An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates

A three-dimensional linear, small deformation theory of elasticity solution by the direct method is developed for the free vibration of simply-supported, homogeneous, isotropic, thick rectangular plates. The solution is exact and involves determining a triply infinite sequence of eigenvalues from a doubly infinite set of closed form transcendental equations. As no restrictions are placed on the thickness variation of stresses or displacements, this formulation yields a triply infinite spectrum of frequencies, instead of only one doubly infinite spectrum by thin plate theory and three doubly infinite spectra by Mindlin's thick plate theory. Further, the present analysis yields symmetric thickness modes which neither of the approximate theories can identify. Some numerical results from the two approximate theories are compared with those from the present solution and some important conclusions regarding the effect of the assumptions made in the approximate theories are drawn. The thickness variations of stresses and displacements are also discussed. The analysis is readily extended for laminated plates of isotropic materials. Numerical results are also given for three-ply laminates, and are used to assess the accuracy of thin plate theory predictions for laminates. Extension to general lateral surface conditions and forced vibrations is indicated.

Effect of matrix cracking and material uncertainty on composite plates

A laminated composite plate model based on first order shear deformation theory is implemented using the finite element method.Matrix cracks are introduced into the finite element model by considering changes in the A, B and D matrices of composites. The effects of different boundary conditions, laminate types and ply angles on the behavior of composite plates with matrix cracks are studied.Finally, the effect of material property uncertainty, which is important for composite material on the composite plate, is investigated using Monte Carlo simulations. Probabilistic estimates of damage detection reliability in composite plates are made for static and dynamic measurements. It is found that the effect of uncertainty must be considered for accurate damage detection in composite structures. The estimates of variance obtained for observable system properties due to uncertainty can be used for developing more robust damage detection algorithms. (C) 2010 Elsevier Ltd. All rights reserved.

Development of a New Finite Element for the Analysis of Sandwich Beams with Soft Core

A new super convergent sandwich beam finite element formulation is presented in this article. This element is a two-nodded, six degrees of freedom (dof) per node (3 dof u(0), w, phi for top and bottom face sheets each), which assumes that all the axial and flexural loads are taken by face sheets, while the core takes only the shear loads. The beam element is formulated based on first-order shear deformation theory for the face sheets and the core displacements are assumed to vary linearly across the thickness. A number of numerical experiments involving static, free vibration, and wave propagation analysis examples are solved with an aim to show the super convergent property of the formulated element. The examples presented in this article consider both metallic and composite face sheets. The formulated element is verified in most cases with the results available in the published literature.

Composite material and piezoelectric coefficient uncertainty effects on structural health monitoring using feedback control gains as damage indicators

This article addresses uncertainty effect on the health monitoring of a smart structure using control gain shifts as damage indicators. A finite element model of the smart composite plate with surface-bonded piezoelectric sensors and actuators is formulated using first-order shear deformation theory and a matrix crack model is integrated into the finite element model. A constant gain velocity/position feedback control algorithm is used to provide active damping to the structure. Numerical results show that the response of the structure is changed due to matrix cracks and this change can be compensated by actively tuning the feedback controller. This change in control gain can be used as a damage indicator for structural health monitoring. Monte Carlo simulation is conducted to study the effect of material uncertainty on the damage indicator by considering composite material properties and piezoelectric coefficients as independent random variables. It is found that the change in position feedback control gain is a robust damage indicator.

Vibration and buckling of hybrid laminated curved panels

Vibration and buckling of curved plates, made of hybrid laminated composite materials, are studied using first-order shear deformation theory and Reissner's shallow shell theory. For an initial study, only simply-supported boundary conditions are considered. The natural frequencies and critical buckling loads are calculated using the energy method (Lagrangian approach) by assuming a combination of sine and cosine functions in the form of double Fourier series. The effects of curvature, aspect ratio, stacking sequence and ply-orientation are studied. The non-dimensional frequencies and critical buckling load of a hybrid laminate lie in between the values for laminates made of all plies of higher strength and lower strength fibres. Curvature enhances natural frequencies and it is more predominant for a thin panel than a thick one.

A new beam finite element for the analysis of functionally graded materials

A new beam element is developed to study the thermoelastic behavior of functionally graded beam structures. The element is based on the first-order shear deformation theory and it accounts for varying elastic and thermal properties along its thickness. The exact solution of static part of the governing differential equations is used to construct interpolating polynomials for the element formulation. Consequently, the stiffness matrix has super-convergent property and the element is free of shear locking. Both exponential and power-law variations of material property distribution are used to examine different stress variations. Static, free vibration and wave propagation problems are considered to highlight the behavioral difference of functionally graded material beam with pure metal or pure ceramic beams. (C) 2003 Elsevier Science Ltd. All rights reserved.

Nonlocal continuum mechanics formulation for axial, flexural, shear and contraction coupled wave propagation in single walled carbon nanotubes

This paper presents the effect of nonlocal scaling parameter on the coupled i.e., axial, flexural, shear and contraction, wave propagation in single-walled carbon nanotubes (SWCNTs). The axial and transverse motion of SWCNT is modeled based on first order shear deformation theory (FSDT) and thickness contraction. The governing equations are derived based on nonlocal constitutive relations and the wave dispersion analysis is also carried out. The studies shows that the nonlocal scale parameter introduces certain band gap region in all wave modes where no wave propagation occurs. This is manifested in the wavenumber plots as the region where the wavenumber tends to infinite or wave speed tends to zero. The frequency at which this phenomenon occurs is called the escape frequency. Explicit expressions are derived for cut-off and escape frequencies of all waves in SWCNT. It is also shown that the cut-off frequencies of shear and contraction mode are independent of the nonlocal scale parameter. The results provided in this article are new and are useful guidance for the study and design of the next generation of nanodevices that make use of the coupled wave propagation properties of single-walled carbon nanotubes.

Thermal vibration analysis of orthotropic nanoplates based on nonlocal continuum mechanics

This paper presents the thermal vibration analysis of orthotropic nanoplates such as graphene, using the two variable refined plate theory and nonlocal continuum mechanics for small scale effects. The nanoplate is modeled based on two variable refined plate theory and the axial stress caused by the thermal effects is also considered. The two variable refined plate theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate, hence it is unnecessary to use shear correction factors. Nonlocal governing equations of motion for the nanoplate are derived from the principle of virtual displacements. The closed form solution for thermal-vibration frequencies of a simply supported rectangular nanoplate has been obtained by using Navier's method of solution. Numerical results obtained by the present theory are compared with available solutions in the literature and the molecular dynamics results. The influences of the small scale coefficient, the room or low temperature, the high temparature, the half wave number and the aspect ratio of nanoplate on the natural frequencies are considered and discussed in detail. It can be concluded that the present theory, which does not require shear correction factor, is not only simple but also comparable to the first-order and higher order shear deformation theory. The present analysis results can be used for the design of the next generation of nanodevices that make use of the thermal vibration properties of the nanoplates. (C) 2012 Elsevier B.V. All rights reserved.

EXTENSIONS OF VECTOR BUNDLES WITH APPLICATION TO NOETHER-LEFSCHETZ THEOREMS

Given a smooth, projective variety Y over an algebraically closed field of characteristic zero, and a smooth, ample hyperplane section X subset of Y, we study the question of when a bundle E on X, extends to a bundle epsilon on a Zariski open set U subset of Y containing X. The main ingredients used are explicit descriptions of various obstruction classes in the deformation theory of bundles, together with Grothendieck-Lefschetz theory. As a consequence, we prove a Noether-Lefschetz theorem for higher rank bundles, which recovers and unifies the Noether-Lefschetz theorems of Joshi and Ravindra-Srinivas.

On Uniform Approximate Solutions in Bending of Symmetric Laminated Plates

A layer-wise theory with the analysis of face ply independent of lamination is used in the bending of symmetric laminates with anisotropic plies. More realistic and practical edge conditions as in Kirchhoff's theory are considered. An iterative procedure based on point-wise equilibrium equations is adapted. The necessity of a solution of an auxiliary problem in the interior plies is explained and used in the generation of proper sequence of two dimensional problems. Displacements are expanded in terms of polynomials in thickness coordinate such that continuity of transverse stresses across interfaces is assured. Solution of a fourth order system of a supplementary problem in the face ply is necessary to ensure the continuity of in-plane displacements across interfaces and to rectify inadequacies of these polynomial expansions in the interior distribution of approximate solutions. Vertical deflection does not play any role in obtaining all six stress components and two in-plane displacements. In overcoming lacuna in Kirchhoff's theory, widely used first order shear deformation theory and other sixth and higher order theories based on energy principles at laminate level in smeared laminate theories and at ply level in layer-wise theories are not useful in the generation of a proper sequence of 2-D problems converging to 3-D problems. Relevance of present analysis is demonstrated through solutions in a simple text book problem of simply supported square plate under doubly sinusoidal load.

A parametric study on the buckling of functionally graded material plates with internal discontinuities using the partition of unity method

In this paper, the effect of local defects, viz., cracks and cutouts on the buckling behaviour of functionally graded material plates subjected to mechanical and thermal load is numerically studied. The internal discontinuities, viz., cracks and cutouts are represented independent of the mesh within the framework of the extended finite element method and an enriched shear flexible 4-noded quadrilateral element is used for the spatial discretization. The properties are assumed to vary only in the thickness direction and the effective properties are estimated using the Mori-Tanaka homogenization scheme. The plate kinematics is based on the first order shear deformation theory. The influence of various parameters, viz., the crack length and its location, the cutout radius and its position, the plate aspect ratio and the plate thickness on the critical buckling load is studied. The effect of various boundary conditions is also studied. The numerical results obtained reveal that the critical buckling load decreases with increase in the crack length, the cutout radius and the material gradient index. This is attributed to the degradation in the stiffness either due to the presence of local defects or due to the change in the material composition. (C) 2013 Elsevier Masson SAS. All rights reserved.

Wave propagation analysis in laminated composite plates with transverse cracks using the wavelet spectral finite element method

This paper presents a newly developed wavelet spectral finite element (WFSE) model to analyze wave propagation in anisotropic composite laminate with a transverse surface crack penetrating part-through the thickness. The WSFE formulation of the composite laminate, which is based on the first-order shear deformation theory, produces accurate and computationally efficient results for high frequency wave motion. Transverse crack is modeled in wavenumber-frequency domain by introducing bending flexibility of the plate along crack edge. Results for tone burst and impulse excitations show excellent agreement with conventional finite element analysis in Abaqus (R). Problems with multiple cracks are modeled by assembling a number of spectral elements with cracks in frequency-wavenumber domain. Results show partial reflection of the excited wave due to crack at time instances consistent with crack locations. (C) 2014 Elsevier B.V. All rights reserved.

Wave propagation analysis in adhesively bonded composite joints using the wavelet spectral finite element method

A wavelet spectral finite element (WSFE) model is developed for studying transient dynamics and wave propagation in adhesively bonded composite joints. The adherands are formulated as shear deformable beams using the first order shear deformation theory (FSDT) to obtain accurate results for high frequency wave propagation. Equations of motion governing wave motion in the bonded beams are derived using Hamilton's principle. The adhesive layer is modeled as a line of continuously distributed tension/compression and shear springs. Daubechies compactly supported wavelet scaling functions are used to transform the governing partial differential equations from time domain to frequency domain. The dynamic stiffness matrix is derived under the spectral finite element framework relating the nodal forces and displacements in the transformed frequency domain. Time domain results for wave propagation in a lap joint are validated with conventional finite element simulations using Abaqus. Frequency domain spectrum and dispersion relation results are presented and discussed. The developed WSFE model yields efficient and accurate analysis of wave propagation in adhesively-bonded composite joints. (C) 2014 Elsevier Ltd. All rights reserved.

Deformation and crystallization of Zr-based amorphous alloys in homogeneous flow regime

The purpose of this study is to experimentally investigate the interaction of inelastic deformation and microstructural changes of two Zr-based bulk metallic glasses (BMGs): Zr41.25Ti13.75Cu12.5Ni10Be22.5 (commercially designated as Vitreloy 1 or Vit1) and Zr46.75Ti8.25Cu7.5Ni10Be27.5 (Vitreloy 4, Vit4). High-temperature uniaxial compression tests were performed on the two Zr alloys at various strain rates, followed by structural characterization using differential scanning calorimetry (DSC) and transmission electron microscopy (TEM). Two distinct modes of mechanically induced atomic disordering in the two alloys were observed, with Vit1 featuring clear phase separation and crystallization after deformation as observed with TEM, while Vit4 showing only structural relaxation with no crystallization. The influence of the structural changes on the mechanical behaviors of the two materials was further investigated by jump-in-strain-rate tests, and flow softening was observed in Vit4. A free volume theory was applied to explain the deformation behaviors, and the activation volumes were calculated for both alloys.

Deformation and failure of a film/substrate system subjected to spherical indentation: Part 1. Experimental validation of stresses and strains derived using Hankel transform technique in an elastic film/substrate system

Our concern here is to rationalize experimental observations of failure modes brought about by indentation of hard thin ceramic films deposited on metallic substrates. By undertaking this exercise, we would like to evolve an analytical framework that can be used for designs of coatings. In Part I of the paper we develop an algorithm and test it for a model system. Using this analytical framework we address the issue of failure of columnar TiN films in Part II [J. Mater. Res. 21, 783 (2006)] of the paper. In this part, we used a previously derived Hankel transform procedure to derive stress and strain in a birefringent polymer film glued to a strong substrate and subjected to spherical indentation. We measure surface radial strains using strain gauges and bulk film stresses using photo elastic technique (stress freezing). For a boundary condition based on Hertzian traction with no film interface constraint and assuming the substrate constraint to be a function of the imposed strain, the theory describes the stress distributions well. The variation in peak stresses also demonstrates the usefulness of depositing even a soft film to protect an underlying substrate.