Crack Tip Enrichment Functions Research Articles

Numerical Simulation of Plane Crack Problems Using Extended Isogeometric Analysis

This paper presents the simulation of plane crack problems using extended isogeometric analysis (XIGA). In XIGA, both geometry and solution are approximated using NURBS basis functions. Discontinuous Heaviside function is used to model the crack face, while crack tip singularity is modeled using asymptotic crack tip enrichment functions. Few plane crack problems are solved in the presence of multiple holes and inclusions using XIGA. These simulations show that the SIFs obtained using XIGA gives more accurate results as compared to those obtained by XFEM.

An edge-based smoothed XFEM for fracture in composite materials

In this work, an edge-based smoothed extended finite element method (ES-XFEM) is extended to fracture analysis in composite materials. This method, in which the edge-based smoothing technique is married with enrichment in XFEM, shows advantages of both the extended finite element method (XFEM) and the edge-based smoothed finite element method (ES-FEM). The crack tip enrichment functions are specially derived to represent the characteristic of the displacement field around the crack tip in composite materials. Due to the strain smoothing, the necessity of integrating the singular derivatives of the crack tip enrichment functions is eliminated by transforming area integration into path integration, which is an obvious advantage compared with XFEM. Two examples are presented to testify the accuracy and convergence rate of the ES-XFEM.

New anisotropic crack-tip enrichment functions for the extended finite element method

In this paper, the extended finite element method (X-FEM) is implemented to analyze fracture mechanics problems in elastic materials that exhibit general anisotropy. In the X-FEM, crack modeling is addressed by adding discontinuous enrichment functions to the standard FE polynomial approximation within the framework of partition of unity. In particular, the crack interior is represented by the Heaviside function, whereas the crack-tip is modeled by the so-called crack-tip enrichment functions. These functions have previously been obtained in the literature for isotropic, orthotropic, piezoelectric and magnetoelectroelastic materials. In the present work, the crack-tip functions are determined by means of the Stroh's formalism for fully anisotropic materials, thus providing a new set of enrichment functions in a concise and compact form. The proposed formulation is validated by comparing the obtained results with other analytical and numerical solutions. Convergence rates for both topological and geometrical enrichments are presented. Performance of the newly derived enrichment functions is studied, and comparisons are made to the well-known classical crack-tip functions for isotropic materials.

Numerical simulation of bi-material interfacial cracks using EFGM and XFEM

In this paper, bi-material interfacial cracks have been simulated using element free Galerkin method (EFGM) and extended finite element method (XFEM) under mode-I and mixed mode loading conditions. Few crack interaction problems of dissimilar layered materials are also simulated using extrinsic partition of unity enriched approach. Material discontinuity has been modeled by a signed distance function whereas strong discontinuity has been modeled by two functions i.e. Heaviside and asymptotic crack tip enrichment functions. The stress intensity factors for bi-material interface cracks are numerically evaluated using the modified domain form of interaction integral. The results obtained by EFGM and XFEM for bi-material edge and center cracks are compared with those available in literature. In order to check the validity of simulations, the results have been obtained for two different ratio of Young’s modulus.

Extended isogeometric analysis for simulation of stationary and propagating cracks

SUMMARYA novel approach based on a combination of isogeometric analysis (IGA) and extended FEM is presented for fracture analysis of structures. The extended isogeometric analysis is capable of an efficient analysis of general crack problems using nonuniform rational B‐splines as basis functions for both the solution field approximation and the geometric description, and it can reproduce crack tip singular fields and discontinuity across a crack. IGA has attracted a lot of interest for solving different types of engineering problems and is now further extended for the analysis of crack stability and propagation in two‐dimensional isotropic media. Concepts of the extended FEM are used in IGA to avoid the necessity of remeshing in crack propagation problems and to increase the solution accuracy around the crack tip. Crack discontinuity is represented by the Heaviside function and isotropic analytical displacement fields near a crack tip are reproduced by means of the crack tip enrichment functions. Also, the Lagrange multiplier method is used to impose essential boundary conditions. Moreover, the subtriangles technique is utilized for improving the accuracy of integration by the Gauss quadrature rule. Several two‐dimensional static and quasi‐static crack propagation problems are solved to demonstrate the efficiency of the proposed method and the results of mixed‐mode stress intensity factors are compared with analytical and extended FEM results. Copyright © 2011 John Wiley & Sons, Ltd.

Stress Intensity Factor for Interfacial Cracks in Bi-Materials Using Incompatible Numerical Manifold Method

Incompatible numerical manifold method (INMM) uses interpolation functions based on the concept of partition of unity, and considers the asymptotic solution and the discontinuity of displacement. This paper describes the application of INMM to bi-material interfacial crack. The two dimensional near-tip asymptotic displacement functions are added to the trial function approximation. This enables the domain to be modeled by manifold elements without explicitly meshing the crack surfaces. The crack-tip enrichment functions are chosen as those that span the asymptotic displacement fields for an interfacial crack. The INMM facilitates the incorporation of the oscillatory nature of the singularity within a conforming manifold element approximation. The complex stress intensity factors for bi-material interfacial cracks are numerically evaluated. Good agreement between the numerical results and the analytical solutions for benchmark interfacial crack problems is realized.

Tailored Extended Finite-Element Model for Predicting Crack Propagation and Fracture Properties within Idealized and Digital Cementitious Material Samples

This paper presents a tailored extended finite-element model (XFEM) to predict crack propagation and fracture properties within idealized and digital cementitious material samples. The microstructure of the idealized cement-based materials includes the cement paste, particles, and interfacial boundaries. The tailored XFEM was developed to allow crack propagation within finite elements by using discontinuous enrichment functions and level-set methods. The Heaviside jump and the elastic asymptotic crack-tip enrichment functions were used to account for the displacement discontinuity across the crack-surface and around the crack-tip. The maximum fracture energy release rate was used as a criterion for determining the crack growth. The shielding effects within the interfacial zone were addressed with a numerical search scheme. The tailored XFEM was implemented with a MATLAB program to simulate the compact tension (CT) and the single-edge notched Beam (SEB) tests. For a homogeneous CT testing sample, the XFEM prediction on stress intensity factors was verified with the fracture mechanics analysis. The idealized samples of cement-based materials were generated with varied microstructures, including particle locations, orientations, and shape factors. The tailored XFEM was applied to investigate the effects of these microparameters on the fracture patterns of the idealized samples under CT loading. The XFEM simulation was also conducted on a homogeneous offset-notched SEB sample to predict the mixed-mode crack propagation. The predicted crack path matches well with refined cohesion fracture modeling from a recent study. Further validation of the tailored XFEM was conducted with fracture simulation of a digital SEB sample generated from the actual tested specimen. The predicted crack path was favorably compared with the fracture pattern of the tested concrete specimen with a middle notch. These simulation results indicated that the tailored XFEM has the ability to accurately predict the crack propagation and fracture properties within idealized and digital cementitious material samples.

XFEM fracture analysis of shells: The effect of crack tip enrichments

In this study the effect of crack tip enrichment functions in the extended finite element analysis of shells is investigated. Utilization of crack tip enrichments leads to reduction of the required number of elements, mesh independency and increased accuracy in computation of fracture mechanics parameters such as the stress intensity factor, the crack tip opening displacement and the crack tip opening angle. The procedure is verified by modeling various shell and plate problems and available benchmark tests. Also, effects of enrichments of in-plane, out-of-plane and rotational degrees of freedom and high order out-of-plane enrichments on different fracture modes are studied. Moreover, reduction of the dependency of crack tip opening angle on the element size in crack propagation problems is discussed.

Modeling delamination in composite laminates using XFEM by new orthotropic enrichment functions

In this research, the extended finite element method (XFEM) is improved for modeling interfacial cracks between two orthotropic media by new set of orthotropic enrichment functions. New set of bimaterial orthotropic enrichment functions are developed and utilized in XFEM analysis of linear elastic fracture mechanics of layered composites. Interlaminar crack-tip enrichment functions are derived from analytical asymptotic displacement fields around a traction free interfacial crack. In this procedure, elements containing a crack tip or strong/weak discontinuities are not required to conform to those geometries. The domain interaction integral approach is also adopted in order to numerically evaluate the mixed-mode stress intensity factors. A number of benchmark tests are simulated to assess the performance of the proposed approach and the results are compared to available reference results.

Partition of unity enrichment for bimaterial interface cracks

AbstractPartition of unity enrichment techniques are developed for bimaterial interface cracks. A discontinuous function and the two‐dimensional near‐tip asymptotic displacement functions are added to the finite element approximation using the framework of partition of unity. This enables the domain to be modelled by finite elements without explicitly meshing the crack surfaces. The crack‐tip enrichment functions are chosen as those that span the asymptotic displacement fields for an interfacial crack. The concept of partition of unity facilitates the incorporation of the oscillatory nature of the singularity within a conforming finite element approximation. The mixed‐mode (complex) stress intensity factors for bimaterial interfacial cracks are numerically evaluated using the domain form of the interaction integral. Good agreement between the numerical results and the reference solutions for benchmark interfacial crack problems is realized. Copyright © 2004 John Wiley & Sons, Ltd.

Vector CTD criterion applied to mixed-mode fatigue crack growth: Li, C. Fatigue Fract. Eng. Mater. Struct. 1989 12, (1), 59–65

In this work, the effect of defects/flaws (holes, inclusions, cracks) on the fatigue life of functionally graded material (FGM) is analyzed by homogenized extended isogeometric analysis (XIGA). In FGM, the gradation in the material properties is taken along the length of the plate. In XIGA, the crack faces are modeled by discontinuous Heaviside jump function, whereas the singularity in stress field at the crack tip is modeled by crack tip enrichment functions. Holes and inclusions are modeled by Heaviside jump function and distance function, respectively. The values of stress intensity factor (SIF) are numerically evaluated using the domain form of interaction integral approach. Paris law of fatigue crack growth is employed for computing the fatigue life. The flaws are modeled in a 30% region near the main crack, while the rest of the region is modeled with an equivalent homogeneous material. Several problems involving discontinuities in 30% region of the domain are solved by XIGA, and the results are compared with those obtained by modeling discontinuities in the entire domain of the plate.