Zakaria Belabed

Bio: Zakaria Belabed is an academic researcher from SIDI. The author has contributed to research in topics: Plate theory & Displacement field. The author has an hindex of 3, co-authored 5 publications receiving 435 citations.

An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates

Abstract: In this paper, an efficient and simple higher order shear and normal deformation theory is presented for functionally graded material (FGM) plates. By dividing the transverse displacement into bending, shear and thickness stretching parts, the number of unknowns and governing equations for the present theory is reduced, significantly facilitating engineering analysis. Indeed, the number of unknown functions involved in the present theory is only five, as opposed to six or even greater numbers in the case of other shear and normal deformation theories. The present theory accounts for both shear deformation and thickness stretching effects by a hyperbolic variation of all displacements across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor. Equations of motion are derived from Hamilton’s principle. Analytical solutions for the bending and free vibration analysis are obtained for simply supported plates. The obtained results are compared with 3-dimensional and quasi-3-dimensional solutions and those predicted by other plate theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of functionally graded plates.

A new 3-unknown hyperbolic shear deformation theoryfor vibration of functionally graded sandwich plate

Abstract: In this work, a simple but accurate hyperbolic plate theory for the free vibration analysis of functionally graded material (FGM) sandwich plates is developed. The significant feature of this formulation is that, in addition to including the shear deformation effect, it deals with only 3 unknowns as the classical plate theory (CPT), instead of 5 as in the well-known first shear deformation theory (FSDT) and higher-order shear deformation theory (HSDT). A shear correction factor is, therefore, not required. Two common types of FGM sandwich plates are considered, namely, the sandwich with the FGM face sheet and the homogeneous core and the sandwich with the homogeneous face sheet and the FGM core. The equation of motion for the FGM sandwich plates is obtained based on Hamilton‟s principle. The closed form solutions are obtained by using the Navier technique. The fundamental frequencies are found by solving the eigenvalue problems. Numerical results of the present theory are compared with the CPT, FSDT, order shear deformation theories (HSDTs), and 3D solutions. Verification studies show that the proposed theory is not only accurate and simple in solving the free vibration behaviour of FGM sandwich plates, but also comparable with the higher-order shear deformation theories which contain more number of unknowns.

Assessment of new 2D and quasi-3D nonlocal theories for free vibration analysis of size-dependent functionally graded (FG) nanoplates

Abstract: In this present paper, a new two dimensional (2D) and quasi three dimensional (quasi-3D) nonlocal shear deformation theories are formulated for free vibration analysis of size-dependent functionally graded (FG) nanoplates. The developed theories is based on new description of displacement field which includes undetermined integral terms, the issues in using this new proposition are to reduce the number of unknowns and governing equations and exploring the effects of both thickness stretching and size-dependency on free vibration analysis of functionally graded (FG) nanoplates. The nonlocal elasticity theory of Eringen is adopted to study the size effects of FG nanoplates. Governing equations are derived from Hamilton\'s principle. By using Navier\'s method, analytical solutions for free vibration analysis are obtained through the results of eigenvalue problem. Several numerical examples are presented and compared with those predicted by other theories, to demonstrate the accuracy and efficiency of developed theories and to investigate the size effects on predicting fundamental frequencies of size-dependent functionally graded (FG) nanoplates.

Effect of homogenization models on stress analysis of functionally graded plates

Abstract: In this paper, the effect of homogenization models on stress analysis is presented for functionally graded plates (FGMs). The derivation of the effective elastic proprieties of the FGMs, which are a combination of both ceramic and metallic phase materials, is of most of importance. The majority of studies in the last decade, the Voigt homogenization model explored to derive the effective elastic proprieties of FGMs at macroscopic-scale in order to study their mechanical responses. In this work, various homogenization models were used to derive the effective elastic proprieties of FGMs. The effect of these models on the stress analysis have also been presented and discussed through a comparative study. So as to show this effect, a refined plate theory is formulated and evaluated. The number of unknowns and governing equations were reduced by dividing the transverse displacement into both bending and shear parts. Based on sinusoidal variation of displacement field trough the thickness, the shear stresses on top and bottom surfaces of plate were vanished and the shear correction factor was avoided. Governing equations of equilibrium were derived from the principle of virtual displacements. Analytical solutions of the stress analysis were obtained for simply supported FGM plates. The obtained results of the displacements and stresses were compared with those predicted by other plate theories available in the literature. This study demonstrates the sensitivity of the obtained results to different homogenization models and that the results generated may vary considerably from one theory to another. Finally, this study offers benchmark results for the multi-scale analysis of functionally graded plates.

Investigation of influence of homogenization models on stability and dynamic of FGM plates on elastic foundations

Abstract: In this paper, the effect of the homogenization models on buckling and free vibration is presented for functionally graded plates (FGM) resting on elastic foundations. The majority of investigations developed in the last decade, explored the Voigt homogenization model to predict the effective proprieties of functionally graded materials at the macroscopic-scale for FGM mechanical behavior. For this reason, various models have been used to derive the effective proprieties of FGMs and simulate thereby their effects on the buckling and free vibration of FGM plates based on comparative studies that may differ in terms of several parameters. The refined plate theory, as used in this paper, is based on dividing the transverse displacement into both bending and shear components. This leads to a reduction in the number of unknowns and governing equations. Furthermore the present formulation utilizes a sinusoidal variation of displacement field across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor. Equations of motion are derived from Hamilton’s principle. Analytical solutions for the buckling and free vibration analysis are obtained for simply supported plates. The obtained results are compared with those predicted by other plate theories. This study shows the sensitivity of the obtained results to different homogenization models and that the results generated may vary considerably from one theory to another. Comprehensive visualization of results is provided. The analysis is relevant to aerospace, nuclear, civil and other structures.

Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories

Abstract: In this work, various higher-order shear deformation plate theories for wave propagation in functionally graded plates are developed. Due to porosities, possibly occurring inside functionally graded materials (FGMs) during fabrication, it is therefore necessary to consider the wave propagation in plates having porosities in this study. The developed refined plate theories have fewer number of unknowns and equations of motion than the first-order shear deformation theory, but accounts for the transverse shear deformation effects without requiring shear correction factors. The rule of mixture is modified to describe and approximate material properties of the functionally graded plates with porosity phases. The governing equations of the wave propagation in the functionally graded plate are derived by employing the Hamilton