Article

# Coupled phase-field and plasticity modeling of geological materials: From brittle fracture to ductile flow

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## 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.

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... For ductile cracking in plastic materials, in particular with a non-associative flow rule, a new formulation has been developed for coupled phase-field and plastic modeling using a rigorous thermodynamics framework. 48 This new formulation is able to conveniently deal with ductile fracturing in plastic geological materials with a non-associative plastic flow rule. However, in most previous studies, only tensile cracks have been taken into account and generally related to tensile strain energy. ...
... In practice, it is quasi impossible to follow individually the evolution of multiple cracks. Therefore, the cracks are approximated by a crack density functional Γ dα so that the total area of crack surfaces is calculated by 48 : ...
... The fulfilment of the non-negative condition for any arbitrary volume V implies that the evolution rate of crack density functional is nonnegative: 48 : ...
Article
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 thermal-hydromechanical behavior of the host Callovo-Oxfordian COx claystone in quasi real conditions, through the international research project DECOVALEX. The emphasis is put on the pore pressure increase generated by the heat released from high level radioactive waste and the consequence on the damage evolution of the host rock. A phase-field method is proposed to describe the evolution of damaged zones around the heating borehole. Both tensile and shear cracks are taken into account. The evolution of damage is coupled with temperature variation, rock deformation, and fluid pressure change. Moreover, the structural anisotropy of elastic properties, permeability, and heat conductivity of the host rock is taken into account. Numerical results are compared with in-situ measurements.
... Furthermore, the shear yield surface plasticity model combining with the Byerlee's rule can be adopted to capture the tensile damage, shear failure and fault sliding behaviors in the brittle-ductile transition process for low porosity rocks [20,21]. The elliptical yield surface plasticity model incorporating with the critical state line theory and the cap plasticity model are used to distinguish the shear dilatant and compactive failure behaviors for high porosity rocks [22][23][24][25][26][27]. It should be noted that the cap plasticity model is constructed by the shear yield surface in the zone of shear deformation and the elliptical yield surface in the zone of compaction, which simultaneously possesses their major advantages and does not incorporate with other critical criterion for evaluating the brittle-ductile failure transition behavior. ...
... However, the conventional coupled damage plasticity models usually involve complicated damage functions that cause great difficulties for model parameter determination and numerical implementation. Additionally, their damage functions also need to be regularized to eliminate the mesh sensitivity in modeling softening related issues [27]. Although many gradient damage models have been developed to alleviate these drawbacks of the coupled damage plasticity models, they are still inconvenient to capture the brittle-ductile failure transition behavior in pressure-sensitive geomaterials. ...
... Until now, many advanced discrete and smeared crack models and methods have been developed to model complex fracture problems [21,27,[33][34][35][36]. Among these models and methods, the phase-field fracture model shows great promising for dealing with nonlocal damage issues [21,27]. ...
Article
A phase-field implicit material point method with the convected particle domain interpolation (PF-ICPDI) is proposed to model the brittle–ductile failure transition in pressure-sensitive geomaterials involving finite deformation. In this method, the phase-field fracture formulation relying on the microforce balance law and the second law of thermodynamics is adopted as a nonlocal damage function for the elastoplastic fracture analysis. A smooth two-yield-surface plasticity constitutive model is utilized to evaluate the brittle–ductile failure transition behavior of pressure-sensitive geomaterials. The coupling effect of phase-field fracture model and cap plasticity model is established by introducing an effective phase-field stress and splitting the total stored energy into elastic and plastic parts. The implicit material point method that can avoid severe mesh distortion and improve numerical stability is then developed to solve the quasi-static elastoplastic fracture finite deformation problem. Furthermore, the convected particle domain interpolation technique is adopted to eliminate numerical noises and improve computational accuracy while material points crossing cell boundaries in modeling the large deformation brittle–ductile failure transition process. The staggered incremental iterative scheme is carried out to solve the coupled discrete governing equations. The accuracy and capability of the proposed PF-ICPDI method are demonstrated through several representative numerical examples, as well as the effects of main material parameters on the phase-field brittle–ductile failure transition modeling of geomaterials are discussed.
... A range of methods and models have been proposed for modelling brittle and ductile crack opening problems: continuum-based models such as the finite element method (FEM) [2], the extended finite element method (XFEM) [2,4], the phase field method [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], damage models [21][22][23][24][25], the numerical manifold method [26] and the material point erosion method [27]. Closed crack problems for modelling brittle materials in compression range from the finite-discrete element method (FDEM) [28] to damage models [29][30][31][32][33], phase field models [34][35][36][37][38] as well as particle-based models [39][40][41][42] and plasticity models [37]. The main advantage of phase field and damage models is that no complex interface tracking is necessary to determine the crack path. ...
... A range of methods and models have been proposed for modelling brittle and ductile crack opening problems: continuum-based models such as the finite element method (FEM) [2], the extended finite element method (XFEM) [2,4], the phase field method [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], damage models [21][22][23][24][25], the numerical manifold method [26] and the material point erosion method [27]. Closed crack problems for modelling brittle materials in compression range from the finite-discrete element method (FDEM) [28] to damage models [29][30][31][32][33], phase field models [34][35][36][37][38] as well as particle-based models [39][40][41][42] and plasticity models [37]. The main advantage of phase field and damage models is that no complex interface tracking is necessary to determine the crack path. ...
... The material is assumed to be isotropic, homogeneous and with no pre-existing flaws. All models are implemented in COMSOL Multiphysics v5.5 using standard physics options: structural mechanics using quadratic Lagrangian shape functions for the elasticity problem of equation (3), domain ODEs compression (ordinary differential equations) using order 4 Gauss point data for the viscous regularisation of equation (13) and the Helmholtz equation using linear Lagrangian shape functions for the smoothing operations of equations (11), (35) and (37). A fully coupled solver is used as this accommodates a larger time step. ...
... The approach developed by Gurtin [53] to derive the Ginzburg-Landau equation as a microforce equilibrium law can also be applied to derive the phase-field governing equation. This was implemented in connection with porous media hydraulic fracturing by, e.g., Wilson and Landis [144], Choo and Sun [32,33], as well as in connection with other models related to phase-field fractures as, e.g., Wilson et al. [145], Borden et al. [17], De Lorenzis et al. [39]. ...
... In this variational consideration with a priori fully dissipative fracture evolution, the stored solid energy functionψ S does not include the fracture energy ψ S frac , see, e.g., [32,90,152]. Therefore, the stored solid energy and its time derivative can be expressed as ...
... We apply in the following the Lagrangian multiplier optimization procedure to determine proper relations for π dis and ξ, see [32]. In this, we start by applying time derivation to the crack surface density function (2 ) 2 , which leads to the following expression: ...
Article
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Motivated by the successful implementation of the phase-field method (PFM) to simulate complicated fracture patterns at moderate computational costs in solid materials, many research groups have started since 2012 applying the PFM to model hydraulic fracturing, especially that occurs in porous geomaterials. These research works have contributed to the development of the PFM from different perspectives, especially in connection with the mathematical formulations of the hydro-mechanical processes and the numerical algorithms to solve the emerging coupled problems. In this regard, the underlying paper aims to review the significant scientific works that utilized the PFM to model fracturing caused mainly by fluid injection in a certain porous domain and, less common, by fluid extraction (e.g., drying) from a certain porous domain. This includes reviewing different approaches for deriving the phase-field evolution formulation (e.g. Ginzburg–Landau approach, thermodynamically consistent approaches, and microforce-based approach) and reviewing several formulations for the stiffness degradation function and that of the crack driving force. Besides, the paper will go through several methods to estimate the crack aperture width, in addition to reviewing different numerical approaches and implementations. The paper will be concluded by presenting a number of open topics and challenges to be addressed in future works.
... Failure in quasi-brittle geomaterials such as rocks and concrete is mostly driven by the growth and coalescence of microcracks. Depending on the confining pressure (among other factors such as temperature and loading rate), distinctive failure modes can be observed, with notably different behavior in tension and compression [1][2][3]. Under tensile loading at low confining pressure, opening microcracks lead to brittle fracture with mode I kinematics. ...
... However, recent works have proposed phase-field models coupled to frictional plasticity, usually of the Drucker-Prager type. For instance, a double phase-field model for tensile, shear, and mixed-mode fracture with plasticity was proposed in You et al. [86], while other studies [3,87] focus on modeling the brittle-to-ductile transition phenomenon. Phase-field models coupled to frictional plasticity have been further considered in the finite strain setting [3,88], as well as in the context of multiphase materials [89] and fluiddriven fracture [90,91]. ...
... For instance, a double phase-field model for tensile, shear, and mixed-mode fracture with plasticity was proposed in You et al. [86], while other studies [3,87] focus on modeling the brittle-to-ductile transition phenomenon. Phase-field models coupled to frictional plasticity have been further considered in the finite strain setting [3,88], as well as in the context of multiphase materials [89] and fluiddriven fracture [90,91]. Moreover, building upon the phase-field modeling of frictional interfaces [77], Bryant and Sun [92] proposed a model that embeds rate-, size-, and temperature-dependent friction. ...
Preprint
Full-text available
This paper presents a framework for modeling failure in quasi-brittle geomaterials under different loading conditions. A micromechanics-based model is proposed in which the field variables are linked to physical mechanisms at the microcrack level: damage is related to the growth of microcracks, while plasticity is related to the frictional sliding of closed microcracks. Consequently, the hardening/softening functions and parameters entering the free energy follow from the definition of a single degradation function and the elastic material properties. The evolution of opening microcracks in tension leads to brittle behavior and mode I fracture, while the evolution of closed microcracks under frictional sliding in compression/shear leads to ductile behavior and mode II fracture. Frictional sliding is endowed with a non-associative law, a crucial aspect of the model that considers the effect of dilation and allows for realistic material responses with non-vanishing frictional energy dissipation. Despite the non-associative law, a variationally consistent formulation is presented using notions of energy balance and stability, following the energetic formulation for rate-independent systems. The material response of the model is first described, followed by several benchmark finite element simulations. The results highlight the ability of the model to describe tensile, shear, and mixed-mode fracture, as well as responses with brittle-to-ductile transition. A key result is that, by virtue of the micromechanical arguments, realistic failure modes can be captured, without resorting to the usual heuristic modifications considered in the phase-field literature. The numerical results are thoroughly discussed with reference to previous numerical studies, experimental evidence, and analytical fracture criteria.
... La littérature est abondante dans ce domaine (Duda et al. [201], Shanthraj et al. [119,202], Diehl et al. [203], Ambati et al. [138,204], Badnava et al. [205], Alessi et al. [206,207], Arriaga et Weisman [208,209], Borden et al. [210], Kuhn et al. [126,211], Miehe et al. [212][213][214], Aldakheel et al. [215], Dittmann et al. [216], de Lorenzis et al. [217], Choo et Sun [218]), et ne sera évoquée que brièvement dans ce qui suit. ...
... où l'écrouissage isotrope est supposé linéaire. De ce fait, l'énergie d'écrouissage intervient dans la force motrice de rupture : [206,207,[212][213][214][215][216]218], faisant notamment intervenir une plasticité non locale à gradient. ...
... Il est donc nécessaire de modifier le modèle de contact unilatéral, ce que Zhang et al. [149], Choo et Sun [218], et Bryant et Sun [159]) ont proposé (voir le paragraphe 3.4.7), en introduisant une ténacité de mode II. ...
Thesis
Ce travail s'inscrit dans la thématique classique en mécanique non-linéaire de la modélisation du comportement à rupture de milieux hétérogènes. On étudie par des moyens numériques la localisation et la propagation de l'endommagement dans un comprimé au TATB (triamino-trinitrobenzène) soumis à diverses sollicitations mécaniques et thermiques. La microstructure polycristalline, qui contient en outre liant et porosités, est partiellement caractérisée à travers des images obtenues par microscopie électronique, ainsi que des essais mécaniques et thermiques, et présente, à l'état initial, des contraintes résiduelles, voir des fissures pré-existantes. On développe dans un premier temps une méthode numérique de calculs par transformées de Fourier rapide pour la prédiction de l'endommagement reposant sur l'utilisation d'un "champ de phase", qui décrit l’endommagement local au sein de la microstructure. Les algorithmes proposés, validés à partir de données issues de la littérature, prennent en compte le caractère irréversible de l'endommagement ainsi que la forte anisotropie des cristaux de TATB, en élasticité maiségalement du point de vue de la fissuration. Dans un second temps, on applique les schémas numériques développés aupolycristal étudié. On s’intéresse en particulier à la fissuration transgranulaire sous cycle thermique à froid, puis on intègre la porosité et la fissuration dans le liant sous chargement thermique et en traction. Ces divers phénomènes sont pris en compte de manière incrémentale et identifiés par méthode inverse sur les données expérimentales disponibles. Les résultats de ce travail montrent d'une part le rôle prépondérant joué par l'endommagement inter et transgranulaire sur le comportement thermique à froid et lors des essais mécaniques. La démarche entreprise permet, en outre, de préciser l'influence relative des divers mécanismes pris en compte : porosité, élasticité dans le liant et dans les grains, mais également anisotropie mécanique des grains ou endommagement du matériau à l'état initial.
... The main driving force for these developments are the possibility to handle complex fracture phenomena within numerical methods in two and three dimensions. In recent years, several brittle [67,188,162,259,270,6,342,285,258,160,79,295,80,119,102,17] and ductile [10,22,64,19,4,110,86,108,8,190,118,186,253] small and large strain deformations, variational formulations, multi-scale/physics problems, mathematical analysis, different decompositions and discretization techniques with many applications in science and engineering. All these examples and the citation therein demonstrate the potential of phase-field for crack propagation. ...
... From a computational point of view, the tracking of sharp crack surfaces provides substantial difficulties and is often restricted to simple crack topologies. This difficulty can be overcome by recently developed phase-field approaches to fracture, which are based on the regularization of sharp crack discontinuities, see [113,230,22,232,64,101,86,14,108,304,338]. ...
... This avoids the use of complex discretization methods for crack discontinuities and can account for multi-branched cracks within the solid skeleton. In recent years, several brittle [188,162,259,270,6,285,258,160,79,295,80,119,17] and ductile [10,22,64,19,4,110,86,108,8,190,118,186,253,104] phase-field fracture formulations have been proposed in the literature. These studies range from the modeling of 2D/3D small and large strain deformations, variational formulations, multi-scale/physics problems, mathematical analysis, different decompositions and discretization techniques with many applications in science and engineering. ...
Book
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The underlying Habilitation aims to contribute to the research on fracture mechanics of solids across the scales. This active research field is driven by the investigation and development of new methods, processes and technologies applicable to engineering problems with complex material behavior of solids at fracture. It includes mathematically precise formulations of theoretical and computational models with emphasis on continuum physics as well as the development of variation methods and efficient numerical implementations tools. In particular, two directions will be considered in this contribution: (i) the construction of advanced multi-scale techniques and (ii) modern element technologies. On the multi-scale techniques, a robust and efficient Global-Local approach for numerically solving fracture-mechanics problems is developed in the first part of this contribution. This method has the potential to tackle practical field problems in which a large-structure might be considered and fracture propagation is a localized phenomenum. In this regard, failure is analyzed on a lower (Local) scale, while dealing with a purely linear problem on an upper (Global) scale. The modeling of crack formation at the Local scale is achieved in a convenient way by continuum phase-field formulations to fracture, which are based on the regularization of sharp crack discontinuities. For this purpose, a predictor-corrector scheme is designed in which the local domains are dynamically updated during the computation. To cope with different element discretizations at the interface between the two nested scales, a non-matching dual mortar method is formulated. Hence, more regularity is achieved on the interface. The development of advanced discretization schemes accounting for meshes with highly irregular shaped elements and arbitrary number of nodes is the main focus in the second part of this work. To this end, a relatively new method - the virtual element method (VEM) - will be presented here that leads to an exceptional efficient and stable formulation for solving a wide range of boundary value problems in science and engineering. The structure of VEM comprises a term in the weak formulation or the potential density functional in which the unknowns, being sought are replaced by their projection onto a polynomial space. This results in a rank-deficient structure, therefore it is necessary to add a stabilization term to the formulation. The performance of the virtual element method is comparable to using finite elements of higher order. It is even more robust than FEM in case of a severe distortion of the element.
... Failure in quasi-brittle geomaterials such as rocks and concrete is mostly driven by the growth and coalescence of microcracks. Depending on the confining pressure (among other factors such as temperature and loading rate), distinctive failure modes can be observed, with notably different behavior in tension and compression [1][2][3]. Under tensile loading at low confining pressure, opening microcracks lead to brittle fracture with mode I kinematics. ...
... However, recent works have proposed phase-field models coupled to frictional plasticity, usually of the Drucker-Prager type. For instance, a double phase-field model for tensile, shear, and mixed-mode fracture with plasticity was proposed in You et al. [86], while other studies [3,87] focus on modeling the brittle-to-ductile transition phenomenon. Phase-field models coupled to frictional plasticity have been further considered in the finite strain setting [3,88], as well as in the context of multiphase materials [89] and fluiddriven fracture [90,91]. ...
... For instance, a double phase-field model for tensile, shear, and mixed-mode fracture with plasticity was proposed in You et al. [86], while other studies [3,87] focus on modeling the brittle-to-ductile transition phenomenon. Phase-field models coupled to frictional plasticity have been further considered in the finite strain setting [3,88], as well as in the context of multiphase materials [89] and fluiddriven fracture [90,91]. Moreover, building upon the phase-field modeling of frictional interfaces [77], Bryant and Sun [92] proposed a model that embeds rate-, size-, and temperature-dependent friction. ...
Article
Full-text available
This paper presents a framework for modeling failure in quasi-brittle geomaterials under different loading conditions. A micromechanics-based model is proposed in which the field variables are linked to physical mechanisms at the microcrack level: damage is related to the growth of microcracks, while plasticity is related to the frictional sliding of closed microcracks. Consequently, the hardening/softening functions and parameters entering the free energy follow from the definition of a single degradation function and the elastic material properties. The evolution of opening microcracks in tension leads to brittle behavior and mode I fracture, while the evolution of closed microcracks under frictional sliding in compression/shear leads to ductile behavior and mode II fracture. Frictional sliding is endowed with a non-associative law, a crucial aspect of the model that considers the effect of dilation and allows for realistic material responses with non-vanishing frictional energy dissipation. Despite the non-associative law, a variationally consistent formulation is presented using notions of energy balance and stability, following the energetic formulation for rate-independent systems. The material response of the model is first described, followed by the numerical implementation procedure and several benchmark finite element simulations. The results highlight the ability of the model to describe tensile, shear, and mixed-mode fracture, as well as responses with brittle-to-ductile transition. A key result is that, by virtue of the micromechanical arguments, realistic failure modes can be captured, without resorting to the usual heuristic modifications considered in the phase-field literature. The numerical results are thoroughly discussed with reference to previous numerical studies, experimental evidence, and analytical fracture criteria.
... To address this limitation, the effect of initial stress state on the gas flow behavior will be considered in this paper by introducing a fictitious strain tensor into the phase field method. In addition, a thermodynamically consistent HM-PF framework will be derived from the theory of thermodynamics proposed by Coussy (2004) for unsaturated porous media and the microforce balance law (Gurtin 1996;Choo and Sun 2018a). ...
... 2, the current research topic involves the complex processes, including the multiphase flow, the fluid-driven fracturing and the effect of interfaces. Compared with the variational approach, the microforce balance law approach may be more flexible to account for these complex physical processes during the fracturing process (Choo and Sun 2018a). Thus, the microforce balance law will be adopted in this paper to develop a thermodynamically consistent phase field model to simulate the gas-driven fracturing in initially saturated bentonite. ...
... Similar to Eq. (3), the microforce on the surface Ω , i.e., , can be expressed as By substituting Eq. (7) into Eq. (6), and neglecting the external microforce, , as done in Choo and Sun (2018a), Na and Sun (2018), the microforce balance law can be derived as As the above integral balance holds in any arbitrary volume, the strong form for the microforce balance law can be derived as ...
Article
Full-text available
A thermodynamically consistent phase field model that accounts for initial stress state is proposed in this paper to simulate the gas migration process in saturated bentonite. The energy contribution due to the fracturing process is included in Coussy’s thermodynamic framework for unsaturated porous media. The possible effect of the interfaces between different phases on the driving force functional for phase field and the effective stress has been identified from the proposed thermodynamic framework. In addition, the initial stress state is innovatively accounted for in the phase field model by introducing a fictitious strain tensor that is calculated from its corresponding initial stress tensor. It is the sum of the fictitious strain tensor and the strain tensor due to elastic deformation that governs the evolution of the phase field. The simulated results showed that the effect of the swelling pressure (regarded as the initial effective stress for a high swelling clay) on the fracture initiation has been well described by the proposed method. Specifically, the effect of either isotropic or anisotropic stress state on the fracturing process can be well reflected by the phase field approach based on Rankine-type fracture criterion. In contrast, the phase field approach based on the Griffith fracture criterion is more appropriate for the isotropic stress state than the anisotropic stress state because of the Poisson’s effect. Moreover, the gas pressure required to trigger the fracturing process needs to exceed the sum of the porewater pressure and the initial stress. The effect of the boundary condition on the evolution of fluid pressure and total stress has been qualitatively captured. It is found that the boundary with higher stiffness leads to a higher gas pressure in the developed fracture and a higher water pressure and total stress in the surrounding porous matrix. In addition, some key experimental findings, such as the preferential gas flow, the build-up of porewater pressure, the almost fully saturated state and the localized consolidation, have been qualitatively captured by the developed phase field model.
... Geological storage of CO 2 and oil recovery often require the injection of the pore fluid in a supercritical state such that the thermal convection may play an important role both for the fluid transport and the fluid-driven fracture. The combination of temperature, pressure, and loading rate are also critical for the brittle-ductile transition of geological materials [166][167][168]. Heat exchange is an important mechanism for selecting the candidate materials for the nuclear waste geological disposal such as clay and salt. ...
... Note that the energy required for crack growth [i.e., W c in Eq. (3.9)] is dissipatve by nature and hence not included in this stored energy function ψ [88,128,168]. Our definition of free energy will be used for constructing the energy balance equations based on the first law of thermodynamics in Section 3.3.1, ...
... which satisfies the following Karush-Kuhn-Tucker condition [77,168]: ...
Thesis
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Global challenges associated with extreme climate events and increasing energy demand require significant advances in our understanding and predictive capability of coupled multi- physical processes across spatial and temporal scales. While classical approaches based on the mixture theory may shed light on the macroscopic poromechanics simulations, accurate forward predictions of the complex behavior of phase-changing geomaterials cannot be made without understanding the underlying coupling mechanisms among constituents at the microstructural scale. To precisely predict the multi-physical behaviors originated by smaller scales, fundamental understandings of the micromechanical interactions among phase constituents are crucial. Hence, this dissertation discusses mathematical and computational frameworks designed to capture coupled thermo-hydro-mechanical-fracture processes in phase-changing porous media that incorporate necessary microscopic details. To achieve this goal, this dissertation aims to introduce a practical way to investigate how phase transition and evolving microstructural attributes at small scales affect the applicability of meso- or macroscopic finite element simulations, by leveraging the phase field method to represent the regularized interfaces of phase constituents. Firstly, a multi-phase-field microporomechanics model is presented to model the growth and thaw of ice lenses. In specific, we extend the field theory for ice lens that is not restricted to one-dimensional space. The key idea is to represent the state of the pore fluid and the evolution of freezing-induced fracture via two distinct phase field variables coupled with balance laws, which leads to an immersed approach where both the homogeneous freezing and ice lensing are distinctively captured. Secondly, a thermo-hydro-mechanical theory for geological media with thermally non-equilibrated constituents is presented, where we develop an operator-split framework that updates the temperature of each constituent in an asynchronous manner. Here, the existence of an effective medium is hypothesized, in which the constituents exhibit different temperatures while heat exchange among the phases is captured via Newton’s law of cooling. Thirdly, an immersed phase field model is introduced to predict fluid flow in fracturing vuggy porous media, where crack growth may connect previously isolated voids and form flow conduits. In this approach, we present a framework where the phase field is not only used as a damage parameter for the solid skeleton but also as an indicator of the pore space, which enables us to analyze how crack growth in vuggy porous matrix affects the flow mechanism differently compared to the homogenized effective medium while bypassing the needs of partitioning the domain and tracking the moving interface. Finally, we present a new phase field fracture theory for higher-order continuum that can capture physically justified size effects for both the path-independent elastic responses and the path-dependent fracture. Specifically, we adopt quasi-quadratic degradation function and linear local dissipation function such that the physical size dependence are insensitive to the fictitious length scale for the regularized interface, which addresses the numerical needs to employ sufficiently large phase field length scale parameter without comprising the correct physical size effect.
... In simulations here, l = 0.04l G = 1 nm such that fracture processes, and overall strength, of small-scale samples are adequately resolved, which will later be confirmed by comparison of results with experiments (Shih et al. 2000;Sarva and Nemat-Nasser 2001;Shin et al. 2012). The absolute size of grains, in the 10-100 micron range, cannot be discretized using elements of nm dimensions (total number of elements becomes intractable for calculations); hence, all spatial dimensions are rescaled downwards consistently to provide accurate predictions of fracture strength (Clayton and Knap 2014; Miehe et al. 2015;Choo and Sun 2018). The length scale l only enters the variational problem through the normalization of energetic contributions to the free energy functional Ψ that depend exclusively on order parameters and their gradients. ...
... For much larger domain sizes, resolution of atomic data is impossible due to mesh constraints, so l must be increased relative to the size of the domain. The fracture surface energy in such cases is often based on a pre-assigned fracture strength and a mesh-controlled l, to avoid spuriously low fracture strength (Miehe et al. 2015;Choo and Sun 2018). A parameter study of effects of l ξ is reported in Clayton and Knap (2015), in the absence of twinning, where a reduction in overall strength of nominally micron-scale polycrystals with increasing l ξ is observed. ...
Article
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Phase field simulations are used to study effects of microstructure features on overall strength and ductility of polycrystalline ceramic composites. The material of present interest is a diamond-silicon carbide (SiC) blend with a grain boundary (GB) phase consisting of much smaller SiC grains, graphitic inclusions, and porosity. Homogenized properties—elastic moduli and surface energy—used in the phase field representation of the GB phase are obtained from a novel approach involving bounds from elastic homogenization theory. Extensive parametric calculations on synthetic microstructures with smoothed polyhedral grains are used to deduce trends in structure-property-performance relations. Results suggest that peak strength, for a fixed fraction of GB phase, can be increased by the following prescriptions: increasing bulk diamond content, reducing porosity, reducing graphite content, and distributing any unavoidable defects uniformly rather than randomly. For the assumed constitutive behavior of the GB phase, graphite appears less deleterious than pores of the same average volume fraction, and median defect concentration appears to be more important than randomization of distributions at low volume fractions. More complex interrelations among microstructures, model features, and material responses are also revealed, for example effects of anisotropic fracture, inelastic dilatation associated with crack opening, and twin variant selection in the SiC phase.
... The first approach by Hillerborg et al. [41] was limited to mode-I fracture. Meanwhile, many models have been developed that are able to handle mixed mode fracture and other complex phenomena including irreversible deformations, stress triaxiality, and rate dependence [42][43][44][45][46][47]. Some CZMs include where ε is defined by: ...
... The method requires a fine mesh along the crack path and a suitable definition for regularization parameters (see a discussion in [128] and recent internal length-insensitive formulations in [175,102]). However, due to the above-mentioned advantages, the phase field method has been widely developed and applied to many problems, such as, among many others: brittle fracture [10,167,189,95], composite delamination [147], dynamic fracture [38,52,98], hydraulic fracture [171,110,179,55], topology optimization for resistance to cracking [176,49], anisotropic material fracture [125,28,187], elastoplastic brittle/ductile fracture [9,8,107,30,6,185], ductile/fragile transition [44], fracture in micro tomography image-based models of microstructures [126,129,127] and more recently adapted in machine learning strategies in [63]. In the following, we review in detail different methods which Chapter Introduction and literature review will be central to this work: (a) quasi-brittle fracture models, (b) elastoplastic brittle/ductile fracture models, and (c) anisotropic fracture models. ...
Thesis
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The objective of this thesis is to develop numerical modeling and simulation techniques to describe the damage in quasi-brittle and elastoplastic composites, which can be obtained by additive manufacturing processes like 3D-printing. We develop phase field methods to fracture and propose several extensions and applications to composites. First, after validating available elastoplastic phase field models on experimental results, we extend these models to interfacial damage, which is central in composites. In a second part, we develop design methodologies for composite microstructures to increase the resistance to cracking, for quasibrittle and elastoplastic composites. For this purpose, we combine the phase field method and topology optimization (SIMP and BESO techniques). Then, models are proposed to describe cracking in polymer-glass fiber composites obtained by 3D printing, and which are characterized by a strong anisotropy. For this purpose, we develop an original anisotropic elastoplastic phase field model for the macro scale. Finally, experimental images obtained by X-Ray micro tomography are used to model the complex cracking process at the microscale of the composites.
... However, these models are limited to purely brittle, pressure-insensitive fracture, neglecting softening behavior and friction effects. Alternatively, Choo and Sun [29] proposed a coupled phase-field and plasticity modeling framework for pressure-sensitive geomaterials. While this modeling framework can well simulate brittle, quasi-brittle, and ductile failures and their transitions, it does not explicitly distinguish between tensile and shear fractures. ...
... For this purpose, we apply the microforce approach in da Silva et al. [46] -adopted by Geelen et al. [38] and Fei and Choo [39] for deriving cohesive and frictional phase-field fracture models, respectively -to doublephase-field modeling of mixed-mode fracture. Although the original phase-field models are formulated based on variational principles for brittle fracture (the seminal work of Francfort and Marigo [47] and its extensions), microforce theory allows one to derive phase-field models for more complex problems for which sound variational principles are unavailable, such as cohesive/frictional fracture (see Choo and Sun [29] for a detailed discussion). It is noted that for the particular case of brittle fracture, the microforce and variational approaches lead to the same phase-field formulation. ...
Article
Cracking of rocks and rock-like materials exhibits a rich variety of patterns where tensile (mode I) and shear (mode II) fractures are often interwoven. These mixed-mode fractures are usually cohesive (quasi-brittle) and frictional. Although phase-field modeling is increasingly used for rock fracture simulation, no phase-field formulation is available for cohesive and frictional mixed-mode fracture. To address this shortfall, here we develop a double-phase-field formulation that employs two different phase fields to describe cohesive tensile fracture and frictional shear fracture individually. The formulation rigorously combines the two phase fields through three approaches: (i) crack-direction-based decomposition of the strain energy into the tensile, shear, and pure compression parts, (ii) contact-dependent calculation of the potential energy, and (iii) energy-based determination of the dominant fracturing mode in each contact condition. We validate the proposed model, both qualitatively and quantitatively, with experimental data on mixed-mode fracture in rocks. The validation results demonstrate that the double-phase-field model-a combination of two quasi-brittle phase-field models-allows one to directly use material strengths measured from experiments, unlike brittle phase-field models for mixed-mode fracture in rocks. Another standout feature of the double-phase-field model is that it can simulate, and naturally distinguish between, tensile and shear fractures without complex algorithms.
... Smeared crack approaches such as phase field fracture (e.g. Miehe et al. (2010); Borden et al. (2012); Choo and Sun (2018);Bryant and Sun (2018); Na and Sun (2018); Bryant and Sun (2021)), nonlocal or gradient damage models Geers et al. (1998);Bažant and Jirásek (2002); Liu and Sun (2020c) provide an alternative to capture the crack branching process without requiring additional criterion to predict the onset of crack branching and the additional implementation effort to embed discontinuity. This ease of implementation provides a great advantage in handling the evolving interfaces. ...
... Note that the material could have melted if (1) the specific heat is low and/or (2) the energy dissipation is large such that the local temperature may rise without significant diffusion (Goldsby and Tullis, 2011; Ma and Sun, 2020). Furthermore, a more profound temperature increase may also trigger the brittleductile transition that affects the mechanical responses and fracture patterns (Choo and Sun, 2018). These mechanisms are not captured in this research but will be considered in future studies. ...
Article
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We propose a material point method (MPM) to model the evolving multi-body contacts due to crack growth and fragmentation of thermo-elastic bodies. By representing particle interface with an implicit function, we adopt the gradient partition techniques introduced by Homel and Herbold 2017 to identify the separation between a pair of distinct material surfaces. This treatment allows us to replicate the frictional heating of the evolving interfaces and predict the energy dissipation more precisely in the fragmentation process. By storing the temperature at material points, the resultant MPM model captures the thermal advection-diffusion in a Lagrangian frame during the fragmentation, which in return affects the structural heating and dissipation across the frictional interfaces. The resultant model is capable of replicating the crack growth and fragmentation without requiring dynamic adaptation of data structures or insertion of interface elements. A staggered algorithm is adopted to integrate the displacement and temperature sequentially. Numerical experiments are employed to validate the diffusion between the thermal contact, the multi-body contact interactions and demonstrate how these thermo-mechanical processes affect the path-dependent behaviors of the multi-body systems.
... While these studies mainly focus on the tensile fracture, some researchers also extended the phase-field formulation to model the mixed-mode (tensile and shear) fracture in rocks (Bryant & Sun, 2018;X. Zhang et al., 2017), and the failure mode transition in the pressure-sensitive geomaterials (Choo & Sun, 2018a). Nevertheless, all these studies employ phase-field formulations originated from linear elastic fracture mechanics (LEFM), on the premise that the fracture is purely brittle. ...
... As introduced earlier, phase-field modeling has emerged as a robust method to handle complex fracture geometry without any explicit representation. After being developed for general solids (Borden et al., 2012;Bourdin et al., 2008;, phase-field modeling has been adopted to simulate rock fracture in a variety of contexts, from hydraulic fracturing to cracking from preexisting flaws (Bryant & Sun, 2018;Choo & Sun, 2018a;Fei & Choo, 2021;Ha et al., 2018;S. Lee et al., 2016;X. ...
Thesis
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Geologic materials contain a wide range of discontinuities and fractures, which are central to many engineering applications and geologic hazards. The fracturing process of geologic materials is characterized by its progressive softening, termed quasi-brittleness. Also, the fractured surfaces exhibit a number of features including frictional contact and roughness effects. To model the discontinuities and fracturing processes in geologic materials, the phase-field method has been increasingly applied, as it has an outstanding ability to handle complex crack geometries without using tracking algorithms. However, few phase-field studies considered the quasi-brittleness of geologic materials. More importantly, all existing phase-field approaches dismissed the frictional contact, let alone the roughness effects. To fill the above research gaps, this thesis develops a suite of phase-field approaches to enable more reliable and systematic modeling of discontinuities and fractures in geologic materials. These approaches focus on four different but interconnected aspects of geologic discontinuities and fractures, namely frictional contact, shear fracture incorporating friction dissipation, mixed-mode rock fracture, and roughness effects of rock fractures. In the first approach, we incorporate the pressure-dependent friction into the phase-field formulation by employing a crack-oriented decomposition of the stress tensor. Each stress component is calculated by identifying the contact condition at the material point of interest. We show that the proposed method can well reproduce the results from the standard and extended finite element method without applying any algorithms to impose contact constraints. Building on the formulation in the first approach, we develop a phase-field approach to model the shear fracture propagation that involves friction dissipation during the fracturing process. The proposed formulation is demonstrably consistent with a fracture-mechanics-based theory. We also devise a new degradation function to avoid the sensitivity of material strengths to the phase-field length parameters, allowing the proposed method to model quasi-brittle materials with prescribed strengths. Next, we introduce a double-phase-field approach to the mixed-mode rock fracture by combining the formulations of cohesive tensile cracks and frictional shear cracks. The proposed formulation is essentially based on three steps: (i) stress decomposition in a crack-oriented coordinate system; (ii) calculation of the total potential energy according to the contact condition; (iii) determination of the dominant fracture mode following an energy-based criterion. We validate the double-phase-field approach through qualitative and quantitative comparisons between the modeling results and the experimental results. Lastly, we introduce a phase-field modeling framework for rock fractures by incorporating roughness effects. The proposed framework aims at transforming a displacement-based constitutive law of rock fractures into a strain-based version without introducing new parameters. In doing so, the continuous phase-field method can accommodate the rough fracture models originally designed for discrete discontinuities. Numerical examples show that the phase-field results have an excellent agreement with the results obtained from the extended finite element method.
... However, as the phase-field method was originally developed for tensile (mode I) fracture, the vast majority of phase-field models in geomechanics have focused on tensile fracture, e.g. [35][36][37][38][39][40]. ...
... For this purpose, it is pos-tulated that the fault propagates in the direction that maximizes the crack driving force, as assumed in the existing phase-field models for quasi-static fractures [42,55,67]. According to Eq. (38), the increment of the crack driving force is proportional to (τ m − τ r − ηV). When the fault starts to grow, V = 0, and thus the radiation damping term in the crack driving force is zero. ...
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Despite its critical role in the study of earthquake processes, numerical simulation of the entire stages of fault rupture remains a formidable task. The main challenges in simulating a fault rupture process include complex evolution of fault geometry, frictional contact, and off-fault damage over a wide range of spatial and temporal scales. Here, we develop a phase-field model for quasi-dynamic fault nucleation, growth, and propagation, which features two standout advantages: (i) it does not require any sophisticated algorithms to represent fault geometry and its evolution; and (ii) it allows for modeling fault nucleation, propagation, and off-fault damage processes with a single formulation. Built on a recently developed phase-field framework for shear fractures with frictional contact, the proposed formulation incorporates rate- and state-dependent friction, radiation damping, and their impacts on fault mechanics and off-fault damage. We show that the numerical results of the phase-field model are consistent with those obtained from well-verified approaches that model the fault as a surface of discontinuity, without suffering from the mesh convergence issue in the existing continuous approaches to fault rupture (e.g. the stress glut method). Further, through numerical examples of fault propagation in various settings, we demonstrate that the phase-field approach may open new opportunities for investigating complex earthquake processes that have remained overly challenging for the existing numerical methods.
... Over the last several decades, various numerical methods, categorized into mesh-based and mesh-free ones, have been proposed and developed to simulate progressive failure processes in geomaterials (Monaghan 1988;Shi and Goodman 1989;Tang 1997;Belytschko and Black 1999;Bouchard et al. 2000;Wells and Sluys 2001;Dolbow et al. 2001;Belytschko 2004, 2007;Potyondy and Cundall 2004;Zhang and Ge 2005;Paluszny and Matthäi 2009;Bui et al. 2011;Zhou and Yang 2012;He et al. 2013;Wang et al. 2014;Zhao 2015;Fan and Li 2017a;Zhou and Cheng 2017;Zhou et al. 2015Zhou et al. , 2018cChoo and Sun 2018;Kou et al. 2019a, b;) based on the classical local continuum mechanics theory. For mesh-based methods, a finiteelement method (FEM) (Tang 1997;Bouchard et al. 2000;Wang et al. 2014), extended finite-element method (XFEM) (Belytschko and Black 1999;Dolbow et al. 2001;Zhou and Yang 2012;Zhou and Cheng 2017), virtual internal bond method (VIBM) (Zhang and Ge 2005), cohesive zone method (CZM) (Wells and Sluys 2001), and phase field method (PFM) Choo and Sun 2018) have been developed to study the failure characteristics of rocks and soils. ...
... Over the last several decades, various numerical methods, categorized into mesh-based and mesh-free ones, have been proposed and developed to simulate progressive failure processes in geomaterials (Monaghan 1988;Shi and Goodman 1989;Tang 1997;Belytschko and Black 1999;Bouchard et al. 2000;Wells and Sluys 2001;Dolbow et al. 2001;Belytschko 2004, 2007;Potyondy and Cundall 2004;Zhang and Ge 2005;Paluszny and Matthäi 2009;Bui et al. 2011;Zhou and Yang 2012;He et al. 2013;Wang et al. 2014;Zhao 2015;Fan and Li 2017a;Zhou and Cheng 2017;Zhou et al. 2015Zhou et al. , 2018cChoo and Sun 2018;Kou et al. 2019a, b;) based on the classical local continuum mechanics theory. For mesh-based methods, a finiteelement method (FEM) (Tang 1997;Bouchard et al. 2000;Wang et al. 2014), extended finite-element method (XFEM) (Belytschko and Black 1999;Dolbow et al. 2001;Zhou and Yang 2012;Zhou and Cheng 2017), virtual internal bond method (VIBM) (Zhang and Ge 2005), cohesive zone method (CZM) (Wells and Sluys 2001), and phase field method (PFM) Choo and Sun 2018) have been developed to study the failure characteristics of rocks and soils. Among mesh-free approaches, the cracking particle method (CPM) Belytschko 2004, 2007), discrete element method (DEM) (Potyondy and Cundall 2004;Paluszny and Matthäi 2009), smoothed particle hydrodynamics (SPH) (Monaghan 1988;Bui et al. 2011), distinct lattice spring method (DLSM) (Zhao 2015), discontinuous deformation analysis (DDA) (Shi and Goodman 1989;Fan and Li 2017a), numerical manifold method (NMM) (He et al. 2013), and general particle dynamics (GPD) (Zhou et al. 2015), for example, have been proposed and developed to investigate the progressive failure behaviors of geomaterials. ...
Article
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Peridynamic (PD) theory is an integral-type nonlocal continuum mechanics theory that reformulates the equation of motion in local continuum mechanics as an integrodifferential equation. PD theory has been used to simulate mechanical responses of various materials with discontinuous structures. During the past two decades, PD theory has been developed to simulate different discontinuous problems and to illustrate various discontinuous phenomena in the diverse fields of engineering and sciences. In this paper, a state-of-the-art review on the investigation of failure processes of geomaterials in a PD framework is performed to illustrate the successful results and potential capability of PD theory in future geotechnical engineering. This review starts with a brief theoretical description of a bond-based peridynamic (BB-PD) model, a state-based peridynamic (SB-PD) model, a hybrid PD–classical continuum mechanics model, and an analytical PD model. Then surveys of PD applications to coupled multiphysical failure problems of geomaterials are conducted, which aim at revealing the associated failure mechanism. The applications of PD theory to simulate real geotechnical engineering are subsequently reported. Finally, some future-oriented research perspectives of PD applications in geotechnical engineering and a brief summary are presented.
... These mechanisms are typically not accounted for in conventional crystal plasticity and crack growth models. Phasefield models (PFM) (Miehe et al., 2010;Clayton and Knap, 2015;Lorenzis et al., 2016;Choo and Sun, 2018;Shahba and Ghosh, 2019;Cheng et al., 2020) have demonstrated significant promise in modeling brittle and ductile fracture, receiving considerable attention from the fracture community in the recent years. These models regularize the sharp crack surface by introducing an auxiliary scalar field, viz. the phase-field order parameter ∈ [0, 1], to represent crack topology. ...
... The regularized crack surface area functional ( ) in the phase-field model has been expressed in the literature (Miehe et al., 2010;Clayton and Knap, 2015;Lorenzis et al., 2016;Choo and Sun, 2018;Shahba and Ghosh, 2019;Cheng et al., 2020) in terms of the order parameter and a length-scale parameter that controls regularization of the discontinuous crack, as: ...
Article
This paper develops a method for physics-based augmentation of the Helmholtz free energy density functionals, used in coupled crystal plasticity phase-field finite element (CP-PF FE) models of fracture in crystalline metallic materials. Specifically, the defect and crack surface energy components are enhanced with terms that mechanistically account for the presence of atomic-scale, crack-tip nucleated dislocations. The additional terms in the free energy representation are motivated and calibrated by energy equivalence between a concurrent atomistic–continuum scale model and the CP-PF FE model. The atomistic domain of the concurrent model incorporates a time-accelerated, molecular dynamics (MD) LAMMPS code, while the continuum domain is modeled by a dislocation-density crystal plasticity FE model. The concurrent model transfers and transforms discrete dislocations in the atomistic domain to dislocation densities in the crystal plasticity domain. Dislocation densities are transported in the continuum domain by solving the advection equation using a particle-based reproducing kernel particle method with collocation. A new form of the defect energy density is proposed by considering the effect of crack-tip nucleated dislocations. Parameters in the augmented defect and surface energies are evaluated by comparing with the concurrent model results. A comparison of crack propagation with and without contributions from the nucleated dislocations demonstrates a significant effect of nucleated dislocations on the evolution of the crack.
... Ductile fracture within the framework of PFF has also garnered interest and shown success for quasistatic configurations and finite strains by relying on a multiplicative decomposition of the stress into its elastic and plastic parts as well as an additive decomposition of the constitutive models into tensile and volumetric components Ambati et al. [2016], Borden et al. [2016], Miehe et al. [2015]. Most recently, geologically-inspired work has explored whether return mapping routines should include plastic degradation functions since they may break assumptions about which stress is used to evaluate the yield condition Choo and Sun [2018]. ...
... One detail that's important to discuss here, however, is that Choo and Sun [2018] recently highlighted a key detail for implementing plastic return mapping when a damage model is being used to degrade stress: an "effective stress" (the stress without degradation) ...
Thesis
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Material fracture surrounds us every day from tearing off a piece of fresh bread to dropping a glass on the floor. Modeling this complex physical process has a near limitless breadth of applications in everything from computer graphics and VFX to virtual surgery and geomechanical modeling. Despite the ubiquity of material failure, it stands as a notoriously difficult phenomenon to simulate and has inspired numerous efforts from computer graphics researchers and mechanical engineers alike, resulting in a diverse set of approaches to modeling the underlying physics as well as discretizing the branching crack topology. However, most existing approaches focus on meshed methods such as FEM or BEM that require computationally intensive crack tracking and re-meshing procedures. Conversely, the Material Point Method (MPM) is a hybrid meshless approach that is ideal for modeling fracture due to its automatic support for arbitrarily large topological deformations, natural collision handling, and numerous successfully simulated continuum materials. In this work, we present a toolkit of augmented Material Point Methods for robustly and efficiently simulating material fracture both through damage modeling and through plastic softening/hardening. Our approaches are robust to a multitude of materials including those of varying structures (isotropic, transversely isotropic, orthotropic), fracture types (ductile, brittle), plastic yield surfaces, and constitutive models. The methods herein are applicable not only to the needs of computer graphics (efficiency and visual fidelity), but also to the engineering community where physical accuracy is key. Most notably, each approach has a unique set of parametric knobs available to artists and engineers alike that make them directly deployable in applications ranging from animated movie production to large-scale glacial calving simulation.
... Initial works on this topic include [14,[26][27][28][29][30][31][32][33] (see [34] for an overview). Phase-field models for ductile fracture were subsequently developed in the context of cohesive-frictional materials [35,36], porous plasticity [37] including thermal effects [38], fiber pullout behavior [39], hydraulic fracture [40][41][42], degradation of the fracture toughness [43], multi-surface plasticity [44] and fatigue [45], among others. The majority of the ductile phase-field models found in the literature are based on local plasticity. ...
Article
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The prediction of crack initiation and propagation in ductile failure processes are challenging tasks for the design and fabrication of metallic materials and structures on a large scale. Numerical aspects of ductile failure dictate a sub-optimal calibration of plasticity- and fracture-related parameters for a large number of material properties. These parameters enter the system of partial differential equations as a forward model. Thus, an accurate estimation of the material parameters enables the precise determination of the material response in different stages, particularly for the post-yielding regime, where crack initiation and propagation take place. In this work, we develop a Bayesian inversion framework for ductile fracture to provide accurate knowledge regarding the effective mechanical parameters. To this end, synthetic and experimental observations are used to estimate the posterior density of the unknowns. To model the ductile failure behavior of solid materials, we rely on the phase-field approach to fracture, for which we present a unified formulation that allows recovering different models on a variational basis. In the variational framework, incremental minimization principles for a class of gradient-type dissipative materials are used to derive the governing equations. The overall formulation is revisited and extended to the case of anisotropic ductile fracture. Three different models are subsequently recovered by certain choices of parameters and constitutive functions, which are later assessed through Bayesian inversion techniques. A step-wise Bayesian inversion method is proposed to determine the posterior density of the material unknowns for a ductile phase-field fracture process. To estimate the posterior density function of ductile material parameters, three common Markov chain Monte Carlo (MCMC) techniques are employed: (i) the Metropolis–Hastings algorithm, (ii) delayed-rejection adaptive Metropolis, and (iii) ensemble Kalman filter combined with MCMC. To examine the computational efficiency of the MCMC methods, we employ the R^-convergence\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{R}{-}convergence$$\end{document} tool. The resulting framework is algorithmically described in detail and substantiated with numerical examples.
... Initial efforts focused on shear fracture [74], while subsequent works [75][76][77] considered mixed-mode fracture, mostly inspired by the so-called F-criterion [72]. Other studies have taken pressure-dependent frictional behavior into account [78], as well as plastic coupling [79][80][81][82]. More recently, a micromechanics-based approach to fracture in geomaterials was proposed [83,84], where the macroscopic crack phase-field and the plastic strain tensor are linked to mechanisms at the microcrack level. ...
Article
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This paper presents a novel variational phase-field model for different fracture processes in fully saturated porous media. As a key feature, the model employs a micromechanics-based theory for the description of brittle-tensile and compressive–ductile fracture. As such, the field variables are linked to physical mechanisms at the microcrack level, with damage emerging as the consequence of microcrack growth. Similarly, plasticity emerges as a consequence of the frictional sliding of closed microcracks. In this way, the evolution of opening microcracks in tension leads to (mode I) brittle fracture, while the evolution of closed microcracks in compression/shear leads to (mode II) ductile fracture. These failure mechanisms are coupled to fluid flow, resulting in a Darcy–Biot–type hydromechanical model. Therein, in the tensile regime, plasticity naturally vanishes, while damage is driven by poroelastic energy, accounting for the pressure in fluid-filled opening microcracks. On the other hand, in the compressive/shear regime, the plastic driving force naturally follows as a Terzaghi-type effective stress in terms of the local stress field acting on the microcrack surfaces, while damage is solely driven by the frictionally blocked free energy. As another important feature, the model includes a non-associative frictional plasticity law. Nevertheless, a thermodynamically consistent variational framework is employed, for which different energetic principles are discussed. Finally, the numerical simulations show that the model captures relevant hydromechanical coupling effects in benchmark problems, including mechanically induced shear fracture and hydraulically induced tensile fracture.
... After being developed for general solids [9][10][11], phase-field modeling of fracture has been adopted to simulate rock fracture in a variety of contexts, from hydraulic fracturing to cracking from preexisting flaws (e.g. [12][13][14][15][16][17]). ...
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Phase-field modeling—a continuous approach to discontinuities—is gaining popularity for simulating rock fractures due to its ability to handle complex, discontinuous geometry without an explicit surface tracking algorithm. None of the existing phase-field models, however, incorporates the impact of surface roughness on the mechanical response of fractures—such as elastic deformability and shear-induced dilation—despite the importance of this behavior for subsurface systems. To fill this gap, here we introduce the first framework for phase-field modeling of rough rock fractures. The framework transforms a displacement-jump-based discrete constitutive model for discontinuities into a strain-based continuous model, without any additional parameter, and then casts it into a phase-field formulation for frictional interfaces. We illustrate the framework by constructing a particular phase-field form employing a rock joint model originally formulated for discrete modeling. The results obtained by the new formulation show excellent agreement with those of a well-established discrete method for a variety of problems ranging from shearing of a single discontinuity to compression of fractured rocks. It is further demonstrated that the phase-field formulation can well simulate complex crack growth from rough discontinuities. Consequently, our phase-field framework provides an unprecedented bridge between a discrete constitutive model for rough discontinuities—common in rock mechanics—and the continuous finite element method—standard in computational mechanics—without any algorithm to explicitly represent discontinuity geometry.
... The phase-field method can be easily implemented in standard finite element codes and has successfully been applied to various engineering problems including multi-physics coupling (Miehe et al., 2015), finite deformation (Borden et al., 2016). It has also been extended to plastic materials (Fang et al., 2019;Choo and Sun, 2018). ...
Article
In the context of feasibility study for geological disposal of radioactive waste, this work focuses on numerical modeling of hydromechanical response and induced damage zones around an experimental gallery at the underground research laboratory (URL) at Bure in France. A new phase-field method is first proposed by considering two independent damage fields in order to easily describe both tensile and shear cracks. The phase-field method is extended to saturated porous media by incorporating the effect of fluid pressure. The proposed method is implemented in a finite element code and applied to the experimental gallery. The evolutions of hydrome-chanical responses and induced damaged zones are analyzed in both excavation and post-excavation phases. Numerical results are compared with experimental observations and measurements. It is found that the proposed method is able to well describe the main features of hydromechanical responses and damage processes such as tensile and shear cracked zones.
... Another appealing aspect of continuum modeling is that it allows one to simulate a wide range of materials -beyond dry, clean granular materials -by taking advantage of various existing material models formulated at finite strains (e.g. [56][57][58][59][60]). These features are highly desired for a large number of practical problems for which it is virtually impossible to model granular media in a discrete way. ...
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Granular impact – the dynamic intrusion of solid objects into granular media – is widespread across scientific and engineering applications including geotechnics. Existing approaches for simulating granular impact dynamics have relied on either a pure discrete method or a pure continuum method. Neither of these methods, however, is deemed optimal from the computational perspective. Here, we introduce a hybrid continuum–discrete approach, built on the coupled material-point and discrete-element method (MP-DEM), for simulating granular impact dynamics with unparalleled efficiency. To accommodate highly complex solid–granular interactions, we enhance the existing MP-DEM formulation with three new ingredients: (i) a robust contact algorithm that couples the continuum and discrete parts without any interpenetration under extreme impact loads, (ii) large deformation kinematics employing multiplicative elastoplasticity, and (iii) a trans-phase constitutive relation capturing gasification of granular media. For validation, we also generate experimental data through laboratory measurement of the impact dynamics of solid spheres dropped onto dry sand. Simulation of the experiments shows that the proposed approach can well reproduce granular impact dynamics in terms of impact forces, intrusion depths, and splash patterns. Further, through parameter studies on material properties, model formulations, and numerical schemes, we identify key factors for successful continuum–discrete simulation of granular impact dynamics.
... for Eq. (13), which provably conserves the angular momentum [25]. Also, following the original APIC [24,25], we employ cubic B-splines for the interpolation functions and the pure PIC method for the velocity update, i.e., set g ¼ 0 in Eq. (15). ...
Article
Full-text available
Granular impact—the dynamic intrusion of solid objects into granular media—is widespread across scientific and engineering applications including geotechnics. Existing approaches to the simulation of granular impact dynamics have relied on either a purely discrete method or a purely continuum method. Neither of these methods, however, is deemed optimal from the computational perspective. Here, we introduce a hybrid continuum–discrete approach, built on the coupled material-point and discrete-element method (MP–DEM), for simulation of granular impact dynamics with unparalleled efficiency. To accommodate highly complex solid–granular interactions, we enhance the existing MP–DEM formulation with three new ingredients: (i) a robust contact algorithm that couples the continuum and discrete parts without any interpenetration under extreme impact loads, (ii) large deformation kinematics employing multiplicative elastoplasticity, and (iii) a trans-phase constitutive relation capturing gasification of granular media. For validation, we also generate experimental data through laboratory measurement of the impact dynamics of solid spheres dropped onto dry sand. Simulation of the experiments shows that the proposed approach can well reproduce granular impact dynamics in terms of impact forces, intrusion depths, and splash patterns. Furthermore, through parameter studies on material properties, model formulations, and numerical schemes, we identify key factors for successful continuum–discrete simulation of granular impact dynamics.
... At intermediate confining pressures, the rock exhibits brittleductile transition behaviour where there is a noticeable large inelastic strain before reaching the peak stress, followed by a gentle decrease of the stress (e.g., Evans et al. 1990;Herrmann et al. 2018).When the confining pressures are high, the rock is ductile where it experiences a very large inelastic strain up to peak stress, and then the stress remains constant (e.g., Nicolas et al. 2017). Figure 1 also displays the different failure modes the rock experiences at failure: axial splitting and shear fracture for brittle failure; shear band for brittleductile transition failure; and ductile flow for ductile failure (e.g., Basu et al. 2013;Choo and Sun 2018). ...
Article
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Understanding the failure behaviour of rocks is very important for many engineering applications. While conceptual models to predict failure behaviour of rocks have been proposed, very few models exist to quantitatively predict the brittle–ductile characteristics of mudstones at high confining pressures. In this study, we performed triaxial compression tests at confining pressures up to 130 MPa on dry indurated mudstones from the Naparima Hill Formation, Trinidad. Experimental results including stress–strain curves, failure modes, and strength parameters of the mudstones were obtained. Analysing the stress–strain curves, shows that the mudstones experienced brittle, brittle–ductile transition, and ductile failure behaviours as the confining pressure increases. The failure behaviour at confining pressures less than 50 MPa is accompanied by axial splitting fractures and shear fractures. At confining pressures greater than 50 MPa, shear band and ductile flow fractures are observed in the mudstones. A failure behaviour model was developed using the failure behaviour and failure strength of the mudstones. This model predicts the brittle, brittle–transition and ductile zones for mudstones at high confining pressures. The brittle to ductile transition is not sudden, but a gradual process that is controlled by the confining pressure and failure strength. The model indicates a narrow brittle-ductile transition zone for weak mudstones as compared to strong mudstones.
... If we define a yield function f (σ ′′ , p c ) ≤ 0, where p c is some plastic internal variable that reflects the hardening/softening response of the material [55,[75][76][77], and if we impose the associative flow rule [34] ϵ p =λ ...
Article
We present a continuum framework to simulate fluid flow through anisotropic elastoplastic media with double porosity. Two effective stress measures σ′ and σ′′ emerge from the thermodynamic formulation, which are energy-conjugate to the elastic and plastic components of strain, respectively. Both effective stress measures can be expressed as a combination of the total Cauchy stress σ and the average pore pressure p¯ in the two pore scales. In the effective stress for elasticity, p¯ is scaled with a rank-2 Biot tensor, whereas the effective stress for plasticity follows the Terzaghi form in which p¯ is scaled by the Kronecker delta. The Biot tensor and storage coefficients are derived as functions of elasticity parameters and porosities. A mixed finite element formulation is introduced to discretize the domain and solve initial boundary-value problems. A stabilization scheme is employed on equal-order interpolation for both displacement and pressure fields throughout the entire range of drainage responses. Numerical simulations reproduce the hydromechanical response of Opalinus shale in one-dimensional consolidation tests throughout the range of primary and secondary consolidation under different external loads. Numerical simulations of the consolidation of a rectangular domain subjected to a strip load demonstrate the efficacy of the proposed stabilization scheme, as well as illustrate the impacts of stress history, mass transfer, and different pore systems on the hydromechanical response.
... It is not only for the entire region, but for the characteristics of fault block oil layer cutting and fine fragmentation, but also for different well groups. To evaluate the effect of water injection development [25,26]. To develop oilfields by water flooding, it is necessary to continuously evaluate the development effect at different stages of development in order to put forward effective adjustment opinions. ...
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Nowadays, people’s demand for underground mineral resources is increasing, and geological disasters have occurred frequently in recent years. Geological disasters refer to geological effects or geological phenomena that are formed under the action of natural or man-made factors, causing loss of human life and property, and damage to the environment; such as landslides, collapses, mudslides, and ground subsidence. Under such a background, people must accelerate the exploration of complex geological structures. This paper is aimed at using the methods and concepts of deep reinforcement learning. Deep learning is to learn the inherent laws and representation levels of sample data. The information obtained during these learning processes is of great help to the interpretation of data such as text and images. In this way, the fine geology of complex fault-block reservoirs is modeled and studied. Geological structures and phenomena are discussed through convolutional neural network models and computer techniques. At the same time, the multitask bird recognition network is used to extract and classify geological images, so as to construct geological model maps with different spatial structures. Finally, the quality of the fault reconstruction model, the calculation of reservoir geological simulation reserves, and the evaluation of the water injection development effect of complex fault blocks are analyzed. In the evaluation of the development effect of water injection in complex fault blocks, comparing the relationship curve between the actual comprehensive water content and the oil recovery factor with the standard curve, the comprehensive water content of the initial block increased rapidly. Through timely and dynamic water allocation and comprehensive management, the water cut rising speed is controlled. The current comprehensive water cut of the reservoir is between 60% and 80%, the actual curve is between 25% and 35%, and the estimated waterflooding recovery is about 30%.
... The present length transformation (l → 10 −3 l, Ω → 10 −9 Ω) produces an energy functional per unit volume Ψ/Ω equivalent to that which would be achieved by increasing the fracture surface energy by the inverse amount (Υ → 10 3 Υ). The latter, essentially choosing the fracture surface energy based on a pre-assigned fracture strength P C and regularization length l, is an analogous approach to avoid spuriously low fracture strength in phase field models when the regularization length l is too large due to minimum mesh size constraints [51,75]. ...
Article
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Diamond-silicon carbide (SiC) polycrystalline composite blends are studied using a computational approach combining molecular dynamics (MD) simulations for obtaining grain boundary (GB) fracture properties and phase field mechanics for capturing polycrystalline deformation and failure. An authentic microstructure, reconstructed from experimental lattice diffraction data with locally refined discretization in GB regions, is used to probe effects of local heterogeneities on material response in phase field simulations. The nominal microstructure consists of larger diamond and SiC (cubic polytype) grains, a matrix of smaller diamond grains and nanocrystalline SiC, and GB layers encasing the larger grains. These layers may consist of nanocrystalline SiC, diamond, or graphite, where volume fractions of each phase are varied within physically reasonable limits in parametric studies. Distributions of fracture energies from MD tension simulations are used in the phase field energy functional for SiC-SiC and SiC-diamond interfaces, where grain boundary geometries are obtained from statistical analysis of lattice orientation data on the real microstructure. An elastic homogenization method is used to account for distributions of second-phase graphitic inclusions as well as initial voids too small to be resolved individually in the continuum field discretization. In phase field simulations, SiC single crystals may twin, and all phases may fracture. The results of MD calculations show mean strengths of diamond-SiC interfaces are much lower than those of SiC-SiC GBs. In phase field simulations, effects on peak aggregate stress and ductility from different GB fracture energy realizations with the same mean fracture energy and from different random microstructure orientations are modest. Results of phase field simulations show unconfined compressive strength is compromised by diamond-SiC GBs, graphitic layers, graphitic inclusions, and initial porosity. Explored ranges of porosity and graphite fraction are informed by physical observations and constrained by accuracy limits of elastic homogenization. Modest reductions in strength and energy absorption are witnessed for microstructures with 4% porosity or 4% graphite distributed uniformly among intergranular matrix regions. Further reductions are much more severe when porosity is increased to 8% relative to when graphite is increased to 8%.
... The main driving force for these developments is the possibility to handle complex fracture phenomena within numerical methods in two and three dimensions. In recent years, several brittle [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27] and ductile [28,29,30,31,32,33,34,35,36,37,38,39,40,41] phase-field fracture formulations have been proposed in the literature. These studies range from the modeling of Figure 1: Offshore Wind Turbine (source: germanoffshorewind.org) ...
Preprint
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A probabilistic approach to phase-field brittle and ductile fracture with random material and geometric properties is proposed within this work. In the macroscopic failure mechanics, materials properties and exactness of spatial quantities (of different phases in the geometrical domain) are assumed to be homogeneous and deterministic. This is unlike the lower-scale with strong fluctuation in the material and geometrical properties. Such a response is approximated through some uncertainty in the model problem. The presented contribution is devoted to providing a mathematical framework for modeling uncertainty through stochastic analysis of a microstructure undergoing brittle/ductile failure. Hereby, the proposed model employs various representative volume elements with random distribution of stiff-inclusions and voids within the composite structure. We develop an allocating strategy to allocate the heterogeneities and generate the corresponding meshes in two- and three-dimensional cases. Then the Monte Carlo finite element technique is employed for solving the stochastic PDE-based model and approximate the expectation and the variance of the solution field of brittle/ductile failure by evaluating a large number of samples. For the prediction of failure mechanisms, we rely on the phase-field approach which is a widely adopted framework for modeling and computing the fracture phenomena in solids. Incremental perturbed minimization principles for a class of gradient-type dissipative materials are used to derive the perturbed governing equations. This analysis enables us to study the highly heterogeneous microstructure and monitor the uncertainty in failure mechanics. Several numerical examples are given to examine the efficiency of the proposed method.
... Initial works on this topic include [21,22,23,14,24,25,26,27] (see [28] for an overview). Phase-field models for ductile fracture were subsequently developed in the context of cohesive-frictional materials [29,30], porous plasticity [31] including thermal effects [32], the virtual element method (VEM) [33], fiber pullout behavior [34], hydraulic fracture [35,36,37], degradation of the fracture toughness [38], multi-surface plasticity [39] and fatigue [40], among others. ...
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The prediction of crack initiation and propagation in ductile failure processes are challenging tasks for the design and fabrication of metallic materials and structures on a large scale. Numerical aspects of ductile failure dictate a sub-optimal calibration of plasticity- and fracture-related parameters for a large number of material properties. These parameters enter the system of partial differential equations as a forward model. In this work, we develop a step-wise Bayesian inversion framework for ductile fracture to provide accurate knowledge regarding the effective mechanical parameters. To this end, synthetic and experimental observations are used to estimate the posterior density of the unknowns. To model the ductile failure behavior of solid materials, we rely on the phase-field approach to fracture, for which we present a unified formulation that allows recovering different models on a variational basis. In the variational framework, incremental minimization principles for a class of gradient-type dissipative materials are used to derive the governing equations. The overall formulation is revisited and extended to the case of anisotropic ductile fracture. Three different models are subsequently recovered by certain choices of parameters and constitutive functions, which are later assessed through Bayesian inversion techniques. To estimate the posterior density function of ductile material parameters, three common Markov chain Monte Carlo (MCMC) techniques are employed: (i) the Metropolis-Hastings algorithm, (ii) delayed-rejection adaptive Metropolis, and (iii) ensemble Kalman filter combined with MCMC. To examine the computational efficiency of the MCMC methods, we employ the R-convergence tool.
... The nucleation, propagation and branching of cracks are represented by the thermodynamically proposed governing equation. 104 Due to its compatibility in combining with conventional coupled HM framework and straightforward manner in describing the fracturing process, the PF has become a popular approach to simulate different failure patterns in rocks, i.e., brittle/quasi-brittle fracture in, [105][106][107][108][109] ductile fracture in [110][111][112] and dynamic fracture in. 113 As another continuous approach to explicitly simulate the fracturing process, the XFEM provided an alternative choice to handle the arbitrary fracture propagation without introducing interface element. ...
Article
Host rocks, as the final impediment to waste migration, play a significant role in the nuclear waste repositories. Modelling of gas migration in saturated host rocks as well as its coupled hydro-mechanical (HM) behaviors is of importance to the assessment of geological disposal facilities. A comprehensive literature review is carried out on the models for simulating gas migration in saturated rock materials. Several aspects have been provided and discussed, including the material properties and experimental interpretations, governing equations, constitutive models for hydraulic and mechanical processes, fracture propagation models. Specifically, these models are discussed in detail with respect to their performance in describing the recorded experimental behaviors. It is found that the visco-capillary two phase flow model with enriched intrinsic permeability is commonly used to describe the advective gas transport in saturated host rocks. The embedded fracture model (EFM) or enriched EFM seems to be the most favored model as it accounts for the fracturing mechanism which is more representative to implicitly simulate the preferential gas pathways. To describe the mechanical behavior of rocks during the gas migration process, linear/nonlinear elastic, elastoplastic and damage models have been included in the mechanical processes. The HM models incorporating plasticity or damage may not be applicable in most experimental studies, which are not competent to simulate the key experimental behavior associated with the development of gas preferential pathways. Current models that can explicitly describe the gas induced micro-fracturing are seldomly reported, the existed ones are not able to represent all the key experimental behaviors related to preferential gas flow. Advanced approaches, i.e., phase field (PF) method, extended finite element method (XFEM), discrete element method (DEM), hybrid finite discrete element method (FDEM) can be integrated with cohesive zone model (CZM) to provide some promising aspects in representing the development of preferential gas pathways. Lastly, the conclusions and recommendations for future modelling are given.
... Moreover, the critical energy release rate G c , which is an important parameter. But its direct measurement by laboratory tests is difficult for rocks, like granites (Choo and Sun 2018). For microscopic numerical model proposed in this paper, the critical energy release rate of different minerals and grain boundaries are the most important parameters. ...
Article
The cracking behavior and mechanical properties of brittle rocks is controlled by its microstructure. In this work, an extended grain-based phase field method (GB-PFM) based on Voronoi tessellation technology is proposed to investigate the mechanical properties and crack propagation in heterogeneous brittle rocks by considering material heterogeneities and microstructure heterogeneities at the grain scale. Then, a trial and error calibration procedure is developed after comprehensive study of uncertain parameters such as grain boundaries thickness and critical energy release rate. It is shown that greater grain boundaries thickness results in a lower tensile strength and lower Young’s modulus, and greater number of intergrain cracks. Then, a group of models with single pre-existing flaw (flaw inclination angles: 0°, 30°, 45°, 60°, 90°) are built according to calibration parameters to study the mechanical properties and cracking behavior of heterogeneous granite under direct tension. The path of the macrocracks obtained from GB-PFM shows a more noticeable tortuous nature as compared with that results by others phase field methods. The variation of flaw inclination has an obviously influence on crack propagation. Failure patterns obtained by GB-PFM revealing the macrocracks are composed of intragrain and intergrain cracks at the microscopic view. Moreover, the stress field distribution obtained by GB-PFM is agree well with the theoretical analyses and other numerical simulations. The above simulated results prove that the GB-PFM provides an applicable numerical method for efficiently reproducing the microstructure and the related crack propagation of brittle crystalline rocks, which greatly expand the application of the phase field methods.
... Choo and Sun [124] combined a pressure-sensitive plasticity model and a phase field model to model fractures in geological materials. Zhou et al. [30] revisited the formulation of Ambati et al. [22] and established a new driving force for the phase field evolution to consider compressive-shear fractures in rocks. ...
Article
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 method approximates discontinuous geometry diffusely using a continuous field variable called the phase field. After being developed for general solids [9][10][11] , phase-field modeling of fracture has been adopted to simulate rock fracture in a variety of contexts, from hydraulic fracturing to cracking from preexisting flaws (e.g., [12][13][14][15][16][17]. ...
Article
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Phase-field modeling—a continuous approach to discontinuities—is gaining popularity for simulating rock fractures due to its ability to handle complex, discontinuous geometry without an explicit surface tracking algorithm. None of the existing phase-field models, however, incorporates the impact of surface roughness on the mechanical response of fractures—such as elastic deformability and shear-induced dilation—despite the importance of this behavior for subsurface systems. To fill this gap, here we introduce the first framework for phase-field modeling of rough rock fractures. The framework transforms a displacement-jump-based discrete constitutive model for discontinuities into a strain-based continuous model, without any additional parameter, and then casts it into a phase-field formulation for frictional interfaces. We illustrate the framework by constructing a particular phase-field form employing a rock joint model originally formulated for discrete modeling. The results obtained by the new formulation show excellent agreement with those of a well-established discrete method for a variety of problems ranging from shearing of a single discontinuity to compression of fractured rocks. It is further demonstrated that the phase-field formulation can well simulate complex crack growth from rough discontinuities. Consequently, our phase-field framework provides an unprecedented bridge between a discrete constitutive model for rough discontinuities—common in rock mechanics—and the continuous finite element method—standard in computational mechanics—without any algorithm to explicitly represent discontinuity geometry.
... To address this issue, Zhang et al. [68] proposed a modified phase field model to account for the distinction between the energy release rates for different failure modes. Choo and Sun [69] presented an expression of g f to consider the non-negligible difference between the uniaxial tension and uniaxial compression. Moreover, Bo et al. reformulated g f to consider the rate dependency for dynamic fracture [11,70]. ...
Article
Phase field approaches have been developed to analyze the failure behavior of ductile materials. In the previous phase field models, a constant critical energy or strain threshold is commonly introduced to the formulation of the driving force, aiming to avoid damage initiation at a low level of elastic and plastic deformations. However, it may not suffice to describe complex ductile fracture behavior of materials subject to various stress states. In this study, a new phase field approach is proposed to consider the effects of stress triaxiality and Lode angle, by incorporating phenomenological ductile fracture criteria. The proposed models are formulated using variational principles and implemented numerically in the finite element framework. Analytical homogeneous solutions for uniaxial tension, simple shear, and equibiaxial tension loads are derived to demonstrate the effectiveness of the proposed models. Three groups of numerical examples, covering a wide range of stress states, are utilized to further examine the proposed models. The results show that the models can reproduce the experimental response of the specimen in terms of force versus displacement curve, crack initiation, and crack propagation under various loading conditions. The proposed models are able to capture the stress-state dependence of fracture behavior of ductile materials.
... If we define a yield function f (σ , p c ) ≤ 0, where p c is some plastic internal variable that reflects the hardening/softening response of the material [31,39,106,107], and if we impose the associative flow rule [19] p =λ ∂f ∂σ , 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Anisotropic elastoplastic response of double-porosity media 13 whereλ ≥ 0 is a nonnegative plastic multiplier, then non-negativity of plastic dissipation requires that ...
Preprint
Manuscript accepted by CMAME, which presents a thermodynamically consistent framework for coupled fluid flow and solid deformation in anisotropic elastoplastic double-porosity media.
... Recently, in an effort to reconcile this inconsistency,Choo and Sun (2018) adopted the postulate that the evolution of the phase-field variable maximizes the energy dissipation. It is questionable whether the proposed derivation honors fracture as a fully dissipative process since it was not explicitly shown that the maximum value of dissipation is indeed non zero. ...
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A phenomenological phase-field model for the formation and growth of fatigue macro-cracks in brittle materials is proposed. Additionally to the classical elastic stiffness degradation in the phase-field approach to overload fracture, a fracture toughness degradation is further introduced. The model can reproduce both total life approaches, such as the Basquin empirical relation, and the Paris law, which lies at the core of most of the defect-tolerant approaches, with exponents characteristic of brittle fatigue crack growth. The numerical implementation of the model in an efficient scheme is described and its ability to handle complex geometries and loading conditions demonstrated.
Conference Paper
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Crack growth and coalescence from discontinuities is the fundamental process that underlies the majority of rock mass failures. Still, however, numerical simulation of this process remains a formidable challenge. The main reasons include that tensile and shear fractures are often mixed in the cracking process, that the cracking patterns depend strongly on the configuration of preexisting discontinuities, and that the cracking process may take place at different scales. The purpose of this paper is to evaluate the capabilities of the recently developed double-phase-field model (Fei and Choo, 2021) for simulating mixed-mode fracture in laboratory- and field-scale rocks with discontinuities. Simulation results show that the double-phase-field model can well reproduce laboratory test data, in terms of not only qualitative mixed-mode cracking patterns but also quantitative stress–strain responses. The results further demonstrate that the model can simulate complex rock fracture processes at the field scale such as slope failure due to crack growth and coalescence from joints.
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This work provides a framework for predicting fracture of catalyst coated membrane (CCM) due to coupled electro-chemo-mechanical degradation processes in proton exchange membrane water electrolysis (PEMWE) cells. Electrolysis in the catalyst layer (CL) bulk, diffusion of Hydrogen proton through the membrane (MEM), mechanical compression at the interface with the porous transport layer (PTL) generate micro-cracks that influence the catalyst degradation. Based on the experimental observations, we propose theoretical formulations along with the constitutive framework to help understanding and providing a reliable description of the stated multi-physics problem. The computational modeling of crack formation in the CL bulk is achieved in a convenient way by continuum phase-field formulations to fracture, which are based on the regularization of sharp crack discontinuities. The model performance is demonstrated through two representative boundary value problems, representing the cell setup and working of the PEMWE cell.
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Cracking in quasi-brittle geomaterials is a complex mechanical phenomenon, driven by various dissipation mechanisms across multiple length scales. While some recent promising works have employed the phase-field method to model the damage and fracture of geomaterials, several open questions still remain. In particular, capturing frictional sliding along the lips of microcracks, incorporating lower scale physics, and calibrating the length scale parameter, are some examples. The present paper addresses these essential problems. By leveraging homogenization-based damage-friction coupling formulations for microcracked solids, the linkage is built between the macroscopic phase-field damage variable and the microcrack density parameter. The phase-field is thus treated not only as an indicator for the location of cracks but also accounts for the density of microcracks. A unified Helmholtz free energy function is then constructed as a sum of the bulk energy (including elastic strain energy and plastic free energy) and the crack surface energy. Furthermore, a new set of degradation functions for the plastic free energy are provided, and a calibration procedure for the length scale parameter is proposed by reflecting a more realistic description of fracture process zone. In addition, an accelerated staggered iteration algorithm is developed to solve the strongly coupled problem more efficiently. Four numerical examples concerning a system of macroscopic cracks are investigated to illustrate the predictive capability of the proposed model in simulating tensile fracture, fault slippage and shear bands.
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Damage models have been successfully employed for many decades in the modelling of tensile failure, where the crack surfaces separate as a crack grows. The advantage of this approach is that crack trajectories can be computed simply and efficiently on a fixed finite element mesh without explicit tracking. The development of damage models for shear failure in compression, where the crack faces slide over each other subject to friction, is a non-trivial extension of this approach. A major difference is that part of the material stiffness in the damaged region must be retained to avoid interpenetration of the crack faces. This problem is resolved here by employing an anisotropic modification to the elastic stiffness tensor in the damaged region. This has the benefit of driving frictional cracks into the correct orientation, according to the Mohr–Coulomb failure criteria, but three issues remain. The first is that the shear discontinuity introduces some spurious stress perturbations around crack regions that are narrow (less than 3-4 elements wide). This is ameliorated by Helmholtz smoothing to allow efficient simulation on a coarse mesh. The second is that the complementarity of shear stress means that the shear stiffness is removed normal to the crack interface as well as parallel to it, and the third is that there are two potentially active failure planes at each point. Both these latter two issues are resolved by the introduction of a novel failure plane selection variable, which regulates either single plane or dual plane failure, and prevents the growth of erroneous cracks normal to the crack face. Both local and non-local models are investigated for linear and exponential strain softening responses. Unlike the non-local model, the local model demonstrates some mesh-size dependence, but it still retains some properties of interest, in that it supports narrower cracks and more rapidly forms a preference for the growth of a single crack when there are a number of competing cracks. The model is implemented in commercial finite element package COMSOL Multiphysics v5.5 and validated against two benchmark simulations: biaxial compression and the failure of a 45° slope. The correct crack angles, stipulated by the Mohr–Coulomb friction angle, are correctly reproduced, as is the post-failure residual frictional force in biaxial compression. The effect of the shear fracture energy on the force–displacement response is investigated, demonstrating successful simulation of the range of material behaviour expected in geological samples, from broad ranged gradual collapse to sharp, almost instantaneous failure.
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This paper presents the mathematical framework and the asynchronous finite element solver that captures the brittle fractures in multi-phase fluid-infiltrating porous media at the mesoscale where the constituents are not necessarily in a thermal equilibrium state. To achieve this goal, we introduce a dual-temperature effective medium theory in which the distinct constituent temperatures are homogenized independently whereas the heat exchange among the constituents is captured via phenomenological heat exchange laws in analog to the dual-permeability theory. To handle the different growth rates of the boundary layers in a stable and computationally efficient manner, an asynchronous time integrator is proposed and implemented in an operator-split algorithm that updates the displacement, pore pressure, phase field, and temperature of each constituent in an asynchronous manner. Numerical examples are introduced to verify the implementation and compare the path-dependent behaviors predicted by the two-temperature and one-temperature models.
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Fracture initiation and propagation in fluid-saturated rocks are controlled by interaction between fluid flow and rock deformation. The description of hydromechanical coupling is essential for modeling the fracture process. In this paper, an improved hydromechanical model is proposed in the framework of the particle flow simulation method. This model provides a better description of hydraulic properties before and after breakage of bonds and can efficiently describe fluid flow through porous rock matrix and along fractures. The efficiency of the proposed model is first assessed by comparisons with analytical solutions and typical experimental evidences. A series of numerical simulations are then realized to investigate effects of some key parameters such as confining stress, fluid injection rate and viscosity on the initiation and propagation of fractures.
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A geometrically nonlinear phase field theory accounting for dissipation, rate effects, nonlinear thermoelasticity, fracture, and other structural changes is constructed in the context of continuum thermodynamics. First, a general framework accommodating arbitrary strain energy potentials, inelastic deformation kinematics, and unlimited order parameters is formulated. Next, the framework is specialized to account for deformation physics pertinent to crystalline ceramics and minerals deformed at high rates and high pressures. Notably, a logarithmic elastic strain tensor referred to an intermediate material configuration enters the nonlinear thermoelastic potential. Order parameters represent fractures, solid–solid phase transformations, deformation twinning, or slip of partial dislocations. Thermodynamically consistent kinetics manifest in equations reminiscent of Ginzburg–Landau dynamics, wherein viscosity coefficients are most generally state- and rate-dependent. Pressure-dependent strength commensurate with frictional resistance is enabled in alternative kinetic equations for dynamic fracture with irreversibility constraints. Linearization of the model suitable for moderate volume changes but small deviatoric elastic strain and rotation is undertaken. The theory is applied to study deformation and failure of polycrystalline forms of boron carbide (B4C), titanium diboride (TiB2), and a B4C-TiB2 ceramic composite. Solutions are derived and evaluated numerically for uniaxial stress tension and compression, uniaxial strain compression, and planar shock compression. The latter analysis yields relationships among viscosity coefficients, gradient regularization lengths, and characteristics of steady plastic waveforms. Results give new insight into high-rate deformation mechanisms previously speculated in these materials.
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Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discretization schemes, and numerical analysis are briefly summarized for both models. Validation against experimental data is essential for all computational methods to demonstrate predictive accuracy. To that end, the Sandia Fracture Challenge and similar experimental data sets where both models could be benchmarked against are showcased. Emphasis is made to converge on common metrics for the evaluation of these two fracture modeling approaches. Both PD and PF models are assessed in terms of their computational effort and predictive capabilities, with their relative advantages and challenges are summarized.
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The phase-field model has proven a promising tool to simulate crack propagation, and special attention has recently been paid to the prediction of crack nucleation. Dynamic shear banding (or the so-called adiabatic shear banding) is a significant ductile failure mechanism in metals and alloys under impact loading, and its nucleation has been considered either strength-like or toughness-like in the literature. In this work, a rate-dependent phase-field model incorporating stored-energy-based shear banding criteria is proposed to simulate dynamic shear band formation within a thermodynamically consistent framework, emphasizing the capability to capture shear band nucleation in all possible modes, be it strength-like, large-scale yielding, or small-scale yielding. The model is first applied on dynamic shear banding under coupled damage and thermal softening, where the phase-field length scale is demonstrated to play an important role in energy dissipation when either the rate- or temperature-dependence during shear band formation is non-negligible, which is significantly different from its interpretation in quasi-static conditions. Then the capability of the proposed model in capturing shear band nucleation is thoroughly investigated by simulating both designed numerical tests and reported physical experiments. Strain-based and stress-based criteria for shear band nucleation can be well retrieved with the energy-based phase-field model. Validation against shear banding experiments of all possible modes over a wide range of specimen geometries, material properties, and loading rates is performed to demonstrate the performance of the model and shed light on a unified understanding of shear band formation in various scenarios.
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Most rocks exhibit time-dependent deformation and failure. Two main mechanisms are generally considered, the progressive growth of cracks and viscoelastic and/or viscoplastic deformation. In this study, cracking process is described by a viscous phase-field method which is coupled with a viscoplastic model. The evolution of crack field is controlled by both elastic and viscoplastic tensile volumetric and deviatoric strains. The threshold of viscoplastic deformation is weakened by the growth of cracks. The efficiency of the proposed model is first assessed by comparing numerical predictions with experimental data in triaxial compression and creep tests. Then, the proposed model is applied to modeling time-dependent deformation and failure process of a high slope section in the left bank of Jinping-I hydropower station in China. Numerical predictions are compared with field measurements.
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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.
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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.
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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.
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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.
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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.
Thesis
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.