MECHANICAL ENGINEERING
A first-order shear deformation finite element model
for elastostatic analysis of laminated composite plates
and the equivalent functionally graded plates
S.S. Alieldin, A.E. Alshorbagy, M. Shaat
*
Mechanical Design and Production Department, Zagazig University, Zagazig 44511, Egypt Received 1 January 2011; revised 18 April 2011; accepted 5 May 2011
Available online 6 August 2011
KEYWORDS
Functional graded plates; Laminated composite plates; First order shear deforma-tion model;
Elastostatic finite element analysis
Abstract In the present paper, the first-order shear deformation plate (FSDT) model is exploited to investigate the mechanical behavior of laminated composite and functional graded plates. Three approaches are developed to transform the laminated composite plate, with stepped material prop-erties, to an equivalent functionally graded (FG) plate with a continuous property function across the plate thickness. Such transformations are used to determine the details of a functional graded plate equivalent to the original laminated one. In addition it may provide an easy and efficient way to investigate the behavior of multilayer composite plates, with direct and less computational efforts. A comparative study has been developed to compare the effectiveness of the three proposed transformation procedures.
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1. Introduction
Composite materials are those formed by combining two or more materials on a macroscopic scale such that they have
bet-ter engineering properties than the conventional mabet-terials, for example, metals. Most man-made composite materials are made from two materials: a reinforcement material called fiber and a base material, called matrix material[1].
The stiffness and strength of fibrous composites come from fibers which are stiffer and stronger than the same material in bulk form. The matrix material keeps the fibers together, acts as a load-transfer medium between fibers, and protects fibers from being exposed to the environment. Matrix materials have their usual bulk-form properties whereas fibers have direction-ally dependent properties.
Functionally graded materials (FGMs) are microscopically inhomogeneous composite materials, in which the volume fraction of the two or more materials is varied smoothly and continuously as a continuous function of the material position along one or more dimensions of the structure. These materials * Corresponding author. Tel.: +20 012 2232469.
E-mail address:[email protected](M. Shaat).
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are mainly constructed to operate in high temperature environments.
In conventional laminated composite structures, homoge-neous elastic lamina are bonded together to obtain enhanced mechanical and thermal properties. The main inconvenience of such an assembly is to create stress concentrations along the interfaces and more specifically when high temperatures are involved. This can lead to delaminations, matrix cracks, and other damage mechanisms which result from the abrupt change of the mechanical properties at the interface between the layers. One way to overcome this problem is to use func-tionally graded materials within which material properties vary continuously. The concept of functionally graded material (FGM) was proposed in 1984 by the material scientists in Ja-pan[2]. The FGM is a composite material whose composition varies according to the required performance. It can be pro-duced with a continuously graded variation of the volume frac-tions of the constituents, which leads to a continuity of the material properties of FGM which is the main difference be-tween such a material and the usual composite one. FGMs are usually made of a mixture of ceramic and metals.
The ceramic constituent of the material provides a high temperature resistance due to its low thermal conductivity, while the ductile metal constituent, on the other hand, prevents the fracture caused by thermal stress due to high temperature gradient in a very short period of time.
The composite plates are studied widely in the literature (a review of the plate theories can be found in Ghugal and Shimpi
[3], and Reddy[1]). Plate models for the functionally graded materials have been developed analytically and numerically. Various approaches have been developed to establish the appropriate analysis of the functionally graded (FG) plates. An analytical model, based on the classical plate theory (CPT) of Love–Kirchhoff has been developed by Chi and Chung[4,5]for the FGM plates. They developed the analytical solution for simply supported FG plates subjected to mechan-ical loads. In practice, this model is not valid for thick plates where there is a comparability for shear deformation energy. Several authors suggested computational models that take into account the transversal shear effect, by using the first-order shear deformation theory (FSDT)[6,7]and higher-order shear deformation theories[8,1]. Praveen and Reddy [9] examined the nonlinear static and dynamic responses of functionally graded ceramic–metal plates using the first-order shear defor-mation theory (FSDT) and the von Karman strain. Zenkour
[10] presented the analysis of FG sandwich plates for the deflection, stresses, buckling and free vibration in[11,12].
The first-order shear deformation plate models for the investigation of the behavior of structures made of function-ally graded materials are proposed by Nguyen, et al. [13]. The identification of transverse shear factors is investigated through these models by energy equivalence. The static analysis of functionally graded, anisotropic and linear mag-neto–electro–elastic plates which has been carried out by semi-analytical finite element method is proposed by Rajesh et al. [14]. The finite element model is derived based on constitutive equation of piezomagnetic material accounting for coupling between elasticity, electric and magnetic effect.
Reddy et al. [15] studied the axisymmetric bending and stretching of FG solid and annular circular plates using the FSDT. The solutions for deflections, force and moment resul-tants were presented in terms of the corresponding quantities
of isotropic plates based on the classical Kirchhoff plate the-ory. Lanhe[16]used the FSDT and derived equilibrium and stability equations of a moderately thick rectangular plate made of FGM under thermal loads. The theoretical formula-tion for bending analysis of funcformula-tionally graded (FG) rotating disks based on first order shear deformation theory (FSDT) is proposed by M. Bayat. Wong[17]. The material properties of the disk are assumed to be graded in the radial direction by a power law distribution of volume fractions of the constituents. The models based on the first-order shear deformation the-ory (FSDT) are very often used owing to their simplicity in analysis and programming. It requires, however, a convenient value of the shear correction factor. In practice, this coefficient has been assumed to be given by 5/6 as for homogeneous plates. This value is a priori no longer appropriate for func-tionally graded material analysis due to the position depen-dence of elastic properties[13].
In the present paper, the first-order shear deformation plate (FSDT) model is exploited to investigate the mechanical behavior of laminated composite and functional graded plates. Three approaches are developed to transform the laminated composite plate with stepped material properties to an equiv-alent FG plate with a continuous property function across the plate thickness. Such transformations are used to deter-mine the details of an FG plate equivalent to the original lam-inated one.
In addition, the transformation procedure may provide an easy and efficient way to investigate the behavior of multilayer composite plates, with direct integration and less computa-tional efforts.
2. Formulation of FSDT finite element model
In this section the first-order shear deformation plate models for modeling structures made of functionally graded materials are proposed. As mentioned previously, the models based on the first-order shear deformation theory (FSDT) are very often used owing to their simplicity in analysis and programming. The straight lines perpendicular to the mid-surface of the plate (transverse normals) before deformation remain straight after deformation but the transverse normals do not experience elongation (they are inextensible), and the transverse normals rotate such that they do not remain perpendicular to the mid surface after deformation. Based on the first assumption, the transverse shear strains and consequently the shear stresses are constant throughout the laminate thickness. In practice, a convenient shear correction factor, equals to 5
6, is assumed
for homogeneous plates (Fig. 1). 2.1. Mathematical model
As mentioned previously it is assumed that the transverse nor-mals rotate such that they do not remain perpendicular to the mid surface after deformation, so the displacement fields are uðx; y; z; tÞ ¼ u0ðx; y; tÞ þ z;xðx; y; tÞ
mðx; y; z; tÞ ¼ m0ðx; y; tÞ þ z;yðx; y; tÞ ð2:1Þ
xðx; y; z; tÞ ¼ x0ðx; y; tÞ
where ðu0;m0;x0;;x;;yÞ are unknown functions to be
deter-mined, t denotes the time andðu0;m0;x0Þ denotes the
displace-ments of the mid plane (z = 0). Note that@u
which indicate that;xand;yare the rotations of a transverse
normal about the y-axis and x-axis, respectively. The strain energy(S.E), is given by
S:E¼ Z m frgTfegdm ¼ Z Z Z frgTfegdx dy dz ð2:2Þ wherefrg ¼ ½Dfeg, [D] is the total material properties matrix. So in Eq.(2.2), we obtain
S:E¼ Z Z Z
fegT½DTfegdx dy dz ð2:3Þ
The strain is the variation of the continuum deformation with respect to its volume, so strain components can be deter-mined from Eq.(2.1)as follows:
exx¼ @u0 @xþ z @;x @x eyy¼ @m0 @yþ z @;y @y cxy¼@u0 @y þ @m0 @xþ z @;x @y þ @;y @x ð2:4Þ cxz¼ @x0 @x þ ;x cyz¼@x0 @y þ ;y
So, the strain components can divided intofebg bending strain,
andfesg shear strain components.
exxðx; y; zÞ eyyðx; y; zÞ cxyðx; y; zÞ 8 > < > : 9 > = > ;¼ 1 0 0 z 0 0 0 1 0 0 z 0 0 0 1 0 0 z 2 6 4 3 7 5 @u0ðx;yÞ @x @m0ðx;yÞ @y @u0ðx;yÞ @y þ @m0ðx;yÞ @x @;xðx;yÞ @x @;yðx;yÞ @y @;xðx;yÞ @y þ @;yðx;yÞ @x 2 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 5 orfebg ¼ ½Zbfe0bg ð2:5Þ cxzðx; y; zÞ cyzðx; y; zÞ ¼ 1 0 0 1 @x0ðx;yÞ @x þ ;xðx; yÞ @x0ðx;yÞ @y þ ;yðx; yÞ " # orfesg ¼ ½Zsfe0sg ð2:6Þ wherefe0
bg is the nodal bending strain and fe0sg is the nodal
shear strain. The nodal strain components are determined in coming sections. Substitute Eq. (2.5) and Eq. (2.6) into Eq.
(2.3), we can obtain: S:E¼ Z Z Z ½fe0 bg T ½zb T ½Db T ½zbfe0bg þ ½fe0 sg T ½zs T ½Ds T ½zsfe0sgdxdydz ð2:7Þ
where the material matrix can be determined from the integra-tion over the material thickness
½DEEb ¼ Z h 2 h 2 ½zb T ½Db T ½zbdz ð2:8Þ
where [DEEb] is the bending material matrix of the plate, so it
can be written in the form ½DEEb ¼ ½A ½B ½B ½D Aij¼ Z h 2 h 2 QijðzÞdz Bij¼ Z h 2 h 2 zQijðzÞdz Dij¼ Z h 2 h 2 z2Q ijðzÞdz
where Aijis the extensional stiffness components, Bijis
bend-ing-extensional coupling stiffness components, and Dijis
bend-ing stiffness components. and½DEEs ¼ Z h 2 h 2 ½zs T ½Ds T ½zsdz ð2:9Þ
where [DEEs] is the shear material matrix of the plate, so it can
be written in the form ½DEEs ¼ ½0 ½0 ½0 ½S Sij¼ Z h 2 h 2 QijðzÞdz
where for isotropic FG materials
½Db ¼ Q11ðzÞ Q12ðzÞ Q16ðzÞ Q12ðzÞ Q22ðzÞ Q26ðzÞ Q16ðzÞ Q26ðzÞ Q66ðzÞ 2 6 4 3 7 5 ð2:10Þ ½Ds ¼ Q44ðzÞ Q45ðzÞ Q45ðzÞ Q55ðzÞ ð2:11Þ Note that QijðzÞ is the equivalent material property
compo-nents as a function of the material thickness direction z. Where in FG plate, its material properties vary smoothly and contin-uously over the thickness of the structure.
2.2. Finite element model
The displacements and normal rotations at any point into a fi-nite element e may be expressed in terms of the n nodes of the element:
Figure 1 Undeformed and deformed geometries of an edge of a plate under the assumptions of the FSDT.
u0ðx; yÞ m0ðx; yÞ x0ðx; yÞ ;xðx; yÞ ;yðx; yÞ 8 > > > > > < > > > > > : 9 > > > > > = > > > > > ; ¼X n i¼1 wei 0 0 0 0 0 wei 0 0 0 0 0 we i 0 0 0 0 0 wei 0 0 0 0 0 wei 2 6 6 6 6 6 4 3 7 7 7 7 7 5 ue 0 me 0 we 0 ;ex0 ;ey 0 8 > > > > > > < > > > > > > : 9 > > > > > > = > > > > > > ; ð2:12Þ
where wei is the Lagrange interpolation function at node i. The Lagrange interpolation function for nine node rectangular ele-ments are given by in terms of the natural coordinates[1].
w1 w2 w3 w4 w5 w6 w7 w8 w9 8 > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > : 9 > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > ; ¼1 4 ð1 nÞð1 gÞðn g 1Þ þ ð1 n2Þð1 g2Þ ð1 þ nÞð1 gÞðn g 1Þ þ ð1 n2Þð1 g2Þ ð1 þ nÞð1 þ gÞðn þ g 1Þ þ ð1 n2Þð1 g2Þ ð1 nÞð1 þ gÞðn þ g 1Þ þ ð1 n2Þð1 g2Þ 2ð1 n2Þð1 gÞ ð1 n2Þð1 g2Þ 2ð1 þ nÞð1 g2Þ ð1 n2 Þð1 g2Þ 2ð1 n2Þð1 þ gÞ ð1 n2Þð1 g2Þ 2ð1 nÞð1 g2Þ ð1 n2Þð1 g2Þ 4ð1 n2Þð1 g2Þ 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ð2:13Þ The nodal bending strain can be written as follow
@u0ðx;yÞ @x @ve 0ðx;yÞ @y @u0ðx;yÞ @y þ @ve 0ðx;yÞ @x @;xðx;yÞ @x @;yðx;yÞ @y @;xðx;yÞ @y þ @;yðx;yÞ @x 8 > > > > > > > > > > > > < > > > > > > > > > > > > : 9 > > > > > > > > > > > > = > > > > > > > > > > > > ; ¼X n i¼1 @we i @x 0 0 0 0 0 @wei @y 0 0 0 @we i @y @we i @x 0 0 0 0 0 0 @wei @x 0 0 0 0 0 @wei @y 0 0 0 @wei @y @we i @x 2 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 5 ue 0 ve 0 xe 0 ;ex 0 ;ey 0 8 > > > > > > < > > > > > > : 9 > > > > > > = > > > > > > ; orfe0 bg ¼ Xn i¼1 ½Bbfu0g ð2:14Þ
where [Bb] is the curvature-displacement matrix. The nodal
shear strain can be written as
@x0ðx;yÞ @x þ ;xðx; yÞ @x0ðx;yÞ @y þ ;yðx; yÞ ( ) ¼X n i¼1 0 0 @wei @x w e i 0 0 0 @wei @y 0 w e i 2 4 3 5 ue 0 me 0 xe 0 ;ex0 ;ey 0 8 > > > > > > < > > > > > > : 9 > > > > > > = > > > > > > ; orfe0 sg ¼ Xn i¼1 ½Bsfu0g ð2:15Þ
Where [Bs] is the shear strain-displacement matrix. So, the
ele-ment stiffness matrices and load vector may be written as follows: ½Ke b ¼ Z A ½Bb T ½DEEb½BbdA ð2:16Þ ½Ke s ¼ Z A ½Bs T ½DEEs½BsdA ð2:17Þ ffeg ¼ Z A ½we i T fqgdA ð2:18Þ
where {q} is the applied force vector.So Eq. (2.14) and (2.15), into Eq. (2.5) and (2.6) the bending and shear strain vectors can be obtained by substitution of.
The normal stress components can be determined as follows rxx ryy rxy 8 > < > : 9 > = > ;¼ Q11ðzÞ Q12ðzÞ Q16ðzÞ Q12ðzÞ Q22ðzÞ Q26ðzÞ Q16ðzÞ Q26ðzÞ Q66ðzÞ 2 6 4 3 7 5 exx eyy cxy 8 > < > : 9 > = > ; orfrbg ¼ ½Dbfebg ð2:19Þ
and the shear stress components are ryz rxz ¼ Q44ðzÞ Q45ðzÞ Q45ðzÞ Q55ðzÞ " # cyz cxz orfrg ¼ ½Dsfesg ð2:20Þ
where the material matrix components of isotropic FG plate are Q11ðzÞ ¼ EðzÞ 1 t2; Q12ðzÞ ¼ tðzÞQ11ðzÞ; Q11ðzÞ Q22ðzÞ; Q66ðzÞ ¼ 1 tðzÞ 2 Q11ðzÞ Q44ðzÞ ¼ Q55ðzÞ ¼ K 1 tðzÞ 2 Q11ðzÞ Q16ðzÞ ¼ Q26ðzÞ ¼ Q45ðzÞ ¼ 0:0
where K is the material shear correction factor, E(z) is the effective young’s modulus, and tðzÞ is the effective Poisson’s ratio of the material through the laminate thickness. The con-venient shear correction factor has been assumed to be given by5
6as for homogeneous plates (Fig. 2)[1].
3. Effective mechanical properties
A functionally graded material is a composite material whose composition and microstructure are locally varied during man-ufacturing so that certain variations of local material (thermal and mechanical) properties are achieved. Comparing to tradi-tional composites merits of FGMs include: (1) small thermal stresses and a designed peak thermal stress location; (2) small stress concentrations at material interfaces in an FGM; (3) strong interfacial bonding; (4) reduced free-edge effects; (5) de-layed plastic yielding; and (6) increased fracture toughness by reducing driving force for crack growth along/across material interfaces[18].
Mechanical properties of an FGP include Young’s modulus E, shear modulus G, Poisson’s ratio t, and mass density q, and
Figure 2 Finite element for shear deformation models of plates (nine node element).
thermal properties include the coefficient of thermal expansion a, the thermal conductivity k, and the specific heat capacity C. q and t are usually linear functions of material volume ratios, but others are nonlinear functions of material volume ratios because they depend on material microstructures [19]. The FG plate theory presented here can work with effective mate-rial properties estimated by using any method. For state of simplicity, the effective property P will be estimated using the simple, Voigt arithmetic method as:
P¼ P1V1þ P2V2; V1þ V2¼ 1
where P1and P2are the properties of the first and second
con-stituent materials (metal and ceramic), and V1and V2are the
volume fractions of the constituent materials. The distribu-tions of volume fracdistribu-tions through the plate thickness are as-sumed to follow the following simple power law:
V2¼
zþ zi
ziþ1 zi
n
where n can be any non-negative real number and zi+1P z P zi. Note that the plate thickness h = zi+1zi,
hence we have the effective material property is given by P¼ P1þ ðP2 P1Þ
zþ zi
ziþ1 zi
ð3:1Þ For detailed modeling of effective material properties of FGMs the reader has to refer to Refs.[19–21].
4. Proposed transformation procedures of a laminated composite plate to an equivalent single-layer FG plate
Many approaches are used to transform a laminated plate, having stepped material properties across the plate thickness, to an equivalent FG plate with continuous material property function across the thickness. A comparative study is devel-oped to check the effectiveness of the different procedure according to the consistency between the behavior of the origi-nal laminated plate and those of the equivalent FG plates.
The first approach is a curve fitting approach which is used to obtain an equivalent function of the FG material property, consistent with the distribution of the composite material property across the plate thickness. This approach provides an FG plate of constituents having nearly the same properties of the original laminated plate. The effective material property of the equivalent FG plate can be obtained by equating the to-tal area under the stepped distribution curve of laminated composite materials properties and the area of the continuous curve concerning the material property function of the equiv-alent FG plate.
The second approach is the effective material property ap-proach. In this approach, the FG material property varies through the plate thickness with the power law given in Eq.
(3.1), and the value of n is calculated by equating the effective material property of the FG material with the effective mate-rial property of the composite matemate-rial. This approach pro-vides an FG plate of the same constituents of the composite plate, and of the same effective material property.
The third approach is the volume fraction approach. In it the FG material property varies through the plate thickness with the power law given in Eq.(3.1), and the value of n is cal-culated by equating the volume fractions of the FG material
constituents with the volume fraction of the composite mate-rial constituents. This approach provides an FG plate of the same constituents of the composite plate, and of the same vol-ume fraction of constituent materials.
To compare between the three approaches, numerical sim-ulations are performed to this aim, these simsim-ulations are dis-cussed in the coming section.
5. Numerical result
In this section we present several numerical simulations, in or-der to assess the behavior of functionally graded plates sub-jected to mechanical loads. To this aim we will consider a simple supported plate. The plate is made by a ceramic mate-rial at the top, a metallic at the bottom and law Eq.(3.1), with n= 0, 0.5, 1, 2 is used for the through-the-thickness variation. 5.1. Verification of the Proposed Model
The analysis of the FGMs plates is performed for a combina-tion of materials of type ceramic-metal. The lower plate sur-face is assumed to be aluminum while the top sursur-face is assumed to be zirconia. Material properties vary with the power law and the values n = 0, 0.5, 1, 2 are considered. Phys-ical material properties are given inTable 1 [22]. A simply sup-ported square plate is considered of side a = 0.2 m and thickness h = 0.01 m. The plate is subjected to a uniformly dis-tributed mechanical normal load on the top surface (SeeFig. 3)
To determine the elastic-static response of FG plates with material constituents (aluminum and zirconia), several simula-tions are performed for different values of the grading param-eter n. The following non-dimensional paramparam-eters are introduced for use within the numerical simulations to be
Table 1 Material properties.
Property Aluminum Zirconia
Young’s modulus EL= 70 GPa EU= 151 GPa
Poisson’s ratio tL= 0.3 tU= 0.3
presented next (here q denotes the intensity of the applied mechanical load and Ebis the Young’s modulus at the bottom
face (aluminum)): Central deflection x¼ x=h Load parameter P¼ a 4q Ebh 4 ð5:1Þ Thickness h¼ h=a
Several simulations were conducted to assess the behavior of functionally graded plates to mechanical load. Focusing on the midpoint deflection x, the objective of the analysis is to compare the response of functionally graded plates with the one of homogeneous single-constituent plates, i.e. those made of aluminum or zirconia only. Toward this goal, several numerical studies are conducted with increasing mechanical load.Fig. 4shows the central deflection x¼ x=h due to a se-quence of mechanical loads for different values of n. The non-dimensional load P takes values in the interval (0,12) and one may see that all the plates with intermediate material proper-ties experience intermediate values of deflection. This was ex-pected since the metallic plate is the one with the lowest stiffness and the ceramic plate is the one with the highest stiff-ness.Table 2, shows the numerical results of the simulations. The results are in excellent agreement with those presented in
[22].
FromFig. 4, the central deflection of the FG plate (in the previous example) decreases by raising the value of the expo-nent parameter (n), because the material stiffness increases. The material stiffness increases due to the volume fraction in-crease of the ceramic by raising the value of (n). Note, this is sufficient for FG material its top surface has young’s modulus higher than its bottom surface.
Fig. 5 represents the axial strain exx at the center of the
square plate along the thickness direction. The plate is as-sumed to be subjected to a uniformly distributed load of inten-sity q = 1 GPa. Based on the formulated model (FSDT), the strain exxof the FGM plate is linear, which is a result of the
assumption that the strain is proportional to z. It is shown also, there is no strain at the mid-plane of the FG plate at (z = 0), while it is well known that the neutral surface moves forward to the top surface of the FGM plate, as the material stiffness decreases. The FSDT of plates assumes that the trans-verse normals to the mid-plane of the plate remain straight and rotate such that they do not remain perpendicular to the mid surface after deformation. This reflects the lack of the theory. The variation of the axial stress rxxat the center of the FGM
plate along the thickness direction for different values of the material parameter n is depicted inFig. 6. The stress of power law FGM (P-FGM) plate can be represented as a function of z of order 3 for material parameter n is assumed to be 2. The maximum tensile stress at the center of the FGM plate is at the bottom edge (z = h/2) and increases as the ratio EU/EL
in-creases. However, the maximum compressive stress is at the top surface (z =h/2) and is high for small EU/EL. For the
ratio EU/EL= 1 in which the FGM plate becomes a
homoge-nous isotropic plate, the stress distribution is a linear function of z and the maximum stress is at the top or bottom surface of the plate.
Figure 4 Non-dimensional center deflection of P-FGM versus load parameter.
Table 2 Non-dimensional deflection of Mid-point versus load parameter.
Load parameter (P) ( w) for
Ceramic n= 0.5 n= 1 n= 2 Metal 2 0.04 0.06 0.06 0.05 0.09 4 0.08 0.13 0.11 0.11 0.18 6 0.13 0.19 0.17 0.16 0.27 8 0.17 0.25 0.23 0.21 0.36 10 0.21 0.32 0.29 0.27 0.45 12 0.25 0.38 0.34 0.32 0.54 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 -0.5 0 0.5 Str a in xx z/h n = 0.5 n = 1 n = 2 ceramic metal
Figure 5 Axial strain distribution through the plate thickness.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Non di ensi onal axial stress z/h Ceramic n = 0.5 n = 1 n = 2 Metal
Figure 6 Non-dimensional axial stress distribution through the plate thickness (*Non-dimensional axial stress ¼ rxxðh
2
a2qÞ
5.2. Effect of the aspect ratio
In order to investigate the effect of the aspect ratio a/b, the center deflection of the P-FGM plate is shown in Fig. 7. It can be observed that the center deflection increases with increasing the aspect ratio for a/b less than 3. This results show good agreement with those obtained by (Chi and Chung)
[4,5,23]. The used material and geometrical properties are E2= 2.1· 106Kg/cm2, t = 0.3, b = 100 cm, h = 2 cm, and
q= 1.0 Kg/cm2. In the figure, E
1/E2denotes the ratio of the
elastic modulus at the top and bottom faces of the plate.
Fig. 7shows that more of E1/E2brings the larger deflection,
because larger E1/E2decreases the stiffness of the FGM plate.
Table 3, shows the numerical results of the simulations. 5.3. Comparison between the multilayer composite and the single-layer FG analyses
The analysis of multilayer composite plates consuming a com-putational effort is considered high relative to the analysis of single-layer FG plates. In multilayer composite plates, the equivalent material property components are computed for each layer and summed together to form the overall material matrix, thus consumes more analysis time and computational effort. Also, the abrupt change in material property from one layer to other in multilayer plates leads to some computa-tional problems such as discontinuity at the layer’s interfaces that could produce stiffness matrix singularity. Transforming the multilayer composite plate to a single-layer FG plate is
per-formed to make a good use of the material continuous change through the plate thickness to overcome the lack of the analy-sis of multilayer composite plates.
To compare between the analysis of multilayer composite plates and the analysis of single-layer FG plates, some simula-tions are performed under the same cause and parameters ex-cept the material parameter. A multilayer composite and single-layer FG plates consisting of the same material constit-uents are used to perform a fair comparison between their analyses. So, the three approaches discussed previously are used to transform the multilayer composite plate to a single-layer FG plate of the same material constituents.
The objective now, is to compute the elastic-static response of a multilayer composite plate and a single-layer FG plate transformed by the three approaches, subjected to the same mechanical loads. To reach this aim a square plate of side length a = 40 m, thickness h = 2 m and subjected to a uni-form distributed load is used. The plate material has four states. In State 1, the plate is multilayered (five layers) and composed of two isotropic materials (Zerconia and Alumi-num) seeFig. 8(a). In States 2,3 and 4, the multi-layer compos-ite plate is transformed to a single layer FG plate by using the numerical curve fitting method approach, the effective material property approach and the volume fraction approach, respec-tively (seeFig. 8(b),(c),(d)).
Fig. 9andTable 4show the non-dimensional central deflec-tion w¼ w=h due to a sequence of mechanical loads for the four states of material.Fig. 10shows the axial strain distribu-tion along the plate thickness for the different States of mate-rial.Fig. 11andTable 5show the non-dimensional axial stress distribution along the plate thickness for the different states of material.
Figs. 9 and 10 show that the volume fraction approach experiences the best transformation from multilayer composite material state to a single-layer FG material state (for the ex-plained simulation). This conclusion could differ for another simulation example, so the three approaches must be applied for each simulation problem to provide the best approach that can be used. Finally, there is no role can be cited to provide the best approach, because the numerical curve fitting method ap-proach depends on the polynomial function, which depends on the lamination scheme of the composite plate. Also, the effec-tive material property approach depends on the effeceffec-tive mate-rial property of the composite matemate-rial, lamination scheme of the composite plate and material constituents of the plate. And, the volume fraction approach depends on the volume fraction of the plate constituents.
FromFig. 11, the axial stress distribution through the plate thickness is linear function of z for multilayer composite plate, while the axial stress distribution through the plate thickness is a polynomial function of z for the three other FG material states. The linearity in the multilayer composite analysis re-flects a big gap between the finite element analysis and reality (the real response of the plate), but the single-layer FG analysis provides close results to reality.
In the multilayer composite plates analysis, the equivalent material property components are computed for each layer and summed together to form the overall material matrix, this consuming more analysis time and computational effort. But, in the single-layer FG plates’ analysis, the overall material ma-trix is formed by computing the equivalent material property components one time. This reflects that the single-layer FG
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 4 6 8 10 Non-Dimensional Center Deflection (w/h)
Aspect ratio a/b E1/E2 = 1
E1/E2= 2 E1/E2 = 10
Figure 7 Non-dimensional center deflection of P-FGM plate versus the aspect ratio for different E1/E2.
Table 3 Non-dimensional center deflection of P-FGM plate versus the aspect ratio for different E1/E2.
a/b Non-dimensional deflection W/h for P-FGM (P = 2)
E1/E2= 1 E1/E2= 2 E1/E2= 10
1 0.13295 0.1662 0.2077 2 0.33015 0.41265 0.51575 4 0.41615 0.52015 0.6502 6 0.42645 0.533 0.66625 8 0.4330 0.54125 0.6765 10 0.4368 0.546 0.68245
analysis provides higher performance than multilayer compos-ite analysis. The single-layer FG analysis reduces the number of the numerical operations required to form the overall mate-rial matrix, this leads to reduction in computational effort and analysis time. For the previous simulation example the save in the analysis time is nearly 65.625%, while this value could be increased for increased number of layers for laminate compos-ite plate.
6. Conclusion
In this study, a first-order shear deformation plate (FSDT) model, exploited for the investigation of the mechanical behav-ior of composite laminated plates and functional graded plates,
is introduced. Three approaches are discussed to transform the multilayer composite plate to an equivalent single-layer FG plate. A comparative study is developed to check the consis-tency between the behavior of the original laminated composite plates and those of the alternatives of the equivalent single-layer FG plate, determined by the proposed transforma-tion procedures. The numerical results lead to the following conclusions:
1. Three approaches are proposed to determine the property details of an FG plate equivalent to the original laminated composite plate. The computational effort of the elastostat-ic analysis of that FG plate is much less than that required for the analysis of the original laminated one.
Figure 8 Material young’s modulus distribution along the plate thickness for (a) Multilayer composite plate (State 1). (b) FGM, Curve fitting approach (State 2). (c) FGM, Effective material property approach n = 0.6 (State 3). (d) FGM, Volume fraction approach n = 2.2 (State 4).
2. The first approach for equating the behavior of the original laminated plate to a corresponding FG plate, with contin-uous property function across the plate thickness, is more consistent with respect to the behaviors of the original plate and the equivalent FG one
3. Prediction of which approach is better to transform the composite plate to an equivalent FG plate depends on the effective material property of the laminate, the volume frac-tion of the laminate material constituents, the laminafrac-tion scheme, and the material properties of each layer.
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Soliman Soliman Soliman Ali-Eldin, associate Professor of Mechanical Design, Faculty of Engineering, Zagazig University. Graduated from Helwan University, Department of Production Engineering ,(Helwan , Cairo) in 1979 and obtained his M.Sc. degree from Zagazig University , Department of Mechan-ical Design and power in 1987. He obtained his Ph.D. degree from Cairo University , Department of Mechanical Design and Production Engineering . His fields of interest include Computational solid mechanics, Design and tribology, Stress analysis of composites, Contact mechanics of deformable solids and Functional graded material science.
AMAL ELHOSSINY ALSHORBAGY,
associate Professor of Mechanical Design, Faculty of Engineering, Zagazig University. Graduated from Zagazig University in 1985 and obtained her M.Sc. degree from the same university in 1990. She obtained her Ph.D. degree from Zagazig University. Her fields of interest include Numerical and Experimental Analyses of Transient Thermo-Elastic Con-tact of Non-Conformal Deformable Bodies, Numerical and Experimental Analyses of bolted Joints in composite materials. Mohammed Ibrahim Shaat, demonstrator in the Faculty of Engineering, Zagazig univer-sity. graduated from Zagazig University in 2007. His fields of interest include analytical and computational models of structures, finite element analysis of FG and composite structures.