Hybrid finite elements for the computation of sandwich plates

In this article, new hybrid finite elements are developed on the basis of the displacements and the Pian and Tong functionals using Lagrange multipliers in order to compute correctly and efficiently interlaminar stresses in sandwich structures. These elements represent the mechanical behaviour of sandwich structures in an accurate way, especially at interfaces, where the force equilibrium state must be ensured. They permit to obtain the values of interlaminar stresses using a coarse mesh through the thickness of the sandwich structure. These hybrid elements are assessed and compared through several examples of static linear problems with solutions found in the literature. Copyright © 2001 John Wiley & Sons, Ltd.     

An invariant eight-node hybrid-stress element for thin and thick multilayer laminated plates

A hybrid-stress eight-node isoparametric element is developed for the analysis of thin or thick multilayer fiber-reinforced composite plates. Transverse shear deformation effects are included by allowing for individual layer cross-section warping for thick laminates, or alternatively, laminate non-normal cross-section rotations for thin to moderately thick laminates. All stress components are included and are interpolated independently within each layer. Interlayer surface traction continuity and appropriate upper/lower surface traction-free conditions are exactly satisfied. The layer stress field is selected on the basis of earlier single-layer element studies so that the resulting element is naturally invariant with respect to co-ordinate translation or rotations, is non-locking in the thin-plate limit, and the element stiffness is of correct rank. An example for which an elasticity solution is available is used to demonstrate the element performance. Schemes for reduction of element stiffness computation time are also presented.     

Hierarchic finite elements for moderately thick to very thin plates

We consider the numerical solution of Reissner-Mindlin plates. The model is widely used for thin to moderately thick plates. Generally low order finite elements (with degree one or two) are used for the approximation. Unfortunately the solution degenerates very rapidly for small thickness (locking phenomenon). Non standard formulations of the problem are usually combined with low order finite elements to weaken or possibly eliminate the locking of the numerical solution. We introduce a family of hierarchic finite elements and we present a set of numerical results for the plate problem in its plain formulation. We show that reliable solutions are achieved for all thicknesses of practical interest by using high order finite elements. Moreover, the hierarchic structure allows a low cost assessment of the quality of the solution.     

Invariant 8-node hybrid-stress elements for thin and moderately thick plates

Eight-node hybrid-stress elements are developed for the analysis of plates ranging from arbitrarily thin to moderately thick. The displacement behaviour is characterized by a transverse displacement and independent cross-section rotations, which are interpolated using the 8-node Serendipity shape functions. All components of stress are included; alternative elements are developed which differe in the form of the inplane distribution of the stresses. Elements are sought for whic the stiffiness is invariant and of correct rank, and whic show on signs of deterioration in the thin-plate limit. A discussion of the prospects for developing a 4-node element with these characteristics is also presented. Example problems are used to compare the performance of the 8-node elements including convergence behaviour, intraelement stress distributions and optimal sampling locations, and range of applicability in terms of plate thickness ratio.     

Two hybrid elements for analysis of thick,thin and sandwich plates

Two general quadrilateral elements for plate bending are developed. Each has 12 degrees of freedom and accounts for transverse shear deformation effects. Each is built from four triangular elements whose properties are derived by the assumed-stress hybrid approach. Matrices needed to generate the stiffness properties of the triangular elements are explicitly stated so as to facilitate their use. Numerical test cases, in which shear deformation effects range from negligible to very important, are used to illustrate the behaviour of the elements. It is concluded that the simpler of the two quadrilaterals is one of the best plate elements currently available.     

Resonance acoustoelasticity measurement of stress in thin plates

A computer-driven, swept-frequency measurement technique is developed on the basis of resonance birefringence acoustoelasticity to evaluate the stresses in thin plates. The resonance frequency depends on the thickness and the elastic wave velocity; they change with stress because of the Poisson effect and the acoustoelastic effect. The resonance frequency is obtained from the spectral response curve in the electric impedance of the piezoelectric transducer. The frequency displacement induced by acoustically coupling the transducer can be minimized by employing the resonance peak closest to the transducer fundamental frequency. To illustrate the method, the residual stress is measured in butt-welded aluminum alloy plates and is compared with the results of conventional methods.     

The determination of three-dimensional stress tensors on multilayer thin films

The three-dimensional residual stress condition of copper sputtered on Kapton films was studied by two nondestructive x-ray diffraction techniques. Four samples were evaluated—one control sample, and three heat-treated samples. Because of concerns about steep stress gradients normal to the specimen surface, an experimental technique described by Dölle⁽¹⁾ was used to obtain the residual strain tensors in the near surface region. The resulting stress tensors were compared to data obtained through a modification of the technique described by Dölle, referred to as the differential method.⁽²⁾     

Three dimensional hybrid‐Trefftz stress finite elements for plates and shells

Three‐dimensional hybrid‐Trefftz stress finite elements for plates and shells are proposed. Two independent fields are approximated: stresses within the element and displacement on their boundary. The required stress field derived from the Papkovitch‐Neuber solution of the Navier equation, which a priori satisfies the Trefftz constraint, is generated using homogeneous harmonic polynomials. Restriction on the polynomial degree in the coordinate measured along the thickness direction is imposed to reduce the number of independent terms. Explicit expressions of the corresponding independent polynomials are listed up to the fifth order. Illustrative applications to evaluate displacements and stresses are conducted by hexahedral hybrid‐Trefftz stress element models. The hierarchical p‐ and h‐refinement strategy are exploited in the numerical tests.     

Influence of stress on the transmission characteristics of multilayer thin films

The relationship between multilayer film strains and film thickness changes was analyzed based on the stress formula and the small flexibility principle of elasticity. A theoretical model of the relationship between changes in substrate curvature and film thickness was established, and the formula of every layer thickness with uneven changes of multilayer films was obtained for the spectra of the before-thinning and after-thinning filter. Both the simulated result for a narrow bandpass filter and experimental results showed that nonuniformities in the film thickness caused by substrate curvature changes are one of the main reasons for spectrum degradation in narrow bandpass filters.     

Computation of membrane and bending stress intensity factors for thin,cracked plates

The crack tip stress fields for plate bending and membrane loading problems are reviewed and the four stress intensity factors that determine these fields are defined. These four stress intensity factors arise from use of Kirchhoff plate theory to account for the bending loads and two dimensional plane stress elasticity to account for the membrane loads. The energy release rate is related to the stress intensity factors and to the stress resultants of plate theory. Virtual crack extension, nodal release and modified crack closure integral methods are discussed for computing components of the energy release rate from finite element analyses of cracked plates. Sample computations of stress intensity factors for single and mixed mode cases are presented for a crack in an infinite plate. Sample computations of stress intensity factors for a double edge notched tension-torsion test specimen are given as well.School of Civil and Environmental Engineering, Cornell University     

Converged stress solutions in thin plates by the extended field method

This paper presents numerical results calculated by the extended field method for the problem of the forced harmonic vibration of two plates—one rectangular and one triangular. The numerical results demonstrate that the extended field method, which is a deflection-based series solution method, is capable of producing converged stress solutions suitable for being bases of comparison for other approximate methods of analysis.     

The hybrid-stress model for thin plates

The assumed-stress hybrid finite element model is examined for application to the bending analysis of thin plates. A hybrid-stress functional is defined by using a Mindlin-type displacement assumption and including all components of stress. The Euler equations and matrix formulation corresponding to this functional are examined to assess the effects of plate thickness, and a rationale is presented for the selection of stress assumptions so that locking is avoided in the thin plate limit. To illustrate these concepts, a series of linear displacement quadrilateral elements are derived and tested, and the best of these elements is identified for suggested implementation in general-purpose computer programs.     

Consistent multiscale analysis of heterogeneous thin plates with smoothed quadratic Hermite triangular elements

A consistent multiscale formulation is presented for the bending analysis of heterogeneous thin plate structures containing three dimensional reinforcements with in-plane periodicity. A multiscale asymptotic expansion of the displacement field is proposed to represent the in-plane periodicity, in which the microscopic and macroscopic thickness coordinates are set to be identical. This multiscale displacement expansion yields a local three dimensional unit cell problem and a global homogenized thin plate problem. The local unit cell problem is discretized with the tri-linear hexahedral elements to extract the homogenized material properties. The characteristic macroscopic deformation modes corresponding to the in-plane membrane deformations and out of plane bending deformations are discussed in detail. Thereafter the homogenized material properties are employed for the analysis of global homogenized thin plate with a smoothed quadratic Hermite triangular element formulation. The quadratic Hermite triangular element provides a complete C¹ approximation that is very desirable for thin plate modeling. Meanwhile, it corresponds to the constant strain triangle element and is able to reproduce a simple piecewise constant curvature field. Thus a unified numerical implementation for thin plate analysis can be conveniently realized using the triangular elements with discretization flexibility. The curvature smoothing operation is further introduced to improve the accuracy of the quadratic Hermite triangular element. The effectiveness of the proposed methodology is demonstrated through numerical examples.     

MITC finite elements for laminated composite plates

Within the framework of the first‐order shear deformation theory, 4‐ and 9‐node elements for the analysis of laminated composite plates are derived from the MITC family developed by Bathe and coworkers. To this end the bases of the MITC formulation are illustrated and suitably extended to incorporate the laminate theory. The proposed elements are locking‐free, they do not have zero‐energy modes and provide accurate in‐plane deformations. Two consecutive regularizations of the extensional and flexural strain fields and the correction of the resulting out‐of‐plane stress profiles necessary to enforce exact fulfillment of the boundary conditions are shown to yield very satisfactory results in terms of transverse and normal stresses. The features of the proposed elements are assessed through several numerical examples, either for regular and highly distorted meshes. Comparisons with analytical solutions are also shown. Copyright © 2001 John Wiley & Sons, Ltd.     

Finite element bending analysis of multilayer plates

The development of a general quadratic multilayer plate element is presented for the analysis of arbitrarily layered curved plates. In the formulation, each layer of the multilayer plate can have different orthotropic properties and can deform locally. Examples of bending problems are presented which demonstrate the applicability of the formulation.     

Construction of a three-dimensional model for calculating multilayer orthotrophic plates

A three-dimensional mathematical model is suggested for layered orthotropic plates by introducing kinematic hypotheses for the distribution of displacements with respect to plate thickness. A feature of the hypotheses is the fact that they are written with respect to functions of displacements on the front surfaces of plates. Approximation of displacements is mutually independent. Equilibrium equations and boundary conditions are obtained on the basis of the Lagrange variation principle. The functions sought have similar operators not exceeding the second order which markedly simplifies their realization. A number of test problems point to the possibility of using this model even with ratios of elasticity moduli for the supporting layers and filler of one hundred with a ratio of four for plate length to height. Translated from Problemy Prochnosti, No. 12, pp. 57–61, December, 1994.