To investigate the mechanical behavior and responses under loading, energy dissipation, and crack propagation in rocks with cross-fractures, this study prepared rock specimens with varying angles of cross-fractures and conducted uniaxial loading tests combined with acoustic emission monitoring to obtain mechanical parameters and acoustic emission characteristics. The experimental results demonstrate that cross-fractures significantly influence the stress–strain curves and mechanical properties of rocks, particularly at fracture angles of 30° and 60°, where pronounced stress drops and local damage phenomena are observed. Energy analysis demonstrates that the specimen with 90° cross fractures possesses the highest capacity for elastic strain energy accumulation, with a peak value of 58.43 KJ·m −3 , while the 30° fractured specimen exhibits the weakest energy storage capacity, reaching a peak of only 31.20 KJ·m −3 . For the 45° fractured specimen, elastic strain energy accounts for merely 70.49% of the total absorbed energy, indicating more sufficient plastic deformation; more work input from external loading is dissipated in the form of plastic damage. Prior to failure, crack propagation triggers a rapid release of elastic strain energy and a dramatic increase in dissipated energy. The acoustic emission ringing count and spatial location maps of events uncover the propagation process of internal cracks in the rocks, which become more pronounced as the degree of rock damage increases. These findings offer a theoretical foundation for the prediction of engineering disasters associated with fractured rocks in the fields of geology and petrology.
A numerical investigation is conducted to analyze the clamping force effect on the fracture strength of holed plates with symmetrical edge cracks, made from Al-Alloy7075-T6, subjected to longitudinal tensile load by quasi-brittle damage evolution, and the results were compared with the available experimental tests. The continuum damage mechanics method and finite element method, along with the Chaboche plasticity hardening model coupled with damage mechanics, were employed to capture accurately the material's mechanical behavior. A quasi-brittle fracture approach was investigated by employing a two-scale damage method with meso-scale volume including a micro-scale volume inside it, in which voids and microcracks develop and cause macroscopic fracture. The meso-scaled volume was considered to have elastic-plastic behavior; however, the micro-scale volume was considered to have elastic-plastic behavior coupled with damage. A finite element package was utilized to simulate bolt clamping and applied load to obtain stress and strain distributions on the holed plate, and the damage was calculated in a code written by the authors. The fracture strength of the bolted cracked plate obtained by the simulation and the code is in good agreement with the experimental results, showing two scale quasi-brittle fracture method coupled with damage can predict the fracture strength in cracked members.
Solder joints on printed circuit boards are constantly subjected to thermal loads. These joints come in various geometries, which can significantly affect their thermomechanical behavior. Previous studies focused on clarifying the thermomechanical behavior resulting from geometric changes by qualitatively analyzing the metallurgical microstructure, stress/strain distribution and strain energy within the solder joint. This research aims to quantify the relationship between joint geometry, and damage caused by thermal loading using a specific parameter. A history-dependent cohesive zone model was developed to calculate the dissipated energy ( DE th ) due to thermal cycling in solder joints with different geometries. The obtained DE th values increased from 75.3 to 147.6 J/m 2 for the Arcan-type specimen and from 90 to 191.2 J/m 2 for the DCB-type specimen as the thermal loading range increased from 25–105 °C to 25–165 °C. These results were subsequently correlated through the constraint parameter, Q . By plotting the DE th - Q diagrams for two specific geometries, a unique geometric locus is observed. Additionally, the variation in the calculated Q values with radial distance remained below 10% for all cases, confirming the numerical robustness of the Q evaluation procedure. The findings suggest that the Q parameter effectively quantifies the geometric effects on the thermomechanical behavior of solder joints. The developed DE th - Q methodology provides a unified and geometry-sensitive approach for thermal damage characterization, reliability assessment, and design optimization of solder joints in electronic packaging applications.
This paper addresses the multiscale moisture absorption and degradation of bamboo fiber–epoxy composites with different fiber fractions (20, 30, and 40 wt.%). The equilibrium moisture absorption of composites has been studied at different relative humidities (30%, 65%, and 80%) using a multiscale homogenization method. The findings reveal that the overall macroscopic property does not obey the classical law of mixtures, which is attributed to the hygro-mechanical coupling arising from the matrix-induced confinement effect. This response is strongly governed by both fiber fraction and humidity levels. The results indicate that the classical Fick model is sufficient to describe the experimental moisture uptake. Interestingly, the predicted hygroscopic stress distributions reveal increasing tensile and compressive stress levels with fiber content, driven by the combined effects of moisture concentration gradients and enhanced hygroscopic expansion. Herein, compressive stress reaches 46.46, 66.91, and 105 MPa in the surface regions during the transient regime for 20, 30, and 40 wt.% fiber composites, suggesting an increased risk of damage initiation even without adding external loads, respectively. More importantly, good retention of the tensile properties is noticed for 20 and 30 wt.% fiber composites after exposure to wet–dry aging, suggesting that fiber expansion and leaching are the dominant mechanisms. However, microstructural damage and active hydrolysis reactions drive the recovery at 40 wt.% fiber fractions. The damage parameter used to reflect the loss of tensile strength reaches values of 0.034, 0.054, and 0.169 at 20, 30, and 40 wt.% fiber fractions, indicating progressive damage with increasing fiber content, respectively.
This study presents a reformulated Gurson-type peridynamic model for the numerical simulation of ductile fracture in high-strength metallic materials. The approach is built within a non-ordinary state-based peridynamics framework and tightly couples the Gurson–Tvergaard–Needleman (GTN) model at the bond level. Specifically, the proposed model directly couples porosity-driven softening with bond degradation based on microscopic damage mechanisms. Then, void-controlled plasticity is embedded into peridynamics, and a bond-level two-way coupling between yield behavior and damage is achieved. Meanwhile, the fracture criterion based on the GTN void volume fraction is established as well, effectively connecting the evolution of microscopic damage with macroscopic fracture behavior, from cavity initiation and growth to crack initiation and propagation. The methodology is validated against several typical examples, illustrating its effectiveness and capacity for tensile, shear, compressive, and impact problems. Overall, the proposed model offers a numerically unified and stable framework for ductile fracture in high-strength metals across diverse stress states and dynamic loading scenarios.
Impact-induced damage in layered structures remains a long-standing challenge in solid mechanics, primarily due to the complex propagation of stress waves across multiple heterogeneous interfaces. Solid oxide fuel cell (SOFC) stacks, as representative layered systems, are particularly vulnerable to core crushing when subjected to severe impact loading in extreme environments. Such damage can result in output voltage degradation and compromise the operational stability of SOFC-supported systems. To efficiently evaluate impact resistance and output voltage stability, this study proposes a multiscale numerical framework for assessing the ultimate load-bearing capacity of SOFC stacks under impact loads. The clamped stack is first homogenized into an equivalent continuum, which significantly enhances computational efficiency. Subsequently, a constitutive model capturing elastic–plastic behavior and dynamic structural failure is developed to predict damage evolution under impact conditions. Validation against the original full-scale detailed model demonstrates that the proposed framework achieves high accuracy while substantially reducing computational cost. This efficiency arises from the layer-number independence of the approach: constitutive parameters calibrated using a reduced-layer stack can be directly applied to full-scale stacks without further modification. Overall, the proposed framework provides an effective analytical tool for evaluating the ultimate load-bearing capacity and functional reliability of SOFC stacks subjected to impact loading.
The coalescence behavior of flaws critically governs rock mass stability. Despite established coalescence classification systems, ambiguities in the mechanisms of morphologically similar crack trajectories and limitations in universality and accuracy persist. To address these issues, the present study employed the bonded particle model to generate a comprehensive range of flaw geometries with fine incremental steps and simulate the coalescence process between parallel flaws under uniaxial compression. By tracking the dynamic evolution of force chains and displacement fields, the physical origin of the long-unexplained compression-induced tensile (CIT) crack is identified. Findings reveal that CIT cracks are driven by the accumulation of nested tensile contact forces within dominant compressive forces; the confinement effect within the rock bridge leads to delayed but abrupt energy release, resulting in sudden internal failure. Furthermore, the present study deciphers the divergent mechanical mechanisms underlying various coalescence patterns in overlapping and stepping geometries, resolving controversies where identical crack morphologies arise from distinct stress states. Additionally, results demonstrate that coalescence patterns are governed not only by macroscopic flaw geometry, but also by stress heterogeneity induced by random particle distribution and dynamic stress redistribution during crack propagation. These insights facilitate a refinement of Wong’s classification system that prioritizes mechanical criteria over geometric morphology. Ultimately, this mechanism-driven framework eliminates previous observational ambiguities, establishing a robust foundation for predicting the failure behavior and stability of jointed rock masses.
A mathematical model of laser-induced phase explosion (LIPE) is established using the finite element method. The main physical processes, including liquid-phase superheating, homogeneous nucleation, and phase explosion (PE), are numerically simulated by coupling temperature and velocity fields. The aluminum target's ablation process, evolving from normal evaporation to superheating and eventually PE, is systematically investigated. A multiwavelength high-speed shadow imaging system captures the ambient air shock wave front (SWF) generated by short pulsed laser (SPL) irradiation to analyze its propagation velocity. Results indicate that the surface temperature rapidly decreases after briefly maintaining a peak of ∼6350 K (0.95Tc), dominated by strong convection of explosion products. With increased laser fluence, LIPE occurrence is advanced and its duration prolonged. Axial temperature distributions reveal PE occurs deeper inside the target, increasing the ablation penetration depth. Far from the surface, the air-borne SWF propagation velocity decays rapidly. This work provides theoretical support for investigating LIPE mechanisms and optimizing laser processing efficiency.
Fatigue damage induced by cyclic loading critically governs the long-term stability of underground rock engineering structures. While the mechanical behavior of rock under cyclic compression has been extensively investigated, the fundamental disparity between tensile and compressive fatigue mechanisms remains elusive, largely due to the lack of a unified experimental framework. In this study, a series of monotonic and constant-amplitude cyclic loading tests were conducted on granite under both uniaxial compression and direct tension. The results highlight fundamentally divergent damage trajectories, where cyclic compression is characterized by elastic shakedown and compaction hardening, with energy dissipation governed primarily by internal friction. Notably, postfatigue monotonic tests reveal a stress-dependent dual mechanism in compression: low-amplitude cycling enhances the residual strength of granite by up to 46.9% (from 97.9 to 143.8 MPa) via mechanically induced closure of preexisting microdefects, whereas high-amplitude cycling precipitates damage accumulation. Conversely, cyclic tension exhibits progressive stiffness degradation driven by direct bond breakage, with negligible hardening effects. To quantify these disparities, a novel damage variable was derived by integrating deformation kinematics and energy principles. Comparative validation confirms that, unlike stiffness, energy, or conventional strain-based indices, this newly derived framework robustly captures the three-stage damage evolution and provides a unified description for both loading modes. Ultimately, this study elucidates the mechanistic transition from friction-dominated compressive fatigue to cleavage-dominated tensile fracture, offering a rigorous quantitative basis for the stability assessment of rock masses under complex cyclic stress states.
This study proposes an innovative negative Poisson's ratio honeycomb metamaterial characterized by a tailored geometric topology, fabricated by laser powder bed fusion (LPBF). By integrating the topological benefits of re-entrant hexagonal honeycomb structures in conjunction with star-shaped lattices, a composite structure was developed that incorporates multi-scale synergistic deformation mechanisms. The LPBF-optimized 316L stainless steel specimens demonstrated high dimensional accuracy. Quasi-static compression and dynamic impact tests demonstrated that the synergistic interaction between plastic hinge propagation within the honeycomb units and branch buckling in the star-shaped units significantly enhances energy absorption efficiency. Notably, under quasi-static loading conditions, the proposed structure exhibits a substantial enhancement in specific energy absorption compared to conventional star-shaped lattices. Finite-element analysis (FEA), corroborated by the digital image correlation technique, provided a detailed characterization of the deformation modes throughout the linear elastic stage, plateau yield region, and densification process. Furthermore, it was observed that internal voids generated during the melting and cooling of the powder contributed to strain-rate softening behavior under dynamic impact loading. This research establishes a robust workflow that integrates “topology optimization–LPBF manufacturing–experimental validation–FEA simulation,” laying a theoretical groundwork for lightweight protective structures applicable in aerospace and biomedical implants. Future endeavors may extend this methodology to encompass functionally graded material design and applications involving multi-physics field coupling.
In this study, carbon-fiber-reinforced polyphenylene sulfide (CF/PPS) laminates were joined by ultrasonic welding using a PPS interlayer to form single-lap joints, and their temperature-dependent fatigue behavior was investigated within an entropy-based fracture framework. A fatigue damage model based on entropy generation was implemented in finite element analysis (FEA), and the responses of bulk PPS and welded joints were compared in relation to local stress concentration and viscoelastic energy dissipation. Static tensile and tensile fatigue tests conducted at multiple temperatures were used to evaluate the model and to examine the effect of temperature on fatigue life, damage localization, and fracture characteristics. The numerical predictions showed good agreement with the experimental results, supporting the applicability of the proposed approach. Although the welded joints exhibited lower fatigue lives than bulk PPS, the main distinction was not limited to the fatigue-life level itself but was also reflected in the different S–N slopes and fracture morphologies, indicating distinct fatigue-damage accumulation mechanisms. These results suggest that the fatigue behavior of welded joints cannot be interpreted solely from bulk material properties and that local entropy generation associated with joint geometry and temperature-dependent dissipation plays an important role in governing fatigue failure.
The loading paths experienced by components during multi-pass forming processes often exhibit significant non-proportionality. However, traditional damage models are predominantly developed and validated under proportional loading conditions, which introduces limitations in their application, as they fail to characterise damage evolution accurately under non-proportional paths. Consequently, developing damage models capable of effectively accounting for non-proportional loading history is paramount. This paper systematically reviews recent advances in damage research under non-proportional loading paths. It encompasses the design of experimental methodologies for non-proportional loading, quantitative characterisation methods for damage, microscopic damage mechanisms and methods for damage modelling. Finally, building upon a summary of existing achievements, future research directions in this field are outlined.
We address in this work unified analysis of phase-field models for cohesive fracture in order to alleviate the difficulty in selecting proper models and for further improvement. Aiming to regularize the Barenblatt's cohesive zone model, all the discussed models are distinguished by three characteristic functions, that is, the geometric function dictating the crack profile, the degradation function for the constitutive relation and the dissipation function defining the crack driving force. The latter two functions coincide in the associated formulation, while in the non-associated one they are designed to be different. Distinct from the counterpart for brittle fracture, in the phase-field model for cohesive fracture the regularization length parameter has to be properly incorporated into the dissipation and/or degradation functions such that the failure strength and traction–separation softening curve are both well-defined. Moreover, the resulting crack bandwidth needs to be non-decreasing during failure in order that imposition of the crack irreversibility condition does not affect the anticipated traction–separation law (TSL). With a truncated degradation function that is proportional to the length parameter, the Conti-Focardi-Iurlano model and the latter improved versions can deal with crack nucleation only in the vanishing limit and capture cohesive fracture only with a particular TSL. Owing to a length scale dependent degradation function of rational fraction, these deficiencies are largely overcome in the phase-field cohesive zone model (PF-CZM). Among many variants in the literature, only with the optimal geometric function, can the associated PF-CZM apply to general non-concave softening laws and the non-associated PF-CZM to (almost) any arbitrary one. Some mis-interpretations are clarified and representative benchmarks are presented.
Polymer-bonded explosive (PBX) is one of the typical heterogeneous composite materials and dynamic damage processes are crucial for understanding its behavior. The present study establishes a comprehensive peridynamic (PD) framework to model dynamic damage of PBX with sensitivity analysis of model parameters. The PD model successfully captured the various characteristics of PBX damage evolution: energetic crystal damage dominated under low-velocity impacts (20 m/s), while interfacial damage contributions increased substantially at higher velocities (40-60 m/s). Moreover, the importance of the total nine PD model parameters belonging to three different materials regarding total bond damage and damage modes (trans-granular/inter-granular) under 20-60 m/s impact loading is quantified based on the established three-dimensional PD model of steel-encased PBX. The Kriging surrogate model from PD model parameters to PBX damage is also constructed to decrease the computational cost. The sensitivity analysis revealed that binder failure strain exerted significant influence on total mechanical damage at different velocities. Notably, Young's modulus of energetic crystal demonstrated strong cross-mode effects at elevated velocities, governing both trans-granular crack initiation and inter-granular debonding, thereby emerging as a critical constitutive parameter for relatively high impact velocity. The current study offers a useful computational framework to evaluate how the uncertainty of PBX properties impacts its dynamic damage response.
Peridynamics (PD) is a relatively new nonlocal method for modeling continua that inherently captures damage evolution, making it an attractive tool for modern engineering applications. Although significant research has focused on isotropic materials since its inception, this review article aims to synthesize studies on modeling material anisotropy within PD. To identify common patterns and approaches to the challenges encountered, key publications from the method's derivation onward have been thoroughly analyzed and categorized. The reviewed papers demonstrate an evolution from simple formulations to more comprehensive models capable of capturing a wide variety of anisotropic phenomena. Fiber composites and general anisotropic models have received the most attention, whereas applications in biomechanics, fluid dynamics, and multiphysics problems remain less explored. This review not only highlights the progress made in modeling anisotropic materials using PD but also identifies gaps in the current literature. This extensive categorization provides a roadmap for addressing the limitations of current models and advancing the practical implementation of PD in various engineering disciplines.
This study reports the synthesis of multilayer Ti3C2Tx MXene through selective etching of Ti3AlC2 and its incorporation into Elium (R) thermoplastic resin at loadings of 0.25-1.0 wt.%. The resulting nanocomposites were characterized by scanning electron microscopy (SEM), X-ray diffraction, thermogravimetric analysis/differential scanning calorimetry, and mechanical tests, including tensile (ASTM D638), flexural (ASTM D790), and fracture toughness (ASTM D5045) analyses. The 0.75 wt.% MXene composite exhibited the most balanced performance, achieving a tensile strength of 69.45 MPa, a flexural strength of 110 MPa, and a 31% improvement in fracture toughness over neat Elium (R). These enhancements were attributed to uniform nanofiller dispersion and effective interfacial interaction that facilitates stress transfer and crack deflection. SEM fractography revealed a transition from smooth, brittle fracture surfaces in the neat matrix to rough, branched crack paths in the MXene-filled composites, indicating improved energy dissipation. Overall, MXene/Elium (R) nanocomposites showed superior strength, toughness, and thermal stability, highlighting their potential as lightweight and recyclable materials for structural applications in automotive, aerospace, and protective sectors.
Short fatigue crack growth is strongly affected by microstructural heterogeneity, crack-size effects, and loading-path dependence, leading to nonlinear, intermittent, and locally sensitive propagation behavior. Conventional long-crack growth models based on linear elastic fracture mechanics are therefore insufficient for describing short-crack propagation under different loading conditions. To address this issue, this study proposes a multi-scale short fatigue crack growth rate model that couples crack-tip strain energy density, crack-size effect, microstructural barrier resistance, and crack closure correction. Compared with the existing short-crack models, the proposed model introduces a microstructural barrier penetration function and a crack-opening correction factor to describe crack arrest, re-initiation, growth-rate fluctuation, and the effective crack-driving force. The model is validated using literature-reported experimental datasets for LZ50 steel, EA4T steel, CuNi 2 Si alloy, and 42CrMo steel under different load sequences, loading frequencies, loading paths, loading modes, and load levels. The results show that the proposed model can reasonably capture the nonlinear evolution of short-crack growth rates under multi-condition loading. Quantitative evaluation shows an average log-scale coefficient of determination of 0.894, with 96.1% of the evaluated data points falling within the factor-of-three error band. These results indicate that the proposed model provides a physically interpretable framework for short fatigue crack growth assessment. Further validation under random or spectrum loading is still required.
Rock masses are inhomogeneous, fractured, anisotropic, and initially stressed in their natural state. They consist of intact rock or rock monolith and macro-damages, which include fractures, bedding planes, faults and other forms of discontinuity. It has been experimentally confirmed that with the increase of the rock mass scale, the mechanical properties of the rock mass decrease as a consequence of discontinuities. The subject of this paper is the analysis and determination of the scale effect on the damage plasticity model parameters, which describe the mechanical behaviour of the rock mass for different loads and scales. The scale effect is analysed on a constitutive model that can simulate the most complex mechanical behaviours in rock masses (elasto-plastic with damage in stiffness and strength). The analysis is based on the results of experimental tests of rock mass, which were carried out for different scales (scale of the rock mass sample in laboratory and scale of the in situ test - shear test), while the tests were performed at the same microlocation. For the considered experimental tests, appropriate finite element models (FEMs) were formed with boundary conditions that correspond to the ones from the experiments. By simulating the performed experiments with a series of FEM analyses with complex optimization algorithms, the parameters of damage plasticity model were determined for both values of the rock mass scale, proving the scale effect numerically. Based on the obtained results, analysis and discussion of the scale effect for a given rock mass were performed.
To represent the rate-dependent mechanical behavior of concrete, a stochastic damage model is proposed that integrates Langevin dynamics within the Micro-Meso Stochastic Fracture framework. In this model, the fracturing process of each micro-spring is described as a barrier-crossing event where the effective barrier height depends on the external loading rate. Evolution of the reaction coordinate follows the Langevin dynamics. Solving the corresponding Fokker-Planck-Kolmogorov equation yields the failure probability of each micro-spring under dynamic loading, from which the dynamic fracture strain is derived. Numerical results demonstrate that the model naturally captures the characteristic two-stage strength enhancement of concrete: a gradual increase of dynamic increase factor at low strain rates, followed by a rapid rise beyond a critical threshold. By linking the static fracture strain of each micro-spring to its unique initial energy barrier, the model assigns distinct rate sensitivities to different material constituents. This approach enables the model to reflect different rate sensitivities in tension and compression without separate empirical adjustments. Finally, simulations of reinforced concrete beams under various loading velocities, including high-rate impact, validate the model's capability to predict structural dynamic responses in practical engineering scenarios.
Hybrid composite materials (HCMs) containing knitted fabric as reinforcement often improve the mechanical properties and retard failure process in the loaded structure. In the present investigation, using numerical methods, was modeled damage accumulation process in a HCM, prepared by uniformly impregnating a knitted fabric of basalt microfiber yarn with epoxy resin having dispersed fine oil shale ash (OSA) particles. The HCM plate was investigated experimentally and performing numerical modelling of damage accumulation in it. The elastic modulus of the polymer matrices at different OSA concentrations was experimentally measured. This data was then used in a finite element model to calculate stresses in the reinforced textile inside the loaded plate. Impregnated yarns in the textile were modeled as macrofibers (MFs) with averaged properties, and their geometry was described using the Leaf-Glaskin approach. Next, the damage accumulation process was simulated under an external unidirectional tensile load. Depending on the applied load, different parts of the MF loops experience overloading. At some point, one of the most overloaded MF cross-sections breaks, as described using the probabilistic Weibull function. The Weibull distribution parameters were accepted using data for single fiber rupture. This leads to increased overload on the cross-sections of adjacent MF loops and eventually their breakage as well. A stochastic numerical model of sequential MF rupture accumulation was developed to describe the probabilities of defects that consist of various numbers of adjacent broken MFs. Analytical and stochastic methods were used to determine the load-carrying capacity and the accumulation of defects of different sizes in the HCM plate.
