Asymptotic Crack-tip Fields Research Articles

Analysis of cracked functionally graded material plates using XIGA based on generalized higher-order shear deformation theory

A generalized higher-order shear deformation theory (GHSDT) along with extended isogeometric analysis (XIGA) is presented for the analysis of cracked functionally graded material (FGM) plates. The material properties of the FGM vary exponentially along the thickness of the plate. The GHSDT ensures a traction-free boundary conditions at the surfaces of the plate without a need of shear correction factor (SCF). To model a crack, the isogeometric analysis (IGA) approximation is enriched with partition of unity (PU) enrichment functions. The C1 continuity requirement of the GHSDT is easily achieved in IGA through Non-uniform rational B-splines (NURBS). The stress intensity factor (SIF) at any location along the thickness of the plate is computed using domain based interaction integral approach. In the interaction integral, 2-D asymptotic crack-tip fields obtained from the LEFM provide a straightforward and simple solution scheme for SIF along the thickness of the FGM plate. The accuracy of the proposed approach is demonstrated by solving various problems. The effect of various parameters such as loading, boundary conditions, crack length and plate thickness is investigated on the SIF. The solutions of several cracked FGM plate problems obtained by GHSDT based XIGA (GHSDT-XIGA) are compared with FSDT based XIGA (FSDT-XIGA) and reference solutions.

Asymptotic Crack Tip Fields in Linear and Nonlinear Materials and Their Role in Crack Propagation

The famous Wieghardt, Griffith and Irwin criteria predict the onset of fracture in linear elastic materials. They have to be supplemented by appropriate criteria for predicting the path that the fracture will follow until the failure of the structure. These require the knowledge of the stress and displacement fields at the front of propagating fracture which depend on the actual loading on the structure and its boundary conditions. In this paper we shall review these fields in brittle and quasi-brittle materials. In the latter materials, a traction-free fracture front often has a large process zone ahead of it in which the material experiences progressive softening. Such a mixed traction-free and process zone is also called a cohesive crack. Over the process zone the material is able to transfer some tractions across the crack faces depending upon how much the faces have separated or slid relative to each other. In the famous Barenblatt model the process zone was very small in comparison with the traction-free crack so that the actual traction-separation relationship in the process zone was not explicitly involved. However, in real quasi-brittle materials the size of the process zone can be commensurate or even larger than the traction-free crack. It is therefore necessary to know this relationship explicitly in order to determine the corresponding stress and displacement fields at the front of the propagating cohesive crack. The asymptotic fields at the front of a crack in brittle materials were obtained by Williams and those for quasi-brittle materials by Xiao and Karihaloo.

Analysis of Semipermeable Crack Growth in Piezoelectric Materials Using Extended Finite Element Method

Piezoelectric materials possess special characteristics of electromechanical coupling behavior and thus have found numerous applications such as transducers, sensors, actuators. Fracture of piezoelectric materials has drawn substantial attention of the research community and is being widely investigated for predicting their failure. Most of the research on piezoelectric materials is based on impermeable crack conditions. In the present study semi-permeable crack boundary conditions has been analyzed using the extended finite element method (XFEM). Combined Mechanical and Electrical loading with quasi-static crack growth has been considered on a pre-cracked rectangular plate with crack at its edge and center. Stress intensity factors have been evaluated by interaction integral approach using the asymptotic crack tip fields. Effect of presence of minor cracks and holes have been analyzed on the intensity factors of semi-permeable major crack.

Creep crack analysis of viscoelastic material by numerical manifold method

In order to gain an insight into creep induced cracking in viscoelastic materials, the creep crack phenomenon in viscoelastic materials are numerically investigated in this paper based on the numerical manifold method (NMM). To more realistically capture the mechanical behaviour in the vicinity of crack tips, the local displacements around crack tips are enriched with the viscoelastic crack-tip asymptotic displacement field. To better represent the viscoelastic materials, an incremental viscoelastic constitutive law based on the generalized Kelvin-Voigt model is incorporated into the NMM. To achieve these treatments in the NMM, an incremental viscoelastic NMM formulation is derived. In addition, an incremental formulation of the mixed mode viscoelastic energy release rate (ERR), which can be tackled in the same process with the incremental viscoelastic constitutive law, is derived and incorporated into the NMM for investigating the variations of the ERR in viscoelastic materials. Based on the developed method, three creep crack problems with different loading conditions and pre-existing crack configurations are simulated, and the crack opening and sliding displacements as well as corresponding mixed mode ERR are analysed. Simulation results show good agreement with the theoretical solutions, which indicate that the proposed NMM is suitable for complex creep crack problems.

Numerical simulation of crack propagation under fatigue loading in piezoelectric material using extended finite element method

Piezoelectric materials due to their electromechanical coupling characteristics are being widely used in actuators, sensor, transducers, etc. Considering wide application it is essential to accurately predict their fatigue and fracture under applied loading conditions. The present study deals with analysis of fatigue crack growth in piezoelectric material using the extended finite element method (XFEM). A pre-cracked rectangular plate with crack at its edge and center impermeable crack-face boundary conditions is considered for simulation. Fatigue crack growth is simulated using extended finite element method under plane strain condition and mechanical, combined (mechanical and electrical) cyclic loading. Stress intensity factors for mechanical and combined (mechanical and electrical cyclic loadings) have been evaluated by interaction integral approach using the asymptotic crack tip fields. Crack propagation criteria have been applied to predict propagation and finally the failure.

Extended radial point interpolation method for dynamic crack analysis in functionally graded materials

Functionally graded materials (FGMs) have been widely used as advanced materials characterized by variation in properties as the dimension varies. Studies on their physical responses under in-serve or external loading conditions are necessary. Especially, crack behavior analysis for these advanced material is one of the most essential in engineering. In this present, an extended meshfree radial point interpolation method (RPIM) is applied for calculating static and dynamic stress intensity factors (SIFs) in functionally graded materials. Typical advantages of RPIM shape function are the satisfactions of the Kronecker’s delta property and the high-order continuity. To assess the static and dynamic stress intensity factors, non-homogeneous form of interaction integral with the nonhomogeneous asymptotic near crack tip fields is used. Several benchmark examples in 2D crack problem are performed such as static and dynamic crack parameters calculation. The obtained results are compared with other existing solutions to illustrate the correction of the presented approach.

Estimation of the crack propagation direction in a mixed-mode geometry via multi-parameter fracture criteria

The presented work introduces a numerical parametric study on the crack propagation direction under mixed-mode conditions (mode I + II). It is conducted for the geometry of an eccentric asymmetric fourpoint bending of a single edge notched beam specimen; various levels of mode-mixity are ensured by modifications in the crack length and crack eccentricity. The direction of crack propagation is estimated semianalytically using both the maximum tangential stress criterion and the strain energy density criterion (implemented as a procedure within the used finite element computational code) as well as numerically (from verification reasons). Multi-parameter fracture mechanics is employed in the presented work for precise analytical evaluation of the stress field in the cracked specimen. This theory is based on description of the stress and deformation fields in the cracked body by means of their approximation using several initial terms of the Williams power series. Recent studies show that utilization of only first term of the series, which corresponds to the stress intensity factor (SIF), the single controlling parameter for the crack initiation and propagation assessment in brittle materials, is insufficient in many crack problems. It appears also in this study that the higher-order terms of the asymptotic crack-tip field are of great relevance for the conducted analysis, similarly to a number of other fracture phenomena (near-crack-tip stress field approximation, non-linear zone extent estimation, etc.).

Mapping Static and Dynamic Crack-Tip Deformations Using Reflection-Mode Digital Gradient Sensing: Applications to Mode-I and Mixed-Mode Fracture

The reflection-mode digital gradient sensing (r-DGS) method is extended for visualizing and quantifying crack-tip deformations in solids under quasi-static and dynamic loading conditions. The r-DGS technique employs digital image correlation principles to quantify two orthogonal surface slopes simultaneously in specularly reflective solids by measuring small deflections of light rays. Here, for the first time, r-DGS is implemented to study both mode-I and mixed-mode (I/II) problems and evaluate fracture parameters. Under dynamic loading conditions, r-DGS is employed in conjunction with high-speed digital photography to map surface slopes in edge cracked plates subjected to one-point impact. The measured surface slopes are used to successfully evaluate instantaneous stress intensity factor histories by combining measurements with the corresponding asymptotic crack-tip fields and performing over-deterministic least-squares analyses. Finite element computations are also used to complement experimental measurements as needed.

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.

Computation of Coefficients of Crack-Tip Asymptotic Fields Using the Weak Form Quadrature Element Method

AbstractCoefficients of crack-tip asymptotic fields are computed using a recently developed weak form quadrature element method (QEM), combined with the subregion generalized variational principle. The variational description of a crack is established by dividing the domain into two regions, the potential energy region and the complementary energy region. Then the weak form QEM is employed to derive a system of algebraic equations. The coefficients are extracted directly from solving the equations. The accuracy, efficiency, and parameter sensitivity of the proposed method are discussed by solving a number of benchmark examples. The computed results are in very good agreement with available analytical or numerical results. The involved parameters can be adjusted according to convergence requirements. Thus, the method enjoys the advantages of straightforwardness and self-adaptivity.

The modified version of strain gradient and couple stress theories in general curvilinear coordinates

Abstract It is well-known that conventional continuum theory is not capable of capturing size effects and therefore, many theories have been proposed. Among these theories, strain gradient and modified strain gradient theories have enjoyed great success so far. In the present study, the modified strain gradient theory is in details derived in general curvilinear coordinates. The modified strain gradient theory involves the modified couple stress theory as a special case and therefore, the modified couple stress theory in curvilinear coordinates is readily obtained. The results are then specialized for two practical orthogonal curvilinear coordinates, i.e. cylindrical and spherical coordinates. The expressions including deformation measures, governing equations, boundary conditions and constitutive equations are expressed in terms of physical components which are more common and convenient. The formulations presented here are general and may be appropriately applied to a broad range of problems in which the use of curvilinear coordinates is unavoidable, such as problems of cylindrical and spherical cavity expansion, the analysis of asymptotic crack tip field and the interpretation of micro/nano indentation tests and also bending or twisting tests on small scales.

A domain-independent interaction integral for linear elastic fracture analysis of micropolar materials

This paper establishes a domain-independent interaction integral (DII-integral) for linear elastic fracture mechanics of micropolar elastic solids. The DII-integral has three amazing features that make it effective for solving the fracture parameters of complex micropolar materials. The first one is that the DII-integral can decouple the stress intensity factors (SIFs) and couple stress intensity factors (CSIFs) both of which are the key fracture parameters charactering the crack-tip asymptotic singular fields. In details, the DII-integral is derived from the J-integral by superimposing an actual field and an auxiliary field. By assigning the fracture parameters in the auxiliary field with different values, the SIFs and CSIFs of different crack opening modes can be obtained separately through the DII-integral. The second important feature is that the DII-integral is domain-independent for material nonhomogeneity and discontinuity. Thanks to this feature, the DII-integral becomes extremely effective for the micropolar materials with arbitrary nonhomogeneous properties or complex interfaces. The third feature is that the DII-integral does not contain any derivatives of material properties, which feature facilitate the practical implementation of the DII-integral on complex micropolar materials. Finally, the DII-integral combined with the extended finite element method (XFEM) is employed to solve four representative crack problems and the results show good validity of the DII-integral for complex micropolar materials.

Dynamic stress intensity factors for suddenly loaded structures using enriched finite elements

In this paper, an efficient finite element formulation for stationary cracks subjected to dynamic impact loading is presented. For impact problems, where wave propagation effects dominate, the onset of rapid crack growth is strongly influenced by inertia effects, including stress wave reflections from geometric boundaries. Dynamic stress intensity factors generally attain maximum values that can be many times greater than their static counterparts. Because of this, there is a strong motivation for developing efficient computational techniques to evaluate this type of engineering problem in a relatively straightforward manner. In order to analyze the dynamic stress intensity factor problem efficiently, a computational technique that does not require a special crack tip mesh is of considerable value. The enriched finite element approach is shown to be a practical and effective technique for obtaining dynamic stress intensity factors, especially for cracks located on bimaterial interfaces, where there is inherent coupling between the mode I and mode II stress intensity factors. The enriched crack tip element approach utilizes the analytic asymptotic crack tip fields to directly compute the stress intensity factors. In this paper, fracture problems known to have two different types of crack tip singularities, subjected to elastic wave propagation effects during impact loading, are given as examples.

Asymptotic mechanical fields at the tip of a mode I crack in rubber-like solids

Asymptotic analyses of the mechanical fields in front of stationary and propagating cracks facilitate the understanding of the mechanical and physical state in front of crack tips, and they enable prediction of crack growth and failure. Furthermore, efficient modelling of arbitrary crack growth by use of XFEM (extended finite element method) requires accurate knowledge of the asymptotic crack tip fields. In the present work, we perform an asymptotic analysis of the mechanical fields in the vicinity of a propagating mode I crack in rubber. Plane deformation is assumed, and the material model is based on the Langevin function, which accounts for the finite extensibility of polymer chains. The Langevin function is approximated by a polynomial, and only the term of the highest order contributes to the asymptotic solution. The crack is predicted to adopt a wedge-like shape, i.e. the crack faces will be straight lines. The angle of the wedge and the order of the stress singularity depend on the hardening of the strain energy function. The present analysis shows that in materials with a significant hardening, the inertia term in the equations of motion becomes negligible in the asymptotic analysis. Hence, there is no upper theoretical limit to the crack speed.

Evaluation of the Higher Order Terms of the Wedge-Splitting Specimen Based on the SBFEM

The scaled boundary finite element method (abbr. SBFEM) is a semi-analytical method developed by Wolf and Song. The analytical advantage of the solution in the radial direction allows SBFEM converge to the Williams expansion. The coefficients of the Williams expansion, including the stress intensity factor, the T-stress, and higher order terms can be calculated directly without further processing. In the paper the coefficients of higher order terms of the crack tip asymptotic field of typical wedge splitting specimens with two different loading arrangements are evaluated using SBFEM. Numerical results show the method has high accuracy and effectiveness. The results have certain significance on determining crack stability of the wedge-splitting specimen.

Effect of T-stress on branch angle of moving cracks

The effects of the T-stress on Yoffe crack propagation are analyzed. Using a maximum kI fracture criterion near the kink of a moving crack tip, a branch angle is determined via asymptotic crack-tip field containing two fracture parameters related to singular and constant terms. Results indicate that crack speeds decrease the T-stress. The crack-tip field and the branch angle depend on the T-stress, especially for higher crack velocities. The critical speed for crack bifurcation is independent of remote transverse loading if neglecting the T-stress. Otherwise, the crack branch speed is reduced or raised, depending on positive or negative transverse loading, respectively.

Analysis of a subinterface crack in piezoelectric bimaterials with the extended finite element method

In this work, a subinterface crack problem in piezoelectric bimaterials is analyzed by the extended finite element method (XFEM). Associated with the level set method, the XFEM enables us to accurately capture the singularities at the crack-tips. The fracture parameters consisting of the mechanical stress intensity factors and the electrical displacement intensity factor are evaluated by using the asymptotic crack-tip fields derived from the generalized Stroh’s formalism and the interaction integral. Numerical examples for an electrically impermeable subinterface crack are presented and discussed to reveal various aspects including the effects of the crack distance to the interface, the crack inclination, the poling direction, the loading conditions, the basis enrichment functions, the enrichment strategies, the domain of the J-integral computation, etc. on the field intensity factors. Convergence study in the energy norm and in the intensity factors is additionally presented. To assess the accuracy of the proposed approach, the results obtained by the XFEM are compared with the analytical reference solutions available in the literature and excellent agreements are found.

The Higher Order Crack Tip Fields for Physical Weak-- Discontinuous Crack of FGM Plate with Reissner’s Effect

The physical weak-discontinuous problem of an interfacial crack between homogeneous material and functionally graded materials (FGMs) is studied based on Reissner’s plates considering transverse shear deformation effect. The crack-tip higher order asymptotic fields of homogeneous materials and FGMs regions are obtained by the asymptotic expansion method, respectively. Finally, the whole crack tip high order fields are assembled and given. The results provide a theoretical basis for solving interfacial crack problems of FGMs plates and their engineering application.

The Crack-Tip Higher Order Field of Power-Law FGMs Plates with Reissner’s Effect for Crack Parallel to Material Property Gradient

Reissner’s plate bending fracture theory with consideration of transverse shear deformation effects is adopted for the crack problem of power-law functionally graded materials (FGMs) plates. Assume that the crack is parallel to the material property gradient. By applying the asymptotic expansion method, the crack-tip higher order asymptotic fields which are similar to Williams’ solutions of homogeneous materials are obtained.

The Higher Order Crack Tip Fields of Exponential Gradient FGMs Plates with Reissner’s Effect

The Reissner’s plate bending theory with consideration of transverse shear deformation effects is adopted to study the fundamental fracture problem in functionally graded materials (FGMs) plates for a crack perpendicular to material gradient. The crack-tip higher order asymptotic fields of FGMs plates are obtained by the asymptotic expansion method. This study has fundamental significance as Williams’ solution.