Stress analysis of elastomeric materials at large extensions using the finite element method

A newly developed finite element method is applied to the stress and strain analyses of stress fields, at the vicinity of a primary crack surrounded by secondary cracks. The results show that the primary crack propagation deviates from the crack axis, when a secondary crack entered the stress fields of the primary crack within the distance of the diameter of the secondary crack. The fundamental unit of surface roughness, the deviation from planarity, will be the diameter of the secondary crack. The roughness generated in real elastomers strongly depends on mechanical hysteresis, and thus the fracture surface energy of the materials.     

Stress analysis around crack tips in finite strain problems using the eXtended finite element method

Fracture of rubber‐like materials is still an open problem. Indeed, it deals with modelling issues (crack growth law, bulk behaviour) and computational issues (robust crack growth in 2D and 3D, incompressibility). The present study focuses on the application of the eXtended Finite Element Method (X‐FEM) to large strain fracture mechanics for plane stress problems. Two important issues are investigated: the choice of the formulation used to solve the problem and the determination of suitable enrichment functions. It is demonstrated that the results obtained with the method are in good agreement with previously published works. Copyright © 2005 John Wiley & Sons, Ltd.     

Stress intensity factor analyses of interface cracks between dissimilar anisotropic materials using the finite element method

Delamination along an interface between dissimilar materials is the primary cause of failure in microstructures like electronic packages, micro-electro-mechanical systems (MEMS), and so on. Fracture mechanics is a powerful tool for the evaluation of delamination. However, many materials used in microstructures such as composite materials and single crystals are anisotropic materials. Stress intensity factors of an interface crack between dissimilar anisotropic materials, which were proposed by Hwu, are useful for evaluating the reliability of microstructures. However, numerical methods that can analyze the stress intensity factors of an interface crack between anisotropic materials have not been developed. We propose herein a new numerical method for the analysis of an interface crack between dissimilar anisotropic materials. The stress intensity factors of an interface crack are based on the generalized plane strain condition. The energy release rate is obtained by the virtual crack extension method in conjunction with the finite element method for the generalized plane strain condition. The energy release rate is separated into individual modes of the stress intensity factors K_I, K_II, and K_III, using the principal of superposition. The target problem to be solved is superposed on the asymptotic solution of displacement in the vicinity of an interface crack tip, which is described using the Stroh formalism. Analyses of the stress intensity factors of center interface cracks between semi-infinite dissimilar anisotropic media subjected to concentrated self-balanced loads on the center of crack surfaces and to uniform loads are demonstrated. The present method accurately provides mode-separated stress intensity factors using relatively coarse meshes for the finite element method.     

Fracture in magnetoelectroelastic materials using the extended finite element method

Static fracture analyses in two‐dimensional linear magnetoelectroelastic (MEE) solids is studied by means of the extended finite element method (X‐FEM). In the X‐FEM, crack modeling is facilitated by adding a discontinuous function and the crack‐tip asymptotic functions to the standard finite element approximation using the framework of partition of unity. In this study, media possessing fully coupled piezoelectric, piezomagnetic and magnetoelectric effects are considered. New enrichment functions for cracks in transversely isotropic MEE materials are derived, and the computation of fracture parameters using the domain form of the contour interaction integral is presented. The convergence rates in energy for topological and geometric enrichments are studied. Excellent accuracy of the proposed formulation is demonstrated on benchmark crack problems through comparisons with both analytical solutions and numerical results obtained by the dual boundary element method. Copyright © 2011 John Wiley & Sons, Ltd.     

Stress transfer analysis of unidirectional composites with randomly distributed fibers using finite element method

A three-dimensional representative volume element (RVE) of unidirectional composites with both randomly distributed fibers and periodic geometry was generated using DIGIMAT-FE software. Finite element analysis of the stress transfer mechanisms around a fiber break in the RVE was performed via ABAQUS/Standard. The influences of distance to the broken fiber, fiber/matrix stiffness ratio and fiber volume fraction on the stress transfer process of intact fibers were discussed for the case of perfect fiber/matrix adhesion. The study shows that the nearest fibers and the second nearest fibers share the stress released from the broken fiber. The stress transfer coefficient and the ineffective stress transfer length of the nearest fibers was found to increase with the increasing distance to the broken fiber and the stiffness ratio, while decrease with the increasing fiber volume fraction. However, the trends in the two stress transfer parameters of the second nearest fibers are slightly different from those of the nearest fibers due to the random distribution of other intact fibers.     

Stress computations for nearly incompressible materials by the p-version of the finite element method

In the case of nearly incompressible elastic materials the strain energy, the shear stress and the difference of normal stresses can be computed accurately by direct methods when the p-version of the finite element method is used. Computation of the sum of the normal stresses requires special procedures. In this paper such procedures are described and examples are presented.     

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.

     

Stress intensity factor analysis of an interface crack between dissimilar anisotropic materials under thermal stress using the finite element analysis

New numerical methods were presented for stress intensity factor analyses of two-dimensional interfacial crack between dissimilar anisotropic materials subjected to thermal stress. The virtual crack extension method and the thermal M-integral method for a crack along the interface between two different materials were applied to the thermoelastic interfacial crack in anisotropic bimaterials. The moving least-squares approximation was used to calculate the value of the thermal M-integral. The thermal M-integral in conjunction with the moving least-squares approximation can calculate the stress intensity factors from only nodal displacements obtained by the finite element analysis. The stress intensity factors analyses of double edge cracks in jointed dissimilar isotropic semi-infinite plates subjected to thermal load were demonstrated. Excellent agreement was achieved between the numerical results obtained by the present methods and the exact solution. In addition, the stress intensity factors of double edge cracks in jointed dissimilar anisotropic semi-infinite plates subjected to thermal loads were analyzed. Their results appear reasonable.     

Mixed finite element method for saturated poroelastoplastic media at large strains

A mixed finite element for hydro‐dynamic analysis in saturated porous media in the frame of the Biot theory is proposed. Displacements, effective stresses, strains for the solid phase and pressure, pressure gradients, and Darcy velocities for the fluid phase are interpolated as independent variables. The weak form of the governing equations of coupled hydro‐dynamic problems in saturated porous media within the element are given on the basis of the Hu–Washizu three‐field variational principle. In light of the stabilized one point quadrature super‐convergent element developed in solid continuum, the interpolation approximation modes for the primary unknowns and their spatial derivatives of the solid and the fluid phases within the element are assumed independently. The proposed mixed finite element formulation is derived. The non‐linear version of the element formulation is further derived with particular consideration of pressure‐dependent non‐associated plasticity. The return mapping algorithm for the integration of the rate constitutive equation, the consistent elastoplastic tangent modulus matrix and the element tangent stiffness matrix are developed. For geometrical non‐linearity, the co‐rotational formulation approach is used. Numerical results demonstrate the capability and the performance of the proposed element in modelling progressive failure characterized by strain localization due to strain softening in poroelastoplastic media subjected to dynamic loading at large strain. Copyright © 2003 John Wiley & Sons, Ltd.     

Elastic-plastic fracture analysis and design using the finite element method

The purpose of analysis in fracture mechanics is to determine characterising parameters which reflect the influence of loading and geometry on the crack tip environment of flawed bodies. Here practical methods are developed which permit the determination of such parameters in general situations. Extensive use of finite element methods has been made to provide relevant field values which are then manipulated to determine the required parameters. However the choice of method to determine field values is arbitrary and is dictated by the ease with which such field values may be found. It is in the manipulation of these values that the fracture mechanics philosophy is introduced.Contributions are made in three areas. First economic methods for the determination of the linear fracture mechanics parameter in general stiuations are developed which are of direct relevance to design procedures. Detailed discussion of the Dugdale model of fracture behaviour is then given and a general method for determining Dugdale model solutions is provided. This method is used to provide solutions for standard specimen geometries and it is suggested that such solutions will enable a rational evaluation of the general applicability of the model. However the method is such that, should sufficient confidence in the model be established, design calculations on its premises may be performed. Finally, it is demonstrated that materials which allow extensive plastic flow at a flaw tip prior to fracture may be analysed using the basic ideas of fracture mechanics. It is shown that a line integral provides a flaw tip environment parameter for materials deforming according to the physically appropriate Prandtl-Reuss laws of plasticity. It is hoped that these results will indicate a rational approach to correlating fracture behaviour in such situations.     

An error calculation method for finite element analysis in large displacements

This article describes an error measurement and mesh optimization method for finite elements in non-linear geometry problems. The error calculation is adapted from a method developed by Ladeveze, based on constructing a local statically admissible stress field. The particular difficulty in non-linear geometry lies in choosing a configuration on which the fields is defined. We propose here the lagrangian or reference configuration. The error is then defined as the value in elastic energy of the difference between the two stresses. The optimization used is the type h version. © 1997 John Wiley & Sons, Ltd.     

Eulerian formulation using stabilized finite element method for large deformation solid dynamics

This paper describes an Eulerian formulation for large deformation solid dynamics. In the present Eulerian approach, an advective equation is solved using the Stream‐Upwind/Petrov–Galerkin finite element method. The Eulerian finite element method is applied to path‐dependent solid analyses such as impact bar and ductile necking problems. These computational results using the Eulerian finite element method are compared with the results obtained from using the Lagrangian finite element method and an Eulerian formulation based on a finite difference method. Copyright © 2007 John Wiley & Sons, Ltd.     

Fracture mechanics analysis using the wavelet Galerkin method and extended finite element method

This paper presents fracture mechanics analysis using the wavelet Galerkin method and extended finite element method. The wavelet Galerkin method is a new methodology to solve partial differential equations where scaling/wavelet functions are used as basis functions. In solid/structural analyses, the analysis domain is divided into equally spaced structured cells and scaling functions are periodically placed throughout the domain. To improve accuracy, wavelet functions are superposed on the scaling functions within a region having a high stress concentration, such as near a hole or notch. Thus, the method can be considered a refinement technique in fixed‐grid approaches. However, because the basis functions are assumed to be continuous in applications of the wavelet Galerkin method, there are difficulties in treating displacement discontinuities across the crack surface. In the present research, we introduce enrichment functions in the wavelet Galerkin formulation to take into account the discontinuous displacements and high stress concentration around the crack tip by applying the concept of the extended finite element method. This paper presents the mathematical formulation and numerical implementation of the proposed technique. As numerical examples, stress intensity factor evaluations and crack propagation analyses for two‐dimensional cracks are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Dispersion free analysis of acoustic problems using the alpha finite element method

The classical finite element method (FEM) fails to provide accurate results to the Helmholtz equation with large wave numbers due to the well-known “pollution error” caused by the numerical dispersion, i.e. the numerical wave number is always smaller than the exact one. This dispersion error is essentially rooted at the “overly-stiff” feature of the FEM model. In this paper, an alpha finite element method (α-FEM) is then formulated for the acoustic problems by combining the “smaller wave number” model of FEM and the “larger wave number” model of NS-FEM through a scaling factor ${a\in [0,1]}The classical finite element method (FEM) fails to provide accurate results to the Helmholtz equation with large wave numbers due to the well-known “pollution error” caused by the numerical dispersion, i.e. the numerical wave number is always smaller than the exact one. This dispersion error is essentially rooted at the “overly-stiff” feature of the FEM model. In this paper, an alpha finite element method (α-FEM) is then formulated for the acoustic problems by combining the “smaller wave number” model of FEM and the “larger wave number” model of NS-FEM through a scaling factor a ? [0,1]{a\in [0,1]} . The motivation for this combined approach is essentially from the features of “overly-stiff” FEM model and “overly-soft” NS-FEM model, and accurate solutions can be obtained by tuning the α-FEM model. A technique is proposed to determine a particular alpha with which the α-FEM model can possess a very “close-to-exact” stiffness, which can effectively reduce the dispersion error leading to dispersion free solutions for acoustic problems. Theoretical and numerical studies shall demonstrate the excellent properties of the present α-FEM.     

A mixed-mode crack analysis of rotating disk using finite element method

This paper presents a rigorous elastodynamic hybrid-displacement finite element procedure for a safety analysis of fast rotating disks with mixed-mode cracks. Based on a modified Hamilton's principle, the finite element model is derived such that the proper crack-tip singularities are taken into consideration and the interelement displacement compatibility conditions are still satisfied. Thus, the specimen can be represented by a finite element assemblage in which “singular” elements are used around the crack-tip and high-order isoparametric “regular” elements are taken elsewhere.To determine the mixed-mode stress intensity factors, the modified $\text{J}\text{?}{}_{\mathrm{k}}$ integrals for rotating cracked disks have been established taking into account the effect of centrifugal force. Using the “strain-energy-density factor” concept, the direction of crack growth of a rotating disk with an arbitrary internal crack is predicted. To provide a method of non-destructive testing in evaluating the integrity of structures, natural vibrations of cracked disk are then studied. Lastly, the influence of inertia effects due to rotating speed changes in determining the dynamic stress intensity factors is examined.For verification purposes, the simple case of a rotating disk with radial cracks is first solved. Excellent correlations between the computed results and available referenced solutions are drawn. New solutions for the circular disk with circumferential or arbitrarily-oriented cracks are also presented.     

Thermal analysis of induction motors using 3D finite element method

Abstract

To design a reliable and economical induction motor, it is necessary to be able to predict accurately the temperature distribution within the motor. In this paper, a 3D thermal model of an induction motor is presented. Except for providing a more accurate representation of the problem, the proposed model can also reduce computer memory and time. The finite element method (FEM) is used to analyze the three dimensional (3D) heat flow equation which describes the thermal model. Galerkin's procedure is used to derive the element equations and first order tetrahedral elements are used to discretize the field region. Galerkin's time‐stepping scheme is employed to treat time differential terms. Values of surface heat transfer coefficients are obtained from the empirical formula and heat losses are revised by the factory test. Application of the proposed method to the analysis of a 9,000 HP induction motor yields temperature distribution very close to the experimental data.     

Dynamic analysis of cracked linear viscoelastic solids by finite element method using singular element

The finite element method for the dynamic problem of cracked linear viscoelastic solids is developed using the singular element where the displacement function is taken from the analytical solution near the crack-tip. The time variation of the dynamic stress intensity factors is determined for a center crack and an oblique crack in standard linear viscoelastic rectangular plates subjected to dynamic loading.

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Résumé La méthode par éléments finis permettant d'aborder le problème dynamique des solides linéaires viscoélastiques fissurés est développée en recourant à un élément singulier pour lequel la fonction de déplacement est prise dans une solution analytique au voisinage del'extrémité de la fissure. La variation dans le temps des facteurs d'intensité de contrainte dynamique est déterminée pour une fissure centrale et pour une fissure oblique dans des plaques rectangulaires standard en matériau linéaire viscoélastique soumises à une sollicitation dynamique.