PSI - Issue 45

Nhan T. Nguyen et al. / Procedia Structural Integrity 45 (2023) 52–59 Nguyen et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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1. Introduction Localised failures in the form of compaction bands are usually encountered in triaxial tests on porous reservoir rocks under high levels of confining pressure (Fossen et al., 2007; Kolyukhin et al., 2010; Renard et al., 2019; Baud et al., 2021). The behaviour inside these bands is intrinsically characterized by the micro-scale process of grain crushing and its interaction with the grain rearrangement, mainly governing the irreversible responses of the whole rock specimen and thus making their deformation much higher than that of the surrounding region. This reflects the inhomogeneous mode of crushed rock samples which controls the macro hardening response and its dependence on the sizes of the localisation band and specimen. The inhomogeneous nature, along with size-dependent behaviour, in the post-localisation regime is rarely explored in classical constitutive models for rocks although they have proven successful in capturing the grain-crushing phenomenon (Einav, 2007a,b; Buscarnera and Einav, 2012; Cil et al., 2019; Xiao et al., 2020). These models are based on assumptions of homogeneous deformation which are broken down when localised failures take place. They lack details on localisation bands (thickness, orientation) and the kinematic enhancement across these bands. Consequently, the adoption of such models in post-localisation regimes is no longer appropriate, given they cannot describe correctly the underlying mechanisms of strain localisation behind the observed macro size-dependent response. This results in ill-posed boundary values problems (BVPs) and discretization-dependent numerical solutions. BVPs involving localised failures can be effectively addressed through advanced higher-order models and numerical methods. For instance, the scale effect can be adequately tackled, while the meaningful convergence of numerical solutions upon discretization can be guaranteed by the theory of nonlocal damage (Chen and Schreyer, 1987; Pijaudier- Cabot and Bažant, 1987) , the gradient regularisation (De Borst and Mühlhaus, 1992), the Cosserat model (Mühlhaus and Vardoulakis, 1987) and the viscous regularisation (Carosio et al., 2000; Pedersen et al., 2008). These approaches use extra parameters of length scale associated with the width of the localisation bands for regularizing BVPs, preserving the ellipticity of the governing partial differential equations in the post-localisation stage. Despite some successes, they are unable to correctly capture the underlying mechanisms of localised failures due to using phenomenological and ad hoc treatments. Furthermore, these approaches are either complex or computationally inefficient for problems at a large (enough) scale since they are usually mixed with the numerical methods for solving BVPs, hence hindering their broad applications. In this paper, we develop a mechanism-based constitutive model to simulate the inhomogeneous phenomenon and size effects induced by compaction bands in porous reservoir rocks, using a rigorous and simple approach. The macro behaviour of this model is governed by the interaction between the elasticity outside and the inelasticity inside the localisation bands. This interaction is established through a kinematically enriched approach accounting for size-effect properties (thickness and orientation of localisation bands, specimen size), while constitutive formulations based on the breakage mechanics are used for capturing the irreversible process inside compaction bands under high confinements. As a consequence, the proposed model can capture the mixed material-structural response and its intrinsic dependence on the mechanism of grain crushing in the post-localisation regime. These promising features are elucidated in some numerical examples simulating the drained triaxial test on sandstone at high confining pressures. 2. Formulation This section presents key constitutive formulations of the two-scale approach reproducing the inhomogeneous behaviour (Nguyen et al., 2012; Nguyen et al., 2014; Nguyen et al., 2016; Nguyen and Bui, 2020) (see sub-section 2.1) and its base model accounting for the grain crushing mechanism inside compaction bands (Einav, 2007a,b) (see sub-section 2.2) 2.1. Post-localisation constitutive structure Discontinuous bifurcation is considered a strain jump that exhibits over two faces of a tabular localisation band while maintaining the kinematic compatibility with the surrounding material. Following this constitutive instability for two different tangent stiffnesses of the inside and outside of a localisation band, the discontinuous bifurcation