Transition finite elements for axisymmetric stress analysis

This paper presents isoparametric finite element formulation for special classes of elements referred to as ‘transition elements’ for axisymmetric stress analysis. The transition elements are necessary for applications requiring the use of both axisymmetric solids and axisymmetric shells. These elements permit transition from the axisymmetric solid portion of the structure to the axisymmetric shell portion. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate the accuracy and applications of such elements.     

Interlaminar stress recovery for three-dimensional finite elements

An accurate evaluation of interlaminar stresses in multilayer composite laminates is crucial for the correct prediction of the onset of delamination. In general, three-dimensional finite element models are required for acceptable accuracy, especially near free edges and stress concentrations. Interlaminar stresses are continuous both across and along layer interfaces. Nonetheless, the continuity of interlaminar stresses is difficult to enforce in C⁰$C^0$ interpolated elements. Nodal values of the stresses are usually retrieved using extrapolation techniques from super-convergent points, if known, or Gauss points inside the element. Stress fields within an element can be deduced using either constitutive relations or variationally consistent procedures. In either case, spurious oscillations in stress fields may be encountered leading to a reduced accuracy of the recovered stresses at nodes. In this paper, an efficient interlaminar stress recovery procedure for three-dimensional finite element formulations is presented. The proposed procedure does not rely on extrapolation techniques from super-convergent or integration points. Interlaminar stress values are retrieved directly at nodes and stress continuity at the inter-element boundary is automatically satisfied. Several benchmark problems were analysed. Comparisons with finite element software and available solutions in the literature confirmed the accuracy of the procedure. Accurate interlaminar stresses were obtained using coarser meshes compared to customary recovery procedures.     

P-version of finite elements for the analysis of stress concentrators

We study the applicability of thep-version of the method of finite elements to the numerical analysis of the concentration of stresses near notches by using a COSMOS/M computer system and the Stress-Check software package. We present the numerical analysis of the stress-strain state of the material near notches of various shapes. The efficiency of the proposed procedure of numerical computations is demonstrated by computing the nominal and plastic stress concentration factors for beam and cylindrical specimens with notches. Department of Mechanics, University of Miskolc, Miskolc, Hungary. Published in Fizyko-Khimichna Mekhanika Materialiv, Vol. 34, No. 5, pp. 108–114, September–October, 1998.     

Plane-wave basis finite elements and boundary elements for three-dimensional wave scattering

Classical finite-element and boundary-element formulations for the Helmholtz equation are presented, and their limitations with respect to the number of variables needed to model a wavelength are explained. A new type of approximation for the potential is described in which the usual finite-element and boundary-element shape functions are modified by the inclusion of a set of plane waves, propagating in a range of directions evenly distributed on the unit sphere. Compared with standard piecewise polynomial approximation, the plane-wave basis is shown to give considerable reduction in computational complexity. In practical terms, it is concluded that the frequency for which accurate results can be obtained, using these new techniques, can be up to 60 times higher than that of the conventional finite-element method, and 10 to 15 times higher than that of the conventional boundary-element method.     

Patch‐averaged assumed strain finite elements for stress analysis

A finite element model for linear‐elastic small deformation problems is presented. The formulation is based on a weighted residual that requires a priori the satisfaction of the kinematic equation. In this approach, an averaged strain‐displacement matrix is constructed for each node of the mesh by defining an appropriate patch of elements, yielding a smooth representation of strain and stress fields. Connections with traditional and similar procedure are explored. Linear quadrilateral four‐node and linear hexahedral eight‐node elements are derived. Various numerical tests show the accuracy and convergence properties of the proposed elements in comparison with extant finite elements and analytic solutions. Specific examples are also included to illustrate the ability to resist numerical locking in the incompressible limit and insensitive response in the presence of shape distortion. Furthermore, the numerical inf‐sup test is applied to a selection of problems to show the stability of the present formulation. Copyright © 2012 John Wiley & Sons, Ltd.     

Rational approach for assumed stress finite elements

A new method for the formulation of hybrid elements by the Hellinger-Reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element. The equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co-ordinates. The resulting element possesses all the ideal qualities, i.e. it is invariant, it is less sensitive to geometric distortion, it contains a minimum number of stress parameters and it provides accurate stress calculations. For the formulation of a 4-node plane stress element, a small perturbation method is used to determine the equilibrium constraint equations. The element has been proved to be always rank sufficient.     

A three-dimensional multilayer composite finite element for stress analysis of composite laminates

A three-dimensional multilayer composite finite element method has been developed based on a composite variational functional which takes three in-plane strains ε_x, ε_x, ε_xy and three transverse stresses σ_z, σ_yz, σ_xz as the basic variables. The continuity of the transverse stresses σ_z, σ_yz, σ_xz across the laminate thickness is assured a priori by introducing a partial stress field parameter α which is associated with the lower and upper surfaces of a lamina in a laminate. A method has been developed to form the partial stress field based on the assumed displacement field. With this method, a three dimensional (3-D) multilayer composite finite element is formulated for stress analysis of composite laminates. A numerical example is given, which shows some advantages of this composite element.     

Generation of two- and three-dimensional finite elements

Many of the known finite elements are based on a variational procedure in a functional basis of monomials. The present paper attempts to formulate completely a finite element procedure which is non-variational on a functional basis which is left to the choice of the user.     

Transient field problems: Two-dimensional and three-dimensional analysis by isoparametric finite elements

The transient field problem of the type encountered in heat conduction problems is formulated in terms of the finite element process using the Galerkin approach. Curved two-dimensional and three-dimensional, isoparametric elements are used in a time-stepping solution and their advantages illustrated by means of several examples.     

Dual analysis of plane stress problems by commonly based finite elements

Finite element models for linear elastic plane stress problems which provide, alternatively, a completely compatible displacement field or a precisely equilibrated stress field are developed. The basis for both models is a biquintic interpolation polynomial representing Airy's stress function over a triangular region. The polynomial coefficients are modified and grouped to establish compatibility while retaining equilibrium within each element. Nodal kinematic parameters are selected and matched to the stress function for the compatible (modified stiffness) model, while nodal stress function parameters are chosen for the equilibrium (modified flexibility) model. Constraints on the global freedoms, enforced by Lagrange multipliers, are introduced to augment nodal connectivity in establishing interelement compatibility in the ‘stiffness’ model and uniform stress transmission in the ‘flexibility’ model. Appropriate boundary conditions are formed for each model. Numerical solutions are obtained and assessed.     

Application of interface finite elements to three-dimensional progressive failure analysis of adhesive joints

ABSTRACT The paper presents a new model for three‐dimensional progressive failure analysis of adhesive joints. The method uses interface elements and includes a damage model to simulate progressive debonding. The interface finite elements are placed between the adherents and the adhesive. The damage model is based on the indirect use of fracture mechanics and allows the simulation of the initiation and growth of damage at the interfaces without considering the presence of initial flaws. The application of the model to single lap joints is presented. Experimental tests were performed in aluminium/epoxy adhesive joints. Linear elastic and elastoplastic analyses were performed and the predicted failure load for the elastoplastic case agrees with experimental results.     

Boundary elements for three-dimensional elastic crack analysis

In this paper 8-node traction singular boundary elements are employed to represent displacement and traction variations in the vicinity of the crack front in three-dimensional geometries. The numerical procedure suggested for evaluating the singular integrals extending over these special elements is described. The efficiency and accuracy of the special elements and integration procedure are demonstrated by the results obtained in a simple test problem whose analytical solution is known. The interaction of two circular coplanar cracks embedded in an infinite medium under uniform tension loading is also analysed. Finally, the stress intensity factor variation computed for a semi-circular inner surface crack in a pressurized cylinder is presented.     

Analysis of three-dimensional interface cracks using enriched finite elements

Many important interface crack problems are inherently three-dimensional in nature, e.g., debonding of laminated structures at corners and holes. In an effort to accurately analyze three-dimensional interface fracture problems, an efficient computational technique was developed that utilizes enriched crack tip elements containing the correct interface crack tip asymptotic behavior. In the enriched element formulation, the stress intensity factors K _I, K _II, and K _III are treated as additional degrees of freedom and are obtained directly during the finite element solution phase. In this study, the results that should be of greatest interest are obtained for semi-circular surface and quarter-circular corner cracks. Solutions are generated for uniform remote tension and uniform thermal loading, over a wide range of bimaterial combinations. Of particular interest are the free surface effects, and the influence of Dundurs’ material parameters on the strain energy release rate magnitudes and corresponding phase angles.     

High order three-dimensional hybrid-stress elements for thick-plate analysis

Two alternative hybrid-stress based three-dimensional elements are presented. These elements are intended for use in the analysis of thick plates for which the through-thickness variation of displacement and stress may be of high order, and in which a single element is to be used through the thickness of the plate. For both elements, the displacement assumption allows for cubic through-thickness variation in the inplane displacements, and quadratic through-thickness variation for the transverse displacement. The elements differ in the stress assumption used. In the first element, the z variations in the stress components are chosen to be consistent with the z variations of the strain distribution calculated from the displacement assumption, and the traction boundary conditions at the upper and lower surfaces are satisfied in the weighted integral sense. In the second element, the z variations in the stress components are chosen to achieve consistency with respect to the equilibrium equations, and the traction-free conditions at the upper and lower surface of the plate are satisfied exactly. The example problem of a semi-infinite, simply-supported thick plate under transverse sinusoidal loading is considered. Results obtained by using each of these elements are compared with available elasticity and approximate solutions, and an assessment of the relative merits and range of applicability of each element is made.     

Improved stress evaluation under thermal load for simple finite elements

The method of initial strain is replaced by a method of initial displacements and applied to simple four-noded membrane and shell elements. The thermal stresses in nodes are improved for quadrilaterals and hexahedrons, which have a discrepancy between interpolation functions for temperature (linear) and strains (constant). To get initial displacements, the singular element stiffness matrix has to be inverted. Minor changes in subroutine MINV³ (enclosed version STDMNV) were made to do this.     

On quarter-point three-dimensional finite elements in linear elastic fracture mechanics

Use of the finite element method for calculating stress intensity factors of two-dimensional cracked bodies has become commonplace. In this study, the more difficult task of applying finite elements to three-dimensional cracked bodies is investigated. Since linear elastic material is considered, square root singular stresses exist along the edge of an embedded crack. To deal with this numerical difficulty, twenty noded, isoparametric, serendipity, quarter-point, singular, solid elements are employed. Examination of these elements is carried out in order to determine the extent of the singular behavior.In addition, the stiffness derivative technique is explored, together with quarter-point elements, to determine an accurate procedure for computing stress intensity factors in three-dimensions. The problem of chosing a proper virtual crack extension is addressed. To this end, the disturbance in the square root singular stresses is examined and compared with a similar disturbance which occurs in two-dimensions. As a numerical example, a pennyshaped crack in a finite height cylinder is considered with various meshes. It is found that stress intensity factors can be calculated to an accuracy within 1 percent when quarter-point cylindrical elements are employed with the stiffness derivative technique such that the crack extension is one in which one corner node is not moved, the other corner node is moved a small distance, and the midside node is moved one-half that distance. This crack extension is analogous to that of a straight crack advance for a brick element. Both of these crack advances disturb the square root singular stresses in a manner similar to that which occurs with the two-dimensional eight noded element in which the crack has been advanced a small distance.     

Simulation of piezoelectric devices by two- and three-dimensional finite elements

A method for the analysis of piezoelectric media based on finite-element calculations is presented in which the fundamental electroelastic equations governing piezoelectric media are solved numerically. The results obtained by this finite-element calculation scheme agree with theoretical and experimental data given in the literature. The method is applied to the vibrational analysis of piezoelectric sensors and actuators with arbitrary structure. Natural frequencies with related eigenmodes of those devices as well as their responses to various time-dependent mechanical or electrical excitations are computed. The theoretically calculated mode shapes of piezoelectric transducers and their electrical impedances agree quantitatively with interferometric and electric measurements. The simulations are used to optimize piezoelectric devices such as ultrasonic transducers for medical imaging. The method also provides deeper insight into the physical mechanisms of acoustic wave propagation in piezoelectric media.