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.     

A transition element for uniform strain hexahedral and tetrahedral finite elements

A transition element is presented for meshes containing uniform strain hexahedral and tetrahedral finite elements. It is shown that the volume of the standard uniform strain hexahedron is identical to that of a polyhedron with 14 vertices and 24 triangular faces. Based on this equivalence, a transition element is developed as a simple modification of the uniform strain hexahedron. The transition element makes use of a general method for hourglass control and satisfies first‐order patch tests. Example problems in linear elasticity are included to demonstrate the application of the element. Copyright © 1999 John Wiley & Sons, Ltd. This paper was produced under the auspices of the U.S. Government and it is therefore not subject to copyright in the U.S.     

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 model updating using crow search algorithm with Levy flight

The accuracy of the finite element model is essential for fatigue analysis and vibration of a flexible body. This paper presents a finite element model updating (FEMU) algorithm based on crows search algorithm incorporated with Levy flight (LFCSA). The proposed algorithm is tested on the updating of two different cases: a simple structure (beam) and a complex structure (gearbox housing). To verify the performance of LFCSA, two optimization algorithm, which is the standard CSA and the particle swarm optimization algorithm with Levy flight (LFPSO), are applied to FEMU of the simple beam. The result of the comparison shows that the LFCSA has sufficient convergence speed and global search ability in the implementation of FEMU. In the case of gearbox housing, the error of frequencies and mode shapes are analyzed to indicate that the LFCSA gives a satisfactory result in a complex structure.     

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.     

Finite element shakedown analysis of two‐dimensional structures

In the present paper, the formulation proposed by Casciaro and Garcea (Comput. Meth. Appl. Mech. Eng., 2002; 191 :5761–5792) and applied to the shakedown analysis of plane frames, is extended to the analysis of two‐dimensional flat structures in both the cases of plane‐stress and plane‐strain. The discrete formulation is obtained using a mixed finite element in which both stress and displacement fields are interpolated. The material is assumed to be elasto‐plastic and a linearization of the elastic domain is performed. The result is a versatile iterative scheme well suited to implementation in general purpose FEM codes. An extensive series of numerical tests is presented showing the reliability of the proposed formulation. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

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.     

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.     

A new finite element approach for solving three‐dimensional problems using trimmed hexahedral elements

In this paper, a novel finite element approach is presented to solve three‐dimensional problems using trimmed hexahedral elements generated by cutting a simple block consisting of regular hexahedral elements with a computer‐aided design (CAD) surface. Trimmed hexahedral elements, which are polyhedral elements with curved faces, are placed at the boundaries of finite element models, and regular hexahedral elements remain in the interior regions. Shape functions for trimmed hexahedral elements are developed by using moving least square approximation with harmonic weight functions based on an extension of Wachspress coordinates to curved faces. A subdivision of polyhedral domains into tetrahedral sub‐domains is performed to construct shape functions for trimmed hexahedral elements, and numerical integration of the weak form can be carried out consistently over the tetrahedral sub‐domains. Trimmed hexahedral elements have similar properties to conventional finite elements regarding the continuity, the completeness, the node–element connectivity, and the inter‐element compatibility. Numerical examples for three‐dimensional linear elastic problems with complex geometries show the efficiency and effectiveness of the present method. Copyright © 2015 John Wiley & Sons, Ltd.     

Finite element analyses of shear localization in rate and temperature dependent solids

The effects of strain hardening, strain rate sensitivity, thermal softening, heat conduction and the imposed strain rate on the shear localization process in plane strain compression are examined. The deformation, stress and temperature fields are computed in an infinite solid which contains a periodic rectangular array of inhomogeneities. The inhomogeneities give rise to non-uniform deformation fields which, under certain conditions, may localize in the form of a shear band. Boundary conditions are prescribed such that the resulting fields possess periodicity with respect to the inhomogeneity distribution. In this manner, attention may be confined to a rectangular region of the solid which surrounds a single inhomogeneity. Full two-dimensional analyses are performed within the context of a viscoplasticity theory which, in the rate independent limit, corresponds to flow theory with combined isotropic and kinematic hardening. Full account is taken of finite strain and rotation effects, but attention is confined to quasi-static loading. The initiation and propagation of shear bands is examined for the bounding theories of isotropic and kinematic hardening. The predicted response depends significantly on the multi-axial hardening characterization of the solid.     

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.