Interaction Integral Method Research Articles

Crack field analysis by optical DIC of short cracks in Zircaloy-4

The prediction of fatigue life is most difficult for short cracks, as their local conditions may differ from what is predicted using the remote applied loading and crack geometry. Digital image correlation (DIC) can be utilized to analyze images from an optical microscope (OM), which facilitates the local characterization of the crack field. This paper presents a Finite-Element-based approach that uses DIC-obtained displacement data to retrieve the crack field and quantify the local crack driving force. With the assumption of linear anisotropic elasticity, the change in both the Mode I and Mode II crack intensity factors over a fatigue cycle can be extracted using the interaction integral method. This allows the determination of the local driving force for short crack propagation. The application of this method is demonstrated by the full-field analysis of short fatigue cracks in blocky alpha Zircaloy-4. The sensitivity of the analysis is investigated to factors that include DIC subset size, uncertainties in crack tip position and lower quality data in the crack vicinity. This technique requires no prior knowledge of theoretical solutions or far-field boundary conditions, and it can be applied to the tip of a tortuous crack by defining an appropriate local frame of reference. The analysis is applied here to cracks under similar mode I loading that propagate on the prism plane at different rates in directions that are parallel and perpendicular to the c-axis of the hexagonal unit cell.

On the combination of Moving Mesh technique and M-integral method for predicting crack propagation mechanisms in Functionally Graded Materials

This work presents an efficient FE modeling approach for predicting crack propagation mechanisms in Functionally Graded Materials (FGMs). The proposed strategy combines the Moving Mesh (MM) technique and the Interaction Integral method (M-integral). The MM is used to reproduce the geometry evolution induced by the crack advance. Specifically, the mesh nodes around the crack tip are moved in such a way to follow crack path evolutions, thus reducing the need for continuous remeshing events that affect negatively the computational efficiency of standard FE modeling strategies. In particular, the proposed scheme employs a moving mesh strategy consistent with the Arbitrary Lagrangian-Eulerian (ALE) formulation, which is suitable for handling the motion of mesh nodes with great flexibility. In addition, the use of proper regularization procedures ensures the consistency of the mesh motion, reducing elevated elements distortions. The motion of the mesh nodes requires a proper definition of crack onset conditions and propagation direction. Hence, a correct evaluation of fracture variables at the crack front (i.e., Stress Intensity Factors and T-stress) is necessary. For this purpose, the proposed method adopts the M-integral approach, which is a widely used method due to its simplicity and accuracy. Since the crack front moves during the crack advance, the M-integral is implemented by using the ALE formulation, thus extracting fracture variables on distorting elements. The validity of the proposed strategy has been validated through comparisons with experimental and numerical data reported in the literature.

Simulation of dynamic fracture in quasi-brittle materials using a finite element modeling approach enhanced by moving mesh technique and interaction integral method

This work presents an efficient modeling approach for predicting dynamic crack propagation mechanisms in quasi-brittle materials. The proposed method consists of a standard Finite Element code enhanced by Moving Mesh (MM) technique consistent with the Arbitrary Lagrangian-Eulerian (ALE) formulation. The MM is used to reproduce the geometry evolution caused by dynamically growing cracks. More precisely, the nodes of the computational mesh around the crack tip change position during the simulation consistently with the growth of the crack front. In such a context, the ALE formulation ensures an effective re-location of mesh nodes inside the computational domain, reducing the overall amount of remeshing events. The motion of the computational mesh takes place according to previsions dictated by classic fracture criteria, which define crack initiation conditions, the direction of propagation, and the velocity of the advancing crack front. Such conditions depend on Dynamic Stress Intensity Factors at the crack front. Hence, accurately predicting these fracture variables is mandatory to ensure consistent mesh motions. To this end, the proposed approach implements the ALE formulation of the M-integral, which enables computing fracture variables on moving elements without losing accuracy. The validity of the proposed modeling strategy has been assessed through comparisons with numerical data and analytical formulations reported in the literature.

Interaction integral method for crack-tip intensity factor evaluations of magneto-electro-elastic materials with residual strain

Magneto-electro-elastic (MEE) materials generally consist of piezoelectric and piezomagnetic constituent phases so that the residual strain often occurs during the manufacturing process. The distributed residual strain and complex interfaces between constituents cause a great challenge to the fracture analysis of MEE materials. Considering the effect of the residual strain, this paper develops an interaction integral (I-integral) method for the extraction of the fracture parameters of an impermeable crack in nonhomogeneous MEE materials. The I-integral method has several merits: 1) it can decouple stress intensity factors (SIFs), electric displacement intensity factor (EDIF) and magnetic induction intensity factor (MIIF); 2) the I-integral is independent of the size of the integration domain, even when the integration domain contains nonhomogeneous and discontinuous material properties. Combined with the extended finite element method (XFEM), the I-integral is adopted to investigate several crack problems subjected to homogeneous, nonhomogeneous and discontinuous residual strains. The I-integral values calculated by different integration domains agree well with each other (relative deviation < 1%) for nonhomogeneous and discontinuous MEE properties. Then, single-crack and multi-crack problems are studied to show the influences of residual strain distributions on the intensity factors. Homogeneous residual strain affects the SIFs dramatically but has a slight influence on the EDIF and MIIF. By comparison, nonhomogeneous residual strains affect the SIFs, EDIF and MIIF significantly. In addition, the residual strain gradients affect the variations of intensity factors with crack location extremely.

Simulation of cryogenic fracturing of rock-like materials using material point method

Cryogenic fracturing is a relatively new technique used in gas and oil extraction. The material point method (MPM) combines the advantages of the Lagrangian and Eulerian methods and effectively solves problems involving fracture propagation and thermomechanical coupling. Hence, in this study, this method was applied to the simulation of cryogenic fracturing in rocks to reveal the mechanism of the process. First, the heat-conduction equation was discretized using the MPM. Subsequently, the thermomechanical-coupling problem was solved under the unified framework of the MPM. The discontinuous fields around the fracture were described, and fracture propagation was predicted using the phantom node method and interaction integral methods in the MPM. The results were then compared with finite-element simulation results to verify the feasibility of these methods. Finally, cryogenic fracturing simulations in the literature were examined, and the simulation results agreed well with the experimental results. Moreover, the distribution characteristics of thermal fractures were explained using the inhibitory interactions between fractures. The simulation results indicated that increasing the convective heat transfer coefficient results in a significant increase in the number of fractures.

An evaluation of the loading condition on mixed-mode stress intensity factors for CTST specimens made of 2024-T351 aluminum alloy

In this paper, the fracture behavior of 2024-T351 aluminum alloy as a key engineering material in the aeronautical industry is investigated under various planar and nonplanar mixed-mode loading conditions involving pure mode-I and pure mode-III loadings. A newly suggested loading setup accompanied by compact tension shearing and tearing (CTST) specimens is utilized to perform fracture tests. Three-dimensional finite element modeling using the interaction integral method is carried out to derive the stress intensity factors (SIFs) and their ratios through the crack front for several mixed-mode configurations. The numerical results reveal that the coupled effect of modes II and III under mixed-mode I/II, I/III and I/II/III loading conditions is remarkable. Moreover, the amounts of SIFs at the center of the specimens are employed to predict the critical fracture loads according to different mixed-mode criteria. Also, scanning electron microscope (SEM) images of the fracture surfaces of CTST specimens are examined to evaluate the effect of different loading angles from a morphological viewpoint. A good consistency can be found among the results of the theoretical solutions of the criteria and the experimental observations for different loading conditions.

Material point method for the propagation of multiple branched cracks based on classical fracture mechanics

The material point method (MPM) combines the advantages of both mesh-free and grid-based methods for simulating complex cracking problems without the limitations of the grid. However, the conventional MPM cannot deal with cracks directly because of the continuous nodal velocity field. Therefore, in this study, the conventional MPM is improved using three new algorithms to simulate the propagation of multiple branched cracks. The first proposed algorithm is the phantom node method using a particle cutting technique, which allows discontinuity of the velocity field around arbitrary cracks, and the second one is the multi-crack contact algorithm to deal with the contact problem between multiple objects cut by multiple cracks, avoiding crack penetration. The last one is particle interaction integral method based on classical fracture mechanics to get the dynamic stress intensity factors (DSIF) and realize the crack propagation. And all algorithms are validated by comparing with the extended finite element method. Subsequently, uniaxial compression tests of three rock samples with different pre-existing cracks are simulated using the extended MPM. Three types of crack coalescence are found, and the simulated results agree with the previous experimental results, which demonstrate the feasibility and flexibility of the proposed extended MPM for simulating the propagation of multiple branched cracks.

A new approach of the interaction integral method with correction terms for cracks in three-dimensional functionally graded materials using the tetrahedral finite element

In this paper, a new approach for evaluating the Stress Intensity Factors (SIFs) using the interaction integral method with correction terms for cracks in three-dimensional Functionally Graded Material (FGM) is presented. The present interaction integral method allows us to adopt finite element models consisting of only tetrahedral finite elements with unstructured mesh arrangements. The proposed interaction integral is tested by numerical examples, for the accuracies under mode I and mixed mode loadings and using unstructured finite element models consisting of tetrahedral finite elements only. The results maintained the so-called path-independent property of the interaction integral method while providing accurate results when compared with available analytical solutions.

Crack propagation under thermo-mechanical loadings based on moving mesh strategy

This paper proposes an effective FE modeling for simulating fracture propagation in linear elastic media under mechanical and thermal loadings. The model joins Moving Mesh (MM) technique with Interaction integral (M−integral) method to simulate crack advancing phenomena. MM reproduces growing cracks by moving the mesh nodes around the crack front, according to proper fracture criteria. In particular, the method adopts the Arbitrary Lagrangian-Eulerian (ALE) Formulation, which handles random cracks, reducing the recourse to remeshing processes. In this context, M−integral determines the fracture quantities at crack front, required to find onset conditions and propagation direction. This takes place during the entire analysis, thus ensuring quite smooth crack paths. Comparison results with other numerical strategies serve as primary tests to check the efficacy of the proposed approach. Besides, simulations with multiple growing cracks in both complex geometries and boundary conditions let checking performance. The results denote that the proposed method is a powerful and versatile tool for reproducing complex crack propagation scenarios.

Design Equations for Mixed-Mode Fracture of Dental Ceramic-Cement Interfaces Using the Brazil-Nut-Sandwich Test.

Dental interfaces are subject to mixed-mode loading. This study provides practical guidance for determining interfacial fracture toughness of dental ceramic systems. We address interfacial fracture of a composite resin cement sandwiched between two dental ceramic materials. Emphasis is placed on sandwich disc specimens with cracks originating from elliptical-shaped flaws near the center, for which analytical fracture mechanics methods fail to predict. The interaction integral method is used to provide accurate finite element solutions for cracks with elliptical-shaped flaws in a Brazil-nut-sandwich specimen. The developed model was first validated with existing experimental data and then used to evaluate the three most widely used dental ceramic systems: polycrystalline ceramics (zirconia), glass-ceramics (lithium disilicate), and feldspathic ceramics (porcelain). Contrary to disc specimens with ideal cracks, those with cracks emanating from elliptical-shaped flaws do not exhibit a monotonic increase in interfacial toughness. Also, interfacial fracture toughness is seen to have a direct relationship with the aspect ratio of elliptical-shaped flaws and an inverse relationship with the modulus ratio of the constituents. The presence of an elliptical-shaped flaw significantly changes the interfacial fracture behavior of sandwich structures. Semi-empirical design equations are provided for fracture toughness and stress intensity factors for interfacial cracks. The developed design equations provide practical guidance for determining interfacial fracture toughness of selected dental ceramic material systems. Those equations take into account four critical factors: size of the elliptical flaw, modulus ratio of constituent materials, loading angle, and applied load.

Multi-patch local mesh refinement XIGA based on LR NURBS and Nitsche’s method for crack growth in complex cracked plates

The objective of this article is to simulate crack growth of complex Mindlin–Reissner plates by developing an adaptive multi-patch extended isogeometric analysis (XIGA). Nitsche’s method is used to treat continuity between multi-patches or the coupling of non-conforming meshes, exactly describing the geometry of complex plates. The computational meshes in XIGA are independent of the cracks by introducing some enrichment functions into the displacement approximation based on the partition of unity, thus it is convenient in modeling evolution of crack. The locally refined non-uniform rational B-splines (LR NURBS), which have the properties of local refinement and exact description of geometry, are used to construct geometric model and taken as the shape functions in XIGA. To implement the adaptive local refinement, a method based on recovery technique by Zienkiewicz and Zhu is proposed. Based on the Mindlin–Reissner plate theory, interaction integral method is employed to calculate stress intensity factors (SIFs) and the maximum circumferential stress criterion is used to determine the crack propagation direction. Several numerical examples are carried out to demonstrate the performance of the adaptive multi-patch XIGA for simulating crack growth in complex plates.

Interaction integral method for computation of crack parameters K–T – A review

A critical overview of past research on the interaction integral (I-integral) method is presented. The I-integral is the two-state mutual energy release rate decided by a designable auxiliary field and the actual field. Due to the designability of the auxiliary field, the I-integral has been used extensively in extracting the individual stress intensity factors (SIFs) and the T-stress of a crack in single materials or at a bi-material interface. The practical implementation of the I-integral mainly experienced three periods. 1) In the last quarter of the twentieth century, the path-independent I-integral was developed for homogeneous materials subjected to static, dynamic and/or thermal loading. 2) In the 2000 s, the domain-form I-integrals were discussed to deal with inhomogeneous materials. 3) In the past decade, the domain-independent I-integral (DII-integral) was proposed for multi-interface materials. After that, we review the I-integrals for functional materials and the extensions of the I-integral to some new fracture problems. Subsequently, diverse practical applications are introduced. Various research works demonstrate that the I-integral is a most accurate and reliable approach to extracting the individual SIFs and T-stress for the materials obeying linear constitutive law or the materials with limited nonlinear effect. Finally, the advantages, limitations, and future issues of the I-integral method are discussed.

Crack propagation modeling in functionally graded materials using Moving Mesh technique and interaction integral approach

This paper presents a novel FE modeling approach based on Moving Mesh technique to reproduce crack propagation mechanisms in Functionally Graded Materials. The moving mesh is consistent with the Arbitrary Lagrangian-Eulerian formulation, which is suited to handle growing random cracks, avoiding extensive remeshing processes. This approach is based on the Interaction Integral Method to extract the mixed-mode Stress Intensity Factors, which are necessary to establish crack onset conditions and propagation direction. Among the different available options for FGM, the incompatibility formulation is adopted. The proposed scheme reproduces the propagation mechanisms by moving the computational nodes around the crack tip, according to standard fracture criteria. Mesh regularization technique based on proper rezoning equations ensures the consistency of the motion, reducing mesh distortion. The reliability of the proposed method is evaluated through comparisons with experimental data and existing numerical approaches. The computational efficiency is checked through parametric analyses on mesh discretization and accuracy in the prediction of the crack path and fracture variables. The results show how the proposed method could represent a valid tool to simulate the propagation mechanisms in FGM, in which heterogeneous macro-properties involve complex crack paths.

Modeling of crack propagation with the quasi-static material point method

Based on the theory of multiple displacement fields, the quasi-static MPM has been extended to simulate crack behavior in this paper. Compared with the existing crack simulation methods, this new method not only avoids the meshing difficulties and complex shape functions but also supports the modeling of crack growth in an arbitrary direction. The key fracture parameters, stress intensity factors, are calculated by the interaction integral method and proved accurate through the example of a cracked plate under uniaxial tension. The crack propagation is predicted with the maximum circumferential stress criterion and applied in the simulation of a compact tension test of clay. Based on this study, the proposed method is suggested as an effective scheme for modeling soil crack behaviors that are not limited to small deformation.

Three‐Dimensional Stress Intensity Factor Correction Method Based on Thickness Size Effect

This paper presents the fatigue crack growth test of 7N01 aluminum alloy specimens with different thicknesses under three‐point bending. Moreover, the effect of specimen thickness on the fatigue crack growth life was analyzed. The influence mechanism of the thickness size effect on the stress intensity factor was explored. A three‐dimensional stress intensity factor calculation model based on the thickness size effect was proposed by combining the finite element method and the interaction integral method. The results show that the fatigue crack growth life of 7N01 aluminum alloy decreases with the increase of specimen thickness. The numerical solutions of the stress intensity factor (the proposed model) are higher than the analytical solutions (the traditional formula) in the steady‐state crack growth stage. The difference between numerical solutions and analytical solutions increases with the increase of crack growth length and specimen thickness. Compared with the analytical solutions, the relationship curve between crack length “a” and cycle number “N” from the numerical solutions is closer to the experimental data. This result proves that the proposed calculation model can correct the stress intensity factor with the thickness size effect.

An effective modeling approach based on the ALE and M-integral for simulating crack propagation under thermo-mechanical loadings

Traditional engineering materials suffer from the presence of internal cracks, originating either during production processes or usage, that tend to grow under the action of thermo-mechanical loading, thus compromising the overall integrity. A proper evaluation of fracture behavior is then essential to limit dramatic failures that may lead to unsafe conditions. The present work proposes an effective modeling strategy for simulating crack propagation mechanisms in linear elastic continuum media subjected to mechanical and thermal loadings. The model involves a classic FE framework enhanced by Moving Mesh (MM) techniques and Interaction Integral Method (M-integral). MM serves as a powerful tool for reproducing geometry evolution caused by crack advance. In particular, the MM uses the Arbitrary Lagrangian-Eulerian (ALE) formulation, which is effective to handle random cracks. In the proposed framework, the ALE moves the computational nodes around the crack front, starting from a referential coordinate system according to classic fracture criteria, which define crack advance conditions and propagation direction. Reliable movements depend upon proper values of Stress Intensity Factors (SIFs) at crack fronts. To this end, the method adopts the M-integral for extracting mixed-mode SIFs. Since the crack front moves during the propagation, the M-integral is expressed by using the ALE formulation, thus evaluating fracture variables on deformed elements without losing accuracy. The use of proper rezoning techniques ensures the consistency of the mesh motion, limiting mesh distortions and remeshing events. The reliability of the proposed model is assessed through comparisons with existing numerical strategies.

Numerical Investigation on Mixed Mode (I-II) Fracture Propagation of CCBD Specimens Under Confining Pressure

A new numerical method, verified by the analytical solution of the weight functions and experimental paths, is developed to evaluate the crack initiation and propagation generally in mixed mode (I-II). This numerical method combining the interaction integral method and the maximum tangential stress (MTS) criterion is based on the finite element method of secondary development. The influence of combined confining pressure and diametric forces on crack propagation trajectories for CCBD specimens are studied. It is indicated that the crack propagation direction independent of the confining pressure keeps the same with the line of original crack as the loading angle is equal to [Formula: see text]. But when the loading angle is greater than [Formula: see text], the curvature of the curve trajectory in the early stage of crack propagation increases with a larger confining pressure. Further, it is found that larger values of the loading angle and relative length will make the effect of confining pressure more significant at the early stage of crack growth.

Thermal fracture analysis in orthotropic materials by XFEM

This paper discusses the thermal fracture analysis of anisotropic materials under thermal shock based on the classic theory of thermoelasticity in the framework of the Extended Finite Element Method (XFEM). A new set of tip enrichment functions for the temperature field in anisotropic materials is proposed. To this end, an analytical solution for the temperature distribution around an insulated crack in an anisotropic medium under uniform heat flux is extracted using the complex variable method. The tip enrichment functions are extracted using the near tip temperature distribution. Generally, the tip enrichment functions of temperature are similar to displacement ones, except that they are dependent on thermal properties. The Stress Intensity Factors (SIFs) are computed using the interaction integral method. The effect of the direction of material properties on the time variation of the temperature distribution in a cracked medium and SIFs is studied. Also, the crack propagation in an orthotropic material under thermal shock is investigated.

Effect of cladding thickness on brittle fracture prevention of the base wall of reactor pressure vessel

In case of a loss of coolant accident, the cooling water is injected into the downcomer of reactor pressure vessel (RPV) and produces a pressurized thermal shock (PTS). Considering that the current cladding thickness is small, thermal shock and neutron irradiation are easy to cause brittle fracture of base material. The transient stress intensity factors of the crack tips at the base-cladding interface are obtained by the interaction integral method. Since the test of the crack propagation on the cladding-base interface is quite difficult and dangerous, the allowable internal pressure values obtained from the K-T curves are verified by XFEM in the thermo-mechanical coupling fields. By increasing the cladding thickness, the resulting temperature fields are compared with each other to demonstrate the different embrittlement effects caused by thermal shock. Then the elastic-plastic fracture mechanics (EPFM) method and the linear elastic interaction integral method are used to study the structural integrity at relatively low and high nil-ductility reference temperatures respectively. Furthermore, the influence of the crack size on the ultimate bearing capacity of the structure is quantitatively analyzed for various thicknesses of cladding and base wall. The size effect of crack decreases obviously with the increase of cladding thickness.

Coupled numerical analysis for in-plane elastic T-stress evaluation in isotropic FG solids

This contribution establishes a coupled numerical method to show its effectiveness in the simulation of LEFM problems. The semi-meshless method uses the MLPG1 meshfree theory to discrete governing equations while geometry is modeled by simple CPS3 triangular cells. The cells help to propose an efficient neighboring method for meshfree interpolations. Three benchmark homogeneous cases are considered whose analytical solutions are available to carefully examine the accuracy of this method and validate the numerical implementation by comparing relative errors in terms of error norms, crack onset angle and also fracture parameters, i.e. KI, KII and T-stress that characterize the crack tip condition. Good computational accuracy is observed in this article. Three FG structures by simple linear, exponential and complex hyperbolic-tangent gradations are taken from the literature and investigated under thermo-mechanical conditions. A thermal barrier coating with an FG structure is also modeled. Then, fracture parameters and crack growth angles with and without considering T-stress are compared with FE solutions. Generalized maximum tangential stress (GMTS) criterion and interaction integral method are used to respectively compute crack growth angle and capture T.