Crack Tip Enrichment Functions Research Articles

Accurate finite element simulation of stresses for stationary dynamic cracks under impact loading

The numerical technique for wave propagation problems recently developed in our papers has been applied to the accurate modeling of stresses in the vicinity of crack tips and the dynamic stress intensity factor (DSIF) for stationary cracks. The numerical technique includes the linear finite elements with reduced dispersion as well as the two-stage time-integration approach that quantifies and filters spurious high-frequency oscillations. Several benchmark problems for stationary cracks at impact loadings have been solved. The accuracy of the stress calculation in the vicinity of the crack tips and the DSIF can be significantly increased by the application of the finite elements with reduced dispersion. Surprisingly, even without a special treatment of singularities at crack tips, the linear finite elements with reduced dispersion (with no crack tip enrichment functions) yield much more accurate results than the XFEM with the special crack tip enrichment functions on comparable meshes. It is also interesting to mention that for the calculation of the DSIF by the finite elements with reduced dispersion there is no necessity in the filtering stage at impact loading (the spurious oscillations in the DSIF are small and decrease with mesh refinement).

Numerical simulation of hydraulic fracturing in orthotropic formation based on the extended finite element method

Shale shows remarkable anisotropy in terms of mechanical properties, and it can often be regarded as an orthogonal anisotropic linear elastic body. In this paper, we present a two-dimensional fluid-solid coupled numerical model based on the extended finite element method (XFEM) to simulate the propagation of hydraulic fracture in orthotropic formations. The interaction between rock deformation and fluid flow within fracture is taken into account. Special crack tip enrichment functions of XFEM are used to model fractures in orthotropic formations, and the maximum circumferential tensile stress criterion is modified to determine the fracture propagation direction. The simulation results show that the hydraulic fracture will deviate from its straight-ahead path if there is an angle (defined as the material angle) between the initial fracture direction and the material axes of orthotropy. The initial deviation angle of fracture changes with the variation of the material angle, and the extent of fracture deviation increases with the Young’s Modulus ratio. The fracture deviation can be avoided only when the initial fracture is parallel to one of the material axes of orthotropy. In the case where the maximum and minimum horizontal stresses are not equal, the direction of fracture propagation is determined by the combined effects of in-situ stress and orthotropy of the material. These findings offer new insights into the hydraulic fracturing design in orthotropic formations, helping to improve the production rates in shale gas reservoirs.

Crack tip enrichment functions for extended finite element analysis of two-dimensional interface cracks in anisotropic magnetoelectroelastic bimaterials

In this paper, the extended finite element method (X-FEM) is employed to present a static fracture analysis of two-dimensional interfacial crack problems in linear magnetoelectroelastic (MEE) bimaterials. Magnetoelectrically impermeable crack-face boundary conditions are adopted and the multi-field coupled effect in MEE bodies is considered. In order to capture the oscillating singularity of the extended stresses near the interfacial crack tip, suitable crack tip enrichment functions for anisotropic and transversely isotropic MEE bimaterials are newly derived and further applied to perform X-FEM analysis. As the fracture parameter, the J-integral is evaluated using the domain form of the contour integral. By comparing yielded results with the analytical and numerical solutions of the corresponding interfacial crack problems, the validity of the proposed formulation is verified. Moreover, it is shown that the results obtained by way of the new enrichment functions are superior to those obtained by the fourfold enrichment functions and twelvefold enrichment functions, especially in the case of topological enrichment. If there is no special requirement for precision, the fourfold enrichment functions with less computational cost can also be used to conduct X-FEM analysis for the present problem.

Fatigue crack growth analysis of an interfacial crack in heterogeneous materials using homogenized XIGA

In this work, the fatigue life of an interfacial cracked plate is evaluated in the presence of flaws by homogenized extended isogeometric analysis (XIGA). In XIGA, the crack faces are modeled by discontinuous Heaviside jump functions, whereas the singularity in the stress field at the crack tip is modeled by crack tip enrichment functions. Holes and inclusions are modeled by Heaviside function and distance function respectively. The discontinuities are modeled in a selected region near the major crack whereas the region away from the crack is modeled by an equivalent homogeneous material. The stress intensity factors (SIFs) for the interface cracks are numerically evaluated using the domain based interaction integral approach. Paris law of fatigue crack growth is employed for computing the fatigue life of an interfacial cracked plate. The results show that the defects/ discontinuities away from the main crack barely influence the fatigue life.

Transient analysis of stationary interface cracks in orthotropic bi-materials using oscillatory crack tip enrichments

The problem of an interface crack between two orthotropic layers under dynamic loading is analyzed. Special crack tip enrichment functions are incorporated into the standard finite element shape functions to exactly reproduce oscillatory stress and displacement fields near the tip of the interface crack. Moreover, kinematics of displacement and its gradients across the crack face and material interface are also modeled by partition of unity enrichments. Special attention is given to extraction of stress intensity factors by utilizing a proper form of the interaction integral for orthotropic bi-materials. Advantages of this method of extracting stress intensity factors over the conventional displacement extrapolation technique are discussed. Several bi-material configurations with both vanishing and non-vanishing oscillatory indexes are solved using the interaction integral and the results are compared with the available data in the literature. Effects of employing oscillatory crack tip enrichments and validation of the path-independent J-integral are also discussed.

Numerical simulations of cracked plate using XIGA under different loads and boundary conditions

ABSTRACTThis article deals with the numerical simulation of cracked plate using extended isogeometric analysis (XIGA) under different loads and boundary conditions. The plate formulation is done using first-order shear deformation theory. The crack faces are modeled by the Heaviside function, whereas the singularity in stress field at the crack tip is modeled by crack tip enrichment functions. The stress intensity factors for the cracked plate are numerically computed using a domain-based interaction integral. The results obtained by XIGA for the center and edge crack plate are compared with extended finite element method and/or literature results for different types of loads and boundary conditions.

A Cell-Based Smoothed XFEM for Fracture in Piezoelectric Materials

This paper presents a cell-based smoothed extended finite element method (CS-XFEM) to analyze fractures in piezoelectric materials. The method, which combines the cell-based smoothed finite element method (CS-FEM) and the extended finite element method (XFEM), shows advantages of both methods. The crack tip enrichment functions are specially derived to represent the characteristics of the displacement field and electric field around the crack tip in piezoelectric materials. With the help of the smoothing technique, integrating the singular derivatives of the crack tip enrichment functions is avoided by transforming interior integration into boundary integration. This is a significant advantage over XFEM. Numerical examples are presented to highlight the accuracy of the proposed CS-XFEM with the analytical solutions and the XFEM results.

Fatigue crack growth in functionally graded material using homogenized XIGA

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.

Interfacial dynamic impermeable cracks analysis in dissimilar piezoelectric materials under coupled electromechanical loading with the extended finite element method

Interfacial dynamic impermeable cracks analysis of dissimilar piezoelectric solids under coupled electromechanical impact loadings by the extended finite element method (X-FEM) is presented. The dynamic X-FEM approach recently developed by the authors is further extended to analyze transient responses of interfacial impermeable cracks in dissimilar piezoelectric structures. The mechanical displacements and electrical potential are approximated by appropriate enrichment functions that are not only for the crack-face and crack-tips, but also for the materials interfaces. In this work, we develop a new set of electro-mechanical enrichment functions for interfacial crack-tip in piezoelectric bimaterials, which is modified from the pure mechanical twelve-fold interfacial crack-tip enrichment functions used for mechanical problems. Unlike the pure mechanical set, the new modified enrichment functions contain the electric-mechanical factor, which is determined from the electro-mechanical fields of interfacial cracks in piezoelectric bimaterials. We also present asymptotic crack-tip fields in piezoelectric bimaterials based on the Stroh’s formalism, and then apply them in the evaluation of generalized dynamic intensity factors (GDIFs) via the interaction integrals. The accuracy of the proposed approach is validated by comparing the obtained GDIFs with the ones derived from boundary element method. Numerical dynamic results of interfacial cracks in piezoelectric bimaterials are investigated and some aspects of the influences of poling angles, interface inclination, impact loading, etc. on the GDIFs are analyzed. There are many important issues have found such as the accuracy of the GDIFs obtained by the present formulation is high; a pure electrical impact immediately induces non-zero GDIFs, while the elastic waves caused by the mechanical impact need some time to reach and excite the crack; the effect of polarization on the GDIFs is significant; the behaviors of the GDIFs are complicated and their peak values decrease with increasing poling angles; the interface inclination angle of cracks greatly alters the GDIFs; and the interfacial crack growth may be delayed either by the effects of polarization or the electrical impacts.

The extended finite element method with new crack-tip enrichment functions for an interface crack between two dissimilar piezoelectric materials

Summary This paper studies the static fracture problems of an interface crack in linear piezoelectric bimaterial by means of the extended finite element method (X-FEM) with new crack-tip enrichment functions. In the X-FEM, crack modeling is facilitated by adding a discontinuous function and crack-tip asymptotic functions to the classical finite element approximation within the framework of the partition of unity. In this work, the coupled effects of an elastic field and an electric field in piezoelectricity are considered. Corresponding to the two classes of singularities of the aforementioned interface crack problem, namely, ϵ class and κ class, two classes of crack-tip enrichment functions are newly derived, and the former that exhibits oscillating feature at the crack tip is numerically investigated. Computation of the fracture parameter, i.e., the J-integral, using the domain form of the contour integral, is presented. Excellent accuracy of the proposed formulation is demonstrated on benchmark interface crack problems through comparisons with analytical solutions and numerical results obtained by the classical FEM. Moreover, it is shown that the geometrical enrichment combining the mesh with local refinement is substantially better in terms of accuracy and efficiency. Copyright © 2015 John Wiley & Sons, Ltd.

Numerical simulation of functionally graded cracked plates using NURBS based XIGA under different loads and boundary conditions

In this paper, extended isogeometric analysis (XIGA) is performed to simulate the cracked functionally graded material (FGM) plates using first order shear deformation theory (FSDT) under different types of loading and boundary conditions. The gradation in material properties of the plate vary exponentially along thickness direction. The top and bottom surfaces of the plate are composed of ceramic and metal respectively. The properties of the equivalent composite are evaluated by rule of mixtures. The crack faces are modeled using Heaviside function, whereas the singularity in the stress field at the crack tip is modeled by employing crack tip enrichment functions. The domain integral approach is used for evaluating the stress intensity factor. The results obtained by XIGA for FGM plate are compared with equivalent composite plate.

A new criterion for modeling multiple discontinuities passing through an element using XIGA

In this paper, a new criterion is proposed for the modeling multiple discontinuities i.e. crack, hole and inclusion passing through an element by XIGA. The modeling of multiple discontinuities passing through an element is done by imposing the additional degrees of freedom at the control points lying inside the influence of elements intersected by the discontinuities. In XIGA, the crack faces are modeled by discontinuous Heaviside jump functions, whereas the singularity in stress field at the crack tip is modeled by crack tip enrichment functions. The modeling of holes and inclusions is performed by Heaviside jump function and distance function respectively. The value of stress intensity factor is computed using domain form of interaction integral approach. Few static plane edge crack problems are analyzed in the presence of holes and inclusions to validate the proposed criterion. The results obtained by XIGA are compared with XFEM.

Thermo-mechanical fracture study of inhomogeneous cracked solids by the extended isogeometric analysis method

Abstract The extended isogeometric analysis method (XIGA) is further developed to study fracture of homogeneous and inhomogeneous materials under mechanical and thermo-mechanical loadings. The domain of the problem is discretized by the knot spans of the isogeometric analysis (IGA) and the same basis functions used for constructing the geometry are employed to discretize the solution. In addition, crack face discontinuity and tip enrichment functions of the extended finite element method (XFEM) are incorporated into the NURBS basis functions of IGA in order to reproduce crack face displacement discontinuity and tip singularity of the stress field. The proper form of the interaction integral method in inhomogeneous materials and thermal conditions is employed to evaluate the mixed mode stress intensity factors. In order to assess the efficacy of the proposed approach, a number of problems with different configurations and loadings are analyzed and the results are verified by comparing with the reference values.

T-spline based XIGA for fracture analysis of orthotropic media

Fracture analysis of orthotropic cracked media is investigated by applying the recently developed extended isogeometric analysis (XIGA) (Ghorashi et al., 2012) using the T-spline basis functions. The signed distance function and orthotropic crack tip enrichment functions are adopted for extrinsically enriching the conventional isogeometric analysis approximation for representation of strong discontinuity and reproducing the stress singular field around a crack tip, respectively. Moreover, by applying the T-spline basis functions, XIGA is further developed to make the local refinement feasible. For increasing the integration accuracy, the ’sub-triangle’ and ’almost polar’ techniques are adopted for the cut and crack tip elements, respectively. The interaction integral technique developed by Kim and Paulino (2003) is applied for computing the mixed mode stress intensity factors (SIFs). Finally, the proposed approach is applied for analysis of some cracked orthotropic problems and the mixed mode SIFs are compared with those of other methods available in the literature.

Enrichment strategies and convergence properties of the XFEM for hydraulic fracture problems

In two recent papers Gordeliy and Peirce (2013) investigating the use of the Extended Finite Element Method (XFEM) for modeling hydraulic fractures (HF), two classes of boundary value problem and two distinct enrichment types were identified as being essential components in constructing successful XFEM HF algorithms. In this paper we explore the accuracy and convergence properties of these boundary value formulations and enrichment strategies. In addition, we derive a novel set of crack-tip enrichment functions that enable the XFEM to model HF with the full range of power law rλ behavior of the displacement field and the corresponding rλ−1 singularity in the stress field, for 12≤λ<1. This novel crack-tip enrichment enables the XFEM to achieve the optimal convergence rate, which is not achieved by existing enrichment functions used for this range of power law. The two XFEM boundary value problem classes are as follows: (i) a Neumann to Dirichlet map in which the pressure applied to the crack faces is the specified boundary condition and the XFEM is used to solve for the corresponding crack width (P→W); and (ii) a mixed hybrid formulation of the XFEM that makes it possible to incorporate the singular behavior of the crack width in the fracture tip and uses a pressure boundary condition away from it (P&W). The two enrichment schemes considered are: (i) the XFEM-t scheme with full singular crack-tip enrichment and (ii) a simpler, more efficient, XFEM-s scheme in which the singular tip behavior is only imposed in a weak sense. If enrichment is applied to all the nodes of tip-enriched elements, then the resulting XFEM stiffness matrix is singular due to a linear dependence among the set of enrichment shape functions, which is a situation that also holds for the classic set of square-root enrichment functions. For the novel set of enrichment functions we show how to remove this rank deficiency by eliminating those enrichment shape functions associated with the null space of the stiffness matrix. Numerical experiments indicate that the XFEM-t scheme, with the new tip enrichment, achieves the optimal O(h2) convergence rate we expect of the underlying piece-wise linear FEM discretization, which is superior to the enrichment functions currently available in the literature for 12<λ<1. The XFEM-s scheme, with only signum enrichment to represent the crack geometry, achieves an O(h) convergence rate. It is also demonstrated that the standard P→W formulation, based on the variational principle of minimum potential energy, is not suitable for modeling hydraulic fractures in which the fluid and the fracture fronts coalesce, while the mixed hybrid P&W formulation based on the Hellinger–Reissner variational principle does not have this disadvantage.

Effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for GFEM/XFEM stress approximations around brittle cracks

The convergence rates of the conventional generalized/extended finite element method (GFEM/XFEM) in crack modeling are similar to the convergence rates of the finite element method (FEM) (Laborde et al., 2005 and Béchet et al., 2005) unless the crack tip enrichment functions are applied in a subdomain with fixed dimension, independently of the mesh parameter h and also demanding some special care for blending along the transition zone (Chahine et al., 2006 and Taracón et al., 2009). Thus, to improve convergence rates, more degrees of freedom (DOF) are generated due to the larger quantity of enriched nodes. This work seeks to identify and understand the advantages of better capturing the information provided by singular enrichments over mesh-based smooth partitions of unity (PoU). Such PoU with higher regularity can be built through the so-called Ck-GFEM framework, following Duarte et al. (2006), based on a moving least square of degree zero and considering mesh-based smooth weighting functions associated with arbitrary polygonal clouds. The purpose herein is to investigate some possible advantages of mesh-based smooth PoU for modeling discontinuities and singularities, in two-dimensional problems of linear elastic fracture mechanics, in such a fashion that the discretization error associated to stress discontinuities inherent in standard C0-continuous GFEM/XFEM approximations is eliminated. The procedure shares features similar to the standard FEM regarding domain partition and numerical integration but, as neither the PoU nor the enrichment functions are defined in natural domains, the integrations are performed using only global coordinates. The approximation capabilities of Ck-GFEM discretizations with different patterns of singular enrichment distribution are investigated by analyzing the convergence rates of the h and p versions, considering global measures in terms of strain energy and L2-norm of displacements. The effects on stability are also verified by analyzing the evolution of the condition number. The effect of smoothness on conditioning is investigated and the eigenvalues distributions are used to identify the several aspects involved: the smoothness of the PoU, the different types of enrichment functions and the pattern of enrichment. The performance of the smooth approximations is compared to the C0 counterparts built using conventional C0 FEM-based PoU. It is shown that smoothness in the presence of extrinsically applied singular enrichments is important, as the information provided by such enrichment is better captured. In addition, there are no stress jumps around singularities, reducing error propagation beyond the neighborhood of the singularity.

Fatigue crack growth analysis of a homogeneous plate in the presence of multiple defects using extended isogeometric analysis

In this work, extended Isogeometric analysis (XIGA) is successfully extended to evaluate the fatigue life of a homogenous finite plate in the presence of multiple defects (cracks, holes and inclusions) under cyclic loading condition. In isogeometric analysis, same basis functions, i.e. non uniform rational B-splines are used for defining the geometry and solution. In XIGA, the crack faces are modeled by discontinuous Heaviside jump functions, whereas the singularity in stress field at the crack tip is modeled by crack tip enrichment functions. The modeling of holes and inclusions is performed by jump function and distance function, respectively. These simulations show that the defects/discontinuities, distributed near to the main crack, have significant effect on the SIF values, whereas the defects/discontinuities away from the main crack have got very small effect on the SIFs.

A direct analytical method to extract mixed‐mode components of strain energy release rates from Irwin's integral using extended finite element method

SUMMARYA new analytical approach, within the extended finite element framework, is proposed to compute mixed‐mode components of strain energy release rates directly from Irwin's integral. Crack tip enrichment functions in extended FEM allow for evaluation of integral quantities in closed form (for some crack configurations studied) and therefore resulting in a simple and accurate method.Several benchmark examples on pure and mixed‐mode problems are studied. In particular, we analyze the effects of high‐order enrichments, mesh refinement, and the integration limits of Irwin's integral. The results indicate that high‐order enrichment functions have significant effect on the convergence, in particular when the integral limits are finite. When the integral limits tend to zero, simpler strain energy release rate expressions are obtained, and high‐order terms vanish. Nonetheless, these terms contribute indirectly via coefficients of first‐order terms.The numerical results show that high accuracy can be achieved with high‐order enrichment terms and mesh refinement. However, the effect of the integral limits remains an open question, with finite integration intervals chosen as h ∕ 2 tending to give more accurate results. Copyright © 2013 John Wiley &amp; Sons, Ltd.

An XFEM crack-tip enrichment for a crack terminating at a bi-material interface

Crack-tip enrichment functions are determined for a crack terminating at a bi-material interface. The stress and displacement components at the crack-tip are expressed considering plane elasticity solution based on Airy functions. The crack-tip characteristic exponents and the enrichment functions are determined using the problem boundary conditions. Single real, double real or complex characteristic exponents are found depending on the crack-angle and the relative mechanical properties of the considered bi-material. Crack-tip enrichment functions are determined for cases where the characteristic exponent is real and complex. To validate the efficiency of the enrichment functions at the crack-tip singularity, the eXtended Finite Element Method (XFEM) is used to solve a mode-I crack terminating at the bi-material interface. Thanks to two Level-set functions, the crack and the bi-material interface are located and the nodes belonging to elements cut by these discontinuities are enriched. The Stress Intensity Factors (SIF) are calculated for a crack perpendicular and terminating at the bi-material interface using the body force method (BFM). The implemented method was tested and good agreement is found when compared to other investigations in the literature.

Fatigue crack growth simulations of bi-material interfacial cracks under thermo-elastic loading by extended finite element method

In this paper, fatigue crack growth simulations of bi-material interfacial cracks have been performed using extended finite element method (XFEM) under thermo-elastic loading. The material discontinuity (interface) has been modelled by a signed distance function whereas a strong discontinuity (crack) has been modelled by two functions i.e. Heaviside and asymptotic crack tip enrichment functions. The values of stress intensity factors are extracted from the XFEM solution by domain based interaction integral approach. Standard Paris fatigue crack growth law is used for the life estimation of various model problems. The results obtained by XFEM for an interfacial edge and centre cracks are compared with those obtained by finite element method based on a remeshing approach.