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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.

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.

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.

The current study aims to simulate fatigue microdamage accumulation in glycated cortical bone with increased advanced glycation end-products (AGEs) using a phase field fatigue framework. We link the material degradation in the fracture toughness of cortical bone to the high levels of AGEs in this tissue. We simulate fatigue fracture in 2D models of cortical bone microstructure extracted from human tibias. The results present that the mismatch between the critical energy release rate of microstructural features (e.g., osteons and interstitial tissue) can alter crack initiation and propagation patterns. Moreover, the high AGEs content through the increased mismatch ratio can cause the activation or deactivation of bone toughening mechanisms under cyclic loading. The fatigue fracture simulations also show that the lifetime of diabetic cortical bone samples can be dependent on the geometry of microstructural features and the mismatch ratio between the features. Additionally, the results indicate that the trapped cracks in cement lines in the diabetic cortical microstructure can prevent further crack growth under cyclic loading. The present findings show that alterations in the materials heterogeneity of microstructural features can change the fatigue fracture response, lifetime, and fragility of cortical bone with high AGEs contents.Cortical bone models are created from microscopy images taken from the cortical cross-section of human tibias. Increased glycation contents in the cortical bone sample can change the crack growth trajectories.

In order to investigate the fatigue performance of orthotropic anisotropic steel bridge decks, this study realizes the simulation of the welding process through elastic-plastic finite element theory, thermal-structural sequential coupling, and the birth-death element method. The simulated welding residual stresses are introduced into the multiscale finite element model of the bridge as the initial stress. Furthermore, the study explores the impact of residual stress on crack propagation in the fatigue-vulnerable components of the corroded steel box girder. The results indicate that fatigue cracks at the weld toe of the top deck, the weld root of the top deck, and the opening of the transverse diaphragm will not propagate under the action of a standard vehicle load. However, the inclusion of residual stress leads to the propagation of these cracks. When considering residual stress, the fatigue crack propagation paths at the weld toe of the transverse diaphragm and the U-rib weld toe align with those observed in actual bridges. In the absence of residual stress, the cracks at the toe of the transverse diaphragm with a 15% mass loss rate are categorized as type I cracks. Conversely, when residual stress is considered, these cracks become I-II composite cracks. Residual stress significantly alters the cumulative energy release rate of the three fracture modes. Therefore, incorporating the influence of residual stress is essential when assessing the fatigue performance of corroded steel box girders in long-span bridges.