Determination of coefficients of the crack tip asymptotic field by fractal hybrid finite elements

This paper applies the fractal finite element method (FFEM) together with 9-node Lagrangian hybrid elements to the calculation of linear elastic crack tip fields. An explicit stabilization scheme is employed to suppress the spurious kinematic modes of the sub-integrated Lagrangian element. An extensive convergence study has been conducted to examine the effects of the similarity ratio, reduced integration and the type of elements on the accuracy and stability of the numerical solutions. It is concluded that (i) a similarity ratio close to unity should be used to construct the fractal mesh, (ii) sub-integrated Lagrangian elements with occasionally unstable behaviour should not be used, and (iii) the good accuracy (with differences less than 0.5% with existing available solutions) and stability over a wide range of numerical tests support the use of fractal hybrid finite elements for determining crack tip asymptotic fields.     

Simulation of stage I-crack growth using a hybrid boundary element technique

Short fatigue crack propagation often determines the service life of cyclically loaded components and is highly influenced by microstructural features such as grain boundaries. A two-dimensional model to simulate the growth of these stage I-cracks is presented. Cracks are discretised by displacement discontinuity boundary elements and the direct boundary element method is used to mesh the grain boundaries. A superposition procedure couples these different boundary element methods to employ them in one model. Varying elastic properties of the grains are considered and their influence on short crack propagation is studied. A change in crack tip slide displacement determining short crack propagation is observed as well as an influence on the crack path.     

Analysis of cracked plates using an iterative hybrid technique of boundary element method and distributed dislocation method

An iterative hybrid technique of boundary element method (BEM) and distributed dislocation method (DDM) is introduced for solving two dimensional crack problems. The technique decomposes the problem into (n + 1) subsidiary problems where n is the number of crack branches. The required solution will be the sum of these (n + 1) solutions. The first subsidiary problem is to find the stress distribution induced in the plate in the absence of the crack using BEM. All of the remaining subsidiary problems, are stress disturbance ones that will be solved using DDM. The results will be added and compared with the boundary conditions of the original problem. Iteration will be performed between the plate boundaries and crack faces until all of the boundary conditions are satisfied.     

Direct evaluation of accurate coefficients of the linear elastic crack tip asymptotic field

An improvement to the extended finite element method (XFEM) and generalised finite element method (GFEM) is introduced. It enriches the finite element approximation of the crack tip node as well as its surrounding nodes with not only the first term but also the higher order terms of the linear elastic crack tip asymptotic field using a partition of unity method (PUM). Numerical results show that together with a reduced quadrature rule to the enriched elements, this approach predicts accurate stress intensity factors (SIFs) directly (i.e. without extra post‐processing) after constraining the enriched nodes properly. However, it does not predict accurately the coefficients of the higher order terms. For that a hybrid crack element (HCE) is introduced which is powerful and convenient not only for directly determining the SIF but also the coefficients of higher order terms in the plane linear elastic crack tip asymptotic field. Finally, the general expressions for the coefficients of the second to fifth terms of the linear elastic crack tip asymptotic field of three‐point bend single edge notched beams (TPBs) with span to depth ratios widely used in testing are extended to very deep cracks with the use of the HCE.     

伴流声场计算的有限元方法及其应用

推导了势流中的声波方程,并运用伽辽金加权余量法建立了相应的有限元(Finite Element Method,FEM)弱形式.对于管道声学问题的计算,给出了所需边界条件的处理方法,通过离散和装配得到有限元矩阵方程.使用自行编写的有限元程序计算分析了 Herschel-Quincke(H-Q)管的消声特性.结果表明,在中...     

Finite element analysis of crack mouth opening displacement compliance in crack length evaluation for clamped single edge tension specimens

In the unloading compliance method developed for clamped single edge tension (SE(T)) specimens, six crack mouth opening displacement (CMOD)‐based compliance equations (i.e. a/W = f(BCE′)) were proposed for the crack length evaluation without clearly clarifying the corresponding predictive accuracies. In addition, the effective elastic modulus (E_e) that reflects the actual state of stress should also be introduced in the crack length evaluation for SE(T) specimens, because the actual state of stress in the remaining ligament of the test specimen is neither plane stress (E) nor plane strain (E′). In this study, two‐dimensional (2D) plane strain and three‐dimensional (3D) finite element analyses (FEAs) are carried out to investigate predictive accuracies of the six compliance equations. In both 2D and 3D FEA, specimens with a wide range of crack lengths and geometric configurations are included. For a given specimen, the value of E_e that presents the equivalent stress state in the remaining ligament is calculated on the basis of 3D FEA data. A set of formulae for the clamped SE(T) specimen is proposed that allows to evaluate E_e from the corresponding CMOD compliance. This approach is verified using numerical data. The observations of the numerical verification suggest that the use of E_e instead of E or E′ in CMOD‐based compliance equations markedly improves the accuracy of the predicted crack length for clamped SE(T) specimens.     

An over‐deterministic method for calculation of coefficients of crack tip asymptotic field from finite element analysis

An over‐deterministic method has been employed for calculating the stress intensity factors (SIFs) as well as the coefficients of the higher‐order terms in the Williams series expansions in cracked bodies, using the conventional finite element analysis. For a large number of nodes around the crack tip, an over‐determined set of simultaneous linear equations is obtained, and using the fundamental concepts of the least‐squares method, the coefficients of the Williams expansion can be calculated for pure mode I, pure mode II and mixed mode I/II conditions. A convergence study has been conducted to examine the effects of the number of nodes used, the number of terms in Williams expansion and the distance of the selected nodes from the crack tip, on the accuracy of the results. It is shown that the simple method presented in this paper, yields accurate results even for coarse finite element meshes or in the absence of singular elements. The accuracy of SIFs and the coefficients of higher‐order terms are validated by using the available results in the literature.     

An effective hybrid displacement function element method for solving the edge effect of Mindlin–Reissner plate

In bending problems of Mindlin–Reissner plate, the resultant forces often vary dramatically within a narrow range near free and soft simply‐supported (SS1) boundaries. This is so‐called the edge effect or the boundary layer effect, a challenging problem for conventional finite element method. In this paper, an effective finite element method for analysis of such edge effect is developed. The construction procedure is based on the hybrid displacement function (HDF) element method [1], a simple hybrid‐Trefftz stress element method proposed recently. What is different is that an additional displacement function f related to the edge effect is considered, and its analytical solutions are employed as the additional trial functions for the first time. Furthermore, the free and the SS1 boundary conditions are also applied to modify the element assumed resultant fields. Then, two new special elements, HDF‐P4‐Free and HDF‐P4‐SS1, are successfully constructed. These new elements are allocated along the corresponding boundaries of the plate, while the other region is modeled by the usual HDF plate element HDF‐P4‐11 β [1]. Numerical tests demonstrate that the present method can effectively capture the edge effects and exactly satisfy the corresponding boundary conditions by only using relatively coarse meshes. Copyright © 2015 John Wiley & Sons, Ltd.     

Probabilistic crack growth analyses using a boundary element model: Applications in linear elastic fracture and fatigue problems

This paper addresses the numerical solution of random crack propagation problems using the coupling boundary element method (BEM) and reliability algorithms. Crack propagation phenomenon is efficiently modelled using BEM, due to its mesh reduction features. The BEM model is based on the dual BEM formulation, in which singular and hyper-singular integral equations are adopted to construct the system of algebraic equations. Two reliability algorithms are coupled with BEM model. The first is the well known response surface method, in which local, adaptive polynomial approximations of the mechanical response are constructed in search of the design point. Different experiment designs and adaptive schemes are considered. The alternative approach direct coupling, in which the limit state function remains implicit and its gradients are calculated directly from the numerical mechanical response, is also considered. The performance of both coupling methods is compared in application to some crack propagation problems. The investigation shows that direct coupling scheme converged for all problems studied, irrespective of the problem nonlinearity. The computational cost of direct coupling has shown to be a fraction of the cost of response surface solutions, regardless of experiment design or adaptive scheme considered.     

Simulation of dynamic crack curving and branching under biaxial loading by a time domain boundary integral equation method

Bifurcation and trifurcation of a fast running crack under various biaxial loading conditions is investigated numerically. The solution procedure for the 2D model in the framework of linear elastodynamics employs a time-domain boundary element method and allows for arbitrary curvilinear crack propagation. Branching events are controlled by the criterion of a critical mode I stress intensity factor while the propagation direction and growth rate of each branch are determined from the criterion of maximum circumferential stress. Numerical results are compared with experimental findings and are discussed with respect to macroscopic and microscopic aspects of dynamic fracture.     

Coefficients of the crack tip asymptotic field for a standard compact tension specimen

The coefficients of the crack tip asymptotic field of a standard compact tension (CT) specimen are computed using a hybrid crack element (HCE). It allows the direct calculation (without post-processing) of not only the stress intensity factor (SIF) but also the coefficients of higher order terms of the crack tip asymptotic field. Approximate closed-form expressions for the first five terms for the CT specimen that are accurate for shallow to very deep cracks are obtained by fitting the computed data. The SIF formula proposed by Brown and Srawley (1966) is shown to be accurate when the crack length to depth ratio () ranges from 0.35 to 0.75. The formula proposed by Newman (1974) and Srawley (1976) is accurate for 0.15. However, the accuracy of available formulas for the second T-term in the literature is quite disappointing. Numerical results also show that, unlike the notched three-point bend beam and the wedge splitting specimen, the second T-term of the CT specimen is always positive.     

Dynamics of charged and conducting drops via the hybrid finite-boundary element method

A numerical study on the dynamic behaviour of a charged and conducting drop, with net electrical charge , is presented here, that is valid for arbitrary initial disturbances. It employs the integral form of Laplace's equation for the calculation of the velocity and electrostatic potentials, which only requires discretization and solution on the surface of the drop. Thus a hybrid method results with the integral equations solved via the boundary element technique, while the Galerkin finite element formulation is used for the kinematic and dynamic condition at the interface as well as for the net charge conservation equation. Recently, the authors followed this approach in their study on the free nonlinear oscillations of inviscid drops, and they were able to optimize time and space discretization as well as the treatment of the integral equation with excellent results.     

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。     

FEM for evaluation of weight functions for SIF, COD and higher-order coefficients with application to a typical wedge splitting specimen

In the evaluation of accurate weight functions for the coefficients of first few terms of the linear elastic crack tip fields and the crack opening displacement (COD) using the finite element method (FEM), singularities at the crack tip and the loading point need to be properly considered. The crack tip singularity can be well captured by a hybrid crack element (HCE), which directly predicts accurate coefficients of first few terms of the linear elastic crack tip fields. A penalty function technique is introduced to handle the point load. With the use of these methods numerical results of a typical wedge splitting (WS) specimen subjected to wedge forces at arbitrary locations on the crack faces are obtained. With the help of appropriate interpolation techniques, these results can be used as weight functions. The range of validity of the so-called Paris equation, which is widely used in the evaluation of the COD from the stress intensity factors (SIFs), is established.     

Delamination analysis of composites by new orthotropic bimaterial extended finite element method

The extended finite element method (XFEM) is further improved for fracture analysis of composite laminates containing interlaminar delaminations. New set of bimaterial orthotropic enrichment functions are developed and utilized in XFEM analysis of linear‐elastic fracture mechanics of layered composites. Interlaminar crack‐tip enrichment functions are derived from analytical asymptotic displacement fields around a traction‐free interfacial crack. Also, heaviside and weak discontinuity enrichment functions are utilized in modeling discontinuous fields across interface cracks and bimaterial weak discontinuities, respectively. In this procedure, elements containing a crack‐tip or strong/weak discontinuities are not required to conform to those geometries. In addition, the same mesh can be used to analyze different interlaminar cracks or delamination propagation. The domain interaction integral approach is also adopted in order to numerically evaluate the mixed‐mode stress intensity factors. A number of benchmark tests are simulated to assess the performance of the proposed approach and the results are compared with available reference results. Copyright © 2011 John Wiley & Sons, Ltd.     

A parameter for quantitative analysis of plasticity induced crack closure

Numerical models have been successfully developed to predict plasticity induced crack closure (PICC). However, despite the large research effort a full understanding of the links between physical parameters, residual plastic wake and PICC has not been achieved yet. The plastic extension of material behind crack tip, Δy_p, obtained by the integration of vertical plastic deformation perpendicularly to crack flank, is proposed here to quantify the residual plastic field. The values of Δy_p and PICC were obtained numerically in a M(T) specimen using the finite element method. An excellent correlation was found between PICC and Δy_p which indicates that this parameter controls the phenomenon, and can be used to quantify the effect of physical parameters. An empirical model was developed to predict PICC assuming that the residual plastic field is a set of vertical plastic wedges, that the linear superposition principle applies and that the influence of a particular wedge exponentially decreases with distance to crack tip. The model was applied successfully to predict PICC for different residual plastic fields which provided an additional validation of Δy_p as the parameter controlling PICC.     

基于快速多极子边界元法的齿轮箱声场分析

齿轮箱是广泛应用的工程机械零部件,准确地模拟其辐射声场对后续的降噪优化设计有着重要作用。边界元方法非常适合分析此类无限域下的声辐射问题。但传统边界元方法有着计算效率低、内存占用高的缺点。该研究发展了宽频的快速多极子边界元方法,并运用该方法计算了齿轮箱在特定频率下的场点声压以及辐射声场。通过对比商用软件的分析结果,验证了所提快速边界元方法的准确性。此外,运用多核并行计算方法,对计算量较大的扫频分析进行加速计算,最终快速、准确地获取了齿轮箱辐射声场的扫频结果。     

The extended/generalized finite element method: An overview of the method and its applications

An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented. This method enables the accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non‐smooth features within elements. This is achieved by enriching the polynomial approximation space of the classical finite element method. The GEFM/XFEM has shown its potential in a variety of applications that involve non‐smooth solutions near interfaces: Among them are the simulation of cracks, shear bands, dislocations, solidification, and multi‐field problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Stress field interaction and strain energy distribution between a stationary main crack and its process zone

The damage process zone developed by brittle materials in front of a macrocrack is simulated by means of a distribution of microcracks. Crack mutual interactions are taken into account by means of a numerical technique, based on a displacement discontinuity boundary element method that is able of considering both the macrocrack–microcrack and microcrack–microcrack interactions inside the process zone. In the frame of linear elastic fracture mechanics the stress field at each crack tip and the related elastic strain energy are calculated. The main features of the interaction phenomena turn out to be almost independent of the microcrack density. Some considerations both on the shielding and amplification effects on the main crack and on the strain energy distribution between cracks give explanation to experimental evidence and prove that crack interaction is not such a short-range effect as sometimes expected.