Meccanica (2012) 47:1441–1453DOI 10.1007/s11012-011-9527-9

Dynamic analysis of thick short length FGM cylinders

Kamran Asemi · Mehdi Akhlaghi ·Manouchehr Salehi

Received: 8 August 2010 / Accepted: 10 November 2011 / Published online: 8 December 2011© Springer Science+Business Media B.V. 2011

Abstract In this paper a thick short length hol-low cylinder made of functionally graded materials(FGMs) under internal impact loading is considered.The inner surface of the cylinder is pure ceramic, theouter surface is pure metal, and the material compo-sition varies continuously along its thickness. FiniteElement Method based on Rayleigh-Ritz energy for-mulation has been applied to study the propagation ofelastic waves in FG thick hollow cylinders. The New-mark direct integration method is applied to solve thetime dependent equations. The time histories of dis-placements, stresses, wave propagation in two direc-tions and velocities of radial stress wave propagationfor various values of volume fraction exponent havebeen investigated. Also by using fast Fourier trans-form, the first natural frequencies for FG cylinderswith simply-simply and clamped-clamped ends con-ditions are illustrated. The model has been comparedwith result of a plane strain FG thick hollow cylinderwhich is subjected to an internal impact loading, andit shows very good agreement.

K. Asemi · M. Akhlaghi (�)Mechanical Engineering Department, AmirkabirUniversity of Technology, Tehran, 1591634311, Irane-mail: makhlagi@aut.ac.ir

M. SalehiMechanical Engineering Department and ConcreteTechnology and Durability Research Centre, AmirkabirUniversity of Technology, Tehran, 1591634311, Iran

Keywords Functionally graded materials · Shortlength cylinder · 2D-wave propagation · Internalimpact loading

1 Introduction

Functionally Graded Materials (FGMs) are compositematerials, microscopically inhomogeneous, in whichthe mechanical properties vary smoothly and continu-ously from one surface to the other. The applicationsof FGMs in engineering designs were first started byLee et al. [1]. These materials are increasingly usedin various applications to maximize strengths and in-tegrities of different engineering structures. Therefore,analysis of FG thick hollow cylinder under impactloading is necessary for practical purposes in order tooptimize their resistance to failure. However investiga-tions into wave propagation and dynamic analysis forFG structures by using numerical and analytical meth-ods are presented in [1–16].

Hosseini and Abolbashari [17] proposed an ana-lytical method for the dynamic response analysis ofinfinite functionally graded thick hollow cylinders un-der impact loading. Some works can be found in theliterature on numerical determination of wave prop-agation in FG cylinders. Houillon et al. [18] studiedthe characterization of propagation constants of re-alistic thin-walled structures with any cross-section.They proposed a propagative approach using finite

1442 Meccanica (2012) 47:1441–1453

element method in order to extract propagation pa-rameters and calculating the dispersion curves of re-alistic thin-walled structures. Manconi and Mace [19]proposed a wave finite element method for the nu-merical prediction of wave characteristics of cylin-drical and curved panels as isotropic, orthotropic andlaminated sandwich constructions. Their method com-bined conventional finite element method and the the-ory of wave propagation in periodic structures. A non-linear Hermitian transfinite element method for tran-sient behavior analysis of hollow functionally gradedcylinders with temperature-dependent materials un-der thermo-mechanical loads has been investigatedby Shariyat [20]. Hosseini et al. [21] obtained thedynamic behavior of cylinders made of functionallygraded materials. The functionally graded cylinderwas assumed to be made of many subcylinders. Shak-eri et al. [22] studied the vibration and radial wavepropagation velocity in FG thick hollow cylinder,the mean velocity of radial stress wave propagationand the first natural frequency for different valuesof the power law exponent were determined. A nu-merical method for calculating guided wave propa-gation in an infinite cylinder composed of function-ally graded materials was presented by Elmaimouniet al. [23]. Nonlinear transient thermal stress andelastic wave propagation analyses for hollow thicktemperature-dependent FGM cylinders subjected todynamic thermomechanical loads have been devel-oped by Shariyat [24]. A numerical method is pro-posed by Han et al. [25] for analyzing transient wavesin cylindrical shells of a functionally graded materialexcited by impact point loads. Azadi and Shariyat [26]studied transient heat transfer of a thick FGM cylinderwith temperature-dependent material properties by us-ing nonlinear transient transfinite element method. El-Raheb [27] has been used Galerkin method to studythe effects on transient waves of circumferential andradial inhomogeneity in a plane-strain hollow cylinder.Honarvar et al. [28] studied wave propagation in trans-versely isotropic cylinders. Asgari et al. [29] obtainedthe dynamic behavior of two-dimensional FG thickhollow cylinder with finite length. Tornabene and Vi-ola [30] used GDQ method to investigate the dynamicbehavior of functionally graded parabolic and circularpanels and shells of revolution.

The main aim of the present paper is to presentdynamic analyses and 2D stress wave propagationfor thick short length FGM cylinders. Investigations

into dynamic analysis of FG thick hollow cylindersare mainly in plane-strain condition, see referencesin [20–27]. The difficulty in obtaining analytical so-lutions for dynamic response of graded material sys-tems is compounded by the dispersive nature of theheterogeneous material systems, which is character-istic of elastodynamic response of materials with thecontinuously varying properties. In consequence, thewave propagation speed is not constant and there isno sharp interface that makes wave reflections. There-fore, analytical or semi-analytical solutions are avail-able only through a number of problems with simpleboundary/initial conditions. Also investigations intoelastic analysis of finite length FG hollow cylinderare mostly under the static loads. For example, Shaoet al. [31, 32] obtained analytically the elastic behaviorof finite length FG hollow cylinders with simply sup-ported end conditions under the hydrostatic pressures.In these publications, in order to satisfy the boundaryconditions, the sinusoidal load along the axial direc-tion can only be used. Therefore to find the solutionfor a FGM cylinder which is subjected to any arbi-trary loading function, especially uniform internal im-pact pressure, finite element method is used. In thispaper, material properties have been interpolated us-ing linear shape functions. Consequently, the materialcomposition varies continuously through the cylinderand artificial wave reflections from the boundary ofhomogeneous elements [21, 22] are avoided. To solvethe time dependent equations, Newmark direct inte-gration method with suitable time step has been used.The time histories of displacements, stresses and ve-locities of radial stress wave propagation for variousvalues of volume fraction exponent have been inves-tigated. Additionally, by using the fast Fourier trans-form, the first natural frequencies for FG cylinderswith simply-simply and clamped-clamped ends con-ditions have been presented.

2 Material gradient and geometry of thick hollowcylinder

Consider a thick hollow cylinder as shown in Fig. 1,where a and b are the radius of inner and outer sur-faces, respectively. L is the length of cylinder; r and z

are the axis of cylindrical coordinate system.The cylinder is made of a combined ceramic-metal

material and the material composition varying contin-

Meccanica (2012) 47:1441–1453 1443

Fig. 1 Geometry of the cylinder

uously along its thickness (r direction). The inner sur-face of the cylinder is pure ceramic, and the outer sur-face is pure metal. The material distribution can be ex-pressed as following, according to [21]

E = Ec + (Em − Ec)

(r − a

b − a

)n

(1a)

ρ = ρc + (ρm − ρc)

(r − a

b − a

)n

(1b)

where E is the Young’s modulus, ρ is the mass density,n is a non-negative volume fraction exponent, sub-scripts “c” and “m” stand for ceramic and metal. Itshould be noted that the Poisson’s ratio is assumed tobe constant through the body. This assumption is rea-sonable because of the small differences between thePoisson’s ratios of basic materials.

3 Governing equations

Consider a thick hollow cylinder as shown in Fig. 1.Axial symmetry is assumed, so the formulation isreduced to two dimensions. The shear stresses andstrains in r–θ and θ–z planes are zero due to axisym-metric geometry and load distribution. By neglectingthe body forces, the equations of motion in axisym-metric cylindrical coordinates could be obtained as

∂σrr

∂r+ (σrr − σθθ )

r+ ∂τrz

∂z= ρ(r)

∂2u

∂t2(2)

∂τrz

∂r+ ∂σzz

∂z+ τrz

r= ρ(r)

∂2v

∂t2(3)

The stress-strain relations [33] from the Hook’s law inmatrix form are

⎧⎪⎪⎨⎪⎪⎩

σrr

σθθ

σzz

τrz

⎫⎪⎪⎬⎪⎪⎭

= [D]

⎧⎪⎪⎨⎪⎪⎩

εrr

εθθ

εzz

εrz

⎫⎪⎪⎬⎪⎪⎭

(4)

where the coefficients of elasticity is [33]

[D] = E

(1 + υ)(1 − 2υ)

×

⎡⎢⎢⎣

1 − υ υ υ 0υ 1 − υ υ 0υ υ 1 − υ 00 0 0 1−2υ

2

⎤⎥⎥⎦ (5)

In (5), E is the Young’s modulus that depends onr-coordinates, and υ denotes Poisson’s ratio which isassumed to be constant.

The strain-displacement relations for this analysisand in matrix form are according to [33]

{ε} = [d]{f } (6a)

{f } ={

u

ν

}(6b)

[d] =

⎡⎢⎢⎢⎣

∂∂r

01r

0

0 ∂∂z

12

∂∂z

12

∂∂r

⎤⎥⎥⎥⎦ (6c)

where u and ν are the displacement components alongthe coordinates r , z.

The boundary conditions for cylinders with sim-ply supported ends and clamped ends are shown in(7a)–(7e).

u(r,0) = u(r,L) = 0 (7a)

u(r,0) = ν(r,0) = u(r,L) = ν(r, l) = 0 (7b)

σrr(rin, z) = pr(t) (7c)

τrz(rin, z) = τrz(rout, z) = σrr(rout, z) = 0 (7d)

σzz(r,0) = σzz(r,L) = 0 (7e)

It should be noted that (7e) is not satisfied for acylinder with clamped ends conditions.

1444 Meccanica (2012) 47:1441–1453

4 Finite element modeling

In order to solving governing equations, finite elementmethod is used. The finite element approximation ofthe domain is in the r–z plane, which is the plane ofrevolution. The section of the cylinder in the r–z planeis considered and divided into a number of simplex lin-ear triangular elements. Assuming shape function N

for the displacement vector, the displacement matrixfor each element (e) in terms of the nodal displace-ment matrix {δ} and shape function [N ] is as follow-ing:

{f }(e) = [N ](e){δ}(e) (8)

Substituting (8) in (6a) gives the strain matrix of ele-ment (e) as

{ε}(e) = [d][N ](e){δ}(e) (9a)

{ε}(e) = [B](e){δ}(e) (9b)

where [B](e) = [d][N ](e). Due to the nature of opera-tor matrix [d] in cylindrical coordinates, matrix [B] isin general a function of the variable r . For the simplexlinear triangular element, the matrix [N ] is

[N ](e) =[Ni 0 Nj 0 Nk 00 Ni 0 Nj 0 Nk

](10)

where Ni , Nj and Nk are the shape functions of trian-gular element. The components of matrix [B](e) and[N ](e) are given in the Appendix.

The finite element model can be derived usingRayleigh Ritz energy formulation. The details of thismethod could be found in different textbooks [33]. Byapplying this method to the governing equations, thestiffness, mass and force element matrices can be writ-ten as

[K](e) =∫

V(e)

[B]T [D][B]dV (11)

[M](e) =∫

V(e)

ρ[N ]T [N ]dV (12)

[F ](e) =∫

A(e)

[N ]T {P(t)}dA (13a)

where

{P(t)} ={

pr(t)

0

}(13b)

where pr(t) is the internal pressure which varies withthe time. V(e), A(e) and {P(t)} are volume of element,area of element under pressure and vector of surfacetractions, respectively. Since the cylinder is under theinternal pressure, the axial component of vector ofsurface tractions in (13b) is assumed to be zero. Totreat the material non-homogeneity inherent in (11)and (12), elements with linearly variation of materialproperties are used. The same shape functions are em-ployed to interpolate the displacements and materialproperties.

Now by assembling the element matrices, theglobal dynamic equilibrium equations for the FG thickhollow cylinder can be obtained as

[M]{δ̈} + [K]{δ} = {F } (14)

Once the finite element equilibrium equation is estab-lished, different numerical methods can be employedto solve (14) in space and time domains. To solve theequilibrium equation, the Newmark direct integrationmethod [33] is used. Newmark integration parameters[33] are taken as: γ = 1

2 and β = 14 , which lead to a

constant average acceleration. This choice of param-eters corresponds to a trapezoidal rule which is un-conditionally stable in linear analyzes. Moreover, toachieve convergent results and to show the propaga-tion of elastic waves properly, the time step is taken as2e−6 (s).

5 Results and discussions

5.1 Validation

The present results can be verified using data of aplane strain FG thick hollow cylinder under the sameloading that were previously presented in [21]. Thecylinder is assumed to be long enough such that the as-sumptions of the plane strain condition holds. The ra-dial displacement at the middle of the thickness is con-sidered here, and the present result for n = 0.5 is com-pared with the published data. Figure 2 shows goodagreement between these results.

5.2 Numerical results and discussions

A simply supported ends FG thick hollow cylin-der with inner radius a = 1 (m) and outer radiusb = 1.5 (m) and length L = 1 (m) is considered

Meccanica (2012) 47:1441–1453 1445

Fig. 2 Time history of radial displacement for middle point of thickness and n = 0.5, compared with [21]

Fig. 3 Time history of radial displacement at r = 1.25 m, z = 0.5 m for different values of n

here. The modulus of elasticity and mass density atthe inner surface (ceramic) are Ec = 380 GPa andρc = 3800 (Kg/m3) and also at the outer surface(metal) are Em = 70 GPa and ρm = 2707 (Kg/m3);Poisson’s ratio is taken as constant value of υ = 0.3.The loading function equation is assumed accordingto [21] as:

P(t) ={

P0t t ≤ 0.005 (s)0 t > 0.005 (s)

where P0 is assumed as 4 GPa/s. The thick hollowcylinder is excited by unloading in t = 0.005 (s). It isobvious that after the unloading, a transient vibration

which is affected by the wave propagation, reflectionand interference would be occurred.

The results for different values of the power lawexponent (n) are presented and discussed as following.

A comparison of the radial displacement responsesbetween the various values of the power law expo-nent are presented in Figs. 3, 4 and 5. Figures 3 and4 show the time histories of the radial displacementsat r = 1.25 m, z = 0.5 m and r = 1.125 m, z = 0.25 mfor three values of the power law exponents. These fig-ures show, while the power law exponent increases,the amplitude of vibration decreases. This is becauseof increasing the modulus of elasticity or increasing

1446 Meccanica (2012) 47:1441–1453

Fig. 4 Time history of radial displacement at r = 1.125 m, z = 0.25 m for different values of n

Fig. 5 Time history of radial displacement at r = 1.25 m, z = 0.5 m for n = 5 and n = 20

the mechanical stiffness of the cylinder. A comparisonbetween Figs. 3 and 4 shows that the amplitude of vi-bration for certain value of the power law exponent fornode located at r = 1.25 (m), z = 0.5 (m) is more thanfor the node located at r = 1.125 (m), z = 0.25 (m).For large values of the power law exponent, the ampli-tude would be constant. This means that the materialdistribution for these values tends toward the behav-ior of the cylinder which is made of full ceramic, seeFig. 5. A comparison between presented radial dis-placement responses in this paper with those one in[21, 22] shows that in a thick short length FG cylinder,the first natural frequencies in contrast to a plane strainFG cylinder are decreased while the pick value of theradial displacement response are increased.

By using the fast Fourier transform, the first natural

frequencies for different values of the power law ex-

ponent for simply-simply and clamped-clamped ends

cylinders are extracted and the results are shown in Ta-

ble 1. The results represent that the first natural fre-

quencies for a clamped ends FG cylinder are higher

than the simply supported one. This is because of in-

creasing the stiffness of the end conditions, therefore

it can be concluded that the amplitude of vibration for

clamped ends cylinder is lower than the simply sup-

ported one. Moreover, results show that by increas-

ing the power law exponent, the natural frequencies

are increasing too. This behavior is due to increase in

Young’s modulus of elasticity.

Meccanica (2012) 47:1441–1453 1447

Table 1 The first natural frequency for different values of n, for simply-simply and clamped-clamped ends FG cylinders

n 0 0.01 0.5 2 5 20

Simply-simply, Freq (Hz) 864.3 881.8 1385 1510.6 1600 1666.66

Clamped-clamped, Freq (Hz) 1060.5 1081.1 1650.5 1855.3 1923.1 2087.7

Fig. 6 Time history of radial stress at r = 1.25 m, z = 0.5 m for two values of n

Fig. 7 Time history of radial stress at r = 1.125 m, z = 0.25 m for two values of n

The time history of radial stress at r = 1.25 (m),z = 0.5 (m) and r = 1.125 (m), z = 0.25 (m) in a veryshort time after the unloading for two values of thepower law exponent are shown in Figs. 6 and 7, re-

spectively. As it can be seen from these figures, themaximum magnitude of radial stress for node locatedat r = 1.25 (m), z = 0.5 (m) is greater than the otherone.

1448 Meccanica (2012) 47:1441–1453

Fig. 8 Time history of tangential stress at r = 1.25 m, z = 0.5 m for different values of n

Fig. 9 Time history of axial stress at r = 1.25 m, z = 0.5 m for two values of n

The time histories of tangential and axial stressesat r = 1.25 (m), z = 0.5 (m) for different values of thepower law exponent are shown in Figs. 8 and 9, respec-tively. Figure 10 shows the time history of shear stressat r = 1.125 (m), z = 0.25 (m) for different values ofthe power law exponent. Results show that the max-imum magnitude of tangential stress in a very shorttime after the unloading is greater than the radial, ax-ial and shear stresses. As it is seen from the results, thetime history of stresses and displacements are stronglyaffected by the power law exponent.

Figures 11a–d show the radial stress wave propaga-tion in the 2D domain. These figures show, while the

power law exponent increases, the mean velocity ofradial stress wave propagation increases too, and moreparts of the domain are affected by the radial stresswave. By computing the time taken by the wave tocover the whole of the cylinder, the approximate ra-dial stress wave propagation velocities are calculatedand the mean results are shown in Table 2. Accordingto the results shown in Table 2, the accuracy of presentsolution is acceptable. This is because the mean veloc-ities of present solution for n = 0 and n = 20 are closerto 5900 (m/s) and 11602.4 (m/s) [22] which belongto the cylinders made of full metal and full ceramic,respectively. The difference between the mean veloci-

Meccanica (2012) 47:1441–1453 1449

Fig. 10 Time history of shear stress at r = 1.125 m, z = 0.25 m for different values of n

Fig. 11 Radial stress distribution through the cylinder at t = 3 × 10−5 for (a) n = 0.01, (b) n = 0.5, (c) n = 5, (d) n = 20

ties of the present solution and those presented in [22]is that in [22] the cylinder is divided into many ho-mogenous subcylinders and this creates artificial wavereflections from the hard interfaces.

Figures 12, 13, 14 and 15 show the distributionof radial, tangential, axial and shear stresses throughthe simply supported cylinder at t = 0.004 (s) forn = 2, respectively. As the variational formulation is

1450 Meccanica (2012) 47:1441–1453

Fig. 12 Radial stress distribution through the cylinder at t = 0.004 (s), for n = 2

Fig. 13 Tangential stress distribution through the cylinder at t = 0.004 (s), for n = 2

Table 2 Mean velocity of radial stress wave propagation fordifferent values of n compared with [22]

n 0 0.01 0.5 5 20

Present (m/s) 6081 6473 8725 10562 11149

Ref. [17] (m/s) 6468 7143 8333 10000 11120

used for deriving the global equilibrium equation, thestress boundary conditions are satisfied during solu-tion process. It can be seen from Fig. 12, the stressboundary conditions which are shown in (7c) and

(7d) are satisfied. It should be noted as the cylinderis subjected to uniform internal pressure, the radialstress at inner surface of the cylinder for nodes whichare away from the boundaries are −16 MPa as ex-pected.

According to the assumed boundary conditionswhich are shown in (7e), the axial stress at the up-per and lower surfaces of the cylinder is zero, Fig. 14.Also it can be seen from Fig. 15, the shear stresses atinner and outer surfaces of the cylinder are zero, ac-cording to (7d). It is seen that the stress distributionshave continuous and smooth variations.

Meccanica (2012) 47:1441–1453 1451

Fig. 14 Axial stress distribution through the cylinder at t = 0.004 (s), for n = 2

Fig. 15 Shear stress distribution through the cylinder at t = 0.004 (s), for n = 2

6 Conclusions

The main objective of the present paper is to givedynamic analysis of functionally graded thick shortlength hollow cylinder. The Finite Element Method,and Rayleigh-Ritz energy formulation is applied. Theproposed method is validated by the result of an in-finite FG thick hollow cylinder under the same load-ing which is extracted from published literature. Thecomparisons between the results show that the presentmethod has a good compatibility with the existing re-sults. The time histories of displacements, stresses,wave propagation in two directions and also the veloc-ity of radial stress wave propagation were investigated

for various values of volume fraction exponent. Addi-tionally, by using the fast Fourier transform, the firstnatural frequencies for two different boundary condi-tions were obtained. Results show that the power lawexponent is a useful parameter from a design pointof view in that it can be tailored for specific appli-cations to control the stress distribution. Advantagesof the present method are its applicability to any FGmaterial model and applying any boundary conditionsand loading. Also results demonstrate that using lin-ear shape functions to interpolate material propertiesprovide smoother and more accurate results than ho-mogeneous elements for dynamic problems.

1452 Meccanica (2012) 47:1441–1453

Appendix

For the simplex linear triangular element the followingformulation is used.

Ni = ai + bir + ciz

2A(A.1)

Nj = aj + bj r + cj z

2A(A.2)

Nk = ak + bkr + ckz

2A(A.3)

The constants a, b and c are defined in terms of thenodal coordinates and A is the area of the element.

ai = rj zk − rkzj (A.4)

bi = zj − zk (A.5)

ci = rk − rj (A.6)

aj = rkzi − rizk (A.7)

bj = zk − zi (A.8)

cj = ri − rk (A.9)

ak = rizj − rj zi (A.10)

bk = zi − zj (A.11)

ck = rj − ri (A.12)

A = 1

2det

⎡⎣ 1 1 1

ri rj rkzi zj zk

⎤⎦ (A.13)

[B](e) = 1

2A

⎡⎢⎢⎢⎢⎣

bi 0 bj 0 bk 02ANi

r0

2ANj

r0 2ANk

r0

0 ci 0 cj 0 ck

ci

2bi

2cj

2bj

2ck

2bk

2

⎤⎥⎥⎥⎥⎦

(A.14)

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