Parametric quadratic programming method for elastic contact fracture analysis

A solution procedure for elastic contact fracture mechanics has been proposed in this paper. The procedure is based on the quadratic programming and finite element method (FEM). In this paper, parametric quadratic programming method for two-dimensional contact mechanics analysis is applied to the crack problems involving the crack surfaces in frictional contact. Based on a linear complementary contact condition, the parametric variational principle and FEM, a linear complementary method is extended to analyze contact fracture mechanics. The near-tip fields are properly modeled in the analysis using special crack tip elements with quarter-point nodes. Stress intensity factor solutions are presented for some frictional contact fracture problems and are compared with known results where available.     

Stress intensity factors for cracks in bolted joints

The mixed-mode stress intensity factors (SIFs) of the bolted joint with single and double cracks were examined. Changes in friction, clearance, applied force and crack angle were included in the nonlinear contact finite element analysis. A fine mesh was made between the contact surface and the crack tip in order to obtain an accurate solution. The least-squares method was used to determine the mixed-mode SIFs. Finite element results indicate that reasonable changes in the applied force, frictional coefficient and the clearance will not make significant changes in the normalized SIFs. The pure opening mode for cracked bolted joints does not occur at the horizontal crack but occurs at the crack with the crack angle between 0° and 22.5°. Nevertheless, using the SIF for a horizontal crack as the maximum opening-crack mode is sufficiently reliable. The maximum mode-II crack is approximately at a crack angle of 45° for both isotropic and orthotropic materials; however, at that angle the maximum mode-II SIF is only about one half of the mode-I SIF. This revised version was published online in August 2006 with corrections to the Cover Date.     

半刚性基层沥青路面反射裂缝扩展过程分析的Williams单元

该文对半刚性基层沥青路面结构采用弹性层状体系平面应变分析模型,利用改进的Williams级数,结合广义参数有限元法和常规等参元,建立了反射裂缝裂尖应力强度因子分析的广义参数Williams单元,并推导了Williams单元的刚度方程,据此研究了正对称荷载和偏载分别作用时,反射裂缝扩展过程中应力强度因子的变化规律;重点分析了偏载作用下路面结构层参数与应力强度因子之间的关系。Williams单元中含有与应力强度因子相关的参数,可以直接获得裂尖应力强度因子。算例分析表明:Williams单元与传统方法的计算结果吻合较好,且格式简单,计算精度高,适用于沥青路面反射裂缝扩展过程分析。     

Stress intensity factor error index for finite element analysis with singular elements

An error index for the stress intensity factor (SIF) obtained from the finite element analysis results using singular elements is proposed. The index was developed by considering the facts that the analytical function shape of the crack tip displacement is known and that the SIF can be evaluated from the displacements only. The advantage of the error index is that it has the dimension of the SIF and converges to zero when the actual error of the SIF by displacement correlation technique converges to zero. Numerical examples for some typical crack problems, including a mixed mode crack, whose analytical solutions are known, indicated the validity of the index. The degree of actual SIF error seems to be approximated by the value of the proposed index.     

Modeling arbitrary crack propagation in coupled shell/solid structures with X‐FEM

In this paper, the extended finite element method (X‐FEM) formulation for the modeling of arbitrary crack propagation in coupled shell/solid structures is developed based on the large deformation continuum‐based (CB) shell theory. The main features of the new method are as follows: (1) different kinematic equations are derived for different fibers in CB shell elements, including the fibers enriched by shifted jump function or crack tip functions and the fibers cut into two segments by the crack surface or connecting with solid elements. So the crack tip can locate inside the element, and the crack surface is not necessarily perpendicular to the middle surface. (2) The enhanced CB shell element is developed to realize the seamless transition of crack propagation between shell and solid structures. (3) A revised interaction integral is used to calculate the stress intensity factor (SIF) for shells, which avoids that the auxiliary fields for cracks in Mindlin–Reissner plates cannot satisfy exactly the equilibrium equations. Several numerical examples, including the calculation of SIF for the cracked plate under uniform bending and crack propagation between solid and shell structures are presented to demonstrate the performance of the developed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Stress intensity factor analysis for an interface crack between dissimilar isotropic materials under thermal stress

Thermal stresses, one of the main causes of interfacial failure between dissimilar materials, arise from different coefficients of linear thermal expansion. Two efficient numerical procedures in conjunction with the finite element method (FEM) for the stress intensity factor (SIF) analysis of interface cracks under thermal stresses are presented. The virtual crack extension method and the crack closure integral method are modified using the superposition method. The SIF analyses of some interface crack problems under mechanical and thermal loads are demonstrated. Very accurate mode separated SIFs are obtained using these methods.     

Effects of CFRP bond locations on the Mode I stress intensity factor of centre‐cracked tensile steel plates

The fatigue life of cracked steel members can be greatly extended by externally attached carbon fibre reinforced plastics (CFRP), which reduces the stress intensity factors (SIFs) at the crack tip. Access to cracks is sometimes limited and the CFRP has to be attached away from the cracks. There is a lack of knowledge on SIFs for such strengthening scheme. This paper presents the effects of CFRP bond locations on the Mode I SIF of centre‐cracked tensile (CCT) steel plate. The Mode I SIF at the crack tip is calculated using the finite element (FE) models. A correction factor is introduced as a function of CFRP bond location and crack length. The FE results are compared and agree well with experimental tests conducted by the authors. By combining with another two factors (one considering CFRP mechanical properties and the other considering CFRP bond width) derived previously by the authors, SIF formulae are proposed for CFRP reinforced CCT steel plates.     

Critical study of existing solutions for a penny-shaped interface crack, comparing with a new boundary element solution allowing for frictionless contact

A numerical study of the fundamental problem of a pressurized penny-shaped crack at the interface of two dissimilar half spaces is carried out allowing for the possibility of frictionless contact between crack faces. A new, highly accurate axi-symmetric formulation of the boundary element method (BEM) for the solution of elastic contact problems is employed. The correctness and accuracy of available predictions of different kinds for several key characteristics of the solution of this problem are checked. First, comparison of the BEM results for the near-tip contact length shows a very good agreement with some existing predictions. Second, the global solution obtained by BEM is compared with existing asymptotic solutions, obtained with both the open and the frictionless contact models. BEM results show that at the closest neighborhood to the crack tip the global solution of the problem is governed by the first term of the asymptotic solution of the frictionless contact model (up to a distance of the order of a fraction of the near-tip contact length). After a small transition region, in an adjacent surrounding zone whose extent is almost independent of the near-tip contact length, the global solution of the problem is governed by the first term of the asymptotic solution of the open model. As a result of the comparison presented, the regions in which the classical fracture parameters, stress intensity factor (SIF) and energy release rate, can be accurately obtained from the global numerical solution of a crack of this kind have been determined. Third, BEM results and previous estimations show certain discrepancies with a recently published closed form solution of the near-tip contact length and the mode II SIF of the frictionless contact model. A new closed form expression of this mode II SIF, derived from the asymptotic solution of the open model, is proposed in this paper.     

Stress intensity factor for center-cracked plate with crack surface interference

This paper presents a simple and physically acceptable analysis of stress intensity factor (SIF) for the center-cracked infinite and finite-width plates. The analysis includes the effect of crack surface interference (i.e., the upper and lower crack surfaces are not allowed to overlap) that influences both the SIF at the tension-side crack tip and the crack opening displacement (COD) profile. For an infinite plate, exact solutions are obtained by superimposing the classical (overlapping) solutions. For a finite-width plate, where the SIF solutions cannot be found in closed form, the solutions are carried out numerically. The overlapping SIF solutions from the weight function method are used. An example is given for the case of a finite-width plate under bending. It was found that the overlapping solutions underestimate the stress intensity factor at the tension-side crack tip up to 15%. The analysis results are also compared with the finite element solutions for verification purpose.     

弯扭组合载荷下圆管半椭圆表面裂纹应力强度因子的有限元分析

对表面裂纹复合型应力强度因子的研究一直是线弹性断裂力学中的重要课题,例如弯扭组合载荷下圆管半椭圆表面裂纹应力强度因子的计算,到现在也没有一个正确的分析解。考虑到裂尖的应力奇异性,在裂纹前沿手动设置三维奇异单元,用三维有限元法中的1/4点位移法计算弯扭组合载荷下圆管表面椭圆裂纹前沿的Ⅰ型、Ⅱ型和Ⅲ型应力强度因子,并分析其随裂纹深度增加时的变化规律。运用该方法计算了有关模型的应力强度因子,并与该模型的实验值进行了比较,计算结果和实验结果吻合良好。     

摩擦接触裂纹问题的扩展有限元法

扩展有限元法(XFEM)是一种在常规有限元框架内求解强和弱不连续问题的新型数值方法。扩展有限元法分析闭合型裂纹时,必须考虑裂纹面间的接触问题。已有文献均采用迭代法求解裂纹面的接触问题。该文建立了闭合型摩擦裂纹问题的扩展有限元线性互补模型,将裂纹面非线性摩擦接触转化为一个线性互补问题求解,不需要迭代求解。算例分析说明了该方法的正确性和有效性,同时表明扩展有限元法结合线性互补法求解接触问题具有较好的前景。     

Computational modeling of strong and weak discontinuities using the extended finite element method

In this article, the extended finite element method is employed to solve problems, including weak and strong discontinuities. To this end, a level set framework is used to represent the discontinuities location, and the Heaviside and Branch function are included in the standard finite element method. The case of two arbitrary curved cracks is solved numerically and stress intensity factor (SIF) values at the crack tips are calculated based on the evaluation of the crack tip opening displacement. Afterwards, J-integral methodology is adopted to evaluate the SIFs for isotropic and anisotropic bi-material interface crack problems. Numerical results are verified with those presented in the literature.     

Determination of Fracture Parameters for Multiple Cracks of Laminated Composite Finite Plate

A predictive method for estimation of stress state at zone of crack tip and assessment of remaining component lifetime depend on the stress intensity factor (SIF). This paper discusses the numerical approach for prediction of first ply failure load (F_L), progressive failure load, SIF and critical SIF for multiple cracks configurations of laminated composite finite plate using finite element method (FEM). The Hashin and Chang failure criterion are incorporated in ABAQUS using subroutine approach user defined field variables (USDFLD) for prediction of progressive fracture response of laminated composite finite plate, which is not directly available in the software. A tensile experiment on laminated composite finite plate with stress concentration is performed to validate the numerically predicted subroutine results, shows excellent agreement. The typical results are presented to examine effect of changing the crack tip distance (S), crack offset distance (H), and stacking fiber angle (θ) on F_L, and SIF .     

Comparative study of time and frequency domain BEM approaches in frictional contact problem for antiplane crack under harmonic loading

Two different boundary element methods (BEM) for crack analysis in two dimensional (2-D) antiplane, homogeneous, isotropic and linear elastic solids by considering frictional contact of the crack edges are presented. Hypersingular boundary integral equations (BIE) in time-domain (TD) and frequency domain (FD), with corresponding elastodynamic fundamental solutions are applied for this purpose. For evaluation of the hypersingular integrals involved in BIEs a special regularization process that converts the hypersingular integrals to regular integrals is applied. Simple regular formulas for their calculation are presented. For the problems solution while considering frictional contact of the crack edges a special iterative algorithm of Udzava's type is elaborated and used. Numerical results for crack opening, frictional contact forces and dynamic stress intensity factors (SIFs) are presented and discussed for a finite III-mode crack in an infinite domain subjected to a harmonic crack-face loading and considering crack edges frictional contact interaction using the TD and FD approaches.     

A new analytical method for stress intensity factors based on insitu measurement of crack deformation under biaxial tension

A new approach for the calculation of stress intensity factors (SIF) for isotropic and orthotropic materials under biaxial tension loading was proposed in this paper. In order to determine SIF from the full-field displacement data, an asymptotic expansion of the crack tip displacement field was performed. The deforming shape and surface residual stress of the crack tip was obtained at the early extended stage of the loading process by using optical microscope and X-ray diffraction measurement. During this stage, a modified Dugdale Model, which takes into account the coupled effect at the crack tip, was proposed for the open displacement of the crack tip. In this paper, the SIFs of two types of silicon steel sheet with isotropic and orthotropic properties were calculated using the modified Dugdale Model based on the biaxial tension experimental data. From the results, it was found that analysis using the modified Dugdale Model is an effective way to evaluate SIF under biaxial stress.     

Mode II stress intensity factors for central slant cracks with frictional surfaces in uniaxially compressed plates

The effect of crack surface friction on mode II stress intensity factor (SIF) of a central slant crack in a plate uniformly loaded in uniaxial compression is quantified. A previously developed two-dimensional finite element analysis was utilised after its modification to accommodate the friction between the crack surfaces. The plane strain state was assumed. A new numerical technique was devised to avoid the iteration procedures, which had to be employed due to the existence of frictional forces.

The crack inclination angle varied between zero and 75° measured from the horizontal direction. The coefficient of friction of the crack surfaces changed from zero to 1. In case of relatively sliding crack surfaces, mode II SIF existed. As is well known, the resulting mode II SIF decreased with increasing the coefficient of friction of the crack surfaces. Further, mode II SIF increased with increasing crack line inclination angle and then decreased after reaching a maximum value. The angle corresponding to that maximum SIF increased as the coefficient of friction of the crack surfaces increased.     

Numerical evaluation of the quarter-point crack tip element

This paper attempts to answer two commonly raised questions during the preparation of a finite element mesh, for the linear elastic fracture analysis of cracked structure: how to set up the finite element mesh around the crack tip, and what level of accuracy is to be expected from such a modelling. Two test problems, with known analytical expressions for their stress intensity factors, are analysed by the finite element method using the isoparametric quadratic singular element. The modified parameters were the order of integration, aspect ratio, number of elements surrounding the crack tip, use of transition elements, the singular element length over the total crack length, the symmetry of the mesh around the crack tip. Based on these analyses, a data base is created and various plots produced. The results are interpreted, the accuracy evaluated and recommendations drawn. Contrary to previous reports, it is found that the computed stress intensity factor (SIF) remains within engineering accuracy (10 per cent) throughout a large range of l/a (singular element length over crack length) for problems with a uniform non-singular stress distribution ahead of the crack tip (i.e. double edge notch), and l/a should be less than 0·1 for problems with a non-singular stress gradient (i.e three-point bend). Also, it is found that the best results are achieved by using at least four singular elements around the crack tip, with their internal angles around 45 degrees, and a reduced (2 × 2) numerical integration.     

Higher order terms of the crack tip asymptotic field for a wedge-splitting specimen

The coefficients of the crack tip asymptotic field of a typical wedge-splitting specimen are computed using a hybrid crack element (HCE), which has the potential to directly calculate not only the stress intensity factor (SIF) but also the coefficients of the higher order terms of the crack tip asymptotic field. The approximate closed-form expression for SIF proposed by Guinea et al. (1996) is calibrated by the results of the HCE. Approximate expressions for the second and third order terms for the wedge-splitting specimen are obtained by fitting the computed data. Numerical results show that the coefficients for terms higher than three are negligibly small, thus the wedge-splitting specimen is more stable than other geometries.     

Stress intensity factor error index for finite element analysis with singular elements

An error index for the stress intensity factor (SIF) obtained from the finite element analysis results using singular elements is proposed. The index was developed by considering the facts that the analytical function shape of the crack tip displacement is known and that the SIF can be evaluated from the displacements only. The advantage of the error index is that it has the dimension of the SIF and converges to zero when the actual error of the SIF by displacement correlation technique converges to zero. Numerical examples for some typical crack problems, including a mixed mode crack, whose analytical solutions are known, indicated the validity of the index. The degree of actual SIF error seems to be approximated by the value of the proposed index.     

Frictional contact analysis of multiple cracks by incremental displacement and resultant traction boundary integral equations

Based on the merits of the dual boundary element technique, a modified dual boundary element technique is extended to deal with the frictional contact of a finite plate with arbitrarily distributed multiple cracks. Besides establishing the incremental displacement boundary integral equation on the outer boundary, the resultant traction boundary integral equation on one of the crack surfaces is also developed. Since the resultant traction instead of incremental traction on the crack surface is introduced, the computed resultant contact tractions under sliding condition satisfy the Coulomb's friction law directly. Hence, as compared with the authors' previous work, only very few computation iterations are required by this method to accurately describe the contact situations of crack surfaces. As a result, not only the linear cracks, but also other types of multiple cracks, for example, curved and kinked cracks, can be tackled. The effects of friction and interaction among cracks on the computation of stress intensity factors are also displayed.