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Coupled Phase-Field and Plasticity Modeling of Geological Materials: from Brittle Fracture to Ductile Flow

Coupled Phase-Field and Plasticity Modeling of Geological Materials: from Brittle Fracture to Ductile Flow

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Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 www.elsevier.com/locate/cma

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

Jinhyun Chooa,b,∗, WaiChing Suna

a Department of Civil Engineering and Engineering , Columbia University, New Work, NY 10027, USA b Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong

Received 12 July 2017; received in revised form 2 October 2017; accepted 4 October 2017 Available online 19 October 2017

Abstract

The failure behavior of geological materials depends heavily on confining and strain rate. Under a relatively low confining pressure, these materials tend to fail by brittle, localized fracture, but as the confining pressure increases, theyshowa growing propensity for ductile, diffuse failure accompanying flow. Furthermore, the rate of often exerts control on the . Here we develop a theoretical and computational modeling framework that encapsulates this variety of failure modes and their brittle–ductile transition. The framework couples a pressure-sensitive plasticity model with a phase-field approach to fracture which can simulate complex fracture propagation without tracking its geometry. We derive a phase-field formulation for fracture in elastic–plastic materials as a balance law of microforce, in a new way that honors the dissipative nature of the fracturing processes. For physically meaningful and numerically robust incorporation of plasticity into the phase-field model, we introduce several new ideas including the use of phase-field effective for plasticity, and the dilative/compactive split and rate-dependent storage of plastic work. We construct a particular class of the framework by employing a Drucker–Prager plasticity model with a compression cap, and demonstrate that the proposed framework can capture brittle fracture, ductile flow, and their transition due to confining pressure and strain rate. ⃝c 2017 Elsevier B.V. All rights reserved.

Keywords: Geomaterials; Phase field; Plasticity; Fracture; Strain localization; Brittle–ductile transition

1. Introduction Geological materials like rocks and may fail in a variety of modes [1–12]. The failure mode ranges from brittle fracture to deformation banding to diffuse plastic flow, and it may undergo transition due to various external factors like stress and temperature fields. Because geological materials in the field are subject to a wide range ofexternal conditions, understanding and predicting this array of failure modes and their transition is crucial for many problems in engineering. Notable examples include geologic hazards, infrastructure failure, subsurface energy production, and disposal of hazardous wastes and greenhouse [13–19].

∗ Corresponding author at: Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong. E-mail addresses: [email protected] (J. Choo), [email protected] (W. Sun). https://doi.org/10.1016/j.cma.2017.10.009 0045-7825/⃝c 2017 Elsevier B.V. All rights reserved. 2 J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32

Fig. 1. The brittle–ductile transition in laboratory samples of a porous under different confining : (a) 0.1 MPa (axial splitting) (b)49 MPa (shear fracture) (c) 98 MPa (shear band) (d) 147 MPa (diffuse plastic flow). After Hoshino et al.[2].

Fig. 2. Effect of displacement rate on brittle behavior of Arkose sandstone in uniaxial compression tests. Numbers in the figure denote displacement rate in mm/s. (The diameter and height of the cylindrical samples were 32 mm and 64 mm, respectively.) After Peng [20].

In this work, we focus on two factors that govern the brittleness/ of geomaterials under non-elevated temperature: confining pressure and strain rate. Geomaterials under a negligible confining pressure are usually brittle because tensile can occur therein. However, most geomaterials in the subsurface are subject to some amount of confining pressure that can inhibit the creation of such tensile fractures. At a relatively low confining pressure, these materials tend to fail by localized shear fractures/deformation bands. This type of failure is usually quasi-brittle, and mainly associated with intergranular microfracture and frictional sliding of grains. However, as the confining pressure increases, these microscopic processes become less activated, and the failure behavior becomes increasingly ductile and diffuse. In this ductile regime, the material manifests compactive plastic deformation instead of dilative strain localization. Microscale mechanisms that lead to this ductile behavior include grain crushing, diffusional mass transfer, and plasticity, among others. Fig.1 shows a laboratory test example of how the failure mode of a porous rock specimen can change by confining pressures. In addition, the rate of deformation often exerts control on the brittleness of a geomaterial. As an example, in Fig.2 we show experimental data of Peng [20] in which the specimens are more brittle at lower displacement rates. Detailed information on the brittle–ductile transition of geological materials can be found from the review paper of Wong and Baud [10] and Chapter 9 of Paterson and Wong [7], among others. J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 3

The purpose of this work is to develop a computational modeling framework that encapsulates this wide range of failure modes at a variety of confining pressures and strain rates. Our purpose is motivated by the factthat most approaches in computational geomechanics have focused on one or two aspects of tensile fracture, shear fracture/localization, and compactive plastic flow. In what follows, we review some of the commonly used approaches to the modeling of brittle and ductile failures of geomaterials. The standard approach to brittle failures of geomaterials under confining pressures is to use a phenomenological criterion for shear strength, such as the Mohr–Coulomb criterion. These failure criteria have also been adopted by the plasticity community, viewing that failure is analogous to the notion of yielding [21]. The resulting pressure- sensitive plasticity models can capture quasi-brittle behavior under low confining pressures, and they have been used in conjunction with stability and bifurcation analysis to mathematically delineate the onset of failure (e.g., [22–27]). These models are, however, unable to consider ductile, compactive behavior under high confining pressures. This limitation can be overcome by employing a cap plasticity model that is designed to capture ductile yielding in the high-pressure regime (e.g., [28–30]). A combination of these two types of plasticity models can address both quasi-brittle and ductile responses and their transition by changes in confining pressure. However, in the quasi-brittle regime whereby softening takes place, plasticity models suffer from pathological mesh sensitivity unless a proper regularization scheme is employed. In addition, plasticity alone cannot take into account the stiffness degradation during cyclic loading, which is significant for many types of geomaterials [31,32]. Lastly, a common drawback of these plasticity models is that they show poor performance for tensile failures. Extrapolation of a shear failure criterion to the tensile regime typically gives unrealistic tensile strengths [33]. Although such a criterion may be modified to fit experimentally measured tensile strengths, it is still inappropriate for capturing responses beyond the onsetof fractures. The use of continuum damage mechanics can overcome some of the aforementioned limitations of plasticity that relate to stiffness degradation and tensile failures. For this reason, a number of studies have proposed coupled damage mechanics and plasticity models (e.g., [34–42]). While these models have shown improved modeling capabilities for brittle responses and associated stiffness degradation, they usually involve complex damage functions that pose significant challenges for parameter determination and numerical implementation. In addition, unless thedamage function is properly regularized, the mesh-sensitivity issue persists. This issue is resolved in a class of gradient damage models that employ nonlocal damage functions. Yet, these nonlocal damage functions are usually more challenging to calibrate, and if defined inappropriately, they can give rise to a non-physical broadening of damage zone[43,44]. can offer a basis for a more mechanically sound, mesh-insensitive approach to brittle failures of geological materials. The application of fracture mechanics to geomechanical problems has been explored since several decades ago (e.g., [45,46]), and it is now an active area of research in various contexts from landslides to fault mechanics to hydraulic fracturing (e.g., [47–54]). Simulating fractures in these geomechanical problems, however, faces theoretical and computational challenges that have not yet been addressed satisfactorily. For example, although the mechanical responses of geological materials strongly depend on confining pressure, they are viewed as pressure independent in fracture mechanics theories. Moreover, it is often very unwieldy to track the geometry of cracks in real geomaterials as most of them are extremely heterogeneous. Phase-field modeling has emerged as an efficient method for computer simulation of fracture in brittle materials (e.g., [55–61]). This method approximates a crack as a diffuse interface – phase field – and captures its inception and propagation by solving a partial differential equation. By doing so, it allows one to handle complex fracture patterns like branching and joining without algorithmic tracking of their geometry. This attractive feature has motivated a number of studies that propose phase-field models of brittle fracture in geomechanical problems (e.g., [62–73]). In most (if not all) of these phase-field models, the material has been idealized as elastic and independent of confining pressure. This idealization, however, can be detrimental to their predictive capabilities for real-world applications whereby geomaterials are subject to some amount of confining pressure. For a more realistic modeling of subsurface fracture processes, a phase-field model must be empowered to accommodate pressure sensitivity inthe failure behavior, including the brittle–ductile transition. Furthermore, because some geomaterials exhibit strong rate dependence as shown in Fig.2, it is also desirable to incorporate possible rate-sensitivity of brittleness. Here we develop a framework that couples a phase-field description of fracture with pressure-sensitive plasticity, for encapsulating a wide range of failure modes of geomaterials from brittle fracture to ductile flow. Particular attention is paid to a realistic capture of the transition of these failure modes with changes in confining pressure and strain rate conditions. In the process of our development, several new contributions are presented to make the two-way coupling 4 J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 between phase-field and plasticity theoretically consistent and numerically robust. They include a microforce-based derivation of a phase-field model that honors the dissipative nature of the fracturing process, the dilative/compactive split and rate-sensitive storage of plastic work, and the use of phase-field effective stress for plasticity. The paper is organized as follows. In Section2, we derive a general formulation for phase-field modeling of elastoplastic materials. Subsequently, in Section3 we develop a constitutive framework for casting a pressure-sensitive plasticity model into the phase-field formulation, and construct a particular class of the framework for illustration purposes. A finite element formulation of the framework is described in Section4. In Section5, we present numerical examples that demonstrate the proposed framework’s capability of capturing an array of failure behaviors of geological materials subjected to various confining pressure and loading rate conditions. The following notations and symbols are used throughout: bold-face letters denote tensors and vectors; the symbol “·” denotes an inner product of two vectors (e.g., a · b = ai bi ), or a single contraction of adjacent indices of two tensors (e.g., c · d = ci j d jk ); the symbol “:” denotes an inner product of two second-order tensors (e.g., c : d = ci j di j ). Following the standard mechanics sign convention, stress is positive in tension and pressure is positive in compression.

2. General formulation The purpose of this section is to derive a general formulation for phase-field modeling of elastoplastic geomaterials susceptible to fractures. Before delving into the details, it is worth reviewing how other researchers have derived phase-field models of fracture and explaining why we pursue a particular direction in thiswork. Previous studies have derived a phase-field model of fracture mainly by two approaches: one relying on variational principles (e.g., [55–59]) and another formulating it as a non-standard balance law of microforce (e.g., [74–76]). The variational approach is built on the framework proposed by Francfort and Marigo [77] which reformulates Griffith’s theory for brittle fracture [78] as an energy minimization problem. This approach has been the common choice of numerous studies that have developed phase-field formulations for brittle fracture in diverse settings (e.g.,[55–59]). For ductile fracture, however, a widely accepted variational description is unavailable yet. While some recent studies have proposed variational frameworks for ductile fracture in [79–81], care must be taken when applying them to geomaterials. The main reason is that the principle of maximum plastic dissipation – which is central to the variational formulation of plasticity – is seldom justified by experimental observations from soils and rocks. Specifically, while the principle of maximum plastic dissipation implies associative flow and associative hardening, geomaterials often exhibit non-associative flow, and an associative hardening form is rarely used in reality [82–84]. For this reason, we choose not to use a variational description of fracture in geomaterials. The second approach, which has been presented in some recent works [74–76], derives the phase-field equation as a non-standard balance law. This approach draws on the procedure developed by Gurtin [85] for deriving the Ginzburg–Landau equation as a microforce balance law. The upshot of this approach is that it allows one to formulate governing equations for a phase-field model of fracture based on the continuum principles of thermodynamics. Then the phase-field equation is identified as a balance law, which is consistent with a tenet in modern that equations should be classified as balance laws or constitutive equations. This type of derivation does notinvolve the principle of maximum dissipation for plasticity, so it is theoretically compatible with a realistic non-associative plasticity theory for geomaterials. Due to this feature, here we adopt this balance law approach to derive a phase-field model of elastoplastic geomaterials. In the literature, these two approaches have led to the same governing equations (at least for brittle fracture), but their thermodynamic implications have been different. In variational principles of phase-field fracture (e.g., [57,58]), the fracturing process is considered fully dissipative in nature. However, in the existing balance law derivations of phase-field fracture [74–76], the fracturing process is not recognized as a dissipative process, unless a rate- dependent regularization is introduced to the phase-field equation. Specifically, these derivations commonly state the energy dissipation due to fracturing is given by β|d˙|, where β is a term vanishes in the rate-independent case. This means that when a rate-independent phase-field equation is used, crack healing does not violate the second law of thermodynamics. But this is inconsistent with real-world observations that cracking is an irreversible process. To reconcile this inconsistency, in this section we present a new balance law derivation of a phase-field fracture formulation that honors the initiation and propagation of fracture is a dissipative process. Our derivation preserves the final form of the phase-field equation, but it leads to a new expression for dissipation inequality in which thefracturing process is fully dissipative. J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 5

In the sequel, we first introduce a Lagrangian description of finite deformation kinematics and a phase-field approximation of fracture surfaces. The balance laws of the , microforce, and internal energy in the material are formulated next. We then derive specific expressions for the microforce terms in the context of phase-field fracture, in a different way from the existing procedure in [74–76]. Lastly, we propose a general form of the stored energy function, with a particular focus on how to consider the stored portion of plastic work that can drive fracture.

2.1. Kinematics

Consider a body whose reference configuration is given by a fixed domain Ω with external boundary ∂Ω. The external boundary is suitably decomposed into the displacement boundary ∂Ωu and the traction boundary ∂Ωt , which satisfy ∂Ω = ∂Ωu ∪ ∂Ωt and ∅ = ∂Ωu ∩ ∂Ωt . The motion of this body is denoted by ϕ(X, t), which maps a point X in the reference configuration at time t to a point x in the current configuration. The displacement vector is then given by u(X, t) = ϕ(X, t) − X. The deformation gradient is defined as ∂ϕ(X, t) F = , (1) ∂X and the Jacobian is defined as dv J = det F = , (2) dV where dV and dv are differential volumes of the reference and current configurations, respectively. Adopting the notation of Gurtin [86], we will use “∇” and “Div” to denote the gradient and divergence operators with respect to a material point X in the reference configuration; and “grad” and “div” to denote these operators with respect to apoint x in the current configuration. When the body undergoes elastoplastic deformation, it is assumed that the deformation gradient can be multiplicatively decomposed into elastic and plastic parts [87], i.e.,

F = FeFp , (3) where the superscripts e and p denote the elastic part and the plastic part, respectively. The same notation will be used throughout. Accordingly, the Jacobian is also decomposed as

J = J e J p , (4) where J e = det Fe and J p = det Fp. In this work, we consider an isotropic material and express the material’s stored energy with the elastic part of the left Cauchy–Green deformation tensor. To begin, the left Cauchy–Green deformation tensor is defined as

b = FFT . (5) The spectral decomposition of b yields

ndim ∑ 2 (A) (A) b = = (λA) n ⊗ n , (6) A=1 (A) (A) where ndim ∈ {2, 3} is the spatial dimensions, λ and n are the principal stretches and the principal axes in the current configuration, respectively. From this decomposition, the principal logarithmic (Hencky) strains canbe calculated as

εA = log(λA) , (7) which reduce to the principal strains of the linearized strain tensor in the limit of infinitesimal deformations. The volumetric logarithmic strain is related to the Jacobian as n ∑dim εv = εA = log(J) . (8) A=1 6 J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32

The multiplicative decomposition of the deformation gradient also enables us to calculate the elastic part of the left Cauchy–Green tensor as

be = FeFeT , (9) of which spectral decomposition leads to

ndim e ∑ e 2 (A) (A) b = = (λA) n ⊗ n . (10) A=1 Again, we can obtain the elastic logarithmic strains as

e e εA = log(λA) , (11) and the elastic volumetric logarithmic strain as

ndim e ∑ e e εv = εA = log(J ) . (12) A=1 The body may contain fractures that are represented by a set of internal discontinuities Γ . The total area of fracture surfaces is given by ∫ AΓ = dA . (13) Γ Because tracking the evolution of Γ is usually extremely unwieldy, we approximate it by introducing a crack density functional Γd, i.e., ∫ ≈ = ∇ AΓ AΓd Γd(d, d) dV , (14) Ω where d ∈ [0, 1] is a phase-field variable which denotes an undamaged state by 0 and a fully damaged (fractured) state by 1. The value of d may also be thought as the density of microfractures which can be calculated by pore-scale models, as done by Tjioe and Borja [88,89]. Note that this approximation transforms an area integral over the evolving domain Γ into a volume integral over the fixed domain Ω. A common form of the crack density functional is (after Ambrosio and Tortorelli [90]) d2 l Γ (d, ∇ d) = + |∇ d|2 , (15) d 2l 2 where l > 0 is a length parameter for the phase-field regularization. For the derivation of governing equation later, we define the rate of crack surface area evolution – which is postulated to befinite–as d ∫ ∫ ˙ = ∇ = ˙ ≥ AΓd Γd(d, d) dV Γd dV 0 , (16) dt Ω Ω where d (d ) Γ˙ = Γ (d, ∇ d) = d˙ + l ∇ d(∇ d˙) ≥ 0 . (17) d dt d l Here, the overdot denotes a material time derivative.

2.2. Governing equations

Consider an arbitrary volume V ∈ Ω with surface boundary ∂V over which the nominal surface traction vector T = P · N acts where P is the first Piola–Kirchhoff stress tensor and N is outward unit normal of ∂V. In this work we consider a quasi-static condition. The balance of linear momentum (equilibrium) over the volume V takes the form ∫ ∫ T dA + ρ0G dV = 0 , (18) ∂V V J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 7 where G denotes the gravity acceleration vector and ρ0 the pull-back density of the . Applying the divergence theorem and noting that V is arbitrary, we obtain

Div P + ρ0G = 0 . (19) Next, following Gurtin [85], we postulate that there exists a microforce system characterized by an internal microforce π and a microforce traction ξ that exerts a surface force ζ = ξ · N on ∂V. The internal and surface microforces are energy-conjugate to the phase-field variable d (referred to as the order parameter in Gurtin [85]). Then the balance of microforce over the volume V is given by [85] ∫ ∫ ζ dA + π dV = 0 . (20) ∂V V Again, by applying the divergence theorem to the first integral and noting the arbitrariness of V, we obtain

Div ξ + π = 0 . (21) Lastly, the balance of energy is stated as ∫ ∫ ∫ ∫ ˙ ρ0e˙ dV = (P · N) ·u ˙ dA + (ρ0G) ·u ˙ dV + (ξ · N) · d dA (22) V ∂V V ∂V where e is the internal energy per unit mass density. Applying the divergence theorem, substituting the balance equations, and noting that V arbitrary, we obtain the following expression of the balance of energy ˙ ˙ ˙ ρ0e˙ = P : F + ξ · ∇ d − πd . (23) Here, the stress power can be expressed by alternative stress and deformation measures

P : F˙ = τ : d , (24) where τ = P · FT is the Kirchhoff stress tensor, and d is the rate of deformation tensor defined as the symmetric part of the spatial velocity gradient l = grad u˙. To get specific expressions of the stress tensor and the microforce variables, we first exploit the secondlawof thermodynamics. Let ψ denote the stored energy density per unit volume. In the absence of heat flux and heat source, the dissipation inequality reads

D = τ : d + ξ · ∇ d˙ − πd˙ − ψ˙ ≥ 0 . (25) As standard in the phase-field modeling of fracture, we consider that d is a damage-like variable that degrades the stored energy of an undamaged material via a degradation function g(d) ∈ [0, 1] satisfying g(0) = 1 and g(1) = 0. A usual choice is g(d) = (1 − d)2, see [57–60] for example. The stored energy density function is assumed to take the form

ψ(be, q, d) = g(d)W(be, q) , (26) where W denotes the energy stored in the undamaged material in which the phase-field variable d is decoupled from ψ (i.e, without stiffness degradation), and q is a vector of strain-like plastic internal variables. We note that unlike previous microforce derivations [74–76], here the stored energy function ψ does not contain the energy used to create fracture surfaces. This is because the fracturing process is considered fully dissipative a priori. Note that the stored energy function is thus related to d only through g(d), same as in the variational frameworks for fracture (e.g., [57,59]). Time differentiation of the stored energy function gives ∂ψ ∂ψ ∂ψ ψ˙ = · b˙e + ·q ˙ + · d˙ . (27) ∂be ∂q ∂d Here, the objective rate of be is given by

˙e e e T e b = l · b + b · l + Lvb , (28) where Lv denotes the Lie derivative. Now we substitute Eqs. (27) and (28) into Eq. (25), and rearranging it with noting that ∂ψ/∂be and be commute and that their tensor product is symmetric. The resulting expression of the dissipation 8 J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 inequality is ( ∂ψ ) ( ∂ψ ) ( 1 ) D = τ − 2 · be : d + 2 · be : − (L be) · be−1 ∂be ∂be 2 v (29) ∂ψ ( ∂ψ ) − ·q ˙ − π + d˙ + ξ · ∇ d˙ ≥ 0 . ∂q ∂d Note that this expression differs from those in [74–76] in that here ξ, which is energy-conjugate to ∇ d˙, is unrelated to the stored energy ψ. This difference is because ∇ d is not an argument of ψ. Applying the standard Coleman–Noll argument to Eq. (29) leads to the following equations: ∂ψ ∂ψ τ = 2 · be = 2be · , (30) ∂be ∂be ∂ψ π = π en + π dis , π en = − . (31) ∂d It is noted that the internal microforce π has been decomposed into the energetic part π en and the dissipative part π dis. This decomposition also appears in [91,92] which use a microforce balance argument to derive other types of models. The reduced dissipation inequality is given by ( 1 ) τ : − (L be) · be−1 + κ ·q ˙ + ξ · ∇ d˙ − π disd˙ ≥ 0 (32) 2 v where κ = −∂ψ/∂q denotes a vector of stress-like plastic internal variables conjugate to q. Clearly, the first two terms of the inequality are related to the plastic deformation, whereas the last two terms are related to the fracture process. Thus we distinguish the plastic and fracture parts of the dissipation, and require that each part is non-negative, i.e., ( 1 ) Dp = τ : − (L be) · be−1 + κ ·q ˙ ≥ 0 , (33) 2 v Df = ξ · ∇ d˙ − π disd˙ ≥ 0 . (34) Let us consider the implication of the plastic dissipation first. Introducing the plastic potential function G(τ, κ), we define the flow rule 1 ∂G − (L be) · be−1 =γ ˙ , (35) 2 v ∂τ where γ is the non-negative plastic multiplier satisfying γ˙ ≥ 0. Then we can rewrite the plastic dissipation inequality as ( ∂G ) Dp = τ : γ˙ + κ ·q ˙ ≥ 0 . (36) ∂τ As long as the flow rule and hardening law satisfy the above inequality, a non-associative plasticity modelcanbe thermodynamically consistent. Indeed, such thermodynamic restrictions for geomechanical plasticity models have been derived (see [82,93] for example), and we will honor them. We now turn our attention to the fracture dissipation inequality Df. Our goal here is to express π dis and ξ in terms of d and ∇ d so that we can solve the microforce balance law Eq. (21). At this point, we note that we will obtain these expressions in a new way, different from previous ones in [74–76]. The previous derivations get expressions for π and ξ from the standard procedure illustrated above, assuming that the fracturing energy is stored in the material. However, because we have treated the fracturing energy as fully dissipative herein, we need to take a new approach. For this purpose, we adopt the postulate in variational frameworks for fracture that the evolution of the phase-field variable maximizes the energy dissipation [59,79–81]. We seek to minimize the negative of the fracture dissipation functional − Df = −ξ · ∇ d˙ + π disd˙ . (37) In doing so, we note that d and ∇ d compose the crack density functional. So their evolution is subject to the constraint (d ) d˙ + l ∇ d(∇ d˙) = Γ˙ , (38) l d J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 9 which was stated in Eq. (17). To solve this constrained minimization problem, we construct a Lagrangian of the form [(d ) ] L(d˙, ∇ d˙, Λ) = −ξ · ∇ d˙ + π disd˙ + Λ d˙ + l ∇ d(∇ d˙) − Γ˙ , (39) l d where Λ is a Lagrange multiplier. From the stationary condition of this Lagrangian, we obtain

δ∇ d˙L = −ξ + Λl ∇ d = 0 → ξ = Λl ∇ d , (40) ( ) ( ) dis d dis d δ ˙L = π + Λ = 0 → π = −Λ . (41) d l l Next, to find the physical implication of Λ, we substitute the above two equations into Df, which is the energy dissipation per unit volume. Integrating this over the domain Ω with the crack surface Γd gives ∫ ∫ [( ) ] ∫ ∫ f d ˙ ˙ ˙ d D dV = Λ d + l ∇ d(∇ d) dV = ΛΓd dV ≈ Λ dA ≥ 0 , (42) Ω Ω l Ω dt Γd where the approximation in the last part is due to the phase-field regularization. From the above we can interpret that the Lagrange multiplier Λ is the energy dissipated by the creation of unit crack surface area—which, by definition, corresponds to the critical fracture energy. Denoting the critical fracture energy by Gc, we then express the scalar microforce as ∂ψ (d ) π = π en + π dis , π en = − = −g′(d)W(be, q) , π dis = −G , (43) ∂d c l and the vector microforce as

ξ = Gcl ∇ d . (44) We now substitute these expressions into the microforce balance equation (21), and obtain (d ) − g′(d)W(be, q) − G − l Div(∇ d) = 0 , (45) c l which, in the absence of the plastic contribution to W, coincides with the equation in the widely used phase-field model of brittle fracture (e.g., [57,58]). Eq. (45) also suggests that a natural extension of the phase-field model to ductile fractures may be to incorporate the energy stored during plastic deformation.

2.3. Thermodynamic implications and crack irreversibility

Our derivation leads to an expression for the dissipation rate of the fracture energy per unit volume as

f ˙ D = GcΓd ≥ 0 . (46) f Since the unit of Γd is one over length and the unit of Gc is energy over area, when we integrate D over a volume ˙ and a time interval, we get a quantity whose unit is energy. Equivalently, the unit of GcΓd is identical to the unit of a stress power. Thus we see that our results are dimensionally consistent. It is also noted that Df has a nonlocal term because ∇ d is an argument of Γd. This observation is supported by the fact that phase-field modeling of fracture does not suffer from mesh sensitivity in the softening regime. ˙ We now focus on how to ensure this dissipation inequality. Since Gc > 0 by definition, Γd ≥ 0 must be satisfied to 2 ensure the second law. Given that the time derivative of |∇ d˙| in Eq. (15) can be expressed as 2|∇ d||∇ d˙| as well, we ˙ can rewrite Γd ≥ 0 as (d ) d˙ + l|∇ d||∇ d˙| ≥ 0 . (47) l From this, we can see that enforcing d˙ ≥ 0 locally can ensure the nonlocal thermodynamic consistency (equivalently, ˙ crack irreversibility) condition given by Γd ≥ 0. Indeed, this agrees with the argument of Miehe et al. [57] that ˙ ˙ imposing d ≥ 0 locally can satisfy the crack irreversibility originally given by a global integral of Γd ≥ 0. In another paper [58] they also proposed an approach for enforcing d˙ ≥ 0, which has proven simple and effective to enforce crack irreversibility in various settings (e.g., [59–61,75]). The approach is to make the energetic force driving the evolution 10 J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 of the phase-field variable d, π en = −g′(d)W(be, q), a non-decreasing function during the course of loading. This can be done by introducing a strain energy history functional H ≥ 0 that satisfies the following Karush–Kuhn–Tucker condition

W+ − H ≤ 0 , H˙ ≥ 0 , H˙ (W+ − H) = 0 , (48)

e where W+ is the portion of W(b , q) that contributes to fracturing (this will be elaborated in the following subsection). In words, H is the maximum of the fracture-driving energy throughout the loading history. Following this, we replace the stored energy term in Eq. (45) by H, and consider a modified phase-field equation of the form G − g′(d)H − c [d − l2 Div(∇ d)] = 0 , (49) l which is the same as the phase-field equation used in many previous studies [58,75]. The foregoing discussion shows that when a phase-field model of fracture is derived in this way, the variational and balance law approaches agree well each other in terms of thermodynamic implications as well as governing ˙ equations. For example, the crack irreversibility condition we obtained from a balance law argument, Γd ≥ 0, is a local version of the same condition in a variational framework (e.g., Eq. (20) of Miehe et al. [57]). This agreement can provide more confidence in the application of the balance law approach to phase-field modeling of fracture inother problems. The balance law approach – which relies on continuum mechanics principles – has been particularly useful for developing a phase-field fracture model for complex problems for which constructing a variational framework is very challenging (e.g., the piezoelectric problem considered in Wilson et al. [74]) or hard to be justified from experimental observations (e.g., non-associative plasticity problems in this work). Yet, rigorous development of a continuum modeling framework for such complex problems often requires a thermodynamic argument (e.g., [94–97]). In this regard, the derivation presented in this work – which leads to a thermodynamic argument consistent with variational theory as well as real-world – can help apply the microforce approach to phase-field modeling of fracture in new complex settings.

2.4. Stored energy

As shown above, in the phase-field model of fracture, the stored energy function governs the fracture initiation and propagation. It is thus crucial to consider an appropriate form of the stored energy function to meaningfully capture pressure- and rate-sensitive failure behaviors. In what follows, we propose a general form of the stored energy function that allows us to model brittle and ductile behaviors under different stress and strain rate conditions. First, we recognize that a crack would not develop in a homogeneous body under purely compressive loading even as it raises the stored energy. A common approach for considering this aspect in the phase-field modeling of brittle fracture is to decompose the stored energy function into two parts, one that drives fracture and another that does not. Extending this approach to ductile fracture, we rewrite ψ(be, q, d) as

e e e ψ(b , q, d) = g(d)W+(b , q) + W−(b , q) , (50) where the subscript + denotes the fracture-driving part and − denotes the other part. Note that both of the two parts may contain contributions from elastic and plastic deformations. Adopting the standard split of the stored energy into elastic and plastic parts, we assume that each part of the non-degraded stored energy function can be additively decomposed into elastic and plastic parts, i.e., e e e p W+(b , q) = W+(b ) + W+(q) , (51) e e e p W−(b , q) = W−(b ) + W−(q) . (52) Equivalently, this means that the elastic and plastic parts of the stored energy are individually decomposed into the fracturing and non-fracturing parts as e e e e e e W (b ) = W+(b ) + W−(b ) , (53) p p p W (q) = W+(q) + W−(q) , (54) with W(be, q) = W e(be) + W p(q). J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 11

Here, W e(be) is identical to the elastic strain energy function commonly considered in phase-field models of brittle fracture (e.g., [56–58,60,61]). Therefore, we adopt an existing approach to the decomposition of W e(be) into fracturing and non-fracturing parts. Two widely used approaches in the literature are: (1) to distinguish tensile and compressive principal strains [57], and (2) to distinguish the volumetric and deviatoric parts of the elastic strain tensor [56]. The major difference between these two is that the former does not allow shear stress to trigger fracture when all principal strains are compressive. In our context, this means that if we adopt the principal-strain based split, compressive loading would lead to fracturing only through plastic deformations. On the other hand, if we adopt the volumetric– deviatoric split, compressive elastic deformation drives fracture as well and thus we open the possibility of purely brittle shear fracture under compressive loading. Another important consideration is that a phase-field approximation limits the maximum admissible stress even in an elastic material, since it eliminates the stress singularity at the crack tip [56,60,75]. This means that the use of the volumetric–deviatoric split along with a plasticity model can create another “shear strength” under compressive loading that is different from the strength. To prevent such existence of two strengths, we adopt the principal-strain based split in this work. The incorporation of the stored energy by plastic deformation W p(q) into the fracture driving force is the fundamental idea shared by recently proposed phase-field formulations of ductile fracture in metals (e.g., [75,76,79– 81,98]). These formulation do not decompose W p(q), presumably because the plastic deformation of metals is virtually isochoric. However, geological materials usually exhibit significant volumetric change during plastic deformations, and fractures would not develop under compactive plastic flow. To consider this aspect, we propose a decomposition of the plastic energy of the form

{ p p p W (q) if J ≥ 1 , W+(q) = (55) 0 if J p < 1 , { p p 0 if J ≥ 1 , W−(q) = (56) W p(q) if J p < 1 , where J p is the plastic part of the Jacobian as defined in Eq. (4). In other words, we postulate that the energy stored by dilative or isochoric plastic flow drives fracture, whereas the energy stored by compactive flow is irrelevant to fracture processes. Given that the plastic deformation of geological materials changes from dilative to compactive with an increase in confining pressure, this split suppresses the development of fracture when the confining pressure is relatively high. Therefore, the proposed dilative–compactive split – which is in the same vein as the tensile– compressive split of the elastic strain energy – is the key to capture the pressure-induced brittle to ductile transition in geological materials. Now we consider how much of the total plastic work may be stored in the material microstructure. The stored portion of plastic work is assumed to have an explicit form when an associative hardening law is used. However, as mentioned before, such associative form of hardening law is far from the behavior of real geomaterials, so we do not attempt to express the stored plastic work explicitly. Instead, because the non-stored portion of plastic work is converted into heat, we draw on the concept of the Taylor–Quinney coefficient which is defined as the fraction of plastic work converted into heat [99]. Let α ∈ [0, 1] denote one minus the Taylor–Quinney coefficient and W p tot denote the total plastic work. Then we may express the stored energy by plastic deformation as W p(q) = αW p tot . (57) The value of α may be much less than unity, given that the Taylor–Quinney coefficient is often assumed to be 0.9 (equivalent to α = 0.1), at least for metals. Also, experimental results have shown that the coefficient can evolve by the deformation and its rate [100]. Specifically, when the strain rate is slower, more plastic work may be stored inthe material. Considering this observation, we assume that for rate-sensitive materials α is a function of the rate of the plastic deformation, i.e., α =α ˜ (F˙ p) , (58) and α becomes greater as the rate of plastic deformation becomes slower. Because α determines W p(q), which drives damage and fracture, the rate-dependence of α gives rise to the rate-dependence of the fracturing process after the onset of plastic deformation. This rate-sensitivity agrees with experimental observations for some geomaterials (see 12 J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32

Fig.2 for example), in which post-peak behavior under compression becomes more brittle at slower strain rates even when pre-peak behavior is insensitive to strain rates. Note that this experimental observation is consistent with our theoretical formulation, because rate-sensitive materials store more plastic work at slower strain rates. To summarize, we consider the stored energy function of the form

ψ(be, q, d) = ψe(be, d) + ψp(q, d) , (59) where e e e e e e ψ (b , d) = g(d)W+(b ) + W−(b ) , (60) p p p ψ (q, d) = g(d)W+(q) + W−(q) ˙ p p tot p tot =α ˜ (F )[g(d)W+ (q) + W− (q)] , (61) are the elastic and plastic parts. Accordingly, the fracture-driving part of the stored energy that defines the history functional H in Eq. (48) is given by e e ˙ p p tot W+ = W+(b ) +α ˜ (F )W+ (q) . (62) So far, we have developed a general framework that can be applied to phase-field modeling of any type of elastoplastic material. Specific expressions for the stored energy functions depend on the choice of a constitutive model. This will be illustrated when we adopt a geomechanical constitutive model in the next section.

3. Constitutive framework This section describes a constitutive framework for elastoplastic susceptible to fracturing described by a phase-field approach. We begin by introducing an approach for incorporating the effect of phase-field fractureon yielding and plastic flow, which borrows the concept of damage effective stress and strain equivalence hypothesis in damage mechanics. We then construct a particular class of the proposed framework by adopting an elastoplasticity model for pressure-sensitive geological materials. We note that this constitutive model is used for illustration purposes, and the framework is not restricted to this particular model.

3.1. Coupling phase-field fracture with elastoplasticity

Recently proposed phase-field models of ductile fracture [75,79–81] as well as those of brittle fracture in elastoplastic solids [59] have employed elastoplastic constitutive models originally developed for non-fracturing solids. Most of them have used a J2 plasticity model with a hardening law, and the plastic behavior remains unchanged during the development of phase-field fracture. In other words, the evolution of the phase-field variable does notaffect the and hardening behavior, even as it degrades elastic behavior through g(d). However, as pointed out by Borden et al. [75], such an approach may lead to unphysical responses, because hardening can make elastic deformations become dominant during ductile fracture. To remedy this problem, Borden et al. [75] multiply a degradation function – which is applied to the fracture-driving part of the stored energy – to a hardening variable as well, so that the degraded stress and yield strength become the same. This approach seems working well for their J2 plasticity model whereby the yield strength is compared with the deviatoric part of the stress tensor. However, this approach may not work well for other plasticity models whereby the volumetric part of the stress tensor – which may or may not be degraded – impacts the yield strength. Also, for plasticity models with multiple yield surfaces/hardening variables, it is unclear whether the degradation function should affect all the hardening variables even as only one of the yield surface may be activated. In fact, plasticity models for geological materials are pressure-sensitive (i.e., the yield surface is a function of the volumetric stress) and they often have multiple yield surfaces and hardening variables (e.g., a Drucker–Prager model with a compression cap). Therefore, the approach of incorporating the degradation function into the yield function may not be an appropriate choice for geological materials. In this work, we propose a different approach to the coupling of a phase-field fracture model and a plasticity model. The approach is motivated by the mathematical correspondence between the phase-field fracture model and the gradient-damage model. Not only the phase-field variable d is naturally interpreted as the damage in the material, but it also has recently been shown that the phase-field model can be regarded as a particular class of gradient-damage J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 13 models [44]. A robust approach for accommodating plasticity in damage mechanics is to postulate that only the undamaged portion of the material can sustain plastic deformation, not the microcracks whose density is given by the damage variable [101,102]. This assumption allows one to compute the strain in the intact portion of the damaged material using the concept of damage effective stress, which is a central concept in damage mechanics. Borrowing the concept of damage effective stress, here we define a phase-field effective stress. In terms of the Kirchhoff stress tensor, the phase-field effective stress is expressed as ∂W e(be) τ¯ = , (63) ∂be where W e(be) is the strain energy density function of the undamaged part, as defined in the previous section. Observe that this phase-field effective stress is different from the homogenized, overall stress τ in that the degradation function is not applied. In this sense, the phase-field effective stress is identical to the damage effective stress, if the phase-field variable d is regarded as a damage variable. However, because the phase-field and damage variables do not refer to the same concept in the literature, here we have coined the term phase-field effective stress to prevent any confusion. Also, one should not confuse this type of effective stress with the effective stress in the contexts of geomechanics and poromechanics, which refers to the stress in the solid matrix in -infiltrated porous media. To avoid this possible confusion, throughout this paper we refer to this stress as a phase-field effective stress. Having introduced the phase-field effective stress, we now adopt the hypothesis of strain equivalence indamage mechanics [103]. The hypothesis states that “the strain associative with a damaged state under the applied stress is equivalent to the strain associative with its undamaged state under the effective stress”. This notion of strain equivalence enables us to reformulate an elastoplasticity model, which was originally developed for undamaged, intact materials, with the phase-field effective stress. This is because the strain obtained by the reformulated plasticity model can be considered the strain in the damaged, fractured material with the homogenized stress. Therefore, we will evaluate the yield and plastic potential functions with the phase-field effective stress, and because the phase-field effective stress is higher than the homogenized stress, yielding would be facilitated by damage processes. A generic algorithm for this procedure can be written as Algorithm1. Note that it is straightforward to adapt this algorithm to models formulated with other stress/strain measures. Algorithm 1 General algorithm for coupling phase-field fracture and plasticity models Require: Displacement field u and phase-field variable d at a given load step. e 1: Compute the phase-field effective stress τ¯ from the undamaged strain energy function W e(b ). 2: Evaluate the yield function F(τ¯ , q) with the phase-field effective stress. 3: if F(τ¯ , q) < 0 then 4: Elastic step. 5: else 6: Plastic step. Perform return mapping with the phase-field effective stress. e e e e e 7: Calculate the homogenized stress τ from the degraded strain energy function ψ(b , d) = g(d)W+(b ) + W−(b ). 8: return Homogenized stress τ. The advantages of this algorithm include that (1) it can be applied to any plasticity model regardless of the number and type of hardening variables, and that (2) it preserves the return mapping algorithm of existing plasticity models so the additional implementation cost is minimal. Our numerical experiments have shown that this approach is numerically robust unless the load increment is inappropriately large.

3.2. An elastoplastic model for pressure-sensitive geomaterials

We now construct a particular class of the developed framework by adopting a specific constitutive model for geological materials. In this work, we consider an elastic strain energy function of the form n 1 ∑dim W e = λ⟨εe⟩2 + µ ⟨εe ⟩2 , (64) ± 2 v ± A ± A=1 where ⟨·⟩± = (· ± |·|)/2 is the operator used for the decomposition into the fracturing and non-fracturing parts, and λ and µ are the Lame´ parameters. This strain energy function is an extension of linear to Hencky 14 J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 elasticity [104], and insensitive to confining pressure. We note that although some strain energy functions have been proposed for pressure-sensitive elasticity [105,106], we do not make use of one of them in this work because they are poorly defined when the confining pressure is zero or tensile. They are thus inappropriate to beusedwitha phase-field model of tensile fracture. Also, while soils may exhibit significant pressure-sensitivity in their elastic responses [107–109], the elastic stiffness of rocks, which we focus on hereafter, is usually reasonably approximated by [8,110]. We thus choose to not accommodate pressure-sensitive elasticity for now, and defer it to future work. From the elastic stored energy function, we can obtain the Kirchhoff stress tensor as

ndim e e e ∑ ∂ψ ∂W+ ∂W− τ = τ n(A) ⊗ n(A) , τ = = g(d) + , (65) A A ∂εe ∂εe ∂εe A=1 A A A and the phase-field effective Kirchhoff stress tensor as

ndim e e e ∑ ∂W ∂W+ ∂W− τ¯ = τ¯ n(A) ⊗ n(A) , τ¯ = = + . (66) A A ∂εe ∂εe ∂εe A=1 A A A Again, note that τ¯ is independent of the phase-field variable d. It would be worth mentioning that, for materials undergoing large plastic deformations, an Eshelby-like stress has recently been proposed by Bennett et al. [111] as a more appropriate stress measure than conventional ones such as the Kirchhoff stress. For practical reasons, however, we resort to the Kirchhoff stress because it has been one of the standard stress measures used for finite strain plasticity modeling of volume-changing geomaterials (e.g., [97,106,112–114]). Next we adopt a plasticity model, reformulating it with the phase-field effective stress. A class of plasticity models widely used for pressure-sensitive geological materials combines two yield surfaces: (1) a conical yield surface that covers quasi-brittle failure at a relatively low confining pressure level, and (2) a cap yield surface that covers ductile flow at a relatively high confining pressure level. In this work, we employ one of such two-surface plasticity models developed by Spiezia et al. [114] for porous rocks, which smoothly connects a conical surface of Drucker–Prager type [115] with a cap surface similar to that proposed by DiMaggio and Sandler [28]. The two yield surfaces are both defined with the two stress invariants √ 1 2 p¯ = tr τ¯ , q¯ = ∥s∥ , (67) 3 3 where s¯ = τ¯ −p ¯1 (1 is the second-order identity tensor) is the deviatoric part of the Kirchhoff stress tensor. The conical yield function Fs is given by

Fs =q ¯ − M p¯ − C ≤ 0 , (68) where M and C are parameters related to and cohesion, respectively. The cap yield function Fc is given by q¯2 (p¯ − p )2 F = + i − 1 ≤ 0 , (69) c B2 A2 where A and B are material parameters, and pi is a hardening variable. One can easily see that the Fs is a linear line in the p¯–q¯ space, whereas Fc is an ellipse in the p¯–q¯ space. To connect the two surfaces smoothly, we impose the constraint 2 2 2 2 2 2 B = M pi − A M + 2MCpi + C . (70) These yield surfaces have been shown to be in good agreement with experimental data of a variety of rocks, see [114]. The hardening law for the cap plasticity is adopted from the equation proposed by Stefanov et al. [116] ( ε∗ )r = v pi pi0 ∗ p (71) εv − εv p p ∗ p where εv = log(J ) is the plastic part of the volumetric logarithmic strain, εv is the value of εv at ultimate compaction, and r is a material parameter. During quasi-brittle yielding (i.e., Fs = 0), we apply a friction hardening law similar to that in [82], given by ( 2λk ) M = M + M , (72) 0 f λ + k J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 15 where M0 and M f are the initial and final values of M, λ is the cumulative plastic multiplier, and k is the material parameter that controls the hardening rate. At this point, it is noted that the plasticity model does not involve a softening law. Thus any softening behavior in the macroscopic response, if exists, should be attributed to the phase-field variable. This point will be illustrated by numerical examples in the next section. As for the flow rule, associative flow is assumed for the cap surface whereas non-associative flow is assumedfor the conical surface. The plastic potential function for the non-associative flow is given by ¯ Gs =q ¯ − M p¯ , (73) where M¯ is the material parameter that determines dilatancy of the plastic flow. It is noted that non-negative plastic dissipation requires M¯ ≤ M, see [82,93]. Also, because the connection of the two potential functions is non-smooth for M¯ ̸= M, a special treatment is required for calculating plastic flow around where the two surfaces meet. The details of this treatment are explained in Spiezia et al. [114]. Lastly, we note that plasticity is considered only when the confining pressure is non-negative, i.e., the yield functions are inactive when the mean normal stress is tensile. The reason is that we have already employed a phase- field model of fracture for capturing tensile failures of geomaterials. This makes the proposed framework degenerates to the standard phase-field model of brittle fracture.

3.3. Critical fracture energy and length parameter

The material property explicitly used in phase-field modeling of fracture is the critical fracture energy Gc, which corresponds to the energy required to create a unit area of fracture surface. This property has a clear physical meaning, but its direct measurement by experiments is indeed very challenging for geomaterials. While the critical fracture energy can be inferred from the critical (fracture toughness) under the assumption of linear elastic fracture mechanics, such assumption is obviously inadequate for this work. Phase-field modeling of fracture further requires one to assign the length parameter l for a diffuse approximation of a sharp fracture surface. In the literature, this parameter has usually been selected with consideration of the size of the domain/mesh (e.g., [61,64,65,72,117–119]). However, although l has a geometric origin, it acts as another material property of phase-field modeling. This is because the length parameter determines the peak stressofa material, together with the critical fracture energy and elasticity parameters [56,60,75]. In other words, as the phase-field regularization removes the stress singularity at the crack tip, the peak stress (called the critical stress in Borden et al. [75]) becomes a function of the length parameter. Analytical expressions for such peak stresses of a homogeneous, elastic material under uniaxial tension and compression have been derived in [56,60,75]. These expressions show that the peak stresses become infinity in the limit of l → 0, whereas they become zero in the limit of l → ∞. Therefore, even if an “exact” Gc were used, use of an inappropriate value for l would produce results that significantly deviate from the real behavior. For brittle materials, some recent studies (e.g.,[120–122]) have evaluated l as a function of the critical fracture energy, Young’s modulus, and the tensile strength. The value of l determined in this way can lead to a peak load close to experimental measurements, but sometimes it can also result in fracture overly diffusive than observations [121,122]. Thus an appropriate selection of l remains an open area of research. For these reasons, here we propose an approach to setting Gc and l such that their resulting peak stress under uniaxial loading is consistent with a shear yield criterion used in plasticity. The approach is motivated by the fact that the notion is “yielding” in a shear yield criterion like Mohr–Coulomb and Drucker–Prager indeed corresponds to “failure” occurring at the peak stress [21]. Thus our idea is to match the peak stresses of phase-field and plasticity models in either uniaxial tension or compression. For a homogeneous, linear elastic body, the peak of the phase-field effective (Cauchy) stress under uniaxial tension σt and the stress under uniaxial compression σc are given by (adapted from [56] considering that their length parameter is a half of our l) √ √ E √ (41 + 14ν) 2E √ σ = G , σ = − G , (74) t 3l c c 96 (1 + ν)l c where E and ν are Young’s modulus and Poisson’s ratio, respectively. We shall “fit” either one of these peak stresses with the Mohr–Coulomb failure criterion. Note that the Drucker–Prager model is a smooth approximation of the 16 J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32

Mohr–Coulomb criterion, which can be fit to either tension or compression corners. In terms of the principal Cauchy stresses, the Mohr–Coulomb criterion is expressed as

(σ1 − σ3) = 2c cos φ − (σ1 + σ3) sin φ , (75) where c and φ are the cohesion and friction angle, which can be related to M and C in Eq. (68), and σ1 and σ3 are the maximum and minimum principal stresses (note that stresses are positive in tension). Matching the phase-field and Mohr–Coulomb models in the uniaxial tension case can be done by substituting σ1 = σt and σ3 = 0 into Eq. (75), whereas matching them in the uniaxial compression case can be done by σ1 = 0 and σ3 = σc. When the length parameter l is predetermined (for example as a function of mesh size), the expressions for Gc are given by ⎧3l ( 2c cos φ )2 ⎪ uniaxial tension , ⎨ E + G = 1 sin φ (76) c (1 + ν)l ( 96(2c cos φ) )2 ⎪ uniaxial compression . ⎩ 2E (41 + 14ν)(1 − sin φ)

From above, we can first see that Gc is estimated to be higher when it is fitted with the Mohr–Coulomb criterion for the uniaxial compression case. Given that a geomaterial fails by shear fracture under compression whereas it fails by tensile fracture under tension, this difference may imply the dependence of critical fracture energies on the fracture mode. A way to incorporate mode-dependent fracture energies of geomaterials has been suggested by Shen and Stephansson [123], whose main idea has been applied to phase-field modeling very recently [68]. However, its further validation and generalization to ductile materials involves significant work which is beyond the scope of this work. Also, while some studies have suggested that Gc of porous rocks might vary with confining pressure and/or porosity (e.g., [124]), no mathematical expression between Gc and the confining pressure is available yet. Thus, for simplicity, we consider Gc constant herein, fitting it to either the uniaxial compressive or tensile strength depending on the mode of loading. We believe that accommodating mode- and pressure-dependent fracture energies is an interesting topic for future research. From Eq. (76), we can also see that larger l requires higher Gc to keep the peak stress constant. This has two implications for phase-field modeling. First, if Gc is estimated to be orders of magnitude larger than physically realistic values, this means that l is too large to give realistic results in boundary-value problems. Second, this implies that use of a realistic critical fracture energy in phase-field modeling requires a sufficiently small parameter for length regularization.

4. Finite element formulation This section presents a finite element formulation for a numerical solution of the proposed framework. Wefirst formulate variational equations and their linearized versions, and then describe an implicit integration scheme for the constitutive model based on return mapping in the principal logarithmic strain space. We also briefly describe a staggered scheme for solving the momentum and phase-field equations in a numerically robust manner.

4.1. Variational form

Consider an initial boundary-value problem whereby governing equations are given by Eqs. (19) and (49). The boundary conditions are

u =u ˆ on ∂Ωu , (77) ˆ P · N = T on ∂Ωt , (78) ∇ d · N = 0 on ∂Ω , (79) and the initial condition is u(X, 0) = u0(X, 0). To develop the variational form of the problem, we define the spaces of trial functions 1 Su = {u | u ∈ H , u =u ˆ on ∂Ω u}, (80) 1 Sd = {d | d ∈ H }, (81) J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 17 where H 1 denotes a Sobolev space of order one. Accordingly, we define the spaces of variations as 1 Vu = {η | η ∈ H , η = 0 on ∂Ω u}, (82) 1 Vd = {φ | φ ∈ H } . (83) Through the standard weighted residual procedure, we obtain the variational equation for balance of linear momentum as ∫ ∫ ∫ ˆ ∇ η : P dV = η · (ρ0G) dV + η · T dA . (84) Ω Ω ∂Ωt Since P = τ · FT, the integrand on the left hand side of Eq. (84) can be expressed in a different form

∇ η : P = grad η : τ . (85) Using this relationship, we can write an alternative form of the balance of linear momentum as ∫ ∫ ∫ ˆ grad η : τ dV = η · (ρ0G) dV + η · T dA . (86) Ω Ω ∂Ωt Similarly, the standard procedure gives the variational equation for the phase-field evolution as ∫ ∫ G φg′(d)H dV + c (φd + l2 ∇ φ · ∇ d) dV = 0 . (87) Ω Ω l

The weak form of the problem is then stated as: Find {u, d} ∈ Su × Sd such that for all {η, φ} ∈ Vu × Vd Eqs. (86) and (87) are satisfied. Because the formulation involves material and geometric nonlinearities, we use Newton’s method to solve the nonlinear system at each time step. To do so we need to consistently linearize the nonlinear terms in the variational equations. Let δ(·) denote the linearization operator. Linearization of the first term of the momentum balance equation gives (∫ ) ∫ δ grad η : τ dV = grad η : a : grad δu dV , (88) Ω Ω where a = α − τ ⊖ 1, with δτ = α : grad δu and (τ ⊖ 1)i jkl = τil δ jk is the so-called geometric stiffness (or initial stress) term, see Chapter 5 of Borja [93] for the same notation. Since ρ0 = Jρ, the second term in the balance of linear momentum equation is linearized as (∫ ) ∫ δ η · (ρ0G) dV = η · δ(J)ρG dV , (89) Ω Ω and the linearization of Jacobian is given by δ J = J div δu. Finite element discretization of Eqs. (86) and (87) and their linearization is straightforward. If the mesh is fine enough to resolve a sharp gradient of the phase-field variable around crack surfaces, a continuous Galerkin method with standard shape functions works well.

4.2. Implicit stress-point integration

We implicitly update stresses and internal variables at integration points via a return mapping algorithm for finite strain plasticity described in Borja [93]. The algorithm performs return mapping in the space of principal logarithmic strains. In what follows, we describe the essence of the algorithm, and refer to Chapters 5 and 6 of Borja [93] for more details. Consider a step from time tn to tn+1 whereby the local displacement field at tn+1 is prescribed. For notational simplicity we shall drop the subscript n + 1 for variables pertaining to the time tn+1, e.g., xn+1 = x. The relative deformation gradient at this time interval is then given by f = ∂x/∂xn. The predictor (trial) elastic left Cauchy–Green deformation tensor is defined and spectrally decomposed as

ndim e tr e T ∑ e tr 2 (A) (A) b = f · bn · f = (λA ) n ⊗ n , (90) A=1 18 J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 where A = 1, 2, 3 for the general case of ndim = 3. From the above equation, we can define the trial principal logarithmic strains as

e tr e tr 2 εA = log(λA ) . (91)

Our task is to compute the stress and internal variables at time tn+1, given the displacement increment 1u = u−un as well as stress and internal variables at time tn. To this end, we additively decompose the principal logarithmic strains into the predictor and corrector parts, as

e e tr ∂G εA = εA − 1γ , (92) ∂τ¯A where 1γ is the increment in the plastic multiplier and τ¯A is the principal phase-field effective Kirchhoff stress defined in Eq. (66). Note, again, that τ¯A ̸= τA whenever d > 0. If a yield surface is not activated in the predictor stage, this step is elastic and the rest procedure is straightforward. Otherwise, we should correct the predictor such that three nonlinear equations emanating from Eq. (92) are satisfied simultaneously. Additionally, we need to satisfy the discrete hardening law, which can be written as ˆ ˆ l(τ1, τ2, τ3, κ, 1γ ) = L(ε1, ε2, ε3, κ, 1γ ) = 0 , (93) and the discrete consistency condition, which can be written as ˆ ˆ f (τ1, τ2, τ3, κ) = F(ε1, ε2, ε3, κ) = 0 . (94) We use Newton’s method to solve this system of nonlinear equations. The residual vector r and the unknown vector x are constructed as ⎧ ⎫ ⎧ ⎫ εe − εe tr + 1γ g εe ⎪ 1 1 1 ⎪ ⎪ 1 ⎪ ⎪ e e tr ⎪ ⎪ e ⎪ ⎪ ε − ε + 1γ g2 ⎪ ⎪ ε ⎪ ⎨⎪ 2 2 ⎬⎪ ⎨⎪ 2 ⎬⎪ = e e tr = e r(x) ε3 − ε3 + 1γ g3 , x ε3 , (95) ⎪ ⎪ ⎪ ⎪ ⎪Lˆ (ε , ε , ε , κ, 1γ )⎪ ⎪ κ ⎪ ⎪ 1 2 3 ⎪ ⎪ ⎪ ⎪ ˆ ⎪ ⎪ ⎪ ⎩ F(ε1, ε2, ε3, κ) ⎭ ⎩1γ ⎭ where gA = ∂G/∂τ¯A for A = 1, 2, 3. The Jacobian of this Newton system at the kth iteration is then given by ⎡ ⎤ c11 c12 c13 1γ ∂g1/∂κ g1 ⎢ ⎥ ⎢ c21 c22 c23 1γ ∂g2/∂κ g2 ⎥ ⎢ ⎥ k = ′ k = ⎢ ⎥ A r (x ) ⎢ c31 c32 c33 1γ ∂g3/∂κ g3 ⎥ , (96) ⎢ ⎥ ⎢∂ Lˆ /∂εe ∂ Lˆ /∂εe ∂ Lˆ /∂εe ∂ Lˆ /∂κ ∂ Lˆ /∂1γ ⎥ ⎣ 1 2 3 ⎦ ˆ e ˆ e ˆ e ˆ ∂ H/∂ε1 ∂ H/∂ε2 ∂ H/∂ε3 ∂ H/∂κ 0 e where cIJ = δIJ + 1γ ∂gI /∂εJ for I, J = 1, 2, 3. The foregoing procedure is general for any finite deformation hyper-elastoplasticity model with an isotropic strain energy function. Specific expressions of r and Ak for the constitutive model we employed can be found in Spiezia et al. [114]. This return mapping scheme also yields a consistent (algorithmic) tangent operator that allows for optimal convergence in the global Newton iteration. We refer to Borja [93] for the detailed procedure of obtaining the global tangent operator.

4.3. Solution scheme

To solve the coupled momentum balance and phase-field equations at discrete time instants, we use the staggered scheme proposed by Miehe et al. [58]. This scheme advances the numerical solution from time tn to tn+1 through the following three substeps:

1. Determine H with the displacement variable u at time tn. 2. With this H, update the phase-field variable d at tn+1 by solving Eq. (87). 3. With the updated d, update the displacement variable u at time tn+1 by solving Eq. (86). J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 19

The primary advantage of this staggered scheme over the monolithic scheme is its superior numerical robustness, which is much needed particularly when the phase-field variable evolves rapidly. However, the load increment in this staggered scheme should be small enough to attain sufficient accuracy, see Ambati et al.[125] for more discussions on this aspect.

5. Numerical examples The purpose of this section is to verify and demonstrate the capability of the proposed framework for modeling a variety of failure modes of geomaterials. For this purpose we simulate a series of laboratory-scale tests under plane strain conditions, similar to a typical experimental program investigating how the failure mode of geomaterials undergoes transition due to changes in confining pressure and strain rate conditions (e.g., [2,6,11,20,126]). Here we particularly focus on conditions where the material is subject to positive confining pressures (compressive stresses). This is because, as noted in Section3, under no confining pressure the proposed framework degenerates to the standard phase-field model of brittle fracture. This degeneration has been confirmed through simulation of the tension andshear examples in Miehe et al. [57] where confining pressure is absent. The numerical simulations are performed using Geocentric, a massively parallel finite element code for geomechanics that has been used in a number of previous studies [13,96,97,127–131]. This code is built upon the deal.II finite element library [132], p4est mesh handling library [133], and the Trilinos project [134].

5.1. Plane strain compression tests at different confining pressures

We first conduct plane strain compression tests under various confining pressures to investigate whetherthe proposed framework can capture pressure-sensitivity in the failure behavior appropriately. Fig.3 depicts the finite element model of the plane strain compression test. The domain is 30 mm wide and 75 mm tall, which is comparable to the dimension of a physical specimen tested in Makhnenko and Labuz [135]. We discretize the domain by 66,570 bilinear quadrilateral elements with a uniform diameter of h = 0.25 mm, which result in 134,274 displacement unknowns and 67,137 phase-field unknowns. Also, to facilitate inhomogeneous deformation, we introduce aweak region at the location drawn in this figure and reduce the cohesive strength C to 98% therein. As for the boundary conditions, the top and bottom boundaries are supported by rollers, except the lower left corner which is fixed by a pin. The top boundary is moving downward by δ(t) > 0 during the simulation. Meanwhile, the lateral boundaries are subject to a constant confining pressure of σc > 0 throughout the simulation. It is noted that one should consider the configuration change of the boundaries to keep the confining pressure constant under finite deformations. We consider a sandstone-like material, whose material parameters are given as follows. Elasticity: λ = 1833 MPa, ¯ µ = 2750 MPa; conical plasticity: C = 55 MPa, M0 = 0.8, M f = 0.9, k = 10, M = 0.5; and cap plasticity: ∗ A = 85 MPa, pi = 190 MPa, r = 1.0, εv = 0.2. Among these parameters, those defining the initial yield surface are calibrated with the experimental data of Adamswiller sandstone reported in Wong et al. [6], see Fig.4. Note that this calibration is just to assign some physically meaningful values, not to simulate the actual experimental data which were obtained from triaxial tests instead of plane strain tests. As for the phase-field model, we compute the critical fracture energy Gc from Eq. (76) (with the uniaxial compression case), and set the length parameter l to 1 mm considering that the choice of l = 4h has been shown to give reasonable accuracy [60,61]. For now we consider a rate-insensitive material, and assign 0.1 to the parameter α in Eq. (57) given that the Taylor–Quinney coefficient, which equals 1-α, is often assumed to be 0.9. Fig.5 presents simulation results of the compression tests under confining pressures of 5, 20, 40, 50, 60, and 150 MPa, in terms of differential stress–axial strain and volumetric strain–axial strain curves (stresses and strains are nominal). This figure clearly shows that the material response undergoes transition from (quasi-)brittle toductile behavior with an increase in the confining pressure. This pattern of pressure dependence agrees well with typical experimental observations for real geomaterials. At lower confining pressures, the specimen shows strain softening after a little amount of hardening. Becausethe plasticity model involves hardening behavior only, the softening response can be mainly attributed to the evolution of the phase-field variable. In other words, the phase-field model naturally plays the role of a softening lawasitmakes the phase-field effective stress higher than the overall, homogenized stress. After the peak stress, the volumeofthe specimen increases, owing to the evolution of the phase-field variable as well as dilative plastic flow. To illustrate how failure takes place during this type of response, in Figs.6 and7 we show equivalent plastic strain and phase-field 20 J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32

Fig. 3. Finite element mesh and boundary conditions of the plane strain compression example. The sample is 30 mm wide and 75 mm tall. A constant confining pressure of σc is applied to the lateral boundaries while δ(t) is increased during the test. In the weak region, the cohesive strength C is reduced to 98%.

Fig. 4. Calibration of the yield surface with the experimental data of Adamswiller sandstone from Wong et al. [6]. variable in the specimen under a confining pressure of 20 MPa, at three instances during the softening phase. At a nominal axial strain of 2.5%, a shear band has formed crossing the weak region, and along this shear band, the phase-field variable has increased. At this point the material inside the shear band may be considered damaged rather than fractured. Upon further loading, fractures has developed progressively inside the shear band, due to intense plastic deformation (more precisely, stored plastic work) therein. It is observed that this failure pattern is common in specimens manifesting a softening response and that the fracturing process becomes slower as confining pressure increases. As the confining pressure becomes higher, the material undergoes brittle–ductile transition. Fig.8 compares plastic strains in the specimens under confining pressures of 40, 50, and 60 MPa, at a nominal axial strain of 11%. Inthe40 MPa case, plastic strain manifests a localized zone wherein some cracks have been developed, but it is more distributed than that in the 20 MPa case. Plastic strain becomes far more diffuse when the confining pressure is increased to 50 MPa. When the pressure is further increased to 60 MPa, the plastic strain becomes nearly homogeneous, despite the weak region inserted to facilitate strain localization. This transition is because cap plasticity – which models diffuse, J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 21

Fig. 5. Differential stress–axial strain and volumetric strain–axial strain curves from plane strain compression tests at different confining pressures. compactive plastic deformation at higher confining pressures – does not contribute to the evolution of the phase-field variable. It is worth noting that in Fig.5 the stress–strain curves of the 40 and 50 MPa cases show mild softening and dilation after the peak stress, whereas the curve of the 60 MPa case shows marked hardening and compaction throughout the loading. Taken these observations together, we can conclude that brittle–ductile transition takes place in between confining pressures of 50 and 60 MPa. For the record, the Adamswiller sandstone samples tested under triaxial conditions also exhibit brittle–ductile transition around this level of confining pressure [6]. These results demonstrate that the proposed framework can capture pressure-induced brittle–ductile transition appropriately. Increasing the confining pressure further to 150 MPa leads to an interesting consequence. The specimen shows strain hardening earlier than the 60 MPa case, because it reaches the cap yield surface earlier. Until the nominal axial strain became about 19%, its stress–strain behavior is qualitatively similar to that of the 60 MPa case. However, after that, it begins to manifest strain softening, as the stress state moves from the cap yield surface to the conical yield surface. Contours of equivalent plastic strains in this specimen at nominal axial strains of 10%, 15%, and 20% are shown in Fig.9. This figure shows that the specimen undergoes distributed plastic deformation up to a substantial amount of compression, but ultimately, the specimen fails by strain localization. Indeed, such strain localization subsequent to significant plastic compaction has been observed for a number of real geomaterials underahigh confining pressure (see [3,7] for example). This type of transitional behavior, often called high-pressure embrittlement, is known to take place once pores collapsed by high pressures. From this case, we have found that coupling plasticity with phase-field fracture offers a way to capture this type of strain localization ensuing diffuse plastic compaction. 22 J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32

p Fig. 6. Equivalent plastic strain ε in the specimen at a confining pressure of 20 MPa, at nominal axial strains εa of 2.5%, 3.0%, and 3.5%. Elements with the phase-field variable d > 0.95 are deleted in the post-processing step.

Fig. 7. Phase-field variable d in the specimen at a confining pressure of 20 MPa, at nominal axial strains εa of 2.5%, 3.0%, and 3.5%.

At this point, we check mesh sensitivity of the coupled phase-field fracture and plasticity model for simulating strain softening behavior. It is well known that a Cauchy (non-polar) continuum strain-softening model without a proper regularization scheme (e.g., a plasticity model with a local softening law, or a local damage model) suffers from pathological mesh sensitivity. This type of mesh sensitivity manifests as a reduction of strength, which in turn leads to vanishing strength upon mesh refinement. Thus one can examine the mesh sensitivity by checking whether the strength depends inappropriately heavily on the mesh refinement. J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 23

p Fig. 8. Equivalent plastic strain ε in specimens at confining pressures of 40, 50, and 60 MPa, at a nominal axial strain εa of 11%. Elements with the phase-field variable d > 0.95 are deleted in the post-processing step.

p Fig. 9. Equivalent plastic strain ε in the specimen at a confining pressure of 150 MPa, at nominal axial strains εa of 10%, 15%, and 20%.

For this purpose, we repeat the 5 MPa confining pressure case with a finer mesh in which the mesh diameter h is halved. The length parameter l is fixed, so l/h is doubled to 8 in the finer mesh case. The results of the l/h = 4 and l/h = 8 cases are compared in Fig. 10. We observe that curves of the two cases are virtually identical except slight differences after many regions in the specimens are cracked. Most importantly, the strength remains unchanged by mesh refinement, which allows us to conclude that there is no spurious mesh sensitivity related tosoftening behavior. Because softening behavior in this model is attributed to the evolution of the phase-field variable, this 24 J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32

Fig. 10. Plane strain compression results under a confining pressure of 5 MPa from two meshes: l/h = 4 and l/h = 8. (l is fixed and h is halved in the latter case). mesh-insensitivity corroborates our phase-field model derivation leading to a thermodynamic analysis that energy dissipation by phase-field evolution is nonlocal.

5.2. Plane strain compression tests at different strain rates

Next, we examine the capability of the proposed framework for capturing rate sensitivity in the softening response. Recall that we have proposed a general form of α as Eq. (58) for materials in which the storage of the plastic work is rate dependent. As a specific expression for Eq. (58), we consider the following form of α:

−n α =α ¯ (1 +γ ˙ /γ˙ref) . (97)

Here, α¯ > 0 is the maximum value of α, γ˙ ≥ 0 is the rate of the plastic multiplier with its reference value γ˙ref, and n −6 is a material parameter. In Fig. 11 we plot how α/α¯ varies with γ˙ in case of n = 1 and γ˙ref = 10 for Eq. (97). This figure shows that as γ˙ becomes smaller, α becomes closer to α¯ , which means that as the rate of plastic deformation becomes slower, the stored portion of plastic work becomes greater. This trend is consistent with the experimental observation of Hodowany et al. [100] for the storage of plastic work in a rate-sensitive material. Since the stored portion of plastic work drives the evolution of phase-field variable in our formulation, the material is expected to become more brittle at a slower rate of plastic deformation. To check whether the rate of plastic deformation exerts control on the brittleness as expected, we perform plane strain compression tests with three different compression rates: ε˙, 10ε˙, and 100ε˙. Here, ε˙ denotes the compression rate in the foregoing example. We set α¯ and the confining pressure to 0.1 and 5 MPa, respectively, to ensure brittle behavior at the compression rate of ε˙. Fig. 12 illustrates differential stress–axial strain curves at the three compression rates, when α varies as in Fig. 11. We see that as the compression rate becomes faster, the specimen fails at a larger strain after more strain hardening, which is in good qualitative agreement with the experimental results in Fig.2. This agreement indicates that the idea of making α rate dependent can be a feasible approach for accommodating rate dependency of brittle failure responses of geomaterials.

5.3. Plane strain extension tests at different confining pressures

Lastly, we turn our attention to failure responses under tensile loading. Experimental studies have shown that the fracture mode of geomaterials in tension undergoes transition with an increase in confining pressure [7,110,136]. Tensile loading on a geomaterial at relatively low confining pressures produces an extension (opening mode) fracture propagating in the direction perpendicular to the maximum principal stress. However, once the confining pressure J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 25

Fig. 11. Variation of α/α¯ with γ˙.

Fig. 12. Differential stress–axial strain and volumetric strain–axial strain curves from plane strain compression tests at three compression rates: ε˙, 10ε˙, and 100ε˙. becomes higher than a certain level, tensile loading gives rise to a shear fracture forming at an acute angle to the minimum principal stress direction. Thus it would be illuminating to see whether the coupled phase-field and plasticity framework can accommodate such pressure-sensitivity in tensile loading as well. For this purpose we simulate a laboratory-scale extension test, changing the specimen geometry to a dog-bone shape as illustrated in Fig. 13. This geometry is analogous to the section of specimens tested in previous experiments [136]. The material parameters remain unchanged from the previous rate-independent example, except that the critical fracture energy is now matched to the uniaxial tensile strength. Given that the length parameter l is 1 mm, we discretize the domain such that l/h is sufficiently large (i.e., l/h ≫ 4) in the middle of the specimen where fractures would develop. This discretization results in 90,400 bilinear quadrilateral elements having 182,158 displacement unknowns and 91,079 phase-field unknowns. It is noted that the node at the bottom left corner isfixedby a pin whereas all other nodes at the top and bottom boundaries supported by rollers: this boundary condition provides mild asymmetry as well as well-posedness of the problem. Fig. 14 presents differential load–displacement curves of the extension tests performed at confining pressures of 20, 35, and 50 MPa. We can see that the specimen is very brittle at a confining pressure of 20 MPa, and it becomes increasingly ductile as the confining pressure increases. The transition of the failure mode is also evident interms of fracture patterns. Figs. 15 and 16 show contours of equivalent plastic strain and phase-field variable in the three 26 J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32

Fig. 13. Finite element mesh and boundary conditions of the plane strain extension example. The sample is 20 mm wide and 40 mm tall. A constant confining pressure of σc is applied to the lateral boundaries while δ(t) is increased during the test. specimens at the time of failure. At a confining pressure of 20 MPa, the specimen fails by an extension fracture developed horizontally at the center. No plastic strain is observed throughout the specimen, which means that the fracture has occurred at a tensile mean stress and handled solely by the phase-field model. On a related note, we observe the same fracture pattern when the confining pressure is absent. By contrast, at higher confining pressures (35 and 50 MPa), extension tests produce plastic deformations prior to failure. This means that an increase in the confining pressure activates another failure mechanism, shear fracture. The failure responses of the 35 and 50 MPa cases show some interesting differences. First, as shown in Fig. 15, fractures in the 50 MPa case have developed inside the localized plastic strain zone, much like those in the compression test performed at a low confining pressure (e.g., Fig.6). However, in the 35 MPa case, fractures are not clearly bounded by a strain localization zone. This implies that not every phase-field evolution may be associated with plastic deformations. The second, and related, difference is that the failure region in the 35 MPa case is less inclined from the horizontal. Indeed, this trend of fracture angle changes by confining pressures is what has been identified by laboratory investigations, see Ramsey and Chester [136] for example. As for the reason for such difference, some geosciences researchers have suggested that there exists a transitional failure mode in which both extension and shear fractures take place in a mixed manner, which they call a hybrid fracture [136,137]. In this sense, the failure response of the 35 MPa case may correspond to a hybrid fracture in geomaterials. These results demonstrate that the proposed framework can also capture pressure-induced transition of failure modes under extension loading.

6. Closure We have developed a framework that couples a phase-field approach to fracture with a pressure-sensitive plasticity for modeling brittle fracture to ductile flow in geomaterials. The framework enjoys advantages of both ofthe phase-field and plasticity approaches: it can capture the inception and propagation of fractures without tracking crack geometry as well as the transition of failure responses due to changes in confining pressure and strain rate conditions. A byproduct from the coupling of these two approaches is mesh-insensitivity in the softening behavior. Numerical examples have demonstrated that the proposed framework can simulate three characteristic failure modes under various stress and loading conditions: (1) brittle, localized failure, (2) ductile, diffuse failure, and (3) strain localization after pore collapse. Simulating all of these failure modes and their transition is beyond the capabilities of J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32 27

Fig. 14. Differential load–displacement curves from the plane strain extension tests at confining pressures of 20, 35, and 50 MPa.

p Fig. 15. Equivalent plastic strain ε in the specimens at confining pressures σc of 20, 35, and 50 MPa, at the time of failure. Elements with the phase-field variable d > 0.95 are deleted in the post-processing step. most existing computational models. As such, the development of this work can be particularly useful for problems whereby geological materials are subject to a variety of stress states and loading conditions, such as those encountered in subsurface energy technologies. Along the way, we have made new contributions for more physically meaningful and computationally efficient coupling of phase-field fracture and plasticity models. They include: (1) a balance law derivation of aphase-field fracture model that honors the dissipative nature of the fracturing process, (2) the use of phase-field effective stress for incorporating degradation to plasticity, and (3) the dilative/compactive split and rate-dependent storage of the plastic work. These contributions are general, and can be applied to future work coupling phase-field and plasticity models for various purposes. The proposed framework can be readily extended to accommodate more complex features of geomechanical behavior. Examples include the effects of temperature and/or pore pressure on the brittle–ductile transition, and differences in the critical fracture energy values with respect to the fracture mode. Also, its computation can be significantly facilitated by employing an adaptive mesh refinement scheme for phase-field modeling (e.g.,that 28 J. Choo, W. Sun / Comput. Methods Appl. Mech. Engrg. 330 (2018) 1–32

Fig. 16. Phase-field variable d in the specimens at confining pressures σc of 20, 35, and 50 MPa, at the time of failure. presented in Heister et al. [117]) in conjunction with an internal variable projection scheme for plasticity modeling (e.g., that proposed by Mota et al. [138]). Research in these directions is underway.

Acknowledgments Financial support for this work has been provided by the Earth Materials and Processes program of the US Army Research Office under Contracts W911NF-15-1-0442 and W911NF-15-1-0581; Sandia National Laboratories under Contract 1557089; the Dynamic Materials and Interactions Program of the Air Force Office of Scientific Research under Contract FA9550-17-1-0169; the Nuclear Energy University Program of the Department of Energy under Contract DE-NE0008534; and the Mechanics of Material program of the National Science Foundation under Contract CMMI-1462760. This support is greatly appreciated. The views and conclusions contained in this document are those of the authors, and should not be interpreted as representing the official policies, either expressed or implied, of the sponsors, including the Army Research Laboratory or the US Government. The US Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

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