A Bending-Gradient model for thick plates. Part I: Theory

Journal of Elasticity, 2016

Mechanics of Advanced Materials and Structures, 2002

This article assesses classical and re ned nite plate elements on bending and vibrations of layered composites and sandwich structures. To this purpose, recent authors' ndings have been extended to dynamics. About 20 plate nite elements have been implemented and compared: classical ones based on displacement assumptions are compared to advanced mixed elements which are formulated on the basis of Reissner's mixed variational theorem. Finite elements which preserve the independence of the number of independent variables from the numbers of the N l layers (equivalent single-layer models) as well as those elements in which the number of the unknown variables remains N l -dependent (layer-wise models) are both considered. Linear up to fourth-order expansions in the thickness direction are treated for the unknown stress and displacement variables. Sandwich beams and cross-ply as well as angle-ply composites plates have been analyzed. Simply supported as well as clamped edges have been considered. Finite-element results have been implemented and compared, where available, to analytical closed form solutions. Mostly the fundamental circular frequency has been used as a test bed to assess the whole implemented multilayered elements.

Mechanics of Composite Materials and Structures, 1995

AIAA Journal, 1999

Three-dimensional deformations of a multilayered, linear elastic, anisotropic rectangular plate subjected to arbitrary boundary conditions at its edges are analyzed by the generalized Eshelby-Stroh formalism. The rectangular laminate consists of anisotropic and homogeneous laminae of arbitrary thicknesses. Perfect bonding is assumed between the adjoining laminae in the sense that both surface tractions and displacements are assumed to be continuous across their interfaces. The analytical solution is in terms of infinite series, and the effect of truncating the series on the accuracy of the solution is scrutinized. The method is also applicable to rectangular laminated plates, with edges of each lamina subjected to different boundary conditions. Results are presented for thick plates with different sets of edge boundary conditions, e.g., two opposite edges simply supported and the other two subjected to eight different conditions or all four edges clamped. F IBER-REINFORCED laminated plates are extensively used in aerospace, automotive, and ship-building industries primarily because of their high strength-to-weight ratio, and their strength and stiffness can be tailored to meet design requirements. The accurate prediction of the response characteristics of such laminated structures is a challenging task because of their intrinsic anisotropy, heterogeneity, and low ratio of the transverse shear modulus to the in-plane Young's modulus. Laminated plates are usually analyzed by use of equivalent single-layer theories based on either the classical laminated plate theory 1 ' 2 (CLPT), which assumes the Kirchhoff-Love hypothesis, or its refinements, such as the first-order shear deformation theory 2 ' 3 (FSDT) and higher-order theories, 2 ' 4 " 6 which include the effect of transverse shear deformations. Accurate prediction of interlaminar stresses is very important since they usually cause delamination failure at the interfaces. A drawback of equivalent single-layer theories is that they allow for discontinuous interlaminar stresses. Layerwise theories 7 " 10 are considerably more accurate than the preceding theories. We refer the reader to Refs. 2, 11, and 12 for a historical perspective and for a review of various approximate theories. The validity of approximate plate theories can be assessed by comparing their predictions with the analytical solutions of the three-dimensional equations of anisotropic elasticity. Pagano, 13 -14 Pagano and Hatfield, 15 Srinivas et al., and Srinivas and Rao 17 obtained analytical solutions for orthotropic simply supported laminates. These benchmark solutions have been used to validate new or improved plate theories and finite-element formulations. 7 " Iltl8 ~22 However, simply supported edge conditions are less frequently realized in practice, and they do not exhibit the well-known boundarylayer effects observed near clamped or free edges. Here we present analytical solutions for the deformations of anisotropic rectangular thick plates subjected to arbitrary boundary conditions. Each lamina may be generally anisotropic with 21 elastic constants and subjected to boundary conditions different from those on the adjoining laminae. The three-dimensional equations of elasticity are solved by a generalization of the Eshelby-Stroh formalism. Thus the governing equations are exactly satisfied, and

Computer Methods in Applied Mechanics and Engineering, 1991

Aerospace Science and Technology, 2008

The purpose of this paper is to critically evaluate two Reissner-Mindlin type theories developed recently for composite laminated plates, namely, VAPAS (Variational Asymptotic Plate And Shell analysis) and EFSDT (Enhanced First-Order Shear-Deformation Theory). The fundamentals of both models are briefly summarized along with their unique features in comparison to most other existing models. The similarities and differences between VAPAS and EFSDT are also examined. Exact solutions of three-dimensional elasticity theory for the cylindrical bending problems are used as the arbiter to assess the accuracy of both models. Such a systematic assessment demonstrates that both models have achieved an excellent compromise between the layer-wise theories, which are accurate but computationally demanding, and the first-order shear deformation theories, which are computationally cheap but not accurate.

Journal of Applied Mechanics, 1987

In order to improve the accuracy of in-plane responses of shear deformable composite plate theories, a new laminated plate theory was developed for arbitrary laminate configurations based upon Reissner’s (1984) new mixed variational principle. To this end, across each individual layer, piecewise linear continuous displacements and quadratic transverse shear stress distributions were assumed. The accuracy of the present theory was examined by applying it to the cylindrical bending problem of laminated plates which had been solved exactly by Pagano (1969). A comparison with the exact solutions obtained for symmetric, antisymmetric, and arbitrary laminates indicates that the present theory accurately estimates in-plane responses, even for small span-to-thickness ratios.

A comprehensive literature review on different theories of laminated plates have been reviewed and discussed thoroughly. It has been found that there are two main theories of laminated plates which are known as linear and nonlinear theories. The two theories are depending on the magnitude of deformation resulting from loading the given plates. The difference between the two theories is that the deformations are small in the linear theory, whereas they are finite or large in the nonlinear theory. In comparisons between FEM and different numerical methods it has been found that FEM can be considered of acceptable accuracy, and can also be applied to different complicated geometries and shapes. Keywords: Theories of laminates, linear and nonlinear, numerical methods, finite element method, small and large deformations. Developments in the theories of laminated plates: From the point of view of solid mechanics, the deformation of a plate subjected to transverse and / or in plane loading consists of two components: flexural deformation due to rotation of cross – sections, and shear deformation due to sliding of section or layers. The resulting deformation depends on two parameters: the thickness to length ratio and the ratio of elastic to shear moduli. When the thickness to length ratio is small, the plate is considered thin, and it deforms mainly by flexure or bending; whereas when the thickness to length and the modular ratios are both large, the plate deforms mainly through shear. Due to the high ratio of in – plane modulus to transverse shear modulus, the shear deformation effects are more pronounced in the composite laminates subjected to transverse and / or in plane loads than in the isotropic plates under similar loading conditions. The three – dimensional theories of laminates in which each layer is treated as homogeneous anisotropic medium (see Reddy [ ]) are intractable. Usually, the anisotropy in laminated composite structures causes complicated responses under different loading conditions by creating complex couplings between extensions and bending, and shears deformation modes. Expect for certain cases, it is inconvenient to fully solve a problem in three dimensions due to the complexity, size of computation, and the production of unnecessary data specially for composite structures.