Stress intensity factor analyses of three-dimensional interface cracks using tetrahedral finite elements

A numerical method is proposed for evaluating the stress intensity factors of a three-dimensional bimaterial interfacial crack using tetrahedral finite elements. This technique is based on the M₁-integral method and employs the moving least-squares approximation. Stress or strain in the M₁-integral equation is automatically approximated from the nodal displacements obtained by the finite element analysis using the moving least-squares method. Therefore, the presented method needs no elemental information from the finite element analysis. In this study, stress intensity factor analyses of some typical three-dimensional interface crack problems using the tetrahedral finite elements are demonstrated.     

Finite element analysis of three dimensional crack growth by the use of a boundary element sub model

A new automated method to model non-planar three dimensional crack growth is proposed which combines the advantages of both the boundary element method and the finite element method. The proposed method links the two methods by a submodelling strategy in which the solution of a global finite element model containing an approximation of the crack is interpolated to a much smaller boundary element model containing a fine discretization of the real crack. The method is validated through several numerical comparisons and by comparison to crack growth measured in a test specimen for an engineering structure.     

Fully plastic analyses for notched bars and plates using finite element limit analysis

In the present work, fully plastic analyses for notched bars and (plane strain) plates in tension are performed, via finite element (FE) limit analysis based on non-hardening plasticity, from which plastic limit loads and stress fields are determined. Relevant geometric parameters are systematically varied to cover all possible ranges of the notch depth and radius. For the limit loads, it is found that the FE solutions for the notched plate agree well with the existing solution. For the notched bar, however, the FE solutions are found to be significantly different from known solutions, and accordingly the new approximation is given. Regarding fully plastic stress fields, it is found that, for the notched plate, the maximum hydrostatic (mean normal) stress overall occurs in the center of the specimen, which strongly depends on the relative notch depth and the notch radius-to-depth ratio. On the other hand, for the notched bar, the maximum hydrostatic stress can occur in between the center of the specimen and the notch tip. The maximum hydrostatic stress for a given notch depth can occur not for the cracked case, but for the notched case with a certain radius. This is true for both bars and plates. For a given notch radius, on the other hand, the maximum hydrostatic stress increases monotonically with the decreasing notch radius.     

Finite element discretization of open-type axisymmetric elements

A series of open-type elements which are compatible with axisymmetric thin shell elements have been dervied. These elements allow the effects of intermediate openings and discrete support systems for rotational shells to be realistically modelled in the dynamic regime. Also, consistent load vectors for several common cases are derived. The availablility of these elements in a public domain computer program enhances the possibility for modelling systems which are basically axisymmetric but which include intermediate or end supports using a rotational shell code.     

Finite element limit analyses of under-matched tensile specimens

This paper quantifies the effect of under-matching on plastic limit loads and fully plastic stress triaxialities for mismatched flat plate (in plane strain and plane stress) and round bar tensile specimens, via parametric finite element (FE) limit analyses based on elastic-perfectly-plastic materials. It is found that the effect of the strength mismatch ratio (the ratio of the yield strength of the weld zone to that of the base material) on plastic limit loads and fully plastic stress triaxialities can be significant for plane strain plate and round bar specimens, but much less significant for plane stress plate specimens. Its effect is dependent significantly on the slenderness of the weld zone (defined by the ratio of the weld zone width to the specimen width). The effect of the slenderness of the weld zone is apparent only when the weld width is less than the specimen width or diameter. In particular, the stress triaxiality in the softer weld zone can increase significantly with decreasing the ratio of the weld zone width to the specimen width (radius). Based on the present limit load results, a simple method to extract intrinsic tensile properties of the under-matched weld zone from test results of under-matched tensile specimens is proposed.     

Finite element solutions of convective diffusion problems

Three weighted residual methods are used to analyse several linear and nonlinear model problems related to the stream function–vorticity formulation of the Navier–Stokes equations. These methods are then used to solve the driven cavity problem. The results obtained at Reynolds numbers of 100 and 400 compare very favourably with those obtained by finite difference and spline methods on meshes using at least four times as many mesh points.     

Finite element recovery techniques for local quantities of linear problems using fundamental solutions

We show that local quantities of interest such as displacements or stresses of a FE–solution can be calculated with improved accuracy if fundamental solutions are employed. The approach is based on Bettis theorem and an integral representation of the local quantities via Greens function. The unknown Greens function is split into a regular part and a fundamental solution so that only the regular part must be approximated on the finite element ansatz space. Some numerical studies for linear elasticity will illustrate our approach.     

Finite element solutions of the non-self-adjoint convective-dispersion equation

Finite element formulations based on the Galerkin and variational principles have been developed for the self-adjoint and non-self-adjoint problems represented respectively by the flow and convective-dispersion equations in the cylindrical polar system of co-ordinates. The formulation based on the variational principle is shown to be restricted to dispersion-dominant transports only. The Galerkin method is demonstrated to be more versatile and free from convergence and stability problems. The computational scheme based on the Galerkin principle is shown to be equally valid for both convection and dispersion dominant transports. The numerical results obtained are verified with known analytical solutions. It is concluded that the suggested scheme can be used in solving a variety of field problems involving groundwater dispersion.     

Static stability of plates using fully conforming element

In this paper the geometric stiffness matrices of a 12 degree-of-freedom rectangular element based on a fully compatible displacement field are developed for the first time and used to solve the elastic stability problem of rectangular plates under combined in-plane forces. The rapid convergence of buckling solutions based on the derived geometric stiffness matrices is demonstrated for plates with various boundary and loading conditions. Comparison of results shows that the present conforming element gives in all cases better results than the well known 12 degree-of-freedom non-conforming rectangular element.     

Finite element analyses of three-dimensional crack problems in piezoelectric structures

In this paper, electromechanical fracture mechanics and finite element techniques for crack analyses are extended to three-dimensional crack configurations. Penny-shaped cracks and elliptical cracks are analyzed, subjected to combined mechanical tension and electric fields. For the penny-shaped crack, exact solutions originating from different resources are compared with numerical results. Some errors in the literature concerning the analytical solution for the elliptical crack are corrected. Numerical results of the stress-intensity factors and energy release rates for these crack configurations are presented.     

A three dimensional global weather prediction model using a finite element scheme for vertical discretization

A global three dimensional model for numerical weather prediction is described. It uses spheric harmonic basis functions with triangular truncation in the horizontal and a finite element discretization for the vertical. Model experiments are used to compare this model with another version, which uses a finite difference scheme for vertical discretization.     

Eroding interface and improved tetrahedral element algorithms for high-velocity impact computations in three dimensions

This paper presents a three-dimensional, eroding interface algorithm which allows Lagrangian computations to be performed for projectile penetration/perforation of thick plates. It also presents an improved tetrahedral element algorithm which provides for very accurate results under mild distortions and allows for continued computations under severe distortions. These algorithms have been incorporated into the EPIC-3 code and examples are presented to illustrate their usefulness.     

Analytical solutions of fully plastic crack-tip higher order fields under antiplane shear

Analytical solutions of higher order fields in a fully plastic power-law hardening material are presented. By the use of hodograph transformation and asymptotic analysis the stress and strain exponents, angular distributions of shear stresses and strains are analytically determined. Special cases, such as linearly elastic, perfectly plastic materials are discussed. Similar characteristics between mode III and mode I plane strain, and mode II plane stress are examined. Comparison of four-term asymptotic solutions with exact and leading term solutions in an infinite strip with a semi-infinite crack under constant displacements along its edges is provided.     

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.     

Reusable intrinsic sample point (RISP) algorithm for the efficient numerical integration of three dimensional curved boundary elements

This paper presents an algorithm that provides an order of magnitude gain in the computational performance of the numerical integration of the boundary integral equations for three dimensional analysis. Existing algorithms for numerical integration have strategically clustered integration sample points based on the relative proximity of the load points to the boundary element being integrated using element subdivision or element co-ordinate transformation. The emphasis in these techniques has been on minimizing the number of sample points required to obtain a given level of accuracy. The present algorithm, while closely following the spirit of these earlier approaches, employs a discrete number of sets of predetermined, customized, near-optimum, sample point quantities associated with the intrinsic boundary element. The ability created by this approach to reuse sample point geometric information of the actual element allows for the realization of substantive computational economy. This algorithm provides accurate and efficient numerical results both when load points are far from, and when they are on the boundary element being integrated. Numerical results are provided to demonstrate the substantial economy achieved through the use of the present algorithm.     

Finite element modelling of plastic instability during ECAP processing of flow-softening materials

A finite element analysis of the equal channel angular pressing (ECAP) of flow-softening materials is presented in this paper. A very fine mesh was used in the simulations, allowing a detailed analysis of the development of localized shear phenomena. Two different flow curves were used in the simulations; one displayed an initial flow-softening followed by perfect plastic behavior, whereas the other followed a constant flow-softening behavior. The flow-softening rate affects the intensity of shear localization. The deformation zone, that is usually concentrated around a fixed shear plane during processing of perfect plastic or strain hardening materials, splits into two parts and its position varies cyclically during the process, leading to oscillations in the punch load during processing. A comparison of the finite element predictions with those from the slip line field theory is also presented.     

Finite element formulations that include particular solutions of governing equations

The correspondence between the homogeneous and particular solutions of differential equations and the related solutions of the discrete equations is brought to light in this study. Guided by this correspondence, the trial functions for finite elements are written in two parts—one the homogeneous and the other a prescribed particular solution. It is shown that this approach allows one to simplify the computations of element matrices and also yields better results, throughout the element, than those obtained by the conventional approach. Two simple examples are provided for illustration.     

Finite element steady-state solutions of the traveling magnetic field problem

The paper is concerned with stability and accuracy of n-order finite element (FE) steady-state solutions of traveling magnetic field problem. It is found that the odd-order FE solutions (nis the odd number) are stable0 < sigma|u|h|nu < f(n) simeq 2.0 + 1.4(n - 1), and that the even-order FE solutions (nis the even number) are unconditionally stable. The consistent domain is also proposed, in which then-order FE solutions are stable and of 2n-order accuracy. Moreover, three-dimensional cases are dealt with, and the comparison with upwind methods is given. The merit and limits of the n-order FE method are finally cleared.     

Some improvements of accuracy and efficiency in three dimensional direct boundary element method

Numerical analysis with the Boundary Element Method (BEM) has been used more and more in various engineering fields in recent years. In numerical techniques, however, there are some problems which have not been fully solved even now. The most essential one is the drop in the accuracy of results for internal points near the boundary of the structure, where the singularity of integrands in the boundary integral equation is too strong to be evaluated with the normal numerical method. For the boundary integral equation of stress, this problem became more serious, and the accuracy can be improved only partly, even though very refined boundary elements are used. In this paper, the boundary integral equation is newly formulated using a relative quantity of displacement. In this way, the singularity of boundary integrals is reduced by the order of 1/r, and the accuracy of solution is improved significantly. Furthermore, in order to integrate it more accurately, two kinds of numerical integral methods are newly developed. By using these methods, both displacement and stress can be obtained with excellent accuracy at almost any point in the structure without any numerical difficulty, although the discretization may be comparatively coarse. The generality and practicability of the present formulation and integral methods are confirmed through some examples of three dimensional elastic problems.