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Latin American Journal of Solids and Structures

versão impressa ISSN 1679-7817versão On-line ISSN 1679-7825

Lat. Am. j. solids struct. vol.14 no.13 Rio de Janeiro  2017

http://dx.doi.org/10.1590/1679-78254096 

Articles

Viscoelastic Substrates Effects on the Elimination or Reduction of the Sandwich Structures Oscillations Based on the Kelvin-Voigt Model

M.M. Alipoura  * 

I. Rajabib 

a Assistant Professor, Department of Mechanical Engineering, University of Mazandaran, Babolsar 47416-13534, IranE-mail: m.mollaalipour@umz.ac.ir, m.m.alipur@gmail.com

b Assistant Professor, Department of Mechanical Engineering, Malek Ashtar University, Shiraz, Iran. E-mail: irajabi@mail.kntu.ac.ir

Abstract

Effects of viscoelastic substrates on the sandwich structures oscillations are examined in this paper. In this regard, dynamic response of sandwich annular panels with FG polar orthotropic face sheets resting on viscoelastic substrate is presented. Young’s modulus in the radial and circumferential direction, shear modulus and density of each face sheet may be varied continuously in the radial direction. The viscoelastic substrate is modeled as Kelvin-Voigt foundation. To describe more accurately response of sandwich structures, the governing dynamical equations are derived based on the layerwise theory and five systems of second order coupled partial differential equations are obtained. The effects of the stiffness and damping coefficients of the foundation on the dynamic behavior of sandwich plate are investigated for various transient loads and boundary condition. Since no available results may be found in literature to demonstrate the efficiency and accuracy of the obtained results, the obtained results are verified by comparison with finite element results based on the three dimensional theory of elasticity for some special cases.

Keywords: Dynamic response; FG polar orthotropic; viscoelasticfoundation; transient loads, Kelvin-Voigt

1 INTRODUCTION

Sandwich structures are widely used in the industries. In practical applications, sandwich structures are subjected to various transient dynamic loads where reduce vibration of these structures underdynamic loads is important in many engineering fields and viscoelastic substrate can be applied to reducethe vibration of structures. In addition, structures resting on viscous, elastic or viscoelastic foundationare extensively used in many engineering fields. However, most of the performed studies arelimited to the static and free vibration analyses of sandwich plates and rare researches are available onthe dynamic analysis especially for plates under viscoelastic foundation.

The dynamic response of a plate of infinite extent on a viscous Winkler foundation subjected to movingtandem-axle loads with amplitude variation was investigated by Kim and McCullough (2003), based onthe classical plate theory. Liang et al. (2014) studied the transient response of functionally graded annularplate under a two parameter viscoelastic foundation based on the three-dimensional theory ofelasticity and using the differential quadrature method, state space method and Laplace transform.Vibration reduction of composite plates by piezoelectric patches was investigated by Bargh and Sadra (2014), based on the classical plate theory and using a modified artificial bee colony algorithm. Basedon the sinusoidal shear deformation theory and by using the Navier's and meshless methods, free vibrationanalyses of viscoelastic double-bonded polymeric nanocomposite plates are investigated byMohammadimehr et al. (2015). Forced vibration analysis of a single layer viscoelastic graphene sheetresting on viscoelastic medium was performed by Hosseini Hashemi et al. (2015). The governing equationis derived using Hamilton’s principle based on the classical plate theory and viscoelastic KelvinVoigt model. Luong-Van et al. (2014) investigated the dynamic responses of sandwich and laminatedcomposite plates resting on viscoelastic foundation based on the first-order shear deformation theoryand finite element method. Arani et al. carried out the free vibration analysis of the coupled system ofdouble-layered rectangular (2012) and annular (2014) graphene sheets embedded in a visco-Pasternak foundation was by. The motion equations were derived based on the Hamilton’s principleand first-order shear deformation theory. The differential quadrature method was applied to obtain thefrequency ratio for various boundary conditions. Lepoittevin and Kress (2011) proposed a method topredict resonance frequencies and modal loss factors of bare and damped samples, using constrainedlayer damping treatment, under free-free boundary condition. Karim and Chen (2012) investigated thesurface damping effects of anchored constrained viscoelastic layers on the flexural response of simplysupported Euler beams or plate strips under base excitations. In order to reduce the structural vibrationsof a mechanical system used in aerospace, a new sketch of eddy current damper (ECD) was proposedby Pan et al. (2016). Zenkour (2016) applied the classical plate theory and Fourier transformmethod for transient thermal analysis of a single layered graphene sheet embedded in viscoelastic medium.The governing dynamical equation was obtained and solved for simply-supported grapheme sheet. By using the Rayleigh-Ritz method, Plattenburg et al. (2016) proposed a new analytical modelfor a thin cylindrical shell that utilizes a homogeneous cardboard liner to increase modal damping. Theproposed theory, incorporated material structural damping along with frequency-dependent viscousand Coulomb interfacial damping formulations for the shell-liner interaction. By using the Galerkinweighted residual method and the classical plate theory, Kiasat et al. (2014) analyzed the transientresponse of viscoelastic beams and plates on viscoelastic medium. Based on Kelvin-Voigt model, freedamped vibration analysis of plates with hybrid material foundation viscoelasticity was performed by Zamani et al. (2017). Free vibration analysis of a simply supported viscoelastic orthotropic nanoplates resting on viscoelastic medium was studied by Pouresmaeeli et al. (2013), using the classical plate theoryand the Navier method. Zhang et al. (2016) analyzed the free transverse vibration of double walledcarbon nanotubes embedded in viscoelastic medium by using the Euler-Bernoulli beam theory. Luo etal. (2016) presented a closed-form solution to a viscoelastically supported Timoshenko beam under aharmonic line load. The differential governing equations were converted into algebraic equations byassuming the deflection and rotation of the beam in harmonic forms with respect to time and space.Some researchers analyzed beams resting on viscoelastic foundations subjected to moving loads. Thevibration instability analysis of an oscillator moving along a Timoshenko beam was performed byMetrikine and Verichev (2001) and Mazilu et al. (2012). The inHluence of a nonlinear foundation on thedynamic response of a periodically supported beam was investigated by Hoang et al. (2016), based on the Euler-Bernoulli beam theory. Ding et al. (2013) investigated the dynamic response of infinitebeams supported by nonlinear viscoelastic foundations. The differential equations were obtained byemploying the Timoshenko beam theory and were solved based on the Adomian decomposition methodand a perturbation method in conjunction with complex Fourier transformation. The dynamic response of finite Timoshenko beamsresting on a six parameter foundation was studied by Yang et al. (2013).

Most of the existing researches were performed based on the equivalent single layer theories, whereusing the equivalent single-layer theories for analysis of sandwich plates may be inaccurate or erroneousin most circumstances. Various theories are presented for analysis of sandwich structure Carrera and Brischetto (2008).Many researchers investigated the laminated composite and sandwich plates based on the sandwichand multilayered structures theories. Thermoelastic bending of functionally graded sandwich plates were analyzed by Houari et al (2013) based on a new higher order shear and normal deformation theory and by Tounsi et al. (2013) based on A refined trigonometric shear deformation theory.Based on layerwise formulation, Alipour (2016a) presented anovel economical analytical method for bending and stress analysis of functionally graded sandwichcircular plates subjected to various loads with general elastic edge conditions. Alipour (2016b) analyzed the effects of elastically restrained edges on FG sandwich annular plates by using a novel solutionprocedure and layerwise method. Alipour (2018) examined the dynamic response of sandwich plate with viscoelastic boundary support. Bending analysis of laminated compositeplates was performed based on a predictor-corrector approach and the zig-zag theory by Lee andCao (1996). Alipour and Shariyat (2015) employed a zigzag-elasticity plate theory for bending andstress analysis of circular/annular sandwich plates with orthotropic composite face sheets and auxeticcores. Free vibration analysis of circular and annular composite sandwich plates with auxetic cores wasperformed by Shariyat and Alipour (2017a).

Although various studies have been presented for dynamic analysis of single layer, laminated compositeand sandwich plates, most of the existing studies were performed for plates subjected to load caseswith constant or specific amplitude. In this study dynamic response of sandwich annular plates withfunctionally graded polar orthotropic face sheets subjected to various dynamic loads are presented, asfirst time. Based on the combination of the Hinite Taylor’s transform (Alipour (2016c), Shariyat and Alipour (2015, 2017b)) and the fourth-order Runge-Kutta procedure, a semi-analytical solution procedureis developed for solution of the relatively complicated second order coupled partial differentialequations. It is worth to be mentioned that modern structures are designed based on the use of compositematerials which are actually anisotropic and inhomogeneous (Peng and Li (2012) and Nie andBatra (2010)). In the presented analysis, Young’s modulus in the radial and circumferential direction, shear modulus and density of each face sheet may be varied continuously in the radial direction. Thesandwich plate may be resting on viscous, elastic or viscoelastic foundation. To describe more accuratelyresponse of sandwich plates, the governing differential equations of motion are derived based on theminimum total potential energy principle by using the layerwise theory. The dynamic responses ofsandwich plate are examined for various stiffness and damping coefficients of the foundation, transientloads and boundary conditions. Since no existing work has been performed on the dynamic analysis ofFG polar orthotropic sandwich plates, accuracy and efficiency of the presented analysis are verified bycomparing the obtained results with results of the three-dimensional theory of elasticity extracted fromthe ABAQUS software for some special cases. The comparisons show that there is a very good agreementbetween present results and results of the three-dimensional theory of elasticity.

2 DESCRIPTION OF THE MATERIAL PROPERTIES, DISPLACEMENT, STRAIN AND STRESS FIELDS

As shown in Figure 1, a three-layer sandwich annular plate with functionally graded polar orthotropic face sheets resting on the viscoelastic substrate which is modeled as Kelvin-Voigt foundation is considered.

Figure 1: Schematic of the sandwich annular plate on viscoelastic substrate. 

Viscoelastic substrate is modeled as a continuously distributed medium with stiffness K w and damping coefficient C t .

Face sheets may be fabricated from functionally graded polar orthotropic materials. Young’s modulus in the radial and circumferential direction, shear modulus and density of each face sheet may be varied continuously in the radial direction according to a power-law fraction, as follows:

{Er(k)(r)Eθ(k)(r)Grz(k)(r)}={E¯r(k)E¯θ(k)G¯rz(k)}[1+α(k)(rro)β(k)] ,ρ(k)(r)=ρ¯(k)[1+γ(k)(rro)η(k)] k=1,3 (1)

vrθEr=vθrEθ (2)

where α (k) , β (k) ,γ (k) and η (k) (k = 1, 3) are inhomogeneity parameters for top (k=1) and bottom (k=3) face sheets.

The three transverse local coordinates ξ (1), ξ (2) and ξ (3) are represented for top, core, and bottom layers, respectively. The local coordinates are measured from the mid plane of the corresponding layer and are positive upward.

Based on the layerwise theory with the piecewise linear local components, after some manipulations and imposing the continuity conditions of the displacement components at the interfaces between face sheets and core, the displacement field of the layers may be written as:

u1=u0+(ξ(1)+h12)ψr(1)+h22ψr(2)u2=u0+ξ(2)ψr(2) u3=u0+(ξ(3)h32)ψr(3)h22ψr(2), hi2 ξ(i)hi2 i=1, 2, 3 (3)

Where u 0 is the radial displacement component of the mid plane of the core and ψr(i) are the local rotation of the layers of the plate.

Cauchy’s strain-displacement relations are:

εr=u,r εθ=ur εrz=u,z+w,r (4)

where the symbol “,” stands for the partial derivative. Based on Hooke's generalized stress-strain law:

σr(k)=Er(k)(r)1vrθ(k)vθr(k)εr(k)+vθr(k)Er(k)(r)1vrθ(k)vθr(k)εθ(k)σθ(k)=vθr(k)Er(k)(r)1vrθ(k)vθr(k)εr(k)+Eθ(k)(r)1vrθ(k)vθr(k)εθ(k)τrz(k)=Grz(k)(r) γrz(k) (5)

where the E, G and υ symbols denote Young’s modulus, shear modulus, and Poisson’s ratio, respectively.

Based on Eqs.(1) to (5), the stress-displacement relations of face sheets and core may be written as:

σr(1)=E¯r(1)[1+α(1)(rro)β(1)]1vrθ(1)vθr(1)[u0,r+vθr(1)ru0+(ξ(1)+h12)(ψr,r(1)+vθr(1)rψr(1))+h22(ψr,r(2)+vθr(1)rψr(2))]σθ(1)=E¯θ(1)[1+α(1)(rro)β(1)]1vrθ(1)vθr(1)[vrθ(1)u0,r+1ru0+(ξ(1)+h12)(vrθ(1)ψr,r(1)+1rψr(1))+h22(vrθ(1)ψr,r(2)+1rψr(2))]τrz(1)=G¯rz(1)[1+α(1)(rro)β(1)] (ψr(1)+w,r)σr(2)=E(2)1v(2)2[u0,r+v(2)ru0+ξ(2)(ψr,r(2)+v(2)rψr(2))]σθ(2)=E(2)1v(2)2[v(2)u0,r+1ru0+ξ(2)(v(2)ψr,r(2)+1rψr(2))]τrz(2)=G(2)(ψr(2)+w,r)σr(3)=E¯r(3)[1+α(3)(rro)β(3)]1vrθ(3)vθr(3)[u0,r+vθr(3)ru0+(ξ(3)h32)(ψr,r(3)+vθr(3)rψr(1))h22(ψr,r(2)+vθr(3)rψr(2))]σθ(3)=E¯θ(3)[1+α(3)(rro)β(3)]1vrθ(1)vθr(1)[vrθ(3)u0,r+1ru0+(ξ(3)h32)(vrθ(3)ψr,r(3)+1rψr(3))h22(vrθ(3)ψr,r(2)+1rψr(2))]τrz(3)=G¯rz(1)[1+α(3)(rro)β(3)] (ψr(3)+w,r) (6)

3 THE GOVERNING EQUATIONS OF MOTION OF THE SANDWICH PLATE WITH FG POLAR ORTHOTROPIC FACE SHEETS

The governing equations of motion of the sandwich annular plate with functionally graded polar orthotropic face sheets on viscoelastic substrate are derived based on using the minimum total potential energy principle:

δΠ=δU+δKδW=0, (7)

where δU, δK and δW are increments of the strain energy, kinetic energy and work done by external applied loads, respectively:

δU=V(σrδεr+σθδεθ+τrzδγrz)dVδK=Vρ(u¨δu+w¨δw)dVδW=APδw dA (8)

For sandwich plate subjected to transient load (q(t)) resting on viscoelastic substrate based on the Kelvin-Voigt model, the distributed transverse load can be defined as follows.

P=q(t)Kww(r,t)Ctw˙(r,t) (9)

In which q(t) is the external dynamic loads, K w and C t are the spring and damper constant of viscoelastic foundation.

Substituting Eq. (8) into Eq. (7) and by using Eqs. (3) and (6), the following five systems of second order coupled partial differential equations are resulted after some manipulations that are not included here for brevity:

k=03(Nr(k)Nθ(k)r+Nr,r(k))=(I0(1)+I0(2)+I0(3))u¨0+h12I0(1)ψ¨r(1)+h22(I0(1)I0(3))ψ¨r(2)h32I0(3)ψ¨r(3) (10)

(1r+r)(h12Nr(1)+Mr(1))1r(h12Nθ(1)+Mθ(1))Qrz(1)=h12I0(1)(u¨0+h22ψ¨r(2))+(I2(1)ψ¨r(1)+h124I0(1)ψ¨r(1)) (11)

(1r+r)(h22Nr(1)+Mr(2)h22Nr,r(3))1rh22(Nθ(1)Nθ(3))Qrz(2)=h22I0(1)(u¨0+h12ψ¨r(1))h22I0(3)(u¨0h32ψ¨r(3))+(h224I0(1)+I2(2)+h224I0(3))ψ¨r(2) (12)

(1r+r)(h32Nr(3)+Mr(3))Qrz(3)=h32I0(3)(u¨0h32ψ¨r(3))+h2h34I0(3)ψ¨r(2)+I2(3)ψ¨r(3) (13)

k=03(1r+r)Qr(k)=qKwwCtw˙+(I0(1)+I0(2)+I0(3))w¨ (14)

where the stress resultants M, N, Q and the higher-order inertias are defined as:

{Ni(k)Mi(k)}=hk2hk2σi(k){1ξ(k)}dξ(k), Qr(k)=hk2hk2τrz(k)dξ(k), k=1, 2, 3 i=r, θ (15)

Ij(k)=hk2hk2ρ(k)(r)ξ(k)jdξ(k), Ij(k)=[1+γ(k)(rro)η(k)]I¯j(k), k=1,3I¯j(i)=hi2hi2ρ¯(i)ξ(i)jdξ(i) i=1,2,3 j=0,2 (16)

Based on Eq. (6), the stress resultants (Eq. (15)) may be rewritten in the following form:

{Ni(1)=Ari(1)(u0,r+h12ψr,r(1)+h22ψr,r(2))+1r Aiθ(1)( u0+h12 ψr(1)+h22ψr(2)),Ni(2)=Ari(2)u0,r+1r Aiθ(2)u0,Ni(3)=Ari(3)(u0,rh32ψr,r(3)h22ψr,r(2))+1r Aiθ(3)(u0h32ψr(3)h22ψr(2)), i=r, θ (17)

Mi(k)=Dri(k)ψr,r(k)+1r Diθ(k)ψr(k), k=1, 2, 3, i=r,θ (18)

Qr(k)=Arz(k)(ψr(k)+w,r), k=1,2,3 (19)

where

{Aii(k)Dii(k)}= h k2h k2Ei(k)(r)1vrθ(k)vθr(k){1ξ(k)2}dξ(k), {Arθ(k)Drθ(k)}= h k2h k2vθr(k)Er(k)(r)1vrθ(k)vθr(k){1ξ(k)2}dξ(k),Arz(k)= hk2h k2Grz(k)(r) dξ(k), k=1,2,3 i=r,θ (20)

{Aij(k)Dij(k)}={A¯ij(k)D¯ij(k)}[1+α(k)(rro)β(k)] , Arz(k)=A¯rz(k)[1+α(k)(rro)β(k)], i,j=r,θ k=1,3 (21)

{A¯ii(k)D¯ii(k)}=hk2hk2E¯i(k)1vrθ(k)vθr(k){1ξ(k)2}dξ(k), {A¯rθ(k)D¯rθ(k)}=hk2hk2vθr(k)E¯r(k)1vrθ(k)vθr(k){1ξ(k)2}dξ(k), A¯rz(k)= hk2hk2G¯rz(k) dξ(k), i=r, θ k=1, 3 (22)

Based on Eqs. (15) to (22), the governing equations (10) to (14) may be rewritten as:

[A¯rr(1)+Arr(2)+A¯rr(3)+A¯rr(1)α(1)(rro)β(1)+A¯rr(3)α(3)(rro)β(3)](u0,rr+u0,rr)[A¯θθ(1)+Aθθ(2)+A¯θθ(3)+A¯θθ(1)α(1)(rro)β(1)+A¯θθ(3)α(3)(rro)β(3)]u0r2+h12[1+α(1)(rro)β(1)][A¯rr(1)(ψr,rr(1)+ ψr,r(1)r)A¯θθ(1)ψr(1)r2]+h22[A¯rr(1)A¯rr(3)+A¯rr(1)α(1)(rro)β(1)A¯rr(3)α(3)(rro)β(3)](ψr,rr(2)+ψr,r(2)r)h22[A¯θθ(1)A¯θθ(3)+A¯θθ(1)α(1)(rro)β(1)A¯θθ(3)α(3)(rro)β(3)]ψr(2)r2h32[1+α(3)(rro)β(3)][A¯rr(3)(ψr,rr(3)+ ψr,r(3)r)A¯θθ(3)ψr(3)r2]+α(1)β(1)(rro)β(1)1[(A¯rr(1)r+A¯rθ(1)r) ( u0+h12 ψr(1)+h22ψr(2))]+α(3)β(3)(rro)β(3)1[(A¯rr(3)r+A¯rθ(3)r) (u0h32ψr(3)h22ψr(2))]=(I¯0(1)+I¯0(2)+I¯0(3))u¨0+γ(1)(rro)η(1)I¯0(1)(u¨0+h12ψ¨r(1)+h22ψ¨r(2))+h12I¯0(1)ψ¨r(1)+γ(3)(rro)η(3)I¯0(3)(u¨0h22ψ¨r(2)h32ψ¨r(3))+h22(I¯0(1)I¯0(3))ψ¨r(2)h32I¯0(3)ψ¨r(3) (23)

h12[1+α(1)(rro)β(1)][A¯rr(1)(u0,rr+u0,rr)A¯θθ(1)u0r2]+(1+α(1)(rro)β(1))[(h124A¯rr(1)+D¯rr(1))(ψr,rr(1)+ ψr,r(1)r)(h124A¯θθ(1)+D¯θθ(1))ψr(1)r2]+h1h24[1+α(1)(rro)β(1)][A¯rr(1)(ψr,rr(2)+ ψr,r(2)r )A¯θθ(1)ψr(2)r2]A¯rz(1)[1+α(1)(rro)β(1)](ψr(1)+w,r)+α(1)β(1)(rro)β(1)1(D¯rr(1)ψr,r(1)+D¯rθ(1)rψr(1))+h12α(1)β(1)(rro)β(1)1[A¯rr(1)(u0,r+h12ψr,r(1)+h22ψr,r(2))+A¯rθ(1)r( u0+h12 ψr(1)+h22ψr(2))]=h12I¯0(1)(u¨0+h12ψ¨r(1)+h22ψ¨r(2))+h12I¯0(1)γ(1)(rro)η(1)(u¨0+h12ψ¨r(1)+h22ψ¨r(2))+I¯2(1)ψ¨r(1)+I¯2(1)γ(1)(rro)η(1)ψ¨r(1) (24)

h22[A¯rr(1)A¯rr(3)+A¯rr(1)α(1)(rro)β(1)A¯rr(3)α(3)(rro)β(3)](u0,rr+u0,rr)h22[A¯θθ(1)A¯θθ(3)+A¯θθ(1)α(1)(rro)β(1)A¯θθ(3)α(3)(rro)β(3)]u0r2+h1h24A¯rr(1)[1+α(1)(rro)β(1)](ψr,rr(1)+ ψr,r(1)r)h1h24A¯θθ(1)[1+α(1)(rro)β(1)]ψr(1)r2+[h224A¯rr(1)+Drr(2)+h224A¯rr(3)+h224A¯rr(1)α(1)(rro)β(1)+h224A¯rr(3)α(3)(rro)β(3)](ψr,rr(2)+ψr,r(2)r)[h224A¯θθ(1)+Dθθ(2)+h224A¯θθ(3)+h224A¯θθ(1)α(1)(rro)β(1)+h224A¯θθ(3)α(3)(rro)β(3)]ψr(2)r2+h2h34A¯rr(3)[1+α(3)(rro)β(3)](ψr,rr(3)+ ψr,r(3)r) h2h34A¯θθ(3)[1+α(3)(rro)β(3)]ψr(3)r2Arz(2)(ψr(2)+w,r)+h22α(1)β(1)(rro)β(1)1[A¯rr(1)(u0,r+h12ψr,r(1)+h22ψr,r(2))+A¯rθ(1)r( u0+h12 ψr(1)+h22ψr(2))]h22α(3)β(3)(rro)β(3)1[A¯rr(3)(u0,rh32ψr,r(3)h22ψr,r(2))+A¯rθ(3)r (u0h32ψr(3)h22ψr(2))]=h22I¯0(1)(u¨0+h12ψ¨r(1)+h22ψ¨r(2))+I¯2(2)ψ¨r(2)h22I¯0(3)(u¨0h22ψ¨r(2)h32ψ¨r(3))+h22I¯0(1)γ(1)(rro)η(1)(u¨0+h12ψ¨r(1)+h22ψ¨r(2))h22I¯0(3)γ(3)(rro)η(3)(u¨0h22ψ¨r(2)h32ψ¨r(3)) (25)

h32A¯rr(3)[1+α(3)(rro)β(3)](u0,rr+u0,rr)+h32A¯θθ(3)[1+α(3)(rro)β(3)]u0r2+h2h34[1+α(3)(rro)β(3)][A¯rr(3)(ψr,rr(2)+ψr,r(2)r)A¯θθ(3)ψr(2)r2]+(D¯rr(3)+h324A¯rr(3))[1+α(3)(rro)β(3)](ψr,rr(3)+ ψr,r(3)r)(D¯θθ(3)+h324A¯θθ(3))[1+α(3)(rro)β(3)]ψr(3)r2A¯rz(3)[1+α(3)(rro)β(3)](ψr(3)+w,r)+α(3)β(3)(rro)β(3)1(D¯rr(3)ψr,r(3)+D¯rθ(3)rψr(3))h32α(3)β(3)(rro)β(3)1[A¯rr(3)(u0,rh32ψr,r(3)h22ψr,r(2))+A¯rθ(3)r (u0h32ψr(3)h22ψr(2))]=h32I¯0(3)(u¨0h22ψ¨r(2)h32ψ¨r(3))h32I¯0(3)γ(3)(rro)η(3)(u¨0h22ψ¨r(2)h32ψ¨r(3))+I¯2(3)ψ¨r(3)+I¯2(3)γ(3)(rro)η(3)ψ¨r(3) (26)

[A¯rz(1)+Arz(2)+A¯rz(3)+A¯rz(1)α(1)(rro)β(1)+A¯rz(3)α(3)(rro)β(3)](w,rr+w,rr)+A¯rz(1)[1+α(1)(rro)β(1)](ψr,r(1)+ψr(1)r)+A44(2)(ψr,r(2)+ψr(2)r)+A¯rz(3)[1+α(3)(rro)β(3)](ψr,r(3)+ψr(3)r)+A¯rz(1)α(1)β(1)(rro)β(1)1(ψr(1)+w,r)+A¯rz(3)α(3)β(3)(rro)β(3)1(ψr(3)+w,r)=q+Kww+Ctw˙+(I¯0(1)+I¯0(2)+I¯0(3))w¨+γ(1)(rro)η(1)I¯0(1)w¨+γ(3)(rro)η(3)I¯0(3)w¨ (27)

For solution of the governing equations (23) to (27), various combinations of the edge conditions may be employed.

  • I. Clamped immovable edge:

{u0 =0ψr(1)=0ψr(2)=0ψr(3)=0w =0 (28)

  • II. Simply-supported movable edge:

{Nr(1)+Nr(2)+Nr(3)=0h12Nr(1)+Mr(1)=0h22Nr(1)+Mr(2)h22Nr(3)=0 h32Nr(3+Mr(3)=0w =0 (29)

  • III. Free edge:

{Nr(1)+Nr(2)+Nr(3)=0h12Nr(1)+Mr(1)=0h22Nr(1)+Mr(2)h22Nr(3)=0 h32Nr(3+Mr(3)=0Qr(1)+Qr(2)+Qr(3)=0 (30)

Also, initial conditions of the sandwich plate are:

u(r,0)=u˙(r,0)=0, ψr(1)(r,0)=ψ˙r(1)(r,0)=0, ψr(2)(r,0)=ψ˙r(2)(r,0)=0, ψr(3)(r,0)=ψ˙r(3)(r,0)=0, w(r,0)=w˙(r,0)=0. (31)

4 SEMI-ANALYTICAL SOLUTION FOR DYNAMIC ANALYSIS OF FG POLAR ORTHOTROPIC SANDWICH PLATE

For solution of the relatively complicated five systems of second order coupled partial differential equations, the semi-analytical solution is developed based on the finite Taylor’s transform and the fourth-order Runge-Kutta procedure.

Based on Taylor’s expansion, the unknown displacement functions can be expressed by the following power series:

{u0(r,t=y Δt)u˙0(r,t=y Δt)u¨0(r,t=y Δt)}=x=0{Ux,yU¯x,yU˜x,y}(rro)x, {ψr(1)(r,t=y Δt)ψ˙r(1)(r,t=y Δt)ψ¨r(1)(r,t=y Δt)}=x=0{Φx,y(1)Φ¯x,y(1)Φ˜x,y(1)}(rro)x,{ψr(2)(r,t=y Δt)ψ˙r(2)(r,t=y Δt)ψ¨r(2)(r,t=y Δt)}=x=0{Φx,y(2)Φ¯x,y(2)Φ˜x,y(2)}(rro)x, {ψr(3)(r,t=y Δt)ψ˙r(3)(r,t=y Δt)ψ¨r(3)(r,t=y Δt)}=x=0{Φx,y(3)Φ¯x,y(3)Φ˜x,y(3)}(rro)x, {w(r,t=y Δt)w˙(r,t=y Δt)w¨(r,t=y Δt)}=x=0{Wx,yW¯x,yW˜x,y}(rro)x (32)

Where ΔT is the time step, y is the time step counter and Ux,y, U¯x,y, U˜x,y, Φx,y(1), Φ¯x,y(1), Φ˜x,y(1), Φx,y(2), Φ¯x,y(2), Φ˜x,y(2), Φx,y(3), Φ¯x,y(3), Φ˜x,y(3), Wx,y, W¯x,y and W˜x,y are the coefficients of series in each time step.

On the other hands, Taylor transform of the functions 1/r and 1/r 2 may be expressed by the following power series whose center is located at r=ro.

1r=s=0(1ro)s+1(rro)s, 1r2=s=0(s+1)(1ro)s+2(rro)s (33)

In practical applications, the transformed form of functions must be expressed by means of finite series.

The transformed form of the governing equations may be obtained by substituting Eqs. (32) and (33) into the governing Eqs. (23) to (27) and performing some manipulations.

x=0X[(A¯rr(1)+Arr(2)+A¯rr(3))[(x+2)(x+1)Ux+2,ys=0xχs+1(xs+1)Uxs+1,y]A¯rr(1)α(1)[(xβ(1)+2)(xβ(1)+1)Uxβ(1)+2,ys=0xβ(1)χs+1(xβ(1)s+1)Uxβ(1)s+1,y]+A¯rr(3)α(3)[(xβ(3)+2)(xβ(3)+1)Uxβ(3)+2,ys=0xβ(3)χs+1(xβ(3)s+1)Uxβ(3)s+1,y]+(A¯θθ(1)+Aθθ(2)+A¯θθ(3))s=0x(s+1)χs+2Uxs,yA¯θθ(1)α(1)s=0xβ(1)(s+1)χs+2Uxβ(1)s,yA¯θθ(3)α(3)s=0xβ(3)(s+1)χs+2Uxβ(3)s,y+h12A¯rr(1)[(x+2)(x+1)Φx+2,y(1)s=0xχs+1(xs+1)Φxs+1,y(1)]+h12A¯rr(1)α(1)[(xβ(1)+2)(xβ(1)+1)Φxβ(1)+2,y(1)s=0xβ(1)χs+1(xβ(1)s+1)Φxβ(1)s+1,y(1)]h12A¯θθ(1)s=0x(s+1)χs+2Φxs,y(1)h12A¯θθ(1)α(1)s=0xβ(1)(s+1)χs+2Φxβ(1)s,y(1)+h22(A¯rr(1)A¯rr(3))[(x+2)(x+1)Φx+2,y(2)s=0xχs+1(xs+1)Φxs+1,y(2)]+h22A¯rr(1)α(1)[(xβ(1)+2)(xβ(1)+1)Φxβ(1)+2,y(2)s=0xβ(1)χs+1(xβ(1)s+1)Φxβ(1)s+1,y(2)]h22A¯rr(3)α(3)[(xβ(3)+2)(xβ(3)+1)Φxβ(3)+2,y(2)s=0xβ(3)χs+1(xβ(3)s+1)Φxβ(3)s+1,y(2)]h22(A¯θθ(1)A¯θθ(3))s=0x(s+1)χs+2Φxs,y(2)h22A¯θθ(1)α(1)s=0xβ(1)(s+1)χs+2Φxβ(1)s,y(2)+h22A¯θθ(3)α(3)s=0xβ(3)(s+1)χs+2Φxβ(3)s,y(2)h32A¯rr(3)[(x+2)(x+1)Φx+2,y(3)s=0xχs+1(xs+1)Φxs+1,y(3)]h32A¯rr(3)α(3)[(xβ(3)+2)(xβ(3)+1)Φxβ(3)+2,y(3)s=0xβ(3)χs+1(xβ(3)s+1)Φxβ(3)s+1,y(3)]+h32A¯θθ(3)s=0x(s+1)χs+2Φxs,y(3)+α(1)β(1)A¯rr(1)(xβ(1)+2)(Uxβ(1)+2,y+h12Φxβ(1)+2(1)+h22Φxβ(1)+2,y(2))+h32A¯θθ(3)α(3)s=0xβ(3)(s+1)χs+2Φxβ(3)s,y(3)α(1)β(1)A¯rθ(1)s=0xβ(1)+1χs+1 ( Uxsβ(1)+1+h12Φxsβ(1)+1,y(1)+h22Φxsβ(1)+1,y(2))+α(3)β(3)A¯rr(3)(xβ(3)+2)(Uxβ(3)+2,yh32Φxβ(3)+2,y(3)h22Φxβ(3)+2,y(2))α(3)β(3)A¯rθ(3) s=0xβ(3)+1χs+1(Uxsβ(3)+1,yh32Φxsβ(3)+1,y(3)h22Φxsβ(3)+1,y(2))(I¯0(1)+I¯0(2)+I¯0(3))U˜x,yγ(1)I¯0(1)U˜xη(1),yγ(2)I¯0(2)U˜xη(2),yγ(3)I¯0(3)U˜xη(3),yh12I¯0(1)Φ˜x,y(1)h22(I¯0(1)I¯0(3))Φ˜x,y(2)+h32I¯0(3)Φ˜x,y(3)γ(1)I¯0(1)(h12Φ˜xη(1),y(1)+h22Φ˜xη(1),y(2))+γ(3)I¯0(3)(h22Φ˜xη(3),y(2)+h32Φ˜xη(3),y(3))](rro)x=0 (34)

x=0X{h12A¯rr(1)[(x+2)(x+1)Ux+2,ys=0xχs+1(xs+1)Uxs+1,y]h12A¯θθ(1)s=0x(s+1)χs+2Uxs,y+h12A¯rr(1)α(1)[(xβ(1)+2)(xβ(1)+1)Uxβ(1)+2,ys=0xβ(1)χs+1(xβ(1)s+1)Uxβ(1)s+1,y]h12A¯θθ(1)α(1)s=0xβ(1)(s+1)χs+2Uxβ(1)s,y+(h124A¯rr(1)+D¯rr(1))[(x+2)(x+1)Φx+2,y(1)s=0xχs+1(xs+1)Φxs+1,y(1)]+(h124A¯rr(1)+D¯rr(1))α(1)[(xβ(1)+2)(xβ(1)+1)Φxβ(1)+2,y(1)s=0xβ(1)χs+1(xβ(1)s+1)Φxβ(1)s+1,y(1)](h124A¯θθ(1)+D¯θθ(1))s=0x(s+1)χs+2Φxs,y(1)(h124A¯θθ(1)+D¯θθ(1))α(1)s=0xβ(1)(s+1)χs+2Φxβ(1)s,y(1)+h1h24A¯rr(1)[(x+2)(x+1)Φx+2,y(2)s=0xχs+1(xs+1)Φxs+1,y(2)]α(1)β(1)D¯rθ(1)s=0xβ(1)+1χs+1Φxsβ(1)+1,y(1)+h1h24α(1)A¯rr(1)[(xβ(1)+2)(xβ(1)+1)Φxβ(1)+2,y(2)s=0xβ(1)χs+1(xβ(1)s+1)Φxβ(1)s+1,y(2)]h1h24A¯θθ(1)s=0x(s+1)χs+2Φxs,y(2)h1h24α(1)A¯θθ(1)s=0xβ(1)(s+1)χs+2Φxβ(1)s,y(2)A¯rz(1)[Φx,y(1)+(x+1)Wx+1,y]A¯rz(1)α(1)[Φxβ(1),y(1)+(xβ(1)+1)Wxβ(1)+1,y]+α(1)β(1)D¯rr(1)(xβ(1)+2)Φxβ(1)+2,y(1)+h12α(1)β(1)A¯rr(1)(xβ(1)+2)(Uxβ(1)+2,y+h12Φxβ(1)+2,y(1)+h22Φxβ(1)+2,y(2))h12α(1)β(1)A¯rθ(1)s=0xβ(1)+1χs+1( Uxsβ(1)+1,y+h12 Φxsβ(1)+1,y(1)+h22Φxsβ(1)+1,y(2))h12I¯0(1)(U˜x,y+h12Φ˜x,y(1)+h22Φ˜x,y(2))h12I¯0(1)γ(1)(U˜xη(1),y+h12Φ˜xη(1),y(1)+h22Φ˜xη(1),y(2))I¯2(1)Φ˜x,y(1)I¯2(1)γ(1)Φ˜xη(1),y(1)}(rr0)x=0 (35)

x=0X{h22(A¯rr(1)A¯rr(3))[(x+2)(x+1)Ux+2,ys=0xχs+1(xs+1)Uxs+1,y]+h22A¯rr(1)α(1)[(xβ(1)+2)(xβ(1)+1)Uxβ(1)+2,ys=0xβ(1)χs+1(xβ(1)s+1)Uxβ(1)s+1,y]h22A¯rr(3)α(3)[(xβ(3)+2)(xβ(3)+1)Uxβ(3)+2,ys=0xβ(3)χs+1(xβ(3)s+1))Uxβ(3)s+1,y]h22(A¯θθ(1)A¯θθ(3))s=0x(s+1)χs+2Uxs,yh22A¯θθ(1)α(1)s=0xβ(1)(s+1)χs+2Uxβ(1)s,y+h22A¯θθ(3)α(3)s=0xβ(1)(s+1)χs+2Uxβ(3)s,y+h1h24A¯rr(1)[(x+2)(x+1)Φx+2,y(1)s=0xχs+1(xs+1)Φxs+1,y(1)]+h1h24A¯rr(1)α(1)[(xβ(1)+2)(xβ(1)+1)Φxβ(1)+2,y(1)s=0xβ(1)χs+1(xβ(1)s+1)Φxβ(1)s+1,y(1)]h1h24A¯θθ(1)s=0x(s+1)χs+2Φxs,y(1)h1h24A¯θθ(1)α(1)s=0xβ(1)(s+1)χs+2Φxβ(1)s,y(1)+(h224A¯rr(1)+Drr(2)+h224A¯rr(3))[(x+2)(x+1)Φx+2,y(2)s=0xχs+1(xs+1)Φxs+1,y(2)]+h224A¯rr(1)α(1)[(xβ(1)+2)(xβ(1)+1)Φxβ(1)+2,y(2)s=0xβ(1)χs+1(xβ(1)s+1)Φxβ(1)s+1,y(2)]+h224A¯rr(3)α(3)[(xβ(3)+2)(xβ(3)+1)Φxβ(3)+2,y(2)s=0xβ(3)χs+1(xβ(3)s+1)Φxβ(3)s+1,y(2)](h224A¯θθ(1)+Dθθ(2)+h224A¯θθ(3))s=0x(s+1)χs+2Φxs,y(2)h224A¯θθ(1)α(1)s=0xβ(1)(s+1)χs+2Φxβ(1)s,y(2)h224A¯θθ(3)α(3)s=0xβ(3)(s+1)χs+2Φxβ(3)s,y(2)+h2h34A¯rr(3)[(x+2)(x+1)Φx+2,y(3)s=0xχs+1(xs+1)Φxs+1,y(3)]+h2h34A¯rr(3)α(3)[(xβ(3)+2)(xβ(3)+1)Φxβ(3)+2,y(3)s=0xβ(3)χs+1(xβ(3)s+1)Φxβ(3)s+1,y(3)]h2h34A¯θθ(3)s=0x(s+1)χs+2Φxs,y(3)h2h34A¯θθ(3)α(3)s=0xβ(3)(s+1)χs+2Φxβ(3)s,y(3)Arz(2)[Φx(2)+(x+1)Wx+1,y]+h22α(1)β(1)A¯rr(1)(xβ(1)+2)(Uxβ(1)+2,y+h12Φxβ(1)+2,y(1)+h22Φxβ(1)+2,y(2))h22α(1)β(1)A¯rθ(1)s=0χs+1( Uxsβ(1)+1,y+h12 Φxsβ(1)+1,y(1)+h22Φxsβ(1)+1,y(2))h22α(3)β(3)A¯rr(3)(xβ(3)+2)(Uxβ(3)+2,yh32Φxβ(3)+2,y(3)h22Φxβ(3)+2,y(2))+h22α(3)β(3)A¯rθ(3)s=0χs+1 (Uxsβ(3)+1,yh32Φxsβ(3)+1,y(3)h22Φxsβ(3)+1,y(2))h22I¯0(1)(U˜x,y+h12Φ˜x,y(1)+h22Φ˜x,y(2))I¯2(2)Φ˜x,y(2)+h22I¯0(3)(U˜x,yh22Φ˜x,y(2)h32Φ˜x,y(3))h22I¯0(1)γ(1)(U˜xη(1),y+h12Φ˜xη(1),y(1)+h22Φ˜xη(1),y(2))I¯2(2)γ(2)Φ˜xη(2),y(2)+h22I¯0(3)γ(3)(U˜xη(3),yh22Φ˜xη(3),y(2)h32Φ˜xη(3),y(3))}(rr0)x=0 (36)

x=0X{h32A¯rr(3)[(x+2)(x+1)Ux+2,ys=0xχs+1(xs+1)Uxs+1,y]h32A¯rr(3)α(3)[(xβ(3)+2)(xβ(3)+1)Uxβ(3)+2,ys=0xβ(3)χs+1(xβ(3)s+1)Uxβ(3)s+1,y]+h32A¯θθ(3)s=0x(s+1)χs+2Uxs,y+h32A¯θθ(3)α(3)s=0xβ(1)(s+1)χs+2Uxβ(3)s,y+h2h34A¯rr(3)[(x+2)(x+1)Φx+2,y(2)s=0xχs+1(xs+1)Φxs+1,y(2)]+h2h34A¯rr(3)α(3)[(xβ(3)+2)(xβ(3)+1)Φxβ(3)+2,y(2)s=0xβ(3)χs+1(xβ(3)s+1)Φxβ(3)s+1,y(2)]h2h34A¯θθ(3)s=0x(s+1)χs+2Φxs,y(2)h2h34A¯θθ(3)α(3)s=0xβ(3)(s+1)χs+2Φxβ(3)s,y(2)+(D¯rr(3)+h324A¯rr(3))[(x+2)(x+1)Φx+2,y(3)s=0xχs+1(xs+1)Φxs+1,y(3)]+(D¯rr(3)+h324A¯rr(3))α(3)[(xβ(3)+2)(xβ(3)+1)Φxβ(3)+2,y(3)s=0xβ(3)χs+1(xβ(3)s+1)Φxβ(3)s+1,y(3)](D¯θθ(3)+h324A¯θθ(3))s=0x(s+1)χs+2Φxs,y(3)(D¯θθ(3)+h324A¯θθ(3))α(3)s=0xβ(3)(s+1)χs+2Φxβ(3)s,y(3)A¯rz(3)(Φx(3)+(x+1)Wx+1)A¯rz(3)α(3)[Φxβ(3)(3)+(xβ(3)+1)Wxβ(3)+1]+α(3)β(3)D¯rr(3)(xβ(3)+2)Φxβ(3)+2,y(3)α(3)β(3)D¯rθ(3)s=0χs+1Φxsβ(3)+1,y(3)h32α(3)β(3)A¯rr(3)(xβ(3)+2)(Uxβ(3)+2,yh32Φxβ(3)+2,y(3)h22Φxβ(3)+2,y(2))+h32α(3)β(3)A¯rθ(3)s=0χs+1 (Uxsβ(3)+1,yh32Φxsβ(3)+1,y(3)h22Φxsβ(3)+1,y(2))I¯2(3)γ(3)Φ˜xη(3),y(3)I¯2(3)Φ˜x,y(3)+h32I¯0(3)(U˜x,yh32Φ˜x,y(3)h22Φ˜x,y(2))+h32I¯0(3)γ(3)(U˜x,yh32Φ˜xη(3),y(3)h22Φ˜xη(3),y(2))}(rro)x=0 (37)

x=0X{(A¯rz(1)+Arz(2)+A¯rz(3))[(x+2)(x+1)Wx+2,ys=0xχs+1(xs+1)Wxs+1,y]+A¯rz(1)α(1)[(xβ(1)+2)(xβ(1)+1)Wxβ(1)+2,ys=0xβ(1)χs+1(xβ(1)s+1)Wxβ(1)s+1,y]+A¯rz(3)α(3)[(xβ(3)+2)(xβ(3)+1)Wxβ(3)+2,ys=0xβ(3)χs+1(xβ(3)s+1)Wxβ(3)s+1,y]+A¯rz(1)(x+1)Φx+1,y(1)+A¯rz(1)α(1)(xβ(1)+1)Φxβ(1)+1,y(1)A¯rz(1)s=0χs+1 Φxs,y(1)A¯rz(1)α(1)s=0χs+1Φxsβ(1),y(1)+A44(2)[(x+1)Φx+1,y(2)s=0χs+1Φxs,y(2)]+A¯rz(3)(x+1)Φx+1,y(3)+A¯rz(3)α(3)(xβ(3)+1)Φxβ(3)+1,y(3)A¯rz(3)s=0χs+1Φxs,y(3)A¯rz(3)α(3)s=0χs+1Φxβ(3)s,y(3)+A¯rz(1)α(1)β(1)Φxβ(1)+1,y(1)+qδ(x)KwWx,yCtW¯x,y+A¯rz(1)α(1)β(1)(xβ(1)+2)Wxβ(1)+2,y+A¯rz(3)α(3)β(3)Φxβ(3)+1,y(3)+A¯rz(3)α(3)β(3)(xβ(3)+2)Wxβ(3)+2,y(I¯0(1)+I¯0(2)+I¯0(3))W˜x,yγ(1)I¯0(1)W˜xη(1),yγ(2)I¯0(2)W˜xη(2),yγ(3)I¯0(3)W˜xη(3),y}(rro)x=0 (38)

Where δ(x) is Kronecker’s delta function.

The transformed forms of the various boundary conditions at the inner (r=r i ) and outer (r=r o ) edges can be obtained by substituting Eq. (32) into the Eqs. (28-30).

u=0x=0XUx,y(rro)x=0 ψr(1)=0x=0XΦx,y(1)(rro)x=0, ψr(2)=0x=0XΦx,y(2)(rro)x=0,ψr(3)=0x=0XΦx,y(3)(rro)x=0,w=0x=0XWx,y(rro)x=0 (39)

Nr(1)+Nr(2)+Nr(3)=0 x=0X{[1+α(1)(rro)β(1)][A¯rr(1)(x+1)(Ux+1,y+h12Φx+1,y(1)+h22Φx+1,y(2))A¯rθ(1)r(Ux,y+h12Φx,y(1)+h22Φx,y(2))]+ [1+α(3)(rro)β(3)][A¯rr(3)(x+1)(Ux+1,yh22Φx+1,y(2)h32Φx+1,y(3))A¯rθ(3)r(Ux,yh22Φx,y(2)h32Φx,y(3))] Arr(2)(x+1)Ux+1,yA¯rθ(2)Ux,yr}(rro)x=0h12Nr(1)+Mr(1)=0 x=0X{h12[1+α(1)(rro)β(1)][A¯rr(1)(x+1)(Ux+1,y+h12Φx+1,y(1)+h22Φx+1,y(2))A¯rθ(1)r(Ux,y+h12Φx,y(1)+h22Φx,y(2))] +[1+α(1)(rro)β(1)](D¯rr(1)ψr,r(1)+1r D¯rθ(1)ψr(1))}(rro)x=0h22Nr(1)+Mr(2)h22Nr(3)=0 x=0X{h22[1+α(1)(rro)β(1)][A¯rr(1)(x+1)(Ux+1,y+h12Φx+1,y(1)+h22Φx+1,y(2))A¯rθ(1)r(Ux,y+h12Φx,y(1)+h22Φx,y(2))] h22[1+α(3)(rro)β(3)][A¯rr(3)(x+1)(Ux+1,yh22Φx+1,y(2)h32Φx+1,y(3))A¯rθ(3)r(Ux,yh22Φx,y(2)h32Φx,y(3))] Drr(2)ψr,r(2)+1r Drθ(2)ψr(2)}(rro)x=0h32Nr(3+Mr(3)=0 x=0X{h32[1+α(3)(rro)β(3)][A¯rr(3)(x+1)(Ux+1,yh22Φx+1,y(2)h32Φx+1,y(3))A¯rθ(3)r(Ux,yh22Φx,y(2)h32Φx,y(3))] +[1+α(3)(rro)β(3)](D¯rr(3)ψr,r(3)+1r D¯rθ(3)ψr(3))}(rro)x=0Qr(1)+Qr(2)+Qr(3)=0 x=0X{A¯rz(1)[1+α(1)(rro)β(1)][Φx,y(1)+(x+1) Wx+1,y]+ Arz(2)[Φx,y(2)+(x+1) Wx+1,y]+ A¯rz(3)[1+α(3)(rro)β(3)][Φx,y(3)+(x+1) Wx+1,y]}(rro)x=0 (40)

The resulting governing differential equations can be solved in terms of time by the fourth-order Runge-Kutta method, numerically.

The unknown displacement, velocity and acceleration parameters can be expressed as following vectors:

F(r, t)=u(r, t), ψr(1)(r, t), ψr(2)(r, t), ψr(3)(r, t), w(r, t)F˙(r, t)=u˙(r, t), ψ˙r(1)(r, t), ψ˙r(2)(r, t), ψ˙r(3)(r, t), w˙(r, t) F¨(r, t)=u¨(r, t), ψ¨r(1)(r, t), ψ¨r(2)(r, t), ψ¨r(3)(r, t), w¨(r, t) (41)

Also, the transformed forms of the unknown parameters may be expressed as following vectors:

Fx,y=Ux,y, Φx,y(1), Φx,y(2), Φx,y(3), Wx,yF¯x,y=U¯x,y, Φ¯x,y(1), Φ¯x,y(2), Φ¯x,y(3), W¯x,yF˜x,y=U˜x,y, Φ˜x,y(1), Φ˜x,y(2), Φ˜x,y(3), W˜x,y (42)

Based on the fourth-order Runge-Kutta method, when the acceleration and velocity parameters at t = y∆t are known as F¯x,y and F˜x,y the displacement parameters at t = y∆t and t = (y + 1) ∆t are known as F x,y and F x,y+1 , the acceleration and velocity parameters at t = (y + 1) ∆t can be found by using the following formulations:

F˜x,y+1=4Δt2(Fx,y+1Fx,yΔt F¯x,y14Δt2F˜x,y) (43)

F¯x,y+1=F¯x,y+Δt2(F˜x,yF˜x,y+1) (44)

By using Eqs. (31) and (32), the initial conditions can be obtained as following transformed forms:

Ux,0=U¯x,0=0, Φx,0(1)=Φ¯x,0(1)=0, Φx,0(2)=Φ¯x,0(2)=0, Φx,0(3)=Φ¯x,0(3)=0, Wx,0=W¯x,0=0, (45)

The unknown functions can be obtained by using the transformed forms of the governing equations Eqs. (34)-(38), boundary conditions Eqs. (39, 40) and initial conditions Eq. (45) and the relations between the acceleration, velocity and displacement parameters Eqs. (43) and (44). From the Eqs. (39) and (40), five equations for each edge must be applied (based on the edge condition at the inner and outer edges).

The unknown functions may be obtained according to the following steps.

  1. Initial values of the acceleration parameters

    (F˜x,0)

    are determined based on the Eqs. (34)-(38) and Eq. (45).

  2. From Eqs. (42) and (43), the acceleration and velocity parameters at the end of each time step

    (F˜x,yandF¯x,y, y=1, 2, 3, ...)

    may be expressed based on the obtained parameters at the previous step

    (Fx,y1, F˜x,y1andF¯x,y1, y=1, 2, 3, ...)

    and unknown displacement parameters at the end of each time step (F x,y , y = 1,2,3...).

  3. The displacements parameters at the end of each time step (F x,y ) are determined based on the obtained results of the acceleration and velocity parameters from step (II), Eqs. (34)-(38) and the edge conditions (Eqs. (39, 40)) corresponding to the end of each time step.

  4. Repeating steps (II) and (III) till the final time instant is reached(t = y ∆T).

5 RESULTS AND DISCUSSIONS

In this section, effects of the viscoelastic foundation on the transient response of the functionally graded polar orthotropic sandwich annular plates are investigated. The presented results cover various dynamic loads, edge conditions and foundation parameters. Dynamic responses of the sandwich structures are extracted for various dynamic loads as q=q^ f(t) , in which q^ is the maximum amplitude and f(t) is an arbitrary unit time-dependency function.

In the presented results, the following dynamic load cases are examined (as shown in Figure 2):

Figure 2: The various kinds of dynamic loads in numerical examples. 

  • Case I) Step loading: f(t) = 1

  • Case II) Pulse loading:

    f(t)={1 for 0 tτ0 for τ t

  • Case III) Sinusoidal loading: f(t) = sin(ωt)

  • Case IV) N-shaped pulse loading:

    f(t)={12tτ for 0 t τ0 for τ t

  • Case V) Sinusoidal loading acting in the same direction:

    f(t)=|sin(π τt)|

  • Case VI) A triangular-shaped pulse loading:

    f(t)={1tτ for 0 t τ0 for τ t

  • Case VII) Exponential loading: f(t) = e -ζ t

  • Case VIII) Repeated rectangular-shaped loading:

f(t)={1for2i τ t(2i+1) τ0for(2i+1) τ t2(i+1) τ i=0,1,2,...

  • Case IX) Repeated triangular-shaped loading:

f(t)=tiττfor2i τ t(2i+1) τ i=0,1,2,...

Sandwich plates under dynamic loads with the maximum amplitude of q^ =1MPa are considered throughout the examples. Also, the following geometric information and material properties for the homogenous core are chosen.

ri=0.2, ro=1, h1=0.1, h2=0.2, h3=0.1, E(2)=20GPa, ρ(2)=950/g N/m3, v(2)=0.25, g=9.81 m/s2

5.1 Transient Response of Heterogeneous Sandwich Plate

In this study, dynamic response of sandwich circular plate with functionally graded orthotropic face sheets are presented as first time and no available results may be found in literature to demonstrate the efficiency and accuracy of the obtained results. For this reason, as a verification example, the obtained results are verified by comparison with finite element results for some special cases. The finite element results are extracted from the ABAQUS software based on the three dimensional theory of elasticity. In this example, sandwich plate with heterogeneous face sheets and asymmetric layups is considered and effects of the damping coefficient on the transient response of heterogeneous sandwich plate are examined. The face sheets have the same stiffness in the radial and circumferential directions as:

{Er(1)(r)=Eθ(1)(r)=310[1(rro)2]GPaρ(1)(r)=1613[1(rro)2]/g N/m3vrθ(1)=vθ r(1)=0.26, {Er(3)(r)=Eθ(3)(r)=310[1+0.5(rro)]GPaρ(3)(r)=1613[1+0.5(rro)] /g N/m3vrθ(3)=vθ r(3)=0.23,G(k)(r)=Er(k)(r)2(1+vrθ(k))

Dynamic responses of the sandwich annular plate with clamped-free boundary condition (clamped inner edge and free outer edge) are shown in Figures 3-5 for the load cases (I), (II) and (III), respectively. In these figures, time variations of the lateral deflection at the outer edge of sandwich plate are plotted for k w =0 and various values of the damping coefficients of viscoelastic foundation C t . Also, the obtained results are compared with results of ABAQUS finite element software when there is no foundation (k w = C t =0). It can be seen that there is an excellent agreement between present results based on the analytical solution (AS) and results of ABAQUS software which are extracted based on the finite element method (FEM). It is shown that when the damping coefficient increases, vibration amplitude becomes smaller, as expected due to the dissipation of system energy. Figure 3 shows that the plate oscillates about the static deflection (W Static ) and for C t =0 (when there is no damping in the foundation) the maximum deflection will be as:W Dynamic-Maximum =2W Static . Also for plate subjected to step loading rested on viscous foundation, as time goes on, the dynamic lateral deflection tends to static response. Time variations of the lateral deflection of the plate under load case (II) are plotted in Figure 4 for time duration τ = 0.5 and 1.5 ms. In this type of loading, the dynamic responses of the plate are separated into two stages as follow:

  • a) Forced vibration (plate under the step loading) t < τ

The plate oscillates about the static deflection of the plate under the step loading.

  • b) Free vibration (the load is removed) t > τ

The plate oscillates about the zero.

Figure 3: Dynamic response of clamped-free sandwich annular plate subjected to an abrupt uniformly distributed transverse load (case (I)). 

Figure 4: Dynamic response of clamped-free sandwich annular plate subjected to an abruptly-imposed uniform transverse load (Case (II)) with time duration: (a) t 0 = 0.5 ms and (b) t 0 = 2 ms. 

Figure 5: Dynamic response of clamped-free sandwich annular plate subjected to a uniform transverse load whose intensity varies according to a sinusoidal function (Case (III)): (a) ω = Ω, (b) ω = Ω/4 (c) ω = Ω/6 (d) ω = Ω/8. 

As expected for plate subjected to a pulse loading rested on viscous foundation, the amplitude decreases and approaches zero as time increases.

Dynamic responses of sandwich plate under sinusoidal loading (case (III)) are illustrated in Figure 5 for various frequency of the harmonic excitation (ω). Value of the parameter ω is chosen as a ratio of the fundamental natural frequency (Ω = 3892.5) of the plate. As it is expected for plate under sinusoidal load with harmonic excitation ω = Ω, the oscillation amplitude becomes unbounded in undamped system (Figure 5(a)). However, there is not an obvious increase in the amplitude of oscillation when the damping coefficient increases up to 3E5 or 8E5.

It can be seen from Figures 5(b), (c) and (d) that for excitation frequencies ω = Ω/ 4, Ω/ 6 and Ω/ 8, influence of other vibration modes are superimposed on the main response in undamped system. On the other hands the dynamic response of plates resting on the viscous foundation (damped system) is smoother and the local oscillations are eliminated. It is observed that the peak deflection slightly decreases as excitation frequency decreases. These figures also show that increasing the damping coefficient of viscous foundation reduces the amplitude and frequency of vibration.

5.2 Transient Response of Functionally Graded Polar Orthotropic Sandwich Plate

In the present section, dynamic responses of circular sandwich plate fabricated from functionally graded polar orthotropic face sheets are investigated for various edge conditions. Also, effects of stiffness and damping coefficients of viscoelastic foundation on the transient response of sandwich plate are examined for various dynamic loads.

Variation of the Young’s modulus (in the radial and circumferential direction), shear modulus and density of each face sheet in the radial direction are chosen as:

  • a) Top face sheet:

{Er(1)(r)Eθ(1)(r)Grz(1)(r)ρ(1)(r)}={170GPa68GPa34GPa1413/g N/m3}[1+0.4(rro)]

  • b) Bottom face sheet:

{Er(3)(r)Eθ(3)(r)Grz(3)(r)ρ(3)(r)}={68GPa170GPa18GPa1413/g N/m3}[1+0.7(rro)]

Also, the Poisson’s ratios are assumed as: vrθ(1)=0.35, vrθ(3)=0.14 where vrθ Erθ=vθrEθr

Dynamic responses of clamped-simply supported sandwich plate subjected to load cases (I), (IV) and (V) are plotted in Figures 6-8, respectively. In these figures, time variations of the lateral deflection are presented for r=0.5. It is evident that the effects of elastic foundation (Winkler-type foundation) can be examined when damping coefficient C t is set to zero. Figure 6 shows that vibration amplitude and frequency of plate under elastic foundation (C t =0 and k w =2E10) decreases and increases, respectively, comparing with those of the plate without foundation. The plate oscillates about the static value of deflection when there is no damping in the foundation (C t =0), in which the vibration amplitude is constant. Also, the vibration amplitude of plate under viscous foundation decreases with time and tends to the static response of plate for C t =0.

Figure 6: Dynamic response of clamped-simply supported sandwich annular plate subjected to an abrupt uniformly distributed transverse load (case (I)). 

Figure 7: Dynamic response of clamped- simply supported sandwich annular plate subjected to an N-shaped pulse loading (case (IV)). 

Figure 8: Dynamic response of clamped-simply supported sandwich annular plate subjected to a sinusoidal loading acting in the same direction (case (V)) with time duration: (a) τ =2 ms and (b) τ = 4 ms. 

It can be seen from Figure 7 that for sandwich plate subjected to an N-shaped pulse loading, the center of oscillation decreases with time and becomes negative when τ < t (forced vibration), when the load is removed (τ > t free vibration) the plate oscillates about the zero. As expected, vibration amplitude of plate without damping (C t =0) is constant in each stage, but for plate subjected to a pulse loading rested on viscous foundation, the amplitude decreases with time and the lateral deflection tends to zero. Also for plate without foundation (C t =k w =0), vibration amplitude in the first stage (τ < t) is higher than that of the second stage (τ > t), but for plate under elastic foundation with C t =0 and k w =2E10, vibration amplitude in the second stage is higher. Indeed, in the first stage, vibration amplitude of plate without foundation is higher than that in the other cases and in the second stage, vibration amplitude of plate with C t =0 and k w =2E10 is higher.

Figure 8 shows that for the plate subjected to load case (V), the response to the loading with τ = 2ms is higher than the response to the loading with τ = 4ms. In the other words, the loading with the higher frequency of excitation leads to higher response.

This figure also shows that influence of other vibration modes which are superimposed on the main response of plate in undamped system, are eliminated for plates resting on the viscous foundation.

Effects of viscoelastic foundation on the dynamic response of clamped-clamped sandwich plate subjected to load cases (I), (VI) and (VII) are presented in Figures 9-11, respectively. In these figures, the results are reported for r=0.5. It can be seen that in comparison with the forgoing example, there is a same trend for the dynamic response of sandwich plate under step loading with clamped-clamped and clamped-simply supported boundary conditions (Figures 6 and 9). However, the vibration amplitude and frequency of clamped-clamped plate decreases and increases, respectively, comparing with those of the clamped-simply supported plate due to increasing the stiffness of plate with clamped-clamped edges.

Figure 9: Dynamic response of clamped-clamped sandwich annular plate subjected to an abrupt uniformly distributed transverse load (case (I)). 

Figure 10: Dynamic response of clamped-clamped sandwich annular plate subjected to a triangular-shaped pulse loading (case (VI)). 

Figure 11: Dynamic response of clamped-clamped sandwich annular plate subjected to an exponential loading (case (VII)): (a) ζ = 100 (b) ζ = 500. 

It is observed that for dynamic response of plate under the triangular-shaped pulse and exponential loadings (Figures 10 and 11), the center of oscillation decreases and approaches zero as time increases. These figures show that the vibration amplitude of plate without foundation (C t =k w =0) is higher than that in the other cases.

Time variations of the lateral deflection at the outer edge of the sandwich plate with clamped-free boundary condition subjected to load cases (VIII) and (IX) are plotted in Figure 12 and 13, respectively. In Figure 12, results of sandwich plate under repeated rectangular-shaped loading (load case (VII)) with time duration of 2ms and 4ms are illustrated.

Figure 12: Dynamic response of clamped-free sandwich annular plate subjected to a repeated rectangular-shaped loading (Case (VIII)) with time duration: (a) τ = 2 ms and (b) τ = 4 ms. 

Figure 13: Dynamic response of clamped-free sandwich annular plate subjected to a repeated triangular-shaped loading (Case (IX)) with time duration: (a) τ = 2 ms and (b) τ = 5 ms. 

It can be seen that in the first duration of loading (t < τ), the peak deflection of plate without foundation is higher than the other cases and then (t > τ) the peak deflection of plate under elastic foundation with K w =5E9 in Figure 12(a) and K w =2E9 in Figure 12(b) is higher. Increasing elastic foundation stiffness value up from 5E9 to 9E9 in Figure 12(a) and 2E9 to 2E10 in Figure 12(b), causes an obvious decrease of vibration amplitude.

It is interesting to note that the viscoelastic foundation causes a reduction in the vibration amplitude of plate in the loading time durations (2 iτ< t < ( 2i + 1)τ i = 0,1,2,...), but in the time durations when the load is removed ((2i+1)τ < t < 2(i + 1)τ i = 0,1,2,...), the vibration amplitude of plate without foundation is smaller and its response is smoother.

Figure 13 shows that the oscillation amplitude of plate subjected to the repeated triangular-shaped loading is unbounded in undamped system. It is seen that the repeated triangular-shaped loading with lower time durations (τ ) increases the amplitude rapidly. Also, vibration amplitude of plate under viscoelastic foundation is bounded in some cases of viscoelastic foundation, where the stiffness and viscosity coefficients of foundation become higher than specific values.

6 CONCLUSIONS

By using the layerwise theory, effects of the viscoelastic foundation on the dynamic response of sandwich annular plates are studied. The viscoelastic substrate is modeled as Kelvin-Voigt foundation. Sandwich plates may be fabricated from functionally graded polar orthotropic face sheets, where Young’s modulus in the radial and circumferential direction, shear modulus and density of each face sheet may be varied continuously in the radial direction. The relatively complicated second order coupled partial differential equations are solved based on a developed semi-analytical solution procedure by using a combination of the finite Taylor’s transform procedure in the space domain and the fourth-order Runge-Kutta method in the time domain. The results are obtained for various dynamic loads and edge conditions. For some special cases, the obtained results based on the proposed solution procedure are compared with finite element results based on the three dimensional theory of elasticity. The comparisons show that there is a very good agreement between present results and results of the three-dimensional theory of elasticity. Results reveal that FG polar orthotropic sandwich plates under various transient loads, edge conditions and foundation parameters may be analyzed by using the proposed solution procedure. Results also show that influence of other vibration modes which are superimposed on the main response of plate in undamped system, are eliminated for plates resting on the viscous foundation.

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Received: June 07, 2017; Revised: August 29, 2017; Accepted: September 04, 2017

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