Finite element Euler computations in three dimensions

An explicit finite element solution procedure for the three dimensional Euler equations is presented. The solution domain is automatically meshed using a tetrahedral mesh generator which is an extension of our previous two dimensional work. Several examples are included to illustrate the performance of the generator and solver. An adaptive mesh regeneration procedure is used for the first time in three dimensions.     

Finite element methodology for path integrals in fracture mechanics

Based on the energy foundation of the path-independent integral in non-linear fracture mechanics, I* integral as the dual form of Rice's J is presented, it is also path-independent and is equivalent to J in value but it relates to the complementary energy. It is proved that, in numerical implementation, the path independence of J and I* can be ensured by using the assumed displacement finite elements and the assumed stress finite elements, respectively. Regarding the bounds of crack parameters, it is demonstrated that the lower bound of J can be estimated by the displacement compatible elements, and the upper bound of I* can be estimated by the stress equilibrium elements. In view of the difficulties in formulating stress equilibrium model, instead of it, a quasi-equilibrium model is proposed, which makes hybrid stress elements be able to estimate the bound of I*, and do not lose the characteristics of stiffness formulation. Two four-node plane elements are suggested; of them, the incompatible one can be used in incompressible/fully plastic fracture analysis, and the penalty-equilibrium one can be implemented to estimate the bound of I*. Furthermore, an incremental formulation is developed for I*, and can be extended into the calculations of ductile fracture under monotonic loading. For attestation, quite a number of numerical experiments is carried out, and some significant results are offered. © 1998 John Wiley & Sons, Ltd.     

拉压异性纤维增强树脂基复合材料弹塑性问题的有限元分析

为了对复杂的非线性问题进行便捷求解,首先提出了考虑拉压异性的纤维增强树脂基复合材料统一非线性本构模型;然后,在此基础上进一步导出了本构模型的三维表现形式,以适用于非线性有限元分析工具的开发;随后,利用有限元软件ABAQUS提供的用户自定义子程序UMAT,自编了在二维和三维情况下的弹塑性应力分析程序;最后,应用程序对复合材料单向板和复合材料斜交板在偏轴拉伸/压缩下应力-应变曲线的预测与测试结果进行了比较,探讨了复合材料悬臂梁的弹塑性问题,并分析和比较了有无考虑拉压异性情况下应力分布和挠度响应的差异。结果表明:运用所提出的本构模型对考虑拉压不对称问题的弹塑性变形分析十分有效,这一本构模型有望成为实用数值分析工具,进而指导工程实践。     

Finite element modelling of crack propagation in elastic-plastic media

Materials which are cyclically stressed by sliding indenters often undergo fatigue wear, as surface breaking vertical cracks and subsurface horizontal cracks propagate causing eventual loss of material. In this study, the authors model crack propagation in an elastic-plastic material using finite element techniques, and consider the influence of friction, elasticity, plasticity and degree of penetration on the J-integral at the tip of a vertical crack. Crack propagation directions are estimated using J-integral maxima as the determining variable. It is found that the J-integral values, as a measure of strain energy release rate, can be used to estimate the crack propagation angle. Its main advantage lies in the fact that it considers both modes (I, II) of crack propagation. Using the J-integral values, one finds that, in the absence of friction between the indenter and the material, the vertical crack is equally prone to propagation at both 45 and 135° angles. However, one notices that the vertical crack favours the direction opposite to the direction of rolling for non-zero values of friction, i.e. 135°. The effects of both the crack depth and the crack tip plasticity are also investigated. It is found that any experimental findings suggestive of crack orientations closer to the horizontal in the direction opposite to the sliding direction are probably a result of shallow vertical asperities or higher crack tip plasticity.     

Finite element analysis of void growth in elastic-plastic materials

Three-dimensional finite element computations have been carried out for the growth of initially spherical voids in periodic cubic arrays and for initially spherical voids ahead of a blunting mode I plane strain crack tip. The numerical method is based on finite strain theory and the computations are three-dimensional. The void cubic arrays are subjected to macroscopically uniform fields of uniaxial tension, pure shear and high triaxial stress. The macroscopic stress-strain behavior and the change in void volume were obtained for two initial void volume fractions. The calculations show that void shape, void interaction and loss of load carrying capacity depend strongly on the triaxiality of the stress field. The results of the finite element computation were compared with several dilatant plasticity continuum models for porous materials. None of the models agrees completely with the finite element calculations. Agreement of the finite element results with any particular constitutive model depended on the level of macroscopic strain and the triaxiality of the remote uniform stress field. For the problem of the initial spherical voids directly ahead of a blunting mode I plane strain crack tip, conditions of small scale yielding were assumed. The near tip stress and deformation fields were obtained for different void-size-to-spacing ratios for perfectly plastic materials. The calculations show that the holes spread towards the crack tip and towards each other at a faster rate than they elongate in the tensile direction. The computed void growth rates are compared with previous models for void growth.

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Résumé On a effectué des calculs par éléments finis à trois dimensions pour l'étude de la croissance de lacunes, initialement sphériques, selon des arrangements en cubes périodiques, et pour des lacunes initialement sphériques se formant en avant de l'extrémité arrondie d'une fissure sollicitée en état de déformation plane et selon un Mode I. La méthode numérique est basée sur la théorie des dilatations finies et les calculs sont effectués en trois dimensions.La disposition en cubes des lacunes est sujette à des champs macroscopiquement uniformes de tractions uniaxiales, de cisaillement pur et de contraintes à haute triaxialité. Le comportement macroscopique en contraintes-dilatations et le changement de volume des lacunes ont été obtenus pour deux fractions du volume initial de cavités.Les calculs montrent que la forme des cavités, leurs interactions et la perte résultante de capacité de change dépendent fortement de la triaxialité du champ de contraintes. On compare les résultats des calculs par éléments finis avec divers modèles de continuum à plasticité dilatante applicable à des matériaux poreux. Aucun de ces modèles n'est en accord complet avec les calculs par éléments finis. L'accord des résultats par éléments finis avec un modèle constitutif particulier dépend du niveau de déformation macroscopique et de la triaxialité du champ des contraintes uniformes agissant à distance. On a supposé des conditions d'écoulement plastique à petite échelle, pour résoudre le problème des lacunes initialement sphériques se trouvant directement en avant de l'extrémité d'une fissure arrondie sollicitée en Mode I et en déformation plane.Les champs de contrainte et de déformation près de la pointe de la fissure sont obtenus pour des matériaux parfaitement plastiques et pour différents rapports dimensions/espacements.Les calculs montrent que les cavités se développent vers l'extrémité de la fissure et l'une vers l'autre à une vitesse plus grande qu'elles ne s'allongent dans la direction de la contrainte de traction.On compare les vitesses calculées de croissance des lacunes avec celles fournies par de précédents modèles.

     

Microstructure and fracture in three dimensions

A high resolution three dimensional (3D) scanning technique called X-ray microtomography was used to measure internal crack growth in small mortar cylinders under compressive loading. Tomographic scans were made at different load increments in the same specimen. 3D image analysis was used to measure internal crack growth during each load increment. Load–deformation curves were used to measure the corresponding work of the external load on the specimen. Fracture energy was calculated using a linear elastic fracture mechanics approach using the measured surface area of the internal cracks created. The measured fracture energy was of the same magnitude that is typically measured in concrete tensile fracture. A nominally bilinear incremental fracture energy curve was measured. Separate components for crack formation energy and secondary toughening mechanisms are proposed. The secondary toughening mechanisms were found to be about three times the initial crack formation energy.     

Energy flux into process region in elastic-plastic fracture problems

Formulas for the fluxes of total and free energies into a finite fracture process region near a crack tip are derived taking account of the effects of the plastic deformation, thermal strain, body forces and inertia forces. Two different limiting procedures of shrinking the size of the process region are considered in order to obtain the formulas for a crack model with an infinitesimal process region. The formulations are extended to three-dimensional crack problems.     

Finite element crack growth algorithm for dynamic fracture

A new finite element crack growth algorithm has been developed to simulate dynamic fracture. In this algorithm, pseudo elements with very high initial density are placed below the crack plane and the density is reduced to zero in a gradual manner as the crack passes the element. A number of linear elastic and elasto-viscoplastic problems have been carried out to test the new algorithm. The results are compared with some of the existing crack growth models.     

Griffith's theory of brittle fracture in three dimensions

The Griffith theory of brittle fracture is extended to the three-dimensional problem of a flat elliptical crack in an otherwise uniform field of tensile and shear stresses. A method for finding the correct expressions of the change in strain energy due to the elliptical crack is developed. This is done by expressing the stresses and displacements in terms of the radius R₀ of a large sphere around the crack and by imposing the condition of equilibrium that the stresses or displacements across the spherical surface should agree with the prescribed boundary conditions as R₀ → ∞. The strain energy due to the presence of the elliptical crack is found to be independent of the tension applied parallel to the crack plane at infinity. On the basis of the thermodynamic argument of Griffith, it is also observed that the critical tensile and shear stresses increase rapidly as the ratio of major to minor semi-axes of the ellipse approaches unity.     

Finite element schemes for non-linear problems in infinite domains

A class of non-linear elliptic problems in infinite domains is considered, with non-linearities extending to infinity. Examples include steady-state heat radiation from an infinite plate, and the deflection of an infinite membrane on a non-linear elastic foundation. Also, this class of problems may serve as a starting point for treating non-linear wave problems. The Dirichlet-to-Neumann (DtN) Finite Element Method, which was originally developed for linear problems in infinite domains, is extended here to solve these non-linear problems. Several DtN schemes are proposed, with a trade-off between accuracy and computational effort. Numerical experiments which demonstrate the performance of these schemes are presented. © 1998 John Wiley & Sons, Ltd.     

A boundary face method for potential problems in three dimensions

This work presents a new implementation of the boundary node method (BNM) for numerical solution of Laplace's equation. By coupling the boundary integral equations and the moving least‐squares (MLS) approximation, the BNM is a boundary‐type meshless method. However, it still uses the standard elements for boundary integration and approximation of the geometry, thus loses the advantages of the meshless methods. In our implementation, here called the boundary face method, the boundary integration is performed on boundary faces, which are represented in parametric form exactly as the boundary representation data structure in solid modeling. The integrand quantities, such as the coordinates of Gauss integration points, Jacobian and out normal are calculated directly from the faces rather than from elements. In order to deal with thin structures, a mixed variable interpolation scheme of 1‐D MLS and Lagrange Polynomial for long and narrow faces. An adaptive integration scheme for nearly singular integrals has been developed. Numerical examples show that our implementation can provide much more accurate results than the BNM, and keep reasonable accuracy in some extreme cases, such as very irregular distribution of nodes and thin shells. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite element techniques for problems of unbounded domains

A general approach for using infinite elements within the context of the finite element method is presented. Based upon Gauss-Laguerre quadrature, a scheme for numerically computing the infinite element characteristic matrices is proposed. The general formulation of the element shape functions is outlined, including that for infinite elements that model the far field of hyperbolic systems. The technique is illustrated with a classical example in elasticity: a vertically loaded circular rigid plate.     

Finite deformation finite element analyses of tensile growing crack fields in elastic-plastic material

Finite deformation finite element analyses of plane strain stationary and quasi-statically growing crack fields in fully incompressible elastic-ideally plastic material are reported for small-scale yielding conditions. A principal goal is to determine the differences between solutions of rigorous finite deformation formulation and those of the usual small-displacement-gradient formulation, and thereby assess the validity of the (nearly all) extant studies of ductile crack growth that are based on a small-displacement-gradient formulation. The stationary crack case with a significantly blunted tip is studied first; excellent agreement in stress characteristics at all angles about the crack tip and up to a radius of about three times the crack tip opening displacement is shown between Rice and Johnson's [1] approximate analytical solution and our numerical solution. Outside this radius, the numerical results agree very well with Drugan and Chen's [2] small-displacement-gradient analytical characteristics solution in the region of principal plastic deformation. Thus we identify accurate analytical representations for the stress field throughout the plastic zone of a blunted stationary crack. For the growing crack case, the macroscopic difference in crack tip opening profiles between previous small-displacement-gradient solutions and the present results is shown to be negligible, as is the difference in the stress fields in plastic regions. The stress characteristics again agree very well with analytical results of [2]. The numerical results suggest—in agreement with a recent analytical finite deformation study by Reid and Drugan [3]—that it is the finite geometry changes rather than the additional spin terms in the objective constitutive equation that cause any differences between the small-displacement-gradient and the finite deformation solutions, and that such differences are nearly indistinguishable for growing cracks.     

Finite element analysis of elastic-plastic layered soil using discrete Fourier series expansion

A method is developed which basically concerns how to reduce the necessary storage and cost of solving a set of algebraic equations arising from applying the initial stress technique to the analysis of a uniformly layered elastic-plastic soil. Advantage is taken of the geometrical symmetry of the problem and the consequent symmetry of the stiffness matrix. The nodal values are expanded in a discrete Fourier series so that a two-dimensional problem is reduced to a number of one-dimensional problems.     

Finite element solution for the continuum traffic equilibrium problems

In this paper, we consider a city with a highly compact Central Business District (CBD), and the commuters’ destinations from the CBD are dispersed over the whole city. The street network is approximated as a continuum and commuters’ movements in the city are measured by the flow intensity, and the local travel cost depends on the location and the traffic flow intensity. We extend the continuum user equilibrium problem to deal with the case of variable demand, in which the traffic demand from any destination in the city to the CBD is assumed to be a function of both the destination location and the total travel cost to the CBD. An equivalent mathematical model is formulated and proved to satisfy the user equilibrium conditions, which is then solved by a finite element solution algorithm. Numerical examples are given to illustrate the effectiveness of the proposed method. © 1998 John Wiley & Sons, Ltd.     

Comparison of finite element solutions with analytical and experimental data for elastic-plastic cracked problems

Non-linear finite element results from a round robin are compared with empirical and experimental data obtained for three types of geometries: compact specimen, three-point bend specimen and a center-cracked panel subject to uniaxial loading. The solution parameters to be compared in various forms are: applied load, clip gauge displacement, RiceJ-integral (with no limit placed on method of calculation),K ₁ crack opening profiles and plastic zone development.

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Résumé Les résultats d'éléments finis non linéaires résultant d'une consultation circulaire sont comparés avec des données empiriques et expérimentales obtenues pour trois types de géométrie: éprouvette compacte, éprouvette de flexion en trois points et panneau à fissure centrale soumis à une charge uniaxiale. Les paramètres de la solution qui doivent être comparés dans les différentes formes sont: la charge appliquée, le déplacement d'un extensiomètre, l'intégraleJ (sans réserve sur sa méthode de calcul), KI, les profils d'ouverture de fissure ainsi que le développement de la zone plastique.

     

Dispersion analysis of the gradient weighted finite element method for acoustic problems in one,two, and three dimensions

This paper reports a detailed analysis on the numerical dispersion error in solving one-, two-, and three-dimensional acoustic problems governed by the Helmholtz equation using the gradient weighted finite element method (GW-FEM) in comparison with the standard FEM and the modified methods presented in the literatures. The discretized system equations derived based on the gradient weighted operation corresponding to the considered method are first briefed. The discrete dispersion relationships relating the exact and numerical wave numbers defined in different dimensions are then formulated, which will be further used to investigate the dispersion effect mainly caused by the approximation of field variables. The influence of nondimensional wave number and wave propagation angle on the dispersion error is detailedly studied. Comparisons are made with the classical FEM and high-performance algorithms. Results of both theoretical and numerical experiments show that the present method can effectively reduce the pollution effect in computational acoustics owning to its crucial effectiveness in handing the dispersion error in the discrete numerical model.     

Finite element method in magnetohydrodynamic channel flow problems

The finite element method has been applied to the steady-state fully developed magnetohydrodynamic channel flow of a conducting fluid in the presence of transverse magnetic field. Simple elements have been used to obtain the numerical values of velocity and induced magnetic field. To test the efficiency of the method, three different geometries, viz., rectangle, circle and triangle, are taken as the section of the pipe whose walls are non-conducting. Comparison is made with those cases in which exact solutions are available. Apart from giving good results, the FEM makes it possible to solve the problem for a pipe with arbitrary cross-section which was not possible by the other methods.