PSI - Issue 68

Nhan T. Nguyen et al. / Procedia Structural Integrity 68 (2025) 91–98 N.T. Nguyen et al. / Structural Integrity Procedia 00 (2025) 000–000

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1. Introduction The Fracture Process Zone (FPZ) is a nonlinear region located ahead of the crack tip in rock materials, where numerous microcracks initiate and coalesce, ultimately merging to form a macroscopic fracture (Ghamgosar and Erarslan, 2016; Dutler et al., 2018; Zhang and Zhou, 2022). Energy dissipation inside the FPZ is composed of two micro-mechanisms related to the fractured surface opening and frictional sliding (Bažant Zdeněk, 1996; Landis et al., 2003; Qiao et al., 2019; Meng et al., 2022; Miao et al., 2024). The contribution of these two mechanisms to the tensile fracture failure has not received sufficient attention in existing studies and frictional dissipation is usually neglected in several damage models. Experimental works by Bažant Zdeněk (1996) and Landis et al. (2003) have resulted in an important conclusion that energy dissipated due to loss of frictional resistance accounts for an equal or even larger proportion of a whole dissipation budget as compared to the creation of fractured surfaces. The distribution of energy dissipation between surface opening and frictional sliding has implications for the strain rate inside the FPZ. In this sense, studies have shown that quasi-brittle materials like rocks and concrete are rate sensitive, as different loading rates would alter their macroscopic behaviour. For example, Cho et al. (2003) and Cao et al. (2023), using split Hopkinson pressure bar testing, observed that higher strain rates generate more microcracks, enhancing peak tensile strength. Similarly, Xu et al. (2023) found that higher loading rates increased volumetric dilation in sandstone, granite, and basalt, while Yan et al. (2020) noted that both energy dissipation density and fragmentation intensity increase with strain rate. Additionally, DIC analyses on Brazilian Disc tests at varying quasi static rates by Xing et al. (2020) concluded that as the quasi-static loading rates increase, the FPZ length is also facilitated, thus enhancing the fracture resistance under mixed-mode (I-II) of the material. Aiming to incorporate strain rate effects in numerical modelling, various approaches have been employed. Models frequently applied a dynamic increase factor (DIF) to compressive or tensile strength values under high strain rates (Du et al., 2014; Jin et al., 2019), or Cho et al. (2003) used displacement increments in finite element simulations to capture dynamic responses, or Zhang et al. (2018) applied peridynamics for investigating meso-mechanical failure mechanisms under uniaxial loads at different quasi-static strain rates. From the existing literature, most studies focused on the FPZ evolution in response to different loading rates, yet only a minority addressed whether strain rates within the FPZ exceed quasi-static levels, especially when overall loading is slow, and the surrounding bulk deforms quasi-statically. Recent studies, however, have started exploring this connection. For instance, Lian et al. (2023) found that crack propagation rates in concrete beams under three-point bending vary between static and dynamic behaviours depending on external loading rates. Similarly, Ngo et al. (2019) showed that increasing loading rates in ultra-high-performance fibre-reinforced concrete (UHPFRC) leads to higher crack velocities, underscoring the strong link between high strain rates and accelerated crack propagation. Chen et al. (2021) introduced a rate-dependent cohesive zone model by the modifying softening function, fracture energy, and FPZ length based on fracture propagation velocity, allowing for a dynamic adjustment of fracture energy to better capture FPZ evolution. This adjustment enables the model to capture the impact of fracture propagation rates on the FPZ’s evolution, where higher fracture growth rates lead to increased FPZ length and decreased material brittleness. These studies focus primarily on the impact of high loading rates on FPZ propagation and expansion, highlighting the speed at which microcracks coalesce and the FPZ lengthens. However, the studies above primarily investigated the relationship between crack propagation rate and loading velocity at very high loading rates, reaching the quasi-static limit and even extending into the dynamic range. Beyond the crack propagation rate, it remains unclear whether reducing the loading velocity would influence the fracture opening rate and, if so, how this would impact the macro response in terms of energy dissipation, peak strength, brittleness, and ductility. In this study, we propose a numerical approach to investigate the strain rate within the FPZ, utilising Smoothed Particle Hydrodynamics (SPH) simulations of a three-point bending test on concrete. This simulation employs a coupled damage-plasticity model proposed by Nguyen et al. (2024), which allows adjustment of the energy dissipation pathways to capture whether fracture opening or frictional sliding dominates the material behaviour. We integrated a Perzyna-type rate-dependent formulation. This rate-dependent enhancement enables the model to effectively represent the constitutive response across various loading velocities, specifically in terms of peak strength and the balance between brittle and ductile behaviour.