Temperature-dependent vibration analysis of a FG viscoelastic cylindrical microshell under various thermal distribution via modified length scale parameter: a numerical solution

Hamed Safarpour; Kianoosh Mohammadi; Majid Ghadiri

doi:10.1515/jmbm-2017-0010

Open Access Published by De Gruyter August 29, 2017

Temperature-dependent vibration analysis of a FG viscoelastic cylindrical microshell under various thermal distribution via modified length scale parameter: a numerical solution

Hamed Safarpour , Kianoosh Mohammadi and Majid Ghadiri

From the journal Journal of the Mechanical Behavior of Materials

https://doi.org/10.1515/jmbm-2017-0010

Abstract

In this article, the vibrational analysis of temperature-dependent cylindrical functionally graded (FG) microshells surrounded by viscoelastic a foundation is investigated by means of the modified couple stress theory (MCST). MCST is applied to this model to be productive in design and analysis of micro actuators and micro sensors. The modeled cylindrical FG microshell, its equations of motion and boundary conditions are derived by Hamilton’s principle and the first-order shear deformation theory (FSDT). For the first time, in the present study, functionally graded length scale parameter which changes along the thickness has been considered in the temperature-dependent cylindrical FG microshell. The accuracy of the present model is verified with previous studies and also with those obtained by analytical Navier method. The novelty of the current study is consideration of viscoelastic foundation, various thermal loadings and size effect as well as satisfying various boundary conditions implemented on the temperature-dependent cylindrical FG microshell using MCST. Generalized differential quadrature method (GDQM) is applied to discretize the equations of motion. Then, some factors are investigated such as the influence of length to radius ratio, damping, Winkler and Pasternak foundations, different temperature changes, circumferential wave numbers, and boundary conditions on natural frequency of the cylindrical FG microshell. The results have many applications such as modeling of microrobots and biomedical microsystems.

Keywords: GDQM; linear and nonlinear thermal loadings; modified length scale parameter; various boundary conditions; viscoelastic foundation

1 Introduction

Because of simple manufacturing and great strength to weight ratio, the use of cylindrical shell structures prevails in many industries. Material of the cylindrical shell model has a direct effect on its vibrational behavior. Cylindrical functionally graded (FG) microshells are being used in different fields of aerospace and mechanical engineering; they can be utilized in environments with high temperatures. Cylindrical FGM shells are also used in such a wide range of applications including pressure vessels, oil pipes, automotive components and nuclear power reactors.

Thermal and mechanical buckling of cylindrical FGM shells under the effect of geometrical imperfections with different types of loadings were investigated by Refs [1], [2], [3], [4]; their equations are based on the first-order shell theory and Sanders’ kinematic equations. Their results demonstrate that the critical temperature difference for the functionally graded cylindrical shell is generally lower than the corresponding value for the isotropic cylindrical shell. This critical temperature difference for the cylindrical shell under nonlinear temperature difference is greater than that under linear temperature difference through the thickness.

Shen [5] studied the post-buckling of a shear deformable functionally graded cylindrical shell with finite length resting on an elastic foundation and under axial compressive loading in thermal environment. Shen also studied [6] torsional buckling and post-buckling of FGM cylindrical shells with thermal loading. Malekzade and Heydarpour [7] investigated free vibration of rotating functionally graded cylindrical shells in thermal environment according to the first order shear deformation theory of shells with temperature-dependent material. Malekzadeh et al. [8] presented a study on free three-dimensional vibration of truncated conical functionally graded shells with temperature dependent material properties investigating the effects of the thickness-to-mean radius ratio, the length-to-mean radius ratio and the power law index. They found that by increasing one of these parameters, the natural frequency increase. Najafizadeh and Isvandzibaei [9], [10] studied free vibration of thin FGM cylindrical shells with ring support using Ritz method based on the first order and higher order shear deformation shell theories. An analytical approach [11] to analyze the vibration and nonlinear dynamic response of imperfect S-FGM thick circular cylindrical shells resting on an elastic foundation using the third order shear deformation shell theory has been presented by Duc et al. and Tornabene et al. [12]. They studied free vibration and thermal buckling behavior of moderately thick functionally graded materials (FGM) structures including plates, cylindrical panels and shells in thermal environments.

Ghadiri and Safarpour [13] investigated free vibration analysis of a functionally graded (FG) porous cylindrical micro shell subjected to a thermal environment. They considered the effects of material length scale parameter, temperature changes, volume fraction of the porosity, FG power index, length, axial and circumferential wave number. Vibration analysis of CNT-reinforced functionally graded composite cylindrical shells in thermal environments presented by Zhang et al [14]. Jooybar et al. [15] modeled functionally graded truncated conical panels to analyze the influences of initial thermal stresses, temperature dependencies of the material properties and parameters of geometry on characteristics of free vibration. Since the viscoelastic foundation is so useful in modeling a lot of structures such as biomechanical-systems, they have been important for researchers in recent years. Daneshmand et al. [16] presented instability analysis of carbon nanotubes conveying fluid surrounded by a linear viscoelastic Winkler foundation. They [17] also considered small-scale effects on vibration of non-uniform carbon nanotubes conveying fluid and subjected to viscoelastic foundation. Under these conditions, they analyzed [18] in plane vibration of curved carbon nanotubes. Lei et al. [19] investigated vibrational characteristics of double-walled carbon nanotubes resting on a viscoelastic medium. Based on the nonlocal Euler-Bernoulli beam theory [20], vibration characteristics were investigated for a horn-shaped single-walled carbon nanotube (SWCNT) exposed to a longitudinal magnetic field and surrounded with a viscoelastic foundation. The novelty of the current study is consideration of viscoelastic foundation, various thermal loadings and size effect as well as satisfying various boundary conditions implemented on the temperature-dependent cylindrical FG microshell using MCST. Because of the efficiency and high accuracy of generalized differential quadrature method (GDQM). This method is employed to solve the governing equations with multiple kinds of boundary conditions. FG cylinders normally are modeled and analyzed along their thickness while the outer surface is metal and the inner surface is ceramic. For the first time, the present study considers the size dependency of the transverse vibration of a temperature-dependent cylindrical FG microshell using the modified couple stress theory. The governing equations and boundary conditions have been developed using Hamilton’s principle and are solved with the aid of GDQM. The results show that, material length scale parameters, stiffness and damping of the visco-Pasternak foundation, length to radius ratio, radius to thickness ratio, FG power index and boundary conditions play important roles on natural frequency of cylindrical FG microshell in thermal environments.

2 Preliminaries

2.1 Temperature-dependent cylindrical FG microshell equations

Figure 1 illustrates a temperature-dependent FG cylindrical microshell subjected to a viscoelastic foundation. The thickness, length and the middle surface radius of cylindrical shell are denoted by h, L and R respectively. The FGM microshell is generally composed of two different materials at inner and outer surfaces. It is assumed that bulk elastic modulus E(z), mass density ρ(z), Poisson’s ratio v(z) and coefficient of thermal expansion α(z) change along the thickness direction based on the power law distribution. Note that, in FG cylindrical microshells, the material length scale parameter l(z) is assumed to be a function of the thickness. The variation of material properties across the FG microshell thickness is determined by power FG index (F). Assuming that the inner surface is ceramic and the outer surface is metal which respectively are denoted by c and m, the mechanical properties for different values of F are expressed as below:

Figure 1:

Geometry of FG cylindrical microshell surrounded by a viscoelastic foundation.

(1)E(z)=(Em−Ec)(0.5+zh)F+Ecρ(z)=(ρm−ρc)(0.5+zh)F+ρcν(z)=(νm−νc)(0.5+zh)F+νcα(z)=(αm−αc)(0.5+zh)F+αcl(z)=(lm−lc)(0.5+zh)F+lc

To observe the behavior of FGM microshell under high temperatures, temperature dependency of material properties must be considered. Equation of thermo-elastic material properties with nonlinear thermal loading is a function of temperature which can be given by [21].

(2)ϒ=ϒ0(ϒ−1T−1+1+ϒ1T+ϒ2T2+ϒ3T3)

where, ϒ₀, ϒ₋₁, ϒ₁, ϒ₂ and ϒ₃ are temperature dependent coefficients which are expressed in Ref [13].

2.2 Modified couple stress theory

According to the modified couple stress theory, the strain energy is defined as a function of gradient of rotation tensor and strain tensor. It also includes length scale parameter and two Lamé parameters. This theory was presented by Yang et al. [22] for the first time that expresses the strain energy as below:

(3)U=12∭V(σijεij+mijsχijs)dV

In Eq. (3), components of symmetric rotation gradient tensor, strain tensor, stress tensor, and higher order stress tensor have respectively been donated by χijs,ε_ij, σ_ij and m_ij which are given as:

(4)εij=12(ui,j+uj,i)

(5)χijs=12(φi,j+φj,i)

(6)mijs=2l2(z)E(z, T)2(1+ν(z, T))χijs, φi=12[curl(u)]i

In Eqs. (4, 5) u_i and φ_i are respectively the components of displacement vector and the infinitesimal rotation vector. The thermomechanical stress–strain relation including the temperature effects is expressed as follows:

(7){σxxσθθσxθσθzσxz}=[E(z, T)1−ν2(z, T)ν(z, T)×E(z, T)1−ν2(z, T)000ν(z, T)×E(z, T)1−ν2(z, T)E(z, T)1−ν2(z, T)00000E(z, T)2(1+ν(z, T))00000E(z, T)2(1+ν(z, T))00000E(z, T)2(1+ν(z, T))]{εxx−αxxΔTεθθ−αθθΔT2εxθ2εθz2εxz}

It is worth mentioning that the two thermal expansion coefficients [αxx(z^, T), αθθ(z^, T)] are equal, due to in-plane uniform distribution of the FGM properties [αxx(z^, T)=αθθ(z^, T)=αee(z^, T)]. In addition, the parameter T₀ is the reference temperature.

2.3 Displacement field for cylindrical microshell

The displacement field of the cylindrical shell along the three directions of x, θ, z based on FSTD is expressed as follows [23]:

(8)U(x, θ, z, t)=u(x, θ, t)+zψx(x, θ, t)V(x, θ, z, t)=v(x, θ, t)+zψθ(x, θ, t)W(x, θ, z, t)=w(x, θ, t)

In Eq. (8), U(x, θ, t), V(x, θ, t) and W(x, θ, t) are neutral axis displacements, and ψ_θ(x, θ, t) and ψ_x(x, θ, t) are rotation of a transverse normal surface about the circumferential and axial directions.

2.4 Governing equations and associated boundary conditions

To extract the equations of motion and boundary conditions of the composed model using FSDT shell model and the modified couple stress theory, components of the displacement field must be inserted in strain equations of the cylindrical FG microshell. So, substituting Eq. (8) into Eq. (4), Eq. (5) and Eq. (6), the components of the deviatory stretch gradient tensor and the strain tensor are achieved as follows:

(9)εxx=∂u∂x+z∂ψx∂xεθθ=1R∂v∂θ+z1R∂ψθ∂θ+wRεxz=12(ψx+∂w∂x)εxθ=12(1R∂u∂θ+∂v∂x)+z2(1R∂ψx∂θ+∂ψθ∂x)εθz=12(ψθ+1R∂w∂θ−vR)

Also the non-zero components of the symmetric rotation gradient tensor are obtained as follows:

(10)χxxs=−12(∂ψθ∂x+1R∂v∂x−1R∂2w∂x∂θ)χθθs=−12R(1R∂u∂θ−∂v∂x−z∂ψθ∂x)−12(1R∂2w∂x∂θ−1R∂ψx∂θ)χzzs=−12(1R∂ψx∂θ−∂ψθ∂x−1R2∂u∂θ)χxθs=−14(1R2∂v∂θ+∂2w∂x2−1R2∂2w∂θ2−∂ψx∂x+1R∂ψθ∂θ)χxzs=−14(1R∂2u∂x∂θ−∂2v∂x2−vR2+1R2∂w∂θ+ψθR)−z4(1R∂2ψθ∂x∂θ−∂2ψθ∂x2)χθzs=−14(1R2∂2u∂θ2−1R∂2v∂x∂θ−1R∂w∂x+ψxR)−z4(1R2∂2ψx∂θ2−1R∂2ψθ∂x∂θ)

For the equations of the motion and boundary conditions, the principle of minimum potential energy states that [24]:

(11)∫t1t2(δT−δU+δW1+δW2−δ˙D)dt=0

Strain energy of the FG shell based on FSDT and using the modified couple stress theory is expressed as follows:

(12)δU=12∭V(σijδεij+mijsδχijs) dV=δU1+δU2δU1=12∭V(σijδεij) dV=∬A{(Nxx∂∂xδu+Mxx∂∂xδψx)+Nθθ(1R∂∂θδv+δwR)+Mθθ1R∂∂θδψθ+Qxz(δψx+∂∂xδw)+Nxθ(1R∂∂θδu+∂∂xδv)+Mxθ(1R∂∂θδψx+∂∂xδψθ)+Qzθ(δψθ+1R∂∂θδw−δvR)}R dx dθδU2=12∭V(mijsδχijs) dV=∬A{(−Yθθ2R2+Yzz2R2)∂∂θδu−(Yθz2R2)∂2∂θ2δu−(Yzx2R)∂2∂θ ∂xδu+(Yθθ2R−Yxx2R)∂∂xδv+(Yxz2)∂2∂x2δv−(Yθx2R2)∂∂θδv+(Yθz2R)∂2∂θ ∂xδv+(Yxz2R2)δv+(Yθz2R)∂∂xδw−(Yθx2)∂2∂x2δw−(Yzx2R2)∂∂θδw+(Yxθ2R2)∂2∂θ2δw+(−Yθθ2R+Yxx2R)∂2∂θ ∂xδw+(Yxθ2)∂∂xδψx+(Yθθ2R−Yxx2R)∂∂θδψx−(ζzx2R)∂2∂θ ∂xδψx−(Yzθ2R)δψx−(Yxθ2R)∂∂θδψθ+(Yθθ2R−Yxx2+Yzz2)∂∂xδψθ+(ζzθ2R)∂2∂θ ∂xδψθ−(ζzθ2R2)∂2∂θ2δψx+(ζxz2)∂2∂x2δψθ−(Yxz2R)δψθ}R dx dθ

where, the classical and non-classical forces and momentums are assumed as follows:

(13)(Nxx, Nθθ, Nxθ)=∫−h/2h/2(σxx, σθθ, σxθ)dz,(Mxx, Mθθ, Mxθ)=∫−h/2h/2(σxx, σθθ, σxθ)zdz,(Qxz, Qzθ)=∫−h/2h/2ks(σxz, σzθ)dz,(Yxx, Yθθ, Yzz, Yxθ, Yxz, Yzθ)=∫−h/2h/2(mxx, mθθ, mzz, mxθ, mxz, mzθ)dz,(ζxx, ζθθ, ζzz, ζxθ, ζxz, ζzθ)=∫−h/2h/2(mxx, mθθ, mzz, mxθ, mxz, mzθ)zdz

Also the kinetic energy of the cylindrical shell can be expressed as:

(14)δT=∫z∬Aρ(z, T) {(∂u∂t+z∂ψx∂t)(∂∂tδu+z∂∂tδψx)+(∂v∂t+z∂ψθ∂t)(∂∂tδv+z∂∂tδψθ)+(∂w∂t)∂∂tδw}R dz dx dθ

For a typical functionally graded microshell which is exposed to high temperatures, the distribution of temperature along its thickness can be assumed linearly. Hence, the first variation of the work done corresponding to temperature changes can be expressed as [25]:

(15)δW1=∬A[N1T∂w∂x∂δw∂x+N2T∂v∂x∂δw∂x]R dx dθ

where, N1T and N2T are thermal resultants. Note that, the two thermal resultants are equal and can be expressed as:

(16)N1T=N2T=∫−h/2h/2E(z, T)α(z, T)(T−T0)dz

In Eq. (16), T₀ is the ambient temperature. Considering a functionally graded microshell that the outer surface (metal-rich) is T_m and the inner surface (ceramic-rich) temperature is T_c, variations of temperature are defined as follows [26]:

(17)T=Tc−ΔT(12+z^ch)αp

where α_p denotes the non-negative power index of temperature variation function and ΔT=T_c−T_m. For example, considering α_p≥2 the temperature variation along the thickness becomes nonlinear and considering α_p=1 the temperature variation becomes linear. The work done by Pasternak foundation acting on the cylindrical FG microshell by the surrounding medium can be given by [24]:

(18)δW2=[Kww+Kp∇2w]δw R dV,∇2=∂2∂x2+1R2∂2∂θ2

in which K_W and K_p are the Winkler and Pasternak coefficients respectively. The work done by dampers on cylindrical functionally graded microshell surrounding by medium can be given by [24]:

(19)δ˙D=∭V{Cd∂w∂tδ˙w}R dV

in which C_d is the damping constant. Substituting Eqs. (12), (14), (15), (18) and (19) into (11) and integrating by parts, equations of motion and boundary conditions using the modified couple stress theory and FSDT shell model can be expressed as follows:

(20)δu:∂Nxx∂x+1R∂Nxθ∂θ+12R2(−∂Yθθ∂θ+∂Yzz∂θ)+12R∂2Yzx∂θ ∂x+12R2∂2Yθz∂θ2=I0∂2u1∂t2+I1∂2ψ1x∂t2δv:∂Nxθ∂x+1R∂∂θNθθ+QzθR+12{1R∂∂x(−Yxx+Yθθ)−1R2∂Yθx∂θ−∂2Yxz∂x2−YxzR2−1R∂2Yzθ∂θ ∂x}+ N2T∂2v∂x2=I0[∂2v∂t2]+I1[∂2ψθ∂t2]δw:∂Qxz∂x+1R∂Qzθ∂θ−NθθR−12R2∂2Yθx∂θ2−12R2∂Yzx∂θ+12R∂Yθz∂x+ ∂2Yxθ2∂x2+N1T∂2w∂x2−12R∂2∂θ ∂x(Yxx−Yθθ)−Kw(w)+Kp∇2(w)− C1d(∂w1∂t−∂w2∂t)=I0(∂2w1∂t2)δψx:∂Mxx∂x+1R∂Mθθ∂θ−Qxz+12∂Yθx∂x−12R∂∂θ(Yzz−Yθθ)+YzzR+ 12R∂2Tzx∂θ ∂x+12R2∂2Tθz∂θ2=I1∂2u∂t2+I2∂2ψx∂t2δψθ:1R∂Mθθ∂θ+∂Mxθ∂x−Qzθ+12∂∂x(Yzz−Yxx+TθθR)−12∂Yθx∂θ+ Yxz2R−12R∂2Tθz∂θ ∂x−12∂2Tzx∂x2=I1(∂2v∂t2)+I2(∂2ψθ∂t2)

Also, associate boundary conditions for each nanotube are as below:

(21)δu=0 or (Nxx+14R∂Yxz∂θ)nx+(Nxθ−Yθθ−Yzz2R+14∂Yxz∂x+12R∂Yθz∂θ)nθ=0,δv=0 or (Nxθ+Yθθ−Yxx2R−12∂Yxz∂x−14R∂Yθz∂θ)nx+(Nθθ−14R∂Yθz∂x−Yθx2R)nθ=0,δw=0 or (Qxz+Yzθ2R+12∂Yxθ∂x+14R∂(Yθθ−Yxx)∂θ)nx+(Qθz−Yzx2R−12R∂Yxθ∂θ+14∂(Yθθ−Yxx)∂x)nθ=0,δψx=0 or (Mxx+14R∂Txz∂θ+Yxθ2)nx+(Mθx+14∂Txz∂x+12R∂Tθz∂θ+(Yθθ−Yzz)2)nθ=0,δψθ=0 or (Mxθ−(Yxx−Yzz)2−14R∂Tθz∂θ−12∂Txz∂x+Tθθ2R)nx+(Mθθ−Yxθ2−14∂Tθz∂x)nθ=0,

For example:

The clamp supported boundary conditions in x=0, L:

(22)u=v=w=ψx=ψθ=0

The simply supported boundary conditions in x=0, L:

(23)v=w=ψθ=0, (Nxx+14R∂Yxz∂θ)=0, (Mxx+14R∂Txz∂θ+Yxθ2)=0.

The parameters that used in Eqs. (20–23) are expressed in Eq. (13).

2.5 Solution procedure

In early 1970s, differential quadrature method was presented by Bellman et al. [27], [28] as a reliable and accurate numerical method. In this method, precision of weighting coefficients adjusted by number of grid points leads to achieving accurate responses. To limit the number of grid points in the introductory version of DQM, an algebraic equation system was used to calculate weighting coefficients. Then, an explicit formulation was derived for weighting coefficients with infinite number of grid points which was developed as GDQ [29]. Shu and Richards [30] introduced a domain decomposition procedure for dealing with multi-domain problems. By this method, the main domain is divided into sub-domains, before discretizing each sub-domain for GDQ. The r-th order derivative of the function f(xi) is expressed as [29]:

(24)∂rf(x)∂xr|x=xp=∑j=1nCij(r)f(xi)

which “n” is the number of grid points along “x” direction, and superscript “r” shows the order of the derivative. Furthermore,“C_ij” is the weighing coefficient and following formulation is used to calculate it. For the first order derivative along x direction:

(25)Cij(1)=M(xi)(xi−xj)M(xj)i, j=1, 2, …, n and i≠jCij(1)=−∑j=1,i≠jnCij(1)i=j

where,

(26)M(xi)=∏j=1,j≠in(xi−xj)

Also C^(r) is given by:

(27)Cij(r)=r[Cij(r−1)Cij(1)−Cij(r−1)(xi−xj)]i, j=1, 2, …, n, i≠j and 2≤r≤n−1Cii(r)=−∑j=1,i≠jnCij(r)i, j=1, 2, …, n and 1≤r≤n−1

Due to geometrical periodicity of the cylindrical shell, the displacement vector for free vibration analysis is expressed as:

(28){u(x, θ, t)v(x, θ, t)w(x, θ, t)ψx(x,θ,t)ψθ(x, θ, t)}=∑n=1∞{U¯(x)cos(nθ)eiωtV¯(x)sin(nθ)eiωtW¯(x)cos(nθ)eiωtΨ¯x(x)cos(nθ)eiωtΨ¯θ(x)sin(nθ)eiωt}

Chebyshev polynomials can be used to discretize the domain [31]. Substituting Eq. (28) into governing equations, following equation is achieved:

(29)([M]{ω2}+[C]{ω}+[K])(βbβd)=0

Then, GDQ is exerted into the equations of motion and the boundary conditions to obtain the mass matrix [M], damping matrix [C] and stiffness matrix [K]. Domains and boundaries are respectively donated by d and b indexes and β also shows the mode shape. A proper way for solving Eq. (29) is to rewrite it as the following first order variable:

(30){Z˙}={A} {Z}

In which, state vector Z and state matrix [A] are defined as:

(31)Z={ddd˙d} and [A]=[[0][I]−[M−1K]−[M−1C]]

In Eq. (31), [0] and [I] are zero and unit (identity) matrices, respectively. Eventually, the natural frequency and its mode shape are obtained.

3 Results and discussion

There are two parts to analyze and describe the results. The first part is verification of proposed model with previous studies and also with those obtained by analytical Navier method. The second one is related to the effect of different parameters including length scale, circumferential wave numbers, length and radius, Winkler and Pasternak foundations, damping, various boundary conditions and thermal loadings on natural frequency of cylindrical FG microshell. However, to the best of authors’ knowledge, there is no research work on cylindrical micro shell with functionally graded length scale parameter which changes along the thickness and has been considered for the first time in the present study.

3.1 Convergence

The effect of the number of grid points on evaluating the convergence of the natural frequency with respect to linear or nonlinear thermal loadings, different damping coefficients, and boundary conditions (B.Cs) has been considered in Table 1. It can be seen that the proper number of grid points to obtain independency and converged responses is n=23. It is worth mentioning, LT and NLT mean linear and nonlinear thermal loadings respectively.

Table 1:

The effect of the number of grid points on evaluating the convergence of natural frequency (GHz) of the FG cylindrical microshell with respect to different damping coefficients, boundary conditions (B.Cs) and L=10 μm, L/R=10, h/R=0.1, l_m=14 μm, l_c=l_m/3, n=1, K_w=2×10¹⁵, K_p=100, ΔT=100, F=0.25.

Boundary condition	Coeff elastic foundation	Thermal loading	n=13	n=15	n=17	n=19	n=21	n=23	n=25
Simply-Simply	C_{_d} =4×10⁶	LT	0.80949	0.80949	0.80949	0.80949	0.80949	0.80949	0.80949
	C_{_d} =6×10⁶	NLT	0.42719	0.42719	0.42719	0.42719	0.42719	0.42719	0.42719
Simply-Clamp	C_{_d} =4×10⁶	LT	0.89206	0.89194	0.89184	0.89179	0.89178	0.89178	0.89178
	C_{_d} =6×10⁶	NLT	0.45725	0.45721	0.45717	0.45715	0.45715	0.45715	0.45715
Clamp-Clamp	C_{_d} =4×10⁶	LT	1.00309	1.00314	1.00320	1.00320	1.00319	1.00319	1.00319
	C_{_d} =6×10⁶	NLT	0.49426	0.49427	0.49429	0.49429	0.49429	0.49429	0.49429
Clamp-Free	C_{_d} =4×10⁶	LT	0.72503	0.72194	0.72039	0.71985	0.71974	0.71974	0.71974
	C_{_d} =6×10⁶	NLT	0.39783	0.39658	0.39595	0.39573	0.39568	0.39569	0.39569

3.2 Validation of the results with other articles

To investigate the accuracy of the first three dimensionless natural frequencies of isotropic homogeneous nanoshells with different thickness, the results have been validated with those have been presented in Table 2 [32] exhibiting the nondimensional frequency of a simply supported cylindrical nanoshell. They are in good agreement with those given by Beni et al. [32]. Table 2 shows the effect of changes in circumferential wave number: while it increases, the natural frequency increases in all values of h/R. If the length scale is zero, the results are drawn employing the classic theory which are in good adaptation with the reference [32], otherwise if the length scale is nonzero, the modified couple stress theory has been utilized. The results reported in this table also verify the analytical results that shows the importance of the study.

Table 2:

Comparison of first three dimensionless natural frequencies of isotropic homogeneous nanoshells, with different thicknesses.

h/R	n	Ref [32] (l=0)	Present study (GDQM) (l=0)	Ref [32] (l=h)	Present study (GDQM) (l=h)
0.02	1	0.1954	0.19536215	0.1955	0.19543206
	2	0.2532	0.25271274	0.2575	0.25731258
	3	0.2772	0.27580092	0.3067	0.30621690
0.05	1	0.1959	0.19542305	0.1963	0.19585782
	2	0.2623	0.25884786	0.2869	0.28543902
	3	0.3220	0.31407326	0.4586	0.45457555

Another validation has been done by Ghadiri and Safarpour [33]. As Figure 2 demonstrates, there is accuracy in obtained natural frequencies of the FG nanoshell especially with increasing the length.

Figure 2:

Comparison of the natural frequency of FG cylindrical shell with the results obtained by Ghadiri and Safarpour [33].

3.3 Validating the results with analytical method

To study the accuracy of the results, the presented responses by GDQM can be validated by the analytical method responses; in order to compare the obtained results by both methods, variations of fundamental frequency versus Pasternak stiffness in the range of 0 to 300 under linear and nonlinear thermal loadings have been reported in Table 3.

Table 3:

Comparison the results of GDQM and exact method for both linear and nonlinear thermal loadings in different Pasternak stiffness.

K_p	Exact method (linear thermal loading)	GDQM (linear thermal loading)	Exact method (nonlinear thermal loading)	GDQM (nonlinear thermal loading)
25	0.40624506	0.40376823	0.42131967	0.42132047
50	0.43143687	0.42910647	0.44565065	0.4456515
75	0.45522079	0.45301368	0.46870473	0.46870563
100	0.47780712	0.47570566	0.49066194	0.49066288
125	0.49935837	0.49734877	0.51166349	0.51166448
150	0.52000324	0.51807446	0.5318226	0.53182364
175	0.53984574	0.53798878	0.55123171	0.55123279
200	0.55897129	0.55717872	0.56996743	0.56996856
225	0.57745115	0.57571672	0.58809412	0.5880953
250	0.59534544	0.59366384	0.60566645	0.60566768
275	0.61270545	0.61107215	0.62273136	0.62273264
300	0.62957538	0.62798645	0.63932946	0.63933079

For another comparison (Figures 3 and 4 ) between GDQM and analytical method, variations of natural frequency versus the FG power index have been indicated for both types of thermal loadings. Figure 3 shows that with considering the coefficient of dampers, there is an exceptional trend by increasing FG power index: at first, frequency increases and reaches to a peak point and then begins to decrease and eventually the natural frequency tends to be constant. Figure 4 shows that without considering the coefficient of dampers and by increasing the FG power index, natural frequency increases and eventually tends to be constant.

Figure 3:

Comparison of the GDQM and analytical method in the model with and L=10 μm, L/R=10, h/R=0.1, l_m=14 μm, l_c=l_m/2, n=1, K_w=2×10¹⁰, K_p=100, ΔT=200, C_d=10⁶ for linear and nonlinear temperature loadings.

Figure 4:

Comparison the GDQM and analytical method in the model with L=10 μm, L/R=10, h/R=0.1, l_m=14 μm, l_c=l_m/2, n=1, K_w=2×10¹⁰, K_p=100, ΔT=200, C_d=0 for linear and nonlinear temperature loadings.

3.4 Parametric results

The materials for this paper are FGMs and it is assumed that they are temperature dependent; in addition, the outer surface is metal (SUS304) and the inner surface is ceramics (Si3N4). The power FG index (F) determines the variations of material properties across the thickness of the cylindrical FG microshell. The temperature-dependent coefficients have been given in Table 4. The length scale parameter of material can be determined experimentally. This needs development of a solution based on the modified couple stress theory. Length scale parameter can be obtained by comparing theoretical results to those achieved from experiments. Adopting this procedure, the length scale parameter for epoxy is determined 17.6 μm. But in the present study, due to inaccessibility to experimental data, we can use approximated values. Length scale parameter value of the metallic phase is specified as l_m=14 μm which has been suggested in the literature [13].

Table 4:

Temperature dependent coefficients of Poisson’s ratio, mass density, thermal expansion coefficient and Young’s modulus for SUS304 and Si₃N₄.

Material	Properties	P₀	P₁	P₂	P₃
Si₃N₄	E(Pa)	348.43e+9	−3.070e−4	2.160e−7	−8.946e−11
	α (K⁻¹)	5.8723e−6	9.095e−4	0	0
	ρ (kg/m³)	2370	0	0	0
	v	0.24	0	0	0
SUS304	E (Pa)	201.04e+9	3.079e−4	−6.534e−7	0
	α (K⁻¹)	12.330e−6	8.086e−4	0	0
	ρ (kg/m³)	8166	0	0	0
	v	0.3262	−2.002e−4	3.797e−7	0

The effect of length and circumferential wave numbers on natural frequency of cylindrical FG microshell in different boundary conditions and thermal loadings

Table 5 represents the frequencies of the first three modes of a FG microshell with different boundary conditions and linear and nonlinear thermal loadings. As it can be seen, with increase in length to radius ratio in all three modes, frequency tends to decrease and this is obvious in all boundary conditions. Frequency increases noticeably in higher modes in all boundary conditions. The case study with clamped-free boundary conditions has the lowest natural frequency while the nanoshell with clamped-clamped boundary conditions has the highest natural frequency because of its rigidity. It is worthy to mention that the influence of boundary condition on the first frequency mode is much stronger than the second mode, and it can be said that the second frequency mode does not have the same sensitivity to boundary condition such as the first mode. This table also states that the effect of nonlinear temperature variations on natural frequency is stronger than linear temperature variations, so with applying nonlinear temperature variations, frequencies are higher in comparison with linear temperature variations which it is observable in all the modes.

Table 5:

The effect of different length to radius ratios and circumferential wave numbers on natural frequency of FG microshell with l_m=14 μm, l_c=l_m/2, n=1, K_w=0, K_p=100, ΔT=100, C_d=10⁶, F=0.5 with different B.Cs and thermal loadings.

	LT		NLT
	Fundamental frequency	Second frequency	Fundamental frequency	Second frequency
Simply-Simply
L/R
8	0.64904673	0.92539502	0.65532067	0.93079032
10	0.48473968	0.90507715	0.49065507	0.90842492
12	0.39914599	0.8964360	0.40443196	0.89860392
14	0.22213244	0.89212083	0.2320852	0.89355647
Simply-Clamp
L/R
8	0.80275775	0.95537184	0.80698359	0.96094347
10	0.603442	0.91964442	0.60770842	0.92324392
12	0.48624575	0.90428343	0.49037063	0.90667583
14	0.41509642	0.89672695	0.41894127	0.89834473
Clamp-Clamp
L/R
8	0.95904831	0.99759566	0.96141484	1.0027600
10	0.72990133	0.94132908	0.73250228	0.94481693
12	0.58592689	0.91622601	0.58864518	0.91861723
14	0.49215032	0.90376117	0.49487374	0.90541712
Clamp-Free
L/R
8	0.40706672	0.91932612	0.40810866	0.92102735
10	0.21487038	0.90350584	0.21655879	0.90469552
12	0.16963173	0.89594765	0.17063477	0.89676681
14	0.14953214	0.8919226	0.15025047	0.8924648

3.5 The effect of Winkler foundation

Figures 5–8 illustrate the effect of Winkler foundation on natural frequency in absence of Pasternak foundation and damping for a microshell with L/R=10, h/R=0.1, l_m=14 μm, l_c=l_m, n=1, F=1 and linear temperature changes. They respectively are related to the clamped-clamped, clamped-simply, simply-simply and clamed-free boundary conditions. As it can be seen in all four figures, with increase in Winkler stiffness, natural frequencies increase. Also, increase in temperature leads to decrease in natural frequencies especially with high temperature variations. The other remarkable point is related to Figure 8, where the effect of temperature on natural frequency of clamed-free boundary conditions is less than three others. It is notable that, in clamped-free boundary conditions, with increasing temperature gradient, decrease of the frequency is less than other boundary conditions.

Figure 5:

The effect of Winkler stiffness on natural frequency with clamped-clamped boundary condition.

Figure 6:

The effect of Winkler stiffness on the natural frequency with clamped-simply boundary condition.

Figure 7:

The effect of Winkler stiffness on the natural frequency with simply-simply boundary condition.

Figure 8:

The effect of Winkler stiffness on the natural frequency with clamped-free boundary condition.

3.6 The effect of Pasternak foundation

Figures 9–12 illustrate the effect of Pasternak foundation on natural frequency in absence of Winkler foundation and damping with L/R=10, h/R=0.1, l_m=14 μm, l_c=l_m, n=1, F=1 and linear temperature changes. They are respectively related to clamped-clamped, clamped-simply, simply-simply and clamed-free boundary conditions; as it is obvious in all figures, with increasing the Pasternak stiffness, the natural frequencies tend to increase. Also, increase in temperature causes to decrease in natural frequencies, especially in high temperature changes. Also this kind of foundation has stronger influence on frequency changes imposed by high temperatures in comparison with Winkler. In clamped-free, frequency variations are modest.

Figure 9:

The effect of Pasternak stiffness on the natural frequency with clamped-clamped boundary condition.

Figure 10:

The effect of Pasternak stiffness on the natural frequency with clamped-simply boundary condition

Figure 11:

The effect of Pasternak stiffness on the natural frequency with simply-simply boundary condition.

Figure 12:

The effect of Pasternak stiffness on the natural frequency with clamped-free boundary condition.

3.7 The effect of damping

Figures 13–16 illustrate the effect of damping on natural frequency in the absence of winkler foundation and Pasternak foundation with L/R=10, h/R=0.1, l_m=14 μm, l_c=l_m, n=1, F=1 and linear temperature changes. Figures 13–16 respectively are related to clamped-clamped, clamped-simply, simply-simply and clamed-free boundary conditions. It is clear in figures which by increasing the temperature gradient, there is a decreasing trend in natural frequency. Also increase in temperature leads to decrease in natural frequencies but the decreasing occurs with a higher rate in foundations with greater stiffness. It is observable that in clamped-free boundary conditions and temperature gradient in the range of 0 to 900, frequency variations curve has a pick which is related to the critical frequency but in other boundary conditions, this phenomenon occurs in higher temperature gradients.

Figure 13:

The effect of damping stiffness on natural frequency with clamped-clamped boundary condition.

Figure 14:

The effect of damping stiffness on the natural frequency with clamped-simply boundary condition.

Figure 15:

The effect of damping stiffness on the natural frequency with simply-simply boundary condition.

Figure 16:

The effect of damping stiffness on the natural frequency with clamped-free boundary condition.

4 Conclusion

This paper represents the temperature-dependent vibration analysis of a cylindrical FG microshell surrounded by viscoelastic foundation. The modified couple stress theory considers size-dependency effect and GDQ method is used to solve the equations of motion. Functionally graded length scale parameter changing along the thickness has been considered for the first time in the present study. The equations of the motion and the non-classic boundary conditions are derived using Hamilton’s principle. The natural frequency of cylindrical FG microshell is investigated regarding length to radius ratio, damping, Winkler and Pasternak foundations, different temperature changes, FG power index, circumferential wave numbers, thermal loadings, and boundary conditions of a cylindrical microshell. In this study, following results can be achieved:

Increase in the length to radius ratio and Winkler and Pasternak foundations coefficients, the natural frequency tends to decrease while an increase in the circumferential wave number and damping coefficient result to increase of the natural frequency of the cylindrical microshell.
By increasing the FG power index, increase or decrease of the natural frequency depends on the damping coefficient.
The effect of nonlinear temperature variations on the natural frequency is more than linear temperature variations; so, applying nonlinear temperature variations, frequencies go higher in comparison with linear temperature variations.
In the first frequency mode, the case with clamped-free boundary conditions has the lowest natural frequency because of its particular condition, while, case study with clamped-clamped boundary conditions has the highest natural frequency.
By increasing the temperature, the natural frequency tends to decrease, this is because, increasing the temperature leads to decreasing the stiffness and so the natural frequency of the cylindrical microshell.

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Published Online: 2017-8-29

Published in Print: 2017-4-25

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Temperature-dependent vibration analysis of a FG viscoelastic cylindrical microshell under various thermal distribution via modified length scale parameter: a numerical solution

Abstract

1 Introduction

2 Preliminaries

2.1 Temperature-dependent cylindrical FG microshell equations

2.2 Modified couple stress theory

2.3 Displacement field for cylindrical microshell

2.4 Governing equations and associated boundary conditions

2.5 Solution procedure

3 Results and discussion

3.1 Convergence

3.2 Validation of the results with other articles

3.3 Validating the results with analytical method

3.4 Parametric results

3.5 The effect of Winkler foundation

3.6 The effect of Pasternak foundation

3.7 The effect of damping

4 Conclusion

References

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