Loading-angle-dependent rate effects on dynamic mixed-mode I/II fracture behavior of coal-measure sandstone

Numerical Investigation of Mixed Mode I-III Fracture Behavior in Sandstone Under Static and Dynamic Loading

Mixed mode I–III fracture commonly occurs in rock masses under complex three-dimensional stress states, yet the combined effects of loading conditions and fracture mode remain unclear. In this study, a heterogeneous sandstone disc model is developed in ABAQUS by coupling the Drucker–Prager elastoplastic model, an equivalent-strain damage model, and cohesive elements. The model is validated against static Brazilian splitting tests and dynamic SHPB tests. ENDB specimens are then employed to investigate mode I, mode III, and mixed mode I–III fracture under static and dynamic loading with offset angles ranging from 0° to 62.5°. As the offset angle increases, crack propagation evolves from straight tensile extension to deflected, twisted, and fragmented patterns dominated by anti-plane shear. Dynamic loading intensifies crack segmentation, localized damage, and transient instability while increasing the peak load, fracture toughness, and fracture energy. In contrast, the effective fracture toughness and fracture energy decrease monotonically with increasing offset angle under both loading conditions. A linear trend consistent with R2 > 0.99 is observed between the mode I–III mixing coefficient and the effective fracture toughness based on single-realization simulations, the effective fracture toughness under dynamic loading is approximately 2.22 times that under static loading based on direct comparison of calculated values. These findings improve the understanding of loading-dependent mixed mode I–III fracture in sandstone and provide guidance for rock mass stability assessment.

Fracture Behavior and Criterion of Granite Under Combined Static–Dynamic Mixed‐Mode Loading

Deep rock masses in underground engineering are often subjected to combined loading conditions, where pre‐existing static tectonic stresses coexist with dynamic disturbances induced by excavation activities. Additionally, fractures in rock engineering are frequently subjected to mixed tension‐shear loading scenarios. To elucidate the cracking mechanisms of rocks under such complex conditions, this study employs a modified split Hopkinson pressure bar (SHPB) to conduct combined static–dynamic (CSD) fracture experiments on notched semi‐circular bend (NSCB) granite samples, with a particular focus on the mixed‐mode I/II and pure mode II loading. These experiments reveal that the effective fracture toughness (FT) decreases as the loading mixity shifts from mode I toward mode II. In contrast, the crack propagation velocity increases as the mode II loading component decreases. Fracture morphology is found to be governed primarily by the loading mixity rather than the loading rate. Furthermore, this study systematically evaluates various fracture criteria and demonstrates that the GMTS fracture criterion, which incorporates the effects of dynamic loading rate on the critical distance, offers preferable predictions of mixed‐mode I/II crack initiation in granite under CSD loading. Notably, an exponential relationship is identified between the critical distance and loading rate. These findings enhance the understanding and predictive capability of mixed‐mode I/II crack initiation in granite under both pure dynamic and CSD loading conditions.

Dynamic Progressive Failure and Energy-Driven Damage Evolution of Coal–Sandstone Composite Specimens Under Impact Loading: Coupling Effects of Component Ratio

Overburden rotation and mining disturbance promote deep coal fracture and spalling, potentially inducing dynamic hazards, while the coal–rock thickness ratio largely governs the composite’s dynamic response and failure severity. Based on the geological conditions of a typical deep coal mine, SHPB tests were performed on coal–sandstone composite specimens with four systematically varied coal-to-rock ratios (C:R = 1:0, 2:1, 1:1, and 1:2). High-speed photography and the digital speckle correlation method (DIC) were integrated to capture displacement, strain, energy, and fragmentation fields throughout progressive dynamic compression. Experimental data revealed four findings: (1) Crack development follows a sequential evolution process of initiation, propagation, and failure. Higher impact air pressure accelerates crack development and coalescence, resulting in a higher degree of specimen fragmentation and fewer large blocks. Within the tested range of coal–rock ratios, an increase in the rock proportion accelerates coal fracture, which is attributed to the significantly higher density and hardness of rock compared to coal. (2) Energy evolution consistently follows three stages: absorption, accumulation, and dissipation. Under identical impact pressure, a higher rock ratio elevates equivalent stiffness and wave impedance, leading to monotonic increases in peak stress, peak strain, absorbed energy, and dissipated energy. (3) The fragmentation degree exhibits a pronounced dependence on impact pressure. Specifically, for the pure coal specimen (C:R = 1:0), when the impact pressure increases from 0.3 MPa to 0.7 MPa, the mass percentage of coarse debris (>30 mm) drops from 73.37% to 18.57%, whereas that of fine particles (<4 mm) rises from 15.69% to 35.24%. (4) Under identical impact conditions, a higher rock proportion leads to increasing trends in all measured indicators, including peak stress, strain, and energy accumulation and dissipation, which are consistent with the superior mechanical properties of the rock. Based on these observations, it can be inferred that the wave impedance mismatch and stiffness ratio at the coal–rock interface play a key role in controlling stress wave transmission/reflection and strain incompatibility; however, the individual contribution of each factor warrants further dedicated investigation.

Mechanical response and failure mechanisms of sandstone under water–rock interaction: a multi-scale study and constitutive modeling

Loading Rate Effects on Crack Initiation and Interaction in Flawed Discs

Mixed Mode Fracture Mechanics and Crack Growth Analysis in Steam Turbine Discs for Remaining Life Assessment under Thermal Stresses and Mechanical Loading

The long‑term performance and structural integrity of steam‑turbine discs are strongly influenced by crack‑growth behavior under combined thermal stresses and mechanical loading. In this study, a comprehensive analytical framework based on finite element analysis (FEA), mixed‑mode fracture‑mechanics evaluation (Modes I, II, and III), and fatigue crack‑growth modeling using the Paris law is presented. The results indicate that Mode II plays the dominant role in the stress‑intensity distribution and crack‑growth behavior of the investigated turbine disc, while Modes I and III also contribute and cannot be neglected. Therefore, safety assessment based solely on Mode I analysis may lead to inaccurate evaluations of structural integrity. The calculated effective stress‑intensity factor was compared with the fracture toughness of 12Cr steel, showing that the turbine disc currently operates within a safe margin against instantaneous fracture; however, gradual fatigue crack growth under thermo‑mechanical loading conditions remains possible. The crack‑growth analysis further demonstrates that a significant portion of the fatigue life is spent during the stable slow‑growth phase as the crack propagates from small initial sizes to intermediate lengths. The proposed fracture‑mechanics‑based approach provides a reliable tool for condition monitoring, inspection planning, and remaining‑life assessment of steam‑turbine discs, thereby contributing to improved operational safety and maintenance decision‑making.