线粘弹性断裂分析的增量加料有限元法

提出了一种用于解决线粘弹性断裂问题的增量加料有限元法。为了反映裂纹尖端的应力奇异性,在裂尖附近的应力奇异区采用若干四边形加料单元和过渡单元,非奇异区采用常规四边形单元,三种单元分区混合使用形成求解域网格划分。加料单元通过引入裂尖渐近位移场,构造出可以较好反映裂尖奇异性的单元位移模式,过渡单元在加料单元基础上引入调整函数构造单元位移模式,用于连接加料单元和常规单元,以消除加料单元和常规单元间位移不协调。基于Boltzmann叠加原理,推导了粘弹性材料的增量型本构关系,进而获得了增量加料有限元列式,并基于节点位移外推法计算粘弹性介质中裂纹应变能释放率。数值算例验证了该文方法的正确性和有效性。     

An enriched element‐failure method (REFM) for delamination analysis of composite structures

This paper develops an enriched element‐failure method for delamination analysis of composite structures. This method combines discontinuous enrichments in the extended finite element method and element‐failure concepts in the element‐failure method within the finite element framework. An improved discontinuous enrichment function is presented to effectively model the kinked discontinuities; and, based on fracture mechanics, a general near‐tip enrichment function is also derived from the asymptotic displacement fields to represent the discontinuity and local stress intensification around the crack‐tip. The delamination is treated as a crack problem that is represented by the discontinuous enrichment functions and then the enrichments are transformed to external nodal forces applied to nodes around the crack. The crack and its propagation are modeled by the ‘failed elements’ that are applied to the external nodal forces. Delamination and crack kinking problems can be solved simultaneously without remeshing the model or re‐assembling the stiffness matrix with this method. Examples are used to demonstrate the application of the proposed method to delamination analysis. The validity of the proposed method is verified and the simulation results show that both interlaminar delamination and crack kinking (intralaminar crack) occur in the cross‐ply laminated plate, which is observed in the experiment. Copyright © 2009 John Wiley & Sons, Ltd.     

单位分解增强自然单元法计算应力强度因子

自然单元法是一种新兴的无网格数值计算方法,但应用于裂纹问题计算时,其近似函数并不能准确反映裂纹尖端附近应力场的奇异性,需要在缝尖附近增大结点布置密度以获得一定的计算精度。在单位分解框架下将缝尖渐近位移场函数嵌入到自然单元法近似函数中,应用伽辽金过程获得平衡方程的离散线性方程,用相互作用能量积分方法计算了混合模式裂纹的应力强度因子。算例分析表明:单位分解增强自然单元法可以方便地处理裂纹问题,在不增加结点布置密度的情况下可有效提高应力强度因子的计算精度。     

Global-local finite element models for calculating fracture parameters

The finite element iterative method (FEIM) is applied to two types of problems for the evaluation of fracture parameters in laminated composite plates. The FEIM is first used to evaluate the order of power-type singularities and the angular variation in the displacement components near the crack tip at an interface between two dissimilar materials. Next, a numerically enriched global-local finite element is formulated that incorporates the asymptotic singular behavior as calculated by the FEIM in the assumed form of the displacements for calculating stress intensity factors. Examples of both approaches yield results that agree very well with existing analytic solutions.     

An alternative complex variable formulation for an inclined crack in an orthotropic medium

Of concern in this paper is the study of the elastostatic fracture response of an orthotropic plate with an inclined crack and subjected at infinity to a biaxial uniform load. To this end an unconventional approach to the derivation of the complex variable expressions of the elastic fields is proposed. The above formulation has been used to solve the boundary value problem as superposition of Mode-I and Mode-II crack problems and it is shown that the near tip asymptotic expressions of stress and displacement fields are affected by non-singular terms originated by load biaxiality. The maximum tensile stress criterion is applied in order to investigate the effects of non-singular terms on the angle of crack extension.     

Direct evaluation of stress intensity factors for curved cracks using Irwin's integral and XFEM with high‐order enrichment functions

This paper presents a comprehensive study on the use of Irwin's crack closure integral for direct evaluation of mixed‐mode stress intensity factors (SIFs) in curved crack problems, within the extended finite element method. The approach employs high‐order enrichment functions derived from the standard Williams asymptotic solution, and SIFs are computed in closed form without any special post‐processing requirements. Linear triangular elements are used to discretize the domain, and the crack curvature within an element is represented explicitly. An improved quadrature scheme using high‐order isoparametric mapping together with a generalized Duffy transformation is proposed to integrate singular fields in tip elements with curved cracks. Furthermore, because the Williams asymptotic solution is derived for straight cracks, an appropriate definition of the angle in the enrichment functions is presented and discussed. This contribution is an important extension of our previous work on straight cracks and illustrates the applicability of the SIF extraction method to curved cracks. The performance of the method is studied on several circular and parabolic arc crack benchmark examples. With two layers of elements enriched in the vicinity of the crack tip, striking accuracy, even on relatively coarse meshes, is obtained, and the method converges to the reference SIFs for the circular arc crack problem with mesh refinement. Furthermore, while the popular interaction integral (a variant of the J‐integral method) requires special auxiliary fields for curved cracks and also needs cracks to be sufficiently apart from each other in multicracks systems, the proposed approach shows none of those limitations. Copyright © 2017 John Wiley & Sons, Ltd.     

New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

A posteriori error estimation and adaptive refinement technique for fracture analysis of 2-D/3-D crack problems is the state-of-the-art. The objective of the present paper is to propose a new a posteriori error estimator based on strain energy release rate (SERR) or stress intensity factor (SIF) at the crack tip region and to use this along with the stress based error estimator available in the literature for the region away from the crack tip. The proposed a posteriori error estimator is called the K-S error estimator. Further, an adaptive mesh refinement (h-) strategy which can be used with K-S error estimator has been proposed for fracture analysis of 2-D crack problems. The performance of the proposed a posteriori error estimator and the h-adaptive refinement strategy have been demonstrated by employing the 4-noded, 8-noded and 9-noded plane stress finite elements. The proposed error estimator together with the h-adaptive refinement strategy will facilitate automation of fracture analysis process to provide reliable solutions.     

Creep crack growth modeling and near tip stress fields

The problem of near tip stress fields in a cracked body subjected to Mode I loading at elevated temperatures is studied. Specifically, the superalloy, IN 718, is examined in the standard compact tension specimen geometry. The simulation is at 650°C. The specimen is assumed to be under dead load conditions. For a stationary crack, the near tip stress fields are calculated and compared with the asymptotic solutions available in the literature. While the results assuming small strains agree very well with the asymptotic solutions, the large strain analysis does not. The results indicate that both the amplitude and the asymptotic exponent are dependent on the applied load level which is in disagreement with the asymptotic predictions. In addition, the zone effected by creep deformation is larger when large strains are considered. An algorithm is developed and tested for the modeling of stable crack growth. Both convergence and stability are investigated. Explicit time integration is used for crack growth studies as it is demonstrated to be computationally more efficient. The algorithm is employed to study the near tip stress fields for a growing crack. The near tip stress fields for a growing crack (with constant velocity) are generated using the developed algorithm. The results demonstrate that the asymptotic behavior of the stress field is load dependent. Comparison is made with the limited analyses available. Recommendations for future research are discussed.     

Fracture mechanics parameters for cracks on a slightly undulating interface

Typical bimaterial interfaces are non-planar due to surface facets or roughness. Crack-tip stress fields of an interface crack must be influenced by non-planarity of the interface. Consequently, interface toughness is affected. In this paper, the crack-tip fields of a finite crack on an elastic/rigid interface with periodic undulation are studied. Particular emphasis is given to the fracture mechanics parameters, such as the stress intensity factors, crack-tip energy release rate, and crack-tip mode mixity. When the amplitude of interface undulation is very small relative to the crack length (which is the case for rough interfaces), asymptotic analysis is used to convert the non-planarity effects into distributed dislocations located on the planar interface. Then, the resulting stress fields near the crack tip are obtained by using the Fourier integral transform method. It is found that the stress fields at the crack tip are strongly influenced by non-planarity of the interface. Generally speaking, non-planarity of the interface tends to shield the crack tip by reducing the crack-tip stress concentration.     

An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields

In this paper, an enriched radial point interpolation method (e-RPIM) is developed for the determination of crack tip fields. In e-RPIM, the conventional RBF interpolation is novelly augmented by the suitable trigonometric basis functions to reflect the properties of stresses for the crack tip fields. The performance of the enriched RBF meshfree shape functions is firstly investigated to fit different surfaces. The surface fitting results have proven that, comparing with the conventional RBF shape function, the enriched RBF shape function has: (1) a similar accuracy to fit a polynomial surface; (2) a much better accuracy to fit a trigonometric surface; and (3) a similar interpolation stability without increase of the condition number of the RBF interpolation matrix. Therefore, it has proven that the enriched RBF shape function will not only possess all advantages of the conventional RBF shape function, but also can accurately reflect the properties of stresses for the crack tip fields. The system of equations for the crack analysis is then derived based on the enriched RBF meshfree shape function and the meshfree weak-form. Several problems of linear fracture mechanics are simulated using this newly developed e-RPIM method. It has demonstrated that the present e-RPIM is very accurate and stable, and it has a good potential to develop a practical simulation tool for fracture mechanics problems.     

Higher order tip enrichment of eXtended Finite Element Method in thermoelasticity

An eXtended Finite Element Method (XFEM) is presented that can accurately predict the stress intensity factors (SIFs) for thermoelastic cracks. The method uses higher order terms of the thermoelastic asymptotic crack tip fields to enrich the approximation space of the temperature and displacement fields in the vicinity of crack tips—away from the crack tip the step function is used. It is shown that improved accuracy is obtained by using the higher order crack tip enrichments and that the benefit of including such terms is greater for thermoelastic problems than for either purely elastic or steady state heat transfer problems. The computation of SIFs directly from the XFEM degrees of freedom and using the interaction integral is studied. Directly computed SIFs are shown to be significantly less accurate than those computed using the interaction integral. Furthermore, the numerical examples suggest that the directly computed SIFs do not converge to the exact SIFs values, but converge roughly to values near the exact result. Numerical simulations of straight cracks show that with the higher order enrichment scheme, the energy norm converges monotonically with increasing number of asymptotic enrichment terms and with decreasing element size. For curved crack there is no further increase in accuracy when more than four asymptotic enrichment terms are used and the numerical simulations indicate that the SIFs obtained directly from the XFEM degrees of freedom are inaccurate, while those obtained using the interaction integral remain accurate for small integration domains. It is recommended in general that at least four higher order terms of the asymptotic solution be used to enrich the temperature and displacement fields near the crack tips and that the J- or interaction integral should always be used to compute the SIFs.     

Stress intensity factors for three‐dimensional surface cracks using enriched finite elements

The analysis of three‐dimensional crack problems using enriched crack tip elements is examined in this paper. It is demonstrated that the enriched finite element approach is a very effective technique for obtaining stress intensity factors for general three‐dimensional crack problems. The influence of compatibility, integration, element shape function order, and mesh refinement on solution convergence is investigated to ascertain the accuracy of the numerical results. It is shown that integration order has the greatest impact on solution accuracy. Sample results are presented for semi‐circular surface cracks and compared with previously obtained solutions available in the literature. Good agreement is obtained between the different numerical solutions, except in the small zone near the free surface where previously published results have often neglected the change in the stress singularity at the free surface. The enriched crack tip element appears to be particularly effective in this region, since boundary conditions can be easily imposed on the stress intensity factors to accurately represent the correct free surface condition. Copyright © 2002 John Wiley & Sons, Ltd.     

Dynamic fields near a crack tip growing in an elastic-perfectly-plastic solid

A complete asymptotic solution is given for the fields in the neighborhood of the tip of a steadily advancing crack in an incompressible elastic-perfectly-plastic solid.For Mode I crack growth in the plane strain condition, the following noteworthy results are revealed: (1) The entire crack tip in steady crack growth is surrounded by a plastic region, and no elastic unloading is predicted by the complete dynamic asymptotic solution. Thus, the elastic unloading region predicted by the result of neglecting the important influence of the inertia terms in the equations of motion. (2) Unlike the quasi-static solution, the dynamic solution yields strain fields with a logarithmic singularity everywhere near the crack tip. (3) The stress field varies throughout the entire crack tip neighborhood, but does display behavior which can be approximated by a constant field followed by an essentially centered-fan field and then by another constant field, especially for small crack growth speeds. Indeed, the stress field reduces to that for the stationary crack, as the crack tip velocity - measured by the Mach number, M - reduces to zero; the strain field, however, does not reduce to that for the static solution, as M vanishes. (4) There are two shock fronts emanating from the crack tip across which certain stress and strain components undergo jump discontinuities. The location of the shock fronts and the magnitude of the jumps depend on the crack growth speed. The stress jump vanishes while the strain jump becomes unbounded, as the crack tip speed goes to zero.Finally, the Mode III steady-state crack growth is reviewed and, on the basis of Mode I and Mode III results, it is concluded that ductile fracture criteria for nonstationary cracks must be based on solutions which include the inertia effects, and that for this purpose, quasi-static solutions may be inadequate. Then, a possible ductile fracture criterion is suggested and discussed.One interesting feature of the complete dynamic asymptotic solution is that, unlike the quasi-static solution, it yields the same strain singularity for all three fracture modes.     

On higher order parameters in cracked composite plates under far‐field pure shear

This paper presents the analytical solution of the crack tip fields as well as the crack parameters in an infinitely large composite plate with a central crack subjected to pure shear loading. To this end, the complex variable method is employed to formulate an asymptotic solution for the crack tip fields in an anisotropic plane. Using a stress‐based definition of the crack tip modes of loading, only the mode II crack parameters are found to be non‐zero under pure shear load. Special focus is given to the determination of the higher order parameters of the crack tip asymptotic field, particularly the first non‐singular term, ie, the T‐stress. Unlike the isotropic materials, in which the T‐stress is zero under pure shear, it is found that the T‐stress is non‐zero for the case of anisotropic materials, being the only material‐dependent crack tip stress parameter. The veracity of our exact crack tip fields is assessed and verified through a comparison made with respect to the finite element (FE) solution. Finally, we demonstrate the significance of the T‐stress on stresses near the crack tip in composite plates under pure shear loads.     

Smart computational fracture of materials and structures

The use of the finite element method for complex engineering problems is now common. To ease the burden on the engineer the development of smart or adaptive computational methods is now required to model complex problems. In this paper we investigate the development of an adaptive finite element method for fracture-related problems. The adaptive method involves various stages which include the finite element analysis, error estimation/indication, mesh refinement and fracture/failure analysis in a loop. Some simple error estimators, based on stress projection, are used to investigate the adaptive finite element process. Element refinement is based on three schemes; the first and second are a simple and hierarchical refinement scheme with transitioning which avoids the need for constraint equations between element boundaries. Another scheme based on constraint equations between elements is also examined. The energy norm is used to estimate the element error. The software has the ability to introduce a discrete fracture in the structure according to standard fracture analysis practice. Crack tip parameters are calculated using a least-squares fit of the displacements into the asymptotic crack tip displacement field. Some simple examples are used to investigate the adaptive process, its behavior and some of the practical problems encountered. The convergence and equilibrium of the adaptive process, in terms of global error in the energy norm, are investigated. In the example the same problem is analyzed using both a fine computational grid and a coarse one. The coarse mesh is then adapted using the three different procedures available. The estimated error in the solution and the stress intensity are shown against the number of elements and number of iterations. Some further areas of research in adaptive finite element analysis are discussed.     

A finite element method for crack growth without remeshing

An improvement of a new technique for modelling cracks in the finite element framework is presented. A standard displacement‐based approximation is enriched near a crack by incorporating both discontinuous fields and the near tip asymptotic fields through a partition of unity method. A methodology that constructs the enriched approximation from the interaction of the crack geometry with the mesh is developed. This technique allows the entire crack to be represented independently of the mesh, and so remeshing is not necessary to model crack growth. Numerical experiments are provided to demonstrate the utility and robustness of the proposed technique. Copyright © 1999 John Wiley & Sons, Ltd.     

The singular function boundary integral method for a two-dimensional fracture problem

The singular function boundary integral method (SFBIM) originally developed for Laplacian problems with boundary singularities is extended for solving two-dimensional fracture problems formulated in terms of the Airy stress function. Our goal is the accurate, direct computation of the associated stress intensity factors, which appear as coefficients in the asymptotic expansion of the solution near the crack tip. In the SFBIM, the leading terms of the asymptotic solution are used to approximate the solution and to weight the governing biharmonic equation in the Galerkin sense. The discretized equations are reduced to boundary integrals by means of Green's theorem and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. The numerical results on a model problem show that the method converges extremely fast and yields accurate estimates of the leading stress intensity factors.     

Modeling crack discontinuities without element‐partitioning in the extended finite element method

In this paper, we model crack discontinuities in two‐dimensional linear elastic continua using the extended finite element method without the need to partition an enriched element into a collection of triangles or quadrilaterals. For crack modeling in the extended finite element, the standard finite element approximation is enriched with a discontinuous function and the near‐tip crack functions. Each element that is fully cut by the crack is decomposed into two simple (convex or nonconvex) polygons, whereas the element that contains the crack tip is treated as a nonconvex polygon. On using Euler's homogeneous function theorem and Stokes's theorem to numerically integrate homogeneous functions on convex and nonconvex polygons, the exact contributions to the stiffness matrix from discontinuous enriched basis functions are computed. For contributions to the stiffness matrix from weakly singular integrals (because of enrichment with asymptotic crack‐tip functions), we only require a one‐dimensional quadrature rule along the edges of a polygon. Hence, neither element‐partitioning on either side of the crack discontinuity nor use of any cubature rule within an enriched element are needed. Structured finite element meshes consisting of rectangular elements, as well as unstructured triangular meshes, are used. We demonstrate the flexibility of the approach and its excellent accuracy in stress intensity factor computations for two‐dimensional crack problems. Copyright © 2016 John Wiley & Sons, Ltd.