Mixed-mode Stress Intensity Factors Research Articles

Numerical analysis of 3D nonplanar hydraulic fracture propagation in fractured-vuggy formations using a hydromechanical coupled XFEM approach

In this paper, a hydromechanical coupled XFEM approach is developed for simulating 3D nonplanar hydraulic fracturing, fracture–vug interactions and natural crack intersection problems. The extended finite element method associated with a high-order tip enrichment function is used for describing rock deformation, and the finite element method is chosen for discretizing fluid flow at the fracture surface. Rock deformation and fluid flow are coupled and solved by the Picard iteration method. The hybrid explicit–implicit representation method is adopted to represent 3D nonplanar fracture surfaces, and the displacement correlation method is chosen for extracting the mixed-mode stress intensity factor. A new mixed-mode failure criterion is introduced for determining 3D nonplanar fracture growth. Analytical solutions and experimental investigations are employed to verify the validity and effectiveness of our model. The simulations of fracture–vug interactions show that fracture surface growth on the side without vugs is constrained, while vugs can aggravate the tortuousness of the fracture surface on the side with vugs. In addition, four new intersection patterns (penetration, partial arrest, partial diversion, full arrest, and vertical propagation) are found in the 3D nonplanar fracture intersection simulations. The numerical examples highlight the considerable impacts of vertical growth and the mixed-mode effect on hydraulic fracturing.

Double inclined cracks overlapping effect on mixed stress intensity factors using XFEM object-oriented implementation

The object of research is the Mixed Mode Stress Intensity Factor (MMSIF) of a two-dimensional (2D) plate. With the emergence of modern technologies and advanced innovations which contribute to the development and improvement of the design, implementation and management of construction projects, it has become easier. However, it is very difficult to manufacture components free from unavoidable defects, such as cracks, which lead to material deterioration and ultimately shorten its service life. Based on the process of local enrichment region using partition of unity concept, the extended finite element method (XFEM) has overcome the limitations of the standard FEM method in terms of modeling and numerical simulation of discontinuities (cracks) while gaining its general advantages. This makes XFEM a powerful and widely used digital tool in recent years. One of the most frequently raised problems in the discontinuities field (cracks) is the phenomenon of juxtaposition of multiple cracks in a cracked isotropic plate, which must be studied to determine the extent of its effect on the crack stress intensity factor in order to obtain higher safety reliability. On this basis, an improved object-oriented programming (OOP) with extended finite elements was used because of its great importance and well-known benefits. In this paper, the MMSIF of a 2D plate is determined to show the effect of the out-of-phase orientation of the angle, as well as the effect of the juxtaposition of two inclined cracks. As a result of the research, it is shown that, the convergence between the results obtained in this study with those reported in the literature, and to theoretical values is remarkable, and their close agreement was noted. In the future, based on the object-oriented approach characteristics represented by flexibility, scalability, and modularity, which were explained in this research, this proposed approach can be enriched to include heterogeneous materials modeling, whether linear or nonlinear, crack propagation in dynamics, in addition to Complex 3D industrial problems.

An effective free-meshing and linear Step-Wise procedure to predict crack initiation and propagation

An automated procedure is proposed to calculate fracture initiation and propagation in two-dimensional structures. The application uses and enriches the Hybrid Semi-Analytical Method to calculate mixed-mode stress intensity factors in linear elastic fracture mechanics framework. The strategy does not necessitate prior knowledge of the starting crack tip location within the structure, resulting in suitable for analyzing structures without initial defects. When neither a priori invitation nor pre-existing notch is provided, the fracture initiation is assumed to occur when the structure reaches its elastic limit, and a non-linear analysis is conducted under the prescribed load to identify the first occurrence of permanent strain and to locate where the crack opening occurs. As the crack propagates, solely linear elastic analyses are performed to calculate the stress intensity factors. Relying upon these parameters, the procedure implements an evolutionary algorithm based on a step-by-step evaluation of the crack increment and direction. A global–local energy-based criterion is employed to drive fracture propagation. Neither local refinement nor mapped meshes around the crack tip are required, resulting in a streamlined and efficient free-meshing strategy.The calculation has been implemented in a standard finite element environment, and several numerical examples have been investigated, both from a qualitative and quantitative point of view. Qualitative crack paths have been shown for a square hollow plate under pure tensile and shear loads, and a modified Single Edge Notched specimen in bending, as a function of the mutual void positions, distance from the plate edges, and different notch lengths. Finally, an effective comparison in terms of forces-displacement curves has been presented between the proposed procedure, and a lab test for a modified Compact Tension specimen under tensile loading, demonstrating the overall accuracy and reliability of the presented strategy.It is felt that the proposed procedure could pave the way for designing hollowed structures and investigating the corresponding evolutionary laws that relate structural geometries to the insurgence and spread of fractures, and envisaging crack initiation and propagation within inhomogeneous media.

Stress intensity factor analysis for mixed-mode fracture behavior of inclined semi-elliptical surface crack in stiffened plate under tension load

This study aims to investigate the mixed-mode fracture behavior of stiffened plates with inclined semi-elliptical surface cracks (ISSCs). For this purpose, a new numerical model is developed for the fracture analysis using FRANC 3D in combination with ABAQUS. Normalized stress intensity factors (SIFs) are used to analyze the fracture behavior at the surface and deepest points of the surface crack. Additionally, an arrest parameter ψ is proposed to quantify the level of crack arrest under mixed-mode conditions. Then, a thorough investigation is conducted to examine the fracture behavior of stiffened plates with ISSCs, focusing on the effects of aspect ratio, relative depth, spacing ratio, stiffness coefficient, crack angle, and loading mode. The findings indicate that the restraint effect of stiffeners on the fracture of large surface cracks is significantly stronger compared to small surface cracks. It is also identified that the restraint effect is more pronounced at the surface point than at the deepest point. Finally, a set of novel formulas is derived to evaluate the mixed-mode SIFs in stiffened plates through statistical analysis of the results.

Improved XFEM for 3D interfacial crack modeling

The conventional extended finite element method (XFEM) usually suffers from the crack tip enrichment for removing the blending elements. This may be inconvenient when treating the three dimensional (3D) and curved crack modeling since the crack tip enrichment is usually defined in the local coordinate system depending on the crack front. To avoid these shortcomings of the XFEM, an improved enrichment scheme is proposed for the 3D interfacial crack modeling. The proposed enrichment scheme removes the crack tip enrichment and two weight functions are introduced to capture the strong and weak discontinuities in the element including the crack front. The new enrichment scheme has advantages for the 3D curved interfacial cracks since all the enrichment functions are defined in the global system and it is unnecessary to introduce additional coordinate systems. Several benchmark problems including the mixed mode stress intensity factors (SIFs), anti-plane load and curved cracks are analyzed by the proposed method with the local refinement technology. Numerical results demonstrate that the proposed method can obtain accurate 3D energy release rate and SIFs. In addition, the proposed method enhances the applications of XFEM and has the potential abilities to the multiple-fields coupling and interfacial crack growth phenomena due to the fact that compatible enrichment functions are utilized.

Calibration of a Test Method to Measure Mode I, Mode II and Mode III Fracture Toughness of Engineering Materials

A refined finite element model and a new post processing sub program to determine mixed mode stress intensity factors and their variation along a surface crack front in components and structures is presented. The proposed finite element model employs a fine mesh of singular isoparametric pentahedral solid element with user specified number and size from one crack face to another and a number of such segments along a surface crack front. A compatible mesh of regular elements namely hexahedral solid element and pentahedral solid element is used to discretize the rest of the domain under consideration. Consistent with the use of the singular element, formulae to compute the Mode I, Mode II and Mode III stress intensity factors using displacements only of flagged nodes on flagged singular elements are implemented in a special purpose post processing sub program named SIF 1-2-3. In the present work the finite element models developed using ANSYS, a commercial FEA program, and the stress intensity factors determined using SIF 1-2-3 are validated using benchmarks. This finite element model is then used to calibrate a proposed test method to measure Mode I, Mode II and Mode III fracture toughness of engineering materials.

U-Net learning for the automatic identification of the sandstone crack tip position to determine mixed-mode stress intensity factors utilizing digital image correlation method

Tracking the crack tip position in crack growth is a key and difficult problem in calculating mixed-mode I–II dynamic stress intensity factors based on displacement field obtained by digital image correlation method (DIC). U-Net in deep learning is used to perform pixel-level semantic segmentation on the experimental images of mixed-mode I–II cracks propagation under the three-point bending of brittle sandstone to identify crack features and locate the crack tip position. On this basis, the digital image correlation displacement data are substituted into the Williams displacement field equation to calculate the dynamic stress intensity factors at different loading stages. The results show that the U-Net network model can achieve image segmentation and process experimental pictures in batches, and accurately detects cracks in images with speckles which has been demonstrated potential in identifying crack propagation paths and tip positions. When the position of the crack tip is known, the solution for the stress intensity factor converts the non-linear equation system to a linear one and solves it with the linear least squares method. This significantly improves the solution accuracy and speed of the mixed-mode I–II dynamic stress intensity factors. Simultaneously, according to the position of the crack tip, the crack propagation path during the loading process can be drawn, which provides a reference for research on the evolution in crack growth.

A data-driven model for predicting the mixed-mode stress intensity factors of a crack in composites

A data-driven model is trained to predict mixed-mode stress intensity factors (SIFs) of composites through an artificial neural network (ANN) method. The model is based on a database generated by combining the interaction integral and the extended finite element method (XFEM). To reduce the input dimensionality and improve predictive performance, feature engineering is performed on the input data, and principal component analysis is conducted. Hyperparameters of the model are adjusted by using K-fold cross-validation and Bayesian optimization algorithm (BOA) to enhance the adaptability and generalization of the model. To overcome the data scarcity issue, an active learning knowledge extraction framework is constructed, which allows for accurate knowledge extraction even with limited data. By utilizing data-driven models to solve mixed-mode SIFs, the computational cost and complexity are greatly reduced compared to numerical simulations, while computational stability and ability to deal with high-dimensional nonlinear problems are significantly improved.

Analysis of a method to compute mixed-mode stress intensity factors for non-planar cracks in three-dimensions

In this work, we present and prove results underlying a method which uses functionals derived from the interaction integral to approximate the stress intensity factors along a three-dimensional crack front. We first prove that the functionals possess a pair of important properties. The functionals are well-defined and continuous for square-integrable tensor fields, such as the gradient of a finite element solution. Furthermore, the stress intensity factors are representatives of such functionals in a space of functions over the crack front. Our second result is an error estimate for the numerical stress intensity factors computed via our method. The latter property of the functionals provides a recipe for numerical stress intensity factors; we apply the functionals to the gradient of a finite element approximation for a specific set of crack front variations, and we calculate the stress intensity factors by inverting the mass matrix for those variations.

Extended wavelet Galerkin method for mixed-mode cracked FGM plate under static and dynamic loads

The extended wavelet Galerkin method (XWGM) is presented for fracture mechanics analysis of functionally graded material (FGM) cracked plates under static and dynamic loads. The nonhomogeneous stress distribution and stress wave transition can be approximated by the piecewise truncated polynomial (B-spline) functions. The steep stress gradients around the crack tip can be captured by superposing the higher resolution wavelet functions with the nature of multiresolution analysis (MRA) in the wavelet theory. Additionally, a trigonometric function and Heaviside function are employed to enrich the wavelet bases for the fracture modeling. The interaction integral method is utilized to evaluate the static and dynamic mixed-mode stress intensity factors taking the inertia and material nonhomogeneity into account. Applicability of XWGM to the FGM crack problems is discussed through the numerical examples.

Determination of mixed-mode stress intensity factors for axial crotch corner cracks at cylindrical vessel to nozzle intersections under internal pressure and nozzle loads

In this study, Stress Intensity Factors (SIF) of quarter circle and semi elliptic cracks at the crotch corner of the intersection between a cylindrical pressure vessel and a radial nozzle have been computed for combinations of the actions of internal pressure and external forces on the nozzle. Until now, practitioners have used simplified solutions for SIFs available from the literature, such as the Newman-Raju equations for the mode I SIF produced by an internal longitudinal semi-elliptical crack in a cylinder, to approximate the SIF produced at a crotch corner crack. These equations do not take into account the effect of the opening of the cylinder for the nozzle and the resulting stress distribution along the cylinder. As a result, there is uncertainty about the validity of using these simplified solutions in engineering critical assessments and a more exact determination of the SIFs for crotch corner cracks would be beneficial.The mode I, II and III stress intensity factors have been determined for crotch corner cracks for a single size of vessel and are presented as a function of variables such as location on the crack front, crack size and aspect ratio (a/c), crack angle with respect to the axis of the vessel, and the magnitude of internal pressure and external forces. The results are presented in graphical form. The study concludes with the recommendation that the analysis be extended to a wider range of vessel and nozzle parameters.

Mixed mode stress intensity factors of single edge cracked/notched plates under different boundary conditions using strain energy approach

This paper presents the mixed mode I/II stress intensity factors (SIFs) of slanted cracks/notches under tension loading. A two-dimensional finite element analysis (FEA) was employed using the ABAQUS program. Based on the literature survey, there is a lack of solutions for SIFs of slanted cracks in plane stress plates. The strain energy approach (SEA) was adopted to analyze the notched cases. Mode I and II SIFs are derived via the SEA method from the strain energy computed using the FEA technique. The results show excellent agreement with other researchers' work. The key contribution of the present paper is the presentation of the effect of the crack/notch length to plate width ratio on the mode I and II SIFs of a slant-edge-cracked/notched plate under different boundary conditions. Sets of new data of mode I and II SIFs are provided herein in tabular and graphical formats.

The sloshing effects on the crack instability of the spherical tank under the simultaneous effect of internal pressure and earthquake

Spherical tanks have an important function in the industry due to the internal pressure of their contents. Also, the supporting structure of the spherical tank controls stability and damage to the tank One of the damages that can lead to a critical situation is an unstable crack in the column-vessel joint. In this study, the sloshing effects on the instability of 10 mm and 20 mm long semi-elliptical cracks with three aspect ratios in the connection zones of the supporting structure under the effect of internal pressure and eleven earthquakes have been investigated. Furthermore, using the finite element sub-modeling method around the crack area and part of the structure, the effect of fluid–structure interaction has been considered for three liquid levels to calculate the stress intensity factors along the crack front. Since the tank is under the simultaneous effect of internal pressure, earthquake, and sloshing motion, a three-dimensional mixed-mode stress intensity factor needed to be considered to study the crack instability. The results show that earthquakes and sloshing motions can have various effects on the equivalent stress intensity factors. For almost two-thirds of the studied cases, the equivalent stress intensity factor for the 50% filled vessel was 10%–70% greater than the 98% filled vessel. This finding shows that the crack located in the connection zones of the supporting structure under the simultaneous effect of internal pressure, earthquake, and sloshing motion, could lead to failure.

Computation of two dimensional mixed-mode stress intensity factor rates using a complex-variable interaction integral

The well-known interaction integral, also known as the M-integral or I-integral, is a method to compute the mixed-mode stress intensity factors (SIFs) for fracture mechanics problems. The capabilities of the M-integral are extended here to compute derivatives of the SIFs with respect to the crack extensions for an isotropic linear elastic material under static loading. These derivatives were calculated using the complex Taylor series expansion (CTSE) numerical differentiation method. The derivatives of the auxiliary fields are calculated by applying CTSE directly to the M-integral formulation. The derivatives of the actual displacement fields are computed using a hypercomplex-variable finite element method (ZFEM) that also employs CTSE. SIF rates with respect to all crack tips are computed in a single analysis. The method is general and can be easily extended to other two-dimensional loading scenarios and different material models in a straightforward manner through the use of the appropriate auxiliary fields. The complex-variable M-integral was implemented within the commercial finite element software Abaqus through a user-defined element subroutine (UEL). Numerical examples demonstrate the high accuracy of the method.

Influence coefficient based fracture parameter modification factor in a cracked T-butt welded joint

An influence coefficient based modification factor (μk) method and its regression formulae are proposed. The μk can be used to calculate the stress intensity factor (SIF) in a cracked T-butt welded joint under arbitrary stress distribution. By multiplying μk with the SIFs of a flat plate obtained by numerical influence function method (NIFM), mixed-mode SIFs (MM-SIFs) for a T-butt joint can be readily evaluated. T-butt joints with semi-circular surface cracks are employed in the numerical examples. The MM-SIFs calculated by the μk method were verified against those computed under direct membrane loading, and under welding residual stress conditions. An adequate agreement was achieved between the solutions given by the proposed μk method and the reference numerical analyses. Therefore, the MM-SIFs of surface-cracked T-butt joints under arbitrary stress distributions can be evaluated using the developed μk formulae.

Rolling contact fatigue crack propagation on contact surface and subsurface in mixed mode I+II+III fracture

Rolling contact fatigue (RCF) is the critical damage from rolling bearings constraining the service, performance, and reliability, which is directly caused by fatigue crack propagation. The crack paths on both the contact surface and subsurface were observed after the bearing fatigue test. The three-dimensional geometry of the RCF crack was characterized and reconstructed based on experimental statistics. Finite element analysis was performed to obtain the mixed mode stress intensity factors (SIFs) along the crack fronts, and the crack propagation behavior was numerically studied. Results indicated that the propagation of cracks on the contact surface and subsurface causes the gradual formation of micropits and the sudden spalling of material respectively. Initial micro-cracks with an inclination angle of 15–30° have the highest growth rate, which should be prevented during the bearing raceway manufacturing process. The propagation paths of the RCF cracks can be more accurately predicted by considering the superposition of fracture mode I + II + III in three-dimensional, and mechanisms of crack propagation on contact surface and subsurface were revealed respectively.

XFEM based thermo-elastic numerical analysis of FGMs with various discontinuities

In this study, mixed-mode stress intensity factor (MMSIF) and crack growth analysis in functionally graded materials (FGMs) with discontinuities (crack, inclusion and void) are presented. The extended finite element method (XFEM) based study with the partition of unity method is utilized under thermo-mechanical loadings. The effect of mechanical and thermal loadings acting individually and combined on cracked FGM plate with other discontinuities is carried out. In mechanical loading, uniform tensile, shear, combination (tensile and shear), point, uniformly varying, and exponential loading are considered, whereas in thermal loadings, isothermal and adiabatic loadings are utilized. The objective of this present study is to find the MMSIF and crack growth analysis of FGM plate with discontinuities under different thermo-mechanical loadings, where the interaction study of discontinuities and different thermo-mechanical loadings are investigated. The present problem is modeled in MATLAB (R 2015a) environment and the results of the present model are compared with the results from the literature.

Effect of edge cracks on critical current degradation in REBCO tapes under tensile stress

The slitting process for manufacturing REBa2Cu3O7-δ (REBCO, RE = Rare earth) tapes of required width significantly improves the production efficiency and reduces production costs. However, edge cracks induced by the slitting process of wide REBCO tapes may cause premature degradation under high tensile stress in high-field magnets. Therefore, it is necessary to evaluate the effect of edge cracks of REBCO tapes on the critical current (Ic) degradation. Firstly, Ic degradation under artificial cracks was measured to validate the applicability of linear elastic fracture mechanics for the REBCO layer. The maximum circumferential stress criterion was used to derive the mixed-mode stress intensity factor of multiple oblique edge cracks. A semi-analytical model considering edge crack properties such as angle β, spacing d, and length a, was built to evaluate the critical load and critical crack. We found that when the stress intensity factor at the crack tip is below KIC=2.3 MPam, edge cracks did not propagate. We examined commercial REBCO tapes manufactured by two different processes, concluding that edge cracks in these tapes will not cause premature degradation.

Investigating the behavior of cracks in welded zones of supporting structure of spherical pressure vessel under seismic loading

In this paper, the numerical studies on the semi-elliptical crack behavior in different locations of welded zones in the supporting structure of a spherical pressure vessel under an earthquake are presented. The cracks in the welded zones of supporting structures under earthquake effects may jeopardize the safety of spherical pressure vessels and result in catastrophic failure. A detailed finite element sub-modeling technique is carried out to compute the mixed-mode stress intensity factors along the crack front. Furthermore, crack behavior with different aspect ratios a/c: 0.25, 0.5, and 0.75 at the weld and the heat-affected zone of the supporting structure is evaluated. The fluid-structure interaction for three liquid levels: 0%, 50%, and 98% filled vessels of a spherical pressure vessel is considered. The results show that seismic loading can have different effects on the mixed-mode stress intensity factors at multiple liquid levels. As an example, for 50% filled vessel, the semi-elliptical crack behavior located at the heat-affected zone of column-vessel joint intersection is more critical than 0%, and 98% filled vessels and is susceptible to crack growth. This finding shows that the safety margin of the structure can be significantly influenced by the presence of cracks under seismic loading.

Three-dimensional non-planar crack growth analysis using enriched finite elements

The paper presents computational mixed mode non-planar fatigue crack growth analyses using Fracture and Crack Propagation Analysis System (FCPAS), which employs enriched crack tip finite elements to calculate the mixed mode stress intensity factors (SIFs) and utilizes from other sub-modules to predict the evolving crack growth trajectories and the resulting fatigue crack growth lives. Numerical applications on different problems selected from the literature; a hollow shaft, an H-shaped specimen and cylindrical rods containing cracks with different initial orientations and sizes and subjected to different fatigue loading types are presented. Comparisons of results from the numerical analyses with the corresponding data from the literature agree well in terms of computed SIFs, predicted evolving crack paths and crack propagation lives.