The surrounding rock around deep underground excavations generally contains fractures and joints that significantly reduce mechanical integrity, particularly in zones where excavation-induced unloading, stress rotation, and anisotropic deformation generate tensile stress concentrations. These tensile regimes govern crack initiation, fracture propagation, spalling, and progressive instability around tunnels, caverns, mines, and wellbores. Although rock grouting is widely used as a reinforcement strategy, its effectiveness in restoring the inherent strength of tension-dominated surrounding rock is often assumed, and the tensile fracture behavior of fracture-grouted rock remains insufficiently understood. This study investigated the evolution of tensile strength (TS) and tensile-induced fracture behavior in fractured surrounding rock before and after fracture grouting, providing mechanistic evidence for assessing strength recovery. In addition to TS, failure modes and total fracture length (TFL) were quantified to assess fracture propagation and grout-reinforcement performance in tension-prone surrounding rock zones. Brazilian disc tests (BDT) were conducted on natural (limestone and dolomite) and synthetic (3D-printed) rock samples, allowing direct comparison of the same samples in ungrouted and fracture-grouted states. To mechanistically interpret and validate the experimental findings, a finite element model employing the cohesive zone method (FEM-CZM) implemented in ABAQUS was developed to simulate the evolution of tensile strength and the initiation and propagation of micro-fracture in the specimens before and after grouting. Results revealed that while fracture grouting does not fully restore the inherent tensile strength of fractured rocks, it significantly altered the tensile failure process. Grout-rock interfaces in the fracture-grouted rocks constrained and redirected crack propagation, reduced total fracture length, and shifted failure modes toward more localized and controlled fracture patterns. A positively correlated TS-TFL relationship observed in ungrouted samples reversed in fracture-grouted samples, indicating that higher tensile resistance in grouted rock mass corresponds to more limited fracture development. The FEM-CZM simulation confirmed the experimentally observed post-grouting delay in rock damage onset with reduced fracture path and an increase in Mode-II energy dissipation, and provided direct visualization of stress concentration, damage evolution, and fracture-path control. These findings demonstrate that surrounding rock control in tensile regimes depends not only on tensile strength recovery but also on the ability of grouting to suppress fracture propagation and damage evolution. The results provide new mechanistic insight into fracture-grouting performance in tension-prone underground environments and demonstrate that TFL, when used alongside tensile strength and other mechanical parameters, is a valuable metric for assessing reinforcement effectiveness. This work advances the understanding of grouting as a surrounding rock control strategy and informs the design of reinforcement systems aimed at stabilizing underground excavations subjected to tensile stress concentrations.

Tensile fatigue evolution in sandstone under lower cyclic stress: Coupled Brazilian disc experiments and enhanced mesoscale damage modeling

Dynamic response and fracture of B₄C/6061 Al composites: experiments and simulations

During fracture propagation in quasi-brittle materials such as rock, stress concentration plays a critical role as a result of stress redistribution, leading to elevated stress levels near crack tips. In classical elastic analytical solutions, stresses at the crack tip tend toward infinity; therefore, conventional elastic analyses are inadequate for describing fractured materials. This study employs Linear Elastic Fracture Mechanics (LEFM) to model this phenomenon in rock, with a focus on representing mode I fracture toughness (KIc). To address these processes, the Continuum Voronoi Block Model (CVBM) is utilized. CVBM is a pseudo-discontinuum approach implemented within a Finite Element Analysis (FEA) framework, combining Voronoi tessellation and the Goodman joint, to represent the rock’s mesoscale structure. The model is first calibrated for deformational properties and tensile strength, and then iteratively adjusted to reproduce experimental KIc values. The calibration is validated under different boundary conditions through numerical simulations of Center-Cracked Tension (CCT) tests, Cracked Straight-Through Brazilian Disc (CSTBD) tests, and Semi-Circular Bending (SCB) tests, successfully reproducing fracture paths and yielding KIc values consistent with the literature.

Accurate simulation of prosthetic materials requires constitutive models that capture pressure sensitivity and tension-compression asymmetry beyond linear elasticity. This study presents a reverse-engineering workflow to calibrate a Drucker-Prager based constitutive model in LS-DYNA using the semi-analytical model for polymers MAT 187L SAMP Light for a resin composite (Brilliant Crios) and a polymer-infiltrated ceramic network (Vita Enamic). Unconfined uniaxial compression, three-point bending, and Brazilian disc tests provide elastic constants and strength measures that serve as inputs and calibration targets. An analytical initialization maps experimentally determined yield stresses to the linear Drucker-Prager yield surface, supplying reliable starting parameters for finite element reverse-engineering optimization. The calibrated model captures the material response in the calibration tests (three-point bending and Brazilian disc) within the pre-peak regime, and an out-of-sample punch-through test confirms the transferability of the parameters without additional tuning. Compared to von Mises characterization approaches, the pressure-dependent characterization was achieved with only one additional test configuration, shifting effort from experiments to numerical computation optimization. Within these limits, the results support pressure-dependent, asymmetric plasticity as a practical basis for predictive finite element analysis of dental restoratives, while highlighting that explicit damage and strain-rate effects should be incorporated in future work to model softening and failure consistently.

To evaluate the effects of freeze–thaw (F–T) cycling on the tensile performance and energy evolution of water-saturated sandstone, this study utilized slope rock specimens collected from a high-altitude open-pit mine in western China. Dynamic Brazilian disc tests were conducted using a split Hopkinson pressure bar (SHPB) system under four distinct strain rate conditions. The specimens were pretreated with varying numbers of F–T cycles before testing. Experimental analysis revealed that under constant F–T conditions, both the tensile strength and energy dissipation ability under impact loading increased with increasing strain rate. In contrast, when the strain rate remained steady, a greater number of F–T cycles led to decreases in the splitting strength and energy dissipation capacity. Moreover, the dynamic increase factor (DIF) increased progressively with increasing strain rate and F–T cycle number. The deterioration in dynamic performance is attributed to internal degradation mechanisms, including the dissolution of cementing materials and the development of microcracks and pores, which reduce the overall integrity of the rock and promote more fragmented macroscopic failure modes. F–T-induced damage is identified as the primary factor responsible for the decrease in impact resistance and increase in the fragmentation of the specimens. Moreover, a scaling law model grounded in weakest-link theory was established to estimate the tensile capacity of F–T-degraded water-saturated sandstone.

Carrying out the constitutive model research of melt-cast explosives is helpful to evaluate the safety of explosives, optimize the performance design, guide the material research and improve the accuracy of engineering design. In this paper, the Karagozian & Case model is applied to melt-cast explosives for the first time. Taking the 3,4-dinitropyrazole-based melt-cast explosive as an example, quasi-static compression mechanical property, dynamic compression mechanical property and quasi-static Brazilian disc tests of its typical formulation were carried out by the universal materials testing machine and the split Hopkinson pressure bar test setup. Based on the test results, the parameters of the Karagozian & Case model were calibrated. Using the calibrated model parameters, the mechanical responses of the 3,4-dinitropyrazole-based melt-cast explosive under dynamic impact, quasi-static compression and triaxial confining pressure were calculated. The results show that the model’s capability in describing the complete mechanical behavior of melt-cast explosives, from elastic deformation through damage evolution to ultimate failure. The Karagozian & Case model effectively reproduces the material’s strain-rate sensitivity and accurately captures the transition from brittle to ductile behavior under confining pressure. Consequently, the Karagozian & Case model is established as an effective tool for predicting the mechanical response of melt-cast explosives under various loading conditions, and valuable insights are provided for safety evaluation and performance optimization in engineering applications.

A pivotal limitation of 3DPC - 3D printed concrete is the weak bond strength between filaments and layers due to the layer-by-layer fabrication process. In this research, the mechanical properties of printing filaments were investigated and developed to enhance bond strength in both the vertical and horizontal directions. Rectangular self-interlocked filaments fabricated by specifically designed nozzles were used to enhance bond strength between vertical layers. The improvement of bond strength between horizontal filaments on the same plane was obtained by controlling printing trajectories. The effectiveness of self-interlocking filaments was evaluated by printing and testing four types of 3DPC specimens: two with flat and straight interlayers, and two with interlocked interlayers. The disc specimens cut from printed beams and blocks were subjected to indirect tensile tests with different angles between loading and printing directions. The present study employed an advanced testing method, AUSBIT - Advanced Universal Snap-Back Indirect Tensile testing method, in conjunction with both AE - Acoustic Emission and DIC - Digital Image Correlation to capture not only the tensile strength but also to investigate the fracture resistance and post-peak behaviour of 3DPC specimens. The results showed that the indirect tensile strength between layers increased by 48.35 % when tested at a loading angle of 0-degree using a rectangular interlocking pattern, while the zigzag interlocking pattern resulted in a 60.26 % improvement in bond strength between horizontal filaments, compared with the control specimens. The use of lateral displacement control in Brazilian disc tests helped stabilise the fracture process and enabled a reliable analysis of the intrinsic mechanism governing the evolution of fracture and its influence on the overall snapback behaviour. The promising results reveal great potential for further investigation in this direction for strengthening 3D-printed structures. • Rectangular interlocking filaments produced by an innovative nozzle led to a 50 % increase in bond strength between layers. • A 56.3 % increase in bond strength between horizontal filaments can be obtained by zigzag printing strategies. • The advanced technique AUSBIT (Advanced Universal Snapback Indirect Tensile testing) effectively helps stabilise the fracture process to obtain a better result on indirect tensile strength.

• Mesostructure-level finite element model of rock based on EBSD data. • Elastic properties of the mineral phases of rock identified via nanoindentations. • Strain rate-dependent cohesive zone model with fatigue damage evolution. • Image treatment method to analyse simulated cracking patterns of rock. • Analysis of fatigue damage in granite caused by alternating current excitations. This paper presents a numerical-experimental approach to investigate the weakening of granite caused by High Voltage Alternating Current (HV-AC) excitations. A two-dimensional, mesoscale finite element (FE) model was developed to simulate the HV-AC-induced damage. The microstructural features of the rock were obtained from Electron Back Scatter Diffraction data, and the micromechanical behavior of each mineral constituent is defined using nanoindentation tests. Cohesive elements account for load transfer between neighboring grains and simulate the damage accumulation during loading. A cohesive zone model was developed to describe the evolving weakening of grain boundaries during cyclic loading, featuring strain rate sensitivity and continuous time-evolving damage. The constitutive cohesive zone model parameters were identified in separate stages using multiple types of experimental data, that is quasistatic compression, dynamic indirect tension of Brazilian disc tests, as well as low speed and high speed (impact) fatigue tests, to isolate the contribution of individual mechanical processes. The model predictions match well with the experimental evidence under all loading conditions, including the resulting reductions of dynamic tensile strength caused by HV-AC excitations, which were around 17% of the strength of the nontreated rock. The cracking patterns predicted by the model match well with the observed experimental patterns. This study provides a quantitative comparison of the simulated cracking patterns of treated and nontreated rocks using a novel image treatment method. The comparison revealed that the HV-AC treatment generates weak spots in the form of small flaws in the rock microstructure. These small flaws act as nucleation sites for fracture propagation during dynamic loading, leading to the coalescence of cracks and facilitating rock breakage.

Investigating dynamic mixed-mode I/II fracture behavior of sandstone using the modified generalized maximum tangential stress criterion

The stability of rock masses is critically influenced by pre-existing cracks, yet the micro-mechanisms governing cracks evolution under varying crack geometries remain inadequately quantified. This work introduces a novel, rock-specific acoustic emission (AE) classification criterion for marble, established through integrated notched semi-circular bend (NSCB) and straight-notched Brazilian disc (SNBD) tests. The proposed criterion for both tensile cracks (AF > 40 RA + 30) and shear cracks (AF < 20 RA + 30) provides a quantitative and material-adapted method for real-time crack-type discrimination. Combining AE monitoring with stress intensity factor (SIF) analysis, 9 groups of uniaxial compression tests were conducted on marble specimens with crack inclinations (15°, 30°, 45°, 60°, 75°, 90°) and lengths (5 mm, 10 mm, 15 mm). The synchronous evolution of SIF and AE parameters across four loading stages validates the AE-based classification. The test results reveal that crack inclination governs the tensile-to-shear transition, while crack length accelerates damage accumulation. It is demonstrated that the coupled AE-SIF analysis captures micro-mechanical crack evolution and bridges microscopic fracture processes with macroscopic failure modes. This study offers a reliable framework for early warning of rock instability.Supplementary InformationThe online version contains supplementary material available at 10.1038/s41598-025-22264-z.

Revisiting the Brazilian disc test with split Hopkinson pressure bar by high-speed digital image correlation analysis

The growing global demand for minerals and metals, coupled with the need for improved energy and water efficiency in resource extraction, has led to the use of numerical modeling, particularly the discrete element method (DEM), to evaluate and optimize comminution processes that account for a significant portion of the energy consumption in mineral and metal extraction. Despite advancements, a significant challenge remains in balancing the local resolution of fractures at the rock particle level, where physics-based material models using the finite element method (FEM) have excelled, with the resolution of industrial-scale total particle interactions within the machine system. This work explores the high-resolution fracture of rock particles using an established material model implemented within FEM as a valuable reference for fractures with a balanced mid-level resolution achieved through a bonded discrete element method applicable to industrial-scale systems. Brazilian tests were performed on two rock types to calibrate the models. Single particle breakage (SPB) experiments employing digital image correlation (DIC) were conducted to evaluate the performance of the models. Finally, the DEM model was demonstrated in an industrial-scale cone crusher application. The results show good agreement for the highly resolved FEM approach (requiring only two material parameters to be determined, which is particularly advantageous for generating virtual particle breakage data across various rock materials, shapes, and sizes) and reasonable agreement for the DEM fracture response, which is attributed to the much coarser mesh used that does not capture the crumbling mechanism (as revealed by the comparison between the two numerical approaches). Despite these discrepancies, the cone crusher predictions fall within the expected ranges for the system response at the machine level. • Single particle breakage evaluated using FEM and DEM. • Rock fracture models calibrated with Brazilian disc test and validated using DIC and scanned particle geometries. • FEM models provide high accuracy and DEM models scalability. • Industrial-scale DEM simulations performed on cone crusher with realistic outcomes.

This study investigates the dynamic tensile behavior of frozen granite under varying strain rates and subzero temperatures using Split Hopkinson Pressure Bar (SHPB) Brazilian disc tests. Tests were performed at temperatures ranging from −30 °C to 20 °C, with high-speed cameras employed to capture the crack initiation and propagation process. The experimental program examined dynamic tensile strength, strain-rate sensitivity, energy absorption characteristics, and fracture morphology. Results show that dynamic tensile strength increases significantly with strain rate, and the strain-rate sensitivity coefficient becomes more pronounced as temperature decreases. A Logistic function was found to accurately describe the relationship between the strain-rate enhancement factor and strain rate. Energy absorption analysis revealed a transition point between 330/s and 420/s, corresponding to a shift in dominant failure mode from single main fracture to multiple radial cracks. At −30 °C and low strain rates, SEM observations identified extensive intergranular cracking and pore expansion, confirming the strong influence of freezing-induced damage. These findings enhance the understanding of low-temperature tensile fracture mechanisms in granite and provide practical guidance for the design, safety evaluation, and maintenance of rock engineering structures in cold-region environments, including applications such as tunneling, slope stability, and Artificial Ground Freezing (AGF) projects.

Abstract Rock drilling is a crucial process in many industries, one example being the mining industry, where it is used for exploration and blasting. In a typical rock drilling process, the rock surface is fractured by dynamic mechanical interaction with a drill bit, resulting in rock fragments detaching from the surface. These cuttings are then transported through the borehole via water or air, and the rock fragment size is important for efficient borehole flushing. In this work, a heterogeneous bonded particle model was calibrated and applied to a laboratory scale rock drilling process. The mineral grain structure was obtained from an electron microscope scan of the rock surface, and the average grain size, volume percentage and stiffness of the three most common minerals were represented in the model. The dynamic mechanical behaviour of the rock material was obtained by conducting uniaxial compression and Brazilian disc tests in a split-Hopkinson pressure bar configuration. The results were used to calibrate the model. After the heterogeneous model was shown to be able to capture the macroscopic strengths and fracture modes of the split-Hopkinson experiments, it was used to simulate the laboratory scale rock drilling experiment, where two tool indentation depths were investigated. Here, the simulation was compared to experimental results in terms of vertical load acting on the tool, machine compliance as well as rock-cutting size distributions. The results from the simulation were in good agreement with the experimental observations.

In this paper, we present a 2D mesomechanical model for describing concrete fracture behavior under dynamic loading. The aggregate-mortar mesostructure of concrete is explicitly described, while the interfacial transition zone is represented as a weak zone of finite elements around the aggregates. Concrete failure is described by a damage-viscoplasticity model based on the Drucker–Prager yield criterion and the Rankine criterion as the tensile cut-off. For the viscoplastic part of the model, the consistency approach is adopted. In the damage model, separate scalar damage variables are applied for tensile and compressive stress regimes. Uniaxial compression and tension tests are simulated as the numerical examples. The model holds some promise because it reproduces the experimental failure modes in tension and compression, and in dynamic Brazilian disc test, and predicts a realistic compressive-to-tensile strength ratio as well as the strain-rate sensitivity effect for concrete.

Tensile cracking is a predominant mode of failure in rocks within underground resource excavation and engineering structures, where rocks are frequently subjected to dynamic disturbances while simultaneously experiencing in-situ stresses. This paper proposes a new dynamic split tension setup utilising a cubic specimen to investigate the dynamic behaviour of rocks across various tensile strain rates and confining pressures. The objective is to extend the applicability of the triaxial Hopkinson bar in studying dynamic behaviour of geomaterials. For comparison, the dynamic Brazilian disc (BD) tests were performed using three rock types (e.g., sandstone, granite and marble) under different strain rates ranging from 10−3∼102 s−1. Besides, the Digital Image Correlation (DIC) technique was adopted to measure full-field real-time tensile strain of rocks and demonstrated that tensile crack initiated at the middle part and split the specimen into two similar halves. Effects of specimen size, geometry, loading rate as well as the confining pressure are investigated in detail. The dynamic fracture behaviours, including dynamic tensile strength, tensile strain, time to fracture and dynamic increase factor (DIF), were characterised for the rocks. It is found that dynamic tensile strength of rock minimal dependence on size and geometry but is significantly influenced by loading rate and confinement. It exhibited a linear increase with strain rate (100∼102 s−1) and demonstrated a nonlinear growth with lateral confinement from 0 to 15 MPa. The nonlinear dependency on confinement can be attributed to the restriction imposed on the opening and propagation of tensile cracks due to the presence of confinement. These findings enhance our understanding of the safety aspects associated with underground rock excavations, particularly in situations where considering in-situ stress is crucial for evaluating the dynamic tensile failure of rocks.

Red sandstone is widely distributed in southern China. Due to the significant difference in mechanical properties before and after hydration and its poor water stability, red sandstone often triggers landslide accidents. In this paper, red sandstone from an open pit slope in Jiangxi Province was taken as the research object. Two variables, namely the initial saturation degree (25%, 50%, 75%, and 100%) and the number of wetting–drying cycles (0, 10, 20, 30, and 40), were set. With the help of nuclear magnetic resonance, the Brazilian disc test, and fractal theory, the relationships among its meso-structure, macroscopic fracture mechanics characteristics, and deterioration mechanism were analyzed. The research results are as follows: (1) Wetting–drying cycles have a significant impact on the pore structure and fracture mechanics characteristics of red sandstone. Moreover, the higher the initial saturation degree, the more obvious the deterioration effect of the wetting–drying cycles on the rock mass. (2) After further subdividing the pores according to their size for research, it was found that sandstone is mainly composed of mesopores, and the deterioration laws of different types of pores after the wetting–drying cycles are different. The porosities of total pores and macropores increase, while the proportions of mesopores and micropores decrease. The fractal dimensions of macropores and total pores of each group of rock samples are all within the range of 2–3, and the fractal dimension value increases with the increase in the number of wetting–drying cycles, showing significant and regular fractal characteristics. Micropores and some mesopores do not possess fractal characteristics. The fractal dimension of rock samples basically satisfies the rule that the larger the pore diameter, the larger the fractal dimension and the more complex the pore structure. (3) Both the type I and type II fracture toughness of rock samples decrease with the increase in the number of cycles, and the decrease is the most significant when the initial saturation degree is 100%. After 40 cycles, the decreases in type I and type II fracture toughness reach 23.578% and 30.642%, respectively. The fracture toughness is closely related to the pore structure. The porosity and fractal dimension of rock samples and their internal macropores are linearly negatively correlated with the type II fracture toughness. The development of the macropore structure is the key factor affecting its fracture mechanics performance. (4) After the wetting–drying cycles, the internal pores of red sandstone continue to develop. The number of pores increases, the pore diameter enlarges, and the proportion of macropores rises, resulting in internal damage to the rock mass. When bearing loads, the expansion and connection of internal cracks intensify, ultimately leading to the failure of the rock mass. The research results can provide important reference for the stability analysis of sandstone slope engineering.

Investigating the tensile properties of rock under varying microwave irradiation directions is essential for understanding the mechanical behavior of microwave-induced rock breaking. A continuous-discrete method is employed to analyze the tensile properties of granite disc samples under three different microwave irradiation directions. The results indicate the variable tensile properties of granite Brazilian disc under different microwave irradiation directions.

Understanding the deformation behaviour of rocks under dynamic loads is crucial in rock mechanics. This study investigates the rate-dependent tensile strength and fracture response of gypsum, a model material for soft rocks, across two different sample porosities. Quasi-static Brazilian disc tests were conducted using a servo-controlled universal testing machine, revealing that tensile strength decreases with increasing porosity. However, in the quasi-static regime, strength remains relatively constant across different strain rates. Dynamic tests were performed using a Split Hopkinson Pressure Bar (SHPB) setup, showing that dynamic tensile strength is consistently higher than quasi-static strength, regardless of porosity. Furthermore, as the dynamic strain rate increases, strength values further rise, indicating a strong rate dependency. Also, due to gypsum’s brittle nature and low tensile strength, fracture resulted in significant powder formation. To capture real-time fracture evolution, high-speed imaging was employed. Full-field strain distribution and fracture evolution has been studied using Digital Image Correlation (DIC). The findings demonstrate a strong correlation between porosity, tensile strength, and fracture behaviour, offering valuable insights into the dynamic response of soft rocks.