Showing papers on "Plate theory published in 2009"

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

Abstract: Modeling of Functionally Graded Materials and Structures Effective Material Properties of FGMs Reddy's Higher Order Shear Deformation Plate Theory Generalized Karman-Type Nonlinear Equations Nonlinear Bending of Shear Deformable FGM Plates Nonlinear Bending of FGM Plates under Mechanical Loads in Thermal Environments Nonlinear Thermal Bending of FGM Plates due to Heat Conduction Postbuckling of Shear Deformable FGM Plates Postbuckling of FGM Plates with Piezoelectric Actuators under Thermoelectromechanical Loads Thermal Postbuckling Behavior of FGM Plates with Piezoelectric Actuators Postbuckling of Sandwich Plates with FGM Face Sheets in Thermal Environments Nonlinear Vibration of Shear Deformable FGM Plates Nonlinear Vibration of FGM Plates in Thermal Environments Nonlinear Vibration of FGM Plates with Piezoelectric Actuators in Thermal Environments Vibration of Postbuckled Sandwich Plates with FGM Face Sheets in Thermal Environments Postbuckling of Shear Deformable FGM Shells Boundary Layer Theory for the Buckling of FGM Cylindrical Shells Postbuckling Behavior of FGM Cylindrical Shells under Axial Compression Postbuckling Behavior of FGM Cylindrical Shells under External Pressure Postbuckling Behavior of FGM Cylindrical Shells under Torsion Thermal Postbuckling Behavior of FGM Cylindrical Shells Appendices

Elastic theory of unconstrained non-Euclidean plates

Abstract: Non-Euclidean plates are a subset of the class of elastic bodies having no stress-free configuration. Such bodies exhibit residual stress when relaxed from all external constraints, and may assume complicated equilibrium shapes even in the absence of external forces. In this work we present a mathematical framework for such bodies in terms of a covariant theory of linear elasticity, valid for large displacements. We propose the concept of non-Euclidean plates to approximate many naturally formed thin elastic structures. We derive a thin plate theory, which is a generalization of existing linear plate theories, valid for large displacements but small strains, and arbitrary intrinsic geometry. We study a particular example of a hemispherical plate. We show the occurrence of a spontaneous buckling transition from a stretching dominated configuration to bending dominated configurations, under variation of the plate thickness.

A Survey With Numerical Assessment of Classical and Refined Theories for the Analysis of Sandwich Plates

Abstract: A large variety of plate theories are described and assessed in the present work to evaluate the bending and vibration of sandwich structures. A brief survey of available works is first given. Such a survey includes significant review papers and latest developments on sandwich structure modelings. The kinematics of classical, higher order, zigzag, layerwise, and mixed theories is described. An exhaustive numerical assessment of the whole theories is provided in the case of closed form solutions of simply supported panels made of orthotropic layers. Reference is made to the unified formulation that has recently been introduced by the first author for a plate/shell analysis. Attention has been given to displacements, stresses (both in-plane and out-of-plane components), and the free vibration response. Only simply supported orthotropic panels loaded by a transverse distribution of bisinusoidal pressure have been analyzed. Five benchmark problems are treated. The accuracy of the plate theories is established with respect to the length-to-thickness-ratio (LTR) geometrical parameters and to the face-to-core-stiffness-ratio (FCSR) mechanical parameters. Two main sources of error are outlined, which are related to LTR and FCSR, respectively. It has been concluded that higher order theories (HOTs) can be conveniently used to reduce the error due to LTR in thick plate cases. But HOTs are not effective in increasing the accuracy of the classical theory analysis whenever the error is caused by increasing FCSR values; layerwise analysis becomes mandatory in this case.

On the linear theory of micropolar plates

Abstract: We discuss the general linear six-parametric theory of plates based on the direct approach. We consider the plate as a deformable surface. Each material point of the surface can be regarded as an infinitesimal small rigid body with six degrees of freedom. The kinematics of the plate is described by using the vector of translation and the vector of rotation as the independent variables. The relations between the equilibrium conditions of a three-dimensional micropolar plate-like body and the two-dimensional equilibrium equations of the deformable surface are established. Using the three-dimensional constitutive equations of a micropolar material we discuss the determination of the effective stiffness tensors appearing in the two-dimensional constitutive equations.

Thickness-dependent bending modulus of hexagonal boron nitride nanosheets

Abstract: Bending modulus of exfoliation-made single-crystalline hexagonal boron nitride nanosheets (BNNSs) with thicknesses of 25-300 nm and sizes of 1.2-3.0 microm were measured using three-point bending tests in an atomic force microscope. BNNSs suspended on an SiO(2) trench were clamped by a metal film via microfabrication based on electron beam lithography. Calculated by the plate theory of a doubly clamped plate under a concentrated load, the bending modulus of BNNSs was found to increase with the decrease of sheet thickness and approach the theoretical C(33) value of a hexagonal BN single crystal in thinner sheets (thickness<50 nm). The thickness-dependent bending modulus was suggested to be due to the layer distribution of stacking faults which were also thought to be responsible for the layer-by-layer BNNS exfoliation.

∞6mixed plate theories based on the Generalized Unified Formulation. Part II: Layerwise theories

Abstract: The generalized unified formulation introduced in Part I for the case of composite plates and Reissner’s Mixed variational theorem is, for the first time in the literature, applied to the case of layerwise theories. Each layer is independently modeled. The compatibility of the displacements and the equilibrium of the transverse stresses between two adjacent layers are enforced a priori. Infinite combinations of the orders used for displacements ux, uy, uz and out-of-plane stresses σzx, σzy, σzz can be freely chosen. ∞6 layerwise theories are therefore presented. The code based on this capability can have all the possible ∞6 theories built-in, thus, making the code a powerful and versatile tool to analyze different geometries, boundary conditions and applied loads. All ∞6 theories are generated by expanding 13 1×1 invariant matrices (the kernels of the Generalized Unified Formulation). How the kernels are expanded and the theories generated is explained. Details of the assembling in the thickness direction and the generation of the matrices are provided.

Vibration analysis of nanoplates under uniaxial prestressed conditions via nonlocal elasticity

Abstract: In this article, nonlocal elasticity theory is applied to investigate the vibration response of nanoplates under uniaxially prestressed conditions. Nonlocal elasticity theory takes into account the small-size effects when dealing with nanostructures. Nonlocal governing equations of the prestressed nanoplate are derived and presented. Differential quadrature method is being utilized and numerical frequency solutions are obtained. Influence of small scale and uniaxial preload on the nonlocal frequency solutions is investigated. It is observed that the frequencies for nanoplates under uniaxially prestressed conditions employing classical plate theory are overestimated compared to nonlocal plate solutions. Considering the nonlocal effects, smaller critical compressive load is required to reach the buckling state of a flexural mode compared to the classical plate theory. The present research work thus reveals that the nonlocal parameter, aspect ratios, boundary conditions, and initial uniaxial prestress have significant effects on vibration response of the nanoplates.

Refined Zigzag Theory for Laminated Composite and Sandwich Plates

Abstract: A refined zigzag theory is presented for laminated-composite and sandwich plates that includes the kinematics of first-order shear deformation theory as its baseline. The theory is variationally consistent and is derived from the virtual work principle. Novel piecewise-linear zigzag functions that provide a more realistic representation of the deformation states of transverse-shear-flexible plates than other similar theories are used. The formulation does not enforce full continuity of the transverse shear stresses across the plate s thickness, yet is robust. Transverse-shear correction factors are not required to yield accurate results. The theory is devoid of the shortcomings inherent in the previous zigzag theories including shear-force inconsistency and difficulties in simulating clamped boundary conditions, which have greatly limited the accuracy of these theories. This new theory requires only C(sup 0)-continuous kinematic approximations and is perfectly suited for developing computationally efficient finite elements. The theory should be useful for obtaining relatively efficient, accurate estimates of structural response needed to design high-performance load-bearing aerospace structures.

Dynamics of plate bending at the trench and slab-plate coupling

Abstract: [1] The bending strength of subducting lithosphere plays a critical role in the Earth's plate tectonics and mantle convection, modulating the amount of slab pull transmitted to the surface and setting the boundary conditions under which plates move and deform. However, it is the subject of a lively debate how much of the potential energy of the downgoing plate is consumed in bending the plate and how the lithospheric strength is defined during this process. We model the subduction of a viscoelastic lithosphere, driven solely by the downgoing plate's buoyancy, freely sinking in a passive mantle, represented by drag forces. To investigate the dynamics of bending, (1) we vary the density and the viscosity profile within the plate from isoviscous, where strength is distributed, to strongly layered, where strength is concentrated in a thin core, and (2) we map the stress, strain, and dissipation along the downgoing plate. The effective plate strength during bending is not a simple function of average plate viscosity but is affected by rheological layering and plate thinning. Earth-like layered plates allow for the transmission of large fractions of slab pull (∼75–80%) through the bend and yield a net slab pull of FSPnet = 1 to 6 × 1012 N m−1, which varies with the buoyancy of plates. In all models, only a minor portion of the energy is dissipated in the bending. Surprisingly, bending dissipation hardly varies with lithospheric viscosity because in our dynamic system, the plates minimize overall dissipation rate by adjusting their bending curvature. As a result, bending dissipation, ΦB, is mainly controlled by the bending moment work rate exerted by slab pull. We propose a new analytical formulation that includes a viscosity-dependent bending radius, which allows for assessment of the relative bending dissipation in the Earth's subduction zones using parameters from a recent global compilation. This yields estimates of ΦB/ΦTOT < 20%. These results suggest that plates on Earth weakly resist bending, yet are able to propagate a large amount of slab pull.

Buckling analysis of plates using the two variable refined plate theory

Abstract: Buckling analysis of isotropic and orthotropic plates using the two variable refined plate theory is presented in this paper. The 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. Governing equations are derived from the principle of virtual displacements. The closed-form solution of a simply supported rectangular plate subjected to in-plane loading has been obtained by using the Navier method. Numerical results obtained by the present theory are compared with classical plate theory solutions, first-order shear deformable theory solutions, and available exact solutions in the literature. 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 shear deformable theory.