FINITE ELEMENT ANALYSIS OF FUNCTIONALLY GRADED PLATES AND SHELLS

Abstract

Functionally graded materials (FGMs) are advanced class of engineering composites constituted of two or more distinct phase materials described by continuous and smooth varying composition of material properties in the required direction. The mechanical properties such as Young’s modulus, Poisson’s ratio, shear modulus, and density are varied according to simple rule of mixture in terms of volume fraction distribution of constituents. Potential reduction of in-plane and transverse through-the-thickness stresses, reduced stress intensity factors, an improved residual stress distribution, and higher fracture toughness are few of the advantages offered by these advanced composites. In addition, these materials are capable of surviving high temperatures and large temperature gradients that may occur within a fraction of seconds in some structures such as aircraft, while preserving their structural integrity. Thin walled rotating blades that could be used in helicopters and turbo machinery fields, thermal barrier coatings, nuclear reactors, micro and nano devices, dental and medical implants, and piezoelectric and thermo electric devices are some of the areas where the concept of FGM has been successfully implemented in the modern era. The proper choice of homogenization approach used in FGM structures should be based on the gradient of gradation relative to the size of a representative volume element (RVE). In literatures, some of the averaging methods that incorporate the homogenous nature at RVE scale and heterogeneous nature at global scale are proposed. Among different methods, rule of mixture is widely employed in many studies, while few assume the Mori-Tanaka and self consistent schemes. Mori-Tanaka based homogenization approach accounts for the interactions among the adjacent inclusions. In the present study, both simple rule of mixture and Mori-Tanaka scheme are employed to homogenize certain mechanical/thermal properties of the constituents. In highly heterogeneous structures like FGMs, membrane-flexure coupling exist due to anti-symmetric nature of material properties. Hence, it seems important to consider the improved structural kinematics in the form of accurate variation of in-plane and transverse displacement components that describe the realistic parabolic distribution of transverse shear deformation. In the present study static, free/forced vibration and buckling responses of FGM skew plate/shell structures are performed by developing an efficient displacement based 2D FE model. A higher order shear deformation theory (HSDT) which ii accounts for realistic parabolic variation of transverse shear deformation is employed in the present thesis work for this purpose. In case of thermal analysis, one-dimensional Fourier heat conduction equation has been solved by imposing appropriate thermal boundary conditions at the top and bottom surfaces. Temperature dependent material properties are also incorporated in the analyses that follow the simple rule of mixture of materials. During the implementation of higher order theory in finite element (FE) method, the problem of C1 continuity is encountered due to the existence of first order derivatives of transverse displacement component in the expression of in-plane fields. In the present FE formulation, this problem has been circumvented by means of appropriate substitution of independent nodal unknowns and thus leading to an efficient C0 FE formulation. A nine node isoparametric Lagrangian element is used to mesh the assumed plate/shell geometry. In case of skew boundary, for nodes lying on the skew edges, suitable transformation rule has been employed to generate the corresponding global element matrices. While incorporating the strain field for FGM shell structures, all three curvature terms i.e., 1/Rx, 1/Ry and 1/Rxy are included for the analysis of different shell forms like hyperbolic paraboloid and hypar shells which are not included in any other studies so far. The conventional power law distribution adopted in most of the literatures leads to the configuration where the top layer will be ceramic rich, and bottom layer will be rich in metal with composite structures in between. But, due to some practical requirements, the design may demand for probable combinations of FGM configuration other than conventional one. To meet this criterion, FGM plate described by a four-parameter power function is incorporated in the present work for static, free vibration and buckling problems. Four-parameters that define the distribution law are suitably assumed to achieve the conventional, symmetric and asymmetric FGM profiles by satisfying the volumetric relationships between the constituents. Due to the large jump of material properties at the layer interfaces, the problem of delamination is generally observed in conventional sandwich structures. The concept of FGM is also employed in such sandwich structures to overcome this problem. In the present research work, an improved structural kinematics that account for realistic variation of transverse shear component and quadratic thickness variation of transverse displacement was employed for static and free vibration analyses of FGM skew sandwich plates/shells. FGM skew sandwich plate/shells are modeled by two kinds of approaches. In the first case, the core part is designated by ceramic component; while top and bottom iii layers are rich in metal component thus leading to an FGM profile at the top and bottom layers. In the later case, the top and bottom layers are defined by ceramic and metal components and hence the core portion is defined by functional grading. By designating the bottom-core-top layer thickness of the plate/shell with respect to overall thickness, different kinds of sandwich plates/shells are modeled in the present study. During their service life, structures may undergo large deformation under mechanical, thermal and thermo-mechanical loading conditions. In such cases, the equilibrium equations should be written in terms of the deformed configuration of the structures by non-linear strain-displacement relationships. Hence, to analyze FGM sandwich structures under large deformation, a non-linear FE formulation is implemented in the frame work of Green-Lagrange form of equations. In this work, FGM skew plates/shells constituted by single/multi layers are analyzed under large deformation. Newton-Raphson iteration scheme is employed to trace the load-displacement path. When in-plane loads are high in magnitude it is necessary to consider the secondary bifurcation stage which is known as post buckling stage of structures. To trace the post buckling equilibrium path of FGM sandwich plates, a direct iterative algorithm is employed to get the converged frequency values under different magnitude of amplitude ratios. Many new results based on various geometric properties such as aspect ratio, thickness ratio, curvature-side ratio, boundary conditions and material properties like volume fraction index and material constituents are accomplished in the present research work to perform linear/non-linear static, free/forced vibration, and buckling analyses of FGM skew plates/shells constituted of single and multiple layers. The various imperative conclusions arrived from the present research work should be useful for researchers, analysts and designers engaged in this area.