability induced by cavitation of the liquid water and gives results according to the time evolution of plastic strains and displacement rate.
This work outlines a rigorous variational-based framework for the phase field modeling of ductile fracture in elastic–plastic solids undergoing large strains. The phase field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with geometric features rooted in fracture mechanics. It has proven immensely successful with regard to the analysis of complex crack topologies without the need for fracture-specific computational structure such as finite element design of crack discontinuities or intricate crack-tracking algorithms. Following the recent work Miehe et al. (2015), the phase field model of fracture is linked to a formulation of gradient plasticity at finite strains. The formulation includes two independent length scales which regularize both the plastic response as well as the crack discontinuities. This ensures that the damage zones of ductile fracture are inside of plastic zones, and guarantees on the computational side a mesh objectivity in post-critical ranges. The novel aspect of this work is a precise representation of this framework in a canonical format governed by variational principles. The coupling of gradient plasticity to gradient damage is realized by a constitutive work density function that includes the stored elastic energy and the dissipated work due to plasticity and fracture. The latter represents a coupled resistance to plasticity and damage, depending on the gradient-extended internal variables which enter plastic yield functions and fracture threshold functions. With this viewpoint on the generalized internal variables at hand, the thermodynamic formulation is outlined for gradient-extended dissipative solids with generalized internal variables which are passive in nature. It is specified for a conceptual model of von Mises-type elasto-plasticity at finite strains coupled with fracture. The canonical theory proposed is shown to be governed by a rate-type minimization principle, which fully determines the coupled multi-field evolution problem. This is exploited on the numerical side by a fully symmetric monolithic finite element implementation. The performance of the formulation is demonstrated by means of some representative examples.
We propose a phase-field model for ductile fracture in a single crystal within the kinematically linear regime, by combining the theory of single crystal plasticity as formulated in Gurtin et al. [1] and the phase-field formulation for ductile fracture proposed by Ambati et al. [2]. The model introduces coupling between plasticity and fracture through the dependency of the socalled degradation function from a scalar global measure of the accumulated plastic strain on all slip systems. A viscous regularization is introduced both in the treatment of plasticity and in the phase-field evolution equation. Testing of the model on two examples for face centred cubic single crystals indicates that fracture is predicted to initiate and develop in the regions of the maximum accumulated plastic strain, which is in agreement with phenomenological observations. A rotation of the crystallographic unit cell is shown to affect the test results in terms of failure pattern and corresponding global and local response.
This work outlines a novel variational-based theory for the phase-field modelling of ductile fracture in elastic-plastic solids undergoing large strains. The phase-field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modelling. It is linked to a formulation of gradient plasticity at finite strains. The framework includes two independent length scales which regularize both the plastic response as well as the crack discontinuities. This ensures that the damage zones of ductile fracture are inside of plastic zones, and guarantees on the computational side a mesh objectivity in post-critical ranges.
In this paper we present a phase field model for a proppant-filled fracture in a poroelastic medium. The formulation of the coupled system involves four unknowns; displacements, phase field, pressure, and proppant concentration. The two-field displacement phase-field system is solved fully-coupled and accounts for crack irreversibility. This solution is than coupled to the pressure equation via a fixed-stress iteration. The pressure is obtained by using a diffraction equation where the phase-field variable serves as an indicator function that distinguishes between the fracture and the reservoir. The transport of the proppant in the fracture is modeled by using a power-law fluid system. The numerical discretization in space is based on Galerkin finite elements for displacements and phase-field, and an enriched Galerkin method is applied for the pressure equation in order to obtain local mass conservation. The concentration is solved with cell-centered finite elements. Nonlinear equations are treated with Newton's method. Our developments are substantiated with several numerical examples in two and three dimensions.
Coupled poromechanical problems appear in a variety of disciplines, from reservoir engineering to biomedical applications. This work focuses on efficient strategies for solving the matrix systems that result from discretization and linearization of the governing equations. These systems have an inherent block structure due to the coupled nature of the mass and momentum balance equations. Recently, several iterative solution schemes have been proposed that exhibit stable and rapid convergence to the coupled solution. These schemes appear distinct, but a unifying feature is that they exploit the block-partitioned nature of the problem to accelerate convergence. This paper analyzes several of these schemes and highlights the fundamental connections that underlie their effectiveness. We begin by focusing on two specific methods: a fully-implicit and a sequential-implicit scheme. In the first approach, the system matrix is treated monolithically, and a Krylov iteration is used to update pressure and displacement unknowns simultaneously. To accelerate convergence, a preconditioning operator is introduced based on an approximate block-factorization of the linear system. Next, we analyze a sequential-implicit scheme based on the fixed-stress split. In this method, one iterates back and forth between updating displacement and pressure unknowns separately until convergence to the coupled solution is reached. We re-interpret this scheme as a block-preconditioned Richardson iteration, and we show that the preconditioning operator is identical to that used within the fully-implicit approach. Rapid convergence in both the Richardson- and Krylov-based methods results from a particular choice for a sparse Schur complement approximation. This analysis leads to a unified framework for developing solution schemes based on approximate block factorizations. Many classic fully-implicit and sequential-implicit schemes are simple sub-cases. The analysis also highlights several new approaches that have not been previously explored. For illustration, we directly compare the performance and robustness of several variants on a benchmark problem.
The paper summarises recent developments in the shear band propagation approach, which enables simplified submarine slope stability analysis to account for catastrophic and progressive failure. This approach covers a wide variety of potential failure mechanisms, such as slab failures, spreadings, ploughings and run-outs, and provides analytical energy balance criteria for predicting their occurrence. The simple form of the resulting criteria enabled their incorporation into deterministic and probabilistic slope stability analysis of offshore developments within the geographical information system. In contrast to conventional limiting equilibrium approaches used in such analysis, the shear band propagation approach is capable of explaining enormous dimensions of observed palaeo-landslides and predicting annual probabilities of failure that are orders of magnitude higher, as validated by reconstructed historical frequencies.
Geomaterials with aggregated structure or containing fissures often exhibit a bimodal pore size distribution that can be viewed as two coexisting pore regions of different scales. The double porosity concept enables continuum model-ing of such materials by considering two interacting pore scales satisfying relevant conservation laws. This paper develops a thermodynamically consistent framework for hydromechanical modeling of unsaturated flow in double porosity media. With an explicit treatment of the two pore scales, conservation laws are formulated incorporating an effective stress tensor that is energy-conjugate to the rate of deformation tensor of the solid matrix. A constitutive framework is developed based on energy-conjugate pairs identified in the first law of thermodynamics, which is then incorporated into a three-field mixed finite element formulation for double porosity media. Numerical simulations of laboratory-and field-scale problems are presented to demonstrate the impact of double porosity on the resulting hydromechanical responses.
Natural geomaterials such as fissured rocks and aggregated soils often exhibit a pore size distribution with two dominant pore scales, usually termed macropores and micropores. High-fidelity descriptions of these materials require an explicit treatment of the two pore regions as double porosity. We develop a finite element framework for coupled solid deformation and fluid diffusion in double porosity media that employs a thermodynamically consistent effective stress. Mixed finite elements that interpolate the solid displacement and pore pressures in the macropores and micropores are used for this purpose. In the limit of undrained deformation, the incompressibility constraint causes unstable behavior in the form of spurious pressure oscillation at the two pore scales. To circumvent this instability we develop a variant of the polynomial pressure projection technique for a twofold saddle point problem. The proposed stabilization allows the use of equal-order (linear) interpolations of the displacement and two pore pressure variables throughout the entire range of drainage condition.
For assessing energy-related activities in the subsurface, it is important to investigate the impact
of the spatial variability and anisotropy on the geomechanical behavior of shale. The Brazilian
test, an indirect tensile-splitting method is performed in this work, and the evolution of strain
field is obtained using digital image correlation. Experimental results show the significant impact
of local heterogeneity and lamination on the crack pattern characteristics. For numerical
simulations, a phase field method is used to simulate the brittle fracture behavior under various
Brazilian test conditions. In this study, shale is assumed to consist of two constituents including
the stiff and soft layers to which the same toughness but different elastic moduli are assigned.
Microstructural heterogeneity is simplified to represent mesoscale (e.g., millimeter scale)
features such as layer orientation, thickness, volume fraction, and defects. The effect of these
structural attributes on the onset, propagation, and coalescence of cracks is explored. The
simulation results show that spatial heterogeneity and material anisotropy highly affect crack
patterns and effective fracture toughness, and the elastic contrast of two constituents significantly
alters the effective toughness. However, the complex crack patterns observed in the experiments
cannot completely be accounted for by either an isotropic or transversely isotropic effective
medium approach. This implies that cracks developed in the layered system may coalesce in
complicated ways depending on the local heterogeneity, and the interaction mechanisms between
the cracks using two-constituent systems may explain the wide range of effective toughness of
shale reported in the literature.
The critical energy release rates for mode I and II fracture for rock-like materials are usually different. In this paper, a modified phase-field model is proposed for simulating mixed mode crack propagation. The model can distinguish between the critical energy release rates for mode I and mode II cracks. For the purpose of validation, rock-like materials with a single flaw or double flaws under compression are studied. The simulated results are compared to experimental data, both qualitatively and quantitatively. It is shown that the proposed model is able to capture the commonly observed propagation pattern of wing crack emergence followed by secondary cracks. Additionally, the typical types of crack coalescence observed in experimental tests are successfully reproduced, including the critical loads at which crack coalescence occurs.
Propagation of fluid-driven fractures plays an important role in natural and engineering processes, including transport of magma in the lithosphere, geologic sequestration of carbon dioxide and oil and gas recovery from low-permeability formations, among many others. The simulation of fracture propagation poses a computational challenge as a result of the complex physics of fracture, and the need to capture disparate length scales. Phase-field models represent fractures as a diffuse interface, and enjoy the advantage that fracture nucleation, propagation, branching or twisting can be simulated without ad hoc computational strategies like remeshing or local enrichment of the solution space. Here, we propose a new quasi-static phase-field formulation for modeling fluid-driven fracturing in elastic media at small strains. The approach fully couples the fluid flow in the fracture (described via the Reynolds lubrication approximation) and the deformation of the surrounding medium. The flow is solved on a lower-dimensionality mesh immersed in the elastic medium. This approach leads to accurate coupling of both physics. We assessed the performance of the model extensively by comparing results for the evolution of fracture length, aperture and fracture-fluid pressure against analytical solutions under different fracture propagation regimes. The excellent performance of the numerical model in all regimes builds confidence in the applicability of phase-field approaches to simulate fluid-driven fracture.
Six argillaceous rocks, six sandstones, two kinds of tuff and two tuffaceous sandstones from various localities in Japan ranging Palaeozoic to Pliocene in age were deformed in the triaxial vessel under 1 to 2500 bars in confining pressure, at room temperature and in strain rate 3.5x10-5/sec. All are dry sample and underwent compression test.The strength of Tertiary argillaceous rocks increases in direct proportion to age. In sandstone the age is important factor for the strength, too. On the other hand, in tuff and tuffaceous sandstone, grain the size seems most important for the strength.As the deformation changes ductile to brittle, the mode of fracturing does wedgelike fracture, singleplane of shear fracture, network of a lots of minute shear fractures, and to flow. The transitional line from the single shear fracture to the network shear fracture occur in lower ductility in both tuff and tuffaceous sandstone, higher in sandstone, and highest in argillaceous rocks.
A stabilized thermo-hydro-mechanical (THM) finite element model is introduced to investigate the freeze-thaw action of frozen porous media in the finite deformation range. By applying mixture theory, frozen soil is idealized as a composite consisting of three phases, i.e., solid grain, unfrozen water and ice crystal.
A generalized hardening rule at finite strain is adopted to replicate how the elasto-plastic responses and critical state evolve under the influence of phase transitions and heat transfer. The enhanced particle interlocking and ice strengthening during the freezing processes and the thawing-induced consolidation at the
geometrical nonlinear regimes are both replicated in numerical examples. The numerical issues due to lack of two-fold inf-sup condition and ill-conditioning of the system of equations are addressed. Numerical examples for engineering applications at cold region are analyzed via the proposed model to predict the impacts of
changing climate on infrastructure at cold regions.
Highlights • A variational based computational framework that combines multiple dissipative phenomena is proposed. • Diffusion induced large plastic deformation and phase field fracture during two-phase lithiation of silicon electrodes is modeled. • The effect of fracture energy release rate, electrode geometry, and geometric constraints on the fracture behavior of silicon electrodes is investigated. Abstract Silicon (Si) is considered to be a promising next-generation anode material for lithium-ion batteries. However, the large volume change during (de)lithiation processes causes fracture of Si electrodes, thereby limiting Si's practical application in lithium-ion batteries. In this work, we formulate a variational-based fully chemo-mechanical coupled computational framework to study diffusion induced large plastic deformation and phase field fracture in Si electrodes. Into this framework we incorporate a recently developed reaction-controlled diffusion model to predict two-phase lithiation for amorphous Si (a-Si) and crystalline Si (c-Si) as well as diffusion induced anisotropic deformation for c-Si. The variational formulation suggests to consider the deformation field, the chemical potential, and the damage field as primary unknowns. The concentration field is considered as a local variable and is recovered from the chemical potential on the element level. We carry out several numerical simulations to show the performance of our computational model and point out the significance of accurately accounting for the presence of the reaction front when modeling diffusion induced fracture problems for both a-Si and c-Si electrodes. In addition, we investigate how the fracture energy release rate, electrode geometry, and geometrical constraints affect the fracture behavior of Si electrodes.
In this work, we present numerical studies of fixed-stress iterative coupling for solving flow and geomechanics with propagating fractures in a porous medium. Specifically, fracture propagations are described by employing a phase-field approach. The extension to fixed-stress splitting to propagating phase-field fractures and systematic investigation of its properties are important enhancements to existing studies. Moreover, we provide an accurate computation of the fracture opening using level-set approaches and a subsequent finite element interpolation of the width. The latter enters as fracture permeability into the pressure diffraction problem which is crucial for fluid filled fractures. Our developments are substantiated with several numerical tests that include comparisons of computational cost for iterative coupling and nonlinear and linear iterations as well as convergence studies in space and time.
We present a simplified model of damaging porous material, obtained through consistent linearization from a recursive-faulting material model described in (Pandolfi et al. 2016). The brittle damage material model is characterized by special planar micro-structures, consisting of nested families of equi-spaced frictional-cohesive faults in an otherwise elastic matrix material. The linear kinematics model preserves the main microstructural features of the finite kinematics one but offers a far better computational performance. Unlike models commonly employed in geo-mechanical applications, the proposed model contains a small number of parameters, to wit, two elastic moduli, three frictional-cohesive parameters, and three hydraulic response parameters, all of which having clear physical meanings and amenable to direct experimental measurement through standard material tests. The model is validated by comparison to triaxial hydro-mechanical experimental data. Despite the paucity of material constants, the salient aspects of the observed behavior are well captured by the model, qualitatively and quantitatively. As an example of application of the model, we simulate the excavation of a borehole in a rocky embankment.
This work outlines a rigorous variational-based framework for the phase field modeling of fracture in isotropic porous solids undergoing large elastic-plastic strains. It extends the recent works Miehe et al., (2015, 2016) to a particular formulation of isotropic porous plasticity. The phase field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with geometric features rooted in fracture mechanics. A gradient plasticity model for porous plasticity with a simple growth law for the evolution of the void fraction is developed, and linked to a failure criterion in terms of the local elastic-plastic work density that drives the fracture phase field. It is shown that this approach is able to model basic phenomena of ductile failure such as cup-cone failure surfaces in terms of only two material parameters on the side of damage mechanics: a critical work density that triggers the onset of damage and a shape parameter that governs the postcritical damage up to fracture. The formulation includes two independent length scales which regularize both the plastic response as well as the crack discontinuities. This allows to design damage zones of ductile fracture to be inside of plastic zones or vice versa, and guarantees on the computational side a mesh objectivity in post-critical ranges. The key aspect that allows to construct a variational theory for porous plasticity at fracture is the use of an Eulerian constitutive setting, where the yield function is formulated in terms of the Kirchhoff stress. Here, we exploit the fact that this stress approximates an effective stress that drives the plasticity in the matrix of the porous solid. The coupling of gradient plasticity to gradient damage is realized by a constitutive work density function that includes the stored elastic energy and the dissipated work due to plasticity and fracture. The latter represents a coupled resistance to plasticity and damage, depending on the gradient-extended internal variables which enter the plastic yield function and the fracture threshold function. The canonical theory proposed is shown to be governed by a rate-type minimization principle, which fully determines the coupled multi-field evolution problem, and provides inherent symmetries with regard to a finite element implementation. The robust computational setting proposed includes (i) a general return scheme of plasticity in the spectral space of logarithmic principal strains and dual Kirchhoff stresses, (ii) the micromorphic regularization of the gradient plastic evolution and (iii) a history-field-driven update of the linear phase field equation.
Phase-field models have been a topic of much research in recent years. Results have shown that these models are able to produce complex crack patterns in both two and three dimensions. A number of extensions from brittle to ductile materials have been proposed and results are promising. To date, however, these extensions have not accurately represented strains after crack initiation or included important aspects of ductile fracture such as stress triaxiality. This work introduces a number of contributions to further develop phase-field models for fracture in ductile materials. These contributions include: a cubic degradation function that provides a stress strain response prior to crack initiation that more closely approximates linear elastic behavior, a derivation of the governing equations in terms of a general energy potential from balance laws that describe the kinematics of both the body and phase-field, introduction of a yield surface degradation function that provides a mechanism for plastic softening and corrects the non-physical elastic deformations after crack initiation, a proposed mechanism for including a measure of stress triaxiality as a driving force for crack initiation and propagation, and a correction to an error in the configuration update of an elastoplastic return-mapping algorithm for flow theory. We also present a heuristic time stepping scheme that facilitates computations that require a relatively long load time prior to crack initiation. A number of numerical results will be presented that demonstrate the effects of these contributions.
We present a stabilized extended finite element formulation to simulate the hydraulic fracturing process in an elasto-plastic medium. The fracture propagation process is governed by a cohesive fracture model, where a trilinear traction-separation law is used to describe normal contact, cohesion and strength softening on the fracture face. Fluid flow inside the fracture channel is governed by the lubrication equation, and the flow rate is related to the fluid pressure gradient by the ‘cubic’ law. Fluid leak off happens only in the normal direction and is assumed to be governed by the Carter's leak-off model. We propose a ‘local’ U-P (displacement-pressure) formulation to discretize the fluid-solid coupled system, where volume shape functions are used to interpolate the fluid pressure field on the fracture face. The ‘local’ U-P approach is compatible with the extended finite element framework, and a separate mesh is not required to describe the fluid flow. The coupled system of equations is solved iteratively by the standard Newton-Raphson method. We identify instability issues associated with the fluid flow inside the fracture channel, and use the polynomial pressure projection method to reduce the pressure oscillations resulting from the instability. Numerical examples demonstrate that the proposed framework is effective in modeling 3D hydraulic fracture propagation. Copyright
Fractures that propagate off of weak slip planes are known as wing cracks, and often play important roles in both tectonic deformation and fluid flow across reservoir seals. Previous numerical models have produced the basic kinematics of wing-crack openings, but generally have not been able to capture fracture geometries seen in nature. Here, we present both a phase-field modeling approach and a physical experiment using gelatin for a wing crack formation. By treating the fracture surfaces as diffusive zones instead of as discontinuities, the phase-field model does not require consideration of unpredictable rock properties or stress inhomogeneities around crack tips. It is shown by benchmarking the models with physical experiments, that the numerical assumptions in the phase-field approach do not affect the final model predictions of wing-crack nucleation and growth. With this study, we demonstrate that it is feasible to implement the formation of wing cracks in large scale phase-field reservoir models.