Mechanical Behavior of a Functionally Graded Rectangular Plate Under Transverse Load: A Cosserat Elasticity Analysis
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TECHNICAL ARTICLE—PEER-REVIEWED
Mechanical Behavior of a Functionally Graded Rectangular Plate Under Transverse Load: A Cosserat Elasticity Analysis Soumen Shaw
Submitted: 5 April 2017 Ó ASM International 2017
Abstract In this article, a static analysis of a functionally graded (FG) rectangular plate subjected to a uniformly distributed load is investigated within the framework of Timoshenko and the higher order shear deformation beam theories. The mechanical behavior of the plate is analysed under the theory of Cosserat elasticity. In the framework of infinitesimal theory of elasticity, the bending of the plate is analyzed subjected to transverse loading. A set of governing equations of equilibrium are obtained based on the method of hypothesis. A semianalytical solution is presented for the governing equations using the approximation theory of Timoshenko. The solutions are validated by comparing the numerical results with their counterparts reported in the literature for classical Timoshenko plate theory. Keywords Functionally graded material Cosserat elasticity Transverse loading Bending of thin plate
Introduction The development of continuum mechanics is closely related to the construction of generalized mathematical models taking into account asymmetric elastic theory. For some specific applications in modern micromechanics and nanotechnology, in the last decade, asymmetric micropolar moment theory has gained an enormous interest. It has been established that, based on field equations, asymmetric theory of elasticity offers a rigorous mathematical S. Shaw (&) Department of Mathematics, IIEST, Shibpur, Howrah, India e-mail: [email protected]
approach to describe the deformations of solids with internal structure or continuously distributed defects. Despite that fact, in the last decade and half, micropolar theory of elasticity is rapidly developing in both theoretical and applied directions [1–5]. Thin rectangular plate is a significant structural component used in numerous engineering applications such as rigid pavements of highways and airports, house and bridge decks. Because of its enormous applications in several fields, in the last one or two decades, thin elastic plates with various boundary conditions have been studied by several authors. Among them we can mention Ciarlet [6, 7], Timoshenko and Woinowsky-Krieger [8], Goldenveizer [9], Neff and Jeong [10], Naghdi [11], etc. After the pioneering work of Kirchhoff, various aspects of classical plate theory have been explored by several scholars. Later on, a number of research articles have been published to give the foundations and methods of presumption of Kirchhoff-Love theory and its possible improvements. Mainly, there are two methods to determine the deflection of thin plate: (a) method of hypothesis (see, Timoshenko and Woinowsky-Krieger [8]) and (b) method of asymptotic expansion (see Goldenveizer [9], Friedrichs and Dressler [12], Ciarlet and Destuynder [13]). There is another important classical theory known as the Reissner-Min
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