Showing papers on "Plate theory published in 2010"

The mechanics of non-Euclidean plates

Abstract: Non-Euclidean plates are plates (“stacks” of identical surfaces) whose two-dimensional intrinsic geometry is not Euclidean, i.e. cannot be realized in a flat configuration. They can be generated via different mechanisms, such as plastic deformation, natural growth or differential swelling. In recent years there has been a concurrent theoretical and experimental progress in describing and fabricating non-Euclidean plates (NEP). In particular, an effective plate theory was derived and experimental methods for a controlled fabrication of responsive NEP were developed. In this paper we review theoretical and experimental works that focus on shape selection in NEP and provide an overview of this new field. We made an effort to focus on the governing principles, rather than on details and to relate the main observations to known mechanical behavior of ordinary plates. We also point out to open questions in the field and to its applicative potential.

Chaotic vibrations of an orthotropic FGM rectangular plate based on third-order shear deformation theory

Abstract: In this paper, an analysis on the nonlinear dynamics and chaos of a simply supported orthotropic functionally graded material (FGM) rectangular plate in thermal environment and subjected to parametric and external excitations is presented. Heat conduction and temperature-dependent material properties are both taken into account. The material properties are graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Based on the Reddy’s third-order share deformation plate theory, the governing equations of motion for the orthotropic FGM rectangular plate are derived by using the Hamilton’s principle. The Galerkin procedure is applied to the partial differential governing equations of motion to obtain a three-degree-of-freedom nonlinear system. The resonant case considered here is 1:2:4 internal resonance, principal parametric resonance-subharmonic resonance of order 1/2. Based on the averaged equation obtained by the method of multiple scales, the phase portrait, waveform and Poincare map are used to analyze the periodic and chaotic motions of the orthotropic FGM rectangular plate. It is found that the motions of the orthotropic FGM plate are chaotic under certain conditions.

Small scale effect on the buckling analysis of single-layered graphene sheet embedded in an elastic medium based on nonlocal plate theory

Abstract: Nonlocal elasticity theory is implemented to investigate the buckling behavior of single-layered graphene sheet (SLGS) embedded in an elastic medium. Nonlocal elasticity theory accounts for the small-size effects when dealing with nanostructures such as graphene sheets. Both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction between the graphene sheet and the surrounding elastic medium. Based on principle of virtual work, governing differential equations for the aforementioned problem are derived. Differential quadrature method is being employed and numerical solutions for the buckling loads of SLGS are obtained. Numerical results show that the buckling loads of SLGS are strongly dependent on the small scale coefficients and the stiffness of the surrounding elastic medium. With elastic medium modeled as Winkler-type foundation, the nonlocal effects are found to have decrease–increase–decrease pattern with increase in stiffness of elastic medium.

Levy Solution for Buckling Analysis of Functionally Graded Rectangular Plates

Abstract: In this article, an analytical method for buckling analysis of thin functionally graded (FG) rectangular plates is presented. It is assumed that the material properties of the plate vary through the thickness of the plate as a power function. Based on the classical plate theory (Kirchhoff theory), the governing equations are obtained for functionally graded rectangular plates using the principle of minimum total potential energy. The resulting equations are decoupled and solved for rectangular plate with different loading conditions. It is assumed that the plate is simply supported along two opposite edges and has arbitrary boundary conditions along the other edges. The critical buckling loads are presented for a rectangular plate with different boundary conditions, various powers of FGM and some aspect ratios.

Guidelines and Recommendations to Construct Theories for Metallic and Composite Plates

Abstract: This work has evaluated the refinement of some classical theories, such as the Kirchhoff and Reissner-Mindlin theories, adding generalized displacement variables (up to fourth-order) to the Taylor-type expansion in the thickness plate direction. Isotropic, orthotropic, and laminated plates have been analyzed, varying the thickness ratio, orthotropic ratio, and stacking sequence of the layout. Higher-order theories have been implemented according to the compact scheme known as the Carrera unified formulation. The results have been restricted to simply-supported orthotropic plates subjected to harmonic distributions of transverse pressure for which closed-form solutions are available. For a given plate problem (isotropic, orthotropic, or laminated), the effectiveness of each employed generalized displacement variable has been established comparing the error obtained accounting for and removing the variable in the plate governing equations. A number of theories have therefore been constructed imposing a given error with respect to the available best results. Guidelines and recommendations that are focused on the proper selection of the displacement variables that have to be retained in refined plate theories are then furnished. It has been found that the terms that have to be used according to a given error vary from problem to problem, but they also vary when the variable that has to be evaluated (displacement, stress components) is changed. Diagrams (errors in terms of geometrical and orthotopic ratios) and graphical schemes have been built to establish the appropriate theories with respect to the data of the problem under consideration.

Stochastic laminate analogy for simulating the variability in modulus of discontinuous composite materials

Abstract: Recent composite technology research and development efforts have focused on discontinuous carbon fiber/epoxy molding systems derived from chopped aerospace-grade unidirectional tape prepreg. Although the average elastic modulus of this material has been shown to be as high as that of the continuous tape quasi-isotropic benchmark, experimental measurement by means of strain gage or extensometer has shown variation as high as 20%. Digital Image Correlation can be used successfully to obtain a full-field strain measurement, and it shows that a highly non-uniform strain distribution exists on the surface of the specimen, with distinct peaks and valleys. This pattern of alternating regions of high and low strain gradients, and which exhibit a characteristic shape and size, can be described in terms of Random Representative Volume Element (RRVE). The RRVE proposed here exhibits random elastic properties, which are assigned based on stochastic distributions. This approach leads to the analysis method proposed here, which is designed to compensate for the fact that traditional methods cannot capture the experimentally observed variation in modulus within a specimen and among different specimens. The method utilizes a randomization process to generate statistical distributions of fractions and orientations of chips within the RRVE, and then applies Classical Laminated Plate Theory to an equivalent quasi-isotropic tape laminate to calculate its average elastic properties. Validation of this method is shown as it applies to a finite element model that discretizes the structure in multiple RRVEs, whose properties are generated independently of the neighboring ones, and then are solved simultaneously. The approach generates accurate predictions of the strain distribution on the surface of the specimen.

A two variable refined plate theory for the bending analysis of functionally graded plates

Abstract: Bending analysis of functionally graded plates using the two variable refined plate theory is presented in this paper. The number of unknown functions involved is reduced to merely four, as against five in other shear deformation theories. The variationally consistent theory presented here has, in many respects, strong similarity to the classical plate theory. It does not require shear correction factors, and gives rise to such transverse shear stress variation that the transverse shear stresses vary parabolically across the thickness and satisfy shear stress free surface conditions. Material properties of the plate are assumed to be graded in the thickness direction with their distributions following a simple power-law in terms of the volume fractions of the constituents. Governing equations are derived from the principle of virtual displacements, and a closed-form solution is found for a simply supported rectangular plate subjected to sinusoidal loading by using the Navier method. Numerical results obtained by the present theory are compared with available solutions, from which it can be concluded that the proposed theory is accurate and simple in analyzing the static bending behavior of functionally graded plates.

Nonlinear dynamic response of a functionally graded plate with a through-width surface crack

Abstract: A nonlinear vibration analysis of a simply supported functionally graded rectangular plate with a through-width surface crack is presented in this paper. The plate is subjected to a transverse excitation force. Material properties are graded in the thickness direction according to exponential distributions. The cracked plate is treated as an assembly of two sub-plates connected by a rotational spring at the cracked section whose stiffness is calculated through stress intensity factor. Based on Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the FGM plate are derived by using the Hamilton’s principle. The deflection of each sub-plate is assumed to be a combination of the first two mode shape functions with unknown constants to be determined from boundary and compatibility conditions. The Galerkin’s method is then utilized to convert the governing equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms under the external excitation, which is numerically solved to obtain the nonlinear responses of cracked FGM rectangular plates. The influences of material property gradient, crack depth, crack location and plate thickness ratio on the vibration frequencies and transient response of the surface-racked FGM plate are discussed in detail through a parametric study.