Available online at www.sciencedirect.com ScienceDirect 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 fracture to ductile flow Jinhyun Chooa,b,∗, WaiChing Suna a Department of Civil Engineering and Engineering Mechanics, 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 pressure and strain rate. Under a relatively low confining pressure, these materials tend to fail by brittle, localized fracture, but as the confining pressure increases, theyshowa growing propensity for ductile, diffuse failure accompanying plastic flow. Furthermore, the rate of deformation often exerts control on the brittleness. Here we develop a theoretical and computational modeling framework that encapsulates this variety of failure modes and their brittle–ductile transition. The framework couples a pressure-sensitive plasticity model with a phase-field approach to fracture which can simulate complex fracture propagation without tracking its geometry. We derive a phase-field formulation for fracture in elastic–plastic materials as a balance law of microforce, in a new way that honors the dissipative nature of the fracturing processes. For physically meaningful and numerically robust incorporation of plasticity into the phase-field model, we introduce several new ideas including the use of phase-field effective stress for plasticity, and the dilative/compactive split and rate-dependent storage of plastic work. We construct a particular class of the framework by employing a Drucker–Prager plasticity model with a compression cap, and demonstrate that the proposed framework can capture brittle fracture, ductile flow, and their transition due to confining pressure and strain rate. ⃝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 soils 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 gases [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 rock under different confining pressures: (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/ductility of geomaterials under non-elevated temperature: confining pressure and strain rate. Geomaterials under a negligible confining pressure are usually brittle because tensile fractures 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 crystal 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]. Fracture mechanics can offer a basis for a more mechanically sound, mesh-insensitive approach to brittle failures of geological materials. The application of fracture mechanics to geomechanical problems has been explored since several decades ago (e.g., [45,46]), and it is now an active area of research in various contexts from landslides to fault mechanics to hydraulic fracturing (e.g., [47–54]). 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