Two-dimensional finite element method for stress intensity factor using adaptive mesh strategy

Finite element analysis (FEA) combined with the concepts of Linear Elastic fracture mechanics (LEFM) provides a practical and convenient means to study the fracture and crack growth of materials. A numerical analysis (FEM) of cracks was developed to derive the SIF for two different geometries, i.e., a rectangular plate with half circle-hole and central edge crack plate in tension loading conditions. The onset criterion of crack propagation is based on the stress intensity factor, which is the most important parameter that must be accurately estimated and facilitated by the singular element. Displacement extrapolation technique (DET) is employed, to obtain the stress intensity factors (SIFs) at crack tip. The fracture is modeled by the splitting node approach and the trajectory follows the successive linear extensions of each crack increment. These comprehensive tests are evaluated and compared with other relevant numerical and analytical results obtained by other researchers.     

A singular finite element for computing time dependent stress intensity factors

A triangular finite element was developed for the purpose of computing time dependent stress intensity factors in cracked panels caused by dynamic loadings. An explicit consistent mass matrix was formulated for use with an existing stiffness matrix developed earlier. The singular finite element and a conventional triangular plane element were used to solve a plane problem with a known solution to evaluate the accuracy. Several other problems with time dependent loadings were solved and discussed.     

Stress intensity factor evaluation using singular finite elements

The classical theory of linear elastic fracture mechanics proposes that the stress and energy field near a crack tip can be accurately evaluated by determining the stress intensity factors. Several recent investigations, however, have demonstrated the previously unrecognized importance of the higher-order terms also present in the series eigenfunction representation of the near-tip crack environment. The finite element method has been shown to quite effectively yield these higher-order coefficients, with the method previously utilized only to determine the first term of the series expansion (the stress intensity factor). By numerically evaluating the higher-order coefficients for several finite geometries, the near-tip environment has been shown to be much more sensitive to variations in these terms, than previously believed. This is a phenomenon that no accurate crack propagation study, regardless of specific propagation theory, should disregard without careful consideration, particularly because of the inherent accumulated error in any incremental propagation study.     

基于有限单元法对动态应力强度因子进行数值计算的研究*

运用ANSYS 商用有限元软件,采用非奇异单元和Newmark 积分算法,通过最小二乘法拟合,准确的获得了多个模型动态应力强度因子的解。所使用的方法适用范围广,这对于运用线弹性断裂动力学解决工程中的实际问题是有益的和必要的。     

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.     

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.     

A mixed finite element method for beam and frame problems

 In this work we consider solutions for the Euler-Bernoulli and Timoshenko theories of beams in which material behavior may be elastic or inelastic. The formulation relies on the integration of the local constitutive equation over the beam cross section to develop the relations for beam resultants. For this case we include axial, bending and shear effects. This permits consideration in a direct manner of elastic and inelastic behavior with or without shear deformation. A finite element solution method is presented from a three-field variational form based on an extension of the Hu–Washizu principle to permit inelastic material behavior. The approximation for beams uses equilibrium satisfying axial force and bending moments in each element combined with discontinuous strain approximations. Shear forces are computed as derivative of bending moment and, thus, also satisfy equilibrium. For quasi-static applications no interpolation is needed for the displacement fields, these are merely expressed in terms of nodal values. The development results in a straight forward, variationally consistent formulation which shares all the properties of so-called flexibility methods. Moreover, the approach leads to a shear deformable formulation which is free of locking effects – identical to the behavior of flexibility based elements. The advantages of the approach are illustrated with a few numerical examples. Dedicated to the memory of Prof. Mike Crisfield, for his cheerfulness and cooperation as a colleague and friend over many years.     

A relaxation technique for evaluating stress intensity factors by the finite element method

A simple two-step corrective technique is presented in this paper for evaluating stress intensity factors in crack problems. In the first step an approximate evaluation of the stress intensity factor was made by considering the cracked plate to be of infinite size. The stresses of the problem were relaxed by the stresses of the infinite body which corresponds to the approximate value of the stress intensity factor. The expected discrepancy in the value of SIF by the infinite plate approximation was corrected in the second step where the existing residual stresses are equilibrated at the cracked plate by using any of the conventional finite element techniques and the corrective value of the stress intensity factor is calculated by using an appropriate collocation formula. The method was applied to three typical plane problems of cracked plates with satisfactory results.     

Estimating stress intensity factors with singular components of the total finite element solution

Displacement formulated singular elements are compared to isoparametric quarter-node elements. The total stress and displacement solution for each element is decomposed into singular and regular components for evaluating stress intensity factors. Specific relationships between nodal displacements and the singular component of stress are presented for the isoparametric element at selected locations within the element. Numerical results for K_I and K_II in the slant crack problem are presented. The singular components are shown to produce superior results when considering crack opening displacements.     

Improvement of stress singular element for crack problems in three dimensional boundary element method

In the paper, a new kind of stress singular element is introduced for crack problems. This kind of element is more simple and widely used than those presented before. In the paper, a cube with embedded circular crack and a first kind Benchmark problem are studied. The study shows that using quarter-point element and the stress singular element can obviously improve the accuracy. The influences of methods estimating stress intensity factor on accuracy are also studied.     

Body-force linear elastic stress intensity factor calculation using fractal two level finite element method

Fractal two level finite element method (F2LFEM) for the analysis of linear fracture problems subjected to body force loading is presented. The main objective here is to show that by employing the F2LFEM a highly accurate and an efficient linear analysis of fracture bodies subjected to internal loading can be obtained as it is hard to find any analytical and exact values of stress intensity factor (SIF) for any kind of geometry subjected to internal loading. Also in this paper, a fast method to transform the body force to the reduced force vector is presented and has been effectively employed. The problems solved here include both the single mode or mixed mode cracks subjected to internal body-force or external loading. In comparison with other numerical algorithms, it seems that with a small amount of computational time and computer storage, highly accurate results can be obtained.     

Subdomain-based error techniques for generalized finite element approximations of problems with singular stress fields

This paper considers four types of error measures, each tailored to the generalized finite element method. Particular attention is given to two-dimensional elasticity problems with singular stress fields. The first error measure is obtained using the equilibrated element residual method. The other three estimators overcome the necessity of equilibrating the residue by employing a subdomain strategy. In this strategy, the partition of unity (PoU) property is used to decompose the error problem into local contributions over each patch of elements. The residual functional of the error problem is the same for the subdomain estimators, but the bi-linear form is different for each one of them. In the second estimator, the bi-linear form is weighted by the PoU functions associated with the patch over which the error problem is stated. No weighting appears in the bi-linear form of the third estimator. The fourth measure is proposed as an alternative strategy, in which the products of the PoU functions and test functions are introduced as weights in the weighted integral statement of the differential equation describing the error problem. The linear form of the local error problem is then identical to that of the other subdomain techniques, while the bi-linear form is stated differently, with the PoU functions directly multiplying the test functions. The goal of this study is to investigate the performance of the four estimators in two-dimensional elasticity problems with geometries that produce singularities in the stress field and concentration of the error in the numerical solution.     

The convergence of a mixed finite element method for steady convective incompressible viscous flow

Some results on the convergence of the assumed deviatoric stress-pressure-velocity mixed finite element method for steady, convective, incompressible, viscous flow are given. An abstract error estimate is proved, which shows that the same LBB conditions for hybrid finite element method for Stokes flow are also applicable to the present method. An unusual term appears in the estimate, the rate of convergence for this term is examined. To make our idea clear, the same finite element method is applied to single elliptic equations first.This work was supported by the Science Foundation of Academia Sinica, No. (84)-103     

Elastodynamic analysis of crack by finite element method using singular element

The finite element method using a singular element near the crack tip is extended to the elastodynamic problems of cracks where the displacement function of the singular element is taken from the solution of a propagating crack. The dynamic stress intensity factor for cracks of mode III or mode I deformations in a finite plate is determined.The results of computation for stationary cracks or propagating cracks under dynamic loadings are compared with the analytical solutions of other authors. It is shown that the present method satisfactorily describes the time variation of the stress intensity factor in dynamic crack problems.

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Résumé La méthode des éléments finis utilisant un élément singulier au voisinage de l'extrémité d'une fissure a été étendue aux problèmes élastodynamiques des fissures tels qu'ils se posent lorsque la fonction de déplacement d'un élément singulier est prise à partir de la solution d'une fissure en cours de propagation. Le facteur d'intensité des contraintes dynamiques correspondant à des fissures de mode III ou des déformations de mode I dans une plaque finie a été déterminé. Les résultats des calculs correspondant à des fissures stationnaires ou des fissures en cours de propagation sous des charges dynamiques sont comparées aux solutions analytiques obtenues par d'autres auteurs. On montre que la méthode présentée décrit de façon satisfaisante la variation en fonction du temps du facteur d'intensité des contraintes dans les problèmes de fissuration dynamique.

     

A finite element convergence study for accelerating flow problems

The convergence properties of several non-linear solution procedures were examined with respect to the accelerated flow of a fluid in a converging channel (the Hamel problem), using two different finite element computer programs with different elemental construction. The Reynolds number varied from that for creeping flow to 1088 without exceeding the radius of convergence. Special attention was given to the successive substitution and Newton–Raphson solution algorithms, with a significant advantage in rate of convergence noted for the latter.     

Determination of stress intensity factors in cracked plates by the finite element method

Stress fields near crack tips in an elastic body can be specified by the stress intensity factors which are closely related to the stress singularities arising from the crack tips. These singularities, however, cannot be represented exactly by conventional finite element models. A new method for the analysis of stresses around cracks is proposed in this paper on the basis of the superposition of analytical and finite element solutions. This method is applied to several two-dimensional problems whose solutions are obtained analytically, and it is shown that their numerical results are in excellent agreement with analytical ones. Sufficiently accurate results can be obtained by the conventional finite element analysis with rather coarse mesh subdivision. Computational efforts are then considerably reduced compared with other methods.     

A simple procedure for the accurate determination of stress intensity factors by finite element method

A simple procedure for the accurate determination of stress intensity factors K_I, K_II by the conventional finite element method is proposed. The first step of the method is to calculate the stress σ₂ of the plate without a crack. The second step is to calculate the stress σ_tip, of the plate with the crack. The value of (σ_tip−σ_g) at the crack tip element is regarded to have the intimate relation with K_I, K_II K_I, and K_II are determined from the value of (σ_tip−σ_g) and a standard solution. It is shown that the results obtained for many problems by the proposed method are in excellent coincidence with the analytical solutions. The error is below 1–3% for the most cases.     

On a mixed finite element method for clamped anisotropic plate bending problems

The paper deals with a new mixed finite element method of solution of the bending problem of clamped anisotropic/orthotropic/isotropic plates with variable/constant thickness. This new mixed method gives simultaneous approximations to displacement u and bending and twisting moment tensor (ψ_ij)_{1 ≤ i, j ≤ 2}. Computer implementation procedures for this mixed method are given along with results of numerical experiments on a good number of interesting problems.