mode-I Stress Intensity Factors Research Articles

Analysis of an interfacial crack between two nonhomogeneous piezoelectric materials using a new domain-independent interaction integral

Considering the difficulties introduced by piezoelectric material non-homogeneity and discontinuities to interfacial crack analysis, a domain-independent interaction integral (DII-integral) is proposed to derive the stress intensity factors (SIFs) and electric displacement intensity factor (EDIF) of an interfacial crack between nonhomogeneous piezoelectrics. For continuously varying material properties, traditional interaction integral (I-integral) needs to compute the derivatives of material parameters with respect to the x1 direction of the local coordinate system, while the proposed DII-integral is demonstrated to be unrelated to any derivatives of material parameters. Furthermore, a significant improvement is achieved that the new formulation of I-integral remains valid even when the integration domains include intricate and multiple interfaces, as long as the material interfaces are perfectly bonded. Firstly, the intensity factors (IFs) calculated by the combination of the DII-integral and the extended finite element method are compared with the analytical solutions, and it is found that the relative errors of the IFs are all less than 1.79 % with various crack lengths, which verifies the accuracy of the proposed method. For the piezoelectric bi-material model with continuously nonhomogeneous properties, the calculated IFs match each other well (relative deviation < 1.3 %) for different sizes of integration domains. In addition, for the perfectly bonded interface with discontinuous electromechanical properties, the IFs are almost identical (relative deviation < 0.85 %) whether or not the integration domain contains the material interface, which further numerically verifies the good domain-independence of the established DII-integral. For an interfacial crack near extra interfaces, the differences in elastic stiffness and piezoelectric coefficient between interface sides have an obvious influence on mode-I SIF, but mode-II SIF and EDIF are affected mainly by piezoelectric coefficient and dielectric permittivity.

Proactive strengthening of U-rib butt-welded joints in steel bridge decks using adhesively bonded Fe-SMA strips

U-rib butt-welded joints are one of typical fatigue-prone details in early aging orthotropic steel bridge decks, which poses a potential threat to the structural behavior and safe operation of steel bridges. Therefore, it is of great importance to seek out an effective way for retrofitting fatigue cracks. This study proposes an efficient prestressing technique using bonded iron-based shape memory alloy (Fe-SMA) to strengthen cracked U-rib butt-welded joints proactively. A series of static and fatigue tests were conducted on the cracked specimens to investigate the feasibility and fatigue improvement of the proposed technique. Results from the experiments demonstrated that applying prestressing force by heating Fe-SMA strips is a convenient and reliable technique. Activated Fe-SMA strips could introduce significant compressive stresses and provide additional stiffness for the cracked components. This led to a reduction of the mean stress and fatigue stress ranges at the U-rib butt-welded joints under cyclic loading. All the cracked specimens strengthened with adhesively bonded Fe-SMA strips presented excellent fatigue performances, with a maximum equivalent fatigue life extension of 4.09 times compared to the specimen before cracking. Numerical simulation was also performed to better understand the test results. It was demonstrated that prestressed Fe-SMA strengthening could decrease the mode-I stress intensity factor (SIF) at the crack tip, indicating the beneficial effects of the prestressing in retarding crack and its further propagation. In conclusion, the use of adhesively bonded Fe-SMA strips for strengthening cracked butt welds is highly effective in improving the fatigue performance, even better than bonding CFRP.

Plane and axisymmetric problems of an interfacial crack in a power-law graded material

Plane and axisymmetric problems of interfacial cracks in a power-law graded infinite medium are examined. It is shown that the models are governed by integral equations of Mellin’s convolution type whose kernels are expressed through the hypergeometric function. Exact solutions are derived by the method of orthogonal Jacobi polynomials in a series form and by the Wiener–Hopf method by quadratures. Mode-I stress intensity factors and the associated weight functions for power-law graded materials in the plane and axisymmetric cases are introduced and evaluated. It is shown that, although the displacement jump and the normal traction component have power singularities at the crack tip different from 1/2, the strain energy variation is proportional to the crack length change as in the case of homogeneous materials. A Griffith-type criterion of crack propagation in power-law graded materials is proposed and results of numerical tests are reported.

Fracture toughness of freestanding plasma sprayed yttria stabilized zirconia coatings via in situ tensile experiments

Tensile response and fracture toughness of the freestanding atmospheric plasma spray 8YSZ coatings was investigated using in situ high resolution optical full-field strain measurement. The elastic properties obtained directly from the tensile experiments were employed in computing the critical mode-I stress intensity factor. Comparison between the fracture toughness values computed from specimen geometry and J-integral via displacement field around the crack tip showed small differences on single edge notched tensile (SENT) specimens to extract their intrinsic mode-I fracture toughness. Two kinds of coatings having different porosity were studied to obtain stress-strain response and compute effect of porosity on mode-I fracture toughness. The samples were fabricated by depositing the coatings on, un-notched and pre-notched substrate templates, and then carefully separating the coatings by acidic dissolution of the substrate. High resolution in situ deformation of the coatings was captured for both tensile and SENT specimens using digital image correlation method combined with an in-house designed microtensile stage. The deformation from the specimen surface was used to obtain elastic material properties under uniaxial tension, which are then used to compute the energy release rate by the path independent J-integral. The critical stress intensity factors calculated from SENT geometry for the low and high porosity coatings were found to be 1.23 ± 0.21 MPa-m0.5 and 0.67 ± 0.12 MPa-m0.5, respectively. In addition, considering the experimental observation of nucleation and propagation of a crack from the notch tip before catastrophic failure, the J-integral based calculation of the critical fracture toughness values were 1.37 ± 0.19 MPa-m0.5 and 0.5 ± 0.1 MPa-m0.5, for the low and high porosity coatings, respectively. As the crack always nucleates from the notch before complete fracture, it can be concluded that the J-integral based method is accurate and suitable for finding the fracture toughness in porous coatings.

Effect of loading rate on the dislocation emission from crack-tip under hydrogen environment

The interplay between H atoms and crack-tip plasticity is essential to understand hydrogen embrittlement. By formulating a novel closed-form energy-based analytical model, the effect of hydrogen on the emission of dislocations in the vicinity of a crack-tip is investigated. Using fcc Ni as model material, the analytical model predicts that the critical radius of the stably developed dislocation loop decreases with applied Mode-I stress intensity factor (SIF) KI. Combined with the transition state theory, it is found that the most probable Mode-I SIF of Ni-H system increases with the H concentration under lower loading rates, but is not sensitive to the H content under higher loading rates. Further atomistic simulations show that, 1) the critical SIF required for dislocation emission does not vary significantly with the H concentration; 2) the dislocation loop is emitted from the crack-tip along the (111) slip plane at ∼ KIe=0.58∼0.64MPam, which is consistent with the present theoretical prediction ∼ 0.63 MPam. These results thus indicate that the crack-tip plasticity is suppressed by the H insertion (i.e. the Song-Curtin nanohydrides formation mechanism) under lower loading rates, but might not be influenced under higher loading rates.

Revisiting classical concepts of Linear Elastic Fracture Mechanics - Part I: The closing ‘mathematical’ crack in an infinite plate and the respective Stress Intensity Factors

This is the first part of a short three-paper series, aiming to revisit some classical concepts of Linear Elastic Fracture Mechanics. The motive of this first paper is to highlight some controversial issues, related to the un­natu­ral overlapping of the lips of a ‘mathematical’ crack in an in­fin­­­ite plate load­ed by specific combinations of principal stresses at in­finity (predicted by the clas­si­c­al solu­tion of the respective first fundamental problem), and the closely as­so­ciated issue of negative mode-I Stress Intensity Factor. The problem is con­­­front­ed by superimposing to the first funda­mental problem of Lin­ear Elastic Frac­ture Mechanics for an in­fin­ite cracked plate (with stress-free crack lips) an ‘in­­verse’ mixed fund­amental problem. This superposition provides naturally ac­­­­­­­­ceptable stress and displacement fields, prohibiting overlapping of the lips (by means of contact stresses generated along the crack lips, which force the over­lapped lips back to naturally accepted position) and, also, non-negative mode-I Stress Intensity Factors. The solu­tions of this first paper form the basis for the next two papers of the series, dealing with the respective prob­lems in fi­­n­ite do­­mains (recall, for example, the cracked Brazil­ian disc con­fig­u­ra­tion) weak­­ened by artificial notches (rather than ‘math­e­mat­ical’ cracks), by far more interesting for practical engineer­ing ap­pli­­ca­tions.

Crack propagation, failure and ultimate load capacity of the grout layer in the prestressed anchor bolt foundation of wind turbine tower

The finite element (FE) analyses based on linear elastic fracture mechanics (LEFM) for the crack propagation and failure of the grout layer in the wind turbine foundation with pre-stressed anchor bolts are presented. Firstly, one half-symmetric planar numerical model is developed. According to the linear elastic analytical results, one initial crack is set to be the zone of the maximum tensile streess and its direction is perpendicular to the direction of the maximum principle tensile stress. The crack will propagate in the direction of the zero mode-II stress intensity factor during the failure of the grout layer. The numerical simulations with different prarmeters are performed. The histories of mode-I stress intensity factor are analyzed and design formulas predicting the failure and ultimate load capacity of the grout layer are proposed based on parametric studies. The increasing width or the decreasing depth of the grout layer can significantly mitigate the failure. One case study of the grout layer with the typical size and the corresponding fatigue analyses are carried out. The numerical results show that the proposed equation can agree well with the cracking status and failure of the grout layer in practical engineering and help predict the fatigue life of the grout layer under the certain intial crack length.

Numerical modelling of brittle fracture using lattice particle method with applications to fluid structure interaction problems via SPH coupling

This paper presents an improved failure model for simulating brittle fracture using the mesh-less Lattice Particle Method (LPM). By modelling the initial crack line using the Remove Bond (RB) approach as outlined in this paper, a new formulation is then developed for predicting the mode-I Stress Intensity Factor (SIF) near the crack tip. Compared to the conventional Remove Particle (RP) approach, it is found that the accuracy of the present SIF formulation based on the RB method is superior. A series of benchmark test cases are simulated to test the numerical accuracy and numerical convergence of the method. Finally, the LPM method is coupled with the Smoothed Particle Hydrodynamics (SPH) method for studying Fluid Structure Interaction (FSI) problems involving solid fracture and free surface. The coupled SPH-LPM method is implemented in the open-source code, DualSPHysics, which has been optimized for both CPU and GPU performances. Upon integrating LPM with SPH, the proposed FSI method is suitable for modelling fracture phenomena caused by natural hazards such as tsunami and flood.

Effect of Residual Stress on Mode-I Stress Intensity Factor: A Quantitative Evaluation and a Suggestion of an Estimating Equation

An extensive literature review was conducted primarily to develop a comprehensive understanding of the quantitative behavior of residual stress (RS) distribution in weld joints. Based on prior data, various levels of the peak RS and distribution profiles were applied to a finite element analyses (FEA) model as an initial loading to evaluate the effect of RS on the Mode-I stress intensity factor (SIF), KI. The RS was not found to have a significant effect on the SIF values when the peak RS was less than 300 MPa. The enhanced effect of RS on the KI values was found to be more pronounced at a lower crack ratio, while the analytical form of the RS distribution had a minor effect. The effective boundary corrections function, FResidual, was derived under the effect of RS for 1T CT specimen for various crack geometries of the crack ratio, (a/W) of 0.2, 0.40, 0.50, 0.60, and 0.75, and with peak RS varying from 100 MPa to 600 MPa. The obtained effective function of KI can be employed to quantitatively evaluate the effect of RS on the fatigue performance of welded joints.

A generalized machine learning framework for brittle crack problems using transfer learning and graph neural networks

Despite their recent success, machine learning (ML) models such as graph neural networks (GNNs), suffer from drawbacks such as the need for large training datasets and poor performance for unseen cases. In this work, we use transfer learning (TL) approaches to circumvent the need for retraining with large datasets. We apply TL to an existing ML framework, trained to predict multiple crack propagation and stress evolution in brittle materials under Mode I loading. The new framework, ACCelerated Universal fRAcTure Emulator (ACCURATE), is generalized to a variety of crack problems by using a sequence of TL update steps including (i) arbitrary crack lengths, (ii) arbitrary crack orientations, (iii) square domains, (iv) horizontal domains, and (v) shear loadings. We show that using small training datasets of 20 simulations for each TL update step, ACCURATE achieved high prediction accuracy in Mode-I and Mode-II stress intensity factors, and crack paths for these problems. We demonstrate ACCURATE’s ability to predict crack growth and stress evolution with high accuracy for unseen cases involving the combination of new boundary dimensions with arbitrary crack lengths and crack orientations in both tensile and shear loading. We also demonstrate significantly accelerated simulation times of up to 2 orders of magnitude faster (200x) compared to an XFEM-based fracture model. The ACCURATE framework provides a universal computational fracture mechanics model that can be easily modified or extended in future work.

A Domain-Independent Interaction Integral for Dynamic Fracture in Nonhomogeneous Magneto-Electro-Elastic Materials

In practice, magneto-electro-elastic (MEE) equipment are inevitably subjected to impact loading during service, resulting in fracture failure of the devices. This paper proposes a dynamic domain-independence interaction integral (DII-integral) to solve the dynamic intensity factors (IFs) at the crack tip of MEE materials. Here, this is a new expression of the interaction integral (I-integral) for dynamic crack study of nonhomogeneous MEE media, and we further extend it to MEE materials containing complex interfaces. Domain-independent property of the I-integral is theoretically demonstrated and excellent numerical results (relative deviation<1%) obtained from the various integration domains again verify the domain-independence for nonhomogeneous and multi-interface material properties without considering the material continuity requirements. Through the combination of the DII-integral and extended finite element method (XFEM), a good agreement is observed by comparing with the published data subjected to different MEE impact loadings. Typical examples reveal that the magnitude of dynamic mode-I stress intensity factor (SIF), dynamic electric displacement intensity factor (EDIF) and dynamic magnetic induction intensity factor (MIIF) decrease with the increase of the poling angle of the MEE material, while the dynamic mode-II SIF shows the opposite change. The peaks of dynamic EDIF and MIIF are more sensitive to the distance between the parallel cracks than the SIF, and all IFs are smaller than single crack due to the shielding effect. Finally, the significant difference on the dynamic IFs is observed with various material discontinuities.

Fracture analysis-based mode-I stress intensity factors of crack under fracture grouting in elastic–plastic soils

For the interaction of crack on the soil–grout interface under fracture grouting, an approximate method to determine the stress intensity factor (SIF) of crack on the soil–grout interface was proposed based on the conservation J2-integral. With this method, the energy release rate of crack propagation under fracture grouting can be defined by the parameters of elastic–plastic soils and the grouting pressure. In order to study the change of strain energy near the crack of elastic–plastic soil under fracture grouting, a mechanical model of elastic–plastic soil with crack was established based on non-associated Mohr–Coulomb criterion model, and the SIF of crack with spring boundary was investigated. The influence of the crack depth ratio and crack aspect ratio on the SIF of cracks under the spring boundary were analyzed, and revealed the rule of crack growth under fracture grouting in elastic–plastic soils. The results showed that the variation of the crack depth ratio and crack aspect ratio had an effect on the change of the SIF of cracks. Increasing the crack depth ratio and crack aspect ratio caused an increase in the SIF of the crack. The results can provide the reference for foundation reinforcement in elastic–plastic soils.

Reduction in Mode-I SIF of an Edge Cracked C-Shaped Specimen Using Piezoelectric Actuator

The focus of the current research is on the feasibility of an adhesively bound piezoelectric actuator in the active repair of an edge-cracked C-shaped specimen. First, only constant uniaxial tensile loading is used to determine the Mode-I stress intensity factor. The Mode-I stress intensity factor is then examined while taking into account the actuation effect provided by the piezoelectric patch when an external voltage is applied. For an edge-cracked C-shaped specimen, the total stress intensity factor is determined analytically using the concept of the superposition principle of the linear-elastic crack problem. The outcome demonstrates a considerable decrease in the Mode-I stress intensity factor following active repair using the piezoelectric actuator. Parametric analysis has been carried out to comprehend repair performance and choose the best-sized actuators for active repair.

Mode-I stress intensity factor by peridynamic stresses

While nodal displacements and/or strain energy have been used to analyze the fracture parameters in previous peridynamic studies, a simple method is here proposed to compute stress intensity factors from the nodal tensile stresses revealed by peridynamics with the virial theorem. The method is demonstrated for the mode I fracture of single edge- and center-cracked plates with a wide range of initial crack length-plate width ratios from 0.1 through 0.5. Crack opening displacement is also studied. δ-convergence is considered. Results are discussed with analytical and finite element methods.

Multiscale modelling of fracture in graphene sheets

Most of the continuum scale processes, such as fracture, plasticity, etc., trace their origin to atomistic scale phenomena. To gain deeper insights into these processes, one needs to understand the behaviour of materials through the lens of multiscale methods. In this manuscript, we study the problem of fracture crack propagation in graphene sheets through a sequential multiscaling technique. The continuum-mechanical smoothed particle hydrodynamics (SPH) is coupled with the atomistic scale molecular dynamics (MD) simulations through proper constitutive modelling — the non-linear material properties and the mechanical equation of state which serve as the inputs to the SPH model are evaluated directly from the MD simulations. Such handshaking ensures that the continuum-scale SPH model is able to faithfully reproduce the atomistic scale stress–strain behaviour until failure. Using a pre-notched continuum scale graphene sheet, we show that the mode-I stress intensity factor obtained from our SPH model agrees well with the published literature. We subsequently study crack propagation in pre-notched graphene sheets, where the influence of the orientation of the notch is evaluated. Lastly, we take the case of a continuum-scale graphene sheet having randomly oriented cracks and identify the changes in the stress–strain behaviour vis-à-vis a pristine graphene sheet.

An Interval Fracture Analysis Method for Cracked Structures with Unknown-But-Bounded Parameters

Abstract An interval fracture analysis method based on the Taylor expansion is presented to predict the stress intensity factor (SIF) bounds for cracked structures with unknown-but-bounded parameters. Traditional probabilistic fracture analysis requires numerous sample points, because large errors will occur when the probability density distribution function of the parameters cannot be described by sufficient sample points. In the present paper, the expression of the SIF crack structure is given using the quarter-point displacement method. Then, the interval expressions of mode-I and mode-II SIF are obtained using the Taylor expansion and the interval finite element method, and the upper and lower limits of an equivalent SIF (ESIF) are determined. Finally, the new method is analytically compared with the interval Monte Carlo method. Numerical examples show that the influence of load uncertainty considerably surpasses that of the dimension uncertainty. The mode-I SIF has a greater impact on the ESIF. As the uncertain level increases, the accuracy of the upper and lower limits of the ESIF calculated by the new method is maintained at a high level.

A pseudo-spring based SPH framework for studying fatigue crack propagation

The existing smoothed particle hydrodynamics (SPH) approaches for propagating fatigue cracks involve either the deletion of the crack front particle or stopping all its interactions in the total Lagrangian form. Here, we adopt the pseudo-spring-based Eulerian form of SPH to model mode-I fatigue crack propagation. For modeling fatigue crack growth, only the interactions between the crack front particle and its neighbors, which display the largest axial stresses in the connected pseudo-springs, are stopped. We show that our framework can determine accurately the mode-I stress intensity factors (SIFs) and capture both the fatigue crack path and the fatigue life of different specimens.

Initiation and Fracture Characteristics of Different Width Cracks of Concretes under Compressional Loading

A stress concentration at a crack tip may cause fracture initiation even under low-stress conditions. The maximum axial stress theory meets the challenges of explaining the fracture propagation of a non-closed fracture of cracked concretes under compressional loading. Uniaxial loading tests of single-crack concrete specimens were carried out and a numerical simulation of fracture propagation under uniaxial compression was performed. The radial shear stress criterion for a mode-II fracture is proposed to examine the stress intensity factor (SIF) of the pre-crack specimens under compressional loading. When the maximum radial shear stress at the crack tip is larger than the maximum axial tensile stress, and the maximum dimensionless SIFs can satisfy frθmax/fθmax &gt; 1 and frθmax/fθmax &gt; KIIC/KIC (fθmax=KIe/σyπa and frθmax=KIIe/σyπa are maximum dimensionless mode-I and mode-II SIFs, respectively), the crack will extend along the direction of the maximum radial shear stress. The influence of the single-crack angle and width on the mechanical properties of the specimens was examined. The experimental and numerical results indicate that the existence of cracks can considerably weaken the strength of the specimen. The distribution and width of the cracks had a significant effect on the specimen strength. The strength of the concrete specimen initially decreased and then increased with increasing fracture angle. The failure mechanism and rupture angle of pre-crack brittle material while considering crack width will be discovered.

Graph neural networks for simulating crack coalescence and propagation in brittle materials

High-fidelity fracture mechanics simulations of multiple microcracks interaction via physics-based models can become computationally demanding as the number of microcracks increases. This work develops a Graph Neural Network (GNN) based framework to simulate fracture and stress evolution in brittle materials due to multiple microcracks’ interaction. The GNN framework is trained on the dataset generated by XFEM-based fracture simulator. Our framework achieves high prediction accuracy on the test set (compared to an XFEM-based fracture simulator) by engineering a sequence of GNN-based predictions. The first prediction stage determines Mode-I and Mode-II stress intensity factors (which can be used to compute the stress evolution by LEFM), the second prediction stage determines which microcracks will propagate, and the final stage actually propagates crack-tip positions for the selected microcracks to the next time instant. The trained GNN framework is capable of simulating crack propagation, coalescence and corresponding stress distribution for a wide range of initial microcrack configurations (from 5 to 19 microcracks) without any additional modification. Lastly, the framework’s simulation time shows speed-ups 6x–25x faster compared to an XFEM-based simulator. These characteristics, make our GNN framework an attractive approach for simulating microcrack propagation and stress evolution in brittle materials with multiple initial microcracks.

A Macro–Micro Damage Model for Rock under Compression Loading

Rock damage caused by its microcrack growth has a great influence on the deformation and strength properties of rock under compressive loading. Considering the interaction of wing cracks and the additional stress caused by rock bridge damage, a new calculation model for the mode-I stress intensity factor at wing crack tip was proposed in this study. The proposed calculation model for the stress intensity factor can not only accurately predict the cracking angle of wing crack, but can also simulate the whole range of variation of wing crack length from being extremely short to very long. Based on the modified stress intensity factor, a macro–micro damage model for rock materials was also established by combining the relationship between microcrack growth and macroscopic strain. The proposed damage model was verified with the results from the conventional triaxial compression test of sandstone sample. The results show that the proposed damage model can not only continuously simulate the stress-strain curves under different confining pressures, but also can better predict the peak strength. Furthermore, the sensitivities of initial crack size, crack friction coefficient, fracture toughness, initial damage and parameter m on the stress-strain relationship are discussed. The results can provide a theoretical reference for understanding the effect of microcrack growth on the progressive failure of rock under the compressive loading.