Analysis of two-dimensional mixed-mode crack problems by finite element alternating method

A finite element alternating method is presented and applied to analyze two-dimensional linear elastic mixed-mode fracture problems with single or multiple cracks. The method involves the iterative superposition of the finite element solution of a bounded uncracked plate and the analytical solution of an infinite two-dimensional plate with a crack subjected to arbitrary normal and shear loadings. The normal and shear residual stresses evaluated at the location of fictitious cracks are fitted by appropriate polynomials through the least-squares method. Based on those coefficients of the determined polynomials, the mixed-mode stress intensity factors can be calculated accurately. The interaction effects among cracks are also considered. This method provides a highly efficient way to deal with two-dimensional fracture problems.     

A strange convergence property of the lobatto-chebyshev method for the numerical determination of stress intensity factors

The Lobatto-Chebyshev method for the numerical solution of the Cauchy type singular integral equation of crack problems in two-dimensional elasticity, plates and shells and the determination of the values of the stress intensity factors at the crack tips is shown to converge for non-differentiable Hölder-continuous or even discontinuous loading distributions as far as the values of the stress intensity factors are concerned. Moreover, in all cases of differentiable loading distributions it is shown to converge more rapidly than believed up to now. The problems of a simple straight crack and a periodic array of cracks loaded by three non-differentiable loading distributions are used for the application of the present results. The displayed numerical results for these problems verify and further corroborate the theoretical results. The extension of the present results to the Gauss-Chebyshev method is also quite possible.     

求解应力强度因子的样条虚边界元交替法

为求解平面裂纹问题的应力强度因子,提出基于Muskhelishvili基本解和样条虚边界元法的样条虚边界元交替法.该方法将平面内带裂纹有限域问题分解成带裂纹无限域问题与不带裂纹有限域问题的叠加.带裂纹无限域问题利用Muskhelishvili基本解法直接得出,不带裂纹有限域问题采用样条虚边界元法求解.利用该方法对复合型中心裂纹方板和I型偏心裂纹矩形板进行分析.数值结果表明该方法精度高且适用性强.     

Dynamic stress intensity factors for ultrasonic fatigue test specimens

Ultrasonic fatigue test methods are increasingly being used to study crack growth and threshold behaviour of metallic materials. In such studies a particular problem is the accurate determination of the dynamic stress intensity factor range, ΔK. A direct time integration finite element analysis of a center cracked specimen tested at its lowest natural frequency is presented. The effects of opening and closing of the crack tip when the test specimen is subjected to a fully reversed load cycle, i.e. zero mean load, is investigated. Numerical results correlate well with experimental data. It is shown that old published data are in error with up to at least 20% due to insufficient evaluation of dynamic stress intensity factors.     

Solution of elastic crack problems by superposition of finite elements and singular fields

A new finite element method is presented for the solution of plane elasticity problems which contain nonremovable stress singularities. Singular stress field are combined with finite element solutions by a superposition technique; an important feature of the method is that use of the singular fields may be restricted to any specified group of elements which include the singular point. It is shown that good estimates for stress intensity factors are obtained when the method is applied to crack problems.     

Application of genetic algorithm in crack detection of beam-like structures using a new cracked Euler–Bernoulli beam element

In this paper, a crack identification approach is presented for detecting crack depth and location in beam-like structures. For this purpose, a new beam element with a single transverse edge crack, in arbitrary position of beam element with any depth, is developed. The crack is not physically modeled within the element, but its effect on the local flexibility of the element is considered by the modification of the element stiffness as a function of crack's depth and position. The development is based on a simplified model, where each crack is substituted by a corresponding linear rotational spring, connecting two adjacent elastic parts. The localized spring may be represented based on linear fracture mechanics theory. The components of the stiffness matrix for the cracked element are derived using the conjugate beam concept and Betti's theorem, and finally represented in closed-form expressions. The proposed beam element is efficiently employed for solving forward problem (i.e., to gain accurate natural frequencies of beam-like structures knowing the cracks’ characteristics). To validate the proposed element, results obtained by new element are compared with two-dimensional (2D) finite element results as well as available experimental measurements. Moreover, by knowing the natural frequencies, an inverse problem is established in which the cracks location and depth are identified. In the inverse approach, an optimization problem based on the new beam element and genetic algorithms (GAs) is solved to search the solution. The proposed approach is verified through various examples on cracked beams with different damage scenarios. It is shown that the present algorithm is able to identify various crack configurations in a cracked beam.     

Elastodynamic fields near running cracks by finite elements

A finite element method is developed for the computation of elastodynamic stress intensity factors at a rapidly moving crack tip. The method is restricted to bodies whose surfaces and two-material interfaces are either parallel to the direction of propagation or are sufficiently remote. The crack tip starts to move at the instant that it is struck by an incident wave. The finite element grid moves undeformed with the crack tip. The main result consists in the fact that the method of non-singular calibrated crack tip elements, in which the stress-intensity factor is determined from its statically calibrated ratio to the crack opening displacement in an adjacent node, is extended to dynamic problems with moving cracks, for both in-plane and anti-plane motions. The dependence of the calibration ratio on the crack tip velocity is established from previously developed analytical solutions for the near-tip displacement fields. Numerical results compare favorably with known analytical solutions for cracks moving in an infinite solid. The grid motion causes an apparent asymmetric additional damping matrix.     

An extended finite element method for modeling crack growth with frictional contact

A new technique for the finite element modeling of crack growth with frictional contact on the crack faces is presented. The eXtended Finite Element Method (X-FEM) is used to discretize the equations, allowing for the modeling of cracks whose geometry are independent of the finite element mesh. This method greatly facilitates the simulation of a growing crack, as no remeshing of the domain is required. The conditions which describe frictional contact are formulated as a non-smooth constitutive law on the interface formed by the crack faces, and the iterative scheme implemented in the LATIN method [Nonlinear Computational Structural Mechanics, Springer, New York, 1998] is applied to resolve the nonlinear boundary value problem. The essential features of the iterative strategy and the X-FEM are reviewed, and the modifications necessary to integrate the constitutive law on the interface are presented. Several benchmark problems are solved to illustrate the robustness of the method and to examine convergence. The method is then applied to simulate crack growth when there is frictional contact on the crack faces, and the results are compared to both analytical and experimental results.     

A hybrid finite element-scaled boundary finite element method for crack propagation modelling

This study develops a novel hybrid method that combines the finite element method (FEM) and the scaled boundary finite element method (SBFEM) for crack propagation modelling in brittle and quasi-brittle materials. A very simple yet flexible local remeshing procedure, solely based on the FE mesh, is used to accommodate crack propagation. The crack-tip FE mesh is then replaced by a SBFEM rosette. This enables direct extraction of accurate stress intensity factors (SIFs) from the semi-analytical displacement or stress solutions of the SBFEM, which are then used to evaluate the crack propagation criterion. The fracture process zones are modelled using nonlinear cohesive interface elements that are automatically inserted into the FE mesh as the cracks propagate. Both the FEM’s flexibility in remeshing multiple cracks and the SBFEM’s high accuracy in calculating SIFs are exploited. The efficiency of the hybrid method in calculating SIFs is first demonstrated in two problems with stationary cracks. Nonlinear cohesive crack propagation in three notched concrete beams is then modelled. The results compare well with experimental and numerical results available in the literature.     

Paired-domination in inflated graphs

The inflation G_I of a graph G with n(G) vertices and m(G) edges is obtained from G by replacing every vertex of degree d of G by a clique K_d. A set S of vertices in a graph G is a paired dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired domination number γ_p(G) is the minimum cardinality of a paired dominating set of G. In this paper, we show that if a graph G has a minimum degree δ(G)2, then n(G)γ_p(G_I)4m(G)/[δ(G)+1], and the equality γ_p(G_I) = n(G) holds if and only if G has a perfect matching. In addition, we present a linear time algorithm to compute a minimum paired-dominating set for an inflation tree.     

Parallel adaptive simulations of dynamic fracture events

Finite element simulations of dynamic fracture problems usually require very fine discretizations in the vicinity of the propagating stress waves and advancing crack fronts, while coarser meshes can be used in the remainder of the domain. This need for a constantly evolving discretization poses several challenges, especially when the simulation is performed on a parallel computing platform. To address this issue, we present a parallel computational framework developed specifically for unstructured meshes. This framework allows dynamic adaptive refinement and coarsening of finite element meshes and also performs load balancing between processors. We demonstrate the capability of this framework, called ParFUM, using two-dimensional structural dynamic problems involving the propagation of elastodynamic waves and the spontaneous initiation and propagation of cracks through a domain discretized with triangular finite elements.     

Topology optimization considering fracture mechanics behaviors at specified locations

As a typical form of material imperfection, cracks generally cannot be avoided and are critical for load bearing capability and integrity of engineering structures. This paper presents a topology optimization method for generating structural layouts that are insensitive/sensitive as required to initial cracks at specified locations. Based on the linear elastic fracture mechanics model (LEFM), the stress intensity of initial cracks in the structure is analyzed by using singularity finite elements positioned at the crack tip to describe the near-tip stress field. In the topology optimization formulation, the J integral, as a criterion for predicting crack opening under certain loading and boundary conditions, is introduced into the objective function to be minimized or maximized. In this context, the adjoint variable sensitivity analysis scheme is derived, which enables the optimization problem to be solved with a gradient-based algorithm. Numerical examples are given to demonstrate effectiveness of the proposed method on generating structures with desired overall stiffness and fracture strength property. This method provides an applicable framework incorporating linear fracture mechanics criteria into topology optimization for conceptual design of crack insensitive or easily detachable structures for particular applications.     

Extended finite element method on polygonal and quadtree meshes

In this paper, we present mesh-independent modeling of discontinuous fields on polygonal and quadtree finite element meshes. This approach falls within the class of extended and generalized finite element methods, where the partition of unity framework is used to introduce additional (enrichment) functions within the classical displacement-based finite element approximation. For crack modeling, a discontinuous function and the two-dimensional asymptotic crack-tip fields are used as enrichment functions. Linearly complete partition of unity approximations are adopted on polygonal (convex and nonconvex elements) and quadtree meshes. Excellent agreement with reference solution results is obtained for mixed-mode stress intensity factors on benchmark crack problems, and crack growth simulations without remeshing are conducted on polygonal and quadtree meshes to reveal the potential of the proposed techniques in computational failure mechanics.     

孔边角疲劳裂纹的扩展仿真及分析

给出了一种利用有限元技术模拟周期性张力载荷作用下孔边角裂纹扩展过程的方法。首先利用一系列点定义裂纹前沿，据此形成包含奇异单元的二维有限元网格，再扩展为三维网格．然后利用有限元法进行应力应变分析，最后使用Paris定律计算局部扩展增量．以此来更新裂纹的形状和尺寸。该方法还能够自动地重复执行扩展仿真。文中还对三个不同半径的四分之一椭圆形边角裂纹扩展过程进行了仿真和分析比较，以此来取得裂纹在扩展过程中的形状变化特征和不同方向上扩展的特征。     

Stability and convergence proofs for a discontinuous-Galerkin-based extended finite element method for fracture mechanics

We prove the optimal convergence of a discontinuous-Galerkin-based extended finite element method for two-dimensional linear elastostatic problems over cracked domains. The method, which we proposed earlier [1], has two distinctive traits: a) it enriches the finite element space with the modes I and II singular asymptotic crack tip fields over a neighborhood of the crack tip termed the enrichment region, and b) it allows functions in the finite element space to be discontinuous across the boundary between the enrichment region and the rest of the domain. The treatment for this discontinuity, generally a non-polynomial function, is facilitated by a specially designed discontinuous Galerkin method based on the Bassi–Rebay numerical flux. The stability of the method is contingent upon an inf–sup condition, which we have proved to hold for any quasiuniform mesh family with sufficiently fine meshes. We have also shown the optimal convergence of the displacement and stress fields, and the convergence of the stress intensity factors extracted as the coefficients of the enrichment functions.     

Investigation of different isoparametric axisymmetric crack tip elements applied to a complete circumferential surface crack in a pipe

Standard isoparametric finite elements can be used as special crack tip elements in fracture mechanical computations by appropriately shifting the middle nodes in the neighbourhood of the crack tip. Such elements have already been applied to several plane and three-dimensional problems so that this method can be considered as commonly well accepted. In this paper the application of isoparametric axisymmetric elements as crack tip elements to a particular axisymmetric problem is studied. For that reason a complete circumferential crack at the inner surface of a pipe under axial tension is considered. The calculated stress intensity factors are compared with results from the literature. The general purpose finite element programs ASKA and ADINA have been used. In the first case triangular and quadrilateral elements were investigated, in the latter case quadrilateral and collapsed quadrilateral elements. In spite of the rather coarse grids good results for the stress intensity factor were found with the only exception of the collapsed quadrilateral elements.     

Numerical modeling of crack reorientation and link-up

The FRANC3D/BES software system has been used to simulate the reorientation and link-up of hydraulic fractures in three-dimensional (3D) problems. The adopted technique only needs to discretize the body surface and the crack surface. The crack propagation direction is determined using the minimum strain energy criterion. Crack propagation amount is calculated using the mode I stress intensity factor. In hydraulic fracturing, the number of multiple cracks for a given number of perforations depends on the resulting interaction of the cracks. The interaction may be expressed by the fracture stiffness which has been obtained for 3D problems in this paper.     

A finite element analysis of the finite width interface mixed mode bi-material fracture problem

One aspect of the terminal crack, mixed mode bi-material fracture mechanics problem is investigated using finite elements. The influence of a finite width bond line interface is considered for one representative material pair combination (E₂/E₁ = 0.10). The stress intensity factors for an inclined crack terminating at a variable thickness interface are established as a function of crack inclination. Since the order of the stress singularity is not the typical r^−1/2 associated with LEFM problems, variable power singular finite elements are used to model the terminating crack tip. Crack tip stress distributions and probable angles of crack extension are presented as a function of crack inclination and bond line thickness. Crack tip stress distributions assuming an interfacial debonding criterion are presented as a function of crack inclination and bond line thickness.     

半椭圆表面裂纹前缘应力强度因子的计算

在断裂力学中,如何求取应力强度因子一直是一个重要的课题.该文通过MSC.Marc提供的断裂力学模块,采用三维J积分法计算含有半椭圆表面裂纹前缘应力强度因子.首先通过MSC.Marc.Mentat建立特定裂纹体有限元模型,假设裂纹前缘处在平面应变状态下.由MSC.Marc计算出裂纹前沿的J积分,再由J积分计算出裂纹前缘的应力强度因子值.最后将计算结果与经验公式得到的结果进行了比较.仿真结果表明,通过MSC.Marc采用三维J积分法计算的应力强度因子具有较高的准确性和可靠性.