Analysis of non symmetric FG sandwich plates under Thermo-mechanical loading using a novel shear deformation theory with stretching effect

The analysis of non-symmetric functionally graded sandwich plates under thermo-mechanical loading is developed using a novel hyperbolic shear deformation theory and considering thickness stretching effects. This theory accounts for adequate distribution of the transverse shear strains in the thickness of the plate and satisfies the traction free boundary conditions on the top and bottom surface of the plates, thus a shear correction factor is not required. The governing equations of equilibrium of non-symmetric functionally graded sandwich plates can be obtained using virtual work principle and the closed form solutions are obtained by using Navier technique. The accuracy of the present results is established by comparing those with well known trigonometric shear deformation theories. The results are presented for deflections and stresses of nonsymmetric simply supported square plates.


Introduction
Composite laminated materials are finding wide application in many engineering fields. Therefore, a number of approximate analytical and numerical methods have been developed in this area. Such composites are made of two or nuclear, automotive fields and ship building industries to eliminate stress concentration and relax residual stresses and enhance bond strength.
Several studies have been performed to this area. Such composites are made of two or more materials to obtain a good structural performance, which the constituent does not show individually. Recently, advanced composite materials known as functionally graded material have attracted much attention in many engineering applications due to their advantages of being able to resist high temperature gradient while maintaining structural integrity [1]. The functionally graded materials (FGMs) are microscopically inhomogeneous, in which the mechanical properties vary smoothly and continuously from Several studies have been performed to analyze the behavior of functionally graded plates and shells. Reddy [2] analyzed the static behavior of functionally graded rectangular plates based on his third-order shear deformation theory of plates. Cheng and Batra [3] related the deflections of a simply supported functionally graded polygonal plate given by the first-order shear de formation theory and a third-order shear de formation theory to those of an equivalent homogeneous Kirchhoff plate. Cheng and Batra [4] also presented results for the buckling and properties vary smoothly and continuously from one surface to the other. They are usually made from a mixture of ceramics and metals to attain the significant requirement of material properties. FGMs are widely used in many structural applications such as mechanics, civil engineering, optical, electronic, chemical, mechanical, biomedical, energy sources, where P m and P c denote the property of the deformation theories are developed using five unknown functions. The well-known higherorder plate theories with five unknown functions are as follows: (i) parabolic shear deformation plate theory (PSDPT) [8], (ii) sinusoidal shear deformation plate theory (SSDPT) [9], and (iii) exponential shear deformation plate theory (ESDPT) [10].
In this study, a novel higher order shear deformation theory for linear analysis of moderately thick functionally graded sandwich plate is examined. In this model the displacement varies as a hyperbolic function where P m and P c denote the property of the bottom and top faces of layer 1, respectively, and vice versa for layer 3 depending on the volume fraction ) (n V (n = 1,2,3). Note that P m and P c are, respectively, the corresponding properties of the metal and ceramic of the FGM sandwich plate.
Couche2 y z displacement varies as a hyperbolic function across the plate thickness and satisfies zero shear stress condition at the top and bottom surfaces of the plate, thus a shear correction factor is not required. The accuracy of the present results is established by comparing those with well known trigonometric shear deformation theory.

Problem formulation
Let us consider the case of a sandwich plate composed of three microscopically

Governing equations for FG sandwich plate
The displacement field for FG sandwich plates is described in the following equations: xz, and xy planes due to bending, respectively.
) (z f represents the shape function determining the distribution of the transverse shear strains and stresses along the thickness. It is given by The linear strain expressions derived from the displacement model are as follows: consideration. The top and bottom layers of non-symmetric sandwich plate are made of an isotropic homogeneous material ceramic and metal, respectively, while the core layer is made of a mixture of (ceramic and metal) materials. Using the principle of virtual work, the following expressions can be obtained: The governing equations of equilibrium can be derived easily from Equation (5) which is expressed as. can be observed that the HSDP overestimates the deflections comparatively to RHSDT and this, is due to the thickness stretching effect. For a sandwich plate, the deflections decrease when the dimensionless ratio ( ⁄ ) increases. Table 2 shows the influence of power law index k , and different plate types on the transverse shear stresses of non-symmetric sandwich plates. As is observed, the increase of the power law index k leads to a decrease of stresses of the non-symmetric sandwich plates, and the thickness stretching effect leads also to a reduction of stresses. for stresses and deflection are obtained and investigated for different plate configurations. The effect of power-law index, span to thickness ratio and aspect ratio are studied. It has been confirmed that the inclusion of thickness stretching effect makes the plate stiffer, and hence, leads to a reduction of deflections and stresses.  Fig.3 : Effect of the aspect ratio b a / on dimensionless center deflection w for sandwich plate type (1-3-1) Figure 2 shows the variation of the center deflection w with side-to-thickness ratio h a / of sandwich plate type (1-3-1) with different volume fraction exponent k. The deflections of the sandwich plates decrease as h a / increases. Figure 3 shows the variation of the center deflection w with the aspect ratio b a / of sandwich plate type (1-3-1). The deflection of the ceramic plate is found to have the smallest magnitude and that of the metallic plate the largest one. The increase of the aspect ratio b a / leads to a decrease of deflections of the homogeneous and FG sandwich plates. [10] S.S. Akavci, Buckling and free vibration analysis of symmetric and antisymmetric laminated composite plates on an elastic foundation, J. Reinf. Plas. Comp, 26, 2007Comp, 26, ,1907Comp, 26, -1919 homogeneous and FG sandwich plates.

Conclusion
A novel hyperbolic shear deformation theory for thermo-mechanical bending of non-symmetric sandwich plates is presented. The theory accounts for the stretching and shear deformation effects without requiring a shear correction factor. The