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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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... This means that shear fractures will initiate and propagate even when all principal strains of the material are compressive. In contrast to this fact, the compressive-shear fractures cannot be predicted by the current PFMs for rock-like solids [21,51,52], where the negative strains and the compressive part of elastic energy are assumed not to contribute to the evolution of phase field. More specifically, the first attempt of the PFM for mixed-mode crack propagation in rock-like materials was proposed by Zhang et al. [21]. ...

... This method was established by introducing the critical energy release rate of mode II and two historic energy references in the model of Miehe et al. [26]. Choo and Sun [51] coupled a pressure-sensitive plasticity model with a phase field approach to capture brittle fracture, ductile flow, and their transition in rocks. Bryant and Sun [52] proposed a kinematic-consistent phase field approach to model mixed-mode fractures in anisotropic rocks. ...

... Bryant and Sun [52] proposed a kinematic-consistent phase field approach to model mixed-mode fractures in anisotropic rocks. On the other hand, the contributions in Zhang et al. [21], Choo and Sun [51], Bryant and Sun [52] also show that the PFMs developed so far do not account for the influence of cohesion and internal friction angle on fracture propagation and the load-displacement curve. This is a missing aspect since it is generally accepted that a rock-like material will have a higher compressive strength if its cohesion or internal friction angle increases. ...

Compressive-shear fracture is commonly observed in rock-like materials. However, this fracture type cannot be captured by current phase field models (PFMs), which have been proven an effective tool for modeling fracture initiation, propagation, coalescence, and branching in solids. The existing PFMs also cannot describe the influence of cohesion and internal friction angle on load-displacement curve during compression tests. Therefore, to develop a new phase field model that can simulate well compressive-shear fractures in rock-like materials, we construct a new driving force in the evolution equation of phase field. Strain spectral decomposition is applied and only the compressive part of the strain is used in the new driving force with consideration of the influence of cohesion and internal friction angle. For ease of implementation, a hybrid formulation is established for the phase field modeling. Then, we test the brittle compressive-shear fractures in uniaxial compression tests on intact rock-like specimens as well as those with a single or two parallel inclined flaws. All numerical results are in good agreement with the experimental observation, validating the feasibility and practicability of the proposed PFM for simulating brittle compressive-shear fractures.

... More recently, only a few researches have been reported on adopting the variational phase-field fracture model to study ductile fracture in porous media [63][64][65][66]. However, the principle of maximum plastic dissipation in the variational formulation of plasticity indicates associative flow and associative hardening, geomaterials such as rocks and soils usually exhibit non-associative flow, care must be taken when using the variational description of fracture in geomaterials [84]. Therefore, it is necessary to develop new explicit phase-field fracture models derived based on the microforce balance law for efficiently solving complex large deformation fracture problems in the poro-elastoplastic media involving non-associative plasticity. ...

... where D represents the energy dissipation, and Ψ denotes the stored energy density per unit volume. Referring to the previous works [71,84], Ψ and its time differentiation are given as ...

... It can be seen that Eq. (19) represents the constitutive relationship for the elastoplastic large deformation analysis of the saturated porous media. In addition, the analytical expressions of ξ and π dis can be obtained by taking a series of algebraic operations on Eq. (21) as [71,84] ...

... The evolution of damage due to void growth at the microscale, described by Gurson-Tvergaard-Needelman (GTN) model (Gurson, 1977;Tvergaard and Needleman, 1984), or other porous plasticity models, has been extensively treated in several contributions (Miehe et al., 2016b;Aldakheel et al., 2018;Dittmann et al., 2020;Krüger et al., 2020;Azinpour et al., 2021;Dittmann et al., 2021;Tao et al., 2022;Chen et al., 2022). The simulation of crack propagation in geological materials of different types has been considered in , Choo and Sun (2018), Kienle et al. (2019), You et al. (2021), Ulloa et al. (2022) and Hu et al. (2022). The combination of phasefield description of damage with multisurface plasticity models has been investigated in Fang et al. (2019a). ...

... The phase-field approach has also been successfully applied to the modeling of cyclic plasticity problems and to low-cycle fatigue problems in the presence of plasticity, see, e.g., Seiler et al. (2020), Aygün et al. (2021), Seleš et al. (2021), Ulloa et al. (2021), Hasan and Baxevanis (2021), Song et al. (2022) and Tsakmakis and Vormwald (2022). Finally, while numerous contributions have been made within the context of small-strain kinematics, there has been an increasing interest in recent years towards the development of formulations within the large-strain plasticity regime (Aldakheel et al., 2014;Miehe et al., 2015Miehe et al., , 2016cMiehe et al., ,a,b, 2017McAuliffe and Waisman, 2015;Borden et al., 2016;Aldakheel, 2017;Shanthraj et al., 2017;Choo and Sun, 2018;Aldakheel et al., 2018;Dittmann et al., 2018;Chu et al., 2019;Kienle et al., 2019;Brepols et al., 2020;Krüger et al., 2020;Shishvan et al., 2021;Dittmann et al., 2021;Eldahshan et al., 2021b;Hu et al., 2021;Proserpio et al., 2021;Talamini et al., 2021;Felder et al., 2022;Han et al., 2022a;Abrari Vajari et al., 2022;Hu et al., 2022;Huber and Asle Zaeem, 2023;Han et al., 2022b). ...

... In contrast, in the nominal stress approach, the nominal stress decreases when the damage-softening branch of the response is entered, while the current yield stress does not decrease (weak plasticity-damage coupling), so that the yield condition cannot be satisfied anymore and the plastic strain evolution stops. In the latter case, the damage evolution becomes purely brittle and a non-physical elastic unloading is observed, as noted by several authors (see, e.g., Borden et al., 2016;Choo and Sun, 2018;Huang and Gao, 2019;Marengo and Perego, 2023). Other models (see, e.g., Alessi et al., 2014Alessi et al., , 2018bUlloa et al., 2016), consider different definitions for the degradation ( ) of stresses and static internal variables (see Section 2.4, Eq. (29), where ( ) ≠ 1 is used for the degradation of the current yield limit). ...

... Also, several models with a similar setup for the damage driving force were reported. Choo & Sun [43] developed a model for geological materials, and Rodriguez et al. [44] proposed a model describing coupling with non-local plasticity. Huang & Gao [45] presented another model by introducing a plastic adjustment function for the damage driving force, and Fang et al. [46] proposed a model to incorporate multi-surface plasticity. ...

... in which we have used the relations in Eqs. (43), (44), (45), (61) as well as P = ∂ F Ψ e and the Gaussian divergence theorem. ...

... Also, if Ψ e cr = 0 is assumed, and if any values are given for Ψ p cr , ζ e and ζ p , the same damage driving force as in Borden [61] is realized by the proposed model. Similarly, our model is capable of realizing the assumptions on the damage driving force made in the other models [33,38,43,44,46,69], as described in Section 3.2. The difference between these models and ours is the origin of the evolution laws of damage variables. ...

Based on the study on characteristic features of existing crack phase-field (PF) models for ductile fracture, we propose a new model endowed with elastic and plastic damage variables. In the review part, the emphasis is placed on the plastic driving force and degrading fracture toughness that enable PF models to represent damage evolution in elastoplastic materials. Attention is also paid to the damage evolution tendency in terms of the definition of the yield function. Based on the reviewed features, we originally formulate the proposed model, which separately accommodates two damage variables for elasticity and plasticity. Its constitutive work density consists of elastic, plastic hardening, and damage hardening components. Evolution laws for plasticity and damage are derived by using separate threshold (yield) functions while having similar formats so that those for damage hold variational and thermodynamic consistencies. Also, the proposed model is equipped with thresholds and coefficients to control the amount of damage driving force. Several numerical examples are presented to demonstrate the characteristic features of the proposed model and the ability to mimic the typical fracture behavior of elastoplastic materials.

... Over the past few decades, the phase-field model for fractures has been identified as a promising alternative to sharp interface approaches to model fracture propagation in rocks and rock-like materials. [13][14][15][16] The fracture, instead of being explicitly represented as an interface, is approximated by a diffuse variable (i.e., damage). The main advantage of this diffuse crack representation is the simplicity of representing complex geometries. ...

... As a consequence, the regularization length becomes a material-specific parameter that should be calibrated based on the tensile and compressive strengths of a given material. 14,32 This length dependency issue has been addressed in the recent works on phase-field models for quasi-brittle materials, 15,29,30 which are inspired by a gradient damage model introduced by Lorentz and Godard. 33 Unfortunately, these models are still derived through an energy minimization process, and as such damage nucleation is governed by a threshold that is energetic. ...

Many geo‐engineering applications, for example, enhanced geothermal systems, rely on hydraulic fracturing to enhance the permeability of natural formations and allow for sufficient fluid circulation. Over the past few decades, the phase‐field method has grown in popularity as a valid approach to modeling hydraulic fracturing because of the ease of handling complex fracture propagation geometries. However, existing phase‐field methods cannot appropriately capture nucleation of hydraulic fractures because their formulations are solely energy‐based and do not explicitly take into account the strength of the material. Thus, in this work, we propose a novel phase‐field formulation for hydraulic fracturing with the main goal of modeling fracture nucleation in porous media, for example, rocks. Built on the variational formulation of previous phase‐field methods, the proposed model incorporates the material strength envelope for hydraulic fracture nucleation through two important steps: (i) an external driving force term, included in the damage evolution equation, that accounts for the material strength; (ii) a properly designed damage function that defines the fluid pressure contribution on the crack driving force. The comparison of numerical results for two‐dimensional test cases with existing analytical solutions demonstrates that the proposed phase‐field model can accurately model both nucleation and propagation of hydraulic fractures. Additionally, we present the simulation of hydraulic fracturing in a three‐dimensional domain with various stress conditions to demonstrate the applicability of the method to realistic scenarios.

... It has been applied to different kinds of materials and problems, such as multi-physics coupling, 15 finite deformation, 16 and plastic damage coupling. 17,18 However, in most previous studies, homogeneous rock materials have been considered. The COx claystone is characterized by a multiscale heterogeneity. ...

... By substituting − for Eqs. (18), the evolution criteria for two crack fields are now expressed as: ...

This study is devoted to numerical modeling of cracking process induced by temperature change in saturated porous rocks in the context of geological disposal of radioactive waste. Effects of material anisotropy and heterogeneity are taken into account. The macroscopic elastic properties are determined from two steps of homogenization by considering pores and mineral inclusions at two different scales. An extended phase-field model is proposed to describe the initiation and propagation of localized cracks. Two damage variables are introduced to conveniently represent both tensile and shear cracks. New damage evolution criteria are defined by incorporating the pore pressure effect. Three application examples are presented. By assuming a random distribution of pores and inclusions, the efficiency of the proposed model for capturing the progressive cracking process is first verified in a triaxial compression test. The thermal cracking process in an anisotropic and heterogeneous sample is then investigated. The respective influences of elastic anisotropy and spatial variability of pores and inclusions are outlined. Finally, the proposed model is applied to a series of real laboratory thermal cracking tests. Both hydromechanical responses and cracking evolution patterns are investigated. Numerical results are compared with experimental measurements. The main mechanisms involved in the thermal cracking process are highlighted.

... Over the past few decades, the phase-field model for fractures has been identified as a promising alternative to sharp interface approaches to model fracture propagation in rocks and rock-like materials [13][14][15][16]. The fracture, instead of being explicitly represented as an interface, is approximated by a diffuse variable (i.e., damage). ...

... This length dependency issue has been thoroughly discussed in the literature (see e.g., [15,[29][30][31]). As a consequence, the regularization length becomes a material-specific parameter that should be calibrated based on the tensile and compressive strengths of a given material [14,32]. This length dependency issue has been addressed in the recent works on phase-field models for quasi-brittle materials [15,29,30], which are inspired by a gradient damage model introduced by Lorentz and Godard [33]. ...

Many geo-engineering applications, e.g., enhanced geothermal systems, rely on hydraulic fracturing to enhance the permeability of natural formations and allow for sufficient fluid circulation. Over the past few decades, the phase-field method has grown in popularity as a valid approach to modeling hydraulic fracturing because of the ease of handling complex fracture propagation geometries. However, existing phase-field methods cannot appropriately capture nucleation of hydraulic fractures because their formulations are solely energy-based and do not explicitly take into account the strength of the material. Thus, in this work, we propose a novel phase-field formulation for hydraulic fracturing with the main goal of modeling fracture nucleation in porous media, e.g., rocks. Built on the variational formulation of previous phase-field methods, the proposed model incorporates the material strength envelope for hydraulic fracture nucleation through two important steps: (i) an external driving force term, included in the damage evolution equation, that accounts for the material strength; (ii) a properly designed damage function that defines the fluid pressure contribution on the crack driving force. The comparison of numerical results for two-dimensional (2D) test cases with existing analytical solutions demonstrates that the proposed phase-field model can accurately model both nucleation and propagation of hydraulic fractures. Additionally, we present the simulation of hydraulic fracturing in a three-dimensional (3D) domain with various stress conditions to demonstrate the applicability of the method to realistic scenarios.

... However, retainment of the bulk modulus in compression, while providing a moderate increase in strength, does not account for the marked difference in compressive versus tensile strength in many brittle solids such as ceramics and those of geologic origin. To address deficiencies such as this, different fracture energy parameters have been prescribed for compressive versus tensile simulations on the same material [12], thereby imposing a stress state-dependent energetic resistance. Other models have proposed non-traditional driving forces for crack nucleation, valid in the context of quasi-statics, that depend in a sophisticated way on stress or deformation state [13][14][15]. ...

... Compared to the inviscid case, simulation execution times were typically 4× to 6× longer when (2.13) was implemented with > 0 and > 0. Solutions of course differ for different viscosities. Twelve simulations (sims) are executed, four for each loading protocol: uniaxial stress tension (1-4), uniaxial stress compression (5)(6)(7)(8), and uniaxial strain compression (9)(10)(11)(12). One of each four invokes isotropic properties. ...

... The terms are responsible for microforces working (see below). The microforce concept is successfully applied in the development of various models of continuum mechanics [22,[29][30][31][32][33][34][35]. ...

... Owing to (30) and the identity ...

Usually, Lagrangian or Eulerian–Lagrangian descriptions of motion are adopted to model interaction of fluids and deformable solid bodies. However, in some cases both approaches lead to serious difficulties. An example is a system with large number of solid bodies; another one is the case where topology of the phases can change. In these situations, an Eulerian description is much more convenient. The presented work is devoted to the development of a phase field mathematical model for description of dynamics of multiphase multicomponent system (mixture) with phases having whether liquid or solid rheology. All the phases and interphase boundaries (interfaces) are directly resolved. Main balance laws of the proposed model are formulated using the Eulerian description. The model is of phase field type: the interphase boundary is diffuse and is described by a thin layer of finite thickness. Mass densities of mixture components are used as order parameters. To describe a stress–strain behavior of the solid phase, we assume that the Helmholtz free energy depends on deformation gradient tensor which is defined as a solution of the corresponding evolution equation. Constitutive relations are derived by means of the well-known Coleman–Noll procedure and the second law of thermodynamics. A distinctive feature of the considered model is its preliminary regularization based on the quasi-hydrodynamic technique, which allows one to improve numerical stability properties when an explicit discretization is applied. A new family of quasi-hydrodynamic closures is obtained.

... This approach is particularly suitable to deal with the progressive transition from diffuse damage to localized cracks, and three-dimensional problems with multiple cracks. The phase-field method was progressively extended and applied to dynamic brittle fracture (Borden et al. 2012), multi-physics problems (Miehe et al. 2015), finite deformation (Borden et al. 2016), plastic materials (Choo and Sun 2018;Fang et al. 2019), etc.. In many previous studies, the accent was put on tensile cracks which are directly driven by the tensile strains and stresses. ...

... The governing equations for the evolution of crack-phase field are classically derived from two main methods: the variational principle (Francfort and Marigo 1998) and the thermodynamics framework (Choo and Sun 2018). In this work, the variational principle is first used to derive the basic equations. ...

A new double-phase-field model is proposed in this paper for modeling cracking processes in rock-like brittle materials under compression-dominating stresses. For this purpose, two crack-phase fields are used to describe tensile and shear cracks respectively. Compared with previous works, a stress-based new criterion is proposed to more physically capture the evolution of shear cracks in rock-like materials. The effects of mean stress and internal friction are taken into account. The proposed model is implemented in the finite element framework. It is applied to investigating cracking processes in a rock sample containing two initial flaws and subjected to uniaxial and biaxial compression. Both the tensile wing and shear cracks as well as crack coalescence observed in laboratory tests are successfully reproduced by the proposed method.

... This model was also inspired by the elliptic regularization method proposed by Ambrosio and Tortorelli (1990) of the functional in image segmentation problems formulated by Mumford and Shah (1989). A novel phase field formulation has been proposed in the framework of irreversible thermodynamics to deal with ductile cracking in plastic materials (Choo and Sun, 2018). With the help of an auxiliary damage variable, sharp cracks surfaces are approximated by a volumetric crack surface density which is a function of damage variable and its gradient. ...

... At the same time, tensile and shear cracks can affect differently the elastic properties of materials. In order to conveniently describe the deterioration of elastic properties, the elastic strain energy is commonly decomposed into several additive parts (Amor et al., 2009;Choo and Sun, 2018;Zhou et al., 2020). Similarly, different decomposition methods are also proposed for stress and strain tensors . ...

In this paper, we first present a new phase-field model for modeling the deformation and progressive failure in saturated and unsaturated porous rocks. Two independent damage variables are used to easily capture tensile and shear cracks. The influences of frictional shear stress and normal stress on the evolution of shear cracks are taken into account. The phase-field model is extended to variably saturated porous rocks by including the effect of pore water pressure. The proposed model is implemented in the framework of finite element method for coupled hydro-mechanical and damage problems. The phase field model is able to describe global stress–strain responses and localized cracking patterns in brittle rocks at the laboratory scale. The onset of localized cracks is directly linked to the non-uniform distribution of porosity. The proposed phase-field model is also applied to the analysis of rainfall-induced landslides. The numerical results of cracking scenarios are consistent with the real field observations in the Mayanpo slope in China. The main physical mechanisms involved in the rainfall induced instability of slopes are analyzed.

... Others consider dynamics and dissipation [3,4]. Subsequent efforts encompass geometric nonlinearity [5][6][7], crystalline anisotropy [8,9], and coupling of fracture with deformation twinning [10], plasticity [11,12], and solid-solid phase transformations [13,14]. Recent comprehensive works outlining many such physical phenomena in the context of phase field modelling include [15,16]. ...

... Irreversibility and threshold initiation constraints [6,12,16] can be included if appropriate. For example, fracture order parameters j j can be constrained to increase irreversibly by replacing the second of (45) and (48) with the following, summing on index n: ...

A phase field theory for crystalline solids accounting for thermoelasticity, fracture, twinning, and limited slip is presented. Residual stresses are incorporated via referencing thermoelastic strain to a reference state that is not always stress-free. Rate dependence, dissipated energy, and residual strain energy (possibly degraded by local fracture) are included in the theory, with physically valid predictions first verified for homogeneous stress states. A variational form of the model is implemented in finite element (FE) calculations of three-dimensional (3-D) polycrystalline aggregates consisting of up to two different crystal constituents and a binding matrix along grain and phase boundaries. Model specifics correspond to constituents of boron carbide-titanium diboride (B4C-TiB2) ceramic composites. Effects of thermal-residual stresses incurred during processing, as well as other local microstructure properties and physical features, on deformation and failure mechanisms are revealed. Peak aggregate strengths observed for different boundary conditions demonstrate a pressure-dependent failure surface.

... Examples of strain energy decompositions formulated with this objective include the volumetric-deviatoric split by Amor et al. [25], the spectral decomposition by Miehe and co-workers [26], and the purely tensile splits (so-called 'no-tension' models) of Freddi and Royer-Carfagni [27,28] and Lo et al. [29]. On the other hand, rising interest in using phase field methods to model fracture in concrete and geomaterials has led to the development of driving force definitions that accommodate nonsymmetric failure surfaces [30]. Zhou et al. [31] and Wang et al. [32] developed new driving force formulations based on Mohr-Coulomb theory. ...

... This load bearing capacity that is retained after reaching the fully cracked state due to dilatancy arises due to two contributions. One is the term 6 in Eq. (30). The second one is the term 1 ( )∕ √ 2 ( ) -as shown in Fig. 5(b), it attains a positive constant value for = 1 and ≠ 0. However, the relation between and 1 ( )∕ √ 2 ( ) is non-linear. ...

Due to its computational robustness and versatility, the phase field fracture model has become the preferred tool for predicting a wide range of cracking phenomena. However, in its conventional form, its intrinsic tension-compression symmetry in damage evolution prevents its application to the modelling of compressive failures in brittle and quasi-brittle solids, such as concrete or rock materials. In this work, we present a general methodology for decomposing the phase field fracture driving force, the strain energy density, so as to reproduce asymmetrical tension-compression fracture behaviour. The generalised approach presented is particularised to the case of linear elastic solids and the Drucker–Prager failure criterion. The ability of the presented model to capture the compressive failure of brittle materials is showcased by numerically implementing the resulting strain energy split formulation and addressing four case studies of particular interest. Firstly, insight is gained into the capabilities of the model in predicting friction and dilatancy effects under shear loading. Secondly, virtual direct shear tests are conducted to assess fracture predictions under different pressure levels. Thirdly, a concrete cylinder is subjected to uniaxial and triaxial compression to investigate the influence of confinement. Finally, the localised failure of a soil slope is predicted and the results are compared with other formulations for the strain energy decomposition proposed in the literature. The results provide a good qualitative agreement with experimental observations and demonstrate the capabilities of phase field fracture methods to predict crack nucleation and growth under multi-axial loading in materials exhibiting asymmetric tension-compression fracture behaviour.

... Amor et al. [25], the spectral decomposition by Miehe and co-workers [26], and the purely tensile splits (so-called 'no-tension' models) of Freddi and Royer-Carfagni [27,28] and Lo et al. [29]. On the other hand, rising interest in using phase field methods to model fracture in concrete and geomaterials has led to the development of driving force definitions that accommodate non-symmetric failure surfaces [30]. Zhou et al. [31] and Wang et al. [32] developed new driving force formulations based on Mohr-Coulomb theory. ...

... This load bearing capacity that is retained after reaching the fully cracked state due to dilatancy arises due to two contributions. One is the term 6B in Eq. (30). The second one is the term I 1 (ε)/ √ J 2 (ε) -as shown in Fig. 5b, it attains a positive constant value for ϕ = 1 and B ̸ = 0. ...

Due to its computational robustness and versatility, the phase field fracture model has become the preferred tool for predicting a wide range of cracking phenomena. However, in its conventional form, its intrinsic tension-compression symmetry in damage evolution prevents its application to the modelling of compressive failures in brittle and quasi-brittle solids, such as concrete or rock materials. In this work, we present a general methodology for decomposing the phase field fracture driving force, the strain energy density, so as to reproduce asymmetrical tension-compression fracture behaviour. The generalised approach presented is particularised to the case of linear elastic solids and the Drucker-Prager failure criterion. The ability of the presented model to capture the compressive failure of brittle materials is showcased by numerically implementing the resulting strain energy split formulation and addressing four case studies of particular interest. Firstly, insight is gained into the capabilities of the model in predicting friction and dilatancy effects under shear loading. Secondly, virtual direct shear tests are conducted to assess fracture predictions under different pressure levels. Thirdly, a concrete cylinder is subjected to uniaxial and triaxial compression to investigate the influence of confinement. Finally, the localised failure of a soil slope is predicted and the results are compared with other formulations for the strain energy decomposition proposed in the literature. The results provide a good qualitative agreement with experimental observations and demonstrate the capabilities of phase field fracture methods to predict crack nucleation and growth under multi-axial loading in materials exhibiting asymmetric tension-compression fracture behaviour.

... presents the post-failure images and schematics of the different failure modes observed in the mudstone specimens tested saturated conditions at 0, 50, and 130 MPa. Four failure modes can be identified: axial splitting fracture, shear fracture, shear bands, and plastic flow(Basu et al. 2013;Choo and Sun 2018). All tested mudstone lithofacies exhibited axial splitting fractures at 0 MPa confining pressure. ...

The influence of water on rock deformation and failure behavior is a critical variable to consider when developing unconventional shale reservoirs. To better understand this effect, we conducted a series of triaxial compression tests on four mudstone lithofacies, namely siliceous calcareous mudstones, siliceous mudstones, calcareous mudstones, and carbonate-rich mudstones from the Naparima Hill Formation, under water-saturated conditions at confining pressures up to 130 MPa. Our results showed that the mudstones displayed brittle, brittle–ductile transition, and ductile failure behaviors as the confining pressure increased. Mudstones with low strength, high porosity, and high silica and clay contents exhibited a strong water-weakening effect, leading to a significant reduction in failure strength. Conversely, high strength, low porosity, and carbonate-rich mudstones showed minor water-weakening effects. We also developed a failure behavior model to predict the brittle, brittle–ductile transition, and ductile zones in the mudstones. The model showed that the transition from brittle to ductile behavior occurs over a wide range of confining pressures, and it occurs at a lower pressure threshold in saturated compared to dry conditions. Additionally, the model suggests that the presence of water results in a wider transition zone between brittle and ductile behavior, as well as a larger ductile zone. Overall, our findings suggest that the water-weakening effect can significantly decrease the depth at which the brittle–ductile transition occurs in mudstones, emphasizing the importance of considering water as a critical variable in rock deformation and failure behavior.

... Extensions such as use of a phase field to represent ductile fracture in an otherwise elastic-plastic continuum have become widespread. 36,37 Enriched theories for structural transformations in nonlinear materials incorporating concepts and tools of Finsler geometry were set forth in 2016, [38][39][40] whereby connections with PFM, micropolar, micromorphic, 41 and other gradient-based models were demonstrated. Use of a Riemannian rather than Finslerian metric enabled reduction of the equilibrium equations derived for certain classes of energy potentials of the Finsler geometric theory 39,42 to those of the PFM. ...

... The starting point in these studies is the definition of a scalar-valued and space-time-dependent phase-field variable in the range [0, 1], which allows for differentiation between the cracked and the intact regions of the domain. With the PFM, the sharp edges of the crack are approximated by diffusive ones, whereas the width of this region is controlled via an internal length scale, see, e.g., [6,12,16,17,21,49,53,58,62,63,70,73,74] among others. In this context, it is worth mentioning that although the PFM is intensively applied for fracture modeling, it is also widely used in the simulation of many other engineering applications, as phase-change materials, see, e.g., [4,[94][95][96]104], for references. ...

This research aims to extend the isothermal continuum mechanical modeling framework of hydraulic fracturing in porous materials to account for the non-isothermal processes. Whereas the theory of porous media is used for the macroscopic material description, the phase-field method is utilized for modeling the crack initiation and propagation. We proceed in this study from a two-phase porous material consisting of thermomechanically interacting pore fluid and solid matrix. The heat exchange between the fluid in the crack and the surrounding porous environment through the diffusive fracture edges is carefully studied, and new formulations here are proposed. Besides, temperature-dependent solid and fluid material parameters are taken into account, which is of particular importance in connection with fluid viscosity and its effect on post-cracking pressure behavior. This continuum mechanical treatment results in strongly coupled partial differential equations of the mass, the momentum, and the energy balance of the thermally non-equilibrated constituents. Using the finite element method, two-dimensional initial-boundary-value problems are presented to show, on the one hand, the stability and robustness of the applied numerical algorithm in solving the emerged strongly coupled problem in the convection-dominated heat transport state. On the other hand, they show the capability of the modeling scheme in predicting important instances related to hydraulic fracturing and the role of the temperature field in this process. Additionally, they show the importance of using stabilization techniques, such as adding an artificial thermo-diffusivity term, to mitigate temperature fluctuations at high flow velocity.

... Of all the existing computational models, phase field modeling (PFM) has proven to be efficient in capturing crack nucleation and tracking crack paths [37,38]. PFM is based on the theoretical aspects of Griffith's fracture criteria and the early works of Bourdin [39], Francfort and Marigo [40]. ...

Elastomers and composites made thereof have wide applications, e.g., in automobile, aerospace, and civil engineering. Predicting fracture in such materials is crucial for efficient design and optimum utilization. These materials are oftentimes hyperelastic and anisotropic in nature and in general subjected to mixed mode loading rather than merely pure modes. Soft biological tissues can also be considered anisotropic hyperelastic materials. Computational modeling helps in studying the role of different sources influencing mixed-mode fracture. A unifying thermodynamically consistent anisotropic phase field formulation for modeling the mixed-mode fracture of hyperelastic soft materials like elastomers, elastomeric fiber-reinforced composites, and soft biological tissues at finite strains is proposed and formulated. To model the mechanical response of anisotropic hyperelastic materials subjected to mixed-mode loading, a coupled Neo-Hookean model with orthotropic anisotropy is adopted considering volumetric-deviatoric and a tension-compression decomposition. For modeling the complex crack initiation and propagation, a phase field method based on a power law criterion is adopted by considering a single order parameter as the damage variable. This model is suitable for capturing the overall response of soft fiber-reinforced elastomeric composites as well as soft biological tissues. The proposed model is validated by conducting fracture tests on (a) silicone elastomers, (b) unidirectional fiber-reinforced elastomeric composites, (c) natural rubber reinforced with black carbon, and (d) brain tissue reinforced with axons. The results obtained are compared with experimental and numerical investigations from literature.

... 21 Wu 22 proposed a phase field regularized cohesive zone model for the quasi-brittle fracture. As of now, the PFM has been extended to the fracture of many materials including geological materials, 23 poroelastic media 24 and polymers. 25 However, the numerical modeling of the dynamic fracture of the rubber-like polymers remains a challenging problem because of the strong nonlinearity of dynamic fracture, especially when it involves contact and impact. ...

The rubber‐like solids, as commonly observed materials in nature and engineering areas, often show high extensibility and weak compressibility, which lead to difficulties for modeling the fracture behavior of these materials. In this paper, we propose a mixed three‐field total Lagrangian material point method (TLMPM) to deal with the fracture of nearly incompressible rubber‐like solids. In this method, the phase‐field method combining with perturbed Lagrangian approach is developed to describe the fracture of nearly incompressible materials. Based on the Lagrangian equation, the governing equations of the displacement, pressure and phase fields are derived for the system considering dynamic effect. Then, a mixed three‐field TLMPM discrete formulation with a remapping strategy is then built to solve the problem involving large deformation. Besides, a staggered solving scheme with explicit time integration is developed for the three‐field problem. The accuracy and capabilities of the proposed method for dealing with the fracture of incompressible materials are demonstrated by modeling several quasi‐static and dynamic numerical problems and comparing the results with other numerical methods and experiments. Moreover, the applicability of the method is further proved by an out‐of‐plane tearing case.

... Several brittle [50,[56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73] and ductile [74][75][76][77][78][79][80][81][82][83][84][85][86][87][88] phase-field fracture formulations have been proposed for the modeling of small and large strain deformations, and multi-scale/physics problems. These phase-field models are also devoted to many applications in engineering, including the brittle failure of metals based on elastic framework [69][70][71][72], the low-cycle fatigue of engineering components based on elastic-plastic framework [15,16,79,85,87], and the adiabatic shear failure of parts under large impact loads based on thermos-elastic-plastic framework [54,[89][90][91][92][93][94]. ...

... However, ductile failure result at high confinement pressures and there is a value of confining pressure, in which the behavior of the soil changes from brittle failure to plastic flow (the transitional state from brittle to ductile). The same results were obtained by Choo and Sun [38], where they found that the failure behavior of geological materials based on the confining pressure and the strain rate. ...

Stabilization of cohesive soils has been practiced for some time by mixing additives, such as cement, lime, and fly ash, into the soil to increase its mechanical strength. However, there is a lack of investigation on the use of natural pozzolana (NP) alone or combined with lime for soil stabilization applications. This work is a part of a research project focused on the evaluation of the effects of adding natural pozzolana as an additive to improve the lime treatment results of local clayey soils. The main purpose of this paper is to present the results of using lime, natural pozzolana, and their combination on shear strength, shear parameters, and failure mode variations of the local clayey soil, classified as fat clay (CH). CH, selected from Tlemcen city in Algeria, is known for its high plasticity and importance in cohesion and compressibility. To achieve this goal, several physicomechanical tests (pH, compaction, undrained unconsolidated triaxial compression test) and microstructural analysis, scanning electron microscope (SEM) have been carried out for the different studied combinations. Natural pozzolana and lime were added to the studied soils at ranges of 0–20% and 0–6%, respectively. The treated samples were cured for 1, 7, and 28 days. The results indicated that the studied properties of clay soil can be considerably improved when treated with lime. The combination of lime and natural pozzolana appears to produce higher shear strength parameters, than when lime or natural pozzolana is used alone. Adding natural pozzolana to clay soil treated with lime produces an additional chemical reaction, especially in the long term, resulting in better flocculation and additional formation of cementing materials. Therefore, the deviatoric stress of the treated soil, with 6% lime and 20% NP, increased up to 200% within 28 days of curing. The treated soil became more brittle, with a significant increase in shear strength, cohesion, and a higher friction angle.

... In Ref [43], Miehe etal developed the thermodynamically consistent phase field model based on the spectral decomposition of the strain tensor in isotropic elastic material, showing great stability and capability in complicated crack simulations. Phase field models have been used to solve brittle fractures [29,2], hydraulic fractures [58,41,61,26], ductile fractures with plasticity [8,7,19,25,1], geomaterials [16,22,12,60], plate and shell [5,48], multiphysics problems [42,39,18] and many others. For a more comprehensive review of the phase field model, the reader is referred to [59,13,62]. ...

A phase field model for ductile fracture considering Hencky strain and finite J2 plasticity is presented using the nonlocal operator method. A variational derivation of J2 plasticity at finite strain with a phase field model is performed. The method includes a logarithmic strain tensor and an exponential mapping in the plasticity evolution. A spectral decomposition based algorithm for computing the first and second order derivatives of the composite matrix function is implemented. A consistent tangential stiffness matrix is derived and used in Newton-Raphson iterations. Several numerical examples are performed to validate the method, including notched single-edged plates with brittle fracture or ductile fracture and necking of a bar with/without phase field model.

... where f n = c e and f s = e , parameter denotes the fraction of plastic work dissipated through heating. 57,58 Because calibrating this parameter is difficult and necessitates specific tests, most approaches just assume a constant value. 59,60 Here, we assume that the energy stored from compressive flow does not contribute to the fracture process, via a plasticity coefficient r p : ...

... Gravitational effects are neglected, and plane strain conditions are assumed. The simulation results are obtained using a parallel finite element code for geomechanics [19][20][21]23], which is built on the deal.II finite element library [1,2], p4est mesh handling library [16], and the Trilinos project [34]. ...

The failure behavior of field-scale rock masses has long been studied indirectly through laboratory compression tests on rock specimens with preexisting flaws. However, little to no attention has been paid to size effects on these cracking processes that may be governed by the relative size between the fracture process zone and the rock structure. Here, we investigate such size effects on the compressive strength and cracking behavior of flawed rocks through high-fidelity simulations of mixed-mode fracture in quasi-brittle materials. We perform a series of numerical uniaxial compression tests on geometrically similar gypsum specimens with single and double flaws, across a wide range of sizes from 0.25 times a standard laboratory specimen size to 16 times. The results suggest strong size effects on both the uniaxial compressive strength and cracking patterns. The size effect on the compressive strength appears qualitatively similar to Bažant's size effect law derived for the tensile strength of notched structures. However, the quantitative changes in the strength deviate from the existing size effect law which does not account for mixed-mode fracture. As for the cracking behavior, three types of cracking patterns per flaw configuration are identified as the specimen size changes. Remarkably, the cracking patterns that emerge at the field scale, where the size of the fracture process zone is negligible, are analogous to those observed from laboratory experiments on highly brittle materials. The findings of this work provide important insights into how to bridge observations on rock fracturing processes across scales.

... In terms of plasticity, we introduce the undamaged effective stress and assume strain equivalence [46]. The commonly used von Mises yield function is adopted here as follows: ...

Crack-direction-based decomposition of elastic strain energy could effectively control the propagation of tensile and shear cracks in a phase field modelling context. The objective of the proposed double-phase field model is to extend the crack-direction-based decomposition strategy from 2D brittle fracture to complex mixed-mode crack modelling in a 3D setting, with plastic deformation incorporated. Both effective (undamaged) stress and plastic strain are split in the crack-orientation-based coordinate system. The decomposed tensile/shear stress contribute to tensile/shear damage evolution, respectively. The plastic contribution is coupled by relating the decomposed tensile/shear plastic strain to the corresponding tensile/shear crack energy release rates. Crack surface normal direction, represented by two orientation variables in 3D spatial domains, is determined by the F-criterion. The proposed model is implemented via ABAQUS subroutines with a staggered scheme for two phase field variables and crack direction. The simulation of a single-edge notch specimen under shear loading demonstrates that the ratio between shear and tensile crack energy release rates plays a significant role in the crack mode and mechanical response. Numerical results of a group of uniaxial tension, simple shear and tension shear specimens show good agreement with the experimental data in terms of the force–displacement curve and crack path, exhibiting the validity of the proposed model for capturing different crack modes. This model has also been proven effective for complex 3D problems via the third Sandia Challenge example.

... This strong strain softening was associated with shear fracture, whereas the weak strain softening was associated with shear fracture, shear band, axial splitting, or diffuse plastic flow (Fig. 3(b)). And this diffusional of mass leads to lower dilation (Choo and Sun, 2018). Additionally, increase in the effective-confining-pressure functioned as radial preloading on the sample and compensated some part of transverse tensile stresses generated due to incremental deviatoric stress. ...

About 1 million tons of sugar-factory-ash generated every year in India. And its usage in geotechnical applications as a stabilizing agent and filling material is gaining momentum. However, a proper assessment of mechanical behavior of sugar-factory-ash as a backfill material and environmental impact of leachates generated during its usage are significantly lacking. In this study, triaxial shear behavior of sugar-factory-ash is investigated. Similarly, leachability tests are carried out to study the environmental impact. Furthermore, a gabion retaining wall was designed with sugar-factory-ash backfill to assess its economic feasibility. Triaxial test results showed that the shear strength (φ′m(p) ≥ 34°) of sugar-factory-ash is comparable with conventional backfill material. The alkaline nature of leachate results in a lower leaching rate (≤2.46) and lower concentration of toxic heavy metals, making it an inert waste (DEFRA, 2010). Its lower monetary value makes the retaining wall up to 41% more economical than conventional material. At the same time, its utilization can save equal amounts of natural resources from being exploited. This study revealed that technically sugar-factory-ash has a high potential as a backfill material and is economically more viable than conventional backfill material.

... 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 [24,19,18,57,51,50,60,29,68,13,30,47,70,45,74,20,56,36,11,80,67,49,81,69,72,48,17,83,71,1,77,22,66,41,27,32,76,46] and ductile [16,6,14,33,73,34,26,35,55,62,54,28,15,25,64,75,2,82,79,53,78] 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. ...

This work addresses a robust and efficient Global-Local approach for numerically solving three dimensional (3D) fracture-mechanics problems. This method has the potential to tackle practical field problems in which a large-structure might be considered and fracture propagation is a localized phenomenon. In this regard, failure is analyzed on a lower (Local) scale, while dealing with a purely linear problem on an upper (Global) scale. The successful application of the method to non-linear problems such as finite strain hydraulic and ductile fracture in 2D leads naturally to the question of its effectiveness and robustness in the third dimension. This work is concerned with the extensions of the Global-Local method to problems of 3D brittle fracture. 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. On the element technologies different mesh types at the Global and Local domains were considered for efficient large scale problems. Several numerical results substantiate our developments.

... Looking at the macroscopic modeling of such behavior of granular geomaterials, few contributions can be retrieved from the literature assuming brittle fracture to be allowed under tensile stresses and shear stresses. We refer among others to Peron et al. (2013) and to those approaches that even model the fracture nucleation/propagation, for instance, the phase-field approaches proposed by Choo and Sun (2018); Cajuhi et al. (2018) and references therein. However, these approaches are not adapted to model the triggering of opening-like fractures under purely compressive effective stresses on the surface of saturated granular materials when subjected to forced drainage or drying boundary conditions. ...

Within the context of immiscible biphasic flow in porous media, when the nonwetting fluid invades the pore spaces which are a priori saturated with the wetting fluid, capillary forces dominate if the pore network is formed by fine-grained soils. Owing to the cohesion-less frictional behavior of such soils, a capillary force–driven fracturing phenomenon has been put forward by some researchers. Unlike the purely mechanistic tensile force–driven mode-I fracturing that typically has been attributed to the formation of desiccation cracks in soils, attempts to model this alternate capillarity-driven mechanism have not yet been realized at a continuum scale. However, the macro-scale counterpart of the capillary energy associated with the various pore-scale menisci is well-established as the interfacial energy characterized by the soil-water retention curve. An investigation of the possible contribution of this interfacial energy in supplying the dissipation related to fracture initiation is the essence of this work, inspired by the vast literature on gradient damage modeling.

... Phase field models for brittle fracture of shells were proposed by Reinoso et al. [20] and Kiendl et al. [21]. To study the failure mechanism of geological materials, Choo and Sun [22] suggested a phase field model considering the effects of confining pressure and strain rate, and Zhou et al. [23] presented a new formulation for compressive-shear fracture in brittle rock-like materials. Multi-physics phase field formulations were proposed by Zhang et al. [24] for modeling fracture in silicon electrodes, and by Cheng et al. [25] for capturing crack patterns induced by thermal spalling in concrete. ...

Phase field models for ductile fracture have gained significant attention in the last two decades due to their ability in implicitly tracking the nucleation and propagation of cracks. However, most crack phase field formulations for elastoplastic solids focus only on the effects of plastic deformation, and do not consider the different multi-axial stress states that may arise in practical designs. In this work, a thermodynamically consistent phase field approach coupled with finite strain plasticity, considering multi-axial stress states is presented. In order to account for the coupling between plasticity and stress states, the Stress-Weighted Ductile Fracture Model (SWDFM) is utilized. The SWDFM represents a criterion for predicting ductile crack initiation under both monotonic and cyclic loadings based on histories of an internal plastic variable, stress triaxiality, and the Lode angle parameter. The excellent performance of the SWDFM for predicting ductile crack initiation motivates for its incorporation into a phase field approach for predicting both crack initiation and propagation through degradation of the fracture toughness. Moreover, based on the second law of thermodynamics, exact requirements are imposed on the rate at which the fracture toughness can evolve. A novel function for degrading the plastic yield surface during the evolution of damage is introduced. This function, in line with experimental observations, leads to an accumulation of plastic deformation in damaged regions of a solid, and avoids numerical instabilities arising from concentrations of large plastic deformations in severely damaged regions. For validating the proposed model, results of computational simulations are compared to data from selected tests considering different multi-axial stress states. Comparisons of the numerical results with data from laboratory experiments demonstrate the capabilities of the proposed framework.

This paper presents an overview of the theories and computer implementation aspects of phase field models (PFM) of fracture. The advantage of PFM over discontinuous approaches to fracture is that PFM can elegantly simulate complicated fracture processes including fracture initiation, propagation, coalescence, and branching by using only a scalar field, the phase field. In addition, fracture is a natural outcome of the simulation and obtained through the solution of an additional differential equation related to the phase field. No extra fracture criteria are needed and an explicit representation of a crack surface as well as complex track crack procedures are avoided in PFM for fracture, which in turn dramatically facilitates the implementation. The PFM is thermodynamically consistent and can be easily extended to multi-physics problem by 'changing' the energy functional accordingly. Besides an overview of different PFMs, we also present comparative numerical benchmark examples to show the capability of PFMs.

This study is part of numerical simulations performed on an in-situ heating test conducted by the French National Radioactive Waste Management Agency (Andra) at the Meuse/Haute-Marne Underground Research Laboratory (URL) to study the thermo-hydromechanical behavior of the host Callovo-Oxfordian COx claystone in quasi real conditions, through the international research project DECOVALEX. We present a numerical study of damage and cracking process in saturated claystone subjected to thermo-hydromechanical coupling by considering material heterogeneity distribution. For this purpose, a macroscopic elastic model is first determined by using two steps of homogenization by taking into account the effects of porosity and mineral inclusions. This model is implemented into a finite element code devoted to solving thermo-hydromechanical coupling problems. The nucleation and propagation of cracks are described by using an extended phase-field method, considering the effects of temperature and fluid pressure on the evolution of phase-field. The proposed model is applied to the numerical analysis of cracking process due to excavation and heating around a group of boreholes (CRQ). The numerical results of the 3D simulation are compared with in-situ measurements of temperature and pore pressure distribution. The excavation damage zone and heating fracture is reproduced and analysed according to the structure of the heating position and the heterogeneity of the rock.

In this study, earth blocks (EB) and compressed earth blocks (CEB) are fabricated and investigated along with the development of a mathematical model of compressive and tensile loads. The investigated specimens are manufactured from a mixture of soil, ground recycled concrete (GRC) powder and water in weight fractions of 4:1:1. EBs are molded and CEBs are obtained by quasi-static compression in wet state. Material samples extracted from the blocks are tested in dry state for compressive strength and three-point flexural strength according to ASTM standards. The results indicate that CEB exhibits 130% higher compressive load capacity and 63% higher tensile strength compared to EB. Furthermore, the proposed and calibrated mathematical model is able to adequately describe the strength and damage behavior of both materials. Finally, microstructural/micromechanical interrelationships with the modeled material response are established based on a characterization of postmortem CEB samples using EDS elemental analysis and SEM micrographs techniques.

The phase-field method has been proven to be a powerful tool for predicting complex crack patterns and widely utilized in many fields. To effectively characterize and depict the complex cracking behaviors of rock-like materials, however, is still a challenge. In this study, a modified phase field model for mixed-mode fractures in rock-like materials is proposed. The effect of plastic free energy on damage is taken into consideration by developing a new phase field evolution equation. A non-associative Drucker–Prager constitutive model is coupled within the framework to capture more accurate stiffness and plastic strain in rock-like materials. To solve the coupled problem more efficiently, an accelerated monolithic iteration method has been developed. Several numerical results are presented to examine the viability of the established phase field model. Additionally, the investigation of the local stress field is done to identify the main driving forces of the different fracture modes.

The Material Point Method (MPM) is well suited to modelling dynamic solid mechanics problems undergoing large deformations with non-linear, history dependent material behaviour. However, the vast majority of existing material point method implementations do not inherit conservation properties (momenta and energy) from their continuum formulations. This paper provides, for the first time, a dynamic updated Lagrangian material point method for elasto-plastic materials undergoing large deformation that guarantees momenta and energy conservation. Sources of energy dissipation during point-to-grid and grid-to-point mappings for FLuid Implicit Particle (FLIP) and Particle In Cell (PIC) approaches are clarified and a novel time-stepping approach is proposed based on an efficient approximation of the Courant-Friedrich-Lewy (CFL) condition. The formulation provided in this paper provides a platform for understanding the energy conservation nature of future/existing features of material point methods, such as contact approaches.

A variational formulation of small strain ductile fracture, based on a phase-field modeling of crack propagation , is proposed. The formulation is based on an effective stress description of gradient plasticity, combined with an AT1 phase-field model. Starting from established variational statements of finite-step elastoplastic-ity for generalized standard materials, a mixed variational statement is consistently derived, incorporating in a rigorous way a variational finite-step update for both the elastoplastic and the phase-field dissipations. The complex interaction between ductile and brittle dissipation mechanisms is modeled by assuming a plasticity driven crack propagation model. A non-variational function of the equivalent plastic strain is then introduced to modulate the phase-field dissipation based on the developed plastic strains. Particular care has been devoted to the formulation of a consistent Newton-Raphson scheme for the case of Mises plasticity, with a global return mapping and relative tangent matrix, supplemented by a line-search scheme, for the solution of the gradient elastoplasticity problem for fixed phase field. The resulting algorithm has proved to be very robust and computationally effective. Application to several benchmark tests show the robustness and accuracy of the proposed model.

Dislocations nucleating from a crack-tip contribute to the plasticity evolution in the vicinity of crack, as well as to the crack propagation process. This paper systematically develops a method for enhancing the defect energy, accounting for crack-tip nucleated dislocations in the Helmholtz free energy density functional, associated with the phase-field formulation of crack evolution, in a coupled crystal plasticity finite element phase field (CPFE-PF) model. A concurrent crystal plasticity FE- hyper-dynamics accelerated molecular dynamics (CPFE-MD) model is created. The model that identifies, transfers, and propagates dislocations from the atomistic to continuum domain, is used to generate a dataset of crack tip dislocation density evolution along with different state variables. The paper focuses on crack tip plasticity mechanisms for the crystalline alloy Ti6Al. Using a Bayesian inference approach the critical state variables that affect the evolution of crack tip nucleated dislocation density are inferred. A functional form of the evolution of dislocation density in terms of the state variables is derived by employing a genetic programming based symbolic regression (GPSR) approach. The contribution of nucleated dislocation densities to effective plastic strain evolution at the crack tip is validated using the CPFE-MD model simulations. A comparison of the crack path and other state variables in the vicinity of the crack with and without contributions from the nucleated dislocations shows the importance of this augmentation on crack evolution.

We present a new phase-field formulation for the formation and propagation of a compaction band in high-porosity rocks. Novel features of the proposed formulation include (a) the effects of inertia on the rate of development of compaction bands, and (b) degradation mechanisms in tension, compression, and shear appropriate for dynamic strain localization problems where disturbances propagate in time in a wave-like fashion to induce micro-cracking, grain crushing, and frictional grain rearrangement in the rock. We also present a robust numerical technique to handle the spatiotemporal formation and evolution of the compaction band. We validate the model by simulating a benchmark problem involving a V-shape notched cylindrical specimen of Bentheim sandstone tested in conventional triaxial compression. The model is shown to reproduce different geometric styles of deformation that include pure compaction, shear-enhanced compaction, and a combination of pure and shear-enhanced compaction, where the combination mechanism consists of a straight primary compaction band surrounded by secondary chevron bands.

A novel explicit phase-field material point method with the convected particle domain interpolation (ePF-CPDI) is proposed for solving large deformation dynamic impact/contact fracture problems in elastoplastic geomaterials. In this method, we derive an explicit rate-dependent phase-field fracture model relying on the microforce balance law and the second law of thermodynamics. A coupled explicit phase-field plasticity model is then developed to describe dynamic elastoplastic fracture responses of geomaterials. Here the explicit time integration strategy and the staggered solution scheme are utilized in this work to solve the coupled-field governing equations based on the material point method. To eliminate the numerical noises caused by material points crossing cell boundaries, the convected particle domain interpolation technique is adopted to improve the computational accuracy in large deformation simulation. Furthermore, the proposed approach combining with the particle to particle contact algorithm is extended to deal with more complicated high-velocity impact and multi-body contact elastoplastic fracture problems. Numerical experiments are employed to validate the high accuracy and excellent capability of the proposed method and discuss the influences of main parameters on the phase-field fracture modeling.

In this work, we shall develop a novel phase-field model for modeling complex cracking processes in rock like materials under various loading paths. Both smooth frictionless and rough frictional cracks are investigated. For smooth cracks, an elastic-damage model is formulated with the unilateral effect on elastic stiffness tensor at the crack opening–closure transition. For rough cracks, an elastic–plastic damage model is developed, also incorporating the unilateral effect. In particular, for closed rough cracks, the damage evolution is explicitly coupled with the frictional sliding along cracks, which results in the macroscopic plastic deformation. The continuity conditions are verified for all energy functions, stress–strain relations and conjugated thermodynamics forces. By incorporating the friction sliding mechanism, the proposed model is able to properly take into account the dependency of mechanical behavior on confining stress of most geological materials under compressive stresses. The efficiency of the novel model is assessed through various cases, including comparisons between numerical results and experimental observations.

In this paper, the microstructure evolution of Ti-3Al-2Fe alloy as fabricated by Vacuum arc remelting(VAR) is studied by simulation combined with experiments. Different parts of the molten pool are affected by the temperature gradient and cause a difference in morphology. The center and non-edge regions of the molten pool (zone1 and znoe2) are mainly equiaxed crystals, and the dendrite width is small. The edge region of the molten pool (zone3) is dominated by columnar crystals, and the dendrite width is large. At the solid-liquid interface, the mass fraction of Al in the solid phase gradually decreases with time, while the mass fraction distribution of Fe increases. At the interface, the partition coefficient of Al decreases slightly and then increases with time, while that of Fe increases first and then decreases. KeywordsTitanium alloySimulationDendriteMicrosegregation

The damage anisotropy of an extruded ZK60 Mg alloy is characterized using tensile tests and scanning electronic microscopy. The accumulation of anisotropic deformations leads to the great differences of the dimple evolution and strains at fracture along different loading directions. To introduce the anisotropic deformation information into the damage constitutive relationship, a thermodynamically consistent phase-field model of ductile damage fully coupled with elastoplastic finite deformations is developed in this study. Using the user-defined constitutive relationship and displacement-temperature coupling element, the finite element simulations are conducted. The results show that: (1) ZK60 Mg alloys presents clear R-value difference in 0°, 45°, and 90° tests of intact specimens. The 45° test possesses the greatest R-value (1.50) and the greatest strain at fracture, however, the R-value for 0° is less than 1, indicating the thinning is preferential. (2) The higher ultimate stress leads to a larger average dimension of the dimples, whereas the higher density correlates with a larger elongation ratio at the fracture. The disappearance of the stress-bearing area indicates that the phase-field assumption on stress degradation is completely compatible with the dimple analysis on fractography. (3) The simulation results of the stress-strain relationships and damage paths correlate well with the experimental ductile damage of magnesium alloys at 200 ℃. Slight errors are basically attributed to the modeling parameters and finite element iteration algorithm. The proposed model presents fine applicability and reliability for the predictions of plastic deformations, ductile damage, and fracture of anisotropic Mg alloys.

Hydraulic and mechanical instabilities in geomaterials refer to a variety of non-linear phenomena that can be triggered by heterogeneities inherent to such materials. While hydraulic instabilities manifest themselves as heterogeneous fluid invasion causing `fingering' phenomenon, mechanical instabilities represent strain localizations and/or fractures. These instabilities and their associated coupling pose a major obstacle for applications involving geomaterials such as Carbon dioxide (CO2) sequestration and contaminant flow in ground waters. Existing classical models lack the required pattern-forming ingredients in their formulation and thus are stable against imposed perturbations. The essence of the current thesis work is to propose and investigate modeling techniques that allow to describe these instabilities. The constitutive approach adopted is that of micro-structured continua, in particular that of enhanced continua with a constitutive law depending on the gradient of so-called phase field variables.In the first part of this work, a fluid-fluid front has been described as a diffused interface by interpreting the presence of two fluids within the pore space as a single non-uniform fluid and the degree of saturation of one of the fluids as the corresponding phase field. While the classical one-to-one relation between capillary pressure and saturation degree describes retention properties of the porous network, an enhanced relation is obtained by prescribing a chemical potential in the spirit of Cahn-Hilliard type modeling of multi-phase fluids. This together with a non-local energy contribution provides the required ingredients required to describe hydraulic instabilites. In a one-dimensional setting, the proposed model allows to replicate experimentally observed non-monotonic saturation profiles during infiltration. Further, a slight non-convexity introduced into the flux function has been shown to allow modeling of drainage fronts, besides imbibition, without employing any additional complexities. A linear stability analysis (LSA) revealing the growth in time of arbitrary perturbations has been done, supplemented by two-dimensional simulations portraying the ability of the proposed model to describe fluid fingering and segregation.In the second part, triggering of a fracture within a drying porous medium has been studied. A prevailing modeling perspective, involving gradient damage modeling, has been first tested for its ability to replicate periodic fracture formation as observed in representative experiments. Further, a new paradigm has been introduced by interpreting the presence of a fracture as a loss of capillary properties, thus allowing passage of non-wetting fluid under vanishing capillary pressure. This is applicable to cohesion-less and unconsolidated fine-grained soils, where resistance against tensile loading is negligible and thus fracturing induced due to development of tensile stresses is not the prevailing phenomenon. Starting from the principles of variational approach, it has been shown that for sufficiently strong desiccation, damage initiates homogeneously on the drying face while progressing into the body with time. The possible occurrence of bifurcations of this base solution, representing initiation of periodic fractures, has been analyzed again in the framework of LSA.This work sets the stage for the study of coupling between the above mentioned instabilities and experimental investigation of unstable flow features such as pinching and coalescence of the wetting phase. Initiation of damage induced due to evolving drainage finger is also of particular interest in the context of earlier mentioned applications.

The shear instability of the bearing outer-ring guiding-surface is investigated within this work when subjected to cyclic impact and sliding actions. The paper combines numerical simulations and experiments. A high-speed bearing oil interruption experiment is carried out for testing the severe damage of the bearing steel at high-speed impact-sliding contacts. A coupled thermo-elasto-plastic phase-field model is established and validated by experimental results. It then allows, by simulating the multi-physics problem, the prediction of damage initiation and propagation for ductile materials at cyclic impact-sliding contacts. To this end, a temperature-dependent isotropic-kinematic hardening model combined with thermal softening, strain hardening, and damage degradation is employed. Various numerical examples are validated experimentally to illustrate the model capability. The results show that under high-speed cyclic impact-sliding conditions, the damage initiated and accumulated at the contact near-surface is accompanied by instantaneous high temperature, plastic deformation, and shear instability. The thermal softening and damage degradation, both play an important role in the shear instability. In addition, the impact velocity, impact frequency, and friction coefficient have significant effects on damage initiation and accumulation. All these effects are discussed in detail in this contribution.

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 the 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.

Compaction bands are tabular regions of localized compressive deformation with little or no shear offset. Often observed in high‐porosity rocks, they can be classified into several subtypes based on their pattern and orientation with respect to the maximum principal stress. The combined effects of material properties and loading conditions on the type of compaction bands that develop are not fully understood, and realistic simulations of their formation are not always successful. In this study, a phase‐field approach for capturing the formation and propagation of compaction bands is proposed. The fracture energy utilized in classic phase‐field formulations is interpreted to be driven by grain crushing and is herein characterized by breakage mechanics theory. A new decomposition of the free energy function is introduced in which the energy stored by plastic compactive flow drives grain crushing. Depending on the value of the energy release rate, the model predicts different styles of compaction bands that are remarkably consistent with those observed in the field. Numerical simulations demonstrate the role of confining pressure, plasticity, and critical breakage energy on the styles of the predicted compaction bands.

We present an extended finite element framework to numerically study competing hydraulic fracture propagation. The framework is capable of modeling fully coupled hydraulic fracturing processes including fracture propagation, elastoplastic bulk deformation and fluid flow inside both fractures and the wellbore. In particular, the framework incorporates the classical orifice equation to capture fluid pressure loss across perforation clusters linking the wellbore with fractures. Dynamic fluid partitioning among fractures during propagation is solved together with other coupled factors, such as wellbore pressure loss (Δpw), perforation pressure loss (Δp), interaction stress (σint) and fracture propagation. By numerical examples, we study the effects of perforation pressure loss and wellbore pressure loss on competing fracture propagation under plane-strain conditions. Two dimensionless parameters Γ=σint/Δp and Λ=Δpw/Δp are used to describe the transition from uniform fracture propagation to preferential fracture propagation. The numerical examples demonstrate the dimensionless parameter Γ also works in the elastoplastic media.

We propose a mechanical and computational model to describe the coupled problem of poromechanics and cracking in variably saturated porous media. A classical poromechanical formulation is adopted and coupled with a phase-field formulation for the fracture problem. The latter has the advantage of being able to reproduce arbitrarily complex crack paths without introducing discontinuities on a fixed mesh. The obtained simulation results show good qualitative agreement with desiccation experiments on soils from the literature.

Phase-field formulations have recently emerged as promising tools to model brittle fracture. Based on the variational approach to fracture, these models aim at overcoming some of the computational challenges found in simulating complex fracture patterns and their evolution due to external or internal loads. Since most applications and validation exercises thus far have been restricted to academic benchmarks, the evaluation of phase-field fracture models against experimental results an practical engineering scenarios remains fragmented.
Here we introduce a straightforward phase-field approach to simulate fluid and mechanically-driven fractures based on energy minimization and thermodynamical principles. We apply our methodology to several laboratory experiments of brittle fracture, and to fracturing processes in two full-scale concrete dams, taking into account the hydraulic forces inside the fractures. We conclude that phase-field models represent a promising computational tool that may be applied to realistic engineering scenarios.

We performed an extensive suite of true triaxial experiments in two porous sandstones, Bentheim (porosity ≈ 24%) and Coconino (17%). The experiments were conducted using a novel loading path, which maintains constant Lode angle (Θ) throughout the test. This path enabled the examination of the effects of Lode angle and mean stress on failure. Our tests covered σ3 magnitudes between 0 and 150 MPa, and of Θ at -30° (axisymmetric extension), -16°, 0°, +11°, +21°, and +30° (axisymmetric compression). Test results revealed the respective contribution of each of the two stress invariants to failure stress, the failure-plane angle, and failure mode. In both sandstones, the shear stress required for failure increases with mean stress but decreases with Θ when shear failure mode dominates. However, the dependence of failure stress on mean stress and Θ is reversed when the compactive failure mode is in control. The compactive failure mode was evident in Bentheim sandstone when compaction bands were observed under high mean stress. The Coconino sandstone did not reach the compactive failure regime within the maximum confinement applied. The failure-plane angle monotonically decreases with increasing mean stress and Θ. For Coconino sandstone, failure-plane angle varies between 80° and 50° for σoct,f between 50 and 450 MPa whereas it drops to 0° as σoct,f, the mean stress at failure, approaches 250 MPa in Bentheim sandstone. We employed the bifurcation theory to relate the stress conditions at failure to the development of failure-planes. The theory is in qualitative agreement with the experimental data.

The phase-field model has been attracting considerable attention due to its capability of capturing complex crack propagations without mesh dependence. However, its validation studies have primarily focused on the ability to predict reasonable, sharply defined crack paths. Very limited works have so far been contributed to estimate its accuracy in predicting force responses, which is majorly attributed to the difficulty in the determination of the length scale. Indeed, accurate crack path simulation can be achieved by setting the length scale to be sufficiently small, whereas a very small length scale may lead to unrealistic force-displacement responses and overestimate critical structural loads. This paper aims to provide a critical numerical investigation of the accuracy of phase-field modelling of brittle fracture with special emphasis on a possible formula for the length scale estimation. Phase-field simulations of a number of classical fracture experiments for brittle fracture in concretes are performed with simulated results compared with experimental data qualitatively and quantitatively to achieve this goal. Furthermore, discussions are conducted with the aim to provide guidelines for the application of the phase-field model.

Hydromechanical interactions between ﬂuid ﬂow and deformation in porous geo- materials give rise to a wide range of societally important problems such as land- slides, ground subsidence, and injection-induced earthquakes. Many geomaterials in these problems possess two-scale porous structures due to fractures, particle ag- gregation, or other reasons. However, coupled hydromechanical processes in these multiscale porous materials, such as ground deformation caused by preferential ﬂow, are beyond the modeling capabilities of classical frameworks.
This thesis develops theoretical and computational frameworks for fully cou- pled hydromechanical modeling of geomaterials with two-scale porous structures. Adopting the concept of double porosity, we treat these materials as a multiscale continuum in which two pore regions of different scales interact within the same continuum. Three major developments are presented.
First, we build a mathematical framework for thermodynamically consistent modeling of unsaturated porous media with double porosity. Conservation laws are formulated incorporating an effective stress tensor that is energy-conjugate to the rate of deformation tensor of the solid matrix. Based on energy-conjugate pairs identiﬁed in the ﬁrst law of thermodynamics, we develop a constitutive framework for hydrological and mechanical processes coupled at two scales.
Second, we introduce a novel constitutive framework for elastoplastic mate- rials with evolving internal structures. By partitioning the thermodynamically consistent effective stress into two individual, single-scale effective stresses, this framework uniquely distinguishes proportional volume changes in the two pore regions under ﬁnite deformations. This framework accommodates the impact of pore pressure difference between the two scales on the solid deformation, which was predicted by thermodynamic principles. We show that the proposed frame- work not only improves the prediction of deformation of two-scale geomaterials, but also simulates secondary compression effects due to delayed pressure dissipa- tion in the less permeable pore region.
Third, we develop a ﬁnite element framework that enables the use of compu- tationally efﬁcient equal-order elements for solving coupled ﬂuid ﬂow and defor- mation problems in double-porosity media. At the core of the ﬁnite element for- mulation is a new method that stabilizes twofold saddle point problems arising in the undrained condition. The stabilized ﬁnite elements allow for equal-order linear interpolations of three primary variables—the displacement ﬁeld and two pore pressure variables—throughout the entire range of drainage conditions.

Rock response to confining pressure and strain rate can change dramatically from very brittle to ductile. Capturing this transition is crucial for a correct prediction of rock mass failure due to blasting, explosion or drilling in mining. In this work, a new constitutive model that accounts for the effects of both confining pressure and strain rate on the nominal strength and post peak behaviour is proposed for dry intact rocks and other similar geological materials. The key features of the proposed constitutive model are the employment of a single loading function that evolves from initial yielding to ultimate failure during damaging and the rate-dependent enhancement so that the strain rate effects can be faithfully described at different confining pressures. The model can adequately capture both the brittle and ductile responses as well as the brittle-ductile transition as a result of both strain rate and confining pressure.

An Arlequin poromechanics model is introduced to simulate the hydro-mechanical coupling effects of fluid-infiltrated porous media across different spatial scales within a concurrent computational framework. A two-field poromechanics problem is first recast as the twofold saddle point of an incremental energy functional. We then introduce Lagrange multipliers and compatibility energy functionals to enforce the weak compatibility of hydromechanical responses in the overlapped domain. To examine the numerical stability of this hydro-mechanical Arlequin model, we derive a necessary condition for stability, the twofold inf–sup condition for multi-field problems, and establish a modified inf–sup test formulated in the product space of the solution field. We verify the implementation of the Arlequin poromechanics model through benchmark problems covering the entire range of drainage conditions. Through these numerical examples, we demonstrate the performance, robustness, and numerical stability of the Arlequin poromechanics model.

Hydraulic fracturing is a big issue in the exploitation of oil and gas resources as well as in the production of heat in deep geothermal energy plants. Investigating hydraulic fracturing processes numerically by means of a finite-element analysis, one has to address the porous solid and its pore content within a fully coupled computational approach. For this purpose, the present article combines the well-established Theory of Porous Media with elements of fracture mechanics, especially, with the phase-field approach to fracture, which has proven as a successful tool for the computation of fracturing processes in the field of standard solid mechanics.
Although hydraulic fracturing is widely applied in practice, this procedure has not yet been investigated adequately by means of a full theoretical and computational framework on the basis of a multicomponent medium tackling a porous solid skeleton and its pore content with their mutual interaction of deformation and fracture, and fluid-driven processes both in the solid bulk and cracking domains. Addressing these features, the article concentrates on a permeable elastic solid skeleton, where the fracturing process is governed by brittle fracture driven either by a prescribed fluid pressure or by a prescribed fluid influx. Two- and three-dimensional numerical examples computed by use of the coupled solver PANDAS exhibit the possibilities of this approach.

We conducted an extensive suite of true triaxial experiments in two porous sandstones, Bentheim (porosity ≈ 24%) and Coconino (17.5%). Our experiments demonstrate that failure of both sandstones is not only a function of σ3, but also of σ2. For a given σ3, σ1 at failure (σ1,peak) increases as σ2 is raised above σ3 between tests. σ1,peak reaches a peak as σ2 is about halfway between σ3 and σ1 and then gradually decreases such that when σ2 ≈ σ1,peak, it approaches its initial magnitude when σ2 = σ3. For a constant σ3, failure-plane angle increases with σ2 by a maximum of less than 10° as σ2 rises from σ2 = σ3 to σ2 = σ1,peak. The effect of σ2 on both failure level and failure-plane angle is stronger in the lower porosity Coconino sandstone than in the Bentheim sandstone. The σ2-dependence of failure mode in the Bentheim is different than Coconino over the same σ3 range. Both sandstones failed dilatantly at low σ3 magnitudes. However, at high σ3 (100-120 MPa), Bentheim sandstone developed shear-enhanced compaction bands, followed by pure compaction bands at σ3 = 150 MPa. Compaction bands were not observed in the Coconino. Microscopic observations via SEM reveals that tensile microcracking is dominant when shear banding occurs (under low σ3), while pervasive grain crushing and pore collapse inside compaction bands are observed at high σ3.

Continuum porous media theories, extended by a diffusive phase-field modeling (PFM) approach, introduce a convenient and efficient tool to the simulation of hydraulic fracture in fluid-saturated heterogeneous materials. In this, hydraulic- or tension-induced fracture occurs in the solid phase. This leads to permanent local changes in the permeability, the volume fractions of the constituents as well as the interstitial-fluid flow. In this work, the mechanical behaviors of the multi-field, multi-phase problem of saturated porous media, such as the pore-fluid flow and the solid-skeleton deformation, are described using the macroscopic Theory of Porous Media (TPM). To account for crack nucleation and propagation in the sense of brittle fracture, the energy-minimization-based PFM procedure is applied, which approximates the sharp edges of the crack by a diffusive transition zone using an auxiliary phase-field variable. Furthermore, the PFM can be implemented in usual continuum finite element packages, allowing for a robust solution of initial-boundary-value problems (IBVP). For the purpose of validation and comparison, simulations of a two-dimensional IBVP of hydraulic fracture are introduced at the end of this research paper.

This paper deals with the hydromechanical modelling of the initiation of failure in soils with particular reference to landslides. To this end, localized and diffused failure modes are simulated with a finite element model for coupled elasto-plastic variably saturated porous geomaterials, in which the material point instability is detected with the second-order work criterion based on Hill’s sufficient condition of stability. Three different expressions of the criterion are presented, in which the second-order work is expressed in terms of generalized effective stress, of total stress and thirdly by taking into account the hydraulic energy contribution for partially saturated materials. The above-mentioned computational framework has been applied to study two initial boundary value problems: shear failure of a plane strain compression test of globally undrained water-saturated dense sand (where cavitation occurs at strain localization) and isochoric grain matter, and the onset of a flowslide from southern Italy due to rainfall (Sarno-Quindici events, May 5–6 1998). It is shown that the second-order work criterion applied at the material point level detects the local material instability and gives a good spatial indication of the extent of the potentially unstable domains in both the localized and diffused failure mechanisms of the cases analyzed, is able to capture the instability induced by cavitation of the liquid water and gives results according to the time evolution of plastic strains and displacement rate.

This work outlines a rigorous variational-based framework for the phase field modeling of ductile fracture in elastic–plastic solids undergoing large strains. The phase field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with geometric features rooted in fracture mechanics. It has proven immensely successful with regard to the analysis of complex crack topologies without the need for fracture-specific computational structure such as finite element design of crack discontinuities or intricate crack-tracking algorithms. Following the recent work Miehe et al. (2015), the phase field model of fracture is linked to a formulation of gradient plasticity at finite strains. The formulation includes two independent length scales which regularize both the plastic response as well as the crack discontinuities. This ensures that the damage zones of ductile fracture are inside of plastic zones, and guarantees on the computational side a mesh objectivity in post-critical ranges. The novel aspect of this work is a precise representation of this framework in a canonical format governed by variational principles. The coupling of gradient plasticity to gradient damage is realized by a constitutive work density function that includes the stored elastic energy and the dissipated work due to plasticity and fracture. The latter represents a coupled resistance to plasticity and damage, depending on the gradient-extended internal variables which enter plastic yield functions and fracture threshold functions. With this viewpoint on the generalized internal variables at hand, the thermodynamic formulation is outlined for gradient-extended dissipative solids with generalized internal variables which are passive in nature. It is specified for a conceptual model of von Mises-type elasto-plasticity at finite strains coupled with fracture. The canonical theory proposed is shown to be governed by a rate-type minimization principle, which fully determines the coupled multi-field evolution problem. This is exploited on the numerical side by a fully symmetric monolithic finite element implementation. The performance of the formulation is demonstrated by means of some representative examples.

We propose a phase-field model for ductile fracture in a single crystal within the kinematically linear regime, by combining the theory of single crystal plasticity as formulated in Gurtin et al. [1] and the phase-field formulation for ductile fracture proposed by Ambati et al. [2]. The model introduces coupling between plasticity and fracture through the dependency of the socalled degradation function from a scalar global measure of the accumulated plastic strain on all slip systems. A viscous regularization is introduced both in the treatment of plasticity and in the phase-field evolution equation. Testing of the model on two examples for face centred cubic single crystals indicates that fracture is predicted to initiate and develop in the regions of the maximum accumulated plastic strain, which is in agreement with phenomenological observations. A rotation of the crystallographic unit cell is shown to affect the test results in terms of failure pattern and corresponding global and local response.

This work outlines a novel variational-based theory for the phase-field modelling of ductile fracture in elastic-plastic solids undergoing large strains. The phase-field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modelling. It is linked to a formulation of gradient plasticity at finite strains. The framework includes two independent length scales which regularize both the plastic response as well as the crack discontinuities. This ensures that the damage zones of ductile fracture are inside of plastic zones, and guarantees on the computational side a mesh objectivity in post-critical ranges.

In this paper we present a phase field model for a proppant-filled fracture in a poroelastic medium. The formulation of the coupled system involves four unknowns; displacements, phase field, pressure, and proppant concentration. The two-field displacement phase-field system is solved fully-coupled and accounts for crack irreversibility. This solution is than coupled to the pressure equation via a fixed-stress iteration. The pressure is obtained by using a diffraction equation where the phase-field variable serves as an indicator function that distinguishes between the fracture and the reservoir. The transport of the proppant in the fracture is modeled by using a power-law fluid system. The numerical discretization in space is based on Galerkin finite elements for displacements and phase-field, and an enriched Galerkin method is applied for the pressure equation in order to obtain local mass conservation. The concentration is solved with cell-centered finite elements. Nonlinear equations are treated with Newton's method. Our developments are substantiated with several numerical examples in two and three dimensions.

Coupled poromechanical problems appear in a variety of disciplines, from reservoir engineering to biomedical applications. This work focuses on efficient strategies for solving the matrix systems that result from discretization and linearization of the governing equations. These systems have an inherent block structure due to the coupled nature of the mass and momentum balance equations. Recently, several iterative solution schemes have been proposed that exhibit stable and rapid convergence to the coupled solution. These schemes appear distinct, but a unifying feature is that they exploit the block-partitioned nature of the problem to accelerate convergence. This paper analyzes several of these schemes and highlights the fundamental connections that underlie their effectiveness. We begin by focusing on two specific methods: a fully-implicit and a sequential-implicit scheme. In the first approach, the system matrix is treated monolithically, and a Krylov iteration is used to update pressure and displacement unknowns simultaneously. To accelerate convergence, a preconditioning operator is introduced based on an approximate block-factorization of the linear system. Next, we analyze a sequential-implicit scheme based on the fixed-stress split. In this method, one iterates back and forth between updating displacement and pressure unknowns separately until convergence to the coupled solution is reached. We re-interpret this scheme as a block-preconditioned Richardson iteration, and we show that the preconditioning operator is identical to that used within the fully-implicit approach. Rapid convergence in both the Richardson- and Krylov-based methods results from a particular choice for a sparse Schur complement approximation. This analysis leads to a unified framework for developing solution schemes based on approximate block factorizations. Many classic fully-implicit and sequential-implicit schemes are simple sub-cases. The analysis also highlights several new approaches that have not been previously explored. For illustration, we directly compare the performance and robustness of several variants on a benchmark problem.

Geomaterials with aggregated structure or containing fissures often exhibit a bimodal pore size distribution that can be viewed as two coexisting pore regions of different scales. The double porosity concept enables continuum model-ing of such materials by considering two interacting pore scales satisfying relevant conservation laws. This paper develops a thermodynamically consistent framework for hydromechanical modeling of unsaturated flow in double porosity media. With an explicit treatment of the two pore scales, conservation laws are formulated incorporating an effective stress tensor that is energy-conjugate to the rate of deformation tensor of the solid matrix. A constitutive framework is developed based on energy-conjugate pairs identified in the first law of thermodynamics, which is then incorporated into a three-field mixed finite element formulation for double porosity media. Numerical simulations of laboratory-and field-scale problems are presented to demonstrate the impact of double porosity on the resulting hydromechanical responses.

Natural geomaterials such as fissured rocks and aggregated soils often exhibit a pore size distribution with two dominant pore scales, usually termed macropores and micropores. High-fidelity descriptions of these materials require an explicit treatment of the two pore regions as double porosity. We develop a finite element framework for coupled solid deformation and fluid diffusion in double porosity media that employs a thermodynamically consistent effective stress. Mixed finite elements that interpolate the solid displacement and pore pressures in the macropores and micropores are used for this purpose. In the limit of undrained deformation, the incompressibility constraint causes unstable behavior in the form of spurious pressure oscillation at the two pore scales. To circumvent this instability we develop a variant of the polynomial pressure projection technique for a twofold saddle point problem. The proposed stabilization allows the use of equal-order (linear) interpolations of the displacement and two pore pressure variables throughout the entire range of drainage condition.

For assessing energy-related activities in the subsurface, it is important to investigate the impact
of the spatial variability and anisotropy on the geomechanical behavior of shale. The Brazilian
test, an indirect tensile-splitting method is performed in this work, and the evolution of strain
field is obtained using digital image correlation. Experimental results show the significant impact
of local heterogeneity and lamination on the crack pattern characteristics. For numerical
simulations, a phase field method is used to simulate the brittle fracture behavior under various
Brazilian test conditions. In this study, shale is assumed to consist of two constituents including
the stiff and soft layers to which the same toughness but different elastic moduli are assigned.
Microstructural heterogeneity is simplified to represent mesoscale (e.g., millimeter scale)
features such as layer orientation, thickness, volume fraction, and defects. The effect of these
structural attributes on the onset, propagation, and coalescence of cracks is explored. The
simulation results show that spatial heterogeneity and material anisotropy highly affect crack
patterns and effective fracture toughness, and the elastic contrast of two constituents significantly
alters the effective toughness. However, the complex crack patterns observed in the experiments
cannot completely be accounted for by either an isotropic or transversely isotropic effective
medium approach. This implies that cracks developed in the layered system may coalesce in
complicated ways depending on the local heterogeneity, and the interaction mechanisms between
the cracks using two-constituent systems may explain the wide range of effective toughness of
shale reported in the literature.

The critical energy release rates for mode I and II fracture for rock-like materials are usually different. In this paper, a modified phase-field model is proposed for simulating mixed mode crack propagation. The model can distinguish between the critical energy release rates for mode I and mode II cracks. For the purpose of validation, rock-like materials with a single flaw or double flaws under compression are studied. The simulated results are compared to experimental data, both qualitatively and quantitatively. It is shown that the proposed model is able to capture the commonly observed propagation pattern of wing crack emergence followed by secondary cracks. Additionally, the typical types of crack coalescence observed in experimental tests are successfully reproduced, including the critical loads at which crack coalescence occurs.

Propagation of fluid-driven fractures plays an important role in natural and engineering processes, including transport of magma in the lithosphere, geologic sequestration of carbon dioxide and oil and gas recovery from low-permeability formations, among many others. The simulation of fracture propagation poses a computational challenge as a result of the complex physics of fracture, and the need to capture disparate length scales. Phase-field models represent fractures as a diffuse interface, and enjoy the advantage that fracture nucleation, propagation, branching or twisting can be simulated without ad hoc computational strategies like remeshing or local enrichment of the solution space. Here, we propose a new quasi-static phase-field formulation for modeling fluid-driven fracturing in elastic media at small strains. The approach fully couples the fluid flow in the fracture (described via the Reynolds lubrication approximation) and the deformation of the surrounding medium. The flow is solved on a lower-dimensionality mesh immersed in the elastic medium. This approach leads to accurate coupling of both physics. We assessed the performance of the model extensively by comparing results for the evolution of fracture length, aperture and fracture-fluid pressure against analytical solutions under different fracture propagation regimes. The excellent performance of the numerical model in all regimes builds confidence in the applicability of phase-field approaches to simulate fluid-driven fracture.

Six argillaceous rocks, six sandstones, two kinds of tuff and two tuffaceous sandstones from various localities in Japan ranging Palaeozoic to Pliocene in age were deformed in the triaxial vessel under 1 to 2500 bars in confining pressure, at room temperature and in strain rate 3.5x10-5/sec. All are dry sample and underwent compression test.The strength of Tertiary argillaceous rocks increases in direct proportion to age. In sandstone the age is important factor for the strength, too. On the other hand, in tuff and tuffaceous sandstone, grain the size seems most important for the strength.As the deformation changes ductile to brittle, the mode of fracturing does wedgelike fracture, singleplane of shear fracture, network of a lots of minute shear fractures, and to flow. The transitional line from the single shear fracture to the network shear fracture occur in lower ductility in both tuff and tuffaceous sandstone, higher in sandstone, and highest in argillaceous rocks.

A stabilized thermo-hydro-mechanical (THM) finite element model is introduced to investigate the freeze-thaw action of frozen porous media in the finite deformation range. By applying mixture theory, frozen soil is idealized as a composite consisting of three phases, i.e., solid grain, unfrozen water and ice crystal.
A generalized hardening rule at finite strain is adopted to replicate how the elasto-plastic responses and critical state evolve under the influence of phase transitions and heat transfer. The enhanced particle interlocking and ice strengthening during the freezing processes and the thawing-induced consolidation at the
geometrical nonlinear regimes are both replicated in numerical examples. The numerical issues due to lack of two-fold inf-sup condition and ill-conditioning of the system of equations are addressed. Numerical examples for engineering applications at cold region are analyzed via the proposed model to predict the impacts of
changing climate on infrastructure at cold regions.

Highlights • A variational based computational framework that combines multiple dissipative phenomena is proposed. • Diffusion induced large plastic deformation and phase field fracture during two-phase lithiation of silicon electrodes is modeled. • The effect of fracture energy release rate, electrode geometry, and geometric constraints on the fracture behavior of silicon electrodes is investigated. Abstract Silicon (Si) is considered to be a promising next-generation anode material for lithium-ion batteries. However, the large volume change during (de)lithiation processes causes fracture of Si electrodes, thereby limiting Si's practical application in lithium-ion batteries. In this work, we formulate a variational-based fully chemo-mechanical coupled computational framework to study diffusion induced large plastic deformation and phase field fracture in Si electrodes. Into this framework we incorporate a recently developed reaction-controlled diffusion model to predict two-phase lithiation for amorphous Si (a-Si) and crystalline Si (c-Si) as well as diffusion induced anisotropic deformation for c-Si. The variational formulation suggests to consider the deformation field, the chemical potential, and the damage field as primary unknowns. The concentration field is considered as a local variable and is recovered from the chemical potential on the element level. We carry out several numerical simulations to show the performance of our computational model and point out the significance of accurately accounting for the presence of the reaction front when modeling diffusion induced fracture problems for both a-Si and c-Si electrodes. In addition, we investigate how the fracture energy release rate, electrode geometry, and geometrical constraints affect the fracture behavior of Si electrodes.

In this work, we present numerical studies of fixed-stress iterative coupling for solving flow and geomechanics with propagating fractures in a porous medium. Specifically, fracture propagations are described by employing a phase-field approach. The extension to fixed-stress splitting to propagating phase-field fractures and systematic investigation of its properties are important enhancements to existing studies. Moreover, we provide an accurate computation of the fracture opening using level-set approaches and a subsequent finite element interpolation of the width. The latter enters as fracture permeability into the pressure diffraction problem which is crucial for fluid filled fractures. Our developments are substantiated with several numerical tests that include comparisons of computational cost for iterative coupling and nonlinear and linear iterations as well as convergence studies in space and time.

We present a simplified model of damaging porous material, obtained through consistent linearization from a recursive-faulting material model described in (Pandolfi et al. 2016). The brittle damage material model is characterized by special planar micro-structures, consisting of nested families of equi-spaced frictional-cohesive faults in an otherwise elastic matrix material. The linear kinematics model preserves the main microstructural features of the finite kinematics one but offers a far better computational performance. Unlike models commonly employed in geo-mechanical applications, the proposed model contains a small number of parameters, to wit, two elastic moduli, three frictional-cohesive parameters, and three hydraulic response parameters, all of which having clear physical meanings and amenable to direct experimental measurement through standard material tests. The model is validated by comparison to triaxial hydro-mechanical experimental data. Despite the paucity of material constants, the salient aspects of the observed behavior are well captured by the model, qualitatively and quantitatively. As an example of application of the model, we simulate the excavation of a borehole in a rocky embankment.

This work outlines a rigorous variational-based framework for the phase field modeling of fracture in isotropic porous solids undergoing large elastic-plastic strains. It extends the recent works Miehe et al., (2015, 2016) to a particular formulation of isotropic porous plasticity. The phase field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with geometric features rooted in fracture mechanics. A gradient plasticity model for porous plasticity with a simple growth law for the evolution of the void fraction is developed, and linked to a failure criterion in terms of the local elastic-plastic work density that drives the fracture phase field. It is shown that this approach is able to model basic phenomena of ductile failure such as cup-cone failure surfaces in terms of only two material parameters on the side of damage mechanics: a critical work density that triggers the onset of damage and a shape parameter that governs the postcritical damage up to fracture. The formulation includes two independent length scales which regularize both the plastic response as well as the crack discontinuities. This allows to design damage zones of ductile fracture to be inside of plastic zones or vice versa, and guarantees on the computational side a mesh objectivity in post-critical ranges. The key aspect that allows to construct a variational theory for porous plasticity at fracture is the use of an Eulerian constitutive setting, where the yield function is formulated in terms of the Kirchhoff stress. Here, we exploit the fact that this stress approximates an effective stress that drives the plasticity in the matrix of the porous solid. The coupling of gradient plasticity to gradient damage is realized by a constitutive work density function that includes the stored elastic energy and the dissipated work due to plasticity and fracture. The latter represents a coupled resistance to plasticity and damage, depending on the gradient-extended internal variables which enter the plastic yield function and the fracture threshold function. The canonical theory proposed is shown to be governed by a rate-type minimization principle, which fully determines the coupled multi-field evolution problem, and provides inherent symmetries with regard to a finite element implementation. The robust computational setting proposed includes (i) a general return scheme of plasticity in the spectral space of logarithmic principal strains and dual Kirchhoff stresses, (ii) the micromorphic regularization of the gradient plastic evolution and (iii) a history-field-driven update of the linear phase field equation.