Two-dimensional analysis of anisotropic crack problems using coupled meshless and fractal finite element method

This paper presents a coupling technique for integrating the element-free Galerkin method (EFGM) with fractal the finite element method (FFEM) for analyzing homogeneous, anisotropic, and two dimensional linear-elastic cracked structures subjected to mixed-mode (modes I and II) loading conditions. FFEM is adopted for discretization of domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprises both the element-free Galerkin and the finite element shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM-FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no structured mesh or special enriched basis functions are necessary and no post-processing (employing any path independent integrals) is needed to determine fracture parameters such as stress-intensity factors (SIFs) and T − stress. The numerical results based on all four orthotropic cases show that SIFs and T − stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions.     

Coupled meshfree and fractal finite element method for mixed mode two‐dimensional crack problems

This paper presents a coupling technique for integrating the element‐free Galerkin method (EFGM) with the fractal finite element method (FFEM) for analyzing homogeneous, isotropic, and two‐dimensional linear‐elastic cracked structures subjected to mixed‐mode (modes I and II) loading conditions. FFEM is adopted for discretization of the domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprise both the EFG and the finite element (FE) shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM–FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no special enriched basis functions or no structured mesh with special FEs are necessary and no post‐processing (employing any path independent integrals) is needed to determine fracture parameters, such as stress‐intensity factors (SIFs) and T‐stress. The numerical results show that SIFs and T‐stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also, a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions. A numerical example on mixed‐mode condition is presented to simulate crack propagation. As in the proposed coupled EFGM–FFEM at each increment during the crack propagation, the FFEM mesh (around the crack tip) is shifted as it is to the new updated position of the crack tip (such that FFEM mesh center coincides with the crack tip) and few meshless nodes are sprinkled in the location where the FFEM mesh was lying previously, crack‐propagation analysis can be dramatically simplified. Copyright © 2010 John Wiley & Sons, Ltd.     

Computation of the stress intensity factors of sharp notched plates by the fractal‐like finite element method

The fractal‐like finite element method (FFEM) is an accurate and efficient method to compute the stress intensity factors (SIFs) of different crack configurations. In the FFEM, the cracked/notched body is divided into singular and regular regions; both regions are modelled using conventional finite elements. A self‐similar fractal mesh of an ‘infinite’ number of conventional finite elements is used to model the singular region. The corresponding large number of local variables in the singular region around the crack tip is transformed to a small set of global co‐ordinates after performing a global transformation by using global interpolation functions. In this paper, we extend this method to analyse the singularity problems of sharp notched plates. The exact stress and displacement fields of a plate with a notch of general angle are derived for plane‐stress/strain conditions. These exact analytical solutions which are eigenfunction expansion series are used to perform the global transformation and to determine the SIFs. The use of the global interpolation functions reduces the computational cost significantly and neither post‐processing technique to extract SIFs nor special singular elements to model the singular region are needed. The numerical examples demonstrate the accuracy and efficiency of the FFEM for sharp notched problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Mode I fatigue crack growth under biaxial loading

This paper is centred on the role of the T-stress during mode I fatigue crack growth. The effect of a T-stress is studied through its effect on plastic blunting at crack tip. As a matter of fact, fatigue crack growth is characterized by the presence of striations on the fracture surface, which implies that the crack grows by a mechanism of plastic blunting and re-sharpening (Laird C. The influence of metallurgical structure on the mechanisms of fatigue crack propagation. In: Fatigue crack propagation, STP 415. Philadelphia: ASTM; 1967. p. 131–68 [8]). In the present study, plastic blunting at crack tip is a global variable ρ, which is calculated using the finite element method. ρ is defined as the average value of the permanent displacement of the crack faces over the whole K-dominance area. The presence of a T-stress modifies significantly the evolution of plastic deformation within the crack tip plastic zone as a consequence of plastic blunting at crack tip. A yield stress intensity factor K_Y is defined for the cracked structure, as the stress intensity factor for which plastic blunting at crack tip exceeds a given value. The variation of the yield stress intensity factor was studied as a function of the T-stress. It is found that the T-stress modifies significantly the yield point of the cracked structure and that the yield surface in a (T, K_I) plane is independent of the crack length. Finally, a yield criterion is proposed for the cracked structure. This criterion is an extent of the Von-Mises yield criterion to the problem of the cracked structure. The proposed criterion matches almost perfectly the results obtained from the FEM. The evolution of the yield surface of the cracked structure in a (T, K_I) plane was also studied for a few loading schemes. These results should develop a plasticity model for the cracked structure taking into account the effect of the T-stress.     

T-stress of cracks loaded by near-tip tractions

T-stress solutions were derived for tractions acting on the crack-faces near a crack tip. Such solutions are of interest for the determination of the leading term of a weight function representation of T-stresses and the computation of an “intrinsic” T-stress for cracks growing in a material with a rising crack growth resistance. First, the type of the Green’s function for T-stresses is theoretically established. Then, results of finite element computations are reported for edge-cracked bars, DCB and CT specimens, which are suited for the determination of the first series term. As an application of the Green’s functions, the T-stresses caused by bridging interactions very close to the crack tip are computed.     

Numerical investigation of crack tip strain localization under cyclic loading in FCC single crystals

In this work, the crack tip strain localization in a face centered cubic single crystal subject to both monotonic and cyclic loading was investigated. The effect of constraint was implemented using T-stress and strain accumulation was studied for both isotropic and anisotropic elastic cases with the appropriate application of remote displacement fields in plane strain. Modified boundary layer simulations were performed using the crystal plasticity finite element framework. The consideration of elastic anisotropy amplified the effect of constraint level on stress and plastic strain fields near the crack tip indicating the importance of its use in fracture simulations. In addition, to understand the cyclic stress and strain behavior in the vicinity of the crack tip, combined isotropic and kinematic hardening laws were incorporated, and their effect on the evolution of yield curves and plastic strain accumulation were investigated. With zero-tension cyclic load, the evolution of plastic strain and Kirchhoff stress components showed differences in magnitudes between isotropic and anisotropic elastic cases. Furthermore, under cyclic loading, ratcheting was observed along the localized slip bands, which was shown to be affected by T-stress as well as elastic anisotropy. Negative T-stress increased the accumulation of plastic strain with number of cycles, which was further amplified in the case of elastic anisotropy. Finally, in all the cyclic loading simulations, the plastic strain accumulation was higher near the \(55^0 \) slip band.     

Interfacial crack-tip constraints and <Emphasis Type="Italic">J</Emphasis>-integral for bi-materials with plastic hardening mismatch

This paper investigates interfacial crack tip stress fields and the J-integral for bi-materials with plastic hardening mismatch via detailed elastic-plastic finite element analyses. For small scale yielding, the modified boundary layer formulation with the elastic T-stress is employed. For fully plastic yielding, plane strain single-edge- cracked specimens under pure bending are considered. Interfacial crack tip stress fields are explained by modified Prandtl slip-line fields. It is found that, for bi-materials consisting of two elastic-plastic materials, increasing plastic hardening mismatch increases both crack-tip stress constraint in the lower hardening material and the J-contribution there. The implication of asymmetric J-integral in bi-materials is also discussed.     

Cracked asphalt pavement under traffic loading – A 3D finite element analysis

An asphalt pavement containing a transverse top-down crack is investigated under traffic loading using 3D finite element analysis. The stress intensity factors (SIFs) and the T-stress are calculated for different distances between the crack and the vehicle wheels. It is found that all the three Modes (I, II and III) are present in the crack deformation. The signs and magnitudes of K_I, K_II, K_III and T are significantly dependent on the location of the vehicle wheels with respect to the crack plane. The magnitude of T-stress is considerable, if compared to the stress intensity factors, when one of the wheels is very close to the crack plane.     

Experimental calculation of mixed-mode notch stress intensity factors for anisotropic materials

This study developed a least-squares method to find the notch stress intensity factors (SIFs) of anisotropic materials using image-correlation experiments without the requirement of smooth procedures. Complex displacement functions are deduced into a least-squares form, and then displacements from the image-correlation experiments are substituted into the least-squares equation to evaluate the notch SIFs for anisotropic materials. Experimental results compared with the H-integral and finite element analyses show that the SIFs evaluated from the current method are acceptably accurate if more than five displacement terms are included. Moreover, the least-squares SIFs are not sensitive to the maximum or minimum radius of the area from which data is included, so experimental data very near the notch tip is not necessary.     

Perturbation method for the solution of a Zener–Stroh crack with a slightly curved configuration

This paper investigates the Zener–Stroh crack with curved configuration in plane elasticity. A singular integral equation is suggested to solve the problem. Formulae for evaluating the SIFs and T-stress at the crack tip are suggested. If the curve configuration is a product of a small parameter and a quadratic function, a perturbation method based on the singular integral equation is suggested. In the method, the singular integral equation can be expanded into a series with respect to the small parameter. Therefore, many singular integral equations can be separated from the same power order for the small parameter. These singular integral equations can be solved successively. The solution of the successive singular integral equations will provide results for stress intensity factors and T-stress at the crack tip. It is found that the behaviors for the solution of SIFs and T-stress in the Zener–Stroh crack and the Griffith crack are quite different. This can be seen from the presented comparison results.     

Mode I cracks subjected to large T-stresses

There are several criteria for predicting brittle fracture in mode I and mixed mode loading. In this paper, the modified maximum tangential stress criterion originally proposed for mixed mode loading, is employed to study theoretically brittle fracture for mode I cracks. In particular, the effect of the non-singular term of stress, often known as the T-stress, on the angle of initiation of fracture and the onset of crack growth is explored. The T-stress component of the tangential stress vanishes along the crack line. Therefore, it is often postulated for linear elastic materials that the effect of T-stress on mode I brittle fracture can be ignored. However, it is shown here that the maximum tangential stress is no longer along the line of initial crack when the T-stress exceeds a critical value. Thus, a deviation in the angle of initiation of fracture can be expected for specimens having a large T-stress. It is shown that the deviation angle increases for larger values of T-stress. Theoretical results show that the apparent fracture toughness decreases significantly when a deviation in angle occurs. Earlier experimental results are used to corroborate the findings. The effect of large T-stresses is also explored for a crack specimen undergoing moderate scale yielding. The elastic-plastic investigation is conducted using finite element analysis. The finite element results reveal a similar deviation in the angle of maximum tangential stress for small to moderate scale yielding.     

Boundary element analysis of three‐dimensional cracks in anisotropic solids

This paper presents a boundary element analysis of linear elastic fracture mechanics in three‐dimensional cracks of anisotropic solids. The method is a single‐domain based, thus it can model the solids with multiple interacting cracks or damage. In addition, the method can apply the fracture analysis in both bounded and unbounded anisotropic media and the stress intensity factors (SIFs) can be deduced directly from the boundary element solutions. The present boundary element formulation is based on a pair of boundary integral equations, namely, the displacement and traction boundary integral equations. While the former is collocated exclusively on the uncracked boundary, the latter is discretized only on one side of the crack surface. The displacement and/or traction are used as unknown variables on the uncracked boundary and the relative crack opening displacement (COD) (i.e. displacement discontinuity, or dislocation) is treated as a unknown quantity on the crack surface. This formulation possesses the advantages of both the traditional displacement boundary element method (BEM) and the displacement discontinuity (or dislocation) method, and thus eliminates the deficiency associated with the BEMs in modelling fracture behaviour of the solids. Special crack‐front elements are introduced to capture the crack‐tip behaviour. Numerical examples of stress intensity factors (SIFs) calculation are given for transversely isotropic orthotropic and anisotropic solids. For a penny‐shaped or a square‐shaped crack located in the plane of isotropy, the SIFs obtained with the present formulation are in very good agreement with existing closed‐form solutions and numerical results. For the crack not aligned with the plane of isotropy or in an anisotropic solid under remote pure tension, mixed mode fracture behavior occurs due to the material anisotropy and SIFs strongly depend on material anisotropy. Copyright © 2000 John Wiley & Sons, Ltd.     

Quasi-static crack tip fields in rate-sensitive FCC single crystals

In this work, the effects of loading rate, material rate sensitivity and constraint level on quasi-static crack tip fields in a FCC single crystal are studied. Finite element simulations are performed within a mode I, plane strain modified boundary layer framework by prescribing the two term (K − T) elastic crack tip field as remote boundary conditions. The material is assumed to obey a rate-dependent crystal plasticity theory. The orientation of the single crystal is chosen so that the crack surface coincides with the crystallographic (010) plane and the crack front lies along [10[`1]][10\overline 1] direction. Solutions corresponding to different stress intensity rates [(K)\dot]\dot{{K}}, T-stress values and strain rate exponents m are obtained. The results show that the stress levels ahead of the crack tip increase with [(K)\dot]\dot{{K}} which is accompanied by gradual shrinking of the plastic zone size. However, the nature of the shear band patterns around the crack tip is not affected by the loading rate. Further, it is found that while positive T-stress enhances the opening and hydrostatic stress levels ahead of crack tip, they are considerably reduced with imposition of negative T-stress. Also, negative T-stress promotes formation of shear bands in the forward sector ahead of the crack tip and suppresses them behind the tip.     

The effective T-stress estimation and crack paths emanating from U-notches

The concept of the T-stress as a local constraint factor has been extended to U-notch tip stress distribution as the effective T-stress. The effective T-stress has been estimated as the average value of the T-stress in the region corresponding to the effective (characteristic) distance ahead of the notch tip. The T-stress is evaluated by finite element method using the experimental load for crack initiation and computing the difference between principal stresses along ligament. A large range of critical effective T-stress values is investigated for different specimen configurations and notch aspect ratios. Crack stabilisation and crack bifurcation for fracture emanating from notches according to the critical effective T-stress is discussed. A model involving the influence of the critical effective T-stress on void growth for ductile failure in the vicinity of the notch tip has been proposed.     

T-stress effects on steady crack growth in a thin, ductile plate under small-scale yielding conditions: Three-dimensional modeling

The non-singular T-stress provides a first-order estimate of geometry and loading mode, e.g. tension vs. bending, effects on elastic–plastic, crack-front fields under mode I conditions. The T-stress has a pronounced effect on measured crack growth resistance curves for ductile metals – trends most computational models confirm using a two-dimensional setting. This work examines T-stress effects on three-dimensional (3D), elastic–plastic fields surrounding a steadily advancing crack for a moderately hardening material in the framework of a 3D, small-scale yielding boundary-layer model. A flat, straight crack front advances at a constant quasi-static rate under near invariant local and global mode I loading. The boundary-layer model has thickness B that defines the only geometric length-scale. The material flow properties and (local) toughness combine to limit the in-plane plastic-zone size during steady growth to at most a few multiples of the thickness (conditions obtainable, for example, in large, thin aluminum components). The computational model requires no crack growth criterion; rather, the crack front extends steadily at constant values of the plane-stress displacements imposed on the remote boundary for the specified far-field stress intensity factor and T-stress. The specific numerical results presented demonstrate similarity scaling of the 3D near-front stresses in terms of two non-dimensional loading parameters. The analyses reveal a strong effect of T-stress on key stress and strain quantities for low loading levels and less effect for higher loading levels, where much of the plastic zone experiences plane-stress conditions. To understand the combined effects of T-stress on stresses and plastic strain levels, normalized values from a simple void-growth model, computed over the crack plane for low loading, clearly reveal the tendency for crack-front tunneling, shear-lip formation near the outside surfaces, and a minimum steady-state fracture toughness for T = 0 loading.     

Stress intensity factors and T-stresses for offset double edge-cracked plates under mixed-mode loadings

A particular weight function method is used in this study to determine the stress intensity factors (SIFs) and T-stresses for offset double edge-cracked plates (ODECPs). By using reference loading conditions prescribed on the crack flanks for finite element analyses, the coefficients of weight functions are derived and compiled in the form of tables. With the weight functions, the SIFs and T-stresses for several loading cases are calculated. The results compare well with those obtained using the displacement method. Applying the derived weight functions, the SIFs and T-stresses for ODECPs under several loading conditions are determined. The results can be used as references for related applications.     

J–Q characterized stress fields of surface-cracked metallic liners – II. Composite overwrapped pressure vessels

Two-parameter J–Q elastic–plastic crack front fields are developed for surface-cracked metallic liners of composite overwrapped pressure vessels (COPV). Uniaxial tensile data from 6061-T6 aluminum coupon specimens for the metallic liner and anisotropic elastic material properties for the filament wound carbon fiber epoxy are used in three-dimensional finite element models. Modified boundary layer (MBL) finite element solutions are used to evaluate near tip dominance and parameterization limits. Semicircular surface cracks of varying depths inserted in the inner liner surface are investigated. J and Q crack front distributions, and the corresponding parameterization limits, and near tip triaxiality trends are obtained and the effects of elastic–plastic material discontinuity of the heterogeneous joint and the biaxiality of stresses are evaluated. J–Q predicted fields maintain accuracy for higher far-field loads and lower near tip deformations compared to the liner only models for angles near the free surface. However, for the critical crack growth region, Q does not maintain a radially independent measure of constraint for loads seen in a typical COPV; therefore, these fracture predictors may not be applicable. In the COPV, large-scale yielding marks a transition where triaxiality is higher as a function of constraint compared to the linear relationship common to homogeneous structures. Results from this study will facilitate the implementation of proof test logic and accurate fracture prediction of COPV liners with emphasis on geometric limits and fracture specimen applicability.     

Effects of non-singular stresses on crack-tip fields for pressure-sensitive materials,Part 1: Plane strain case

Mode I near-tip stress fields for elastic perfectly plastic pressure-sensitive materials under plane strain and small-scale yielding conditions are presented. A Coulomb-type yield criterion described by a linear combination of the effective stress and the hydrostatic stress is adopted in the analysis. The finite element computational results sampled at the distance of a few crack opening displacements from the tip show that, as the pressure sensitivity increases, the magnitudes of the normalized radial and hoop stress ahead of the tip decrease, the total angular span of the singular plastic sectors decreases, and the angular span of the elastic sectors bordering the crack surfaces increases. When non-singular T stresses are considered along the boundary layer of the small-scale yielding model, the near-tip stresses decrease as the T stress decreases. The plastic zone shifts toward the crack surfaces as the T stress increases. When the discontinuities of the radial stress and the out-of-plane normal stress along the border between the plastic sector and the elastic sector are allowed, the angular variations of the asymptotic crack-tip fields agree well with those of the finite element computations. Variation of the Q stresses for pressure-sensitive materials can be found from the asymptotic solutions when the plastic zone size ahead of the tip is relatively larger than the crack opening displacement. In addition the T stress is shown to have strong effects on the plastic zone sizes and shapes which could affect the toughening of pressure-sensitive materials.     

求解双材料界面裂纹应力强度因子的扩展有限元法

基于双材料界面裂纹尖端的基本解,构造扩展有限元法(eXtended Finite Element Methods, XFEM)裂尖单元结点的改进函数。有限元网格剖分不遵从材料界面,考虑3种类型的结点改进函数:弱不连续改进函数、Heaviside改进函数和裂尖改进函数,建立XFEM的位移模式,给出计算双材料界面裂纹应力强度因子(Stress Intensity Factors, SIFs)的相互作用积分方法。数值结果表明:XFEM无需遵从材料界面剖分网格,该文的方法能够准确评价双材料界面裂纹尖端的SIFs。     

A finite element modelling for directly determining intensity factors of piezoelectric materials with cracks

Recently, authors(Cao et al., Acta Aeronautica et Astronautica Sinica 25(5): 470–472, 2004) extended the singular crack element originally introduced by Wang et al. (Eng Fract Mech 37(6):1195–1201, 1990) for evaluating the stress intensity factors (SIFs). Extensive studies have proved the versatility and accuracy of the element. This study is to show the versatility of the element for piezoelectric materials. In this paper, electric potential and displacement fields near a crack tip of piezoelectric materials are first used to construct a finite element version for directly determining intensity factors of piezoelectric materials with cracks. A singular finite element is constituted and a new method to calculate intensity factors of piezoelectric materials with cracks is obtained without any post-processing procedures. Detailed derivations are given and the results obtained with present method are good agreement with those of theoretical results, the FEM data by ANSYS and singular electromechanical crack tip elements. The results to the different selections of the structural dimensions are carried out. Numerical examples demonstrate the accuracy and validity of the novel element of present method.