Mesh generation for complex three-dimensional non-planar crack surface using singular elements is a time-consuming task for stress intensity factor evaluation. This study demonstrates that a newly developed virtual grid stress recovery (VGSR) method provides an effective alternative to the well-established singular element approach. The VGSR method achieves accuracy comparable to that of the singular elements approach while offering the advantage of simplified mesh generation procedures. The effectiveness of the VGSR method is demonstrated through computational results for four benchmark crack geometries with different boundary and loading conditions.
The total fracture energy arises from the energy dissipated within the fracture process zone, comprising the work of necking and the work of fracture, and the energy dissipated in the surrounding plastic zone, referred to as plastic work. The essential work of fracture methodology expresses the fracture toughness in terms of these energy contributions. Although the influence of the geometric parameters, namely thickness and ligament length, on the cracking resistance is well established in the literature, their effect on the different energy contributions are less understood. This work investigates how the different energy contributions evolve with sheet thickness and ligament length, using existing theoretical framework. The ligament length effect was studied experimentally and numerically under the scope of the essential work of fracture, covering both conventional and small ligament regimes, whereas the effect of the thickness was investigated numerically. A new local quantity was introduced to quantify the energy dissipated within the fracture process zone, at a distance from the crack plane, allowing the estimation of necking height. The results show that the necking work per unit area increases with thickness up to a critical value, beyond which it decreases. While thinner specimens undergo more severe necking, thicker specimens dissipate greater necking energy before the critical thickness. Furthermore, the linear extrapolation of the essential work of fracture in the small-ligament regime yields a fracture energy whose value depends on the sheet thickness.
Fracture-resistant, light-weight Inconel-718 components are important for aerospace applications. This work studies the single-edge notch bending (SENB) fracture properties of lattice-structured Inconel 718 parts manufactured by laser powder bed fusion (LPBF) additive manufacturing. The lattice structures incorporated in the SENB test specimens include three new strut-based unit cells, named ULS-1, ULS-2, and ULS-3, which were designed to have positive Maxwell numbers and stretch-dominant behaviour. Their flexural strength and fracture strength parameters were estimated using the ASTM-E1820 standard for SENB in the three-point bending test configuration and compared with those of lattice-structured SENB specimens with two existing unit cells, namely, Kagome and Octet, and one triply periodic minimal surface (TPMS) lattice structure, namely, the Primitive. The ULS-1, ULS-2, and ULS-3 were found to have higher flexural modulus, specific yield load, fracture toughness, specific fracture toughness, fracture energy release rate, and specific fracture energy release rate compared to those for the Primitive, Kagome, and Octet by about 13
Pearlitic carbon steel, a primary material widely employed in railway engineering, frequently undergoes cyclic loading in service. Nevertheless, the intricate microstructure of pearlite complicates the understanding of its influence on fatigue crack initiation and propagation in such steels. To address this, the present study employs a modeling framework that integrates a fracture phase-field approach with a crystal plasticity model to examine the microstructure-sensitive fracture response of pearlite. Within this framework, the crystal plasticity model captures the deformation behavior of ferrite, whereas the deformation of cementite is represented by an isotropic plasticity formula. The fracture behavior of pearlite is analyzed by a fracture phase field model considering the influence of fatigue damage. Utilizing this coupled methodology, simulations are conducted to evaluate the effects of ferrite orientation, cementite lamellar angle, and interlamellar spacing on the fatigue fracture characteristics of pearlite. The findings indicate that ferrite orientation alters the distribution of cumulative plastic slip, thereby influencing both fatigue crack initiation and propagation. The lamellar angle markedly governs the accumulation rate of plastic slip, which in turn affects crack initiation. Variations in lamellar spacing are shown to significantly alter the fatigue crack propagation path in pearlite. This study advances the mechanistic understanding of fracture in pearlitic microstructures and provides insights for the design of pearlitic materials with enhanced fatigue resistance.
The Sandia Mechanics Challenge (SMC) provides the solid-mechanics community a forum for assessing its ability to predict mechanical behavior in structures and materials through a blind, round-robin format. Computationalists are asked to predict the behavior of an unfamiliar geometry given experimental calibration data; their predictions are compared to experimental measurements of the SMC scenario, and then the participants assess and compare their approaches, documenting their findings. The SMC broadens the scope of Sandia-hosted benchmarking problems, which previously focused on ductile failure through the Sandia Fracture Challenges, enabling an enduring, community-wide self-assessment of predictive capabilities for various mechanics topics. The topic of the SMC2023 is threaded fastener joints, specifically the deformation and failure of a structure with a threaded fastener joint in a drop scenario. These analyses required modeling of several phenomena, including multi-axial joint mechanics, material deformation and failure at intermediate strain rates, and contact mechanics. The experiments of the Challenge geometry revealed two competing failure mechanisms as the drop velocity increased: material failure of a flanged cantilever and fastener failure. While all teams were able to predict aspects of the fastener failure, certain modeling choices obscured the material failure of the flanged cantilever, in some cases completely. Incorporation of shear behavior of the fasteners, not just the tensile behavior typically considered, improved the predictions of fastener behavior. This SMC called attention to these diverse and challenging analyses, rallying the solid-mechanics community to identify and address potential shortcomings.
Brittle fracture modeling is framed as a coupled evolution of spatial and material configurations. The displacement field between these configurations drives the fracture process, and the crack path alters the geometry. This interplay has been expressed in the variational formalism of Francfort and Marigo. However, numerical models based on this formalism generally introduce a regularization to circumvent the need for geometry modification. Here, we propose to combine explicit sharp crack modeling with a variational formalism. Both mesh and displacement field are tuned to minimize the functional. Consequently, the crack path appears naturally. In this work, we first present the minimization of the fracture functional in a r-adaptive scheme and the advantage it presents. Afterwards, we present a staggered optimization procedure to construct the spatial and material configurations of a body undergoing brittle fracture. Afterwards, the method is demonstrated through several numerical examples. Namely, we recover theoretical behavior for a manufactured solution and present the crack path of single edge notch specimen. This method is a direct implementation of the variational formalism of fracture mechanics. Therefore, it provides a straightforward way to compute the energy release rate and crack path. Moreover, it does not require any regularization and the crack is directly represented in the geometry of the specimen. However, it is computationally expensive as the mesh needs to be optimized at each step of the loading.
The assumption of a straight crack path is generally made when studying crack initiation at V-notch tips under mixed-mode loading using the Coupled Criterion (CC). This work investigates whether the optimal initiation configuration could rather consist of curved cracks instead of straight ones. Curved crack initiation is studied through the Matched Asymptotic approach of the CC. It is found that slightly curved cracks may be optimal depending on the mode mixity and the material properties. The initiation loading corresponding to curved crack initiation remains close to the one obtained for straight cracks. The loading magnitude and crack initiation angle predicted using the CC well represent the ones obtained experimentally on V-notched Arcan specimens.
We present a compact finite-element framework for incipient mixed-mode crack kinking in plane strain, in which the physical crack is augmented by a short, zero-thickness cohesive continuation. A thin auxiliary channel is used only for mesh generation and is collapsed after T6 conversion, so that the analysis is carried out on the physical interface with coincident node pairs. The cohesive law is potential-based, differentiable, and characterized by finite normal and tangential cohesive strengths; the corresponding cohesive potential yields an energetically consistent formulation with a symmetric Hessian tangent and robust Newton convergence. The construction of the interface degrees of freedom, jump operator, line-integration operators, and consistent linearization is detailed explicitly. Two complementary selectors are considered for the kink direction: a global load-based criterion and a local energetic mouth criterion. Numerical studies show rapid convergence of the critical load, stable determination of the kink angle, and close agreement between bulk tractions and cohesive tractions reconstructed from the displacement jumps along almost the entire cohesive leg. The framework is examined deliberately in a near-LEFM regime. In this setting, the predicted kink direction remains close to the classical elastic benchmark, whereas the critical load exhibits a clearer sensitivity to the finite cohesive zone and to the traction–separation law. A calibrated sensitivity study of the auxiliary cohesive-leg length shows that, when the active cohesive zone is kept well contained within the trial segment, the directional prediction depends only weakly on this auxiliary choice. The method is therefore best suited to quasi-static mixed-mode crack kinking in a near-LEFM regime, where it provides a controlled finite-process-zone counterpart to classical elastic predictors.
Understanding the behaviour of masonry under various loading conditions is crucial for the structural design, assessment, and preservation of masonry structures. High-fidelity computational modelling offers a powerful method for investigating the complex mechanical interactions within masonry composites, complementing and reducing the need for extensive experimental programs. To this end, this research presents a numerical framework using the discrete element method (DEM), a discontinuum-based approach, to simulate the compressive behaviour of masonry prisms. The modelling approach represents masonry (i.e., brick units and cement mortar) as rigid, irregular triangular blocks, enabling explicit simulation of cracking and damage evolution. Building on previous developments, this study provides an in-depth examination of the advantages and limitations of the proposed framework. Particular attention is given to the influence of geometrical parameters such as block size and regularity and the stochastic variations associated with the block generation process. The findings indicate that incorporating block irregularity is crucial for achieving realistic compressive responses, while variations in block size have only a minor impact on the predicted capacity within the examined range. The study also demonstrates that the stochastic nature of the tessellation process leads to some variability in predicted strength, underscoring the value of analyzing multiple realizations to obtain representative trends. Overall, the framework provides reliable predictions of behaviour and fracture patterns, supporting its use in the assessment of masonry composites under compressive loading.
Dynamic fragmentation of brittle materials involves complex crack nucleation, propagation, and interaction under high-rate loading. Traditional cohesive zone models (CZM) accurately capture discrete cracks, but suffer from mesh dependency and constrained crack paths, whereas continuous damage models offer mesh independence at the expense of sharp-crack representation. This paper presents the Cohesive Lipschitz (CLIP) model, which unifies interface-based cohesive damage with a Lipschitz-projected bulk damage field to enforce spatial regularization. Analytical equivalence with a linear CZM yields closed-form degradation and dissipation functions, and a regularization parameter governs energy redistribution between cohesive interfaces and diffuse zones. Quasi-static validation confirms exact reproduction of CZM behavior under tensile loading. An explicit dynamic finite-element implementation simulates one-dimensional fragmentation of alumina across a wide range of strain rates. The CLIP model successfully predicts the transition from sparse to fine fragmentation, producing fragment statistics in agreement with established theoretical and numerical studies, thereby demonstrating robust fracture predictions with computational efficiency and a clear physical interpretation.
Glass matrix particulate composites, known for their high strength-to-weight ratio and superior optical and thermal stability, have emerged as promising materials for applications in missile protection systems, brake pads, and biomedical implants. However, their inherent brittleness and flaw sensitivity pose significant challenges for crack initiation and propagation. In this work, we investigate the fracture mechanisms of these composites using the phase-field model, in which the crack and material interfaces are approximated by two distinct phase-field variables: c and α . The material’s fracture toughness is expressed using the phase-field variable α , with distinct values assigned to the interface compared to the bulk. A series of numerical simulations is performed to investigate the roles of Poisson’s ratio and interfacial fracture toughness in the fracture of glass matrix particulate composite and the associated fracture mechanisms. We explore crack deflection and attraction by fiber, investigating these effects with different Young’s modulus and Poisson’s ratio mismatches and interfacial fracture toughness. The present analysis provides valuable insights into designing crack-propagation paths and improving the fracture toughness of the WG/a-SiO _2 and WG/BZBa composites by tuning the interface fracture toughness.
In tunnel blasting engineering, drilling and blasting method remains a widely adopted and efficient technique for excavating hard rock masses. Precise control of blast-induced dynamic responses is crucial for both project safety and construction efficiency. Smoothed particle hydrodynamics (SPH) and peridynamics (PD) are widely used to simulate fluid-structure interactions (FSI). This study proposes an SPH-PD FSI model to investigate gas-rock interactions under blast loading. An index-acceleration algorithm is proposed to optimize computational efficiency during the preprocessing stage. The proposed model offers advantages in algorithmic simplicity, computational efficiency, and adaptability to significant particle spacing differences. The model was validated through representative cases. The displacement trend line (DTL) analysis and the quantitative relative displacement method were applied to elucidate the blast-induced crack initiation and propagation mechanisms. Numerical results reveal the influence of prefabricated crack angles on cracking patterns. This study offers theoretical insights into the damage evolution of rocks with prefabricated cracks under blast loading, advancing understanding of crack propagation mechanisms.
Delayed Hydride Cracking (DHC) is a hydrogen embrittlement phenomenon that may affect Zircaloy-4 fuel claddings. An experimental procedure was previously developed to measure the fracture toughness of this material using notched C-ring specimens with a precrack, for both in presence of DHC ( K_I_DHC ) and without ( K_I_C ) (François et al. 2024). Based on these experiments, and on additional experimental results on notched C-ring specimens without a precrack, a finite element model was developed to numerically reproduce the DHC phenomenon. This model couples the mechanical behavior of the material with the presence of hydrogen in solid solution and hydrides, considering the kinetics of hydrogen diffusion, and the nucleation, growth and dissolution of hydrides (HNGD model). A cohesive zone model was used for crack propagation. The numerical model successfully reproduces the experimental results and is consistent with the experimental values of K_I_DHC , crack propagation rate and incubation time at 150, and 200 ^∘C for precracked specimens and 250 ^∘C for both precracked and notched specimens. In addition, this study highlights the great influence of the swelling induced by the presence of hydrogen in solid solution and precipitated hydrides on the fracture of the material in case of DHC, and the importance to take it into account to model this phenomenon.
We introduce an improved peridynamic (PD) model for dynamic fracture in fiber reinforced composites (FRCs). We verify it on elastodynamic and elastostatic problems and test it against the intersonic crack propagation experiment by Coker and Rosakis (2001). When a notched unidirectional FRC lamina is loaded by asymmetric impact, cracks can reach intersonic speeds of propagation. Using the PD model we explain the mechanism of crack initiation and propagation. We predict experimentally observed crack patterns, the crack propagation speed behavior, and the shockwaves generated by the propagating crack. When the PD horizon size is of a similar scale with the actual notch size used in the experiments, the match between computed and experimental results becomes quantitative. This is related to the crack nucleation process and is strength driven. The new PD composite model is calibrated with a homogenized classical model but maintains the sharp distinction between longitudinal and transverse bonds (jump discontinuity in their elastic stiffness), preserving this microstructure information (anisotropy) of the composite. We show that PD models for FRCs in which the micro-scale variation of PD bonds moduli mimics the continuous tension surface of homogenized composite cannot capture the observed failure behavior. Preservation of some essential features pertaining to failure initiation/behavior from the micro-scale, which the present PD model does, appears to be critical in predicting dynamic failure in FRCs using homogenized models.
Microcracks are prevalent defects in mechanical structures, and their propagation behavior under cyclic loading dominates structural integrity and service life. Conventional numerical methods for three-dimensional fatigue crack growth analysis suffer from high computational complexity and poor convergence, limiting their engineering applicability. To address this challenge, this study proposes a Kriging-based surrogate model for high-efficiency prediction of microcrack propagation behavior, validated by the consistency between experimental fatigue data and finite element analysis (FEA) results. A five-dimensional feature space is constructed to map the nonlinear relationship among fatigue parameters, stress ratio, residual stress, load amplitude, and crack growth rate, enabling direct mathematical characterization of crack evolution. Numerical verification shows that the model achieves a prediction error of less than 5
Hydrogel coatings integrate the high water content and softness of hydrogels with engineered surface functionalities, enabling diverse applications in soft robotics, stretchable electronics, and biomedical devices. However, their high water content and softness, thin-film geometry, and substrate confinement make them susceptible to delamination, cracking, fatigue, and wear, which compromise long-term reliability. This review summarizes the current understanding of fracture in hydrogel coatings. We first overview the experimental methods commonly used to characterize adhesion toughness, adhesion fatigue threshold, and adhesion strength. We then discuss fracture phenomena that are particularly relevant to coating geometries, including thickness-dependent adhesion, swelling- or stimulus-induced debonding, sliding-induced wear, and fatigue fracture under cyclic loading. Finally, we review emerging toughening and fatigue-resistant strategies for robust hydrogel coatings, including double-network designs, hierarchical architectures, nanocrystalline domains, and phase-separated structures. We conclude by outlining key challenges and opportunities for translating laboratory concepts into durable hydrogel coatings for practical applications.
Laser Repair (LR) via Directed Energy Deposition (DED) is a key additive manufacturing technology for high-value applications. However, LR introduces microstructural heterogeneity and interfaces that complicate fatigue assessment, especially in hydrogen-rich environments which exacerbate fatigue damage. This study employs the crack tip localization method and Crack Opening Rate (COR) parameter via BSL 3D DIC to investigate hydrogen-accelerated fatigue crack propagation in laser-repaired (LR) GH4169 superalloy. The fatigue lives and crack closure levels of pure substrate (PS), pure deposited (PD), and LR specimens were compared. Hydrogen charging significantly accelerated crack growth, reducing fatigue life by 23
This paper proposes a method for calculating microdamage in metastable austenitic steel based on changes in elastic moduli associated with the formation of micropores, microcracks, and an increase in the volume fraction of strain-induced martensite under fatigue. The crack shape factor for low- and high-cycle fatigue is calculated using a micromechanical model of the relationship between microdamage and Young’s modulus. Three stages of elastic moduli change, corresponding to the rate of strain-induced martensitic transformation, are identified. It is shown that the change in the rate of elastic moduli can be used to identify the transition from the low-cycle to the high-cycle region. It was found that the relationship between microdamage and cycle ratio can be described by power-law dependence similar to the expression in the nonlinear Marco–Starkey damage accumulation model, which makes it possible to predict fatigue failure.
This study presents an analysis of the influence of T-stresses on the stress field near the tip of an interfacial crack under plane strain conditions, specifically examining the initial stage of crack kinking from the interface. The analysis focuses on the formation of a small-scale process zone within the less crack-resistant elastic material of a bimaterial joint. The process zone is modeled as a discontinuity line of normal displacement, where the normal stress is assumed to be equal to the failure stress of the respective material. T-stresses are incorporated into the model by including their contribution to the asymptotic stress field near the crack tip. This asymptotic field is subsequently used to formulate the condition at infinity for the corresponding boundary value problem within the theory of elasticity. The parameters of the process zone (its length and angle of inclination) are calculated by solving the boundary value problem using the Wiener-Hopf method. From the derived solution, an equation is obtained for calculating the length and angle of inclination of the process zone. This calculation is based on the criterion of maximum potential energy accumulated within the zone. Furthermore, the energy release rate and crack opening displacement are also determined, providing essential metrics for formulating crack initiation conditions based on energy or deformation criteria. A numerical analysis was conducted to investigate the dependence of the zone’s parameters on the applied load and to specifically study the effect of T-stresses on its orientation. Finally, a comparative analysis of the model’s predictions for the kinking angles was performed against values reported in existing theoretical and experimental literature.
A steady-state, finite element analysis of brittle, dynamic crack propagation in PMMA is presented. The steady-state assumption means that all time derivatives are translated into a spatial derivative instead. The bulk PMMA is modelled as a linear elastic, isotropic material. The crack is modelled by use of a non-standard cohesive zone model that ensures that material stability is maintained. The cohesive zone contains two lengths, which allows for a regularisation of the crack problem. The boundary conditions were adjusted so that the results could be compared to experimental studies. The stress and strain fields at the crack tip resulting from the numerical analysis were shown, and the possible implications for damage evolution and crack branching were discussed. The study supports the idea that microcracks are initiated at some distance from the crack plane and then grow and join the main crack. The study suggests that the propagating crack goes from a 'simple crack' to an unstable crack when the peak in the maximum principal strain - which appears at some distance from the crack plane - exceeds the dynamic fracture strain of the material.
