Approximate Plate Theories Research Articles

3D nonlinear variable strain-rate-dependent-order fractional thermoviscoelastic dynamic stress investigation and vibration of thick transversely graded rotating annular plates/discs

In the present article, the idea of using the variable-order fractional-derivative thermoviscoelastic constitutive laws in dynamic stress and vibration analysis of the engineering structures, the required implementation backgrounds, and the relevant numerical solution procedures are investigated for the first time. In this regard, dynamic 3D stress and displacement fields and radial/transverse vibrations of transversely graded viscoelastic spinning thick plates/discs exposed to sudden thermoelastic loads are investigated. Instead of using the approximate plate theories, the exact thermoviscoelasticity theory is employed in the development of the governing equations. Since the variable fractional order is dependent on the localized deformation rates, the resulting thermoviscoelastic integro-differential equations are nonlinear. These equations are solved by utilizing a combination of the second-order backward/central/forward finite difference discretization of the spatial and time domains, numerical evaluation and updating of the Caputo-type fractional derivatives, updating the growing number of terms of the governing equations, and Picard's iterations. Various edge conditions are considered. Finally, comprehensive sensitivity analyses and various 3D plots are presented and discussed regarding the effects of the variable fractional order of the constitutive law, time variations of the nonuniformly distributed transverse loads, and edge conditions on the distributions and damping of the resulting displacement and stress components.

Shear post buckling analysis of FGM annular sector plates based on three dimensional elasticity for different boundary conditions

In this paper, shear post-buckling analysis of functionally graded annular sector plates is considered. In-plane shear loads have been applied to either radial, circumferential, or all edges of annular sector plates. Moreover, post-buckling of annular sector plates subjected to in-plane shear load at upper/lower surfaces is investigated for the first time. Material properties are graded asymmetrically in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents while the Poisson’s ratio is assumed to be constant. The governing equations are based on three dimensional theory of elasticity in conjunction with non-linear Green strain tensor instead of the approximate plate theories and Von Karman assumptions. The governing equations are developed based on the principle of virtual work and solved based on graded finite element method. Non-linear equilibrium equations are solved based on iterative Newton–Raphson method. The effects of material gradient exponent, different sector angles, thickness ratio and four different boundary conditions on the post-buckling behavior of FGM annular sector plates have been investigated. Results denote that due to weak coupling between applied shear loads and the resulting twisting moments, shear post-buckling behavior of simply supported FGM plates is of the bifurcation-type buckling following a stable post-buckling path.

Uniaxial and biaxial post-buckling behaviors of longitudinally graded rectangular plates on elastic foundations according to the 3D theory of elasticity

Longitudinally varying the material properties may be an adequate optimization solution when there are restrictions in varying cross sections of the load carrying elements but leads to distorted and variable length buckling and post-buckling deformation waves. In the present paper, uniaxial and biaxial post-buckling behaviors of the longitudinally graded plates are investigated. Neither post-buckling nor even buckling of the longitudinally graded plates have been studied so far. Moreover, effects of the elastic foundation that significantly alter the post-buckling deformations and strength of the plate, are studied. Due to occurrence of large deformations in the post-buckling region, the exact 3D elasticity theory is employed here instead of the approximate plate theories. Furthermore, the full form of Green’s strain tensor is adopted. To establish a stress field that is continuous at the mutual nodes of the adjacent elements, a C1-continuous 3D Hermitian element is employed to discretize the plate. The nonlinear post-buckling equilibrium equations are solved by Crisfield–Ramm arc-length method in conjunction with the modified Newton–Raphson technique. Results show that heterogeneity of the material properties may lead to abrupt deflection decreases or even reversals and the elastic foundation leads to increases in both the buckling strength and number of the deformation waves.

Three-dimensional biaxial post-buckling analysis of heterogeneous auxetic rectangular plates on elastic foundations by new criteria

Post-buckling behavior of the auxetic (with negative Poisson ratio) structures has not been investigated so far, especially for structures fabricated from heterogeneous materials. In the present research, effects of auxeticity of the materials on uniaxial and biaxial post-buckling behaviors of the FGM plates are investigated using the exact three-dimensional theory of elasticity, instead of the approximate plate theories. Furthermore, influence of the Winkler-type elastic foundation on the post-buckling responses of the plate is investigated. In this regard, a full compatible C1-continuous 3D Hermitian element that in contrast to the conventional elements, satisfies the stress continuity condition at the mutual nodes of the adjacent elements, is employed to solve the governing equations of the post-buckling. The full Green’s strain tensor is employed instead of using the approximate von Karman strain–displacement expressions to enable tracing behavior of the plate for quite large deformations. The governing system of equations is solved by a condensation-based incremental under-relaxation Newton–Raphson procedure, due to sensitivity and high numerical compliance of the resulting highly non-linear finite element system of equations. Furthermore, new criteria are proposed for tracing the post-buckling behavior. Results reveal that the elastic foundation significantly affects the post-buckling mechanism and deformation pattern and post-buckling behavior of the plate is enhanced drastically by more negative Poisson’s ratios.

Guided wave scattering analysis for a plate with arbitrarily shaped elastic inclusions using the T-matrix method

Guided wave scattering by an arbitrarily shaped elastic inclusion in a flat plate is analyzed by using the three-dimensional elasticity. An inclusion is decomposed into a number of small subscatterers and the field scattered by the subscatterers is solved by using a multiple scattering theory. The transition matrix relating the scattering field by a single subscatterer and an incident field is utilized for the multiple scattering analysis, but near-field interactions among subscatterers must be considered because of the closely packed arrangement of subscatterers. This calls for an analysis of nonpropagating wave modes and requires consideration of their influences on near-field interactions. Most earlier guided wave scattering analyses were based on an approximate plate theory and the wavefunction expansion techniques based on the three-dimensional elasticity were developed only for a cavity with a small range of shapes and sizes. Therefore, this investigation presents new results applicable for wave scattering by an elastic inclusion with a wide range of shapes and sizes in a plate. Several numerical tests were conducted to ensure stable solution convergence; the scattered wave fields generated by using the developed method for some case studies were compared with those of finite element analysis.

High-Frequency Guided Wave Scattering by a Partly Through-Thickness Hole Based on 3D Theory**Supported by the National Natural Science Foundation of China under Grant Nos 11474195, 11274226 and 61171145.

We present a theoretical investigation of the scattering of high frequency S0 Lamb mode from a circular blind hole defect in a plate based on the 3D theory. The S0 wave is incident at the frequency above the A1 mode cut-off frequency, in which the popular approximate plate theories are inapplicable. Due to the non-symmetric blind hole defect, the scattered fields will contain higher order converted modes in addition to the fundamental S0 and A0 modes. The far-field scattering amplitudes of various propagating Lamb modes for different hole sizes are inspected. The results are compared with those of lower frequencies and some different phenomena are found. Two-dimensional Fourier transform (2DFT) results of transient scattered Lamb and SH wave signals agree well with the analytical dispersion curves, which check the validity of the solutions from another point of view.

Three-dimensional magneto-elastic analysis of asymmetric variable thickness porous FGM circular plates with non-uniform tractions and Kerr elastic foundations

In the present paper, a complicated and general problem of a functionally graded porous variable thickness circular plate subjected to non-axisymmetric and non-uniform shear and normal tractions and a magnetic actuation, and supported by a non-uniform Kerr elastic foundation is analyzed. Although such a combination of complexities may rarely be observed in the practical applications, the developed solution may identically be employed for many practical problems, as special cases. The governing equations are derived based on the three-dimensional theory of elasticity instead of using the approximate plate theories. The solution is proposed based on the differential quadrature and state space vector techniques in the radial and thickness directions, respectively. A comprehensive parametric study is presented to evaluate effects of the material, loading, boundary, and elastic foundation specifications on the resulting displacement, stress, Lorentz force, electromagnetic stress, and magnetic perturbation quantities.

Three dimensional graded finite element elasticity shear buckling analysis of FGM annular sector plates

In the present paper, shear buckling analysis of functionally graded annular sector plates is investigated for the first time. A three-dimensional elasticity approach is employed instead of the approximate plate theories. In-plane shear loads have been applied to radial, circumferential, or all edges of annular sector plates. Moreover, buckling of annular sector plates subjected to in-plane shear load at upper/lower surfaces is investigated for the first time. Three different boundary conditions have been examined: (1) Movable simply supported edges. (2) Immovable simply supported radial edges and free circumferential edges. (3) Immovable simply supported circumferential edges and free radial edges. The material properties are assumed to have transverse heterogeneity according to a power law distribution while Poisson's ratio is assumed to be constant. Results are extracted based on the principle of minimum total potential energy and a graded finite element method. Buckling loads are obtained based on a generalized geometric stiffness concept. In addition to different loading and boundary conditions, the effects of material gradient exponent, sector angles, aspect ratio and thickness ratio on the shear buckling loads and mode shapes of FGM annular sector plates have been investigated in detail.

The High-frequency Scattering of the S0 Lamb Mode by a Circular Blind Hole in a Plate

The scattering problem of an incident high-frequency S0 Lamb wave in a plate with a circular blind hole is investigated. The study is not limited to the low-frequency range of this wave, in which the popular approximate plate theories are inapplicable. A 3D analytical approach is used where the wave fields are expanded in all possible Lamb modes, including propagating and evanescent modes. Due to the non-symmetric blind hole defect, the scattered fields will contain higher order converted modes in addition to the fundamental S0 and A0 modes. The far field scattering amplitudes of various propagating Lamb modes for different hole sizes are inspected. The results are compared with those of low frequencies and some different phenomena are found.

Ultrasonic wave propagation in composite laminates by numerical simulation

The role of transverse shear is crucial in modeling ultrasonic wave propagation in composites. The approximate plate theories have been extremely useful in providing solutions to problems involving static and dynamic loadings of uniform or laminated plates of finite dimensions. Moreover, since their simple formulations, they can also be useful for solving wave propagation problems in laminates of finite thickness and large lateral dimensions. A four-node plate bending element based on first order shear deformation theory is used in this work to predict the propagation velocity of first antisymmetric Lamb wave mode in graphite/epoxy composite plates by numerical simulation. Calculations are made using material properties obtained from a unidirectional panel via mechanical tests. For investigation, propagation along multiple directions is evaluated. Experimental and numerical results are compared at different angles of propagation and agreement between experimental and numerical approaches is found.

Large Amplitude Free Vibrations of Plates Resting on a Pasternak Type Elastic Foundation

In this paper following Berger's approximate plate theory for large deflections, large amplitude free vibrations of Rectangular Plate, isosceles right-angled triangular plate, Equilateral triangular plate and circular plate resting on a Pasternak-type elastic foundation have been discussed.

Three-dimensional non-linear elasticity-based 3D cubic B-spline finite element shear buckling analysis of rectangular orthotropic FGM plates surrounded by elastic foundations

In the present paper, shear buckling analysis of the orthotropic heterogeneous FGM plates is investigated for the first time. Moreover, influence of the Winkler-type elastic foundation is considered. The material properties are assumed to have in-plane orthotropy and transverse heterogeneity. The most accurate approach, i.e., the three-dimensional elasticity is employed instead of using the approximate plate theories. In contrast to all of the available displacement-based buckling analyses that have employed C0-continuous commercial finite element codes or semi-analytical methods, present formulations are C2-continuous due to using the proposed 3D cubic B-spline element. Results are derived based on principle of minimum potential energy and a non-linear finite element procedure utilizing a Galerkin-type 3D cubic B-spline solution algorithm. Buckling loads are detected based on a generalized geometric stiffness concept. In this regard, effects of both the prebuckling and buckling states are considered. To present a better imagination and more detailed discussions, details of the buckling mode shapes and the foundation interaction are discussed for plates with simply-supported edges. In addition, the more practical free and clamped edge conditions are also considered.

Reliability assessment of different plate theories for elastic wave propagation analysis in functionally graded plates

The importance of elastic wave propagation problem in plates arises from the application of ultrasonic elastic waves in non-destructive evaluation of plate-like structures. However, precise study and analysis of acoustic guided waves especially in non-homogeneous waveguides such as functionally graded plates are so complicated that exact elastodynamic methods are rarely employed in practical applications. Thus, the simple approximate plate theories have attracted much interest for the calculation of wave fields in FGM plates. Therefore, in the current research, the classical plate theory (CPT), first-order shear deformation theory (FSDT) and third-order shear deformation theory (TSDT) are used to obtain the transient responses of flexural waves in FGM plates subjected to transverse impulsive loadings. Moreover, comparing the results with those based on a well recognized hybrid numerical method (HNM), we examine the accuracy of the plate theories for several plates of various thicknesses under excitations of different frequencies. The material properties of the plate are assumed to vary across the plate thickness according to a simple power-law distribution in terms of volume fractions of constituents. In all analyses, spatial Fourier transform together with modal analysis are applied to compute displacement responses of the plates. A comparison of the results demonstrates the reliability ranges of the approximate plate theories for elastic wave propagation analysis in FGM plates. Furthermore, based on various examples, it is shown that whenever the plate theories are used within the appropriate ranges of plate thickness and frequency content, solution process in wave number-time domain based on modal analysis approach is not only sufficient but also efficient for finding the transient waveforms in FGM plates.

On the three-dimensional undulation instability of a rectangular viscoelastic composite plate in biaxial compression

Within the frame work of the second version of small precritical deformation in the three-dimensional linearized theory of stability of deformable bodies (TDLTSDB), the undulation instability problem for a simply supported rectangular plate made of a viscoelastic composite material is investigated in biaxial compression in the plate plane. The corresponding boundary-value problem is solved by employing the Laplace transformation and the principle of correspondence. For comparison and estimation of the accuracy of results given by the TDLTSDB, the same problem is also solved by using various approximate plate theories. The viscoelasticity properties of the plate material are described by the Rabotnov fractional-exponential operator. The numerical results and their discussion are presented for the case where the plate is made of a multilayered viscoelastic composite material. In particular, the variation range of problem parameters is established for which it is necessary to investigate the undulation instability of the viscoelastic composite plate by using the TDLTSDB.

An investigation into the propagation characteristics of AE signals in PE/PE composite laminates

Theoretical and experimental methods were presented to study the propagation characteristics of elastic waves in PE/PE composite laminates. Approximate plate theories were explored to predict the dispersion of flexural mode of the elastic waves. Arrival times of the waves at different frequency were determined on the base of Gabor wavelet transform. A pencil lead break generated the elastic waves in orthotropy and quasi-isotropy PE/PE composite laminates. Experimental results illustrated more serious attenuation of the waves. The comparison between the experimental measurements and predictions showed good agreement.

Torsional vibrations of circular elastic plates with thickness steps

This paper presents a theoretical study of torsional vibrations in isotropic elastic plates. The exact solutions for torsional vibrations in circular and annular plates are first reviewed. Then, an approximate method is developed to analyze torsional vibrations of circular plates with thickness steps. The method is based on an approximate plate theory for torsional vibrations derived from the variational principle following Mindlin's series expansion method. Approximate solutions for the zeroth- and first-order torsional modes in the circular plate with one thickness step are presented. It is found that, within a narrow frequency range, the first-order torsional modes can be trapped in the inner region where the thickness exceeds that of the outer region. The mode shapes clearly show that both the displacement and the stress amplitudes decay exponentially away from the thickness step. The existence and the number of the trapped first-order torsional modes in a circular mesa on an infinite plate are determined as functions of the normalized geometric parameters, which may serve as a guide for designing distributed torsional-mode resonators for sensing applications. Comparisons between the theoretical predictions and experimental measurements show close agreements in the resonance frequencies of trapped torsional modes.

Analysis of the acoustical edge flexural mode in a plate using refined asymptotics

The dispersion properties of bending wave, localized near the stress-free edge of a thin isotropic plate, are investigated for relatively high frequency. In order to clarify the nature of the wave, attention is focused to the sensibility to the improved boundary conditions and to the iterations of the main differential operator when describing by two-dimensional (2-D) approximate plate theories. As shown, the boundary conditions of next asymptotic order do not change too much the leading part of the desired wave, but the corrections of the main operator lead to dramatical changes. Finally, beginning with its simplest Kirchooff’s plate model only the leading part of wave may be subjected to asymptotic correction. For applications the wave speed calculation is reduced to two simple analytical formulas. The results agree well with experiments and finite element testing. The alternative approach related to the Timoshenko–Reissner–Mindlin theory is discussed.

Analysis of transient Lamb waves generated by dynamic surface sources in thin composite plates

A theoretical analysis is carried out in an effort to understand certain unusual properties of transient guided waves produced in a thin unidirectional graphite/epoxy composite plate by a localized dynamic surface load. The surface motion is calculated using an approximate plate theory, called the shear deformation plate theory (SDPT), as well as a recently developed finite element analysis (FEA), for their mutual verification. The results obtained by the two methods are shown to have excellent agreement. An interesting, nearly periodic “phase reversal” of the signal with propagation distance is observed for each propagation direction relative to the fiber direction. For clarification, a closed form analytical expression for the vertical surface displacement in an aluminum plate to an impulsive point force is obtained using the steepest descent method. It is found that the strong dispersion of the first antisymmetric waves at low frequencies is the main reason behind the phase reversal. This is verified further by measuring the surface response of a relatively thick aluminum plate to a pencil lead break source. The understanding developed in the paper is expected to be helpful in detecting and characterizing the occurrence of damage in composite structures.

Three-dimensional thermomechanical deformations of functionally graded rectangular plates

Abstract Three-dimensional thermomechanical deformations of simply supported, functionally graded rectangular plates are studied by using an asymptotic method. The locally effective material properties are estimated by the Mori–Tanaka scheme. The temperature, displacements and stresses of the plate are computed for different volume fractions of the ceramic and metallic constituents, and they could serve as benchmark results to assess two-dimensional approximate plate theories.

Midfrequency acoustical modeling of a coated plate

A steel plate with an attached thick viscoelastic layer exhibits ‘‘midfrequency’’ characteristics at relatively low frequencies, certainly below 1 kHz for a 1-in. steel plate with thicker coating. The low-shear stiffness of the coating leads to a cut-on frequency associated with a shear thickness resonance. Classical plate theory or composite plate theory cannot capture this phenomenon. Although it is not difficult to compute the exact dispersion and modal properties of this system, an approximate plate theory is required in order to examine how the coated plate interacts dynamically with uncoated regions (reflection, transmission), and for modeling scattering from structure incorporating these elements. A hybrid plate theory is presented, based on a classical Kirchhoff assumption for the steel, and a Timoshenko/Mindlin model for the coating. This leads to a three degrees of freedom model, which exhibits the same dispersive properties as the exact theory. Comparisons and differences are discussed. The effects of fluid loading are also presented. [Work supported by ONR.]