Isogeometric Analysis of Functionally Graded Plates using High-Order Theories

A meshless study of antisymmetric angle-ply laminates using high-order shear deformation theories

An attempt has been made to study the effect of cross‐sectional warping in the symmetrically laminated anisotropic composite sandwich plates for transient loads. A higher‐order shear deformation theory (HOST) is used in conjunction with the simple displacement based C° finite element method (FEM). As is well‐known, the classical first‐order theories hitherto considered were inadequate to describe the propagation of waves in the highly orthotropic sandwich laminates. The present theory, which is more accurate than the Reissner‐Mindlin theory, is applied herein, for the evaluation of plate response to different types of dynamic loads. An explicit central difference scheme is employed for the integration of dynamic equations of equilibrium with a diagonalized mass matrix obtained by a special procedure applicable to quadrilateral isoparametric elements. The numerical results of the present investigation have been compared with the first‐order shear deformation theory (FOST) and the differences between HOST and FOST are examined. The results presented here should be useful in obtaining better correlation between theory and experiment, and to numerical analysts in verifying their results.

In this paper, a novel simple shear deformation theory for buckling analysis of single layer graphene sheet is formulated using the nonlocal differential constitutive relations of Eringen. The present theory involves only three unknown and three governing equation as in the classical plate theory, but it is capable of accurately capturing shear deformation effects, instead of five as in the well-known first shear deformation theory (FSDT) and higher-order shear deformation theory (HSDT). A shear correction factor is, therefore, not required. Nonlocal elasticity theory is employed to investigate effects of small scale on buckling of the rectangular nano-plate. The equations of motion of the nonlocal theories are derived and solved via Navier\'s procedure for all edges simply supported boundary conditions. The results are verified with the known results in the literature. The influences played by Effects of nonlocal parameter, length, thickness of the graphene sheets and shear deformation effect on the critical buckling load are studied. Verification studies show that the proposed theory is not only accurate and simple in solving the buckling nanoplates, but also comparable with the other higher-order shear deformation theories which contain more number of unknowns.

Improved refined plate theory accounting for effect of thickness stretching in functionally graded plates

Assessment of mixed and displacement-based models for static analysis of composite beams of different cross-sections

Analytical solutions are presented for the bending of clamped laminated plates using a higher-order shear deformation theory. Unlike the first-order shear deformation theory, the higher-order theory does not require a shear correction factor. The displacement field of the higher-order theory contains the same number of dependent variables as in the first-order shear deformation theory, and also accounts for the cubic variation of the in-plane displacements through the thickness. Closed-form solutions are not tractable for clamped composite plates due to the boundary conditions and anisotropy of the plate and hence approximate methods are required. In the present work, a method based on Lagrange multipliers is used to enforce the boundary conditions not satisfied by the assumed series. Results are presented for antisymmetric cross-ply and angle-ply plates to study the influence of anisotropy, aspect ratio, and loadings on the deflections and stresses of clamped plates.

Auxetic materials are solids that exhibit negative Poisson's ratio. Due to its uniqueness, it is expected that auxetic solids and structures manifest certain mechanical responses that are distinct from conventional materials. So far classical plate theory (CPT) has been used to investigate the behavior of thin auxetic plates and, for thick auxetic plates, the first-order shear deformation theory (FSDT) has been recently employed. The third-order shear deformation theory (TSDT) is adopted herein as it employs a realistic transverse shear stress and strain distribution across the plate cross section, and therefore removes the need for a correction factor. The results obtained herein reveal that as a plate material's Poisson's ratio becomes more negative, the plate deflection characteristic mimics that of (a) reduced plate thickness, (b) reduced plate aspect ratio, and (c) reduced load waviness. The results of this investigation support (i) the use of auxetic materials in applications where there is a need to reduce the extent of transverse shear deformation, and (ii) the use of the simpler CPT instead of any shear deformation plate theory if the Poisson's ratio of the plate material is sufficiently negative.

Abstract. In the present study, a static response of functionally graded plate and spherical shells is investigated using higher-order trigonometric shear and normal deformation theory. A need of the shear correction factor is obviated and the effect of actual cross-sectional warping has been considered to get the realistic behaviour of transverse shear stresses across the thickness of the shell. The Navier solution technique has been used to analyse the simply-supported boundary conditions of the shell. To verify the theory, the numerical results obtained using the present theory are compared with other higher-order shear deformation theories available in the literature. The numerical results are obtained with and without considering the effects of transverse normal strain (εz).

Introduction. The theory of plates with shear deformation has been developed for several decades beginning from the studies [13, 17] (first-order shear deformation theory or Reissner–Mindlin theory [2–4, 20]). The dynamics of plates with shear compliance and rotary inertia was for the first time studied in [6]. Later, extensive studies in this area were performed in [1, 5, 8, 10–12, 16, etc.], where the effect of shear compliance on the natural frequencies was numerically analyzed in detail using various theories. The following circumstance is noteworthy. The first studies on the dynamics of plates and rods with shear deformation ([6], etc.) discovered two spectra of natural frequencies (two very different (low and high) natural frequencies correspond to the same total deflection mode shape). However, the sense of these two branches is still unclear. It is usually stated that the first branch is related to flexural vibrations and waves, whereas the second branch to shear vibrations and waves. Since this is about the ratio between the bending and shear components, which are present in both spectra, the difference between the two spectra appears to be purely quantitative. Some authors cast doubt on the physical sense of the second spectrum for both Timoshenko beams and for plates with shear compliance and proposed to reject the associated results [7, 17, 18]. A literature review indicates that there has yet been no consensus on the sense of the second eigenvalue spectrum. An important fact is pointed out in [19]: for the first spectrum of a Timoshenko beam, the shear and bending angles have the same phase and are added to produce the total deflection angle, whereas for the second spectrum, the shear and bending angles are opposite in phase, and the total angle is equal to their difference. This observation for beams was later confirmed in [9, 17]. As far as we know, such statements, even as illustrative calculations, have not been made for plates. Here we will use dimensionless equations of motion of a plate (in first-order shear deformation theory) that incorporate only one generalized dimensionless shear compliance parameter to rigorously substantiate the qualitative difference between the two natural frequency spectra (for a hinged plate as an example) and carry out a generalized parametrical analysis. 1. Problem Formulation. Basic Equations. There are various ways of deriving the basic equations of the first-order shear deformation theory of plates (discussed in [2, 3, 20, etc.]). The simplest method is given below. Consider an isotropic plate of constant thickness h with the coordinates x and y in the midsurface and the coordinate z running along the normal. Let w x y ( , ) denote the total deflection (resulting from bending and shear), x and y the angles of rotation of normal sections due to bending, and x and y the shear angles (averaged over the thickness of the plate). The total angles of rotation of the normal to the mid-surface in two planes are given by International Applied Mechanics, Vol. 50, No. 2, March, 2014

A beam made of composite material moving longitudinally over its supports is analyzed based on a higher-order shear deformation theory. The dynamic behavior of the beam is studied using a finite element formulation based on a variational principle. The essential constraints are applied via Lagrange multipliers and this method is effective for the moving beam problem in which the support locations relative to the beam change with time and do not always fall exactly at the nodes. An initially deformed overhang beam moving over two simple supports is used for the analysis. The first flexural mode shape of the beam is used as the initial shape of the beam. The finite element equations of motion are then solved using Newmark's method. The results are presented in terms of time history of tip deflections. The performance of the higher-order shear deformation model is compared with that of the firstorder shear deformation theory and classical laminate plate theory.

Two dimensional kinematic models for CNT reinforced sandwich cylindrical panels with accurate transverse interlaminar shear stress estimation

In the present investigation the higher-order and conventional first-order shear deformation theories are used to study the impact response of doubly curved laminated composite shells. The formulation is based on the Donnell's shallow shell theory. Nine noded Lagrangian elements are used for the finite element formulation. Modified Hertzian contact law is used to calculate the contact force. The results obtained from the present investigation are compared with those existed in the open literature. Numerical results are presented to study the effects of span to thickness ratio, boundary condition, impact point, stacking sequence and velocity of the impactor for cylindrical and spherical shells.

This paper is concerned with the theoretical treatment of transient thermoelastic problem of a cross-ply laminated cylindrical shell due to axisymmetrical heat supply. We analyze the transient thermoelastic problem of a cross-ply laminated cylindrical shell under a simply supported condition using a higher-order shear deformation theory. Some numerical results for the temperature change, the displacements and the stresses are shown in figures. Furthermore, numerical results from this formulation are compared with those obtained using classical and first-order shear deformation theories.

Nowadays, the use of composite laminated structures in the automobile, railroad, civil, aeronautical, space and naval industries is growing at a huge rate. Recently, there has been a renewed interest in sandwich laminated structures, whose bending capability and performance is much better when compared to classical laminates. Sandwich structures, which are mainly used as bending components, are formed by materials with very different stiffness in the faces and in the core. Therefore the behaviour of such structures under bending conditions does not fit the classical laminate theories. Despite this, sandwich structures are simple enough to allow simplified analysis, whose mathematical accuracy depends greatly on the structure itself. In this paper the goal is to study and validate the application in bending of a developed sandwich beam element with transverse compressibility of the core. This element uses a mixed layerwise approach by considering a higher-order shear deformation theory (HSDT) to represent the displacement field of the core and a first-order shear deformation theory (FSDT) for the face layers. Results are compared with three-dimensional finite element solutions and validation is also conducted with values retrieved experimentally.

In this research, the dynamic instability region (DIR) of the sandwich nano-beams are investigated based on nonlocal strain gradient elasticity theory (NSGET) and various higher order shear deformation beam theories (HSDBTs). The sandwich piezoelectric nano-beam is including a homogenous core and face-sheets reinforced with functionally graded (FG) carbon nanotubes (CNTs). In present study, three patterns of CNTs are employed in order to reinforce the top and bottom face-sheets of the beam. In addition, different higher-order shear deformation beam theories such as trigonometric shear deformation beam theory (TSDBT), exponential shear deformation beam theory (ESDBT), hyperbolic shear deformation beam theory (HSDBT), and Aydogdu shear deformation beam theory (ASDBT) are considered to extract the governing equations for different boundary conditions. The beam is subjected to thermal and electrical loads while is resting on Visco-Pasternak foundation. Hamilton principle is used to derive the governing equations of motion based on various shear deformation theories. In order to analysis of the dynamic instability behaviors, the linear governing equations of motion are solved using differential quadrature method (DQM). After verification with validated reference, comprehensive numerical results are presented to investigate the influence of important parameters such as various shear deformation theories, nonlocal parameter, strain gradient parameter, the volume fraction of the CNTs, various distributions of the CNTs, different boundary conditions, dimensionless geometric parameters, Visco-Pasternak foundation parameters, applied voltage and temperature change on the dynamic instability characteristics of sandwich piezoelectric nano-beam.