Interaction Integral Approach Research Articles

A combined displacement discontinuity-interaction integral method for computing stress intensity factors and T-stress

In this paper, the displacement discontinuity method (DDM) is combined with interaction integral to simultaneously evaluate stress intensity factors and T-stress for analyzing two-dimensional mixed-mode crack problems. The displacement discontinuity method has been proved to be one of the most efficient boundary element methods for solving crack problems with boundary-only discretization character. However, there is a lack of efficient methods collaborating with the displacement discontinuity method to evaluate fracture parameters. The available geometrical extrapolation method and J-integral technique for calculating fracture parameters in combination with the displacement discontinuity method are imprecise and difficult to be implemented into mixed-mode crack analysis. The present combined displacement discontinuity method and interaction integral approach can easily extract stress intensity factors and T-stress for mixed-mode crack problems without needing any decomposition of elastic field into symmetric and antisymmetric components. The fundamental basis lies in introducing auxiliary fields for the proper defined single mode crack, then calculating fracture parameters by evaluating the formulated interaction integral in terms of displacement discontinuity solutions and auxiliary elastic fields. In this paper, the basis of the displacement discontinuity method is illustrated firstly, then the explicit expressions for auxiliary fields of displacement gradients are derived. Next, the interaction integral can be determined by being converted to an equivalent area integral. Finally, several numerical examples are examined to demonstrate the correctness of the present method.

Fracture mechanics approach to TRISO fuel particle failure analysis

Weibull stress-based methods for failure probability assessment have been developed and analyzed to assess the integrity of tristructural isotropic (TRISO) fuel particles during fuel life cycles and accident operating conditions. While simple, these methods entail a number of drawbacks when stress concentrates near crack tips, including finite element mesh size dependency when the Weibull stress is averaged over the finite element domain. Fracture mechanics approaches involving the use of interaction integrals eliminate this lack of mesh convergence and produce consistent fracture predictions. In this work, we use an interaction integral approach to computing stress intensity factors for a crack in the inner pyrolytic carbon layer perpendicular to the silicon carbide layer, which is simplified representation of a failure mode in TRISO particles. The interface between these two TRISO layers has been shown to become porous, which we simulate by considering a transition of mechanical properties over such porous length, i.e. the layers are modeled as a functionally graded material. Aspects such as porosity and thermal and irradiation eigenstrains are considered in computing the stress intensity factor from a fracture mechanics approach and compared with the known Weibull stress failure approach. The methodology introduced in this paper enables a more general fracture probability assessment in TRISO particles and eliminates the need to identify best suited parameters when using local or averaged stress-based failure criterion. Finally, the numerical sensitivity studies show how parameters such as the porous transition zone length, the material stiffness, and creep affect the probability of TRISO fuel particle failure.

Analysis of cracked Reissner-Mindlin plate using an extended meshfree method

An extended meshfree method for analyzing cracked plates based on Reissner-Mindlin theory is presented in this paper. Among a variety of meshfree formulations, the radial point interpolation method (RPIM) is chosen in this study due to the satisfaction of the Kronecker delta property. The essential boundary conditions, therefore, are easily imposed in the RPIM. The shape function derived from RPIM is employed to interpolate the field variables. An extended RPIM formulation is used to model the crack segment without explicitly defining it in the discretized domain. The discontinuity due to the crack is defined by extrinsic enriched functions, particularly, the jump in the displacement field on two sides of the crack is modelled by the Heaviside function, and the stress singularity near the crack tip is described by the asymptotic enriched function. In this study, the stress resultant intensity factors (SRIFs) are evaluated through the interaction integral approach. The obtained SRIFs are shown in the paper through many numerical examples for comparison purposes. The trending variation of SRIFs is also observed from the numerical results. It can be remarked that the SRIFs depend on many factors: the number of cracks, crack orientation, load type and boundary conditions. The numerical examples show the accuracy of the present approach. The obtained results are compared with analytical solutions and other numerical methods.

Interaction integral approach for the Evaluation of Crack-front Singularities in a Solid Cantilever Beam under Magneto-Electro-Elastic loading using XFEM

In the piezoelectric energy harvesting devices, the most preferred is the cantilever beam orientation that can produce maximum deflection. Therefore, in this paper, three-dimensional fracture analysis of a cracked magneto-electro-elastic cantilever beam is studied using extended finite element methods (XFEM). The crack-front singularities like stress intensity factors (SIFs), electric displacement intensity factor (EDIF), and magnetic induction intensity factor (MIIF) are evaluated using the MEE interaction integral approach. The contour J-integral in domain form is solved with the asymptotic crack front fields as auxiliary fields using the Stroh formalism. The numerical results showed good agreement with the benchmark problem. Few case studies for the crack front singularities are presented using the proposed method. The results provided give useful guidance in the designing and manufacturing of smart devices.

Fracture analysis of functionally graded material reinforced with MWCNT

Computational model based on extended isogeometric analysis is implemented for the fracture analysis of functionally graded material (FGM) reinforced with multi-walled carbon nanotube (MWCNT), i.e., FGM-MWCNT structure under mechanical and thermo-mechanical loading conditions. The FGM body taken for the study is composed of metal (Ti–6Al–4V) on the left side and ceramic (ZiO2) on right side. For the gradation of FGM properties that change throughout the domain length, exponential law is assumed. Moreover, the FGM is reinforced with 0–5% of MWCNT to obtain FGM-MWCNT composite. In order to perform this fracture analysis, first the equivalent mechanical and thermal properties of the FGM-MWCNT composite are evaluated using some of the micromechanics models cited in the literature. Using the interaction integral approach, SIFs are extracted. The results of the test show that when the volume percentage of MWCNTs increases, it enhances the failure resistance of the composite.

Three dimensional extended finite element simulation of cracked functionally graded pipe and pipe bend

In this work, extended finite element method (XFEM) is used to simulate the fracture mechanics problems of part-through semi-elliptical circumferential crack in functionally graded pipe and pipe bend. Part-through semi-elliptical circumferential crack present in functionally graded pipe and pipe bend at different location has been subjected to internal pressure. Higher order shape function is used to model the crack with level set functions. The inner surface of the functionally graded pipe/pipe bend contains 100% steel alloy and outer surface contains 100% alumina ceramic. The material property gradation is followed by exponential law from inner surface to outer surface of the pipe and pipe bend in radial direction. Domain-based interaction integral approach has been used to determine the stress intensity factor at different location of the crack front.

A new framework based on XFEM to study the role of electrostatic tractions in semipermeable piezoelectric material

In this work, a crack opening model is proposed to study the role of electrostatic tractions in a cracked semipermeable piezoelectric material. The extended finite element method (XFEM) is used in conjunction with six-fold electro-mechanical enrichment functions to capture the singularity and the discontinuous surface independent of the background mesh. The electric displacement intensity factor (EDIF) is evaluated using an electromechanical interaction integral approach in the presence of a dielectric medium inside the crack cavity. The accuracy of the proposed XFEM based approach is verified by comparing the numerical results with available analytical solutions. Further, a systematic numerical study is done to study the influence of electrostatic tractions on the EDIF using different crack configurations under far-field loading.

Smoothed floating node method for modelling 2D arbitrary crack propagation problems

In this work, Floating Node Method (FNM), first developed for fracture modelling of laminate composites, is coupled with cell-wise strain Smoothed Finite Element Method (SFEM) for modelling 2D linear elastic fracture mechanics problems. The proposed method is termed as Smoothed Floating Node Method (SFNM). In this framework, FNM is used to represent the kinematics of crack and the crack front inside the domain without the requirement of remeshing and discontinuous enrichment functions during crack growth. For smoothing, a constant smoothing function is considered over the smoothing domains through which classical domain integration changes to line integration along each boundary of the smoothing cell, hence derivative of shape functions are not required in the computation of the field gradients. The values of stress intensity factor are obtained from the SFNM solution using domain based interaction integral approach. Few standard fracture mechanics problems are considered to check the accuracy and effectiveness of the proposed method. The predictions obtained with the proposed framework improves the convergence and accuracy of the results in terms of the stress intensity factors and energy norms.

Thermo-elastic analysis of cracked functionally graded materials using XIGA

The thermo-elastic analysis of thermally induced crack in functionally graded materials (FGMs) is performed using extended isogeometric analysis (XIGA). For the study, two types of cracks, i.e., adiabatic and isothermal, are considered in the FGM body. The suitable enrichment functions are employed to capture the discontinuity (i.e., displacements and temperature) along the crack face and singular fields around the crack tip. The material property gradation in FGM (composed of aluminum and ceramic) is assumed to be governed by the exponential law. The domain-based interaction integral approach evaluates the stress intensity factors (SIFs) at the crack tips. Several numerical experiments are performed to verify the accuracy of XIGA for the FGM body under the action of the thermo-elastic load. Further, the effect of defects (i.e., voids and inclusions) is examined on the SIFs of the FGM body. Moreover, the impact of different material properties of inclusion on the SIFs is also investigated.

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.

Investigation of fatigue crack growth in engineering components containing different types of material irregularities by XFEM

The extended finite element (XFEM) has been applied to investigate fatigue crack growth in engineering components containing different types of material irregularities. The standard approximations of finite element method are enriched with additional functions by employing the partition of unity approach. In order to keep track of discontinuities in the domain, level set method is employed. Fatigue crack growth in presence of different material irregularities, such as elliptical, rectangular, hexagonal and octagonal discontinuities, has been the main focus of the study. The effect of geometry, position and material properties of these discontinuities on fatigue behavior of the cracked specimen has been addressed. The fatigue life of the cracked component has been estimated by the generalized Paris law. The mixed mode stress intensity factors have been evaluated by the domain-based interaction integral approach. Several numerical problems are solved by XFEM to study the fatigue crack growth behavior of cracked specimens containing different types of material irregularities.

Stress intensity factor computation of inclined cracked tension plate using XFEM

One of the major successes in the field of Linear Elastic Fracture Mechanics (LEFM) is the groundwork of the stress intensity factor (SIF) computation. The approaches used to carry out SIF values may be analytical, semi-analytical, experimental or numerical. Each one of the above methods has its own benefits however the use of numerical solutions has become the most frequent and popular. Numerous schemes for the numerical computation of SIF have been developed, the J-integral method being the most popular one. In this article we examine the SIFs of an edge cracked two dimensional (2-D) steel plate subjected to tensile loading. Extended finite element (XFEM) computational scheme has been employed to estimate the values of SIF. The SIF values of cracks with different lengths and inclination angles (different configurations) have been examined by utilizing the domain based interaction integral approach. The effect of crack inclination and crack position on SIFs (KI and KII) has also been studied. The results obtained in this study were compared with those from literature and theoretical values and observed that they are in close agreement.

A thermo-mechanical fracture analysis of linear elastic materials using XIGA

In this work, the extended isogeometric analysis (XIGA) is presented for the analysis of cracks in elastic material subjected to thermal, mechanical, and thermo-mechanical loading. The comparative study on the effect of mechanical and combined thermo-mechanical load on stress intensity factors (SIFs) is performed. Two types of cracks, i.e., isothermal and adiabatic crack, is considered for the simulation. The crack is modeled using suitable enrichment functions. A modified domain-based interaction integral approach is used to extract the SIFs. Several simulations are performed to validate the accuracy of the results. Moreover, the effect of discontinuities (e.g., voids) on SIFs is analyzed.

Effect of parametric uncertainties on fracture behavior of cortical bone using XIGA

Bone tissues are heterogeneous composites that consist of various microstructural constituents at different length scales. These microstructural constituents and their heterogeneous distribution significantly affect the fracture behavior of bones. Thus, the effect of parametric uncertainties on the fracture analysis of a cortical bone is presented in this study. A 2D model of the cortical bone is generated with the help of micro-CT image of a cortical bone and the fracture analysis is performed for the developed model with the help of an Extended Isogeometric Analysis (XIGA) using linear elastic fracture mechanics. The values of stress intensity factor are calculated with the help of interaction integral approach and the direction of crack propagation after each step is evaluated using the maximum principle stress criterion. The obtained results are compared with the finite element results using Abaqus software and found in good agreement. Further, uncertainties in the values of osteon Young’s modulus, cement line thickness and porosity percentage are taken into consideration for stochastic analysis. The effect of variation in the values of input parameters on the stress intensity factor values and the crack path trajectories is illustrated, and observed that the variation in osteon Young’s modulus and porosity percentage is more pronounced than the cement line thickness.

A polygonal finite element approach for fatigue crack growth analysis of interfacial cracks

The fatigue crack growth analysis of an interfacial cracked structure plays an essential role in the design process as well as the safety assessment for various engineering fields in real life. This paper proposes a novel and effective computational approach based on polygonal meshes for crack growth simulation of interfacial crack problems. Polygonal meshes are fully exploited to provide greater flexibility in mesh generation for complicated domains as well as obtain more accuracy in terms of the numerical results. The twelve crack tip enrichment functions to capture the near-tip oscillating singular field are first implemented for polygonal meshes to improve the obtained results. The Wachspress shape function is applied for arbitrary polygonal elements. The strong discontinuities e.g. crack face, crack tip can be modeled, respectively, via Heaviside and asymptotic crack tip enrichment functions while the material interface discontinuities are defined by signed distance functions. The stress intensity factors (SIFs) of the interfacial crack problems can obtain by using the domain based interaction integral approach while the fatigue life is estimated via the Paris law. The accuracy and efficiency of the present approach are demonstrated through several numerical investigations regarding the crack growth problems in both homogeneous materials and bi-materials. The results which are generated from the proposed approach are compared with other earlier ones as well as the conventional XFEM solutions.

A semi-homogenized extended isogeometric analysis approach for fracture in functionally graded materials containing discontinuities

In this paper, we present a semi-homogenized extended isogeometric analysis approach for assessment of stress intensity factor of center cracked functionally graded domain containing discontinuities such as holes and inclusions. In the semi-homogenized approach, the presence of discontinuities is considered in the 30% region in the vicinity of crack, and the remaining part of the domain is modeled using equivalent homogenous material properties. The Heaviside and singular crack tip enrichment functions are employed to model the center crack, whereas crack topology is tracked using level set functions. The domain form of interaction integral approach is employed to estimate the stress intensity factors at the tips of the center crack. Several problems containing discontinuities in 30% portion of the domain are investigated using the semi-homogenized approach. The efficiency and accuracy of the developed method are examined by comparing the computed results with those attained by modeling discontinuities in the entire domain. The advantages of the semi-homogenized extended isogeometric analysis approach are highlighted in terms of higher computational efficiency.

Crack growth propagation modeling based on moving mesh method and interaction integral approach

This study presents a novel modeling strategy for predicting crack propagation phenomena in linear elastic continuum media under external loading conditions. The model combines the Moving Mesh method with Interaction integral approach in a FE framework. The former is used to reproduce the changes in the geometry caused by crack evolution, whereas the latter is adopted to evaluate the stress intensity factors (SIFs) to predict crack growth. The computational nodes are moved starting from a fixed referential coordinate system on the basis of a crack growth criterion, which predicts the direction of crack propagation at tip front. The mesh frame is changed, ensuring limited distortions of the elements by using a mesh regularization technique based on proper rezoning equations. This strategy also reduces re-meshing processes, which frequently occur in standard crack propagation FE procedures. The validity of the proposed model is verified through comparisons with existing experimental data and other advanced numerical approaches.

Modeling of embedded and edge cracks in steel alloys by XFEM

This work aims at the modelling and simulating of the embedded and edge cracks in steel alloys by employing extended finite element method. Unlike FEM where the crack growth modelling is cumbersome due to the conformal meshing and mesh refinement, this technique allows modelling of all types of discontinuities independent of the mesh. Appropriate functions called enrichment functions are included to the standard FEM approximations so as to accommodate the influence of different discontinuities present in the domain. The level set method has been employed to keep track of discontinuities. The stress intensity factors have been obtained by employing the domain based interaction integral approach. Accurate and effective MATLAB codes have been developed on XFEM to model various types of cracks in engineering materials. Finally some numerical problems have been solved by the extended finite element method to reveal the capabilities of the proposed approach in modelling fracture mechanics problems.

Failure analysis of orthotropic composite material under thermo-elastic loading by XFEA

Present work describes the crack growth characteristics in orthotropic composite materials by utilizing extended finite element analysis under combined thermal–mechanical loading conditions. In combined system, temperature and displacement approximations are extrinsically governed by additional functions to define discontinuity profile across crack surface. In present work, two step simulation methodologies are adopted where thermal discrete equations are solved for thermo-elastic problems to obtain temperature distribution in domain. Thereafter, these results acts as load input criterion for elastic discrete equations. Post process of simulation, SIFs at crack tips are calculated using interaction integral approach for few orthotropic composites with preexisted crack.

Numerical Simulation of Tri-layer Interface Cracks in Piezoelectric Materials Using Extended Finite Element Method

Piezoelectric materials have wide applications due to their special electromechanical coupling characteristics but are vulnerable to sudden failure because of their brittle nature. Researchers have been profoundly interested in investigating the fracture of piezoelectric materials under different applied conditions. In the present work, tri-layer plates with interface cracks in piezoelectric materials have been analyzed using XFEM. Impermeable static cracks at the interfaces of PZT-5H and PZT-4 have been investigated under mechanical and electromechanical loading conditions. Numerical results show that the ɛ (oscillating) singularity vanishes and $$\kappa$$ class singularity exists for the current combination of materials. Enrichment functions for $$\kappa$$ class are applied to obtain the stress and electric fields near the crack tip. An interaction integral approach is used to evaluate fracture parameters. An effect on stress and electric displacement intensity factors due to variation in mechanical and electric loading has also been investigated.