The complete work presented in this thesis has been organized into seven chapters. Chapter 1 is devoted to the literature survey and the proposed objectives of the present work. An overview of the contents of the remaining six chapters is presented below.
Chapter 2 presents an analytical 2D elasticity solution for static and free vibration analysis of axially functionally graded beams subjected to arbitrary boundary conditions using the multi- term extended Kantorovich method (EKM). Further, the static and free vibration solution for the arbitrarily supported homogeneous beam are also obtained as a special case of the present study. The material properties of the beam are considered to vary linearly along its axial (x) direction. Modified Hamilton’s principle is applied to derive the weak form of coupled governing equations in which all the stresses and displacements act as primary variables. Further, the extended Kantorovich method is employed to reduce the governing equations into sets of ordinary differential equations (ODEs) along the axial (x) and thickness (z) directions. The system of ODEs along the z-direction has constant coefficients which is solved analytically. But, the system of ODEs along x-direction has variable coefficients which is solved using a modified power series method. Interface continuity and boundary conditions are satisfied in exact point-wise manner which ensures the same order of accuracy for all field variables (stresses and displacements). Efficacy and accuracy of the present methodology are verified thoroughly with existing literature and 2D finite element solution.
Benchmark numerical results are presented for various cases of material property variations under different type of boundary conditions. The influence of the axial gradation, aspect ratio and boundary conditions on the static and dynamic behavior of the beam are investigated. Significant effect of axial gradation on the static and dynamic behavior of the beam is observed.
Chapter 3, contains 3D elasticity based EKM solution for the static and free vibration analysis of in-plane functionally graded rectangular plate subjected to Levy-type boundary conditions. A linear variation of material compliances is assumed along its longitudinal (x) direction. Bench- mark numerical results are presented for different sets of boundary conditions, aspect ratios and configurations by considering the various cases of material property gradation. For the first time, numerical results for a two-layered plate having different in-plane material gradation in each layer
ORGANISATION OF THE THESIS
are presented along with single layer plate results. The numerical study reveals that the EKM solution converges within just one/two terms for both single layer and two-layer FGM plates. It is observed that the static and dynamic response of plate is influenced greatly by the extent of mate- rial gradation along the in-plane direction. Effects of in-plane property gradation in the adhesive interlayer on deflections and stresses of the plate are also investigated under different boundary conditions. The present analytical solution can serve as a benchmark for assessing the accuracy of the 2D or 3D numerical solutions. The current research will also be beneficial to model rectangular plate structures and adhesive bonded rectangular plates in which material properties are degraded due to some environmental effect.
In Chapter 4, the generalized 3D EKM solution is presented for the static and free vibration analysis of longitudinally functionally graded angle-ply flat panels subjected to arbitrary boundary conditions. Benchmark numerical results are presented for single layered and multi-layered in-plane functionally graded angle-ply flat panels. Numerical results are validated thoroughly by comparing with 3D finite element (FE) results. The influence of property variation on the deflections and stresses are studied and discussed comprehensively for arbitrary sets of boundary conditions and configurations. The present method provided benchmark results to assess the validity and accuracy of different plate theories and computational models for the analysis of axially functionally graded angle-ply flat panels. The current research will also be beneficial to model real-life panel structures in which its material properties deteriorate due to some environmental effect.
In Chapter 5, a 3D piezoelectricity based analytical solution is developed for free vibration analysis of the angle-ply elastic and piezoelectric flat laminated panels under arbitrary boundary conditions. The present analytical solution is applicable to composite, sandwich and hybrid pan- els having arbitrary angle-ply lay-up, material properties and boundary conditions. The modified Hamilton’s principle approach has been applied to derive the weak form of governing equations where stresses, displacements, electric potential, and electric displacement field variables are con- sidered as primary variables. After that multi-term multi-field extended Kantorovich approach (MMEKM) is employed to transform the governing equation into two sets of algebraic-ordinary differential equations (ODEs), one along in-plane (x) and other along with the thickness (z) di- rection, respectively. These ODEs are solved in closed-form manner which ensures the same order of accuracy for all the variables (stresses, displacements, and electric variables) by satisfying the boundary and continuity equations in exact manner. A robust algorithm is developed for extracting the natural frequencies and mode shapes. The numerical results are reported for various config- urations such as elastic panels, sandwich panels and piezoelectric panels under arbitrary sets of boundary conditions. The accuracy and efficacy of the present method have been established by
comparing the present numerical results with the results available in the literature and with the 3D FE results of ABAQUS. The effect of ply-angle and thickness to span ratio (s) on the dynamic behavior of the panels are also investigated. The presented 3D analytical solution will be helpful in the assessment of various 1D theories and numerical methods.
In Chapter 6, a novel 2D analytical free vibration solution is developed for axially function- ally graded (AFG) beams integrated with piezoelectric layers and subjected to arbitrary support boundary conditions. The material properties of the elastic layers are considered to vary linearly along the axial (x) direction of the beam. Further, the free vibration solution for the arbitrarily supported piezo-laminated beams is also obtained as a special case of the present study. New benchmark numerical results are presented for a laminated piezoelectric beams and axially func- tionally graded beams integrated with piezoelectric layers. The influence of the axial gradation, aspect ratio and boundary conditions on the natural frequencies of the beam are also investigated.
Moreover, the longitudinal variation of displacements and stresses (mode shapes) for various cases are also plotted for different support conditions. These numerical results can be used for assessing 1D beam theories and numerical techniques.
Finally, the major conclusions of this work and suggestions for future research are summarized in Chapter 7.
Chapter 2
Functionally Graded Beams- 2D Elasticity Analysis
An analytical two-dimensional (2D) elasticity solution for static and dynamic analysis of arbitrarily supported axially functionally graded (AFG) beams is presented first time in this chapter using the multi-term extended Kantorovich method (EKM). The material properties of the beam are consid- ered to vary linearly along the axial (x) direction, as given in Sec. 2.1 . The modified Hamiltons principle is applied to derive the mixed form of governing equations in which stresses and displace- ments act as primary variables, as explained in Sec. 2.2. Further, the extended Kantorovich method is employed to reduce the governing equation into sets of ordinary differential equations (ODEs) along the axial (x) and thickness (z) directions, as explained in Sec. 2.3, 2.4 and 2.5. The ODEs along thez-direction have constant coefficients which are solved exactly in closed form manner for static and dynamic case in Sec. 2.4.1 and Sec. 2.4.1, respectively. Where the ODEs alongx-direction have variable coefficients which are solved using the modified power series method, as explained in Sec. 2.5.1 and 2.5.2 for static and dynamic case, respectively. The boundary and continuity equations are satisfied in exact point-wise manners which ensures the same order of accuracy for all the variables. For free vibration case, the coefficient of final ODEs contains the frequency ω which is obtained by first bracketing the root and then using the bi-section method. It is found that the single-term solution is sufficient enough for obtaining the natural frequencies. Where the two-term solution is needed to predict the variation of stresses at very near to clamped and free edges. New benchmark numerical results are presented for a single-layered AFG beam and laminated AFG beams. Numerical results are presented for various variation cases to studied the effect of in-plane variation of material properties on the static and free vibration response of the beams in Sec. 2.8 and Sec. 2.9, respectively. The influence of the boundary conditions, configuration and aspect ratio on the static and free vibration response of the beam are also investigated. These numerical results can serve as a benchmark for assessing 1D beam theories and other semi-analytical or numerical techniques.
2.1 BASIC ASSUMPTION
Consider a cross-ply multilayered axially graded beam (Fig. 2.1) with span length a along x- direction and total thicknessh alongz-direction. It can have any arbitrary boundary conditions at x= 0 and a, and is subjected to uniformly distributed pressure loads ofp1 and p2 applied on the bottom and top surfaces, respectively. The laminated AFG beam is made of L perfectly bonded layers of orthotropic materials with the principal material axis x3 oriented along the z-direction.
The mid-plane of the plate is chosen as the xy-plane. The principal material axis x1 of the kth layer numbered from the bottom is at an angle θk to the x-axis. The thickness of the kth layer is t(k), and the z-coordinate of its bottom surface is denoted as zk−1. The interface between the kth and the (k+ 1)th layer is named as the kth interface. The layer superscript is omitted unless needed for clarity.
L
L-th layer K-th layer
h/2
h/2
k-1
1st layer 0
K0th layer FGM layers
Fig. 2.1: Geometry of the AFG laminated beam.
The compliance and density of the elastic layers are assumed to vary linearly and continuously along the axis (x) of beam
¯
sm1j(ξ) = ¯s1j(1 +δ1ξ) =⇒s¯1j + ˆs1j for j= 1,3
¯
sm55(ξ) = ¯s55(1 +δ2ξ) =⇒s¯55+ ˆs55 (2.1) ρm(ξ) =ρ(1 +δpξ) =⇒ρ+ ˆρ
where ξ non-dimensional parameter along x (ξ = x/a). δ1 and δ2 are variation indexes which control the gradation in elastic properties, andδp is the variation index for density. These variation indexes can have any arbitrary value.
Various devices/equipment are subjected to various types of loading, boundary conditions and under chemical and gaseous environments such as hydrogen gas exposure, erosive and corrosive environment, under cryogenic conditions etc. Metals, composite materials, fibers, are adsorb hy- drogen, moisture and other chemical. From the preliminary studies, it has been found that elastic
GOVERNING EQUATIONS FOR FGM BEAMS
properties of composites change specifically along the span due to the diffusion of hydrogen atoms and other environmental effect [288–290]. Similarly, In biomedical sciences also, it has been found that stiffness of major bones in our body varies along the span of bone. To address the above issues, such type of variation is considered where theδ1,δ2 andδp may be positive or negative.
Comparison of present gradation model with other probabilistic gradation models, like Rule of Mixture, Mori-Tanaka scheme and Exponential function gradation, is presented in Appendix A. It is observed that the type of gradation profiles obtained from other probabilistic gradation models, can also be achieved in the present gradation model by selecting the proper gradation parameters (δ1,δ2 andδp ).