Journal of Stress AnalysisVol. 3, No. 2, Autumn − Winter 2018-19

The Effect of Grading Index on Two-dimensional Stressand Strain Distribution of FG Rotating Cylinder Restingon a Friction Bed Under Thermomechanical Loading

M. Omidi bidgoli, A. Loghman∗, M. ArefiDepartment of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Iran.

Article info

Article history:Received 2 December 2018Received in revised form11 March 2019Accepted 13 March 2019

Keywords:Grading indexFG rotating cylinderStress and strainThermomechanical loadingFriction bed

Abstract

This paper presents two-dimensional stress and strain behavior of a FGrotating cylindrical shell subjected to internal-external pressure, surface shearstresses due to friction, an external torque, and constant temperature field.A power law distribution was considered for thermomechanical materialproperties. First order shear deformation theory (FSDT) was used to definethe displacement and deformation field. Energy method and Euler equationwere employed to derive constitutive differential equations of the rotatingshell. Systems of Six differential equations were achieved. Eigenvalue andeigenvector methods were used to solve these equations. It was found thatthe material grading index has a significant effect on stresses and strainsof a rotating functionally graded material cylindrical shell in radial andlongitudinal directions.

Nomenclaturer Radius of an arbitrary layer of cylinder Ri Inner radiusz Coordinate of arbitrary layer of cylinder

respect to middle surfaceVθ Circumferential component of deforma-

tionR Radius of mid-surface of cylinder Ti Inner temperatureUr Radial component of deformation T0 Outer temperatureWx Axial component of deformation εrr Radial strainR0 outer radius εxx Axial strainu Displacement component of radial defor-

mationw Displacement component of axial defor-

mationεθθ Circumferential strain γrx Shear strain in rx planev Displacement component of circumferen-

tial deformationϕz Rotational component of radial deforma-

tionγrθ Shear strain in rθ plane γxθ Shear strain in xθ planeϕx Rotational component of axial deforma-

tionϕθ Rotational component of circumferential

deformationσrr Radial stress σxx Axial stressE Modulus of elasticity σθθ Circumferential stress

∗Corresponding author: A. Loghman (Professor)E-mail address: aloghman@kashanu.ac.irhttp://dx.doi.org/10.22084/jrstan.2019.17619.1073ISSN: 2588-2597

75

U Total energy τrx Shear stress in rx planePi Inner pressure τrθ Shear stress in rθ planeP0 Outer pressure τxθ Shear stress in xθ planeF General potential function σeff Effective stress

1. Introduction

Functionally graded materials (FGMs) are advancedcomposites in which element gradation changes con-tinuously from metal to ceramic phase or vice versadepending on the requirements. Continuous changesin the FGM composition results in improving the me-chanical and thermal properties. Bewar and Duwez[1] first explained and introduced the FGM concepttheoretically. However, FGM was successfully devel-oped by Japanese scientists in 1984 for an aerospaceapplication to attain a thermal gradient of 1000Kalong 10mm cross-section [2]. Since then FGM hasdrawn more attention in most of the engineering utili-ties such as carriage systems, medical application, nu-clear components, space vehicle components, energystorage systems, aero engine components, thermal pro-tection packages for high temperature environmentsand etc. [3]. Shells are structural elements widely usedin applications such as mechanical, civil, aeronautical,and marine engineering. Shell structures are used asroofs, liquid storage vessels, nuclear plant accessories,piping structures, and pressure vessels [4].

Hollow composite cylindrical shells have many en-gineering uses because composite materials have tailor-ing properties, less weight, low maintenance, and highperformance with increased service life. Functionallygraded material is a type of composite appropriate forbulk and shell material application. With increasingusage of cylindrical shells and FGM in various applica-tions, it is necessary to investigate the performance ofFGM cylindrical shells at different working load condi-tions. Cylindrical shells mainly fail due to axial com-pressive stresses, buckling loads and large deformationsdue to internal pressure.

Various problems of FGM have attracted consid-erable attention in recent years. That is an impor-tant topic in engineering because of many rigorous ap-plications. The study on the stresses and strains inrotating hollow cylinders has never stopped becauseof the importance of these basic elements in severalmechanical, building, power and computer engineer-ing applications. Loghman and Wahab [5] studiedthe thermo-elasto-plastic and residual stresses in thick-walled cylindrical pressure vessels of strain hardeningmaterial. Horgan and Chan [6] solved the classic prob-lem of stress distribution in an inhomogeneous isotropicrotating solid disc and pressurized hollow cylinder.

Moradi et al. [7] investigated reverse yielding andthe Baushinger effect on residual stresses in thick-walled cylinders. Tutuncu and Ozturk [8] calculatedthe stress distribution in an axisymmetric structure.

They derived closed-form solutions for the stresses anddeformations of functionally graded cylindrical andspherical shells under internal pressure. Ghorbanpouret al. [9] presented the Bauschinger and hardening ef-fect on residual stresses in thick-walled cylinders of SUS304. A computational study on functionally gradedrotating solid shafts was carried out by argeso andEraslan [10]. Displacements and stresses of rotatingFGM thick hollow cylindrical shell under internal pres-sure and thermal load was studied by Zamani nejadand Rahimi [11].

Using plane theory of elasticity and procedure ofcomplementary functions, Tutuncu and Temel [12] de-termined axisymmetric deformations and stresses infunctionally graded hollow cylinders, disks, and spheresunder uniform internal pressure. Eipakchi [13] de-rived stresses and displacements of a thick conicalshell with variable thickness subjected to distributednonuniform internal pressure analytically using third-order shear deformation theory (TSDT). Azturk andGalgec [14] studied elastic-plastic stress in a long func-tionally graded solid cylinder with fixed ends subjectedto uniform heat generation. Khorshidvand and Javadi[15] investigated deformation and stresses in FG ro-tating hollow disk and cylinder Subjected to Thermaland Mechanical Load. Ghannad et al. [16] inves-tigated elastic behavior of pressurized thick cylindri-cal shells with variable thickness made of functionallygraded materials using FSDT. Zamani nejad et al. [17]studied the Effect of exponentially-varying propertieson Displacements and Stresses in Pressurized Func-tionally Graded Thick Spherical Shells using IterativeTechnique. Fatehi and Zamani nejad [18] consideredthe effects of material gradients on the onset of yieldin fgm rotating thick cylindrical shells. Zamani nejadand Gharibi [19] considered the effect of material grad-ing index on stresses of thick FGM spherical pressurevessels with exponentially-varying properties.

Jabbari et al. [20] studied the effect of material gra-dient on stresses of FGM rotating thick-walled cylindri-cal shell with longitudinal variation of properties undernon-uniform internal and external pressure. Arefi etal. [21] investigated the effect of axially variable ther-momechanical loads on the 2D thermo-elastic responseof FG cylindrical shell. Singh et al. [22] investigatedstress and deformation of rotating cylindrical pressurevessel of functionally graded material modeled by Mori-Tanaka scheme. Habibi et al. [23] evaluated the stressintensity factor (SIF) in FGM thick-walled cylindricalvessel. Jabbari et al. [24] considered the analysis ofstress in rotating thick truncated conical shells withvariable thickness under thermomechanical loads.

The Effect of Grading Index on Two-dimensional Stress and Strain Distribution of FG Rotating CylinderResting on a Friction Bed Under Thermomechanical Loading: 75–82 76

Fig. 1. A rotating cylinder made of FGM material and the selected coordinate system.

From the abovementioned literature, one can real-ize that a rotating cylinder under an applied externaltorque on a friction bed has not yet been considered.

The main objective of this paper is stress and strainanalysis of a finite length rotating FG cylinder sub-jected to a distributed shear stress due to outer surfacefriction, an external driving torque, internal-externalpressure and a uniform temperature distribution usingFSDT.

2. Temperature Distribution

For the cylindrical shell in this study a steady statesymmetrical conduction heat transfer without heatgeneration was considered. The reduced heat conduc-tion equation in this case is written [25] as follows:

d

dr

(kr

dT

dr

)= 0 (1)

Using the power law distribution for the material ther-mal conductivity coefficient, (kT = rβk) the aboveequation can be written as:

d

dr

(rβ+1k

dt

dr

)= 0 ⇒ rβk

dT

dr+ rβ+1k

d2T

dr2= 0,

(2)T (r) = −A1r

−β +A2 (3)

A1 =Ti − T0

−r−βi + r−β

0

, A2 =Tir

βi − T0r

β0

rβi − rβ0, (4)

T (r) = − Ti − T0

−r−βi + r−β

0

r−β +Tir

βi − T0r

β0

rβi − rβ0(5)

where Ti and T0 are the inner and outer temperaturesat ri and r0 respectively.

Fig. 2. Temperature distribution versus radius for fivegrading index.

3. Formulation Based on the FSDTTechnique

In the FSDT, the assumption is that the planes nor-mal to the mid-plane remain plane after deformationbut not necessarily perpendicular to it after loadingand the consequent deformations. In this case, shearstrain and stress are considered. In the classical theoryof shells, it is assumed that the planes normal to themid-plane remain plane even after deformation occurs.

According to the selected coordinate system (r, x, θ)one can write:

r = R+ z, −h

2≤ z ≤ +

h

2(6)

where h and L are the shell thickness and length of thecylinder.

The general axisymmetric displacement field(Ur,Wx, Vθ) according to Mirsky-Hermann’s first-ordertheory is expressed on the basis of axial and radial dis-

Journal of Stress Analysis/ Vol. 3, No. 2/ Autumn − Winter 2018-19 77

placements, as follows:

Ur(x, θ) = U(x, θ) + Zϕr(x, θ)

Wx(x, θ) = W (x, θ) + Zϕx(x, θ) (7)Vθ(x, θ) = V (x, θ) + Zϕθ(x, θ)

where ϕr, ϕx and ϕθ are the middle surface rotationcomponents. Furthermore, U , W and V are the func-tions used to define the displacement field. The strain-displacement formulas in the cylindrical coordinate sys-tem are:

εrr =∂u

∂z+ z

∂ϕz

∂z+

∂z

∂zϕz = ϕz

εθθ =1

r(u+ zϕz) =

u

R+ z+ z

ϕz

R+ z

εxx =∂w

∂x+ z

∂ϕx

∂x

γrθ =1

r

∂uz

∂θ+

∂vθ∂z

− vθr

= ϕθ −v

r− z

ϕθ

r

γrx =∂wx

∂z+

∂vz∂x

= ϕx +∂u

∂x+ z

∂ϕz

∂x

γxθ =1

r

∂wx

∂θ+

∂vθ∂x

=∂v

∂x+ z

∂ϕθ

∂x

(8)

Stress-strain relations are written as follows:

σrr =E

(1 + v)(1− 2v)

[(1− v)εrr + v(εθθ + εxx)− (1 + v)αT ]

σθθ =E

(1 + v)(1− 2v)

[(1− v)εθθ + v(εrr + εxx)− (1 + v)αT ]

σxx =E

(1 + v)(1− 2v)

[(1− v)εxx + v(εrr + εθθ)− (1 + v)αT ]

τrθ = KE

2(1 + v)γrθ, τrx = K

E

2(1 + v)γrx,

τθx = KE

2(1 + v)γθx

(9)

According to the principle of virtual work, the vari-ations of strain energy must be set equal to the varia-tions of the external work as follows:

δU = δW (10)

where δU is the variation of total strain energy of theelastic body and δW is the variation of total externalwork due to internal, external pressure, friction force,and centrifugal body force. The strain energy is thenwritten as:

U =

∫∫∫v

udv =

∫ 2

0

π

∫ L

0

∫ +h/2

−h/2

ur dr dx dθ

= 2π

∫ L

0

∫ +h/2

−h/2

u(R+ z) dz dx

(11)

where:

⇒ u =1

2

E

(1 + v)(1− 2v)[[(1− v)](ε2rr + ε2θθ + ε2xx) + 2v(εrrεθθ + εrrεxx + εθθεxx)

−(1 + v)αT (εrr + εθθ + εxx) +K(1− 2v)

2[γ2

rθ + γ2rx + γ2

θx]

](12)

The external work is the sum of works due to internal,external pressure (W1), centrifugal body force (W2)and friction force (W3):

W = W1 +W2 +W3 (13)

where:

W1 =

∫ L

0

(d1u+ d2ϕz) dx

W2 =

∫ L

0

(H1u+H2ϕz) dx

W3 =

∫ L

0

(I1v + I2ϕz) dx

(14)

That:

d1 = 2π

[Pi(x)

(R− h

2

)− P0(x)

(R+

h

2

)]

d2 = 2πh

2

[−Pi(x)

(R− h

2

)− P0(x)

(R+

h

2

)]

H1 = 2πρ0ω2

(R2h+

R2h3

2+

h5

80

)(15)

H2 = 2πρ0ω2

(R2h3

3+

Rh5

40

)I1 = −2πµP0(x)

I2 = −2π

(h

2

)µP0(x)

Taking variation from energy relation we have:

U =

∫ L

0

(Us − UT ) dx−∫ L

0

(W1 +W2 +W3) dx

=

∫ L

0

F (u,w, v, ϕz, ϕx, ϕθ) dx (16)

Among the 6 variables in the relations obtained,Euler’s equations are:

∂F

∂qi− ∂

∂x

∂F

∂

(∂qi∂x

) = 0,

qi(i = 1, 2, 3, 4, 5, 6) = u, ϕr, w, ϕx, v, ϕθ

(17)

where F is general potential function.

The Effect of Grading Index on Two-dimensional Stress and Strain Distribution of FG Rotating CylinderResting on a Friction Bed Under Thermomechanical Loading: 75–82 78

By applying the Euler equations and using equa-tions written in appendix A system of differential equa-tions governing the problem is obtained as follows:

[G1]d2

dx2y+ [G2]

d

dxy+ [G3]y = F

y = u, ϕr, w, ϕx, v, ϕθT(18)

Matrix G1, G2, G3 and F are written in appendix B.Solving the differential equations, general and par-

ticular solutions are written as follows:

y = yg+ yp (19)

That general solution is represented as follows:

yg = viemx (20)

Finally, by substituting special values, the general so-lution is written as follow:

yg =10∑i=1

civiemix + c11x+ c12 (21)

c11x + c12 term is due to a pair of zero roots. Giventhe presence of mixed roots, these roots need to betransformed into a true form to continue solving [26]:

λi,i+1 = a± bi (22)

Then the special vectors derived from these roots willbe in the following form:

λi,i+1 = Γ± Ωi (23)

Particular solution consists of thermal and mechanicalcomponents as:

yp = yp1 + yp2 (24)

Finally, with determination of unknown coefficients thesolution is obtained as:

y =

10∑i=1

civiemix + c11x+ c12 + yp (25)

Substituting the solution into Eqs. (9), (10) thestresses and strains can be calculated where the for-mulation of effective stress and strain in terms of stressand strain components is as follows:

σe =1√2

[(σrr − σθθ)

2 + (σrr − σxx)2

+(σθθ − σxx) + 6τ2rθ + 6τ2rx + 6τ2θx

]12 (26)

εe =

√2

3

[(εrr − εθθ)

2 + (εrr − εxx)2

+(εθθ − εxx) + 6ε2rθ + 6ε2rx + 6ε2θx

]12 (27)

4. Results and Discussion

In this section the numerical results for effective stressand strain components are presented in terms of grad-ing index. Power law distribution for radial dependentproperties of FGM [3] is written as follows:

P = P0

(r

Ri

)β

(28)

where P represents a property and the P0 is the refer-ence material property.

The reference properties, geometry and loadingdata used in this paper for rotating FG hollow cylinderare assumed to be:

Table 1Basic material properties, geometry and loading data used inthis paper.E0 220GPav 0.3

α0 1.2× 10−06 1C

Ri 0.04mR0 0.06mPi 80MPaP0 30MPaTi 150CT0 70Cl 1m

4.1. Comparison and Validation

Before presentation of full numerical results, a compar-ison with other results is required. For this aim, the nu-merical analysis based on Abaqus package was selected.Shown in Fig. 3 is comparison between present resultsusing analytical method and corresponding results us-ing the Abaqus software for a case study. This com-parison indicates that the present results are in goodagreement with results of numerical analysis. Type ofelement that was used is C3D8T and number of ele-ments for Convergence were 50160.

Fig. 3. Comparison between the present results usinganalytical method and corresponding results using theAbaqus software.

Journal of Stress Analysis/ Vol. 3, No. 2/ Autumn − Winter 2018-19 79

Effective stress distribution in radial direction isshown in Fig. 4. For negative values of grading in-dex and for homogeneous material (β = 0) maximumvalues of effective stresses are located at the inner sur-face of the shell and their minimum values are locatedat the outer surface of the shell. However, there are nosignificant changes throughout thickness for positivevalues of grading index. In Fig. 5 shear strain ver-sus radius is shown and the minimum absolute valueof which belongs to a material identified by the grad-ing index β = +2 and the maximum absolute valuebelongs to β = −2.

In Figs. 6 and 7 shear strains in longitudinal di-rection are shown. Except for the end condition thereis no significant changes for different material proper-ties. Figs. 8, 9, and 10 present radial, longitudinal andtangential displacements, respectively.

Fig. 4. Effective stress (σeff ) versus radius.

Fig. 5. Shear strain (γrx) versus radius.

Fig. 6. Shear strain (γxθ) versus length.

Fig. 7. Shear strain (γrx) versus length.

Fig. 8. Radial displacement (Ur) versus radius.

Fig. 9. Longitudinal displacement (Wx) versus radius.

Fig. 10. Tangential displacement (Vθ) versus radius.

The Effect of Grading Index on Two-dimensional Stress and Strain Distribution of FG Rotating CylinderResting on a Friction Bed Under Thermomechanical Loading: 75–82 80

Fig. 11. Tangential displacement (Vθ) versus lengthfor external torque for (β = 0).

5. Conclusions

In this work, formulation of 3D thermo-elastic anal-ysis of an FG rotating cylinder subjected to inter-nal/external pressure and shear stresses due to fric-tion bed was performed using first-order shear defor-mation theory (FSDT). The mechanical properties ex-cept Poisson’s ratio were variable along the radial di-rection according to a power law distribution. By us-ing the boundary conditions, constant coefficients ofthe six differential equations were obtained. It is con-cluded that the grading index has a significant effecton stresses and strains.

References

[1] M.B. Bever, P.E. Duwez, Gradients in compositematerials, J. Mater. Sci. Eng., 10 (1972) 1-8.

[2] M.S. EL-Wazery, A.R. EL-Desouky, A reviewon functionally graded ceramic-metal materials,Mater. Environ. Sci., 6 (2015) 1369-1376.

[3] Y. Miyamoto, W.A. Kaysser, B.H. Rabin, A.Kawasaki, R.G. Ford, Functionally Graded Ma-terials: Design, Processing and Applications,Springer Publisher, (1999).

[4] A. Alibeigloo, A.M. Kani, M.H. Pashaei, Elastic-ity solution for the free vibration analysis of func-tionally graded cylindrical shell bonded to thinpiezoelectric layers, Int. J. Press. Vessels Pip., 89(2012) 98-111.

[5] A. Loghman, M.A. Wahab, Thermoelastoplasticand residual stresses in thick walled cylindricalpressure vessels of strain hardening material, Adv.Eng. Plast. Appl., (1993) 843-850.

[6] C.O. Horgan, A.M. Chan, The pressurized hollowcylinder or disk problem for functionally graded

isotropic linearly elastic material, J. Elast. 55(1)(1999) 43-59.

[7] M. Moradi, A. Ghorbanpour, H. Khademizadeh,A. Loghman, Reverse yielding and bauschinger ef-fect on residual stresses in thick-walled cylinders,Pakistan J. Appl. Sci., (2001) 44-51.

[8] N. Tutuncu, M. Ozturk, Exact solution for stressesin functionally graded pressure vessels, Compos.Part B: Eng., 32(8) (2001) 683-686.

[9] A. Ghorbanpour Arani, A. Loghman, H. Khadem-izadeh, M. Moradi, The bauschinger and harden-ing effect on residual stresses in thick-walled cylin-ders of SUS 304, Trans. Can. Soc. Mech. Eng.,26(4) (2003) 361-372.

[10] H. Argeso, A.N. Eraslan, A computational studyon functionally graded rotating solid shafts, Int.J. Comput. Methods Eng. Sci. Mech., 8(6) (2007)391-399.

[11] M. Zamani Nejad, G.H. Rahimi, Deformationsand stresses in rotating FGM pressurized thickhollow cylinder under thermal load, Sci. Res. Es-say, 4(3) (2009) 131-140.

[12] N. Tutuncu, B. Temel, A novel approach to stressanalysis of pressurized FGM cylinders, Disks andSpheres, Compos. Struct., 91(3) (2009) 385-390.

[13] H.R. Eipakchi, Third-order shear deformation the-ory for stress analysis of a thick conical shell underpressure, J. Mech. Mater. Struct., 5(1) (2010) 1-17.

[14] A. Ozturk, M.U. GUlgec, Elastic-plastic stressanalysis in a long functionally graded solid cylin-der with fixed ends subjected to uniform heat gen-eration, Int. J. Eng. Sci., 49(10) (2011) 1047-1061.

[15] A.R. Khorshidvand, M. Javadi, Deformation andstresses analysis in FG rotating hollow disk andcylinder subjected to thermal and mechanicalload, App. Mech. Mater., 187 (2012) 68-73.

[16] M. Ghannad, G.H. Rahimi, M. Zamani Nejad,Elastic analysis of pressurized thick cylindricalshells with variable thickness made of functionallygraded materials, Compos. Part B: Eng., 45(1)(2013) 383-396.

[17] M. Zamani Nejad, A. Rastgoo, A. Hadi, Effect ofexponentially-varying properties on displacementsand stresses in pressurized functionally gradedthick spherical shells with using iterative tech-nique, J. Solid. Mech., 6(4) (2014) 366-377.

[18] P. Fatehi, M. Zamani Nejad, Effects of materialgradients on onset of yield in FGM rotating thickcylindrical shell, Int. J. Appl. Mech., 6(4) (2014)1-20.

Journal of Stress Analysis/ Vol. 3, No. 2/ Autumn − Winter 2018-19 81

[19] M. Zamani Nejad, M. Gharibi, Effect of materialgradient on stresses of thick FGM spherical pres-sure vessels with exponentially-varying properties,J. Adv. Mater. Process., 2(3) (2014) 39-46.

[20] M. Jabbari, M. Zamani Nejad, M. Ghannad, Ef-fect of material gradient on stresses of FGM ro-tating thick-walled cylindrical pressure vessel withlongitudinal variation of properties under non-uniform internal and external pressure, J. Adv.Mater. Process., 4(2) (2016) 3-20.

[21] M. Arefi, R. Koohi Faegh, A. Loghman, The effectof axially variable thermal and mechanical loadson the 2D thermoelastic response of FG cylindri-cal shell, J. Therm. Stresses., 39(12) (2016) 1539-1559.

[22] R. Singh, L. Sondhi, A. Kumar Thawait, Stressand deformation analysis of rotating cylindrical

pressure vessel of functionally graded materialmodeled by mori-tanaka scheme, J. Exp. Appl.Mech., 8(3)(2017) 1-7.

[23] N. Habibi, S. Asadi, R. Moradikhah, Evaluationof SIF in FGM thick-walled cylindrical vessel, J.Stress Anal., 2(1) (2017) 57-68.

[24] M. Jabbari, M. Zamani Nejad, M. Ghannad,Stress analysis of rotating thick truncated coni-cal shells with variable thickness under mechani-cal and thermal loads, J. Solid Mech., 9(1) (2017)100-114.

[25] R. Hetnarski B., M.R. Eslami, Thermal Stresses-Advanced Theory and Applications, SpringerPublisher, (2009).

[26] T. Myint-U, L. Debnath, Linear Partial Differ-ential Equations for Scientists and Engineers,Birkhauser, Boston Publisher, (2007).

The Effect of Grading Index on Two-dimensional Stress and Strain Distribution of FG Rotating CylinderResting on a Friction Bed Under Thermomechanical Loading: 75–82 82