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 finite element analysis of the finite width interface mixed mode bi-material fracture problem

One aspect of the terminal crack, mixed mode bi-material fracture mechanics problem is investigated using finite elements. The influence of a finite width bond line interface is considered for one representative material pair combination (E₂/E₁ = 0.10). The stress intensity factors for an inclined crack terminating at a variable thickness interface are established as a function of crack inclination. Since the order of the stress singularity is not the typical r^−1/2 associated with LEFM problems, variable power singular finite elements are used to model the terminating crack tip. Crack tip stress distributions and probable angles of crack extension are presented as a function of crack inclination and bond line thickness. Crack tip stress distributions assuming an interfacial debonding criterion are presented as a function of crack inclination and bond line thickness.     

Finite element analysis of shell structures

Summary A survey of effective finite element formulations for the analysis of shell structures is presented. First, the basic requirements for shell elements are discussed, in which it is emphasized that generality and reliability are most important items. A general displacement-based formulation is then briefly reviewed. This formulation is not effective, but it is used as a starting point for developing a general and effective approach using the mixed interpolation of the tensorial components. The formulation of various MITC elements (that is, elements based on Mixed Interpolation of Tensorial Components) are presented. Theoretical results (applicable to plate analysis) and various numerical results of analyses of plates and shells are summarized. These illustrate some current capabilities and the potential for further finite element developments.     

Dynamic finite element analysis of nonaxisymmetric structures

A dynamic finite element method of analysis is developed for structural configurations which are derived from axisymmetric geometries but contain definite nonaxisymmetric features in the circumferential direction. The purpose of the analysis is to develop a method which will take into consideration the fact that the stress and strain conditions in these geometries will be related to the corresponding axisymmetrie solution. This analysis is an extension of previously published work in which a similar approach was developed for static structural problems. The analysis is developed in terms of a cylindrical coordinate system r, θ and z. As the first step of the analysis, the geometry is divided into several segments in the r-θ plane. Each segment is then divided into a set of quadrilateral elements in the r-z plane. The axisymmetric displacements are obtained for each segment by solving a related axisymmetric configuration. A perturbation analysis is then performed to match the solutions at certain points between the segments, and obtain the perturbation displacements for the total structure. The total displacement is then the axisymmetric displacement plus the perturbation displacement. The analysis allows for elastic-plastic materials with orthotropic yield criterion based on Hill's yield function and kinematic strain hardening. The finite element dynamic equations are solved by finite differences by dividing the time domain into incremental steps. The solution has been programmed on a computer and applied to a number of examples.     

Nonlinear finite element analysis of shell structures using the semi-loof element

The Semi-Loof Shell element originally developed by Irons [2] for linear elastic analysis of thin shell structures is formulated to include large deflection and plastic deformation effects. In this paper the details of the finite element formulation of the problem using total Lagrangian coordinate systems are presented and different element matrices are given. For plastic materials following the Prandtl-Reuss flow rule with isotropic strain hardening a multi-layer approach using a subincremental technique is employed. Numerical results on the performance of the element for a variety of applications are presented. These computer studies include complete load-deflection curves into the post-buckling range and comparisons are made with other existing results. Current experience with the element indicates that it is a reliable and competitive element for nonlinear analysis of shells of general geometry.     

Simulation and experimental analysis of thermo-mechanical behavior of microresonators under dynamic loading

Simulation, finite element analysis and experimental investigations of the dynamical response of a microresonator under electrostatic actuation are presented in this paper. The scope of this paper is to characterize the influence of thermo-mechanical behavior of the material on the frequency response, amplitude and velocity of oscillations under continuous actuation. The effect of the thermoelastic damping on vibrating structures is experimentally investigated by measuring the loss in amplitude and velocity of oscillations as a function of time and the changes in quality factor. The tests are performed in ambient conditions and in vacuum in order to separate the extrinsic damping of beam by the intrinsic effect given by the thermoelastic damping. The vibrating structure under investigation is a polysilicon clamped–clamped beam.     

Finite element analysis of maneuvering spacecraft truss structures

A finite element modeling and solution technique capable of determining the time response of flexible spacecraft truss structures undergoing large angle slew maneuvers has been developed. The elastic deformations of the structure are coupled with large nonsteady translational and rotational motions with respect to an inertial reference frame. The governing equations of motion of the system are derived using momentum conservation principles and the principle of virtual work. The finite element approximation is applied to the equations of motion and the resulting set of nonlinear second order matrix differential equations is solved timewise by an iterative direct numerical integration scheme based on the trapezoidal rule. The solution technique is tested on both planar and three-dimensional maneuvering spacecraft truss structures.     

Minimization of sound radiation from vibrating bi-material structures using topology optimization

Up to now, work on topological design optimization of vibrating structures against noise radiation has mainly addressed the maximization of eigenfrequencies and gaps between consecutive eigenfrequencies of free vibration, and minimization of the dynamic compliance subject to harmonic loading on the structure. In this paper, we deal with topology optimization problems formulated directly with the design objective of minimizing the sound power radiated from the structural surface(s) into a surrounding acoustic medium. Bi-material elastic continuum structures without material damping are considered. The structural vibrations are excited by time-harmonic external mechanical loading with prescribed frequency and amplitude. It is assumed that air is the acoustic medium and that a feedback coupling to the structure can be neglected. Certain conditions are assumed that imply that the sound power emission from the structural surface can be obtained in a simpler way than by solving Helmholz’ integral equation. Hereby, the computational cost of the structural-acoustical analysis is substantially reduced. Several numerical results are presented and discussed for plate- and pipe-like structures with different sets of boundary and loading conditions.     

Non-linear finite element dynamic analysis of two-dimensional concrete structures

This paper presents a numerical procedure for predicting the non-linear dynamic response of plane and axisymmetric reinforced concrete structures. Isoparametric elements with special embedded axial members are used to discretize concrete and steel in space. A summary of a rate and history dependent constitutive model for progressive failure analysis of concrete is given in which the compression behaviour is modelled as a strain rate sensitive elasto-viscoplastic material and in tension as strain rate dependent linear elastic strain softening material. The different rales governing the pre-failure and post-failure behaviour in compression and tension are developed in which the strain rate dependency is included. Steel is modelled as a strain rate dependent uniaxial elasto-viscoplastic material. Explicit central difference scheme in conjunction with an energy balance check is employed for time integration of equations of motion. A computer program for linear and non-linear dynamic analysis of concrete structures is described. Finally, some numerical applications are presented.     

On the finite element creep rupture analysis of structures

The analysis of problems involving creep rupture is considered. Two creep material failure criteria are employed, i.e. the Kachanov-Rabotnov damage relations and a new, in conjunction with the finite element method, energy criterion.Calculations are reported for titanium notched tensile specimens, where the plastic strains are evaluated by the Ramberg-Osgood formulas.     

Finite element analysis of dynamic quasi-bifurcations in elastic-plastic structures

Some numerical results related to the critical behavior of inelastic structures subjected to dynamic step loading are given. The underlying theory of dynamic quasi-bifurcations is adapted to the analysis of elastic-plastic solids. Two examples, a 2D truss and a plane stress column-like structure, are discussed at length.     

Boundary element analysis of linear viscoelastic problems using realistic relaxation functions

This paper presents an efficient method for the stress analysis of realistic viscoelastic solids by the time-domain boundary element method. The fundamental solutions and stress kernels are obtained using the elastic-viscoelastic correspondence principle. Since it is inconvenient to obtain the Laplace transform of the relaxation functions of realistic viscoelastic solids, the method of collocation has been employed and the relaxation function has been expanded in a sum of exponentials. Numerical results of example problems show the effectiveness and applicability of the proposed method.     

Finite element analysis of lateral buckling for beam structures

The application of finite element analysis to lateral buckling problems, locating the critical points and tracing the postbifurcation path, is treated on the basis of a geometrically nonlinear formulation for a beam with small elastic strain but with possibly large rotations. The existing finite element formulations for thin beams are examined in the aspect of application to bifurcation problems, such as lateral buckling, and the choice of an appropriate rotation parameter for representing incremental or variational rotations in finite element formulations is discussed in relation to locating bifurcation points. This is illustrated through several numerical examples and followed by appropriate discussion.     

Boundary element analysis of elastic contact problems using gap finite elements

Gap finite elements provide a practical approach for dealing with elastic contact problems, and it is not possible to derive something similar with boundary elements. This work introduces a simple technique for the analysis of elastic contact problems by coupling a gap finite element subregion with boundary element subregions. The developed algorithm proves to be accurate and reliable, combining the advantages within contact problems of both the gap finite element and the boundary element method.     

A family of homogeneous analysis models for the design of scalable structures

Research in optimum structural design has shown that mathematical programming techniques can be employed efficiently only in conjunction with explicit approximate constraints. In the course of time a well-established approximation for homogeneous functions (scalable structures) has emerged based on the linear Taylor expansion of the displacement functions in the compliance design space (Reciprocal approximation). It has been shown that the quality of this approximation is based on the property that homogeneity of the constraints is maintained and consequently the approximation is exact along the scaling line.The present paper presents a new family of approximations of homogenous functions which have the same properties as the Reciprocal approximation and which produce more accurate models in most of the tested cases. The approximations are obtained by mapping the direct linear Taylor expansion of the constraints unto a space spanned by intervening variables (original design variables to a powerm). Taking the envelope of these constraints along the scaling line yields a new family of approximations governed by the parameterm. It is shown that the Reciprocal approximation is a particular member of this family of approximations (m = –1).The new technique is illustrated with classical examples of truss optimization. An optimal plate design is also reported. A parametric study of the results for various values of the exponentm is presented. It is shown that for special values of the exponentm the new approximations usually yield better models than those based on the Reciprocal approximation.     

Boundary element stress analysis for copper-based dies in pressure die casting

It has recently been demonstrated that, for pressure die casting, high rates of heat extraction are possible with the use copper–alloyed dies suitably protected with a thermally sprayed steel layer. The structural integrity of dies of this type is paramount as the thermally sprayed layer has the potential to de-bond necessitating repair or retirement of the dies.In this paper, an efficient three-dimensional elastostatic stress model for the pressure die casting process is described. The collocation based boundary element method is used for the prediction of transient stress fields over a thermally stabilised casting cycle. A peculiar feature of the pressure die casting process is that transient thermal penetration into the die is limited to regions close to the surface of the die cavity and nozzle. The presence of a transient thermal field necessitates the evaluation of domain integrals in the boundary element stress formulation. Two methods for evaluating these integrals are presented in the paper: (i) a simplex method on a mesh local to the cavity surface; (ii) a modified reciprocity method utilising Gaussian radial-basis functions, interpolating on a perturbed thermal field. The simplex method is shown to be superior, providing high accuracy, stability and computational efficiency. The dies present a multi-domain environment for stress predictions making it necessary to utilise a suitably constructed coarse preconditioner to enhance numerical stability and provide for efficient computation. A multiplicative Schwarz method is presented that enables parameter matrix accelerated GMRES to be applied on each domain. Numerical experiments are performed to demonstrate the computational effectiveness of the approach. Predicted strain fields are compared with strain gauge measurements obtained on a purpose built rig designed to be representative of the casting process.     

Postbuckling analysis of elastic structures by the finite element method

An asymptotic method based on Koiter's elastic stability theory is presented for geometrically nonlinear structure analysis; these structures are subjected to a proportional applied load system. An approach that makes possible the choice of the modes which are necessary to obtain a good representation of the reduced energy is developed as an iterative process. The approach is transferred into the framework of the finite element method. Applied to the study of thin walled structures through some sample tests, the method gives, at low cost, numerical results which are in good agreement with the preceding studies.     

Nonlinear finite element analysis of concrete structures using new constitutive models

Nonlinear finite element analysis was applied to various types of reinforced concrete structures using a new set of constitutive models established in the fixed-angle softened-truss model (FA-STM). A computer code FEAPRC was developed specifically for application to reinforced concrete structures by modifying the general-purpose program FEAP. FEAPRC can take care of the four important characteristics of cracked reinforced concrete: (1) the softening effect of concrete in compression, (2) the tension-stiffening effect by concrete in tension, (3) the average (or smeared) stress–strain curve of steel bars embedded in concrete, and (4) the new, rational shear modulus of concrete. The predictions made by FEAPRC are in good agreement with the experimental results of beams, panels, and framed shear walls.     

A versatile panel element for the analysis of shear wall structures

A panel element substructure is developed, defined in terms of boundary node arrangement and intended for use in efficient static and dynamic analysis of shear wall and box-type structures. The panel substructuring scheme is described as a multi-level super finite element technique based on a series of static condensations. A stressdisplacement matrix, compatible with the reduced representation of panel stiffness and mass, allows full stress recovery within the element. A catalogue of element formations is created to allow selection of the desired substructure nodal arrangement with a minimum of computer input compared to the corresponding fully assembled finite element model. Several numerical examples are presented to demonstrate the versatility and accuracy of the panel elements when used for either static or dynamic analysis.     

A simple and effective element for analysis of general shell structures

A simple flat three-node triangular shell element for linear and nonlinear analysis is presented. The element stiffness matrix with 6 degrees-of-freedom per node is obtained by superimposing its bending and membrane stiffness matrices. An updated Lagrangian formulation is used for large displacement analysis. The application of the element to the analysis of various linear and nonlinear problems is demonstrated.