An accurate two-dimensional elasticity analytical solution is presented for static and free vibra- tion analysis of axially functionally graded beam subjected to the arbitrary boundary condition using extended Kantorovich method. Further, static and free vibration results for the arbitrary supported homogeneous beam are obtained as a special case of present development. New bench- mark results are presented for the axially functionally graded beams as well as a homogeneous beam subjected to arbitrary support conditions. For static case, single term (n=1, iter.1) gives accurate prediction for deflections and stresses for simply-supported case whereas two-term solu- tion (n=2, iter.2) is required for the other boundary conditions. For free vibration case, the single term solution is sufficient enough to obtaining the accurate flexural frequencies for all the cases and boundary conditions. The influence of material properties variation on the static and dynamic re- sponse of axially FG beam, as compared to the homogeneous beam is investigated comprehensively by considering various material property variation cases. It is observed that the bending and free vibration behaviour of beam affected significantly by the axial gradation of material properties.
SUMMARY
The percentage effect is almost increased by 1.5 times as the gradation indexes are increasing from 0.5 to 1. Further, the effects of axial gradation on the behavior of the beam depend significantly on boundary conditions of the beams. This development has shown that, by controlling the axial variation of material properties, the desired response of beam can be achieved for specific appli- cations. The current research will also be beneficial to modeled real-life beam structures in which material properties of beam deteriorate due to some environmental effect. The present 2D elasticity analytical solution can be used for assessing the validity and accuracy of different beam theories and computational models for analysis of axially graded beams.
Functionally Graded Rectangular Plates- 3D Elasticity Analysis
A three-dimensional analytical solution is developed for static [296] and free vibration analysis of longitudinally functionally graded plate subjected to Levy-type boundary conditions. A linear vari- ation of material compliances and density are assumed along thex-direction, as given in Sec. 3.1.1.
The Hamilton-type weak-form of governing equations, which has been developed in Chapter 2 for two-dimensional beams, is now extended to three-dimensional rectangular plates, as described in Sec. 3.1.2. The multi-term extended Kantorovich approach(EKM), as developed in Chapter 2, is applied to solve the governing equations in mixed form. By employing EKM, a set of ODEs with constant coefficient is obtained along the z-direction which is solved analytically in closed form, as explained in Sec. 3.3. Another set of ordinary differential equations (ODEs) with varying coefficients are obtained along the x-direction, which is solved by using the recently developed modified power series method in Sec. 3.4. Numerical results are presented for various configuration, aspect ratio, and boundary conditions under static bending in Sec. 3.6 and for free vibration case in Sec. 3.7. The present method is validated by comparing the results with those obtained using the 3D FE (ABAQUS). Effect of variation of the material property on the longitudinal variation of deflection, stresses, and natural frequencies is studied by considering various gradation cases in conjunction with the arbitrary support conditions at the ends of the plate. For the first time, numerical results for a two-layered plate having a different in-plane material variation in each layer are also presented along with single-layer plate results. Through-the-thickness variation of various entities for particular material indexes is also presented. Benchmark results for static and dynamic analysis are presented for various cases which can be used to validate approximate two-dimensional solutions and 3D numerical solutions.
The effects of longitudinally varying adhesive properties on the interfacial and inter-laminar stress distribution are also studied for the adhesively bonded rectangular plate in Sec. 3.6.3. The elastic properties of adhesive interlayer are assumed to vary linearly along the x-direction. A specific
Chapter 3
predefined in-plane variation in adhesive mechanical property is assumed and their influence on bending response of bonded plate subjected to mechanical loading is investigated. The present method can be beneficial to modeled adhesive structure in which adhesive interlayer properties degraded due to some environmental effect.
3.1 THEORETICAL FORMULATION FOR FGM PLATES 3.1.1 In-plane Material Property Variation
The compliances of the layers are assumed to vary linearly alongx-direction as:
¯
sm1j = ¯s1j(1 +δ1ξ1) for j= 1,2,3;
¯
sm55= ¯s55(1 +δ2ξ1); s¯m66= ¯s66(1 +δ2ξ1) (3.1) ρm(ξ) =ρ(1 +δpξ)
where δα is a parameter that governs the material properties variation, ξ1 = x/a, ξ2 = y/b are non-dimensional quantities. For the present case, δ1 is variation index for Young’s modulus, δ2
variation index for shear modulus or modulus of rigidity and δp is the variation index for density.
The variation indexes (δ1, δ2 and δp ) can be positive or negative. The property variation along the in-plane direction may be due to diffusion of chemicals such as hydrogen, corrosion or cyclic exposure to temperature, etc. 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. The gradation profiles obtained from present gradation models and other probabilistic gradation models have a similar nature.
3.1.2 Basic Governing Equation for 3D Plate
The geometry of a multilayered FGM laminated plate as shown in Fig. 3.1 is considered for the analysis. The plate has length a, b and total thickness h along x, y and z-direction, respectively.
Levy-type support conditions are assumed for the present case, i.e., simply-supported aty= 0 and band arbitrary support end conditions at x= 0 anda. Perfect bonding is considered between the interface of two layers. The thickness of thekth layer and thez-coordinate of its upper surface is defined as t(k) and zk. The layer superscript over the entities is omitted unless needed for clarity.
The infinitesimal strain tensor is used εij = 1
2(ui,j+uj,i) (i, j=x, y, z) (3.2) whereux=u,uy =v,uz =w and a subscript comma denotes differentiation.
L
L-th layer K-th layer
h/2
h/2 k
1st layer 0
K0th layer FGM layers
b
Fig. 3.1: Geometry of the FGM laminated plate.
The three-dimensional constitutive equations for a linear orthotropic lamina are given as
ε= ¯Smσ (3.3)
whereε= [εx εy εz γyz γxz γxy]T, σ = [σx σy σz τyz τxz τxy]T
S¯m =
[S¯Nm 0 0 S¯mS
]
S¯Nm =
¯
sm11 s¯m12 ¯sm13
¯
sm12 s¯22 ¯s23
¯
sm13 s¯23 ¯s33
S¯Sm=
¯
s44 0 0 0 ¯sm55 0 0 0 s¯m66
whereσ andεare the normal stress and strain components, respectively,τ andγ denote the shear stress and strain components, respectively. ¯Sm is transformed elastic compliance.
The weak-form of governing equations using the Hamilton’s-type mixed variational principle for a linear elastic medium without body force are expressed as,
∫
t
∫
a
∫
b
∫
h
[δu(τxz,z+σx,x+τxy,y−ρmu) +¨ δv(τyz,z+τxy,x+σy,y−ρmv)¨ +δw(σz,z+τzx,x+τyz,y−ρmw) +¨ δσx(¯sm11σx+ ¯sm12σy+ ¯sm13σz−u,x)
+δσy(¯sm12σx+ ¯s22σy+ ¯s23σz−v,y)−δσz(w,z−s¯m13σx−¯s23σy−¯s33σz) (3.4)
−δτyz(v,z+w,y−s¯44τyz)−δτzx(u,z+w,x−¯sm55τzx)
+δτxy(¯sm66τxy−v,x−u,y)] dzdydxdt= 0 ∀ δui, δσi, δτij
The non-dimensional local thickness parameter ζ(k) for the kth layer is defined as ζ(k) = (z− zk−1)/t(k) such that 0≤ζk ≤1 forzk−1≤z≤zk. The top and bottom surfaces of FGM plate are
Chapter 3
subjected to following boundary conditions,
z= (−h/2, h/2),: σz =−pα, τyz = 0, τzx= 0 (3.5) where pα (α = 1,2) is uniformly distributed pressure load. Following variables satisfy continuity conditions at thekth interface between kth and (k+ 1)th layers:
[(u, v, w, σz, τyz, τzx)|ζ=1](k)= [(u, v, w, σz, τyz, τzx)|ζ=0](k+1) (3.6) In the present solution methodology, the system of differential equations along thexandz-direction are solved analytically without dividing the plate into fictitious sub-layers. Therefore, displacements and equilibrium conditions for the fictitious interface are not required. Further, two ends of FGM plate (ξ1 = 0 and 1) can be subjected to simply-supported (S), clamped (C) and free (F) support conditions, e.g.
Simply-supported (S) : v= 0, w= 0, σx= 0
Clamped (C) : u= 0, v= 0, w= 0 (3.7)
Free (F) : σx= 0, τxy = 0, τxz = 0
The boundary conditions at the edgesξ2= 0 and 1 are assumed to be simply-supported:
u= 0, w= 0, σy = 0 (3.8)
3.2 FOURIER SERIES-GENERALIZED EKM SOLUTION FOR PLATE