A variable power singular element for analysis of fracture mechanics problems

A variable power displacement formulated singular element is developed. The stress and displacement fields within the element are expressed in terms of singular and regular components. The singular components of these fields are shown to provide a more accurate representation of the singularity than the total field solutions. The problem of a crack terminating at a bi-material interface is solved for two extreme ratios of singular element radius to crack length. The solution shows that accurate results can be achieved for either case.     

A singular cell-based smoothed radial point interpolation method for fracture problems

A cell-based smoothed radial point interpolation method (CS-RPIM) is developed for fracture problems. The strain smoothing is performed over background triangular cells. The stiffness matrix is calculated based on the weakened weak formulation, using shape functions obtained by radial point interpolation method. A layer of five-node singular elements are used to simulate the singularity around the crack tip. Different schemes are devised in the five-node elements to perform the strain smoothing. Several examples are presented to validate the newly developed method. The results are found in excellent agreement with the exact (or reference) solutions.     

Elastic-plastic nonlinearities considering fracture mechanics

The solution of fracture mechanics type problems including material and geometric non-linearities along with time-independent and time-dependent constitutive relations is discussed. The effect of solution tolerances on a fracture type problem are discussed and the use of Green's strain tensor and the Piola-Kirchhoff stress tensor is examined in a 1-D analysis and a 2-D fracture problem. A comparison of large and small displacement analysis for a center-cracked panel with elastic-plastic material is made. Small displacement, viscoplastic analysis results also are presented for a center-cracked panel.     

基于奇异值分解理论的钢轨断裂检测方法

为了检测出钢轨断裂点的准确位置,首先基于传输线理论建立钢轨断裂时的机车分路电流幅值包络仿真模型,分析了钢轨断裂对分路电流幅值包络的影响,然后利用对分路电流幅值包络进行多分辨奇异值分解后细节信号的奇异性特征检测钢轨断裂点的准确位置。实验结果表明,该方法可以有效检测出不同断裂点等效阻抗值下钢轨断裂点的准确位置,从而弥补了目前监测方法的不足。并且该方法的数据可以由机车自身的记录器提供,不需要增加其他检测设备,因此能够在降低检测成本的同时能满足检测及时性方面的要求。     

Self-adaptive finite elements in fracture mechanics

A new finite element capability which permits the analyst to vary the order of polynomial approximation over each finite element is discussed with reference to its potential for application to stress intensity factor computations in linear elastic fracture mechanics. Computational experiments, in which polynomial orders ranging from 1 to 8 were used, indicated strong and monotonie convergence of the strain energy release rate even for very coarse finite element meshes as the order p of the approximating polynomial was increased. Pointwise convergence of stresses was achieved by averaging approximations of different polynomial orders. The strong and monotonie convergence of K_I factors with respect to increasing p provides a new method for computing stress intensity factors. The main advantage of this method is that the accuracy of approximation can be established without mesh refinement or the use of special procedures.     

Semi-radial singularity mapping theory for line singularities in fracture mechanics

Fundamental theory of the semi-radial singularity mapping dealing with line singularities in the stress and strain is presented. Explicitly the brick-type semi-radial singularity mapping element as well as the wedge-type one are proposed. Element familization is then described based on the conventional polynomial interpolations for the cube and simplex, with examinations of the elaborate trial function spaces.     

A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics

We introduce a novel enriched Boundary Element Method (BEM) and Dual Boundary Element Method (DBEM) approach for accurate evaluation of Stress Intensity Factors (SIFs) in crack problems. The formulation makes use of the Partition of Unity Method (PUM) such that functions obtained from a priori knowledge of the solution space can be incorporated in the element formulation. An enrichment strategy is described, in which boundary integral equations formed at additional collocation points are used to provide auxiliary equations in order to accommodate the extra introduced unknowns. In addition, an efficient numerical quadrature method is outlined for the evaluation of strongly singular and hypersingular enriched boundary integrals. Finally, results are shown for mixed mode crack problems; these illustrate that the introduction of PUM enrichment provides for an improvement in accuracy of approximately one order of magnitude in comparison to the conventional unenriched DBEM.     

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

Two aspects of the mixed mode bi-material fracture mechanics problem are investigated using finite elements. The stress intensity factors for an inclined crack at various distances from a bi-material interface are established as a function of inclination for two material pair combinations. The probable angle of crack extension is established for this problem using the maximum hoop stress criterion. The inclined terminal crack problem is studied using variable power singular elements at the interface. Crack tip stress distributions and probable angle of crack extension are presented as functions of crack inclination and material pair combinations. Crack tip stress distributions assuming an interfacial debonding criterion are also presented as functions of crack inclination and material pair combinations.     

A one field full discontinuous Galerkin method for Kirchhoff–Love shells applied to fracture mechanics

In order to model fracture, the cohesive zone method can be coupled in a very efficient way with the finite element method. Nevertheless, there are some drawbacks with the classical insertion of cohesive elements. It is well known that, on one the hand, if these elements are present before fracture there is a modification of the structure stiffness, and that, on the other hand, their insertion during the simulation requires very complex implementation, especially with parallel codes. These drawbacks can be avoided by combining the cohesive method with the use of a discontinuous Galerkin formulation. In such a formulation, all the elements are discontinuous and the continuity is weakly ensured in a stable and consistent way by inserting extra terms on the boundary of elements. The recourse to interface elements allows to substitute them by cohesive elements at the onset of fracture.The purpose of this paper is to develop this formulation for Kirchhoff–Love plates and shells. It is achieved by the establishment of a full DG formulation of shell combined with a cohesive model, which is adapted to the special thickness discretization of the shell formulation. In fact, this cohesive model is applied on resulting reduced stresses which are the basis of thin structures formulations. Finally, numerical examples demonstrate the efficiency of the method.     

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.     

基于概率断裂力学的管道疲劳寿命分析

采用奇异单元模拟裂纹尖端应力场的奇异性,计算裂纹尖端的应力强度因子和张开应力.以概率论为基础,结合确定性疲劳断裂力学估算方法,考虑参数的不确定性和随机性,应用蒙特卡洛模拟法分析管道的疲劳寿命.结果表明：通过J积分计算得到的裂纹尖端张开应力与计算得到的管道工作应力基本相等.采用蒙特卡洛模拟法进行的一定可靠度和置信度下的疲劳寿命预测能反映评定参数的不确定性,较传统的断裂力学计算结果更安全.     

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.     

Axisymmetric shell optimization under fracture mechanics and geometric constraint

This paper describes a problem of axisymmetric shell optimization under fracture mechanics and geometric constraints. The shell is made from quasi-brittle materials, and through crack arising is admitted. It is supposed that the shell is loaded by cyclic forces. A crack propagation process related to the stress intensity factor is described by Paris fatigue law. The problem of finding the meridian shape and the thickness distribution (geometric design variables) of the shell having the smallest mass subject to constraints on the cyclic number for fatigue cracks and geometrical constraint on the shell volume is investigated. Special attention is devoted to different possibilities of problem transformation and analytical methods of their solution. Using minimax approach, optimal shapes of the shells and their thickness distributions have been found analytically.     

On adaptive strategies and error control in fracture mechanics

Finite element approximations in elastic fracture mechanics are traditionally carried out on a priori constructed meshes with singularity elements that surrounds the crack tip. In this contribution we discuss an adaptive algorithm based on goal oriented error measures using ordinary elements and a p-refinement for the linearization of the secant forms. The numerical experiments show that good results can be obtained without imposition of hands before the computations starts. Furthermore, we note that it is often possible, for engineering goal quantities, to linearize the data to the dual problem such that additional discretization error is avoided, and the error representation formula is in a sense trivial.     

A stabilized multidomain approach for singular perturbation problems

We present a new technique of stabilization for finite difference or spectral approximations of singular perturbation problems. Here we allow the artificial viscosity to be constant and independent of the step size. The results are generalized to variable coefficient problems. Suitable multigrid components are proposed. Numerical results are presented which substantiate the usefulness of our technique.