A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates
Introduction
Sandwich construction is one of the most functional forms of composite structures developed by the composite industry. It has attained broad acceptance in aerospace and many other industries and it is widely employed in aircraft and space vehicles, ships, boats, cargo containers, and residential constructions. Sandwich plates revolutionized the aerospace industry over 40 years ago, making aircraft lighter, stronger and faster, and allowing them to carry more weight and improve fuel efficiency. Sandwich composite construction offers great potential for large civil infrastructure projects such as industrial buildings and vehicular bridges. Sandwich structures represent a special form of a layered structure that consists of two thin stiff and strong face sheets separated by a thin and a relatively thick, lightweight, and compliant core material. In modern sandwich structures the faces are usually made of metal or laminated composite materials, and typically a compliant compressible core made of a low-strength honeycomb type material or polymeric foam. The faces and the core are joined by adhesive bonding, which ensures the load transfer between the sandwich constituent parts.
However, the demand for improved structural efficiency in many engineering fields has resulted in the development of a new class of materials, called functionally graded materials (FGMs) [14], [15], [36]. FGMs are heterogeneous materials in which the material properties are varied continuously from point to point. This is achieved by varying the volume fraction of the constituents, for example, of ceramic and metal in a predetermined manner. This continuously varying composition eliminates interface problems, and thus, the stress distributions are smooth. In the simplest FGMs, two different material ingredients change gradually from one to the other. Discontinuous changes such as a stepwise gradation of the material ingredients can also be considered an FGM. The most familiar FGM is compositionally graded from a refractory ceramic to a metal. Typically, FGMs are made from a mixture of ceramic and metal or a combination of different materials. The ceramic in an FGM offers thermal barrier effects and protects the metal from corrosion and oxidation, and the FGM is toughened and strengthened by the metallic composition. FGMs are now developed for general use as structural elements in extremely high temperature environments and different applications. Several studies have been performed to analyze the behavior of FGM plates and shells [4], [5], [6], [19], [20], [23], [25], [30], [31], [40], [42].
Some so-called equivalent single layer models (ESLMs) have been developed for the analysis of FGM plates and shells, such as classical plate/shell theories [18], first-order shear deformation theories [22], [34] and higher-order shear deformation theories [17], [28]. These ESLMs have been applied to a variety of mechanical problems related to FGM plates and shells. However, these theories do not satisfy the continuity conditions of transverse shear stress at layer interfaces. Although the discrete layer approximation theories are accurate, but they are rather complex in solving problems because the order of their governing equations depends on the number of layers. In layer-wise theories, displacement components are continuous through the laminate thickness but the derivatives of the displacements thickness direction may be discontinuous at various points through the thickness, thereby allowing for the possibility of continuous transverse stresses at layer interfaces separating dissimilar materials [29], [33]. Barbero et al. [2] developed analytical solutions for displacements and stresses in composite laminates using a layer-wise theory. Lee and Saravanos [16] investigated a smart thermo-piezoelectric plate structures using a generalized discrete layer mechanics. Carrera [3] presented a historical review of zig-zag theories of multilayered plates and shells. Cho and Oh [8] developed a higher-order zig-zag plate theory to refine the prediction of the mechanical, thermal, and electric behaviors fully coupled. Gherlone and Di Sciuva [12] presented a plate mixed finite element based on a refined zig-zag plate model for the analysis of multilayered composite plates subjected to thermal and mechanical loads. Some layer-wise theories [29] have also been used extensively for the analysis of smart FGM plates. Based on a discrete layer theory, Ramirez et al. [26], [27] investigated the static behaviors of functionally graded (FG) elastic plates and the free vibration responses of FG magneto-electro-elastic plates, respectively. Tahani and Mirzababaee [37] used a layer-wise theory to analyze analytically displacements and stresses in functionally graded composite plates in cylindrical bending subjected to thermomechanical loadings.
The FGM sandwich can alleviate the large interfacial shear stress concentration because of the gradual variation of material properties at the facesheet–core interface. The effects of FGM core were studied by Venkataraman and Sankar [41], and Anderson [1] on the shear stresses at the facesheet–core of FGM sandwich beam. Pan and Han [24] analyzed the static response of the multilayered rectangular plate made of functionally graded, anisotropic, and linear magneto-electro-elastic materials. Das et al. [9] studied a sandwich composed of a single FGM soft core with relatively orthotropic stiff facesheets by using a triangular plate element. Shen [35] considered two types of FGM hybrid laminated plates, one is with FGM core and piezoelectric ceramic facesheet and the other is with FGM facesheet and piezoelectric ceramic core.
Various higher-order shear deformation theories involve use of five unknown functions. The well-known higher-order plate theories are as follows: (i) parabolic shear deformation plate theory (PSDPT) [28], (ii) sinusoidal shear deformation plate theory (SSDPT) [39], and (iii) exponential shear deformation plate theory (ESDPT) [13]. The present paper presents a new trigonometric shear deformation theory involving only four unknown functions, as against five in case of other shear deformation theories. The trigonometric function in terms of thickness coordinate is used in the displacement field to account for shear deformation. The novel feature of the theory is that it does not require shear correction factor, satisfying the shear stress free boundary conditions at top and bottom of the plate and has strong similarities with the CLPT in some aspects such as governing equation.
In this article, the present refined trigonometric shear deformable plate theory (RTSDT) is used for thermoelastic bending of FGM sandwich plates. This theory captures only sinusoidal through-thickness shear strain/stress distributions. The effects of temperature field on the dimensionless axial and transverse shear stresses and deflection of the FGM sandwich plate are studied. Numerical examples are presented to illustrate the accuracy and efficiency of the present theory by comparing the obtained results with those computed using various other theories.
Section snippets
Problem formulation
Consider the case of a uniform thickness, rectangular FGM sandwich plate composed of three microscopically heterogeneous layers referring to rectangular coordinates () as shown in Fig. 1. The top and bottom faces of the plate are at , and the edges of the plate are parallel to axes x and y.
The sandwich plate is composed of three elastic layers, namely, “Layer 1”, “Layer 2”, and “Layer 3” from bottom to top of the plate (Fig. 2). The vertical ordinates of the bottom, the two
Exact solution for a simply supported FGM sandwich plate
Rectangular plates are generally classified in accordance with the type of support used. We are here concerned with the exact solution of Eqs. (23a), (23b), (23c), (23d) for a simply supported FGM plate. The following boundary conditions are imposed at the side edges for RTSDT: To solve this problem, Navier presented the transverse temperature loads , , and in the form of a double trigonometric series as
Numerical results and discussion
In order to prove the validity of the present refined trigonometric shear deformation theory (RTSDT), results were obtained for isotropic plates and compared with the existing ones in the literature. The exact solutions of Timoshenko and Woinowsky-Krieger [38], Das and Rath [10], Zenkour [43] and finite element solutions of Reddy and Hsu [32] for fully elastic rectangular plates under uniform temperature are used to assess the improvement of the present theory. Table 1 contains bending
Conclusion
A refined trigonometric shear deformation theory (RTSDT) is developed for the thermoelastic bending response of FGM sandwich plates. The number of primary variables in this theory is even less than that of first- and higher-order shear deformation plate theories, and moreover, it obviates the need for a shear correction factor. All comparison studies demonstrated that the deflections and the thermal stresses obtained using the present refined theory (with four unknowns) and other higher-order
References (43)
A 3-D elasticity solution for a sandwich composite with functionally graded core subjected to transverse loading by a rigid sphere
Compos. Struct.
(2003)- et al.
Three-dimensional thermoelastic deformations of a functionally graded elliptic plate
Composites, Part B
(2000) - et al.
Exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plates
J. Sound Vib.
(2000) - et al.
Mechanical behavior of functionally graded material plates under transverse load—Part II: Numerical results
Int. J. Solids Struct.
(2006) - et al.
Thermo-mechanics of undamaged and damaged multilayered composite plates: a sub-laminates finite element approach
Compos. Struct.
(2007) - et al.
Mechanical behavior of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity
Int. J. Solids Struct.
(2003) FGM activities in Japan
Composites, Part B
(1997)- et al.
Generalized finite element formulation for smart multilayered thermal piezoelectric composite plates
Int. J. Solids Struct.
(1997) - et al.
Vibration of functionally graded cylindrical shells
Int. J. Mech. Sci.
(1999) - et al.
Exact solution for functionally graded and layered magneto-electro-elastic plates
Int. J. Eng. Sci.
(2005)