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

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... The stress variables that are commonly used for unfrozen saturated or unsaturated soils are Terzaghi effective stress, 6 Bishop stress 7 and net stress 8 . Three types of counterparts for frozen soils were proposed as the primary stress variables, which are ( ) the solid-phase stress by considering ice crystals as a part of the solid phase, 3,9 ( ) Bishop-type stress 10,11 and ( ) net-akin stress 12 (or termed as effective stress 13,14 ) by using ice pressure as a substitute for air pressure. Regarding the second stress variable for frozen soils, Nishimura et al. 13 introduced cryo-suction ( − , with and being ice and water pressures, respectively) followed by Ghoreishian Amiri et al 9 and Na and Sun 11 . ...

... Three types of counterparts for frozen soils were proposed as the primary stress variables, which are ( ) the solid-phase stress by considering ice crystals as a part of the solid phase, 3,9 ( ) Bishop-type stress 10,11 and ( ) net-akin stress 12 (or termed as effective stress 13,14 ) by using ice pressure as a substitute for air pressure. Regarding the second stress variable for frozen soils, Nishimura et al. 13 introduced cryo-suction ( − , with and being ice and water pressures, respectively) followed by Ghoreishian Amiri et al 9 and Na and Sun 11 . Instead of using cryo-suction, on the other hand, Zhou 14 used temperature as the second independent variable through a liquid-crystal equilibrium condition, resembling the Clausius-Clapeyron equation that is widely used to define the relation between cooling temperature and cryo-suction. ...

... It is recognized that the presence of pore ice is the main cause for strengthening of freezing soils, while vanishing of pore ice causes weakening of thawing soils. In the previous models for frozen soils, the effect of ice bonding is usually addressed by a second stress variable or an independent variable directly or indirectly related to a measure of ice, such as pore ice ratio 3 , temperature 14 and cryo-suction 9,11,13 . The relation between the pore ice ratio and temperature can be obtained from a soil-freezing characteristic curve (SFCC), while the relation between temperature and cryo-suction is generally established through liquid-crystal equilibrium equation. ...

The need for unified constitutive models for frozen and unfrozen soils in simulating frost heave and thaw consolidation has been recognized. However, a reliable and easy-to-implement model is not yet available. This paper extends the one-dimensional elasto-plastic model proposed by Yu et al. to solve multi-dimensional problems. In the extended model, total and effective stresses are taken as the stress variables for frozen and unfrozen soils, respectively. A critical-state-based mode, Clay And Sand Model (CASM), is employed for soil at unfrozen states. The emphasis is placed on how a basic constitutive model for unfrozen soils is extended to describe the elasto-plastic behavior for both unfrozen and frozen soils in two steps: (1) extension of the constitutive model from unfrozen to frozen soils by considering ice-bonding effect, and (2) integration of descriptions for soils at unfrozen and frozen states using the concept of residual stress (by Nixon and Morgenstern). The freeze-thaw cycling effect is naturally incorporated by the introduction of residual stress line that is used to determine the initial stress state and preconsolidation pressure of thawed soil. After being implemented into FORTRAN subroutines for ABAQUS, the performance of the proposed model is demonstrated by simulating triaxial compression tests on frozen sand at different temperatures under confining pressures, and freeze-thaw cycling processes of silty clay under loading.

... Modeling the frost heave within the thermomechanical framework has also been conducted with finiteelement approaches [8]. The more general numerical approaches for modeling combined thermo-hydro-mechanical effects in frozen soil have been proposed [9], by simulating freeze-thaw cycles [10] or by incorporating the finite strain theory [11]. Associated with the climate changes that bring in more extreme weather in the permafrost region, the demand for understanding the freeze-thaw action and its interaction with underground infrastructures is further intensified. ...

... Specifically, the new expressions of cryosuction, the concept of negative pressure in freezing soils resulting from transformation of liquid water to ice in the soil pores, are derived including the gas phase with the unsaturated poromechanics concept on top of the solid-ice-water phases, which is new and unique in modeling frozen soils. The formulations are implemented within the implicit finite element framework by leveraging the previous stabilization scheme [11]. The balance principles (linear momentum, mass, and energy) are explicitly formulated and solved to account for the coupled multiphysics effect. ...

... The temperature is gradually increased up to 1°C. As identified from the freezing characteristic equation (11), and Figure 1, dramatic phase transition occurs when the temperature of the soil is between 0 to −2°C. Therefore, not only the mechanical but also the hydraulic impact might be expected from this simulation, which is presented in Figure 7. Figure 7 shows the contour of temperatures around the pipelines 20 hours after the boundary temperature is increased to 1℃. ...

This paper presents a thermo-hydro-mechanics theory and corresponding computational framework to capture the freezethaw action of frozen porous media and associated frost action under chilled gas pipelines. Based on the mixture theory, frost-susceptible soils are formulated to capture the Darcy flux and thermal actions below the pipelines. Constitutive models that combine the cryo-suction are presented to reproduce the changes in volume, strength, and thermal characteristics of solid grain, pore water, and ice crystal. A generalized hardening rule is adopted to replicate the elasto-plastic responses which strengthens the frozen porous media due to ice crystallization. Changes in permeability and thermal diffusivity are also incorporated by considering the phase transitions of pore water and ice crystal. Numerical examples for pipeline applications are designed to analyze the influence of the freezing and melting process around the pipelines.

... This has given impetus to different modifications of these criteria [3]. The thermo-hydro-dynamical models of soil freezing which take into account plastic strains are presented in [4][5][6][7]. In [4][5][6], the so-called Barcelona Basic Model [8] is used to describe the plastic deformations of saturated frozen soils. ...

... The thermo-hydro-dynamical models of soil freezing which take into account plastic strains are presented in [4][5][6][7]. In [4][5][6], the so-called Barcelona Basic Model [8] is used to describe the plastic deformations of saturated frozen soils. According to this model, the stress-strain state of soils is determined by analyzing two state variables -effective stress and a parameter that describes the moisture migration effect. ...

... The thermo-hydro-dynamical models based on this approach can be employed to consider the influence of moisture migration and mechanical pressure on the plastic deformation of frozen soils. In [6], the Barcelona Basic Model is generalized to the case of large deformations. Another way [7] has been proposed to derive the constitutive relations for describing the elastoplastic behavior of frozen soils in the framework of a thermodynamic approach. ...

A simple analytical relations are commonly used in engineering practice to calculate ice wall thickness. One of them is Vyalov’s relation that takes into account the features of a real technological process of tubing lining and the inelastic deformation associated with frozen soil creep. An estimation of applicability and margin of safety of this equation is an issue of engineering mechanics. In this paper, we propose a mathematical model for description of ice wall deformation under natural external loading and present the results of the computational experiments in which an optimal thickness for the ice wall is determined. Based on this simulation, we modify the existing analytical relation, which makes it possible to calculate the thickness of an ice wall of unlimited height.

... Meanwhile, within the geomechanics and geotechnical engineering community, a number of theories and numerical modeling frameworks have been proposed based on the mixture theory and thermodynamics principles [67][68][69][70] with a variety of complexities and details. By adopting the premelting theory and considering the frozen soil as a continuum mixture of the solid, unfrozen water, and ice constituents, the freezing retention behavior of frozen soil can be modeled in a manner similar to those for the unsaturated soil. ...

... In this section, we introduce the ingredients necessary to derive the field theory for the phase field modeling of frozen soil presented later in Section 2.3. Similar to the treatments in [67], [68], and [69], we first assume that the frozen soil is fully saturated with either water or ice and therefore idealize the frozen soil as a three-phase continuum mixture that consists of solid, water, and ice phase constituents whereas the ice lens is a special case in which the solid skeleton no longer holds bearing capacity. This treatment enables us to formulate a multi-phase-field approach to employ two phase field variables as indicator functions for the state of the pore fluid (in ice or water form) [74,78,79] and that of the solid skeleton (in damage or intact form) [75][76][77]. ...

... Clearly, Eq. (2.12) alone cannot capture the deviatoric stress induced by the deformation of the ice lens. Previous efforts on modeling frozen soil often relies on an extension of critical state theory that evolves the yield function according to the degree of saturation of ice (and therefore introduces the dependence of the tensile and shear strength on the presence of ice) [67,69]. However, this treatment is not sufficient to consider the soil that may become brittle at low temperature due to the low moisture content and the influence of ice lens on the elasticity. ...

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.

... Meanwhile, within the geomechanics and geotechnical engineering community, a number of theories and numerical modeling frameworks have been proposed based on the mixture theory and thermodynamics principles [Nishimura et al., 2009, Zhou and Meschke, 2013, Na and Sun, 2017, Michalowski and Zhu, 2006] with a variety of complexities and details. By adopting the premelting theory and considering the frozen soil as a continuum mixture of the solid, unfrozen water, and ice constituents, the freezing retention behavior of frozen soil can be modeled in a manner similar to those for the unsaturated soil. ...

... In this section, we introduce the ingredients necessary to derive the field theory for the phase field modeling of frozen soil presented later in Section 3. Similar to the treatments in [Nishimura et al., 2009], [Zhou and Meschke, 2013], and [Na and Sun, 2017], we first assume that the frozen soil is fully saturated with either water or ice and therefore idealize the frozen soil as a three-phase continuum mixture that consists of solid, water, and ice phase constituents whereas the ice lens is a special case in which the solid skeleton no longer holds bearing capacity. This treatment enables us to formulate a multi-phase-field approach to employ two phase field variables as indicator functions for the state of the pore fluid (in ice or water form) [Warren and Boettinger, 1995, Boettinger et al., 2002, Sweidan et al., 2020 and that of the solid skeleton (in damage or intact form) [Bourdin et al., 2008, Miehe et al., 2010a, Borden et al., 2012. ...

... For all the numerical examples presented in Section 5, we adopt the same values used in [Na andSun, 2017, Bai et al., 2018] ...

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.

... Porous media with pores partially filled with water may exhibit path-dependent responses that are distinctive to those of the single-phase solid and fully saturated porous media [207,235,284,297,321]. 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. ...

... which is obtained by the material time derivative following the solid motion. An explicit relation of the solid volume fraction n S as a function of the solid matrix deformation u S can then be derived based on the assumption that the solid constituent is incompressible (see, [64,128,207,247,272]). Having F S as the solid deformation gradient, the related Jacobian J S := det(F S ) = dv S /dV S is introduced as a mapping of a referential differential volume element dV S into a current differential volume element dv S . ...

... On the other hand, reduction of the saturation degree, i.e. drying, in clay and clayey soils might result in shrinkage-induced cracking and, thus, significantly affects the thermo-hydro-mechanical properties of geological systems. As references to variably-saturated porous media modeling within small and large deformations together a handful of applications, such as coupled behavior during water escape from unsaturated domains or slope stability with the elastoplastic material response, we refer to, e.g., [39,86,107,207,217,235,247,263,272,284,288,291,297,321] among others. ...

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.

... Porous media with pores partially filled with water may exhibit path-dependent responses that are distinctive to those of the single-phase solid and fully saturated porous media [Prévost, 1980, Zienkiewicz et al., 1999, Sun et al., 2011, Song and Borja, 2014, Wang and Sun, 2015, 2016, Na and Sun, 2017. At low confining pressure and room temperature, sub-critical 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. ...

... An explicit relation of the solid volume fraction n S as a function of the solid matrix deformation u S can then be derived based on the assumption that the solid constituent is incompressible (cf. Sanavia et al. [2002], Song and Borja [2014], , Na and Sun [2017], Choo and Sun [2018]). Having F S as the solid deformation gradient, the related Jacobian J S := det(F S ) = dv S /dV S is introduced as a mapping of a referential differential volume element dV S into a current differential volume element dv S . ...

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.

... Koniorczyk [17] established a mathematical model for heat and water transport of deformable materials considering the kinetics of water phase change. Na and Sun [18] applied the volume averaging theory, built a stabilized thermohydro-mechanical finite element model to investigate the freezethaw action of frozen porous media. All of these abovementioned thermo-hydro-mechanical coupled models, especially the models established through the volume averaging theory, can well simulate the interaction of water, heat and stress in the Nomenclature a; b; m testing constants [-] a w ; a w;0 water activity for a given salinity, and at freezing temperature [ Generally salt exists in two phases in porous medium: dissolved salt and salt crystals. ...

... Based on the concept of equivalent water content, it is suggested that moisture in pores consist of unfrozen water, ice and crystal water during the freezing process of saturated sulfate saline soil. Therefore, neglecting the inertial and viscous effects [36], the moisture conservation equation can be established from the equation of linear momentum balance for the liquid and the mass balance equation of water, which can be expressed as [18,19]: Based on Darcy's law, the equation of linear momentum balance for the liquid is given by: ...

The freezing of saturated saline soil is a dynamic hydro-thermal-salt-mechanical (THSM) interacting process. One-side freezing experiment of saturated sulfate saline soil in an open system with no-pressure water supplement is carried out. The coupling mechanism of water and salt migration and the soil deformation in the freezing process has been investigated by the one-side freezing experiment. Based on the crystallization kinetics theory, a hydro-thermal-salt-mechanical coupled mathematical model for saturated frozen sulfate saline soil with the effect of phase change is proposed. Moreover, the influence of solute on the physical and mechanical properties of soil during the freezing process is considered. To solve the nonlinear equations, the finite element algorithm is applied to solve the general form of governing equations. Finally, numerical simulation is implemented with the assistance of COMSOL. Validation of the model is illustrated by comparisons between the simulation and experimental results. From this study, it is found that, (1) the phenomenon of macroscopic crystallization can be well illustrated by the microscopic crystallization kinetics theory on the basis of the concepts of the water activity and the solution supersaturation. (2) The pore pressure due to the effect of phase change is main driving force for water and salt migration as well as the deformation of porous medium. It is concluded that the positive pore pressure is the main factor for soil deformation, and the negative pore pressure is the driving force for water migrates to the frozen zone. (3) Salt migrates with water and is rejected into the unfrozen water in the process of ice formation, and the rate of salt rejection gradually increases with the decrease of cooling rate. Therefore, salt crystals with layer distribution are formed in the frozen zone during the freezing process, and the largest salt crystals distribution zone is formed near the freezing front due to the effect of solute diffusion. (4) The calculated results are well agreement with the experimental data, demonstrating that the proposed hydro-thermal-salt-mechanical coupling model can well clarify the mechanism of heat and mass transfer in saturated frozen sulfate saline soil, and predict the deformation due to the effects of frost heave and salt expansion.

... The volume averaging theory opens up an even broader horizon for THM coupling [54] . Based on this theory, Tong et al. and Na and Sun (2017) [55,56] proposed the multiphase flow model for THM coupling and the porous medium model for stable THM coupling. Based on the volume average theory, the above models can reasonably simulate the freezing process where moisture migration, heat conduction, stress, and strain interact with each other in the soil in engineering construction. ...

... The volume averaging theory opens up an even broader horizon for THM coupling [54] . Based on this theory, Tong et al. and Na and Sun (2017) [55,56] proposed the multiphase flow model for THM coupling and the porous medium model for stable THM coupling. Based on the volume average theory, the above models can reasonably simulate the freezing process where moisture migration, heat conduction, stress, and strain interact with each other in the soil in engineering construction. ...

This paper reviews the history of the research development on the coupling mechanism of the multiphysical field, e.g., thermo-hydro-mechanical (THM), for frozen soil. The objective is to deepen the current understanding of the theories and mechanism of multiphysical field coupling in the frozen soil and the dynamic changes in the temperature, moisture, and stress fields during soil freezing. A new differential equation of the coupling of temperature field and moisture field is proposed. Based on the DiscreteFrechetDist algorithm, a fitting method of evaluating a curve is proposed. The paper is expected to help understand the soil freezing process in cold regions and enhance the innovativeness of the research methodologies dealing with multifield coupling for the frozen soil.

... On the other hand, recent contributions have employed coupled water and heat transfer models associated with soil deformation based on thermo-hydro-mechanical (THM) analysis. The general method to simultaneously simulate frost heave and thaw consolidation is to establish a unified constitutive model for frozen and unfrozen soils (e.g., [32][33][34][35][36]. Nevertheless, the thermo-mechanical coupling was established by means of poroelasticity, or the soil freezing characteristic function associated with unfrozen water contents and cry-suction processes was approximated by a simple relationship like the van Genuchten type, which is not coupled with the mechanical aspect. ...

... Interested readers about monolithic solvers, THM analysis, and governing equations related to soil freezing phenomenon are referred to Na and Sun. 34 ...

Many numerical models for simulating freezing and thawing phenomena of soil have been developed due to emerging geotechnical issues in cold regions. In particular, coupled thermo‐hydro‐mechanical (THM) analysis is used to evaluate complicated deformation, thermal, and moisture transport behavior of freezing–thawing soils. This study proposes a soil‐freezing characteristic curve (SFCC) that is robust and adaptive with various computational frameworks, including the THM approach. The proposed SFCC can also account for different soil types by incorporating the particle size distribution. Here an automatic regression scheme is adopted to update the SFCC associated with deformation and thermal changes. In addition, a smoothing algorithm is adopted to prevent a sharp change of the SFCC due to phase transition between the liquid water and crystal ice. Based on experimental works in the literature, the applicability of our model is demonstrated when the initial water contents and soil particle distribution differ. We further investigate the performance of the proposed SFCC as a constitutive model within a simplified THM framework. Our results show that the proposed model captures the desired behavior of different soil types in the freezing process, such as freezing temperature depreciation, the effect of compaction, and mechanical loading on unfrozen water content.

... Poroelasticity is a sub-discipline of poromechanics problem that focuses on the path independent response of porous media. It has important applications across multiple disciplines including seismology (Cocco and Rice, 2002;Chambon and Rudnicki, 2001), hydraulic fracture (Detournay and Cheng, 1993;Detournay, 2016), petroleum engineering, reservoir management, geological disposal (Sun, 2015;Na and Sun, 2017), and biomechanics modeling for soft tissues and bones (Cowin, 1999). ...

... For completeness, we provide a concise review of the poroelasticity model, which consists of two major components, i.e., the field theory that provides the necessary constraints for the field variables in the space-time domain and the material laws that provides the local constitutive updates for both the solid skeleton and the fluid constituents. Interest readers may refer to Prevost (1985); Borja and Alarcón (1995); Zienkiewicz et al. (1999); Coussy (2004); Sun et al. (2013aSun et al. ( , 2014a; Na and Sun (2017); Na et al. (2019) for a more comprehensive treatment for the topic. ...

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.

... In the literature, different THM models within porous media frameworks have been developed to capture the multi-physical phenomena related to soil freezing, see, e.g., [28,56,68,79,80,82]. Additionally, several THM numerical models were developed within the AGF context, to capture the volume expansion of water transforming into ice and the contribution of the micro-cryo-suction mechanism responsible for driving liquid water toward the frozen zone, see, e.g., [23,112,124,125]. ...

... In this work, to take this suction effect in the soil freezing continuum model into account, a phenomenological formulation similar to the soil/water retention curve (SWRC) in unsaturated soils is implemented. More details about frost heave physics and modeling within a continuum mechanical framework can also be found in [79,87,88,104]. Starting with the mechanical equilibrium at the microscopic pore scale, the relationship between the pressures of the liquid water and the ice crystals can be described by the Young-Laplace's law as ...

This research work presents an experimental and numerical study of the coupled thermo-hydro-mechanical (THM) processes that occur during soil freezing. With focusing on the artificial ground freezing (AGF) technology, a new testing device is built, which considers a variety of AGF-related boundary conditions and different freezing directions. In the conducted experiments, a distinction is made between two thermal states: (1) The thermal transient state, which is associated with ice penetration, small deformations, and insignificant water suction. (2) The thermal (quasi-) steady state, which has a much longer duration and is associated with significant ice lens formation due to water suction. In the numerical modeling, a special focus is laid on the processes that occur during the thermal transient state. Besides, a demonstration of the micro-cryo-suction mechanism and its realization in the continuum model through a phenomenological retention-curve-like formulation is presented. This allows modeling the ice lens formation and the stiffness degradation observed in the experiments. Assuming a fully saturated soil as a biphasic porous material, a phase-change THM approach is applied in the numerical modeling. The governing equations are based on the continuum mechanical theory of porous media (TPM) extended by the phase-field modeling (PFM) approach. The model proceeds from a small-strain assumption, whereas the pore fluid can be found in liquid water or solid ice state with a unified kinematics treatment of both states. Comparisons with the experimental data demonstrate the ability and usefulness of the considered model in describing the freezing of saturated soils.

... In a thawing process, frost heaving still occurs in the frozen zone due to water migration from the thawed zone (Cheng 1983). Currently, the general method to simultaneously simulate frost heave and thaw consolidation is to establish a unified constitutive model for frozen and unfrozen soils (Nishimura et al. 2009; Thomas et al. 2009;Zhang and Michalowski 2015;Na and Sun 2017). In this method, an expression for a unified stress is employed and the Clausius-Clapeyron equation is required to calculate the pore-water pressure within the frozen zone based on an assumed distribution of ice pressure. ...

This paper presents a thermo-hydro-mechanical framework to model frost heave and (thaw) consolidation simultaneously, in which effective and total stresses are taken as the stress variables for unfrozen and frozen soils, respectively. “Effective/total stresses-void ratio-permeability” relations are proposed to interpret the frost heave behavior of soil in different cooling modes, (thaw) consolidation processes and changes in key parameters induced by freeze-thaw cycles. The water flux function proposed by Yu et al. (2019) is used to calculate frost heave in the frozen zone, and to determine the moving boundary of the unfrozen zone during thaw consolidation. Compared with conventional methods, two other modifications are made to characterize the effect of residual stress and the influence of freeze-thaw on permeability in the thaw consolidation analysis. After the governing equations developed in Lagrangian coordinates are implemented in a finite-element system, the framework is firstly verified by the comparison with both small- and large-strain thaw consolidation theories in terms of simulating a semi-infinite thaw consolidation case, and then examined focusing on the three modifications one-by-one. Following that, the framework is assessed by two numerical examples that reasonably reproduce the freeze-thaw cycling processes in both seasonal frost and permafrost regions.

... When the defects are partially or fully filled with pore fluid, the size and the geometric features of the defects may both impose significant changes on both the effective stiffness and permeability of the materials as well as the Biot's coefficient [Wang and Sun, 2017] and the undrained and drained shear strength Sun, 2019a, Sun, 2013]. For instance, carbonate rocks and limestone often contains pores of profoundly different sizes [Coussy, 2004, Sun et al., 2011, Na and Sun, 2017, Choo and Sun, 2018, Sun and Wong, 2018. ...

This study presents a phase field model for brittle fracture in fluid-infiltrating vuggy porous media. While the state-of-the-art in hydraulic phase field fracture considers Darcian fracture flow with enhanced permeability along the crack, in this study, the phase field not only acts as a damage variable that provides diffuse representation of cracks or cavities, but also acts as an indicator function that separates the domain into two regions where fluid flows are governed by Stokes and Darcy equations, respectively. Since the phase field and its gradient can be respectively regarded as smooth approximations of the Heaviside function and Dirac delta function, our new approach is capable of imposing interfacial transmissibility conditions without explicit interface parametrizations. In addition, the interaction between solid and fluid constituents is modeled by adopting the concept of mixture theory, where the fluid velocities in Stokes and Darcy regions are considered as relative measures compared to the solid motion. This model is particularly attractive for coupled flow analysis in geological materials with complex microstructures undergoing brittle fracture often encountered in energy geotechnics problems, since it completely eliminates the needs to generate specific enrichment function, integration scheme, or meshing algorithm tailored for complex geological features.

... When the defects are partially or fully filled with pore fluid, the size and the geometric features of the defects may both impose significant changes on both the effective stiffness and permeability of the materials as well as the Biot's coefficient [30] and the undrained and drained shear strength [25,32]. For instance, carbonate rocks and limestone often contain pores of profoundly different sizes [5,7,17,27,28]. ...

This study presents a phase field model for brittle fracture in fluid-infiltrating vuggy porous media. While the state-of-the-art in hydraulic phase field fracture considers Darcian fracture flow with enhanced permeability along the crack, in this study, the phase field not only acts as a damage variable that provides a diffuse representation of cracks or cavities but also acts as an indicator function that separates the domain into two regions where fluid flows are governed by Stokes and Darcy equations, respectively. Since the phase field and its gradient can be respectively regarded as smooth approximations of the Heaviside function and Dirac delta function, our new approach is capable of imposing interfacial transmissibility conditions without explicit interface parametrizations. In addition, the interaction between solid and fluid constituents is modeled by adopting the concept of mixture theory, where the fluid velocities in Stokes and Darcy regions are considered as relative measures compared to the solid motion. This model is particularly attractive for coupled flow analysis in geological materials with complex microstructures undergoing brittle fracture often encountered in energy geotechnics problems since it completely eliminates the needs to generate specific enrichment function, integration scheme, or meshing algorithm tailored for complex geological features.

... Moreover, poromechanical deformations are poroelastic when they are controlled by the reversible storage and release of elastic energy. Menel In the last few decades, the mechanics of porous media has been of great interest due to its potential application in many geological and biological systems across a wide range of scales such as civil engineering [7,13,20,21,32,40,43,45], energy and environmental technologies [14,17,26,37,39,51], material science [29] and biophysics [18,30], where poromechanics plays an important role in modelling bones and soft tissues [1,15,41]. In physical chemistry, poromechanical processes include mass and heat transfer [53]. ...

Poroelasticity theory can be used to analyse the coupled interaction between fluid flow and porous media (matrix) deformation. The classical theory of linear poroelasticity captures this coupling by combining Terzaghi’s effective stress with a linear continuity equation. Linear poroelasticity is a good model for very small deformations; however, it becomes less accurate for moderate to large deformations. On the other hand, the theory of large-deformation poroelasticity combines Terzaghi’s effective stress with a nonlinear continuity equation. In this paper, we present a finite element solver for linear and nonlinear poroelasticity problems on triangular meshes based on the displacement-pressure two-field model. We then compare the predictions of linear poroelasticity with those of large-deformation poroelasticity in the context of a two-dimensional model problem where flow through elastic, saturated porous media, under applied mechanical oscillations, is considered. In addition, the impact of introducing a deformation-dependent permeability according to the Kozeny-Carman equation is explored. We computationally show that the errors in the displacement and pressure fields that are obtained using the linear poroelasticity are primarily due to the lack of the kinematic nonlinearity. Furthermore, the error in the pressure field is amplified by incorporating a constant permeability rather than a deformation-dependent permeability.

... Scholars have done lots of research on frost heave [6][7][8][9][10][11]. Since the 1960s, scholars had put upward a large number of frost heave theories [12][13][14][15][16][17][18][19][20][21] and mathematical models [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], revealing the mechanism of frost heave and predicting the frost heave caused by soil freezing. The rapid development of frost heave theory and mathematical models have laid a solid foundation for the research of frost heave. ...

Frost heave is an important factor affecting the safety and practicability of buildings in cold regions or artificial freezing engineering. In order to reduce frost heave, frost-susceptible silty clay was used in a one-dimensional frost heave testing system in three different freezing modes. The results show that, compared with the continuous freezing mode, frost heave in the intermittent freezing mode and the continuous-intermittent freezing mode is reduced by 14.4% and 43.6%, respectively. These results clearly demonstrate that frost heave can be restrained in the continuous-intermittent freezing mode more effectively than in the other two freezing modes. The periodic step growth on the frost heave curves in the continuous-intermittent freezing mode is the main reason for this, as explained by the frost heave theory in this paper. To acquire appropriate settings on the cold end temperature, frost heave tests were carried out at different amplitudes and periods of temperature change in the continuous-intermittent freezing mode. The frost heave decreases with the increase of the amplitude of temperature change and period of temperature change. The power function growth, periodic step growth and periodic polyline growth are shown on the frost heave curves at different periods of temperature change of 2, 4, and 8 h, respectively. Due to the good inhibition effect of frost heave, periodic step growth will be a better way to reduce frost heave, which is of great significance to the life cycle safety of buildings.

... This numerical validation proved to be robust in practice. In particular, the inf-sup test has been used successfully to assess the stability of formulations for incompressible materials [46], a contact algorithm with the Lagrange multiplier method [18], a shell element [19], a thermo-hydro-mechanics problem [134], and a four-field formulation of fluid flow in porous media [86], among others. We will present it in details for our problem in Section 3.6.2. ...

In this thesis a flexible and general stable displacement–Lagrange multiplier mixed formulation is developed to model distributed cracking in cohesive grain-based materials in the framework of the cut finite element method. The displacement field is discretized on each grain separately, and the continuity of the displacement and traction fields across the interfaces between grains is enforced by Lagrange multipliers. The design of the discrete Lagrange multiplier space is detailed for bilinear quadrangular elements with the potential presence of multiple interfaces/discontinuities within an element. We give numerical evidence that the designed Lagrange multiplier space is stable and provide examples demonstrating the robustness of the method. Relying on the stable discretization, a cohesive zone formulation equipped with a damage constitutive model expressed in terms of the traction is used to model the propagation of multiple cracks at the interfaces between grains. To prevent the crack faces from self-penetrating during unloading, a contact condition is enforced. The solutions for the mechanical fields and the damage field are separately obtained and an explicit damage update algorithm allows using a non-iterative approach. The damage formulation couples the normal and tangential failure modes, accounts for different tension and compression behaviours and takes into account a compression-dependent fracture energy in mixed mode. The framework is applied to complex 2D problems inspired by indirect tension tests and compression tests on heterogeneous rock-like materials.

... A majority of numerical simulations of coupled poromechanics have employed mesh-based Lagrangian methods that trace solid material points (e.g. [9][10][11][12][13][14][15][16][17][18]). These Lagrangian methods can straightforwardly incorporate loading and deformation history in the solid material, which is a crucial advantage over Eulerian methods because many porous materials such as soils show strong nonlinearity and history-dependence. ...

The material point method (MPM) has been increasingly used for the simulation of large-deformation processes in fluid-infiltrated porous materials. For undrained poromechanical problems, however, standard MPMs are numerically unstable because they use low-order interpolation functions that violate the inf–sup stability condition. In this work, we develop stabilized MPM formulations for dynamic and quasi-static poromechanics that permit the use of standard low-order interpolation functions notwithstanding the drainage condition. For the stabilization of both dynamic and quasi-static formulations, we utilize the polynomial pressure projection method whereby a stabilization term is augmented to the balance of mass. The stabilization term can be implemented with both the original and generalized interpolation material point (GIMP) methods, and it is compatible with existing time-integration methods. Here we use fully-implicit methods for both dynamic and quasi-static poromechanical problems, aided by a block-preconditioned Newton–Krylov solver. The stabilized MPMs are verified and investigated through several numerical examples under dynamic and quasi-static conditions. Results show that the proposed MPM formulations allow standard low-order interpolation functions to be used for both the solid displacement and pore pressure fields of poromechanical formulations, from undrained to drained conditions, and from dynamic to quasi-static conditions.

... Constitutive or material laws that provide the explicit relations among strain history, state variables and stress are the key component to supplement hard constraints such as balance principles and thermodynamic laws for single-phase solid mechanics problems [Kirchdoerfer and Ortiz, 2016, Eggersmann et al., 2019, Wang et al., 2019. These constitutive laws are often proposed based on abstractions and generalizations of phenomenological and experimental observations [Drucker, 1950, Green, 1972, Schofield and Wroth, 1968, Sun, 2013, Na and Sun, 2017, Bryant and Sun, 2019. However, the recent success in machine learning and mining on big data has become an impetus for an alternative data-driven approach, where these manual abstractions and generalization processes are either bypassed through a new optimization procedure that minimizes the distance between points in dataset and the constraint (equilibrium and compatibility) (cf. ...

This paper examines the frame-invariance (and the lack thereof) exhibited in simulated anisotropic elasto-plastic responses generated from supervised machine learning of classical multi-layer and informed-graph-based neural networks, and proposes different remedies to fix this drawback. The inherent hierarchical relations among physical quantities and state variables in an elasto-plasticity model are first represented as directed graphs, where three variations of the graph are tested. While feed-forward neural networks are used to train path-independent constitutive relations (e.g., elasticity), recurrent neural networks are used to replicate responses that depends on the deformation history, i.e. or path dependent. In dealing with the objectivity deficiency, we use the spectral form to represent tensors and, subsequently, three metrics, the Euclidean distance between the Euler Angles, the distance from the identity matrix, and geodesic on the unit sphere in Lie algebra, can be employed to constitute objective functions for the supervised machine learning. In this, the aim is to minimize the measured distance between the true and the predicted 3D rotation entities. Following this, we conduct numerical experiments on how these met-rics, which are theoretically equivalent, may lead to differences in the efficiency of the supervised machine learning as well as the accuracy and robustness of the resultant models. Neural network models trained with tensors represented in component form for a given Cartesian coordinate system are used as a benchmark. Our numerical tests show that, even given the same amount of information and data, the quality of the anisotropic elasto-plasticity model is highly sensitive to the way tensors are represented and measured. The results reveal that using a loss function based on geodesic on the unit sphere in Lie algebra together with an informed directed graph yield significantly more accurate rotation prediction than the other tested approaches.

... We start with the kinematic and constitutive relations for size-dependent micro-polar elastic material undergoing infinitesimal deformation, followed by the principle of minimum potential energy that yields the governing equations to be solved. The corresponding variational form is recovered by the standard procedure while we adopt the Taylor-Hood finite element space for the displacement and microrotation fields that satisfies the Ladyzhenskaya-Babuška-Brezzi (LBB) stability condition [Chapelle and Bathe, 1993, Babuška and Narasimhan, 1997, Sun, 2015, Na and Sun, 2017. The regularization length insensitive phase field formulation is then extended to the plane micropolar elasticity which enables us to simulate cohesive fracture for the higher-order continua. ...

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.

... Accompanying water migration from the unfrozen zone, consolidation is usually triggered by the cryogenic suction developed in the frozen zone (Konrad and Nixon 1994;Tiedje and Guo 2011;Zhang et al. 2016;Wang et al. 2018). Various models for frost heave, at least seven groups, have been developed: (i) capillary theory based on the Laplace surface tension formula (Taber 1930) for primary frost heave; (ii) rigid ice model for the secondary frost heave characterized by the existence of a partially frozen zone (Miller 1972(Miller , 1978; (iii) hydrodynamic model and the corresponding extended thermohydromechanical models (e.g., Harlan 1973;Thomas et al. 2009;Liu and Yu 2011;Li et al. 2015); (iv) poromechanics-based models (Coussy and Monteiro 2007;Zhou and Meschke 2013;Na and Sun 2017); (v) models based on the concept of segregation potential, which is defined as the ratio of the water intake rate and temperature gradient (Konrad and Morgenstern 1980); (vi) porosity rate function, which characterizes frost susceptibility as a property of the soil-ice-water composite (Michalowski 1993;Michalowski and Zhu 2006;Zhang and Michalowski 2015); and (vii) theory of premelting dynamics that focuses on the essen-tial physical interactions (Rempel et al. 2004;Rempel 2007). By considering the intermolecular interactions across thin films being the driving force for frost damage, the theory of pre-melting dynamics provides a powerful tool that takes into account the physical process involved in freezing of soil before developing a mathematical model. ...

Although much effort has been made to develop various frost heave models in the past decades, a simple yet versatile model is still needed for engineering applications. This paper presents a method to estimate frost heave in frozen soil using a macroscopic water flux function that extends the segregation potential to make it applicable for both steady state and transient freezing/thawing states. The formation of individual ice lens is modelled by combining previously developed stress and strain criteria. The water flux function, which includes various factors in accordance with the porosity rate function, can describe the growth of both new and old ice lenses. More importantly, every component of the water flux function is physically explained by the theory of premelting dynamics, where all the influencing factors are traced back to their impacts on the ice volume distribution. The performance of the model is demonstrated via simulations on one-dimensional freezing/thawing processes after the model is validated by a specific case from previous literature. Although adequate data are not available for a more strict experimental verification of the model, we observe that the simulations predict the general course of events together with significant specific features that were identified in previous experimental studies.

... The hardening law for the Cam-Clay type models accommodates the bilogarithmic relationship be-133 tween the specific volume and the preconsolidation presssure under the condition of 0 < c r < c c (e.g., 134 Borja and Tamagnini [1998], Borja [2013], Borja et al. [1997], Na and Sun [2017], Semnani et al. [2016], 135 White and Borja [2008]), that is, ...

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.

... In the literature, the most common choice might be two-field mixed continuous Galerkin (CG) finite elements (FE), in which both the solid deformation and fluid flow problems are approximated by the CG method. See [16][17][18][19][20][21] for some of the recent works that have used mixed CG/CG elements for poromechanical problems and their extensions. In this combination, the use of the CG method for the solid deformation problem is a natural and well-justified choice. ...

Local (element-wise) mass conservation is often highly desired for numerical solution of coupled poromechanical problems. As an efficient numerical method featuring this property, mixed continuous Galerkin (CG)/enriched Galerkin (EG) finite elements have recently been proposed whereby piecewise constant functions are enriched to the pore pressure interpolation functions of the conventional mixed CG/CG elements. While this enrichment of the pressure space provides local mass conservation, it unavoidably alters the stability condition for mixed finite elements. Because no stabilization method has been available for the new stability condition, high-order displacement interpolation has been required for mixed CG/EG elements if undrained condition is expected. To circumvent this requirement, here we develop stabilized formulations for the mixed CG/EG elements that permit equal-order interpolation functions even in the undrained limit. We begin by identifying the inf–sup condition for mixed CG/EG elements by phrasing an enriched poromechanical problem as a twofold saddle point problem. We then derive two types of stabilized formulations, one based on the polynomial pressure projection (PPP) method and the other based on the fluid pressure Laplacian (FPL) method. A key finding of this work is that both methods lead to stabilization terms that should be augmented only to the CG part of the pore pressure field, not to the enrichment part. The two stabilized formulations are verified and investigated through numerical examples involving various conditions ranging from 1D to 3D, isotropy to anisotropy, and homogeneous to heterogeneous domains. The methodology presented in this work may also help stabilize other types of mixed finite elements in which the constraint field is enriched by additional functions.

... The implementation of the spatial discretization is done using the finite element library deal.ii [81], whereas the implicit nonlinear PDE solver, including the assembly procedure of the residuals and the corresponding tangents, and the Newton-Raphson scheme are modified from the software code base geocentric [79,82,83,126,130,151]. ...

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.

... When the defects are partially or fully filled with pore fluid, the size and the geometric features of the defects may both impose significant changes on both the effective stiffness and permeability of the materials as well as the Biot's coefficient [30] and the undrained and drained shear strength [25,32]. For instance, carbonate rocks and limestone often contain pores of profoundly different sizes [5,7,17,27,28]. ...

This study presents a phase field model for brittle fracture in fluid-infiltrating vuggy porous media. While the state-of-the-art in hydraulic phase field fracture considers Darcian fracture flow with enhanced permeability along the crack, in this study, the phase field not only acts as a damage variable that provides diffuse representation of cracks or cavities, but also acts as an indicator function that separates the domain into two regions where fluid flows are governed by Stokes and Darcy equations, respectively. Since the phase field and its gradient can be respectively regarded as smooth approximations of the Heaviside function and Dirac delta function, our new approach is capable of imposing interfacial transmissibility conditions without explicit interface parametrizations. In addition, the interaction between solid and fluid constituents is modeled by adopting the concept of mixture theory, where the fluid velocities in Stokes and Darcy regions are considered as relative measures compared to the solid motion. This model is particularly attractive for coupled flow analysis in geological materials with complex microstructures undergoing brittle fracture often encountered in energy geotechnics problems, since it completely eliminates the needs to generate specific enrichment function, integration scheme, or meshing algorithm tailored for complex geological features.

... As a general-purpose tool, the proposed framework is equally applicable for addressing engineering challenges beyond 3D printing. Tangible problems of important engineering significance that can be tackled by the proposed framework include methane hydrate exploitation [108], soil freeze and thaw in permafrost regions [109], snow avalanches [110], and glacial ice melting [111]. Methodologically, the high computational cost incurred by the multiphase, multiway coupling magnified by fine temporal and spatial resolutions may pose a major drawback restricting its practical applications for large-scale simulations, the solution of which mandates efficient parallel computing techniques on of high-performance computing facilities. ...

Laser powder bed fusion represents the future for metal additive manufacturing. Advance of this emerging technology is bottlenecked by the unavailability of high-fidelity prediction tools for cost-effective optimization on printing design. Simulations of selective laser melting of metals must tackle a complex granular solids and multiphase fluids system that undergoes intra-and inter-phase interactions and thermal-induced phase changes, including melting, vaporization, and solidification, which are challenging to model. We develop a high-fidelity computational tool to provide high-resolution simulations of the multiphase, multiphysics processes of selective laser melting (SLM). Key to this tool is a multi-phase, semi-coupled resolved Computational Fluid Dynamics (CFD) and Discrete Element Method (DEM). It contains innovative features including (1) a fully resolved immersed boundary CFD with fictitious particle domain coupling with DEM for resolving mechanical interactions and heat transfers between solid particles and surrounding fluid; (2) An evaporation model in consideration of the Knudsen layer implemented in the volume of fluid (VOF) method which is enriched by two sharp interface capture schemes, isoAdvector and MULES, for accurate identification of the vaporization process and phase boundaries of fluids with different Courant numbers; and (3) a ray tracing model compatible with the VOF method for high-resolution of absorbed laser energy. We demonstrate the proposed method can quantitatively reproduce key observations from synchrotron experiments and captures critical interdependent physics involving melt pool morphology evolution, vapor-driven keyhole dynamics and powder motions. This new computational tool opens a new avenue for quantitative design and systematic optimization of laser powder bed fusion and may find wider engineering applications where thermal induced phase changes in a multi-phase system are important.

... Permafrost covers almost a quarter of the land area in the Northern Hemisphere. Many people around the world live in permafrost and seasonal frozen areas in Alaska, Canada, and Russia (Na and Sun [1], Yu et al. [2], Tounsi et al. [3], Marchenko et al. [4], Knoblauch et al. [5]). The ground freezing and thawing process can change the shape of the land surface and damage constructions and buildings in the permafrost area (Zhou and Meschke [6], Sweidan et al. [7], Xu et al. [8]). ...

In this work, we consider a mathematical model and finite element implementation of heat transfer and mechanics of soils with phase change. We present the construction of the simplified mathematical model based on the definition of water and ice fraction volumes as functions of temperature. In the presented mathematical model, the soil deformations occur due to the porosity growth followed by the difference between ice and water density. We consider a finite element discretization of the presented thermoelastic model with implicit time approximation. Implementation of the presented basic mathematical model is performed using FEniCS finite element library and openly available to download. The results of the numerical investigation are presented for the two-dimensional and three-dimensional model problems for two test cases in three different geometries. We consider algorithms with linearization from the previous time layer (one Picard iteration) and the Picard iterative method. Computational time is presented with the total number of nonlinear iterations. A numerical investigation with results of the convergence of the nonlinear iteration is presented for different time step sizes, where we calculate relative errors for temperature and displacements between current solution and reference solution with the largest number of the time layers. Numerical results illustrate the influence of the porosity change due to the phase-change of pore water into ice on the deformation of the soils. We observed a good numerical convergence of the presented implementation with the small number of nonlinear iterations, that depends on time step size.

... This type of material strongly intensifies the significance of internal permafrost defects compared with normal permafrost in terms of both the relative quantity and influence on the soil mechanical properties [8][9][10]. The elastic modulus of frozen soil is assumed to be constant in numerical calculations owing to its high instantaneous strength [11][12][13][14]. However, for warm frozen soil, the deterioration of the frozen soil stiffness is more apparent because of the increased unfrozen water content at higher temperatures [15]. ...

Under the background of rising environmental temperature, the state of warm frozen soil near the phase transition is extremely unstable. In order to explore the relationship between temperature and mechanical properties of warm frozen soil, a damage model of warm frozen soil structure under the coupling of stress and temperature is established based on the strain equivalent theory of damage mechanics. Based on the Mohr-Coulomb criterion, the nominal stress is used to represent the stress damage of frozen soil elements, the initial elastic modulus is used to represent the temperature damage, and a composite damage factor is introduced to describe their coupled relationship. Through a triaxial compression experiment of frozen soil, the experimental data and stress-strain curve are obtained. The full-fitting method based on the experimental data (method 1) and the semitheoretical semifitting method based on the characteristic points of the stress-strain curve (method 2) are used to obtain the shape parameters and scale parameters of the stress-temperature coupled damage model corresponding to different fitting methods. Based on the triaxial compression tests of frozen sand and frozen silty clay, the reliability of the stress-temperature coupled damage model results obtained by the two parameter determination methods under the conditions of strain softening and strain hardening is verified. The results show that both methods are applicable under the condition of strain softening and strain hardening and method 2 is better than method 1 under the condition of strain softening. Compared with the prediction results of the single stress damage model, the stress-temperature coupled damage model can effectively reduce the influence of the parameter estimation error on the results and improve the overall stability of the model.

... Nishimura et al. [14] developed a formulation of the coupled THM finite element code to study freeze and thaw in water-saturated soils. Na and Sun [15] presented a finite strain formulation for frozen porous media to investigate the freeze-thaw action of frozen porous media by applying multiplicative kinematics. Yasuhara et al. [16] developed a coupled THMC model for examining the long-term change in the permeability of porous sedimentary rocks, predicting the expected stress and temperature conditions. ...

This paper presents an experimental investigation on the properties of hydraulic conductivity and permeability of conglomerates under different temperatures and confining pressures with integrated samples and samples with shear failure. Constant head tests were carried out in a temperature-controlled triaxial cell with samples obtained from the Zhuxianzhuang Coal Mine. Five levels of temperatures (10°C, 20°C, 28°C, 35°C, and 50°C) and three levels of confining pressures (3 MPa, 5 MPa, and 7 MPa) were chosen for the tests. The results show that there is a negative relationship between hydraulic conductivity and confining pressure with both original and shear failure samples. An inflection point of 35°C is found in the relationship between the flow rate and temperature. However, with increasing temperature conditions, hydraulic conductivity first increases and then decreases at 50°C with the intact sample, while hydraulic conductivity first decreases from 20°C and then increases with the shear failure sample. Finally, nonlinear regression equations of hydraulic conductivity and temperature were obtained under different confining pressures.

... The material states that sufficiently represent the history that leads to the initial yielding and subsequent plastic deformation can be described by a set of selected descriptors, including stress, strain (e.g. strain-based plasticity, and damage models (Naghdi, 1990;Iai et al., 1992)), chemical potential (Ulm et al., 1999;Hueckel, 1997;, dislocation density (Gurtin et al., 2007;Ma et al., 2021), porosity (Schofield and Wroth, 1968;Paterson and Wong, 2005;Wang and Sun, 2019), volume fractions of constituents (Clayton, 2009;Na and Sun, 2017), as well as internal variables that cannot directly be observed (Rice, 1971). ...

Elastoplasticity models often introduce a scalar-valued yield function to implicitly represent the boundary between elastic and plastic material states. This paper introduces a new alternative where the yield envelope is represented by a manifold of which the topology and the geometry are learned from a set of data points in a parametric space (e.g. principal stress space, pi-plane). Here, deep geometric learning enables us to reconstruct a highly complex yield envelope by breaking it down into multiple coordinate charts. The global atlas that consists of these coordinate charts in return allows us to represent the yield surface via multiple overlapping patches, each with a specific local parametrization. This setup provides several advantages over the classical implicit function representation approach. For instance, the availability of coordinate charts enables us to introduce an alternative stress integration algorithm where the trial stress may project directly on a local patch and hence circumvent the issues related to non-smoothness and the lack of convexity of yield surfaces. Meanwhile, the local parametric approach also enables us to predict hardening/softening locally in the parametric space, even without complete knowledge of the yield surface. Comparisons between the classical yield function approach on the non-smooth plasticity and anisotropic cam-clay plasticity model are provided to demonstrate the capacity of the models for highly precise yield surface and the feasibility of the implementation of the learned model in the local stress integration algorithm.

Most existing thermal–hydromechanical (THM) models used to describe the process of frost heave assumed the freezing soil to be elastic. However, an inelastic constitutive model capable of reflecting the viscous constitutive behaviour of freezing soils should be considered. Based on the existing mathematical model, this study presented an improved mathematical model of coupled water, heat, and stress for saturated freezing soil, in which the soil was assumed to be elastic-viscoplastic and its viscoplasticity was modelled by means of a simple (linear) Norton–Hoff’s law. In addition, solid–fluid interface energy was considered to formulate the effective water and ice pressures and liquid–crystal equilibrium condition which can be used to explain the micro-cryosuction mechanism was adopted to replace Clapyron equation which were used in most existing models. To solve the nonlinear governing equations, numerical simulations were performed using COMSOL software. Finally, the improved model was validated by comparing its simulation results with reference model and the distribution curves of frost heave, temperature, water content and flux rate were discussed.

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.

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.

We present a variational approach towards identifying conditions for the stability of Galerkin methods for multi-field saddle point problems in continuum mechanics and continuum thermodynamics with free energy functions that have positive second variations at critical points. The framework aims to generalize the discrete inf–sup theory and its numerical verification in the context of general problems in an arbitrary number of fields for both linear and nonlinear problems. In utilizing a linearized second derivative test the proposed scheme is purely based on uniqueness properties of a mixed Lagrangian around the solution, thus providing a purely variational requirement for the well-posedness of Galerkin approximations. In maintaining the dependence on parameters and in incorporating the notion of multifold inf–sup conditions the scheme provides a generalization of the classical LBB theory that allows the study of multi-field saddle point problems in the limit of vanishing parameters and their connection to generalized numerical eigenvalue tests. The proposed generalization is trusted to provide a helpful tool for the development of mixed methods that arise in many novel engineering problems due to the coupling of multiple irreversible phenomena or the incorporation of Lagrange multipliers in advanced methods. An emphasis is given with regard to time-dependent problems in irreversible thermodynamics that arise from Biot-type variational formulations if conjugate variables are employed by means of a Legendre–Fenchel transformation. Examples are given for a two-field variational principle in finite deformation poroelasticity in the undrained limit, a three-field variational principle for elasticity in the incompressible limit, and a recent four-field variational principle in gradient-extended plasticity.

Heat generation and transfer in a granular material can be intricately coupled with their mechanical responses, playing a key role in causing excessive large deformation, flow and failure of the material. The coupling may manifest in various forms, including thermal induced stress, mechanically induced heat and thermally induced melting in granular media. We propose a novel hierarchical multiscale modeling framework, TM-DEMPM, to model the coupled thermo-mechanical behavior in granular media which may undergo large deformation and flow. Material Point Method (MPM) is hierarchically coupled with Discrete Element Method (DEM) to offer physics-based, natural scale-crossing simulations of thermo-mechanical granular responses without assuming complicated phenomenological con-stitutive models. To offer speedup for the numerical solution, hybrid OpenMP and GPU-based parallelization is proposed to take advantage of the hierarchical computing structure of the framework. The proposed framework may provide an effective and efficient pathway to next-generation simulation of engineering-scale large-deformation problems that involve complicated thermo-mechanical coupling in granular media.

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.

Для освоения криолитозоны необходимо строить и численно реализовывать математические модели мультифизичных процессов термоупругости с фазовыми переходами первого рода в основаниях инженерных сооружений и зданий. Численная реализация таких моделей связана с вычислительными трудностями из-за наличия в прикладных задачах различного вида неоднородностей и нелинейности определяющих уравнений, для чего требуются очень мелкие сетки, увеличивающие вычислительные затраты. На основе обобщенного многомасштабного метода конечных элементов разработан численный метод решения задачи термоупругости с фазовыми переходами, главная идея которого состоит в построении многомасштабных базисных функций, учитывающих неоднородности среды. Аппроксимация на мелкой сетке проводится с помощью метода конечных элементов со стандартными линейными базисными функциями. Для проверки точности метода численно найдены решения двумерной и трехмерной задач в неоднородных средах. Результаты показывают, что многомасштабный метод может обеспечить хорошую аппроксимацию решения задачи термоупругости с фазовым переходом на мелкой сетке при значительном сокращении размерности дискретной задачи.

This paper presents a combined experimental-modeling effort to interpret the coupled thermo-hydro-mechanical behaviors of the freezing soil, where an unconfined, fully saturated clay is frozen due to a temperature gradient. By leveraging the rich experimental data from the microCT images and the measurements taken during the freezing process, we examine not only how the growth of ice induces volumetric changes of the soil in the fully saturated specimen but also how the presence and propagation of the freezing fringe front may evolve the anisotropy of the effective media of the soil-ice mixture that cannot be otherwise captured phenomenologically in the isotropic saturation-dependent critical state models for plasticity. The resultant model is not only helpful for providing a qualitative description of how freezing affects the volumetric responses of the clayey material, but also provide a mean to generate more precise predictions for the heaving due to the freezing of the ground.

In frozen soils, a portion of pore water remains unfrozen due to the effects of capillarity,
adsorption, and possibly solute. The variation of the amount of unfrozen water and ice in a
frozen soil, which is primarily influenced by subzero temperature, has great impacts on the
physical and mechanical behavior of the soil and is critical for broad applications ranging from
engineering to climate change. In the present study, the various methods that have been used
for determining unfrozen water (and ice) content are comprehensively reviewed. Their principles,
assumptions, advantages, and limitations are discussed. It is noted that there is yet no
perfect way to accurately quantify unfrozen water content in frozen soils. In addition, the soilfreezing
characteristic curve (SFCC) of a typical volcanic soil sampled in the Hokkaido prefecture
of Japan is investigated. The unfrozen water content of the prepared soil specimens was
measured using a cheap moisture sensor, which is based on the frequency domain reflectometry
technique. The temperature of the specimens was determined by a rugged temperature
sensor. Different numbers of freeze-thaw (F-T) cycles and different freezing/thawing methods
(i.e., one- and three-dimensional) were considered, and their effects on the SFCC were investigated.
The experimental results suggest that neither the F-T cycles nor the freezing/thawing
methods had significant influence on the measured SFCC. The presented comprehensive review
and experimental investigations are of importance for both the scientific and engineering
communities.

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.

A hierarchical multiscale coupling of the finite element method (FEM) and the discrete element method (DEM) is proposed to model coupled thermo-mechanical behavior of granular materials. The DEM is employed to model the thermo-mechanical responses of a repre- sentative volume element (RVE, a granular assembly) embedded at a Gauss (quadrature) point of the FEM. The material responses derived from each Gauss point feed two super-imposed FEM to find global solutions subject to two concurrent boundary value problems (BVPs), i.e., heat conduction and mechanical deformation. The two concurrent FEMs exchange information on temperature change and fabric variation at their commonly shared Gauss points. The proposed approach is benchmarked by two examples of transient and steady-state thermal conduction where analytical solutions are available. It is further applied to investigating the thermo-mechanical responses of confined granular columns under cyclic thermal loads with emphasis placed on the effect of boundary condition and inherent anisotropy of a granular column. The proposed approach offers a novel multiscale pathway to model thermo-mechanical responses of granular media based on sound physics.

The description of the freezing characteristics of porous media is one of the most conspicuous ingredients in flow and heat transport models that involve freezing and thawing processes. Unfrozen liquid content (ULC) shows strong hysteresis during freezing and thawing cycles in different types of soils and other porous media. We discuss the possible mechanisms of hysteresis in porous media and develop a numerical model for the unfrozen liquid content that is capable of describing the hysteresis phenomenon in freezing and thawing cycles. We present a coupled finite element model as the framework for the numerical simulation of fluid flow and heat transport in partially frozen porous media. The implementation aspects of the ULC model as well as its integration into numerical codes are discussed in detail. We investigate the potential impact of the hysteresis phenomenon on the numerical simulation of transport processes in porous media through benchmark examples and validate the behavior of the model against available laboratory measurement data.

Coupled thermo-hydro-mechanical models are commonly used to model the evolution of temperature, pore pressure, and stress in a wide range of geotechnologies such as geothermal applications or around canisters of high-level radioactive waste in deep underground storage facilities. Their numerical modeling is often computationally highly demanding, especially if parameter identification, sensitivity analyses or uncertainty quantification require many model evaluations. Often, the thermally driven pore pressure evolution and the subsequently altered flow processes are the primary targets of an analysis. To benefit from the computational efficiency of hydro-thermal (HT) models while maintaining the accuracy of the thermo-hydro-mechanical (THM) model, we derived two cases of a simplified representation of mechanical deformations in a coupled hydro-thermal model. Deformations induced by pressure as well as temperature changes are consistently incorporated into the mass balance storage terms. We demonstrate the exact coincidence of THM and modified TH formulations in isotropic and orthotropic materials as long as the basic assumptions like constant hydrostatic stress conditions or uniaxial strain hold. By modeling of a point heat source in isotropic or anisotropic porous media it is shown that a good agreement between TH and THM models can be maintained even though the assumptions underlying the simplification are no longer valid exactly. On our test-machine, a significant speed-up could be achieved by the reduction of the problem size when transitioning from a THM to a TH model. The highest speed-ups were achieved when Taylor-Hood elements were employed in order to avoid the problem of spurious pressure oscillations in the fully coupled THM model.

This research work introduces a novel phase-field thermo-hydro-mechanical (P-THM) modeling approach that allows to deeply understand and model the freezing–thawing cyclic process in a fluid-saturated porous medium. In this, a biphasic macroscopic, non-isothermal porous media model, augmented by the phase-field method (PFM), is applied to account for the temperature development, the interstitial pore-fluid flow, and the volumetric deformations due to ice formation (phase change). Utilizing the theory of porous media (TPM) in the continuum mechanical formulation provides a well-founded basis for the description of deformable, fluid-saturated, non-isothermal porous solid materials. Of particular importance in the underlying work is the unified kinematics treatment of the ice and water constituents as a single pore-fluid, where the PFM is employed for the description of the phase transition between both constituents. The PFM is a diffuse-interface approach that relies on the specification of the free energy density function as the main driving force in the phase transition. It employs a scalar-valued, phase-field variable to indicate the state of the pore-fluid, i.e., a solid (ice) or a liquid (water). A significant virtue of using the PFM approach is its viability in the implementation within standard finite element frameworks, as no need to explicitly track the moving boundaries (interfaces) of the phase-change constituent. The numerical examples and comparisons presented at the end of the manuscript demonstrate the ability, reliability and usefulness of the proposed modeling framework in describing the freezing–thawing process in a saturated porous solid under thermal loading within an elastic deformation limit.

Supervised machine learning via artificial neural networks (ANN) has gained significant popularity for many geomechanics applications that involves multi-phase flow and poromechanics. For unsaturated poromechanics problems, the multi-physics nature and the complexity of the hydraulic laws make it difficult to design the optimal setup, architecture, and hyper-parameters of the deep neural networks. This paper presents a meta-modeling approach that utilizes deep reinforcement learning (DRL) to automatically discover optimal neural network settings that maximize a pre-defined performance metric for the machine learning constitutive laws. This meta-modeling framework is cast as a Markov Decision Process (MDP) with well-defined states (subsets of states representing the proposed neural network (NN) settings), actions, and rewards. Following the selection rules, the artificial intelligence (AI) agent, represented in DRL via NN, self-learns from taking a sequence of actions and receiving feedback signals (rewards) within the selection environment. By utilizing the Monte Carlo Tree Search (MCTS) to update the policy/value networks, the AI agent replaces the human modeler to handle the otherwise time-consuming trial-and-error process that leads to the optimized choices of setup from a high-dimensional parametric space. This approach is applied to generate two key constitutive laws for the unsaturated poromechanics problems: (1) the path-dependent retention curve with distinctive wetting and drying paths. (2) The flow in the micropores, governed by an anisotropic permeability tensor. Numerical experiments have shown that the resultant ML-generated material models can be integrated into a finite element (FE) solver to solve initial-boundary-value problems as replacements of the hand-craft constitutive laws. Keywords Deep reinforcement learning, neural network settings, unsaturated porous media, retention curve, anisotropic permeability.

Phase field modeling of coupled crystal plasticity and deformation 1 twinning in polycrystals with monolithic and splitting solvers 2 Ran Ma · WaiChing Sun 3 4 Abstract For some polycrystalline materials such as austenitic stainless steel, magnesium, TATB, and HMX, 6 twinning is a crucial deformation mechanism when the dislocation slip alone is not enough to accommodate 7 the applied strain. To predict this coupling effect between crystal plasticity and deformation twinning, 8 we introduce a mathematical model and the corresponding monolithic and operator splitting solver that 9 couples the crystal plasticity material model with a phase field twining model such that the twinning 10 nucleation and propagation can be captured via an implicit function. While a phase-field order parameter 11 is introduced to quantify the twinning induced shear strain and corresponding crystal reorientation, the 12 evolution of the order parameter is driven by the resolved shear stress on the twinning system. To avoid 13 introducing an additional set of slip systems for dislocation slip within the twinning region, we introduce a 14 Lie algebra averaging technique to determine the Schmid tensor throughout the twinning transformation. 15 Three different numerical schemes are proposed to solve the coupled problem, including a monolithic 16 scheme, an alternating minimization scheme, and an operator splitting scheme. Three numerical examples 17 are utilized to demonstrate the capability of the proposed model, as well as the accuracy and computational 18 cost of the solvers. 19

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.

To uilize the shallow geothermal energy, heat pumps are often coupled
with borehole heat exchangers (BHE) for heating and cooling
buildings. In cold regions, soil freezing around the BHE is a potential
problem which can seriously influence the underground soil temperature
distribution, inlet and outlet fluid temperature of the BHE, and
thus the efficiency of the whole GSHP system. The influence of the
freezing process on the overall system performance is investigated by
comparing different BHE configuration with and without latent heat
effect from the frozen groundwater. The coefficient of performance
(COP) of the heat pump will alter when freezing process in taken into
account and lead to various electricity consumption. Except for the
efficiency aspect, the freezing behavior can also lead to the redistribution
of pore pressure and fluid flow, and in some extreme cases
can even result in frost damage to the BHEs. A fully coupled thermohydro-
mechanical model is required for advanced system design and
scenario analyses. Based on the framework of the Theory of Porous
Media, a triphasic freezing model is derived and solved with the finite
element method. Ice formation in the porous medium results from
a coupled heat and mass transfer problem with phase transition and
is accompanied by volume expansion. The model is able to capture
various coupled physical phenomena through the freezing process including
the latent heat effect, groundwater flow with porosity change
and mechanical deformation. With this kind of THM freezing model,
we are also able to solve different kinds of engineering problem, e.g.
geotechnics, construction engineering and material engineering.

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

Northern peatlands contain a large terrestrial carbon pool that plays an important role in the Earth's carbon cycle. A considerable fraction of this carbon pool is currently in permafrost and is biogeochemically relatively inert; this will change with increasing soil temperatures as a result of climate warming in the 21st century. We use a geospatially explicit representation of peat areas and peat depth from a recently-compiled database and a geothermal model to estimate northern North America soil temperature responses to predicted changes in air temperature. We find that, despite a widespread decline in the areas classified as permafrost, soil temperatures in peatlands respond more slowly to increases in air temperature owing to the insulating properties of peat. We estimate that an additional 670 km<sup>3</sup> of peat soils in North America, containing ~33 Pg C, could be seasonally thawed by the end of the century, representing ~20% of the total peat volume in Alaska and Canada. Warming conditions result in a lengthening of the soil thaw period by ~40 days, averaged over the model domain. These changes have potentially important implications for the carbon balance of peat soils.

This paper provides an overview of the new features of the finite element library dealii version 8.2.

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.

The responsibility for the scope of geotechnical engineering lies strongly in the hands of geotechnical engineers with a vision for the discipline. These engineers will need to come from consulting practices, the construction industry, government agencies and universities.

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

The objective of this research is to use grain-scale numerical simulations to analyze the evolution of stress anisotropy exhibited in wetted granular matters. Multiphysical particulate simulations of unsaturated granular materials were conducted to analyze how the interactions of contact force chains and liquid bridges influence the macroscopic responses under various suction pressure and loading history. To study how formation and rupture of liquid bridges affect the mechanical responses of wetted granular materials, a series of suction-controlled triaxial tests were simulated with two grain assemblies, one composed of large particles of similar sizes, another one composed of a mixture of large particles with significant amount of fines. Our results indicate that capillary stress are anisotropic in both sets of specimens, and that the stress anisotropy is more significant in granular assemblies filled with fine particles. A generalized tensorial Bishop's coefficient is introduced to analyze the connections between microstructrual attributes and macroscopic responses. Numerical simulations presented in this paper indicate that the principal values and directions of this Bishop's coefficient tensor are path dependent.

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

The authors start with an introduction to the concepts involved in physics giving the equations of flow through porous media and the deformation characteristics of soils and rocks. Succeeding chapters deal with the practical implications of these phenomena and explain the application of theory in both experimental and field work. Details are given of actual incidents, such as the subsidence experienced in Venice and Ravenna. The authors have also formulated a consolidation code, which is detailed at the end of the book, and provide instructions on how to modify the given program.

Since An Overview of the Trilinos Project [ACM Trans. Math. Softw. 31(3) (2005), 397-423] was published in 2005, Trilinos has grown significantly. It now supports the development of a broad collection of libraries for scalable computational science and engineering applications, and a full-featured software infrastructure for rigorous lean/agile software engineering. This growth has created significant opportunities and challenges. This paper focuses on some of the most notable changes to the Trilinos project in the last few years. At the time of the writing of this article, the current release version of Trilinos was 10.12.2.

A soil freezing characteristic (SFC) represents the relationship between the quantity and the energy status of liquid water in frozen soil The SFC is the analogue to the soil moisture characteristic (SMC) and is essential to modeling the transport of water, heat, and solutes in frozen soil. This paper presents a new, automated technique to measure an SFC in situ, for which there has previously been no method. Liquid water content in frozen soil was measured with time domain reflectometry. The corresponding energy status was inferred from accurate soil temperature measurements with a generalized form of the Clapeyron equation. Since both SFC and SMC describe water retention properties in soil, their similarity was investigated. The SMC and SFC agreed to within 1% moisture content across a wide range of matric potentials. Determination of the SMC is reliable at high matric potentials but becomes increasingly inaccurate and time consuming as soil dries. By contrast, the SFC determination becomes more accurate and rapid at lower matric potentials. We thus propose that water retention properties at high matric potentials are best obtained from draining and at low matric potentials from freezing.

Miscible multiphasic materials like classical mixtures as well as immiscible materials like saturated and partially saturated porous media can be successfully described on the common basis of the well-founded Theory of Mixtures (TM) or the Theory of Porous Media (TPM). In particular, both the TM and the TPM provide an excellent frame for a macroscopic description of a broad variety of engineering applications and further problems in applied natural sciences. The present article portrays both the standard and the micropolar approaches to multiphasic materials reflecting their mechanical and their thermodynamical frameworks. Including some constitutive models and various illustrative numerical examples, the article can be understood as a reference paper to all the following articles of this volume on theoretical, experimental and numerical investigations in the Theory of Porous Media.

We generalize the multiscale overlapped domain framework to couple multiple rate-independent standard dissipative material models in the finite deformation regime across different length scales. We show that a fully coupled multiscale incremental boundary-value problem can be recast as the stationary point that optimizes the partitioned incremental work of a three-field energy functional. We also establish inf-sup tests to examine the numerical stability issues that arise from enforcing weak compatibility in the three-field formulation. We also devise a new block solver for the domain coupling problem and demonstrate the performance of the formulation with one-dimensional numerical examples. These simulations indicate that it is sufficient to introduce a localization limiter in a confined region of interest to regularize the partial differential equation if loss of ellipticity occurs.

Frost heave is the process by which the freezing of water-saturated soil causes the deformation and upward thrust of the ground surface. We describe the fundamental interactions between phase change and fluid flow in partially frozen, saturated porous media (soils) that are responsible for frost heave. Water remains only partially frozen in a porous medium at temperatures below $0\,^\circ$C owing both to the depression of the freezing temperature at curved phase boundaries and to interfacial premelting caused by long-range intermolecular forces. We show that while the former contributes to the geometry of fluid pathways, it is solely the latter effect that generates the forces necessary for frost heave. We develop a simple model describing the formation and evolution of the ice lenses (layers of ice devoid of soil particles) that drive heave, based on integral force balances. We determine conditions under which either (i) a single ice lens propagates with no leading frozen fringe, or (ii) a single, propagating ice lens is separated from unfrozen soil by a partially frozen fringe, or (iii) multiple ice lenses form.

Thaw consolidation of ice-rich permafrost is a typical problem in cold regions engineering. This paper proposes a three dimensional analysis of large strain thaw consolidation for post-thawed zone of permafrost, which is defined by a moving thawing boundary problem with phase changes. The theory is implemented in a numerical code and the numerical results are compared with thaw consolidation tests. For problems with low water contents, the small and large strain methods provide virtually the same results. For problems with high water contents, however, the large strain theory shows a much better performance.

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.

Northern peatlands contain a large terrestrial carbon pool that plays an important role in the Earth's carbon cycle. A considerable fraction of this carbon pool is currently in permafrost and is biogeochemically relatively inert; this will change with increasing soil temperatures as a result of climate warming in the 21st century. We use a geospatially explicit representation of peat areas and peat depth from a recently-compiled database and a geothermal model to estimate northern North America soil temperature responses to predicted changes in air temperature. We find that, despite a widespread decline in the areas classified as permafrost, soil temperatures in peatlands respond more slowly to increases in air temperature owing to the insulating properties of peat. We estimate that an additional 670 km3 of peat soils in North America, containing ~33 Pg C, could be seasonally thawed by the end of the century, representing ~20 % of the total peat volume in Alaska and Canada. Warming conditions result in a lengthening of the soil thaw period by ~40 days, averaged over the model domain. These changes have potentially important implications for the carbon balance of peat soils.

A new method is presented to account for phase changes in a fully implicit numerical model for coupled heat transport and variably saturated water flow involving conditions both above and below zero temperature. The method is based on a mixed formulation for both water flow and heat transport similar to the approach commonly used for the Richards equation. The approach enabled numerically stable, energy- and mass-conservative solutions. The model was evaluated by comparing predictions with data from laboratory column freezing experiments. These experiments involved 20-cm long soil columns with an internal diameter of 8 cm that were exposed at the top to a circulating fluid with a temperature of −6°C. Water and soil in the columns froze from the top down during the experiment, with the freezing process inducing significant water redistribution within the soil. A new function is proposed to better describe the dependency of the thermal conductivity on the ice and water contents of frozen soils. Predicted values of the total water content compared well with measured values. The model proved to be numerically stable also for a hypothetical road problem involving simultaneous heat transport and water flow. The problem was simulated using measured values of the surface temperature for the duration of almost 1 yr. Since the road was snow-plowed during winter, surface temperatures varied more rapidly, and reached much lower values, than would have been the case under a natural snow cover. The numerical experiments demonstrate the ability of the code to cope with rapidly changing boundary conditions and very nonlinear water content and pressure head distributions in the soil profile.

A new and relatively simple equation for the soil-water content-pressure head curve is described. The particular form of the equation enables one to derive closed-form analytical expressions for the relative hydraulic conductivity, when substituted in the predictive conductivity models of N. T. Burdine or Y. Mualem. The resulting expressions contain three independent parameters which may be obtained by fitting the proposed soil-water retention model to experimental data. Results obtained with the closed-form analytical expressions based on the Mualem theory are compared with observed hydraulic conductivity data for five soils with a wide range of hydraulic properties.

A fully coupled thermo-hydro-mechanical (THM) finite element (FE) formulation is presented that considers freezing and thawing in water-saturated soils. The formulation considers each thermal, hydraulic and mechanical process, and their various interactions, through fundamental physical laws and models. By employing a combination of ice pressure, liquid pressure and total stress as state variables, a new mechanical model has been developed that encompasses frozen and unfrozen behaviour within a unified effective-stress-based framework. Important frozen soil features such as temperature and porosity dependence of shear strength are captured inherently by the model. Potential applications to geotechnics include analysis of frost heave, foundation stability or mass movements in cold regions. The model's performance is demonstrated with reference to the in situ pipeline frost heave tests conducted by Slusarchuk et al. Detailed consideration is given to FE mesh design, the influence of hydraulic parameters, and the treatment of air/ground interface boundary conditions. The THM simulation is shown to reproduce, with fair accuracy, the observed pipeline heave and the porosity growth driven by water migration.

The linearity of e-log p' curves as a description of the change of volume of a soil with variations in mean effective stress (p') has become one of the central empirical relationships of particulate mechanics. But it has important shortcomings, and this technical note explores an alternative approach. The new compression law is applied to the Cam Clay model. -Keith Clayton

A class of two-invariant stored energy functions describing the hyperelastic characteristics of soils is coupled with a critical-state plasticity model. The functions include constant as well as pressure-dependent elastic shear modulus models, and automatically satisfy the requirement that the elastic response for any loading path be energy conserving. The elastic responses predicted by the hyperelastic model are compared with measured undrained elastic responses of an overconsolidated clay in order to assess, both qualitatively and quantitatively, the predictive capability of the hyperelastic model. The importance of the pressure-dependent nature of the elastic shear modulus is assessed within the context of elastic and plastic responses. An energy-conserving model provides a fundamentally correct description of elastic material behavior even in the regime of plastic responses.

The present work discusses a finite-strain plasticity model for soft clays. To motivate such a model, the infinitesimal-strain assumption is shown to be inadequate for the constitutive description of soft clays. Hence, assuming the multiplicative elasto-plastic decomposition of the deformation gradient, a finite-strain Cam-clay model is presented. In particular, with respect to the original Cam-clay formulation, this model improves the description of the isotropic compression behaviour as well as of the elastic shearing response. The constitutive laws are discussed and their implications are pointed out. The physical meaning of the model parameters is carefully addressed. Finally, the ability to properly match some experimental results available in the literature is assessed.

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.

We study unconditionally stable sequential methods for the all-way coupled thermoporomechanical problems. We first propose two sequential methods: The undrained-adiabatic split that combines the undrained split in poromechanics with the adiabatic split in thermomechanics, and the extended fixed- stress split. We perform new stability and convergence analysis for the undrained-adiabatic and extended fixed-stress split methods, introducing a new extended norm for nonlinear stability analysis, which can cover all-way coupled thermoporomechanical problems. In this study we show that the two methods are unconditionally stable (i.e., contractive and B-stable), when we take implicit time stepping. We also perform spectral analysis in order to investigate convergence of the two methods when linearizing the coupled problem. The spectral analysis will be useful for designing reliable pre-conditioners of the monolithic method. The spectral analysis shows that the two sequential methods are convergent and that the extended-fixed stress split is more accurate than the undrained-adiabatic split for strong coupling. We show numerical examples, which support the a-priori stability and convergence estimates.

The Harbin-Dalian high-speed railway in northeastern China has a significant portion of track foundation built on seasonally frozen ground. Wide-spread frost heave was observed during the first winter of its operation and the heave occurred mainly in coarse fills that were considered not susceptible to frost heave. This paper first presents the field data of frost heave and frost depth observed along the railway. It then analyses alternative mechanisms that have been considered to have caused the observed frost heave. The three most likely mechanisms are poor quality control of fine content in the coarse fill, the top-down water supply mechanism, and the bottom-up water supply mechanism. The likelihoods of these mechanisms are analysed against observed field data, using a one-dimensional frost heave model. The results indicate that the most likely explanation for the unexpected frost heave is a combined action of different mechanisms. © 2016, National Research Council of Canada. All Rights Reserved.

Shear banding, as an unstable process of localization, is a common precursor to fracture in materials under high strain rate loadings, making the detection of the instability point after which localization will occur of significant importance. Stability analysis based on the perturbation method or the acoustic tensor have been employed. However, these methodologies are limited to certain class of problems and are difficult to generalize. In this work we propose an alternative for identifying the instability point by employing the concept of generalized stability analysis. In this framework, a stability measure is obtained by computing the instantaneous growth rate of the vector tangent to the solution. Such an approach is more appropriate for non-orthogonal problems and is easier to generalize to difficult dynamic fracture problems. Under conditions where the local instability triggers the non-homogeneous solution growth, i.e. problems that are homogeneous until the appearance of local instability, the generalized stability analysis and the modal stability analysis will closely match. Therefore, the non-homogeneous growth can be approximated by the Rayleigh Quotient of the vector tangent to the solution, which is easier to compute. We show that for a particular class of problems that respect the aforementioned conditions, in 1D and 2D examples, both quantities successfully find the instability point predicted analytically and validated experimentally in past literature results. This methodology is general and can be applied to a wide array of dynamic fracture problems, for which instability that leads to localization is important.