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... In order to model the fracture flow in a fluid-infiltrating porous media, we adopt the permeability enhancement approach that approximates the water flow inside the fracture as the flow between two parallel plates [138][139][140][141]: ...

... In addition, we introduce a set of heterogeneous material properties that solely depends on the spatial distribution of initial porosity ϕ 0 . Specifically, we adopt a phenomenological model proposed by [155] for the shear modulus G, while we use a power law for the critical energy release rate G d similar to [140,156]: conditions. Based on this setting, the water is supplied from the bottom during the freezing ...

... In this study, the fluid flow within both undamaged porous matrix and fracture is assumed to be similar to the laminar flow of a Newtonian fluid that possesses a low Reynolds number. This approach has been widely accepted in modeling hydraulic fracture in porous media [139,140,142,241,242] such that the laminar fracture flow is approximated as the flow between two parallel plates which leads to an increase in permeability along the flow direction (i.e., cubic law). Hence, we assume that the fluid flow inside both the host matrix and the fracture obeys the generalized Darcy's law, i.e., (3.66) ...

Global challenges associated with extreme climate events and increasing energy demand require significant advances in our understanding and predictive capability of coupled multi- physical processes across spatial and temporal scales. While classical approaches based on the mixture theory may shed light on the macroscopic poromechanics simulations, accurate forward predictions of the complex behavior of phase-changing geomaterials cannot be made without understanding the underlying coupling mechanisms among constituents at the microstructural scale. To precisely predict the multi-physical behaviors originated by smaller scales, fundamental understandings of the micromechanical interactions among phase constituents are crucial. Hence, this dissertation discusses mathematical and computational frameworks designed to capture coupled thermo-hydro-mechanical-fracture processes in phase-changing porous media that incorporate necessary microscopic details. To achieve this goal, this dissertation aims to introduce a practical way to investigate how phase transition and evolving microstructural attributes at small scales affect the applicability of meso- or macroscopic finite element simulations, by leveraging the phase field method to represent the regularized interfaces of phase constituents.
Firstly, a multi-phase-field microporomechanics model is presented to model the growth and thaw of ice lenses. In specific, we extend the field theory for ice lens that is not restricted to one-dimensional space. The key idea is to represent the state of the pore fluid and the evolution of freezing-induced fracture via two distinct phase field variables coupled with balance laws, which leads to an immersed approach where both the homogeneous freezing and ice lensing are distinctively captured. Secondly, a thermo-hydro-mechanical theory for geological media with thermally non-equilibrated constituents is presented, where we develop an operator-split framework that updates the temperature of each constituent in an asynchronous manner. Here, the existence of an effective medium is hypothesized, in which the constituents exhibit different temperatures while heat exchange among the phases is captured via Newton’s law of cooling. Thirdly, an immersed phase field model is introduced to predict fluid flow in fracturing vuggy porous media, where crack growth may connect previously isolated voids and form flow conduits. In this approach, we present a framework where the phase field is not only used as a damage parameter for the solid skeleton but also as an indicator of the pore space, which enables us to analyze how crack growth in vuggy porous matrix affects the flow mechanism differently compared to the homogenized effective medium while bypassing the needs of partitioning the domain and tracking the moving interface. Finally, we present a new phase field fracture theory for higher-order continuum that can capture physically justified size effects for both the path-independent elastic responses and the path-dependent fracture. Specifically, we adopt quasi-quadratic degradation function and linear local dissipation function such that the physical size dependence are insensitive to the fictitious length scale for the regularized interface, which addresses the numerical needs to employ sufficiently large phase field length scale parameter without comprising the correct physical size effect.

... At low confining pressure and room temperature, subcritical crack growth is influenced by the change of the degree of water saturation in the brittle regime such that both drying and wetting may both induce fractures. In the petroleum engineering industry, hydraulic fracturing (fracking) is widely applied to extract special forms of natural gas including coal seam gas, shale, and tight gas [130,200,299]. In enhanced geothermal systems (EGS), high-pressure water is injected into deep hot rock layers of very low permeability in order to enhance their permeability and, thus, improve the efficiency of the energy system. ...

... The material time derivative of an arbitrary vector quantity (•) with respect to the motion of ' α is mentioned to in the following as (•) α . Using v β with β ∈ {L, G} to denote the fluid velocities, the motion of the liquid and gas phases are expressed in Eulerian settings with reference to the solid motion [39,86,285,287,299,302,321] . Therefore, we apply the material time derivative with respect to the motion of the solid constituent as ...

... However, a modified anisotropic permeability tensor relation K frac is introduced in the phase-field fracture problem, where the Poiseuille's flow in the crack region, similarly to the flow between two parallel plates (i.e. Stokes' hypothesis is applied, see, [64,194,299,312]), is incorporated as a material law for the damaged porous media. This leads by an analytical solution of the Stokes equation to a modified Darcy's law for the flow in the crack region, commonly referred as the cubic law. ...

Fracture mechanics counts to the most emerging and promising fields of engineering mechanics. In the last few decades, the topics of crack initiation and propagation in solid and porous materials have attracted numerous theoretical, experimental, and numerical studies. This was driven by many challenges and necessities in engineering fields, such as the bad need for designing safe, reliable, and sustainable structures that withstand all types of expected natural and human actions, or the promising application of fracture tools in sectors like energy production, geothermal systems, soil science, and geotechnical engineering. From a mechanical and computational point of view, the fracturing of solid and porous materials presents a challenging multi-scale multi-phase problem, which includes possible several simultaneous physical processes and many sources of numerical instability. For a holistic understanding as well as efficient and accurate fracture modeling, the underlying monograph will address fracture mechanics and related processes across the scales, i.e. nanoscale, microscale, and macroscale. This includes, first, utilization of Molecular Dynamics (MD) simulations to understand fracture mechanism and conclude material parameters of brittle solid materials on the nanoscale, second, embedding the phase-field modeling (PFM) approach in continuum mechanics for fracture modeling on the macroscopic scale, and, third, embedding the PFM approach in continuum porous media mechanics (PM) to model hydraulic fracturing in saturated and unsaturated porous media, i.e. PM-PFM combined procedure. In conventional approaches in the mechanics of materials, such as in fracture mechanics, solid mechanics, or porous media mechanics, the constitutive modeling provides explicit mathematical expressions, which are based on phenomenological observations or experimental data. These models can further be subjected to hard constraints, such as the balance equations or the thermodynamics restrictions. To avoid the constitutive model's complexity and the increase of the number of required material parameters to an impractical level, these material models partially or entirely overlook microscopic information. This might lead, however, to deterioration of the model's accuracy, especially in the description of multi-scale and time- or path-dependent responses like in crystal plasticity or in nonlinear anisotropic porous media flow. This paves the way for the implementation of data-based artificial neural networks (ANN) to generate machine-learning (ML)-material models, which are capable to extract complex dependencies on micro-geometry and time or path dependencies without the need to explicitly determine the material parameters. Therefore, the fourth aim of the underlying monograph will be utilizing the capabilities of Machine Learning, via using deep neural networks (DNN) and deep reinforcement learning (DRL) to generate ML-based material models, which rely on microstructural information in the training datasets. The aforementioned approaches backed by powerful computational capacities and experimental data give the ability to reliably simulate and understand complicated real multi-phase and multi-scale problems out of solid and porous media mechanics.
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Kurzfassung:
Die Bruchmechanik zählt zu den aufstrebenden und vielversprechendsten Gebieten der Ingenieurmechanik. In den letzten Jahrzehnten wurden zahlreiche theoretische, experimentelle und numerische Studien zu den Themen Rissentstehung und -ausbreitung in festen und porösen Materialien durchgeführt. Dies wurde durch viele Herausforderungen und Notwendigkeiten in den Ingenieurbereichen vorangetrieben, wie z.B. die dringende Notwendigkeit, sichere, zuverlässige und nachhaltige Strukturen zu entwerfen, die allen Arten von erwarteten natürlichen und menschlichen Einwirkungen standhalten können, oder die vielversprechende Anwendung von Bruchmethoden in Bereichen wie Energieproduktion, geothermische Systeme, Bodenkunde und Geotechnik. Aus mechanischer und rechnerischer Sicht stellt das Bruchverhalten fester und poröser Materialien ein anspruchsvolles mehrskaliges Mehrphasenproblem dar, das mehrere mögliche gleichzeitige physikalische Prozesse und viele Quellen numerischer Instabilität umfasst. Um ein ganzheitliches Verständnis sowie eine effiziente und genaue Bruchmodellierung zu ermöglichen, wird sich die zugrundeliegende Monographie mit der Bruchmechanik und den damit verbundenen Prozessen auf allen Skalen, d.h. auf der Nano-, Mikro- und Makroskala, befassen. Dies beinhaltet erstens die Nutzung von Molekulardynamik-Simulationen (engl. Molecular Dynamics, Abk. MD) zum Verständnis des Bruchmechanismus und zur Bestimmung von Materialparametern spröder Festkörperwerkstoffe auf der Nanoskala, zweitens die Einbettung des Phasenfeld-Modellierungsansatzes (engl. Phase-Field Modeling, Abk. PFM) in die Kontinuumsmechanik für die Bruchmodellierung auf der makroskopischen Skala und drittens die Einbettung des PFM-Ansatzes in die Kontinuumsmechanik poröser Medien (engl. Porous Media, Abk. PM) zur Modellierung des hydraulischen Bruchs in gesättigten und ungesättigten porösen Medien, d.h. PM-PFM kombiniertes Verfahren. In konventionellen Ansätzen der Werkstoffmechanik, wie z.B. der Bruchmechanik, der Festkörpermechanik oder der Mechanik poröser Medien, liefert die konstitutive Modellierung explizite mathematische Ausdrücke, die auf phänomenologischen Beobachtungen oder experimentellen Daten beruhen. Diese Modelle können darüber hinaus strengen Einschränkungen unterworfen werden, wie z.B. den Bilanzgleichungen oder den thermodynamischen Einschränkungen. Um die Komplexität des konstitutiven Modells und die Erhöhung der Anzahl der erforderlichen Materialparameter auf ein unpraktisches Niveau zu vermeiden, übersehen diese Materialmodelle zum Teil oder ganz mikroskopische Informationen. Dies kann jedoch zu einer Verschlechterung der Genauigkeit des Modells führen, insbesondere bei der Beschreibung von mehrskaligen und zeit- oder pfadabhängigen Reaktionen wie bei Kristallplastizität oder bei der nichtlinearen anisotropen Strömung in porösen Medien. Dies ebnet den Weg für die Implementierung von datenbasierten künstlichen neuronalen Netzen (engl. Artificial Neural Networks, Abk. ANN) zur Erzeugung von maschinell lernenden (ML)-Materialmodellen, die in der Lage sind, komplexe Abhängigkeiten von Mikrogeometrie und Zeit- oder Pfadabhängigkeiten zu extrahieren, ohne dass die Materialparameter explizit bestimmt werden müssen. Daher wird das vierte Ziel der zugrundeliegenden Monographie darin bestehen, die Fähigkeiten des maschinellen Lernens zu nutzen, indem tiefe neuronale Netze (engl. Deep Neural Networks, Abk. DNN) und tiefes Verstärkungslernen (engl. Deep Reinforcement Learning, Abk. DRL) verwendet werden, um ML-basierte Materialmodelle zu generieren, die auf mikrostrukturellen Daten in den Trainingsdatensätzen beruhen. Die oben genannten Ansätze, unterstützt durch leistungsstarke Rechenkapazitäten und experimentelle Daten, ermöglichen es, komplizierte reale Mehrphasen- und Mehrskalenprobleme aus der Mechanik fester und poröser Medien zuverlässig zu simulieren und zu verstehen.

... Pressurized and fluid-filled fractures using phase-field modeling was subject in numerous papers in recent years. These studies range from mathematical modeling [28][29][30][31][32][33][34][35], mathematical analysis [36][37][38][39][40], numerical modeling and simulations [41][42][43][44][45][46][47][48][49][50][51][52][53], and up to (adaptive) global-local formulations [54] (see here in particular also [55] and [27] for non-pressurized studies) and high performance parallel computations [56,57]. Extensions towards multiphysics phase-field fracture in porous media were proposed in which various phenomena couple as for instance proppant [35], two-phase flow formulations [58] or given temperature variations [32]. ...

... Herein, i refers to the layer number within the domain, see Fig. 13b, with (β a , χ a ) and (β g , χ g ) are corresponding to the a and g, respectively, see (30) and (47). Thus, if ξ i > 1 means a is stiffer than g then crack orientation is in direction parallel to a. Otherwise, if ξ i < 1 means g is stiffer than a then crack orientation is in direction parallel to g. ...

... Thus, if ξ i > 1 means a is stiffer than g then crack orientation is in direction parallel to a. Otherwise, if ξ i < 1 means g is stiffer than a then crack orientation is in direction parallel to g. To formulate the fracture process, the stiffer fiber is set with a larger value of the penalty-like parameter in the (30) and (47). Therefore, in the following, we consider four different cases. ...

In this work, we employ a Bayesian inversion framework to fluid-filled phase-field fracture. We develop a robust and efficient numerical algorithm for hydraulic phase-field fracture toward transversely isotropic and orthotropy anisotropic fracture. In the fluid-driven coupled problem, three primary fields for pressure, displacements, and the crack phase-field are solved while direction-dependent responses (due to the preferred fiber orientation) are enforced via penalty-like parameters. A new crack driving state function is introduced by avoiding the compressible part of anisotropic energy to be degraded. Next, we use a successful extension of the anisotropic hydraulic phase-field fracture as a departure point for Bayesian inversion to estimate material parameters. To this end, we employ the delayed rejection adaptive Metropolis–Hastings (DRAM) algorithm to identify the parameters. The focus is on uncertainties arising from different variables, including elasticity modulus, Biot’s coefficient, Biot’s modulus, dynamic fluid viscosity and Griffith’s energy release rate in the case of the isotopic hydraulic fracture while in the anisotropic setting, we will have additional penalty-like parameters to be identified. Several numerical examples substantiate our algorithmic developments.

... Stokes' hypothesis is applied, cf. Miehe et al. [2015a], Wilson and Landis [2016], Wang and Sun [2017], Choo and Sun [2018]), is incorporated as a material law for the damaged porous media. This leads by an analytical solution of the Stokes equation to a modified Darcy's law for the flow in the crack region, commonly referred as the cubic law. ...

... This leads by an analytical solution of the Stokes equation to a modified Darcy's law for the flow in the crack region, commonly referred as the cubic law. Similarly to works on fully saturated porous media-phase-field fracture, such as in Miehe et al. [2015a], Wilson and Landis [2016], Wang and Sun [2017], Choo and Sun [2018], the intrinsic permeability tensor of a representative elementary volume can be partition into two components, K poro which governs the deformation-dependent bulk hydraulic responses that remains isotropic and K frac , a rank-one tensor that represents the crack opening and closure both as a fluid conduit for the plane orthogonal to the normal vector of the crack, i.e. ...

... Chukwudozie et al. [2019]) to introducing phenomenological relations to introduce an effective anisotropic permeability model based on Eq. (21) (cf. Wang and Sun [2017], , Heider et al. [2018]). While both approaches may capture the anisotropic nature of the fluid conduit effect, the validity of the former approach can be viewed as questionable, as this implies the removal of solid mass in the damaged zone. ...

This manuscript introduces a unified mathematical framework to replicate both desiccation-induced and hydraulic fracturing in low-permeable unsaturated porous materials observed in experiments. The unsaturated porous medium is considered as a three-phase solid-liquid-gas effective medium of which each constituent occupies a fraction of the representative elementary volume. As such, an energy-minimization-based phase-field model (PFM) is formulated along with the Biot's poroelasticity theory to replicate the sub-critical crack growth in the brittle regime. Unlike hydraulic fracturing where the excess pore liquid pressure plays an important role at the onset and propagation of cracks, desiccation cracks are mainly driven by deformation induced by water retention. Therefore, the wettability of the solid skeleton may affect the evolution of the capillary pressure (suction) and change the path-dependent responses of the porous media. This air-water-solid interaction may either hinder or enhance the cracking occurrence. This difference of capillary effect on crack growth during wetting and drying is replicated by introducing retention-sensitive degradation mechanisms in our phase field fracture approach. To replicate the hydraulic behaviors of the pore space inside the host matrix and that of the cracks, the path-dependent changes of the intrinsic permeability due to crack growth and porosity changes are introduced to model the flow conduit in open and closed cracks. Numerical examples of drying-induced and hydraulic fracturing demonstrate the capability of the proposed model to capture different fracture patterns, which qualitatively agrees with the fracture mechanisms of related experiments documented in the literature.

... An interesting aspect of this variational formalism is that multiple ways are available to introduce regularization for the fracture energy functional (Schmidt et al. 2009;Borden et al. 2012;Geelen et al. 2018). By taking different forms of the nonlocal fracture energy functional, mesh size independence can be achieved, as shown in the cases of the phase-field model and the eigenfracture scheme, see Schmidt et al. (2009), Pandolfi andOrtiz (2012), Bourdin et al. (2008), Wang and Sun (2017), Choo and Sun (2018), Bryant and Sun (2018), Na and Sun (2018). In both cases, regularization is provided by introducing a finite length scale in the expression of the gradient or nonlocal integral of the damage parameter, a technique well known in classical damage mechanics (Bažant and Jirásek 2002;Liu et al. 2016;de Borst and Verhoosel 2016). ...

... Gravouil et al. 2002;Sun et al. 2011a, b). However, in the family of variational fracture models, including phase-field fracture and the thick levelset fracture approaches, damage is smeared around the cracks rather than localized (Bourdin et al. 2008;Wang and Sun 2017;Cazes and Moës 2015;Navas et al. 2018). As a result, enrichment functions do not need to be introduced to capture the strong discontinuity. ...

... Pandolfi and Ortiz 2012;Stochino et al. 2017). As the size of the -neighbourhood is predefined and independent of the mesh size, the energy dissipated due to crack growth is directly associated with the size of the element and, hence, the resultant simulations are less sensitive to the mesh size than the classic element-deletion models (Wang and Sun 2017;Song et al. 2008). ...

This article introduces and compares mesh r-and h-adaptivity for the eigenfrac-ure model originally proposed in [1, 2], with the goal of suppressing potential mesh bias due to the element deletion. In the r-adaptive approach, we compute the configurational force at each incremental step and move nodes near the crack tip parallel to the normalized configurational forces field such that the crack propagation direction can be captured more accurately within each incremental step. In the h-adaptive approach, we introduce mesh refinement via a quad-tree algorithm to introduce more degrees of freedom within the nonlocal-neighborhood such that a more refined crack path can be reproduced with a higher mesh resolution. Our numerical examples indicate that the r-adaptive approach is able to replicate curved cracks and complex geometrical features, whereas the h-adaptive approach is advantageous in simulating sub-scale fracture when the nonlocal regions are smaller than the unrefined coarse mesh.

... A future extension may include distinction between the tensile and compression degradation mechanism as those in Gültekin, Dal, and Holzapfel [127], where a different degradation function is applied on the compressive stress and the compressive strain energy to reflect the difference in load-bearing capacities. Meanwhile, Wang and Sun [136] has associated the degradation of compression as a result of anti-crack propagation that could be triggered by a higher anti-crack fracture energy to examine the propagation of compaction band. These extensions will be considered in future study but is out of the scope of the current work. ...

... However, this naive approach may lead to a vanishing gradient field inside the damaged zone if the length scale parameter of the phase field is sufficiently larger than the mesh size [136,212]. While this issue can be suppressed by using a small length scale parameter respect to the mesh size, the reduction of the ratio between the length scale and mesh size may make it difficult for the solver to recover the sharp gradient of phase field. ...

We present a computational framework for modeling geomaterials undergoing failure in the brittle and ductile regimes. This computational framework introduces anisotropic gradient regularization to replicate a wide spectrum of size-dependent anisotropic constitutive responses exhibited in layered and sedimentary rock. Relevant subsurface applications include oil/gas wellbore completions, caprock evaluation for carbon sequestration in saline aquifers, and geothermal energy recovery. Considered failure modes are mixed-mode fracture, shear band formation due to plastic strain localization, and rate-dependent frictional slip along the propagated fracture's rock surface, subsequent to fracture closure.
Our nonlocal modeling framework extends the state-of-the-art gradient-enhanced plasticity and damage mechanics for frictional materials with a special treatment that injects bias for the regularization for different orientations. A novel contribution is that the formulations not only contains a regularization, but that the regularization also provides a method to introduce size-dependent anisotropies. Consequently, this treatment provides a new means to create non-associative flow via a variational framework while introducing different anisotropic responses for specimens of different sizes (introduced in Chapter 1). These anisotropic regularization modeling techniques are then applied to three classes of common geomechanics problems: critical state plasticity of clay and shale rock (Chapter 2), brittle fracture of rock (Chapter 3), and the plastic slip of interfaces and cracks (Chapter 4).
This combination, of established rock physics, local anisotropy, and size-dependent anisotropy enfranchised with diffusive regularization, is investigated. For instance, experimentation on uniaxially compressed specimens failing in the brittle regime reveals a repeatable typology of wing- and coalescent-crack patterns, broadly taken to indicate a mixed-mode fracture phenomenon particular to rock-like materials. In the ductile regime, biaxially compressed shale rock displays orientation-dependence of the plastic deformation difficult to capture merely by attributing anisotropy to the elastic response, with localization at or near the critical state. We numerically capture both these phenomena. Verification and/or validation is provided for proposed constitutive relations.

... To this end, we substitute the results of Eqs. (70) and (71) into D f in Eq. (66), which is the energy dissipation per unit volume. Integrating the resulting dissipation density over the domain Ω with the crack surface Γ d gives ...

... Given these considerations, here we take an approach that augments an anisotropic permeability tensor describing Poiseuille flow to the absolute permeability tensor in Eq. (48). Such an approach has been advocated by Miehe and Mauthe [40,49] and Wang and Sun [70], among others. We now write the absolute permeability tensor as ...

Cracking and damage from crystallization of minerals in pores center on a wide range of problems, from weathering and deterioration of structures to storage of CO2 via in situ carbonation. Here we develop a theoretical and computational framework for modeling these crystallization-induced deformation and fracture in infiltrated porous materials. Conservation laws are formulated for coupled chemo-hydro-mechanical processes in a multiphase material composed of the solid matrix, liquid solution, gas, and crystals. We then derive an expression for the effective stress tensor that is energy-conjugate to the strain rate of a porous material containing crystals growing in pores. This form of effective stress incorporates the excess pore pressure exerted by crystal growth—the crystallization pressure—which has been recognized as the direct cause of deformation and fracture during crystallization in pores. Continuum thermodynamics is further exploited to formalize a constitutive framework for porous media subject to crystal growth. The chemo-hydro-mechanical model is then coupled with a phase-field approach to fracture which enables simulation of complex fractures without explicitly tracking their geometry. For robust and efficient solution of the initial-boundary value problem at hand, we utilize a combination of finite element and finite volume methods and devise a block-partitioned preconditioning strategy. Through numerical examples we demonstrate the capability of the proposed framework for simulating complex interactions among unsaturated flow, crystallization kinetics, and cracking in the solid matrix.

... One of the most important but 4 often overlooked aspects in fracture modeling is the effect of internal micro-structure on 5 crack propagation. On the contrary, materials exhibiting size effects due to the internal 6 micro-structures are often manufactured to attain certain desirable engineering properties 7 [1,2]. The classical continuum mechanics assumes continuous distribution of matter over Ω = Ω ∪ ∂Ω. ...

Realizing the limitations of classical phase-field fracture models and the need for the development of non-local phase-field models that can capture the experimentally observed size effects in brittle and quasi-brittle materials, we propose thermodynamically consistent volumetric-deviatoric and spectral decomposition-based phase-field models (PFMs) for brittle fracture in micro-polar continua. We have provided open-source finite element (FE) implementation of the proposed micro-polar phase-field models using Gridap in Julia. For developing the proposed models, we have considered the micro-rotation as an additional kinematic descriptor, and have derived the balance equations by invoking the virtual power principle. On satisfying the thermodynamic laws, we have determined the constitutive relations for the thermodynamic fluxes and demonstrated that any dissipative effect can easily be incorporated into the proposed models. We have also carried out a comparative study for the proposed volumetric-deviatoric and spectral decomposition-based micro-polar phase-field models to investigate their advantages over one another in certain conditions. We have considered a set of numerical examples with varying micro-polar material constants to demonstrate the capability and the limitations of the developed phase-field models.

... Although the Griffith criterion is presented in a generalized form in Equation 4, its numerical implementation is not straightforward because of the discontinuous and evolving crack set Γ. Among available methods (Schmidt et al., 2009;Wang & Sun, 2017), a regularization method via Gamma-convergence (Bourdin et al., 2000; Figure B2. Comparisons of the computation of an energy release rates with results from the solutions of (a) He and Hutchinson (1989) and (b) Sneddon and Lowengrub (1969 Chambolle, 2004) has become the standard implementation method for Equation 4 in the last decade (e.g., Ambati et al., 2014;Borden et al., 2012;Freddi & Royer-Carfagni, 2010) and is now typically referred to as the phase-field model. ...

Plain Language Summary
Hydraulic fractures may form complex patterns as they grow outward from a wellbore by turning or deflecting when they interact with preexisting discontinuities in rocks. Because complex fractures enhance the permeability of rock formations more effectively than planar fractures, many studies have investigated how a fracture interacts with a preexisting discontinuity such as a natural fracture. The fate of a growing fracture at a discontinuity—whether it penetrates or deflects—is typically judged based on the in situ subsurface stress, and the characteristics of the discontinuity. However, we observed in experiments that fractures deflected more often at discontinuities (grain boundaries) as they propagated farther away from the wellbore, which cannot be explained by the conventional criteria. To explain these observations, we analyzed the energy expenditure of a deflecting fracture and showed that it becomes energetically more favorable for a fracture to deflect at a discontinuity as it grows farther away from the wellbore. We confirmed this insight by using numerical simulations. We thus caution that the conventional criteria may not be applicable in the near wellbore region, and we suggest that energetic stability, rather than the local stress at the fracture tip, should be analyzed to determine fracture paths.

... Additionally, the material time derivative of an arbitrary vector quantity (•) with respect to the motion of is expressed as (•) ′ ,. One example is the description of the fluid velocity " where the motion of the fluid phases is expressed in Eulerian configuration with reference to the solid motion 78,[83][84][85] . Hence, the material time derivative is applied with respect to the motion of the solid constituent as ...

In this paper, we present a reliable micro‐to‐macroscale framework to model multiphase fluid flow through fractured porous media. This is based on utilizing the capabilities of the lattice Boltzmann method (LBM) within the phase‐field modeling (PFM) of fractures in multiphase porous media. In this, we propose new physically motivated phase‐field‐dependent relationships for the residual saturation, the intrinsic as well as relative permeabilities. In addition, an anisotropic, phase‐field‐dependent intrinsic permeability tensor for the fractured porous domains is formulated, which relies on the single‐ and multiphasic LBM flow simulations. Based on these results, new relationships for the variation of the macroscopic theory of porous media (TPM)–PFM model parameters in the transition zone are proposed. Whereby, a multiscale concept for the coupling between the multiphasic flow through the crack on one hand and the porous ambient, on the other hand, is achieved. The hybrid model is numerically applied on a real microgeometry of fractured porous media, extracted via X‐ray microcomputed tomography data of fractured Berea Sandstone. Moreover, the model is utilized for the calculation of the fluid leak‐off from the crack to the intact zones. Additionally, the effects of the depth of the transition zone and the orientation of the crack channels on the amount of leakage flow rates are studied. The outcomes of the numerical model proved the reliability of the multiscale model to simulate multiphasic fluid flow through fractured porous media.

... First and most importantly, both PD and PF are appearing to be the most prominent approaches for free fracture modeling. Other notable approaches are the displacement-discontinuity method [19], cohesive-zone models [20], boundary elements [21], XFEM/GFEM [22][23][24], and the eigen-erosion framework [25][26][27][28]. A comparative review between XFEM (extended finite elements), mixed FEM, and phase-field models appeared in [29]. ...

Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discretization schemes, and numerical analysis are briefly summarized for both models. Validation against experimental data is essential for all computational methods to demonstrate predictive accuracy. To that end, the Sandia Fracture Challenge and similar experimental data sets where both models could be benchmarked against are showcased. Emphasis is made to converge on common metrics for the evaluation of these two fracture modeling approaches. Both PD and PF models are assessed in terms of their computational effort and predictive capabilities, with their relative advantages and challenges are summarized.

... In order to model the fracture flow in a fluid-infiltrating porous media, we adopt the permeability enhancement approach that approximates the water flow inside the fracture as the flow between two parallel plates [Miehe and Mauthe, 2016, Mauthe and Miehe, 2017, Wang and Sun, 2017, Suh and Sun, 2021b: ...

This article presents a multi-phase-field poromechanics model that simulates the growth and thaw of ice lenses and the resultant frozen heave and thaw settlement in multi-constituent frozen soils. In this model, the growth of segregated ice inside the freezing-induced fracture is implicitly represented by the evolution of two phase fields that indicate the locations of segregated ice and the damaged zone, respectively. The evolution of two phase fields are driven by the driving forces that capture the physical mechanisms of ice and crack growths respectively, while the phase field governing equations are coupled with the balance laws such that the coupling among heat transfer, solid deformation, fluid diffusion, crack growth, and phase transition can be observed numerically. Unlike phenomenological approaches that indirectly captures the freezing influence on the shear strength, the multi-phase-field model introduces an immersed approach where both the homogeneous freezing and the ice lens growth are distinctively captured by the freezing characteristic function and the driving force accordingly. Verification and validation examples are provided to demonstrate the capacities of the proposed models.

... In recent years, several pressurized [69,242,240,313,162,163,290,257] and fluid-filled [239,317,196,235,234,115,160,12,200,310,198,75,199,88,161,318,15] phase-field fracture formulations have been proposed in the literature. These studies range from modeling of pressurized and fluid-filled fractures, mathematical analysis, numerical modeling and simulations up to high-performance parallel computations. ...

The underlying Habilitation aims to contribute to the research on fracture mechanics of solids across the scales. This active research field is driven by the investigation and development of new methods, processes and technologies applicable to engineering problems with complex material behavior of solids at fracture. It includes mathematically precise formulations of theoretical and computational models with emphasis on continuum physics as well as the development of variation methods and efficient numerical implementations tools. In particular, two directions will be considered in this contribution: (i) the construction of advanced multi-scale techniques and (ii) modern element technologies. On the multi-scale techniques, a robust and efficient Global-Local approach for numerically solving fracture-mechanics problems is developed in the first part of this contribution. This method has the potential to tackle practical field problems in which a large-structure might be considered and fracture propagation is a localized phenomenum. In this regard, failure is analyzed on a lower (Local) scale, while dealing with a purely linear problem on an upper (Global) scale. The modeling of crack formation at the Local scale is achieved in a convenient way by continuum phase-field formulations to fracture, which are based on the regularization of sharp crack discontinuities. For this purpose, a predictor-corrector scheme is designed in which the local domains are dynamically updated during the computation. To cope with different element discretizations at the interface between the two nested scales, a non-matching dual mortar method is formulated. Hence, more regularity is achieved on the interface. The development of advanced discretization schemes accounting for meshes with highly irregular shaped elements and arbitrary number of nodes is the main focus in the second part of this work. To this end, a relatively new method - the virtual element method (VEM) - will be presented here that leads to an exceptional efficient and stable formulation for solving a wide range of boundary value problems in science and engineering. The structure of VEM comprises a term in the weak formulation or the potential density functional in which the unknowns, being sought are replaced by their projection onto a polynomial space. This results in a rank-deficient structure, therefore it is necessary to add a stabilization term to the formulation. The performance of the virtual element method is comparable to using finite elements of higher order. It is even more robust than FEM in case of a severe distortion of the element.

... To handle the geometrical nonlinearity during fracture, Moutsanidis et al. (2019) incorporate the phase field fracture model in a material point method (MPM). Meanwhile, Zhang et al. (2020) enhance the MPM with eigenerosion (Pandolfi and Ortiz, 2012;Li et al., 2015;Wang and Sun, 2017b;Qinami et al., 2019) to simulate dynamic fracturing. Recently, Homel and Herbold (2017) employ a damage scalar field to present the fractures in the material point method and use this scalar field to detect contacts. ...

We propose a material point method (MPM) to model the evolving multi-body contacts due to crack growth and fragmentation of thermo-elastic bodies. By representing particle interface with an implicit function, we adopt the gradient partition techniques introduced by Homel and Herbold 2017 to identify the separation between a pair of distinct material surfaces. This treatment allows us to replicate the frictional heating of the evolving interfaces and predict the energy dissipation more precisely in the fragmentation
process. By storing the temperature at material points, the resultant MPM model captures the thermal advection-diffusion in a Lagrangian frame during the fragmentation, which in return affects the structural heating and dissipation across the frictional interfaces. The resultant model is capable of replicating the crack growth and fragmentation without requiring dynamic adaptation of data structures or insertion of interface elements.
A staggered algorithm is adopted to integrate the displacement and temperature sequentially. Numerical experiments are employed to validate the diffusion between the
thermal contact, the multi-body contact interactions and demonstrate how these thermo-mechanical processes affect the path-dependent behaviors of the multi-body systems.

... prescribed defects, in the model and cracks and fractures initiated over time. Another notable approach with the same feature is the eigen-erosion framework [309][310][311][312]. ...

Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discretization schemes, and numerical analysis are briefly summarized for both models. Validation against experimental data is essential for all computational methods to demonstrate predictive accuracy. To that end, The Sandia Fracture Challenge and similar experimental data sets where both models could be benchmarked against are showcased. Emphasis is made to converge on common metrics for the evaluation of these two fracture modeling approaches. Both PD and PF models are assessed in terms of their computational effort and predictive capabilities with their relative advantages and challenges are summarized.

... In this model, bulk and shear moduli are related to diagonal terms of Hessian matrix, 2 nd derivative of energy functional with respect to volumetric and shear strains, with the following relations: Figure 24 plots bulk and shear moduli as exponential functions fitted to curves provided in (Andrä et al., 2013a), see Fig. 2(a) and Fig. 4(a) in the reference paper. These porosity dependent modului can be related to volumetric strain by φ = (1 + v )φ 0 for the small deformation analysis (Wang and Sun, 2017). Therefore, for each computational element with a distinct initial porosity φ 0 we are able to calibrate model parameters p 0 , c r , c mu , and v0 to match bulk and shear moduli Eqs. ...

We present a hybrid model/model-free data-driven approach to solve poroelasticity problems. Extending the data-driven modeling framework originated from Kirchdoerfer and Ortiz (2016), we introduce one model-free and two hybrid model-based/data-driven formulations capable of simulating the coupled diffusion-deformation of fluid-infiltrating porous media with different amounts of available data. To improve the efficiency of the model-free data search, we introduce a distance-minimized algorithm accelerated by a k-dimensional tree search. To handle the different fidelities of the solid elasticity and fluid hydraulic constitutive responses, we introduce a hybridized model in which either the solid and the fluid solver can switch from a model-based to a model-free approach depending on the availability and the properties of the data. Numerical experiments are designed to verify the implementation and compare the performance of the proposed model to other alternatives.

... Next, further improvements of the present research are exposed. First, the eigensoftening approach can be extrapolated to rock fracture in the same manner the eigenerosion was [36], taking into account that the volumetric stiffness remains in order to support bulk loads. Secondly, the algorithm would have the ability to simulate of crack patterns in composite material such as steel or carbon fibre reinforced concretes by means of empirical definitions of curve [29]. ...

This work aims to introduce an alternative technique to address the fracture process of brittle and quasi-brittle materials under the Material Point Method (MPM) framework. With this purpose the eigensoftening algorithm, developed originally for the Optimal Transportation Meshfree (OTM) approximation scheme, is extended to the MPM to present a suitable alternative to the existing fracture algorithms developed for such a methodology. The good fitting in the predictions made by the eigensoftening algorithm against both analytical and experimental results proves the excellent performance of the method when challenging applications are to model.

... Particularly, two different solvers have been implemented to perform the time-space integration: the first considers Taylor-Hood elements together with an implicit Euler scheme, the second adopts equal-order elements and a semi-explicit-implicit scheme. A stabilization procedure based on the pressure projection method is used to circumvent the lack of inf-sup condition and resolve the sharp pore pressure gradient [29][30][31][32][33][34]. Comparisons are performed by taking into account literature examples and efficiency features are analyzed. ...

The triggering and spreading of volumetric waves in soils, namely pressure (P) and shear (S) waves, developing from a point source of a dynamic load, are analyzed. Wave polarization and shear wave splitting are innovatively reproduced via a three-dimensional Finite Element research code upgraded to account for fast dynamic regimes in fully saturated porous media. The mathematical-numerical model adopts a u-v-p formulation enhanced by introducing Taylor-Hood mixed finite elements and the stability features of the solution are considered by analyzing different implemented time integration strategies. Particularly, the phenomena have been studied and reconstructed by numerically generating different types of medium anisotropy accounting for (i) an anisotropic solid skeleton, (ii) an anisotropic permeability tensor, and (iii) a Biot's effective stress coefficient tensor. Additionally, deviatoric-volumetric coupling effects have been emphasized by specifically modifying the structural anisotropy. A series of analyses are conducted to validate the model and prove the effectiveness of the results, from the directionality of polarized vibrations, the anisotropy-induced splitting, up to the spreading of surface waves.

... In recent years, several pressurized [1][2][3][4][5][6][7][8] and fluid-filled [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] phase-field formulations within hydraulic fracturing setting have been proposed in the literature. These studies range from modeling of pressurized and fluid-filled fractures, mathematical analysis, numerical modeling and simulations up to high-performance parallel computations. ...

In this work, phase-field modeling of hydraulic fractures in porous media is extended towards a Global–Local approach. Therein, the failure behavior is solely analyzed in a (small) local domain. In the surrounding medium, a simplified and linearized system of equations is solved. Both domains are coupled with Robin-type interface conditions. The fractures inside the local domain are allowed to propagate and consequently, both subdomains change within time. Here, a predictor–corrector strategy is adopted, in which the local domain is dynamically adjusted to the current fracture pattern. The resulting framework is algorithmically described in detail and substantiated with some numerical tests.

... Pressurized and fluid-filled fractures using phase-field modeling was subject in numerous papers in recent years. These studies range from mathematical modeling [16,17,18,19,20,21,22,23], mathematical analysis [24,25,26,27,28], numerical modeling and simulations [29,30,31,32,33,34,35,36,37,38,39,40], and up to (adaptive) global-local formulations [41] (see here in particular also [42] and [15] for non-pressurized studies) and high performance parallel computations [43,44]. Extensions towards multiphysics phase-field fracture in porous media were proposed in which various phenomena couple as for instance proppant [23], two-phase flow formulations [45] or given temperature variations [20]. ...

In this work, a Bayesian inversion framework for hydraulic phase-field transversely isotropic and orthotropy anisotropic fracture is proposed. Therein, three primary fields are pressure, displacements, and phase-field while direction-dependent responses are enforced (via penalty-like parameters). A new crack driving state function is introduced by avoiding the compressible part of anisotropic energy to be degraded. For the Bayesian inversion, we employ the delayed rejection adaptive Metropolis (DRAM) algorithm to identify the parameters. We adjust the algorithm to estimate parameters according to a hydraulic fracture observation, i.e., the maximum pressure. The focus is on uncertainties arising from different variables, including elasticity modulus, Biot's coefficient, Biot's modulus, dynamic fluid viscosity, and Griffith's energy release rate in the case of the isotropic hydraulic fracture while in the anisotropic setting, we identify additional penalty-like parameters. Several numerical examples are employed to substantiate our algorithmic developments.

... In recent years, several pressurized [11,52,51,61,37,38,58,55] and fluid-filled [50,65,39,48,47,24,35,36,43,60,41,15,42,16,66,6] phase-field fracture formulations have been proposed in the literature. These studies range from modeling of pressurized and fluid-filled fractures, mathematical analysis, numerical modeling and simulations up to high-performance parallel computations. ...

In this work, phase-field modeling of hydraulic fractures in porous media is extended towards a global-local approach. Therein, the failure behavior is solely analyzed in a (small) local domain. In the surrounding medium, a simplified and linearized system of equations is solved. Both domains are coupled by Robin-type interface conditions. The fracture(s) inside the local domain are allowed to propagate and consequently both subdomains change within time. Here, a predictor-corrector strategy is adopted in which the local domain is dynamically adjusted to the current fracture pattern. The resulting framework is algorithmically described in detail and substantiated with some numerical tests.

... The size effect and the corresponding length scale parameter associated with the phase field fracture model for brittle or quasi-brittle materials have been a subject of intensive research in recent years [Francfort and Marigo, 1998, Bourdin et al., 2008, de Borst and Verhoosel, 2016, Wang and Sun, 2017, Aldakheel et al., 2018, Wu and Nguyen, 2018, Geelen et al., 2018, Bryant and Sun, 2018, Choo and Sun, 2018b, Qinami et al., 2019. Due to the fact that the phase field approach employ regularized (smoothed) implicit function to represent sharp interface, the physical interpretation of the length scale parameter (and in some cases, the lack thereof) has become a hotly debated topic among the computational fracture mechanics community. ...

While crack nucleation and propagation in the brittle or quasi-brittle regime can be predicted via variational or material-force-based phase field fracture models, these models often assume that the underlying elastic response of the material is non-polar and yet a length scale parameter must be introduced to enable the sharp cracks represented by a regularized implicit function. However, many materials with internal microstructures that contain surface tension, micro-cracks, micro-fracture, inclusion, cavity or those of particulate nature often exhibit size-dependent behaviors in both the path-independent and path-dependent regimes. This paper is intended to introduce a unified treatment that captures the size effect of the materials in both elastic and damaged states. By introducing a cohesive micropolar phase field fracture theory, along with the computational model and validation exercises, we explore the interacting size-dependent elastic deformation and fracture mechanisms exhibits in materials of complex microstructures. To achieve this goal, we introduce the distinctive degradation functions of the force-stress-strain and couple-stress-micro-rotation energy-conjugated pairs for a given regularization profile such that the macroscopic size-dependent responses of the micropolar continua is insensitive to the length scale parameter of the regularized interface. Then, we apply the variational principle to derive governing equations from the micropolar stored energy and dissipative functionals. Numerical examples are introduced to demonstrate the proper way to identify material parameters and the capacity of the new formulation to simulate complex crack patterns in the quasi-static regime.

... The phase field or other nonlocal variation of variational fracture models (e.g. eigenfracture [Schmidt et al., 2009, Wang and Sun, 2017, Qinami et al., 2019 and thick level set [Moës et al., 2011, Cazes andMoës, 2015]) provide an simple treatment to introduce size effect for the damage mechanics. By employing the smooth implicit function to represent sharp crack interface, this family of models regularize the sharp crack surfaces by diffusive damage zones and therefore bypass the need to embed strong discontinuities [Bourdin et al., 2008, Kuhn and Müller, 2010, Miehe et al., 2010a,b, Hofacker and Miehe, 2013, Choo and Sun, 2018, Bryant and Sun, 2018. ...

A micropolar phase field fracture model is implemented in an open-source library FEniCS. This implementation is based on the theoretical study in Suh et al. [2020] in which the resultant phase field model exhibits the consistent micropolar size effect in both elastic and damage regions identifiable via inverse problems for micropolar continua. By leveraging the automatic code generation technique in FEniCS, we provide documentation of the source code expressed in a language very close to the mathematical expressions without comprising significant efficiency. This combination of generality and interpretability, therefore, enables us to provide a detailed walk-through that connects the implementation with the regularized damage theory for micropolar materials. By making the source code open source, the paper will provide an efficient development and educational tool for third-party verification and validation, as well as for future development of other higher-order continuum damage models.

... Practically, it can be considered as a regularised non-local element-erosion method which does not involve extra equations with auxiliary fields (like the phase-field). An elaborate presentation of this eigen-erosion was reported in Wang and Sun [2017] for poromechanics problems. Even though the method has not yet been studied as extensively as the Bourdin et al. [2000] model, impressive results for problems involving fragmentation, contacts and phase transition were obtained [Pandolfi et al., 2014, Li et al., 2015a. ...

Fracture is one of the most commonly encountered failure modes of engineering materials and structures. Prevention of cracking-induced failure is, therefore, a major constraint in structural designs. Computational modelling of fracture constitutes an indispensable tool not only to predict the failure of cracking structures but also to shed insights into understanding the fracture processes of many materials such as concrete, rock, ceramic, metals and biological soft tissues. This manuscript provides an extensive overview of the literature on the so-called phase field fracture (PFF) models, particularly, for quasi-static and dynamic fracture of brittle and quasi-brittle materials, from the points of view of a computational mechanician. PFF models are the regularised versions of the variational approach to fracture which generalises Griffith's theory for brittle fracture. They can handle topologically complex fractures such as intersecting and branching cracks in both two and three dimensions with a quite straightforward implementation. One of our aims is to justify the gaining popularity of PFF models. To this end, both theoretical and computational aspects are discussed and extensive benchmark problems (for quasi-static and dynamic brittle/cohesive fracture) that are successfully and unsuccessfully solved with PFF models are presented. Unresolved issues for further investigations are also documented.

... Constitutive responses of interfaces are important for a wide spectrum of problems that involve spatial domain with embedded strong discontinuity, such as fracture surfaces [Rice, 1968, Park et al., 2009, Wang and Sun, 2017, Bryant and Sun, 2018, slip lines [Rabczuk andAreias, 2006, Borja andFoster, 2007], joints [Elices et al., 2002] and faults [Ohnaka and Yamashita, 1989, Wang and Sun, 2018, Sun and Wong, 2018. While earlier modeling efforts, in particular those involving the modeling of cohesive zones , often solely focus on mode I kinematics, the mixed mode predictions of tractionseparation law relations are critical for numerous applications, ranging from predicting damage upon impacts [Ortiz and Pandolfi, 1999], to predicting seismic events [Rudnicki, 1980]. ...

This paper presents a new meta-modeling framework to employ deep reinforcement learning (DRL) to generate mechanical constitutive models for interfaces. The constitutive models are conceptualized as information flow in directed graphs. The process of writing constitutive models are simplified as a sequence of forming graph edges with the goal of maximizing the model score (a function of accuracy, robustness and forward prediction quality). Thus meta-modeling can be formulated as a Markov decision process with well-defined states, actions, rules, objective functions, and rewards. By using neural networks to estimate policies and state values, the computer agent is able to efficiently self-improve the constitutive model it generated through self-playing, in the same way AlphaGo Zero (the algorithm that outplayed the world champion in the game of Go)improves its gameplay. Our numerical examples show that this automated meta-modeling framework not only produces models which outperform existing cohesive models on benchmark traction-separation data but is also capable of detecting hidden mechanisms among micro-structural features and incorporating them in constitutive models to improve the forward prediction accuracy, which are difficult tasks to do manually.

... A future extension may include distinction between the tensile and compression degradation mechanism as those in Gültekin et al. [2016], where a different degradation function is applied on the compressive stress and the compressive strain energy to reflect the difference in load-bearing capacities. Meanwhile, Wang and Sun [2017] has associated the degradation of compression as a result of anti-crack propagation that could be triggered by a higher anti-crack fracture energy to examine the propagation of compaction band. These extensions will be considered in future study but is out of the scope of the current work. ...

Under a pure tensile loading, cracks in brittle, isotropic, and homogeneous materials often propagate such that pure mode I kinematics are maintained at the crack tip. However, experiments performed on geo-materials, such as sedimentary rock, shale, mudstone, concrete and gypsum, often lead to the conclusion that the mode I and mode II critical fracture energies/surface energy release rates are distinctive. This distinction has great influences on the formation and propagation of wing cracks and secondary cracks from pre-existing flaws under a combination of shear and tensile or shear and compressive loadings. To capture the mixed-mode fracture propagation, a mixed-mode I/II fracture model that employs multiple critical energy release rates based on Shen and Stephansson, IJRMMS, 1993 is reformulated in a regular-ized phase field fracture framework. We obtain the mixed-mode driving force of the damage phase field by balancing the microforce. Meanwhile, the crack propagation direction and the corresponding kinematics modes are determined via a local fracture dissipation maximization problem. Several numerical examples that demonstrate the mode II and mixed-mode crack propagation in brittle materials are presented. Possible extensions of the model capturing degradation related to shear/compressive damage, as commonly observed in sub-surface applications and triaxial compression tests, are also discussed.

... As such, the work done on the salt grain may lead to increase in the elastic strain energy or can be dissipated via crack growth or plastic flow. Previously, small and finite strain models that couple phase field fracture and cap-plasticity have been proposed to capture the brittle-ductile transition in isotropic materials [1,2,6]. Meanwhile, relationship between effective critical energy release rate and the spatial heterogeneity is explored in Na et al. [4]. ...

... Practically, it can be considered as a regularised non-local element-erosion method which does not involve extra equations with auxiliary fields (like the phase field). An elaborated presentation of this eigen-erosion was reported in Wang and Sun [2017] for poromechanics problems. Even though the method has not studied as extensively as the Bourdin et al. [2000] model, impressive results for problems involving fragmentation, contacts, phase transition were obtained [Li et al., 2012, Pandolfi et al., 2014. ...

Fracture is one of the most commonly encountered failure modes of engineering materials and structures. Prevention of cracking-induced failure is, therefore, a major constraint in structural designs. Computational modelling of fracture constitutes an indispensable tool not only to predict the failure of cracking structures but also to shed insights into understanding the fracture processes of many materials such as concrete, rock, ceramic, metals and biological soft tissues. This manuscript provides an extensive overview of the literature on the so-called phase field fracture (PFF) models, particularly, for quasi-static and dynamic fracture of brittle and quasi-brittle materials, from the points of view of a computational mechanician. PFF models are the regularised versions of the variational approach to fracture which generalises Griffith's theory for brittle fracture. They can handle topologically complex fractures such as intersecting and branching cracks in both two and three dimensions with a quite straightforward implementation. One of our aims is to justify the gaining popularity of PFF models. To this end, both theoretical and computational aspects are discussed and extensive benchmark problems (for quasi-static and dynamic brittle/cohesive fracture) that are successfully and unsuccessfully solved with PFF models are presented. Unresolved issues for further investigations are also documented.

... The EG discretization of the fluid flow equation can also help address more complex poromechanical problems whereby the permeability field evolves dramatically with respect to anisotropy as well as magnitudes. Notable problems involving anisotropic permeability evolution include fluid-driven fracture in porous media (e.g., [93][94][95][96][97][98][99][100]). Applications of the EG finite element method to such challenging poromechanical problems will be reported in future publications. ...

... Furthermore, the viscous dissipation of fluid flow leads to a reduction of pressure as the distance increases from the source so that the fractures close to the fluid source tend to grow faster [79]. Understanding this twoway coupled hydromechanical effect of fluid-driven fracture is important for hydrocarbon recovery and the related geological applications, such as carbon dioxide storage [14,22,39,59,65,68,73,74]. ...

We present a numerical analysis on injection-induced crack propagation and coalescence
in brittle rock. The DEM-network coupling model in PFC is modified to consider the evolution of fracture geometry. An improved fluid flow model for fractured porous media is proposed and coupled with a bond-based DEM model to simulate the interactions among cracks induced by injecting fluid in two nearby flaws at identical injection rates are investigated numerically. The material parameters are calibrated based on the macro-properties of Lac du Bonnet granite and KGD solution. A grain- based model, which generates larger grains from assembles of particles bonded together, is calibrated to identify the microscopic mechanical and hydraulic parameters of Lac du Bonnet granite that yields a consistent ratio of compressive strength to tensile strength with experiments. The simulations of fluid injection reveal that the initial flaw direction plays a crucial role in crack interaction and coalescence pattern. When two initial flaws are aligned, cracks show relatively higher propagation rate. Some geometrical measures from graph theory are used to analyze the geometry and connectivity of the crack network. The results reveal that initial flaws in the same direction may lead to a well connected crack network with high global efficiency.

... We then update the phase-field variables as in (76). To 552 simplify the implementation, the temporal discretization of plastic dissipation and structural heating, Sun [77,122,123], this semi-implicit approach can be effective if used properly. Finally, it should 555 be noticed that one may choose other partition strategies to solve the same system of equations. ...

Rock salt is one of the major materials used for nuclear waste geological disposal. The
desired characteristics of rock salt, i.e., high thermal conductivity, low permeability, and self-healing
are highly related to its crystalline microstructure. Conventionally, this microstructural effect is
often incorporated phenomenologically in macroscopic damage models. Nevertheless, the thermomechanical behavior of a crystalline material is dictated by the nature of crystal lattice and micromechanics (i.e., the slip-system). This paper presents a model proposed to examine these fundamental mechanisms at the grain scale level. We employ a crystal plasticity framework in which single-crystal halite is modeled as a face-centered cubic (FCC) structure with the secondary atoms in its octahedral holes, where a pair of Na+ and Cl− ions forms the bond basis. Utilizing the crystal plasticity framework, we capture the existence of an elastic region in the stress space and the sequence of slip system activation of single-crystal halite under different temperature ranges. To capture the anisotropic nature of the intragranular fracture, we couple a crystal plasticity model with a multi-phase-field simulation that does not require high-order terms for the phase field. Numerical examples demonstrate that the proposed model is able to capture the anisotropy of inelastic and damage behavior under various loading rates and temperature conditions.

... After removal of a beam, there will be no tensile force exerted on the contact pair and the compressive forces are exerted only if two particles are overlapping and therefore lead to closure of cracks. This path-dependent behavior is enforced by labeling the contact pair in a damaged state, a technique commonly used in element-deletion methods in finite element fracture models [51,52]. Thus, forces are in this case given by ...

Fluid-driven fractures of brittle rock is simulated via a dual-graph lattice model. The new discrete hydromechanical
model incorporates a two-way coupling mechanism between the discrete element model and the
flow network. By adopting an operator-split algorithm, the DEM-network coupling model is able to replicate
the transient poroelasticity coupling mechanism and the resultant Mandel-Cryer hydro-mechanical coupling
effect in a discrete mechanics framework. As crack propagation, coalescence and branching are all pathdependent,
irreversible processes, capturing this transient effect is important for capturing the essence of the
fluid-driven fracture in simulations. Injection simulations indicate that the onset and propagation of fractures
is highly sensitive to the ratio between the injection rate and the effective permeability. Furthermore, we
show that in a permeable rock the borehole breakdown pressure, the pressure at which fractures start to
grow from the borehole, depends on both the given ratio between injection rate and permeability and the
Biot’s coefficient.

... Fracture mechanics can offer a basis for a more mechanically sound, mesh-insensitive approach to brittle failures of geological materials. The application of fracture mechanics to geomechanical problems has been explored since several decades ago (e.g., [45,46]), and it is now an active area of research in various contexts from landslides to fault mechanics to hydraulic fracturing (e.g., [47][48][49][50][51][52][53][54]). Simulating fractures in these geomechanical problems, however, faces theoretical and computational challenges that have not yet been addressed satisfactorily. ...

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.

This article presents a multi-phase-field poromechanics model that simulates the growth and thaw of ice lenses and the resultant frozen heave and thaw settlement in multi-constituent frozen soils. In this model, the growth of segregated ice inside the freezing-induced fracture is implicitly represented by the evolution of two phase fields that indicate the locations of segregated ice and the damaged zone, respectively. The evolution of two phase fields is induced by their own driving forces that capture the physical mechanisms of ice and crack growths respectively, while the phase field governing equations are coupled with the balance laws such that the coupling among heat transfer, solid deformation, fluid diffusion, crack growth, and phase transition can be observed numerically. Unlike phenomenological approaches that indirectly capture the freezing influence on the shear strength, the multi-phase-field model introduces an immersed approach where both the homogeneous freezing and the ice lens growth are distinctively captured by the freezing characteristic function and the driving force accordingly. Verification and validation examples are provided to demonstrate the capacities of the proposed models.

This paper presents the mathematical framework and the asynchronous finite element solver that captures the brittle fractures in multi-phase fluid-infiltrating porous media at the mesoscale where the constituents are not necessarily in a thermal equilibrium state. To achieve this goal, we introduce a dual-temperature effective medium theory in which the distinct constituent temperatures are homogenized independently whereas the heat exchange among the constituents is captured via phenomenological heat exchange laws in analog to the dual-permeability theory. To handle the different growth rates of the boundary layers in a stable and computationally efficient manner, an asynchronous time integrator is proposed and implemented in an operator-split algorithm that updates the displacement, pore pressure, phase field, and temperature of each constituent in an asynchronous manner. Numerical examples are introduced to verify the implementation and compare the path-dependent behaviors predicted by the two-temperature and one-temperature models.

We present a hybrid model/model-free data-driven approach to solve poroelasticity problems. Extending the data-driven modeling framework originated from \citet{kirchdoerfer2016data}, we introduce one model-free and two hybrid model-based/data-driven formulations capable of simulating the coupled diffusion-deformation of fluid-infiltrating porous media with different amounts of available data. To improve the efficiency of the model-free data search, we introduce a distance-minimized algorithm accelerated by a k-dimensional tree search. To handle the different fidelities of the solid elasticity and fluid hydraulic constitutive responses, we introduce a hybridized model in which either the solid and the fluid solver can switch from a model-based to a model-free approach depending on the availability and the properties of the data. Numerical experiments are designed to verify the implementation and compare the performance of the proposed model to other alternatives.

This paper presents an immersed phase field model designed to predict the fracture-induced flow due to brittle fracture in vuggy porous media. Due to the multiscale nature of pores in vuggy porous material, crack growth may connect previously isolated pores which lead to flow conduits. This mechanism has important implications for many applications such as disposal of carbon dioxide and radioactive materials, hydraulic fracture and mining. To understand the detailed microporomechanics that causes the fracture-induced flow, we introduce a new phase field fracture framework where the phase field is not only used as an indicator function for damage of the solid skeleton, but also as an indicator of the pore space. By coupling the Stokes equation that governs the fluid transport in the voids, cavities and cracks, and the Darcy's flow in the deformable porous media, our proposed model enables us to capture the fluid-solid interaction of the pore fluid and solid constituents during the crack growth. Numerical experiments are conducted to analyze how presence of cavities affects the accuracy of the predictions based on homogenized effective medium during crack growth.

Cracks, veins, joints, faults, and ocean crusts are strong discontinuities of different length scales that can be found in many geological formations. While the constitutive laws for the frictional slip of these interfaces have been the focus of decades-long geophysical research, capturing the evolving geometry such as branching, coalescence, and the corresponding interplay between frictional slip and the mode II crack growth in compression remains a challenging task. This work employs a phase field framework for frictional contact originated in Fei and Choo [2020] , s.t. these strong discontinuities are represented by implicit functions and the frictional responses of the transitional damage zone is approximated by a diffusive constitutive law that captures the coupling between the bulk and interfacial plasticity. To replicate the rate dependence and size effects commonly exhibited in frictional interfaces, we propose a regularized constitutive law for the slip weakening/strengthening at different loading rates and temperature regimes. Numerical examples are provided to show that the regularized model may converge into the strong-discontinuity counterpart and capture the frictional response along geometrically complex interfaces.

While crack nucleation and propagation in the brittle or quasi-brittle regime can be predicted via variational or material-force-based phase field fracture models, these models often assume that the underlying elastic response of the material is non-polar and yet a length scale parameter must be introduced to enable the sharp cracks represented by a regularized implicit function. However, many materials with internal microstructures that contain surface tension, micro-cracks, micro-fracture, inclusion, cavity or those of particulate nature often exhibit size-dependent behaviors in both the path-independent and path-dependent regimes. This paper is intended to introduce a unified treatment that captures the size effect of the materials in both elastic and damaged states. By introducing a cohesive micropolar phase field fracture theory, along with the computational model and validation exercises, we explore the interacting size-dependent elastic deformation and fracture mechanisms exhibits in materials of complex microstructures. To achieve this goal, we introduce the distinctive degradation functions of the force-stress-strain and couple-stress-micro-rotation energy-conjugated pairs for a given regularization profile such that the macroscopic size-dependent responses of the micropolar continua are insensitive to the length scale parameter of the regularized interface. Then, we apply the variational principle to derive governing equations from the micropolar stored energy and dissipative functionals. Numerical examples are introduced to demonstrate the proper way to identify material parameters and the capacity of the new formulation to simulate complex crack patterns in the quasi-static regime.

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L'Institut de Radioprotection et de Sûreté Nucléaire (IRSN) s'intéresse à l'étude des réactions de gonflement interne, dont les Réactions Sulfatiques, et à leur impact sur l'évolution des propriétés du matériau cimentaire. Les Réactions Sulfatiques sont caractérisées par la précipitation de l'ettringite, dans les pores du matériau durci entraînant des gonflements locaux et une fissuration par déformations différentiées. Les fissures créées constituent alors le lieu privilégié de la précipitation d'ettringite et accélèrent le transport des espèces chimiques au sein du milieu poreux. La modification locale des phénomènes de transport induit une accélération de la dégradation du matériau.Ce travail de thèse modélise à l'échelle mésoscopique d'une collection de granulats, le gonflement du béton par les Réactions Sulfatiques et la cinétique de dégradation. Un modèle chimio-mécanique basé sur une description du transport réactif (diffusion d'espèces et réactions chimiques) et mécanique (Modèle de Zones Cohésives) dans un milieu poreux fissuré est proposé et résolu à l'aide d'un couplage étagé générique.Les paramètres chimiques et mécaniques initiaux sont estimés par un calcul d'hydratation et d'homogénéisation analytique.La modélisation chimio-mécanique tridimensionnelle est validée de façon modulaire et appliquée aux Réactions Sulfatiques Externe et Interne. Les effets de la composition du béton et des conditions environnementales chimiques sur la cinétique d'expansion et le faciès de rupture sont étudiés. Les applications mettent en évidence l'influence des granulats et des fissures dans la répartition spatiale inhomogène des zones de précipitation de l'ettringite et les contraintes de gonflement associées.

This paper presents the application of a fast Fourier transform (FFT) based method to solve two phase field models designed to simulate crack growth of strongly anisotropic materials in the brittle regime. By leveraging the ability of the FFT-based solver to generate solutions with higher-order and global continuities, we design two simple algorithms to capture the complex fracture patterns (e.g. sawtooth, and curved crack growth) common in materials with strongly anisotropic surface energy via the multi-phase-field and high-order phase-field frameworks. A staggered operator-split solver is used where both the balance of linear momentum and the phase field governing equations are formulated in the periodic domain. The unit phase field of the initial failure region is prescribed by the penalty method to alleviate the sharp material contrast between the initial failure region and the base material. The discrete frequency vectors are generalized to estimate the second and fourth-order gradients such that the Gibbs effect near shape interfaces or jump conditions can be suppressed. Furthermore, a preconditioner is adopted to improve the convergence rate of the iterative linear solver. Three numerical experiments are used to systematically compare the performance of the FFT-based method in the multi-phase-field and high-order phase-field frameworks.

We introduce a mesh-adaption framework that employs a multi-physical configurational force and Lie algebra to capture multi-physical responses of fluid-infiltrating geological materials while maintaining the efficiency of the computational models. To resolve sharp changes of both displacement and pore pressure, we introduce an energy-estimate-free re-meshing criterion by extending the configurational force theory to consider the energy dissipation due to the fluid diffusion and the gradient-dependent plastic flow. To establish new equilibria after re-meshing, the local tensorial history-dependent variables at the integration points are first decomposed into spectral forms. Then, the principal value and direction are projected onto a smooth field interpolated by the basis function of the finite element space via the Lie-algebra mapping. Our numerical results indicate that this Lie algebra operator, in general, leads to a new trial state closer to the equilibrium than the ones obtained from the tensor component mapping approach.
A new configurational force for dissipative fluid-infiltrating porous materials that exhibit gradient-dependent plastic flow is introduced such that the re-meshing may accommodate the need to resolve the sharp pressure gradient as well as the strain localization. The predicted responses are found to be not influenced by the mesh size due to the micromorphic regularization, while the adaptive meshing enables us to capture the width of deformation bands without the necessity of employing fine mesh everywhere in the domain.

We present a computational framework for the treatment of shear localization in metallic materials under dynamic loading, based on the integration of a shear band tracking strategy into an explicit 3D finite element formulation with embedded weak discontinuities. Within this computational framework, an embedded shear band's mid-surface is represented by an iso-surface of a level set function, which is obtained by solving a heat-conduction type boundary value problem (BVP). The solution of this BVP is carried out either globally over the entire problem domain, or locally at the level of individual elements in the mesh. In this paper, we present a detailed comparison of these global and local algorithmic implementations of the shear band tracking strategy. Numerical results obtained using these two implementations, and using a simplified formulation without shear band tracking, are presented and compared. Moreover, we compare the computational efficiency and parallel scaling performance of the local and global implementations. This comparative study shows that both implementations can simulate the initiation of two independent shear bands and their propagation past each other without merging, whereas only the global implementation can successfully simulate the merging of two branches of a single shear band. This study also confirms that the global implementation is more computationally intensive, since it requires a global system of linear equations associated with the level set BVP to be solved at each time step. Both implementations exhibit very good scalability in domain decomposition-based parallel simulations.

This paper presents a new meta-modeling framework to employ deep reinforcement learning (DRL) to generate mechanical constitutive models for interfaces. The constitutive models are conceptualized as information flow in directed graphs. The process of writing constitutive models is simplified 8 as a sequence of forming graph edges with the goal of maximizing the model score (a function of accuracy, robustness and forward prediction quality). Thus meta-modeling can be formulated as a Markov decision process with well-defined states, actions, rules, objective functions and rewards. By using neural networks to estimate policies and state values, the computer agent is able to efficiently self-improve the constitutive model it generated through self-playing, in the same way AlphaGo Zero (the algorithm that outplayed the world champion in the game of Go) improves its gameplay. Our numerical examples show that this automated meta-modeling framework not only produces models which outperform existing cohesive models on benchmark traction-separation data, but is also capable of detecting hidden mechanisms among micro-structural features and incorporating them in constitutive models to improve the forward prediction accuracy, which are difficult tasks to do manually.

Many engineering applications and geological processes involve embedded discontinuities in 6 porous media across multiple length scales (e.g. rock joints, grain boundaries, deformation bands and 7 faults). Understanding the multiscale path-dependent hydro-mechanical responses of these interfaces across 8 length scales is of ultimate importance for applications such as CO 2 sequestration, hydraulic fracture and 9 earthquake rupture dynamics. While there exist mathematical frameworks such as extended finite element 10 and assumed strain to replicate the kinematics of the interfaces, modeling the cyclic hydro-mechanical 11 constitutive responses of the interfaces remains a difficult task. This paper presents a semi-data-driven 12 multiscale approach that obtains both the traction-separation law and the aperture-porosity-permeability 13 relation from micro-mechanical simulations performed on representative elementary volumes in the finite 14 deformation range. To speed up the multiscale simulations, the incremental constitutive updates of the 15 mechanical responses are obtained from discrete element simulations at the representative elementary vol-16 ume whereas the hydraulic responses are generated from a neural network trained with data from lattice 17 Boltzmann simulations. These responses are then linked to a macroscopic dual-permeability model. This 18 approach allows one to bypass the need of deriving multi-physical phenomenological laws for complex 19 loading paths. More importantly, it enables the capturing of the evolving anisotropy of the permeabilities 20 of the macro-and micro-pores. A set of numerical experiments are used to demonstrate the robustness of 21 the proposed model. 22

One of the goals of the Journal of Elasticity: The Physical and Ma- ematical Science of Solids is to identify and to bring to the attention of the research community in the physical and mathematical sciences extensive expositions which contain creative ideas, new approaches and currentdevelopmentsinmodellingthebehaviourofmaterials. Fracture has enjoyed a long and fruitful evolution in engineering, but only in - cent years has this area been considered seriously by the mathematical science community. In particular, while the age-old Gri?th criterion is inherently energy based, treating fracture strictly from the point of view of variational calculus using ideas of minimization and accounting for the singular nature of the fracture ?elds and the various ways that fracture can initiate, is relatively new and fresh. The variational theory of fracture is now in its formative stages of development and is far from complete, but several fundamental and important advances have been made. The energy-based approach described herein establishes a consistent groundwork setting in both theory and computation. While itisphysicallybased,thedevelopmentismathematicalinnatureandit carefully exposes the special considerations that logically arise rega- ing the very de?nition of a crack and the assignment of energy to its existence. The fundamental idea of brittle fracture due to Gri?th plays a major role in this development, as does the additional dissipative feature of cohesiveness at crack surfaces, as introduced by Barenblatt. Thefollowinginvited,expositoryarticlebyB. Bourdin,G. Francfort and J. -J. Marigo represents a masterful and extensive glimpse into the fundamentalvariationalstructureoffracture.

Fluid flow in fractures that pre-exist or propagate in a porous medium can have a major influence on the deformation and flow characteristics. With the aim of carrying out large-scale calculations at reasonable computing costs, a sub-grid scale model has been developed. While this model was originally embedded in extended finite element methods, thereby exploiting some special properties of the enrichment functions, we will herein show that, using proper micro-macro relations, in particular for the mass balance, sub-grid scale models can be coupled to a range of discretisation methods at the macroscopic scale, from standard interface elements to isogeometric finite element analysis.

A micropolar discrete-continuum coupling model is proposed to link the collectively particulate mechanical simulations at high-order representative elementary volume to field-scale boundary value problems. By incorporating high-order kinematics to the homoge-nization procedure, contact moment and force exerted on grain contacts are homogenized into a non-symmetric Cauchy stress and higher-order couple stress. These stress measures in return become the constitutive updates for the macroscopic finite element model for micropolar continua. Unlike the non-lcoal weighted averaging models in which the intrinsic length scale must be a prior knowledge to compute the nonlocal damage or strain measures, the proposed model introduces the physical length scale directly through the higher-order kinematics. As a result, there is no need to tune or adjust the intrinsic length scale. Furthermore, since consti-tutive updates are provided directly from micro-structures, there is also no need to calibrate any high-order material parameters that are difficult to infer from experiments. These salient features are demonstrated by numerical examples. The classical result from Mindlin is used as a benchmark to verify the proposed model.

Hydraulic fractures represent a particular class of tensile fractures that propagate in solid media under pre-existing compressive stresses as a result of internal pressurization by an injected viscous fluid. The main application of engineered hydraulic fractures is the stimulation of oil and gas wells to increase production. Several physical processes affect the propagation of these fractures, including the flow of viscous fluid, creation of solid surfaces, and leak-off of fracturing fluid. The interplay and the competition between these processes lead to multiple length scales and timescales in the system, which reveal the shifting influence of the far-field stress, viscous dissipation, fracture energy, and leak-off as the fracture propagates.

The variational approach to fracture is effective for simulating the
nucleation and propagation of complex crack patterns, but is computationally
demanding. The model is a strongly nonlinear non-convex variational inequality
that demands the resolution of small length scales. The current standard
algorithm for its solution, alternate minimization, is robust but converges
slowly and demands the solution of large, ill-conditioned linear subproblems.
In this paper, we propose several advances in the numerical solution of this
model that significantly improve its computational efficiency. We reformulate
alternate minimization as a nonlinear Gauss-Seidel iteration and employ
over-relaxation to accelerate its convergence; we compose this accelerated
alternate minimization with Newton's method, to further reduce the time to
solution; and we formulate efficient preconditioners for the solution of the
linear subproblems arising in both alternate minimization and in Newton's
method. We demonstrate the improvements in efficiency on several examples of
physical relevance.

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.

This paper describes an approach to account for the multi-scale nature of soil by using a multi- scale hierarchical Monte Carlo simulation framework. The behavior of particulate media, such as sands, is encoded at the granular-scale, and so methods for accurately predicting soil behavior must rely on methods for up-scaling such behavior across relevant scales of interest. Multi-scale analysis is known to be especially important under strain localization, penetration or liquefaction conditions, where a classical constitutive description may no longer apply. A probabilistic framework across multiple scales is needed to efficiently model and simulate multiscale fields of spatially varying material properties and to consistently compute the behavior of the material in a multi-scale model. From a material modeling standpoint, the multi-scale framework is facilitated here using a hierarchical conditional simulation procedure. With this approach, a more accurate material description at finer scales is pursued only when needed, such as in the presence of strong inhomogeneities. Monte Carlo simulation is used to simulate material properties at an initial coarse scale, and that initial simulation is adaptively refined at finer scale materials whenever necessary, conditional upon previously simulated coarse scale data. Here the background of the multiscale geomechanics motivation is summarized, the mathematics of this simulation approach is developed, and then several example calculations are shown to bring insights regarding the approach and its potential application in problems where multi-scale effects are important. Details regarding open-source software documenting these calculations are also provided.

Phase field theory is developed for solids
undergoing potentially large deformation and fracture.
The elastic potential depends on a finite measure of
elastic strain. Surface energy associated with fracture
can be anisotropic, enabling description of preferred
cleavage planes in single crystals, or isotropic, applicable
to amorphous solids such as glass. Incremental
solution of the Euler–Lagrange equations corresponds
to local minimization of an energy functional for the
solid, enabling prediction of equilibrium crack morphologies.
Predictions are in close agreement with analytical
solutions for pure mode I or pure mode II loading,
including the driving force for a crack to extend
from a pre-existing plane onto a misoriented cleavage
plane. In an isotropic matrix, the tendency for a crack
to penetrate or deflect around an inclusion is shown
to depend moderately on the ratio of elastic stiffness
in matrix and inclusion and strongly on their ratio of
surface energy. Cracks are attracted to (shielded by)
inclusions softer (stiffer) than the surrounding matrix.
The theory and results apparently report the first fully
three-dimensional implementation of phase field theory
of fracture accounting for simultaneous geometric
nonlinearity, nonlinear elasticity, and surface energy
anisotropy.

Hydraulic fracturing is the method of choice to enhance reservoir permeability and well efficiency for extraction of shale gas. Multi-stranded non-planar hydraulic fractures are often observed in stimulation sites. Non-planar fractures propagating from wellbores inclined from the direction of maximum horizontal stress have also been reported. The pressure required to propagate non-planar fractures is in general higher than in the case of planar fractures. Current computational methods for the simulation of hydraulic fractures generally assume single, symmetric, and planar crack geometries. In order to better understand hydraulic fracturing in complex-layered naturally fractured reservoirs, fully 3D models need to be developed. In this paper, we present simulations of 3D non-planar fracture propagation using an adaptive generalized FEM. This method greatly facilitates the discretization of complex 3D fractures, as finite element faces are not required to fit the crack surfaces. A solution strategy for fully automatic propagation of arbitrary 3D cracks is presented. The fracture surface on which pressure is applied is also automatically updated at each step. An efficient technique to numerically integrate boundary conditions on crack surfaces is also proposed and implemented. Strongly graded localized refinement and analytical asymptotic expansions are used as enrichment functions in the neighborhood of fracture fronts to increase the computational accuracy and efficiency of the method. Stress intensity factors with pressure on crack faces are extracted using the contour integral method. Various non-planar crack geometries are investigated to demonstrate the robustness and flexibility of the proposed simulation methodology. Copyright © 2014 John Wiley & Sons, Ltd.

The mathematical analysis of the flow of a single-phase Newtonian fluid
through a rough-walled rock fracture is reviewed, starting with the
Navier-Stokes equations. By a combination of order-of-magnitude
analysis, appeal to available analytical solutions, and reanalysis of
some data from the literature, it is shown that the Navier-Stokes
equations can be linearized if the Reynolds number is less than about
10. Further analysis shows that the linear Stokes equations can be
replaced by the simpler Reynolds lubrication equation if the wavelength
of the dominant aperture variations is about three times greater than
the mean aperture. However, this criterion does not seem to be strongly
obeyed by all fractures. The Reynolds equation (i.e., the local cubic
law) may therefore suffice in estimating fracture permeabilities to
within a factor of about 2, but more accurate estimates will require
solution of the Stokes equations. Similarly, estimates of mean aperture
based on inverting transmissivity data may have errors of a factor of
two if any version of the local cubic law is used to relate
transmissivity to mean aperture.

We present a method for the simulation of 3-D hydraulic fracturing in fully saturated porous media. The discrete fracture(s) is driven by the fluid pressure. A cohesive fracture model is adopted where the fracture follows the face of the elements around the fracture tip which is closest to the normal direction of the maximum principal stress at the fracture tip. No predetermined fracture path is needed. This requires continuous updating of the mesh around the crack tip to take into account the evolving geometry. The updating of the mesh is obtained by means of an efficient mesh generator based on Delaunay tessellation. The governing equations are written in the framework of porous media mechanics theory and are solved numerically in a fully coupled manner. An examples dealing with a concrete dam is shown.

A stabilized enhanced strain finite element procedure for poromechanics is fully integrated with an elasto-plastic cap model to simulate the hydro-mechanical interactions of fluid-infiltrating porous rocks with associative and non-associative plastic flow. We present a quantitative analysis on how macroscopic plastic volumetric response caused by pore collapse and grain rearrangement affects the seepage of pore fluid, and vice versa. Results of finite element simulations imply that the dissipation of excess pore pressure may significantly affect the stress path, and thus alter the volumetric plastic responses.

The elastic strain energy released per unit advance of a compaction band in an infinite layer of thickness h is used to identify and assess quantities relevant to propagation of isolated compaction bands observed in outcrop. If the elastic moduli of the band and the surrounding host material are similar and the band is much thinner than the layer, the energy released is simply sigma+xih$\epsilon$p where sigma+ is the compressive stress far ahead of the band edge, xih is the thickness of the band and $\epsilon$p is the uniaxial inelastic compactive strain in the band. Using representative values inferred from field data yields an energy release rate of 40 kJ/m2, which is roughly comparable with compaction energies inferred from axisymmetric compression tests on notched sandstone samples. This suggests that a critical value of the energy release rate may govern propagation, although the particular value is likely to depend on the rock type and details of the compaction process.

[1] Detailed observations of compaction bands exposed in the Aztec Sandstone of southeastern Nevada indicate that these thin, tabular, bounded features of localized porosity loss initiated at pervasive grain-scale flaws, which collapsed in response to compressive tectonic loading. From many of these Griffith-type flaws, an apparently self-sustaining progression of collapse propagated outward to form bands of compacted grains a few centimeters thick and tens of meters in planar extent. These compaction bands can be idealized as highly eccentric ellipsoidal bodies that have accommodated uniform uniaxial plastic strain parallel to their short dimension within a surrounding elastic material. They thus can be represented mechanically as contractile Eshelby inclusions, which generate near-tip compressive stress concentrations consistent with self-sustaining, in-plane propagation. The combination of extreme aspect ratio (∼10−4) and significant uniaxial plastic strain (∼10%) also justifies an approximation of the bands as anticracks: sharp boundaries across which a continuous distribution of closing mode displacement discontinuity has been accommodated. This anticrack interpretation of compaction bands is analogous to that of pressure solution surfaces, except that porosity loss takes the place of material dissolution. We find that displacement discontinuity boundary element modeling of compaction bands as anticracks within a two-dimensional linear elastic continuum can accurately represent the perturbed external stress fields they induce.

Breakouts are valuable indicators of the direction of the minimum compressive stress orthogonal to the axis of the borehole. Their shapes may provide information about the magnitudes of both the maximum and minimum stresses relative to the strength of the rock. Borehole breakouts also may be impediments to drilling and to in situ measurement techniques, such as hydraulic fracturing. Observations and analyses of borehole breakouts raise three important questions. First, how does the shape of the borehole breakout evolve? Second, why are breakout shapes stable despite the very high compressive stress concentrations that they produce? Third, how is the shape of the breakout related to the magnitudes of the stresses in the rock? The stresses outside the stable breakout are found to be everywhere less than the limiting values of shear strength given by a Mohr-Coulomb criterion. In the regions of great stress concentrations at the ends of a breakout cross section, which have a pointed shape, the state of stress approaches that of equal biaxial compression in plane strain, as it does ahead of a mathematical crack or notch. The fact that the stresses around a breakout are less than the relevant strength establishes both the stability of the final breakout cross section and the appropriateness of an elastic analysis of the stresses. According to this model, the cross-sectional shapes of stable breakouts are not related uniquely to the state of stress and the strength of the rock. -from Authors

A method of measuring mean mechanical aperture of fractures based on gas volume balance is introduced. The effects of shear displacement and normal stress on mechanical and hydraulic behaviour of fractures are also investigated. The results obtained from tests conducted on granite samples from Olympic Dam (Central Australia) are compared with those calculated from existing shear dilation theories. It is found that hydraulic aperture is considerably narrower than the measured mean mechanical aperture. This highlights the need to consider tortuosity and surface roughness of fractures in the calculation of hydraulic aperture. It is also found that shear dilation angle decreases linearly with increases in confining pressure, as opposed to more rapid decreases predicted by existing empirical models. From the results of this study, a range of data describing the relationships between confining pressure, shear displacements, hydraulic aperture and permeability are presented, which could help to develop stimulation programs for geothermal reservoirs.

A compaction band is modelled as a thin, ellipsoidal heterogeneity with an imposed inelastic compactive strain and different elastic moduli from the surrounding matrix. Previously published results are used to determine the stress state in the band. For a wide variation of properties, stress conditions, and inelastic strain, the stress state in the band for aspect ratios observed in the field, 10(-3) - 10(-4), is indistinguishable from the result in the zero aspect ratio limit. In this limit, the compressive stress immediately adjacent to the band tip is roughly 10100 times the far-field stress for parameters representative of field conditions. This value is relatively insensitive to the elastic mismatch between the band and the surrounding material, and is primarily controlled by the ratio of the far-field stress to twice the shear modulus times the inelastic compactive strain. This ratio is inferred to be about 0.02-0.05 from published field results, but may be several times larger for laboratory specimens. The ratio of tip to far-field stress increases with decrease of band shear modulus and becomes unbounded if both the shear modulus and aspect ratio go to zero. A combined anti-crack-dislocation model, in which a compactive relative displacement 2h is specified in the centre of the band and uniform traction elsewhere, predicts that for growth at constant energy release rate h is proportional to root L where L is the half-length of the band. For an energy release rate of 40 kJ m(-2), inferred in an earlier study from field observations and comparable with compaction energies inferred from laboratory tests on circumferentially notched compression samples, the constant of proportionality is consistent with that inferred from laboratory observations and earlier field data.

Gradient-enhanced damage models and phase-field models are seemingly very disparate approaches to fracture. Whereas gradient-enhanced damage models find their roots in damage mechanics, which is a smeared approach from the onset, and gradients were added to restore well-posedness beyond a critical strain level, the phase-field approach to brittle fracture departs from a discontinuous description of failure, where the distribution function is regularised, leading to the inclusion of spatial gradients as well. Herein, we will consider both approaches, and discuss their similarities and differences. The averaging (diffusion) equations for the averaging field and the phase-field will be compared, and it is shown that the diffusion equation for the phase-field can be conceived as a special case of the averaging equation of a gradient-damage model where the damage is averaged. Further, the role of the driving force is examined, and it is shown that subtle differences in the degradation functions commonly adopted in damage and phase-field approaches are key to the observation that, different from damage mechanics, the fracture process zone does not broaden in the wake of the crack tip.

This work presents phase field fracture modeling in heterogeneous porous media. We develop robust and efficient numerical algorithms for pressure-driven and fluid-driven settings in which the focus relies on mesh adaptivity in order to save computational cost for large-scale 3D applications. In the fluid-driven framework, we solve for three unknowns pressure, displacements and phase field that are treated with a fixed-stress iteration in which the pressure and the displacement-phase-field system are decoupled. The latter subsystem is solved with a combined Newton approach employing a primal-dual active set method in order to account for crack irreversibility. Numerical examples for pressurized fractures and fluid filled fracture propagation in heterogeneous porous media demonstrate our developments. In particular, mesh refinement allows us to perform systematic studies with respect to the spatial discretization parameter.

A phase field theory incorporating both fracture and deformation twinning behaviors in crystalline solids is described and implemented in finite element calculations. A variational approach is used to derive governing equations for quasi-static loading. The constitutive theory accounts for possible anisotropy of surface energy of fracture, enabling preferential cleavage on intrinsically weak crystallographic plane(s). Both linear elastic and nonlinear elastic models for bulk material behavior are addressed, the latter via compressible neo-Hookean elasticity. Numerical implementation is undertaken via the finite element method, wherein nodal degrees of freedom are displacement components and order parameters associated with twinning shear and local elastic stiffness reduction from fracture. Three dimensional simulations are reported, with solutions obtained via incremental energy minimization subjected to appropriate boundary and irreversibility constraints. Two sets of calculations are considered: a single crystal with a geometric notch, from which a crack and/or twin may extend upon mode I or mode II loading, and simple tension of a polycrystal consisting of grains with various lattice orientations. Results from the first set of calculations demonstrate a tendency for fracture before twinning when surface energies of the two mechanisms are equal, and a tendency for twinning to delay fracture when the fracture energy substantially exceeds the twin boundary energy. Results from the second set demonstrate effects of relative orientations of cleavage planes to habit planes (parallel or perpendicular), effects of initial orientation distributions, and effects of secondary grain boundary phases differing in strength and stiffness from surrounding crystals.

The thermo-hydro-mechanical (THM) coupling effects on the dynamic wave propagation and strain localization in a fully saturated softening porous medium are analyzed. The characteristic polynomial corresponding to the governing equations of the THM system is derived, and the stability analysis is conducted to determine the necessary conditions for stability for both non-isothermal and adiabatic cases. The result from the dispersion analysis based on the Abel-Ruffini theorem reveals that the roots of the characteristic polynomial for the thermo-hydro-mechanics problem can not be expressed algebraically. Meanwhile, the dispersion analysis on the adiabatic case leads to a new analytical expression of the internal length scale. Our limit analysis on the phase velocity for the non-isothermal case indicates that the internal length scale for the non-isothermal THM system may vanish at the short wavelength limit. This result leads to the conclusion that the rate-dependence introduced by multiphysical coupling may not regularize the THM governing equations when softening occurs. Numerical experiments are used to verify the results from the stability and dispersion analyses.

Field observations are combined with microscopic analyses to investigate the mechanism of formation of wiggly compaction bands (CBs) in the porous Jurassic aeolian Aztec Sandstone exposed at Valley of Fire State Park, Nevada. Among the three types of CBs (T1, T2, and T3), we focused on the wiggly CBs (T3), which show a chevron (T31) or wavy (T32) pattern with typical corner angles of approximately 90° or 130°, respectively. Where corner angles of wiggly CBs increase to 180°, they become straight CBs (T33). Image analyses of thin sections using an optical microscope show host rock porosity increases downslope in this dune, and the predominant type of wiggly CBs also varies from chevron to straight CBs. Specifically, band type varies continuously from chevron to wavy to straight where the porosity and grain sorting of the host rock increase systematically. Based on the crack and anticrack models, we infer that the change from chevron to straight CBs is due to increasing failure angle of the sandstone and this may correlate with increasing grain sorting. Wavy CBs with intermediate failure angle and host rock porosity are an intermediate stage between chevron and straight CBs. Previous sedimentological studies also have suggested that grain size and sorting degree increase downslope on the downwind side of sand dunes due to a sieving process of the wind-blown grains. Therefore, the transition of wiggly CB types in this regard correlates with increasing sorting and perhaps with increasing porosity downslope.

The prediction of fluid- and moisture-driven crack propagation in deforming porous media has achieved increasing interest in recent years, in particular with regard to the modeling of hydraulic fracturing, the so-called “fracking”. Here, the challenge is to link at least three modeling ingredients for (i) the behavior of the solid skeleton and fluid bulk phases and their interaction, (ii) the crack propagation on not a priori known paths and (iii) the extra fluid flow within developed cracks. To this end, a macroscopic framework is proposed for a continuum phase field modeling of fracture in porous media. It provides a rigorous geometric approach to a diffusive crack modeling based on the introduction of a constitutive balance equation for a regularized crack surface and its modular linkage to a Darcy–Biot-type bulk response of hydro-poro-elasticity. The approach overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies including branching. A modular concept is outlined for linking of the diffusive crack modeling with the hydro-poro-elastic response of the porous bulk material. This includes a generalization of crack driving forces from energetic definitions towards threshold-based criteria in terms of the effective stress related to the solid skeleton of a fluid-saturated porous medium. Furthermore, a Poiseuille-type constitutive continuum modeling of the extra fluid flow in developed cracks is suggested based on a deformation-dependent permeability, that is scaled by a characteristic length. This proposed modular model structure is exploited in the numerical implementation by constructing a robust finite element method, based on an algorithmic decoupling of updates for the crack phase field and the state variables of the hydro-poro-elastic bulk response. We demonstrate the performance of the phase field formulation of fracture for a spectrum of model problems of hydraulic fracture. A slight modification of the framework allows the simulation of drying-caused crack patterns in partially saturated capillar-porous media.

A series of laboratory drilling experiments were conducted on two arkosic sandstones (Tenino and Tablerock) under polyaxial far-field stress conditions (σ
h
≠ σ
H
≠ σ
v
). V-shaped breakouts, aligned with the σ
h
direction and revealing stress-dependent dimensions (width and length), were observed in the sandstones. The microscale damage pattern leading to the breakouts, however, is different between the two, which is attributed to the difference in their cementation. The dominant micromechanism in Tenino sandstone is intergranular microcracking occurring in clay minerals filling the spaces between clastic grains. On the other hand, intra- and transgranular microcracking taking place in the grain itself prevails in Tablerock sandstone. To capture the grain-scale damage and reproduce the failure localization observed around the borehole in the laboratory, we used a discrete element (DE) model in which a grain breakage algorithm was implemented. The microparameters needed in the numerical model were calibrated by running material tests and comparing the macroscopic responses of the model to the ones measured in the laboratory. It is shown that DE modeling is capable of simulating the microscale damage of the rock and replicating the localized damage zone observed in the laboratory. In addition, the numerically induced breakout width is determined at a very early stage of the damage localization and is not altered for the rest of the failure process.

This chapter aims to model deformable porous media with weak and strong discontinuities using an enriched finite element (FE) model based on the extended finite element method (X-FEM) technique. It illustrates a fully coupled numerical model on the basis of the X-FEM technique for the hydro-mechanical analysis of hydraulic fracture propagation in porous media. The X-FEM is employed to model the hydraulic fracturing process in saturated two phase porous media. The weak form of the equilibrium and flow continuity equations of the porous medium is used to derive the discrete form of governing equations by discretizing the governing equations first in the spatial domain employing the X-FEM and then in the time domain applying the generalized Newmark scheme. The contact condition in a fractured porous medium is modeled using the X-FEM, and the effect of contact on the fluid phase is employed by considering no leak-off from/into the porous medium.

A new type of concrete, named metaconcrete, has been developed for the attenuation of shock waves induced by dynamic excitation. Inspired by the metamaterials used for the manipulation of electromagnetic and acoustic waves, this new metamaterial for the mitigation of shock waves utilizes the activation of resonance within engineered inclusions. Metaconcrete replaces the standard stone and gravel aggregates of regular concrete with spherical inclusions consisting of a heavy core coated in a compliant outer layer. Finite-element analyses of metaconcrete slabs for the case of purely elastic constituents reveal trapping of the supplied energy within the inclusions and a reduction in mortar stress, indicating the presence of resonance behavior within the aggregates. Mortar is, however, a brittle material and the fracture properties under dynamic loading should also be considered. Thus, the models used in the elastic analyses are extended by incorporating brittle fracture through the use of an eigenerosion scheme, which erodes elements satisfying an energy-based fracture criterion. The effect of different fracture parameters on the performance of the slab is investigated through parametric studies, looking at the change in slab behavior caused by various aggregate geometry and material configurations. These studies indicate that mechanical energy is captured by the aggregates, reducing the transmission of energy through the slab, the extension of the zone damaged by fracture, and the longitudinal stress within the mortar matrix. The understanding gained from these analyses incorporating fracture characteristics will enable more informed design of metaconcrete aggregates for dynamic loading applications, such as blast shielding and impact protection.

An adaptively stabilized monolithic finite element model is proposed to simulate the fully coupled thermo-hydro-mechanical behavior of porous media undergoing large deformation. We first formulate a finite-deformation thermo-hydro-mechanics field theory for non-isothermal porous media. Projection-based stabilization procedure is derived to eliminate spurious pore pressure and temperature modes due to the lack of the two-fold inf-sup condition of the equal-order finite element. To avoid volumetric locking due to the incompressibility of solid skeleton, we introduce a modified assumed deformation gradient in the formulation for non-isothermal porous solids. Finally, numerical examples are given to demonstrate the versatility and efficiency of this thermo-hydro-mechanical model.

In this paper, the crack growth simulation is presented in saturated porous media using the extended finite element method. The mass balance equation of fluid phase and the momentum balance of bulk and fluid phases are employed to obtain the fully coupled set of equations in the framework of \(u{-}p\) formulation. The fluid flow within the fracture is modeled using the Darcy law, in which the fracture permeability is assumed according to the well-known cubic law. The spatial discritization is performed using the extended finite element method, the time domain discritization is performed based on the generalized Newmark scheme, and the non-linear system of equations is solved using the Newton–Raphson iterative procedure. In the context of the X-FEM, the discontinuity in the displacement field is modeled by enhancing the standard piecewise polynomial basis with the Heaviside and crack-tip asymptotic functions, and the discontinuity in the fluid flow normal to the fracture is modeled by enhancing the pressure approximation field with the modified level-set function, which is commonly used for weak discontinuities. Two alternative computational algorithms are employed to compute the interfacial forces due to fluid pressure exerted on the fracture faces based on a ‘partitioned solution algorithm’ and a ‘time-dependent constant pressure algorithm’ that are mostly applicable to impermeable media, and the results are compared with the coupling X-FEM model. Finally, several benchmark problems are solved numerically to illustrate the performance of the X-FEM method for hydraulic fracture propagation in saturated porous media.

In the modeling of pressurized fractures using phase-field approaches, the irreversibility of crack growth is enforced through an inequality constraint on the temporal derivative of the phase-field function. In comparison to the classical case in elasticity, the presence of the pressure requires the additional constraint and makes the problem much harder to analyze. After temporal discretization, this induces a minimization problem in each time step over a solution dependent admissible set. To avoid solving the resulting variational inequality corresponding to the first order necessary conditions, a penalization approach is used, commonly, to remove the inequality constraint. It is well-known that for large penalty parameters the algorithm suffers from numerical instabilities in the solution process. Consequently, to avoid such a drawback, we propose an augmented Lagrangian algorithm for the discrete in time and continuous in space phase-field problems. The final set of equations is solved in a decoupled fashion. The proposed method is substantiated with several benchmark and prototype tests in two and three dimensions.

In many geomaterials, particularly rocks and clays, permeability is greatly enhanced by the presence of fractures. Fracture sets create an overall permeability that is anisotropic, enhanced in the directions of the fractures. In modeling the fractures via a finite element method, for example, meshing around these fractures can become quite difficult and result in computationally intensive systems. In this article, we develop a relatively simple method for including the fractures within the elements. Flow through the bulk medium is assumed to be governed by Darcy’s law, and the flow on the fracture by a generalization of the law. This model is embedded in a finite element framework, with the fractures passing through the elements. In this formulation, elements with fractures are given an enhanced permeability in the direction of the fractures. With these enhancements, the material essentially becomes anisotropically more permeable in the direction of fracture sets.

The present work is concerned with the verification and validation of a variant of the eigenfracture scheme of Schmidt et al. (2009) based on element erosion, which we refer to as eigenerosion. Eigenerosion is derived from the general eigenfracture scheme by restricting the eigendeformations in a binary sense: they can be either zero, in which case the local behavior is elastic, or they can be equal to the local displacement gradient, in which case the corresponding material neighborhood is failed or eroded. When combined with a finite‐element approximation, this scheme gives rise to element erosion, i.e., the elements can be either intact, in which case their behavior is elastic, or be completly failed, or eroded, and have no load bearing capacity. We verify the eigenerosion scheme through comparisons with analytical solutions and through convergence studies for mode I fracture propagation, both in two and three dimensions and for structured and random meshes. Finally, by way of validation, we apply the eigenerosion scheme to the simulation of mixed modes I–III experiments in poly‐methyl methacrylate plates. Copyright © 2012 John Wiley & Sons, Ltd.

In contrast to discrete descriptions of fracture, phase-field descriptions do not require numerical tracking of discontinuities in the displacement field. This greatly reduces implementation complexity. In this work, we extend a phase-field model for quasi-static brittle fracture to the dynamic case. We introduce a phase-field approximation to the Lagrangian for discrete fracture problems and derive the coupled system of equations that govern the motion of the body and evolution of the phase-field. We study the behavior of the model in one dimension and show how it influences material properties. For the temporal discretization of the equations of motion, we present both a monolithic and staggered time integration scheme. We study the behavior of the dynamic model by performing a number of two and three dimensional numerical experiments. We also introduce a local adaptive refinement strategy and study its performance in the context of locally refined T-splines. We show that the combination of the phase-field model and local adaptive refinement provides an effective method for simulating fracture in three dimensions.

The localized deformation is a ubiquitous phenomenon in geomaterials. It occurs over a vast range of size scales, from the microscale level of grains to faults extending over hundreds of kilometers. It occurs in a variety of forms: a concentration or coalescence of cracks; a distinct, planar frictional surface; a gouge zone of finely comminuted material; or simply a region of higher shear strain. In geomaterials, the severe shearing in regions of localized deformation may be accompanied by dilatancy (inelastic volume increase) or compaction (inelastic volume decrease) and by chemical alteration. Localization can even occur purely by compaction without any evident shear. If the material is fluid-saturated, as is frequently the case, inelastic volume changes can induce the flow of fluid or changes in pore pressure that affect the response. Localization occurs under a variety of conditions. Although most often associated with the formation of shear bands or faults under nominally brittle conditions, localization can also occur by cataclastic flow of rocks at higher mean stresses and by ductile shearing at temperatures and pressures typical of depths of 10 km to 15 km in the earth's crust.

The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

When subjected to nonhydrostatic, compressive stresses, some porous sandstones exhibit nonuniform compaction. The compaction occurs as a localization process, analogous to shear localization, but results in a thickening, tabular zone of compaction as opposed to culminating in a shear fracture. We report the results of several triaxial compression experiments done at a confining pressure of 45 MPa on Castlegate sandstone, measuring simultaneously, stress, strain, acoustic emission locations, and permeability. A major result is that compaction localization produces up to a 2 order-of-magnitude decrease in permeability. Correlation of local strain measurements and acoustic emission locations made on the same specimen show that the compaction process proceeds as a propagating front approximately 20 mm thick. A model of the compaction process was developed that incorporates the moving boundary between compacted, low-permeability regions and uncompacted, higher-permeability regions, and compaction-induced fluid injection at the boundaries. Because of the inhomogeneous nature of compaction produced by compaction localization, and its temporal evolution, a number of phenomena related to fluid flow are predicted by the model: locally increased pore pressures and spatial changes in the effective permeability. Experimental results are reported that show the evolution of effective permeability to be linear with respect to the distance the compaction fronts propagated as predicted by the model. Implications of the results for future experimentation and for reservoirs are briefly discussed; in particular, the interaction between compaction-induced fluid pressure and compaction localization should lead to a phenomenon analogous to dilatancy hardening, impeding the propagation of compaction bands.

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. Following our recent work [C. Miehe, F. Welschinger, M. Hofacker, Thermodynamically-consistent phase field models of fracture: Variational principles and multi-field fe implementations, International Journal for Numerical Methods in Engineering DOI:10.1002/nme.2861] on phase-field-type fracture, we propose in this paper a new variational framework for rate-independent diffusive fracture that bases on the introduction of a local history field. It contains a maximum reference energy obtained in the deformation history, which may be considered as a measure for the maximum tensile strain obtained in history. It is shown that this local variable drives the evolution of the crack phase field. The introduction of the history field provides a very transparent representation of the balance equation that governs the diffusive crack topology. In particular, it allows for the construction of a new algorithmic treatment of diffusive fracture. Here, we propose an extremely robust operator split scheme that successively updates in a typical time step the history field, the crack phase field and finally the displacement field. A regularization based on a viscous crack resistance that even enhances the robustness of the algorithm may easily be added. The proposed algorithm is considered to be the canonically simple scheme for the treatment of diffusive fracture in elastic solids. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples.