Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 1, pp. 173-189, Warsaw 2010 COMPARISON BETWEEN HPM AND FINITE FOURIER SOLUTION IN STATIC ANALYSIS OF FGPM BEAM UNDER THERMAL LOAD Ahad Armin Amirkabir University of Technology, Department of Mechanical Engineering, Tehran, Iran e-mail: aarmin84@gmail.com; ahad armin mec84@yahoo.com Iman Shafieenejad K.N. Toosi University of Technology, Department of Aerospace Engineering, Tehran, Iran e-mail: shafiee iman@yahoo.com Nima Moallemi Iran University of Science and Technology, Department of Mechanical Engineering, Tehran, Iran e-mail: nima.moallemi@yahoo.com Alireza B. Novinzadeh K.N. Toosi University of Technology, Department of Aerospace Engineering, Tehran, Iran e-mail: novinzadeh@kntu.ac.ir Linear and nonlinear phenomena play important role in applied mathematics, physics and also in engineering problems in which any parameter may vary depending on different factors. In recent years, the homotopy perturbationme- thod (HPM) is constantly being developed and applied to solve various linear and nonlinear problems. In this paper, static analysis of functionally graded piezoelectric beams based on the first-order shear deformation theory under thermal loads has been investigated. The beamwith a functionally graded pie- zoelectric material (FGPM) is graded in the thickness direction and a simple power law index governs the piezoelectric material properties. The electric po- tential is assumed linear across the beam thickness. The governing equations are obtained using potential energy and Hamilton’s principle andmay lead to a system of differential equations. We suggest two methods to solve this pro- blem, the homotopyperturbation and analytical solution obtained by the finite Fourier transformation. The homotopy perturbation method and a proper al- gorithm are suggested to solve simultaneous differential equations. The results are presented for different power law indexes under uniform thermal gradient. The results are compared with the analytical solution obtained by the finite Fourier transformation for simply supported boundary conditions. Key words: functionally graded piezoelectricmaterial, thermal load, first-order shear deformation theory, homotopy perturbationmethod 174 A. Armin et al. 1. Introduction Since the piezoelectric phenomenonwas first discovered, which is widely used tomanufacture smart structures, various piezoelectric sensors, actuators, con- ductors and transducers are implemented to operate in structural design pro- blems. Piezoelectric actuators and sensors have novel applications for micro- electromechanical systems and smartmaterial systems, especially in themedi- cal and aerospace industries (Chen et al., 2004;Takagi et al., 2003). Sincemost aerospace applications involve operations in changing thermal environments, increased interests in piezothermoelasticity during recent years have addressed the thermo- and electromechanical behaviour of such materials. Nowadays, the use of functionally graded materials (FGM) has gained intensive atten- tion especially in extreme high temperature environments and reduction high thermal stresses. FGMs are inhomogeneous materials the material properties of which vary continuously in one (or more) direction(s). This is achieved usually by gradually changing the composition of constituent materials along the thickness to obtain smooth variation of material properties and optimum responses to externally applied thermo-mechanical loadings. FGMs are now developed for the general use as structural components in high temperature environments and being strongly considered as potential structural material candidates for future high-speed spacecraft. Typical FGMs are made of a mi- xture of ceramic and metal, or a combination of different metals or different ceramics that are appropriate to achieve the desired objective. Another kind of FGMs, called functionally graded piezoelectricmaterials (FGPM), obtained even more attention in the recent years (Wu et al., 2002). It is well known that piezoelectric materials have been widely used as sensors and actuators in control systems. Smart structures or elements ma- de of these so-called FGPMs are usually superior to conventional sensors and actuators and are often made of the uni-morph, bi-morph and multimorph materials. For a piezoelectric laminate with homogeneous material properties in layers, large bending displacements, high stress concentrations, creep at high temperature and failure from interfacial bonding are usually presented at the layer interfaces under a mechanical or electric loading. These effects may lead to lifetime limitations and reliability reduction. To reduce the draw- backs, piezoelectricmaterials and structureswith functionally gradedmaterial properties along the layer-thickness direction have been introduced and fabri- cated. The additional advantage of FGPMactuators is that no bonding agent is needed to bond the piezoelectric ceramic plates, because the piezoelectric layers and the inner electrode can be formed together by the sintering process. Comparison between HPM and finite Fourier solution... 175 The functionally graded sensors and actuators play an important role in the field of micro structural engineering (Pin et al., 2006; Liao, 1995). This paper presents static analysis of an FGPM beam based on the first- order shear deformation theory under thermal load. The analysis is based on the homotopy perturbation method (HPM) proposed by Liao (1995, 1997). The results are compared with the finite Fourier transformation for different power law indexes.Using thismethod, one concludes that it is apropermethod to overcome FGPM’s difficulties in engineering problems. 2. Derivation of governing equations Consider a functionally graded piezoelectric beam as shown in Fig.1. The material properties change functionally between the upper and lower surfaces across the beam thickness. Fig. 1. FGPMbeam and coordinates In themodel that is used in this paper, thematerial properties are expres- sed as P(z)= Pul (2z +h 2h )n +Pl Pul = Pu −Pl (2.1) where z is coordinate along the thickness direction of the beam, Pu, Pl are properties of theupper surface and the lower surface, respectively; and h is the thickness of the FGPM beam. The power n is the volume fraction exponent. FGPMs are particularly effective in high-temperature environments. In the present analysis, constant surface temperatures are imposed at the upper and lower surfaces. The variation in temperature is assumed to occur along the thickness direction only. Thus, the steady-state heat transfer equation is reduced to a one-dimensional equation as − d dz ( K(z) dT dz ) =0 (2.2) 176 A. Armin et al. with boundary conditions such that T = Tu at z = h/2 and T = Tl at z = −h/2. Here, T is the temperature; Tu and Tl are the applied tempera- tures on the upper and lower surfaces, respectively; z is the coordinate in the thickness direction and K is the thermal conductivity which varies according to the profile given by Eq. (2.2). The solution to Eq. (2.3) can be obtained by means of the polynomial series (Lanhe et al., 2007) T(z)= Tl + ∆T Λ [2z +h 2h + ∑ i=1 (−1)iKiul (ni+1)Ki l (2z +h 2h )ni+1] (2.3) with Λ =1+ ∑ i=1 (−1)iKiul (ni+1)Ki l (2.4) Using the first-order shear deformation theory, the displacement components are u(x,z) = u0(x)−zψ(x) w(x)= w0(x) (2.5) where u is the axial displacement, w is the transverse displacement in the z direction and ψ is the rotation angle of the cross-section with respect to the longitudinal axis. The subscript zero denotes themiddle surface displacement. In terms of the displacement components, the normal and shear strains are given by εx = u0,x −zψ,x γxz = w0,x −ψ (2.6) where the comma in subscript denotes partial differentiation. The constitutive relationships describing electrical andmechanical interac- tions for piezoelectric materials are given as (Wang and Noda, 2001) σij = cijklεkl −elijEl −βijθ Di = eiklεkl +ηilEl +piθ (2.7) here, σij and εkl are the stress and strain tensors, respectively, Di is the electrical displacement vector, El = −ϕ,l is the electrical field vector and ϕ is the electrical potential, cijkl is the elasticity matrix, eikl – piezoelectric constant matrix, ηil – dielectric permittivity coefficient matrix, βij = cijklαkl with αkl being the thermal expansion coefficients, pi denotes the pyroelectric constants and θ = T −T0, where T0 is the reference temperature. For an FGPM beam with a small width, assume the plane state of stress (σy = σyz = σxy =0), neglect the transverse normal stress (σz ≈ 0) and assu- me the axial and transverse displacements u, w and the electric potential ϕ to be independent of y. The electric field Ex is considered non-zero, as itmay be induced by the piezoelectric coupling.With these assumptions, the general Comparison between HPM and finite Fourier solution... 177 linear constitutive equations for the stresses and the electric displacements reduce to (Kapuria et al., 2004) σx = Ê(z)εx −e31(z)Ez − Ê(z)α(z)θ = Ê(z)(u0,x −zψ,x)+e31(z)ϕ,z + −Ê(z)α(z)θ σxz = Ĝ(z)γxz −e15(z)Ex = Ĝ(z)(w0,x −ψ)+e15(z)ϕ,x (2.8) Dx = e15(z)γxz +η11(z)Ex = e15(z)(w0,x −ψ)−η11(z)ϕ,x Dz = e31(z)εx +η33(z)Ez +p3(z)θ = e31(z)(u0,x −zψ,x)−η33(z)ϕ,z + +p3(z)θ where Ê and ĜareYoung’smodulusand shearmodulus, respectively.Forma- thematical simplification, the potential ϕ is assumed linear across the FGPM beam thickness as (Trindade and Benjeddou, 2006) ϕ(x,z) = z h [ϕ+(x)−ϕ−(x)]+ 1 2 [ϕ+(x)+ϕ−(x)] (2.9) where ϕ+ and ϕ− are the electric potentials on the upper and lower surfaces of the FGPM beam, respectively. For static analysis of the beam, the variational formulationmaybewritten using the virtual work principle extended to piezoelectric media (Kapuria et al., 2004) δW − δH =0 (2.10) where δH and δW are the virtual works of electromechanical internal and appliedmechanical forces, respectively. The virtual work done by the electro- mechanical internal forces in the FGPMbeam is δH = ∫ v (σxδεx +σxzδγxz −DxδEx −DzδEz) dv (2.11) Using constitutive relations Eq. (2.7) for the strain tensor, Eqs. (2.8) and the electric field relations of Eq. (2.9), and substituting into Eq. (2.11) and carrying the variational formulation, the governing equations are obtained as δu0 : A ′ 1u0,xx +A ′ 2ψ,xx +A ′ 3ϕ + ,x +A ′ 4ϕ − ,x =0 δψ : B′1u0,xx +B ′ 2ψ+B ′ 3ψ,xx +B ′ 4w0,x +B ′ 5ϕ + ,x +B ′ 6ϕ − ,x =0 δw0 : C ′ 1ψ,x +C ′ 2w0,xx +C ′ 3ϕ + ,xx +C ′ 4ϕ − ,xx =0 δϕ+ : D′1u0,x +D ′ 2ψ,x +D ′ 3w0,xx +D ′ 4ϕ ++D′5ϕ −+D′6ϕ + ,xx +D ′ 7ϕ − ,xx=D ′ 8 δϕ− : E′1u0,x +E ′ 2ψ,x +E ′ 3w0,xx +E ′ 4ϕ ++E′5ϕ −+E′6ϕ + ,xx +E ′ 7ϕ − ,xx = E ′ 8 (2.12) where A′, B′, C′, D′, and E′ are given in Appendix A. 178 A. Armin et al. 3. Solution procedure To solve the simultaneous governing equations, dimensionless values are defi- ned as u0 = u0 l w0 = w0 l x = x l ϕ+ = e31u Êul ϕ+ ϕ− = e31u Êul ϕ− (3.1) where l is the length of the beam, Êu and e31u are Young’s modulus and piezoelectric constant of the upper surface of the FGPM beam, respectively. The sign (−) indicates the dimensionless value. Using the dimensionless parameters, governing Eqs. (2.12) are given as a′1u0,xx +a ′ 2ψ,xx +a ′ 3ϕ + ,x +a ′ 4ϕ − ,x =0 b′1u0,xx + b ′ 2ψ+ b ′ 3ψ,xx + b ′ 4w0,x + b ′ 5ϕ + ,x + b ′ 6ϕ − ,x =0 c′1ψ,x + c ′ 2w0,xx + c ′ 3ϕ + ,xx + c ′ 4ϕ − ,xx =0 (3.2) d′1u0,x +d ′ 2ψ,x +d ′ 3w0,xx +d ′ 4ϕ ++d′5ϕ −+d′6ϕ + ,xx +d ′ 7ϕ − ,xx = d ′ 8 e′1u0,x +e ′ 2ψ,x +e ′ 3w0,xx +e ′ 4ϕ ++e′5ϕ −+e′6ϕ + ,xx +e ′ 7ϕ − ,xx = e ′ 8 where a′i, b ′ i, c ′ i, d ′ i, and e ′ i are dimensionless constants. In the present analysis, an analytical solution is obtained for the simply supported FGPM beamwith the following boundary conditions as σx =0 → u,x = ψ,x =0 x =0,1 w0 =0 x =0,1 ϕ =0 → ϕ+ = ϕ− =0 x =0,1 (3.3) 3.1. Finite Fourier transformation To solve the system of equations (3.2), a finite Fourier transformation can be used as u0m = 1∫ 0 u0(x)cos(mπx) dx ψm = 1∫ 0 ψ0(x)cos(mπx) dx w0m = 1∫ 0 w0(x)sin(mπx) dx ϕ + m = 1∫ 0 ϕ+m(x)sin(mπx) dx (3.4) ϕ−m = 1∫ 0 ϕ−m(x)sin(mπx) dx Comparison between HPM and finite Fourier solution... 179 Formulas for the inverse of transformation of Eqs. (3.4) are obtained by using a relationship from the theory of Fourier series u0(x)= 2 ∑ m u0mcos(mπx) ψ(x)= 2 ∑ m ψmcos(mπx) w0(x)= 2 ∑ m w0m sin(mπx) ϕ +(x)= 2 ∑ m ϕ+m sin(mπx) (3.5) ϕ−(x)= 2 ∑ m ϕ−m sin(mπx) m =1,3,5, . . . , r = mπ Solution (3.5) automatically satisfies boundary conditions (3.3). Applying transformation (3.4) to Eqs. (3.2) the following expressions are obtained −r2a′1u0m −r 2a′2ψm +ra ′ 3ϕ + m +ra ′ 4ϕ − m =0 b′1u0m +(b ′ 2−r 2b′3)ψm +rb ′ 4w0m +rb ′ 5ϕ + m +rb ′ 6ϕ − m =0 −rc′1ψm −r 2c′2w0m −r 2c′3ϕ + m −r 2c′4ϕ − m = 2 r c′5 (3.6) −rd′1u0m −rd ′ 2ψm −r 2d′3w0m +(d ′ 4−r 2d′6)ϕ + m +(d ′ 5−r 2d′7)ϕ − m = 2 r d′8 −re′1u0m −re ′ 2ψm −r 2e′3w0m +(e ′ 4−r 2e′6)ϕ + m +(e ′ 5−r 2e′7)ϕ − m = 2 r e′8 Tofindthe mthFourier components of u0m,ψm,w0m,ϕ + m andϕ − m, the system of Eqs. (3.6) must be solved based on the choice of m. Using above Fourier components and applying into Eqs. (3.5), the series solution is determined. 3.2. Homotopy perturbation method Various perturbationmethods have beenwidely applied to solve linear and nonlinearproblems.Unfortunately, the traditional perturbation techniques are based on the assumption that a small parametermust exist, which is too over- strict to find wide application (He, 2003). To illustrate the basic ideas of the newmethod, we consider the following differential equation A(u)+f(r)= 0 r ∈ Ω (3.7) with boundary conditions B ( s, ∂s ∂q ) =0 r ∈ Γ (3.8) where A is a general differential operator, B is a boundary operator; f(r) is a known analytic function, Γ is the boundary of the domain Ω (He, 1999; 180 A. Armin et al. Sajid et al., 2006). The operator A can, generally speaking, be divided into twoparts L and N, where L is linear, while N nonlinear, Eq. (3.7), therefore, can be rewritten as follows L(s)+N(s)−f(r)= 0 (3.9) By the homotopy technique proposed by Liao (1995, 1997), we construct a homotopy of Eq. (3.7) g(r,m): Ω × [0,1]→ℜwhich satisfies J(g,m) = (1−m)[L(g)−L(s0)]+m[A(g)−f(r)]= 0 m ∈ [0,1], r ∈ Ω (3.10) or J(g,m) = L(g)−L(s0)+mL(s0)+m[N(g)−f(r)]= 0 (3.11) where m ∈ [0,1] is an embedding parameter and s0 is the initial approxima- tion which satisfies the boundary conditions. Here the embedding parameter is introduced much more naturally, unaffected by artificial factors; further it can be considered as a small parameter for 0¬ m ¬ 1. So it is very natural to assume that the solution of Eq. (3.7) and Eq. (3.8) can be expressed as g = g0+mg1+m 2g2+ . . . (3.12) The approximate solution to Eq. (3.7), therefore, can be readily obtained s = lim g→1 g = g0+g1+g2+ . . . (3.13) The convergence of the series of Eq. (3.13) has been proved in Takagi et al. (2003). We construct a homotopy Ω × [0,1]→ℜwhich satisfies L1(g)−L1(s0)+mL1(s0)+m[N1(g)−f1(r)] = 0 L2(g)−L2(s0)+mL2(s0)+m[N2(g)−f2(r)] = 0 L3(g)−L3(s0)+mL3(s0)+m[N3(g)−f3(r)] = 0 (3.14) L4(g)−L4(s0)+mL4(s0)+m[N4(g)−f4(r)] = 0 L5(g)−L5(s0)+mL5(s0)+m[N5(g)−f5(r)] = 0 Comparison between HPM and finite Fourier solution... 181 where L1(g) = u0,xx f1(r)= 0 N1(g)= a2 a1 ψ,xx + a3 a1 ϕ+,x + a4 a1 ϕ−,x L2(g) = ψ,xx f2(r)= 0 N2(g)= b1 b3 u0,xx + b2 b3 ψ+ b4 b3 w,x + b5 b3 ϕ+,x + b6 b3 ϕ−,x L3(g) = w0,xx f3(r)= 0 (3.15) N3(g)= c1 c2 ψ+ c3 c2 ϕ+ ,xx + c4 c2 ϕ− ,xx L4(g) = ϕ + ,xx f4(r)= d8 d6 N4(g)= d1 d6 u0,x + d2 d6 ψ,x + d2 d6 w,xx + d4 d6 ϕ++ d5 d6 ϕ−+ d7 d6 ϕ−,xx L5(g) = ϕ − ,xx f5(r)= e1 e7 N5(g)= e1 e7 u0,x + e2 e7 ψ,x + e3 e7 w,xx + e4 e7 ϕ++ e5 e7 ϕ−+ e6 e7 ϕ+,xx The initial approximation of Eq. (3.2) based on boundary conditions is assu- med as u00 = 1 3 x3− 1 2 x2 ψ 0 = 1 3 x3− 1 2 x2 w00 = x 3−x2 ϕ+0 = x3−x2 ϕ−0 = x3−x2 (3.16) Suppose the solution to Eq. (3.14) has the form g = g0+mg1+m 2g2+ . . . (3.17) Substituting Eq. (3.17) into (3.14), and equating the terms identically m0u0 : U0,xx = u 0 0,xx m1u0 : U1,xx =− ( u00,xx + a2 a1 ψ 0 ,xx + a3 a1 ϕ+0,x + a4 a1 ϕ−0,x ) m2u0 : U2,xx =− (a2 a1 ψ 1 ,xx + a3 a1 ϕ+1,x + a4 a1 ϕ−1,x ) m3u0 : U3,xx =− (a2 a1 ψ 2 ,xx + a3 a1 ϕ+2 ,x + a4 a1 ϕ−2 ,x ) m0ψ : Ψ0,xx = ψ 0 ,xx m1ψ : Ψ1,xx =− ( ψ 0 ,xx + b1 b3 u0,xx + b2 b3 ψ 0 + b4 b3 w0,x + b5 b3 ϕ+0,x + b6 b3 ϕ−0,x ) 182 A. Armin et al. m2ψ : Ψ2,xx =− (b1 b3 u1,xx + b2 b3 ψ 1 + b4 b3 w1,x + b5 b3 ϕ+1,x + b6 b3 ϕ−1,x ) m3ψ : Ψ3,xx =− (b1 b3 u2,xx + b2 b3 ψ 2 + b4 b3 w2,x + b5 b3 ϕ+2,x + b6 b3 ϕ−2,x ) m0w0 : W0,xx = w 0 0,xx m1w0 : W1,xx =− ( w00,xx + c1 c2 ψ 0 ,x + c3 c2 ϕ+0,xx + c4 c2 ϕ−0,xx ) m2w0 : W2,xx =− (c1 c2 ψ 1 ,x + c3 c2 ϕ+1,xx + c4 c2 ϕ−1,xx ) (3.18) m3w0 : W3,xx =− (c1 c2 ψ 2 ,x + c3 c2 ϕ+2,xx + c4 c2 ϕ−2,xx ) m0ϕ+ : Φ + 0,xx = ϕ +0 ,xx m1ϕ+ : Φ + 1,xx =− ( ϕ+0,xx + d1 d6 u0,x + d2 d6 ψ 0 ,x + d3 d6 w0,xx + d4 d6 ϕ+0+ d5 d6 ϕ−0+ + d7 d6 ϕ−0,xx ) m2ϕ+ : Φ + 2,xx =− (d1 d6 u1,x + d2 d6 ψ 1 ,x + d3 d6 w1,xx + d4 d6 ϕ+1+ d5 d6 ϕ−1+ d7 d6 ϕ−1,xx ) m3ϕ+ : Φ + 3,xx =− (d1 d6 u2,x + d2 d6 ψ 2 ,x + d3 d6 w2,xx + d4 d6 ϕ+2+ d5 d6 ϕ−2+ d7 d6 ϕ−2 ,xx ) m0ϕ− : Φ − 0,xx = ϕ −0 ,xx m1ϕ− : Φ − 1,xx =− ( ϕ−0,xx + e1 e7 u0,x + e2 e7 ψ 0 ,x + e3 e7 w0,xx + e4 e7 ϕ+0+ e5 e7 ϕ−0+ + e6 e7 ϕ+0,xx ) m2ϕ− : Φ − 2,xx =− (e1 e7 u1,x + e2 e7 ψ 1 ,x + e3 e7 w1,xx + e4 e7 ϕ+1+ e5 e7 ϕ−1+ e6 e7 ϕ+1,xx ) m3ϕ− : Φ − 3,xx =− (e1 e7 u2,x + e2 e7 ψ 2 ,x + e3 e7 w2,xx + e4 e7 ϕ+2+ e5 e7 ϕ−2+ e6 e7 ϕ+2,xx ) Consequently, solving the above equations, the components of the homotopy perturbation solution for the system of Eqs. (3.2) are derived. 4. Results The numerical results are presented using the theory of Fourier series in this paper. The present study considers functionally graded materials composed of two piezo-electric constitutive materials. The bottom surface of the FGPM Comparison between HPM and finite Fourier solution... 183 beam is Platinum whereas the top surface of the beam is PZT-4-rich. The material properties of PZT-4 and Platinum are shown in Table 1. Table 1.Material properties of PZT-4 and platinum PZT-4 Platinum Ê =74GPa, K =9W/(mK), Ê =168GPa, α =4.4 ·10−61/K, e31 =−0.9C/m 2, K =77.8W/(mK), e15 =4.6C/m 2, η11 =8.26 ·10 −11N/m2, α =9 ·10−61/K, e31 =0, η33 =9.03 ·10 −11N/m2, e15 =0C/m 2, η11 =0N/m 2, p3 =3 ·10 −6C/(Km2) η33 =0N/m 2, p3 =0C/(Km 2) The FGPM beam of length of 0.2m and height 0.0025m with simply supported conditions is assumed.TheFGPMbeam is studied under a thermal gradient through its thickness direction. The temperature of the top PZT-4- rich surface is fixed at 400K and that of bottom Platinum surface is kept constant at 300K. It is assumed that the reference temperature is T0 =295K. The temperature field through the thickness of the beam is shown in Fig.2. Fig. 2. Distribution of temperature through dimensionless height of the FGPMbeam According toEqs. (3.16) and (3.18), the first few components of the homo- topy perturbation solution for the system of Eqs. (3.2), for n =0, are derived as W0 = x 3 −x2 W1 =−0.28571x 4 −0.812x3 +0.833x2 W2 =0.001241x 6 −0.0148x5 +0.2742x4 −0.212x3 +0.1971x2 (4.1) W3 =0.000051x 8 +0.000034x7 −0.001156x6 +0.01791x5 +0.008331x4 + +0.0189x3 −0.02493x2 184 A. Armin et al. Thus we have a fourth order of approximation for the dimensionless lateral deflection and the dimensionless electric potentials of Eqs. (3.2) W = W0+W1+W2+W3+ . . . Φ = Φ+0 +Φ + 1 +Φ + 2 +Φ + 3 + . . . (4.2) Φ0 = Φ − 0 +Φ − 1 +Φ − 2 +Φ − 3 + . . . And so on, in thismanner the rest of components of the homotopy pertur- bation solution for the system of Eqs. (3.2) can be obtained for n = 0. The solution is given by W =5.1 ·10−5x8+3.4 ·10−5x7−8.5 ·10−5x6+3.111 ·10−3x5+ (4.3) −3.179 ·10−3x4−5.1 ·10−3x3+5.168 ·10−3x2 Table 2 is presented to show the effect of the power law index of the func- tionally graded piezoelectric beam for distribution of dimensionless deflection through the dimensionless length. The results obtained by the homotopy per- turbation method are compared with the finite Fourier transformation for different power law indexes. Table 2 shows that with the increase of n, when the beam constituent materials change form the PZT-4-rich to the Platinum- rich, the midpoint deflection of the FGPM beam decreases accordingly. It is seen that for larger values of n, the deflection of the beam is changed and decreased evidently. The top and bottom surfaces variation of dimensionless electric potential through the dimensionless length of the FGPMbeam caused by thermal load for different power law indexes are shown in Figs.3-7. The figures show that for most Platinum-rich FGPM beam, (higher values of n), the electric potential distribution decreases in value due to less piezoelectric constants. Good agreements are observed between the homotopy perturbation method and the finite Fourier transformation. Table 2.Maximum deflection in simply-supported FGPMbeam n Finite Fourier method HPMmethod Error 0 5.762 ·10−4 5.521 ·10−4 4.18% 2 3.462 ·10−5 3.31 ·10−5 4.3% 5 1.01 ·10−6 0.96 ·10−6 4.98% 10 0.453 ·10−6 0.432 ·10−6 4.63% Comparison between HPM and finite Fourier solution... 185 Fig. 3. Top surface variation of dimensionless electric potential through dimensionless length of the FGPMbeam caused by thermal load for n =2 Fig. 4. Top surface variation of dimensionless electric potential through dimensionless length of the FGPMbeam caused by thermal load for n =5 Fig. 5. Bottom surface variation of dimensionless electric potential through dimensionless length of the FGPMbeam caused by thermal load for n =0 186 A. Armin et al. Fig. 6. Bottom surface variation of dimensionless electric potential through dimensionless length of the FGPMbeam caused by thermal load for n =2 Fig. 7. Bottom surface variation of dimensionless electric potential through dimensionless length of the FGPMbeam caused by thermal load for n =5 5. Conclusions In the present paper, the static analysis of an FGPMbeambased on the first- order shear deformation theory under thermal load is investigated. The beam is subjected to constant surface temperatures on the upper and lower surfa- ces. Boundary conditions of the beamare taken to be simply supported at the ends of the beam. To solve the problem, the numerical homotopy perturba- tionmethod is used.Moreover, the results are comparedwith thefiniteFourier transformation for different power law indexes. They show that for larger va- lues of power law indexes which provide less PZT-4 rich FGPM, the lateral deflection of the FGPMbeamdecreases constantly due to the applied thermal load. By increasing the metal share in the FGPM beam, the maximum value Comparison between HPM and finite Fourier solution... 187 of electric potential decreases on the top andbottombeam surfaces.Moreover, the obtained results show the accuracy of this approach and reveal that the proposed homotopy perturbation method is very effective and simple to the problem like this. It can be predicted thatHPM is a suitablemethod for other FGPM problems in which exact solutions are not easily achieved. Appendix A A′1 = ∫ Ê(z) dz A′2 = ∫ −zÊ(z) dz A′3 = ∫ e31(z) h dz A′4 = ∫ − e31(z) h dz B′1 = ∫ −zÊ(z) dz B′2 = ∫ Ĝ(z) dz B′3 = ∫ z2Ê(z) dz B′4 = ∫ −Ĝ(z) dz B′5 = ∫ [ − ze31(z) h − (z h + 1 2 ) e15(z) ] dz B′6 = ∫ [ze31(z) h − (z h − 1 2 ) e15(z) ] dz C′1 = ∫ −Ĝ(z) dz C′2 = ∫ Ĝ(z) dz C′3 = ∫ (z h + 1 2 ) e15(z) dz C ′ 4 = ∫ ( − z h + 1 2 ) e15(z) dz D′1 = ∫ e31(z) h dz D′2 = ∫ [ − ze31(z) h − (z h + 1 2 ) e15(z) ] dz D′3 = ∫ (z h + 1 2 ) e15(z) dz D ′ 4 = ∫ − η33(z) h2 dz D′5 = ∫ η33(z) h2 dz D′6 = ∫ −η11(z) (z2 h2 + z h + 1 4 ) dz D′7 = ∫ η11(z) (z2 h2 − 1 4 ) dz D′8 = ∫ − p3(z)θ(z) h dz E′1 = ∫ − e31(z) h dz E′2 = ∫ [ze31(z) h + (z h − 1 2 ) e15(z) ] dz E′3 = ∫ − (z h − 1 2 ) e15(z) dz E ′ 4 = ∫ η33(z) h2 dz E′5 = ∫ − η33(z) h2 dz E′6 = ∫ η11(z) (z2 h2 − 1 4 ) dz E′7 = ∫ −η11(z) (z h − 1 2 )2 dz E′8 = ∫ p3(z)θ(z) h dz 188 A. Armin et al. References 1. 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Sajid M., Hayat T., Asghar S., 2006, On the analytic solution of steady flow of a fourth grade fluid,Physics Letter A, 355, 18-24 Comparison between HPM and finite Fourier solution... 189 Porównanie rezultatów analizy statycznej belki FGPM otrzymanych metodą HPM i za pomocą skończonej transformacji Fouriera Streszczenie Zjawiska liniowe i nieliniowe odgrywają ważną rolę w dziedzinie matematyki sto- sowanej, fizyki, a także zagadnieniach inżynierskich, w których dowolny parametr może ulegać zmianie podwpływem różnych czynników.W ostatnich latach perturba- cyjna metoda homotopii (HPM) ulegała ciągłemu rozwojowi i znalazła zastosowanie w rozwiązywaniu różnorodnych liniowych i nieliniowych zadań.Wtej pracy zaprezen- towano wyniki analizy statycznej belki wykonanej z gradientowegomateriału zawie- rającego frakcję piezoelektryczną i obciążonej termicznie otrzymanych przy pomocy teorii odkształceń postaciowych pierwszego rzędu. Belka z materiału funkcjonalne- go (FGPM) ma strukturę gradientową, tj. posiada właściwości materiałowe zmienne w sposób ciągły wzdłuż grubości tej belki, zgodnie z założonym rozkładem wykład- niczym zawartości aktywnej frakcji piezoelektryka w całym materiale. Założono, że potencjał elektrycznyma rozkład liniowywzdłuż grubości belki. Różniczkowe równa- nia ruchu układu otrzymano, używając wyrażenia na energię potencjalną i stosując zasadę Hamiltona. Do ich rozwiązania zaproponowano dwie metody: perturbacyjną homotopii i analitycznąw drodze skończonej transformacji Fouriera.Wmetodzie ho- motopii zasugerowanoodpowiedni algorytmrozwiązywaniaukładu równań różniczko- wych.Wyniki przedstawionodla różnychrozkładówaktywnej frakcjipiezoelektrycznej przy utrzymaniu jednorodnego gradientu temperatury. Wyniki porównano z rozwią- zaniem analitycznym otrzymanym za pomocą skończonej transformacji Fouriera dla warunków brzegowych belki odpowiadających swobodnemu podparciu. Manuscript received April 6, 2009; accepted for print July 2, 2009