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This study investigated the relationship between fracture toughness (Kc) and energy release rate (Gc) through fracture morphology analysis, emphasizing the critical role of fractal dimensions in accurately characterizing fracture surfaces. Traditional linear elastic fracture mechanics (LEFM) models relate Gc to Kc by combining energy principles with the nominal area of the fracture surface. However, real materials often exhibit plasticity, and their fracture surfaces are not regular planes. To address these issues, this research applied fractal theory and introduced the concept of ubiquitiform surface area to refine the calculation of fracture surfaces, leading to more accurate estimates of Gc and Kc. The method was validated through standard compact tensile specimen tests on a nickel-based superalloy at 550 °C. Additionally, the analysis of fractal dimension differences and dispersion in various fracture regions provides a novel perspective for evaluating the fracture toughness of materials.

Coal burst hazards seriously restrict the safe mining of deep coal resources. Establishing reasonable indexes from the perspective of inherent material properties is crucial for assessing the risk of coal burst. Coal burst is an issue concerning dynamic fracture, and the generation of kinetic energy is attributed to the mismatch between the energy release rate and the energy demand of dynamic fracture. In view of the above fact, a new realistic energy release rate (RERR) index for assessing the risk of coal burst was proposed in this study. This index considers the contribution of post-peak energy to coal burst and the time effect of failure, and reflects the intensity of energy release during post-peak dynamic fracture. Subsequently, uniaxial compression tests were performed on anisotropic coal specimens with different bedding angles, during which acoustic emission (AE) signals and fracture process images were acquired. Based on the test results, the influence of bedding angle on the mechanical properties and burst behaviors of coal was comprehensively explored. Meanwhile, the reasonableness and effectiveness of the RERR index were verified through a comparison with the actual failure situation (sound, fragment ejection, and AE response) and the results obtained on the basis of the conventional indexes (uniaxial compressive strength USC, elastic strain energy index W ET , bursting energy index K E , and duration of dynamic fracture D T ). The comparison reveals that the RERR index is highly linearly correlated with the conventional indexes. Furthermore, the RERR-based classification standard of coal burst proneness grade was given. In addition, the dynamic realistic energy release rate (DRERR) index can effectively characterize the stress drop events during the whole loading (seismogenic) process. The DRERR index exhibits an exponential correlation with the cumulative AE energy during stress drop. Highlights A new realistic energy release rate (RERR) index is proposed as a criterion for assessing coal burst proneness, and a classification standard of proneness grade is given. The effect of bedding angle on coal burst proneness and mechanical properties is comprehensively explored , and the effectiveness of the RERR index is verified by comparison. The dynamic realistic energy release rate (DRERR) index can characterize the stress drop events during the whole loading (seismogenic) process.

Diabetes is associated with increased fracture risk in human bone, especially in the elderly population. In the present study, we investigate how simulated advanced glycation end-products (AGEs) and materials heterogeneity affect crack growth trajectory in human cortical bone. We used a phase field fracture framework on 2D models of cortical microstructure created from human tibias to analyze crack propagation. The increased AGEs level results in a higher rate of crack formation. The simulations also indicate that the mismatch between the fracture properties (e.g., critical energy release rate) of osteons and interstitial tissue can alter the post-yielding behavior. The results show that if the critical energy release rate of cement lines is lower than that of osteons and the surrounding interstitial matrix, cracks can be arrested by cement lines. Additionally, activation of toughening mechanisms such as crack merging and branching depends on bone microstructural morphology (i.e., osteons geometrical parameters, canals, and lacunae porosities). In conclusion, the present findings suggest that materials heterogeneity of microstructural features and the crack-microstructure interactions can play important roles in bone fragility.

We have advanced phase-field simulation of hydraulic fracturing with consideration of thermoporoelasticity and discretization based on the mixed finite element in temperature, pressure, and the phase field. The key application is intended for hydraulic fracturing by water and by CO2 in hot dry rock. In geothermal fracturing, the injection fluid may have much lower temperature than the hot-volcanic rock and consideration of thermoporoelasticity may have a significant effect. We provide numerical simulations and comparison with laboratory data and examine the effect of thermoporoelasticity on breakdown pressure and fracture intensity. The thermal effect is more pronounced under unconfined conditions, especially for CO2 fracturing. The change of granite rock strength in the Brazilian tests at different temperatures without specific fluid confinement may not apply to high stress boundary conditions. Based on simulation of hydraulic fracturing experiments using water in heated and unheated granite, we conclude that the critical energy release rate Gc which is a key parameter of the phase field is not affected by temperature in the range of 20–300 °C. In that respect, there is similarity on the independency of Young’s modulus from temperature. The critical stress is, however, known to be a function of temperature. An important observation relates to simulation of fracturing by water and CO2 in a domain larger than laboratory scale. CO2 fills the created fractures quickly. Filling of created fractures by water takes time, and as a result fractures propagate in many stages. We observe from simulations that fracture intensity from CO2 is higher than by water in line with laboratory measurements. Higher fracture intensity and fracture surface area is an important consideration in renewable energy production from geothermal formations due to low thermal conductivity in volcanic rocks.HighlightsThe phase-field model predicts a single long fracture by water in granite at low temperature, and vast fracture network at geothermal conditions.The phase-field predicts, in line with experiments, lower breakdown pressure and higher fracture density by CO2 than by water.Phase-field simulation of geothermal formulations is advanced to capture extensive branching observed in laboratory scale.The critical energy release rate Gc of granite-water may not be affected by temperature in the range of 20–300 °C.In large scale, fracturing by water may go through a cycle of stop and go, while continuous CO2 fracture propagation is more likely.

Soft materials are an important class of materials. They play critical roles both in nature, in the form of soft tissues, and in industrial applications. Quantifying their mechanical properties is an important part of understanding and predicting their behavior, and thus optimizing their use. However, there are often no agreed upon standards for how to do so. This also holds true for quantifying their fracture toughness; that is, their resistance to crack propagation. The goal of our work is to fill this knowledge gap using blood clot as a model material. In total, we compared three general approaches, some with multiple different implementations. The first approach is based on Griffith’s definition of the critical energy release rate. The second approach makes use of the J-Integral. The last approach uses cohesive zones. We applied these approaches to 12 pure shear experiments with notched samples (some approaches were supplemented with unnotched samples). Finally, we compared these approaches by their intra- and inter-approach variability, the complexity of their implementation, and their computational cost. Overall, we found that the simplest method was also the most consistent and the least costly one: the Griffith-based approach, as proposed by Rivlin and Thomas in 1953.

The construction of the relation between the critical energy release rate, GIc, and the mode I fracture toughness, KIc, is of great significance for understanding the fracture mechanism and facilitating its application in engineering. In this study, fracture experiments using NSCB and CCCD specimens were conducted. The effects of specimen sizes, loading rate and lithology on the relation between GIc and KIc were studied. GIc was calculated by integrating the load–displacement curve according to Irwin’s approach. Based on the measured KIc and GIc of the rock specimens, a relation between GIc and KIc was found to be different from the classical formula under linear elasticity. It was found that both specimen size and loading rate do not influence this relation.

Delayed rockburst experiments with different numbers of unloading surfaces (DNUS) were performed using an independently developed true triaxial multisurface unloading rockburst experimental system. Based on the rockburst excess energy theory, the energy storage characteristics, excess energy, excess energy release rate (EERR), and crack evolution characteristics of rockbursts with DNUS were studied, and the following main conclusions were obtained. The occurrence of rockbursts is mainly due to the generation of an excess energy Δ E . Δ E depends on the elastic strain energy stored in the rock before the rockburst, the energy input by the equipment after the peak, and the residual elastic strain energy. As the DNUS increases, Δ E gradually decreases, but the EERR value increases, and the rockburst becomes increasingly severe; Rapid unloading of the specimen under true triaxial high-pressure loading will produce an unloading platform in the stress–strain curve, causing unloading damage. The damage is mainly concentrated near the free surface in the form of tension failure, and the unloading damage gradually increases with increasing DNUS; Tensile cracks play a dominant role in the damage and destruction of sandstone. In the final rockburst stage, the slope of the shear crack curve was greater than that of the tensile cracks, indicating that shear cracks were a critical factor affecting the instability and failure of the specimen.

In engineering practice, layered rock masses often display obvious anisotropy while deforming and failing, and the failure mode directly impacts the engineering construction stability. In this study, the fracture failure load, fracture toughness, crack deflection angle, and failure mode of a layered rock mass under different fracture modes were analyzed by utilizing improved asymmetric semi-circular disc specimens. According to the constitutive model of transversely isotropic materials, the maximum tensile stress (MTS), maximum energy release rate (MERR), and maximum strain energy density (MSED) calculation formulas were modified, and the calculation formulas of the three prediction criteria under anisotropic materials were derived. The calculation results were compared with the experimental results. The results show that the fracture toughness and crack deflection angle were significantly affected by the weak bedding plane. As a result of applying the MTS criterion, the results are closer to the experimental results, providing a solid foundation for engineering deformation, failure, and fracture analyses.

In recent years, manufacturers have increasingly embraced sustainable practices by employing eco-friendly, recycled, and recyclable materials along with greener production processes. Adhesive bonding has become a preferred method for joining composites, as it forms strong joints without disrupting structural continuity, unlike fasteners such as rivets and screws that cause fibre-matrix discontinuities and stress concentrations. Aerospace and automotive industries have also begun integrating natural fibres due to their low weight, renewability, and reduced environmental impact. However, studies involving failure modes in bonded joints of these structures are still minimal. This study presents crack propagation Resistance curves (R-curves) under Mode I loading using Double Cantilever Beam (DCB) tests according to ASTM D5528 and D3433, manufactured with castor oil polyurethane matrix as adhesive. Three configurations were evaluated: Metal Substrate Bonded Joints (MSBJ) with secondary bonding; jute fibre Composite Substrate Bonded Joints (CSBJ) also with secondary bonding; and Delamination Failure (DF) specimens, where the interface was generated during curing in a co-curing process. The Compliance-Based Beam Method (CBBM) for was compared with J-integral evaluation, as specimens may display inelastic response. Results revealed that the Fracture Process Zone (FPZ) was smaller for metallic substrates than for natural fibre composites. Fractographic analysis by Scanning Electron Microscopy (SEM) showed cohesive failure in all specimens. Despite sharing the same interface constituent, MSBJ and DF exhibited higher average critical SERR values ( and kJ/m ) compared to CSBJ ( ) using CBBM. This work contributes to the development of bonded joints through a fully renewable composite system with both natural fibres and matrix under Mode I interlaminar fracture. Furthermore, comparison of data reduction schemes for R-curve values reinforces the methodology and provides experimental data applicable to numerical models for sustainable joints.

Double-cantilever beams (DCBs) are widely used to study mode-I fracture behavior and to measure mode-I fracture toughness under quasi-static loads. Recently, the authors have developed analytical solutions for DCBs under dynamic loads with consideration of structural vibration and wave propagation. There are two methods of beam-theory-based data reduction to determine the energy release rate: (i) using an effective built-in boundary condition at the crack tip, and (ii) employing an elastic foundation to model the uncracked interface of the DCB. In this letter, analytical corrections for a crack-tip rotation of DCBs under quasi-static and dynamic loads are presented, afforded by combining both these data-reduction methods and the authors’ recent analytical solutions for each. Convenient and easy-to-use analytical corrections for DCB tests are obtained, which avoid the complexity and difficulty of the elastic foundation approach, and the need for multiple experimental measurements of DCB compliance and crack length. The corrections are, to the best of the authors’ knowledge, completely new. Verification cases based on numerical simulation are presented to demonstrate the utility of the corrections.