Thermoelastic analysis of a functionally graded rotating disk

Composite Structures 79 (2007) 508–516 www.elsevier.com/locate/compstruct

Thermoelastic analysis of a functionally graded rotating disk S.A. Hosseini Kordkheili *, R. Naghdabadi

1

Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran Available online 20 March 2006

Abstract A semi-analytical thermoelasticity solution for hollow and solid rotating axisymmetric disks made of functionally graded materials is presented. The radial domain is divided into some virtual sub-domains in which the power-law distribution is used for the thermomechanical properties of the constituent components. Imposing the necessary continuity conditions between adjacent sub-domains, together with the global boundary conditions, a set of linear algebraic equations are obtained. Solution of the linear algebraic equations yields the thermoelastic responses for each sub-domain as exponential functions of the radial coordinate. Some results for the stress, strain and displacement components along the radius are presented due to centrifugal force and thermal loading. Results obtained within this solution are compared with those of a finite element analysis in the literature. Based on the results, it is shown that the property gradation correlates with thermomechanical responses of FG disks. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Functionally graded materials; Thermoelasticity; Rotating disk

1. Introduction In functionally graded materials (FGMs), the material properties are varied continuously through the thickness according to a power-law distribution of volume fraction of the constituents. Due to their functional gradation for optimized design, in recent years interests to making components out of compositionally graded material are increased. The main application of FGMs is in high temperature Aerospace environments, such as skin structures and turbine disks, which operate under complex thermal and mechanical loading conditions. As the use of FGMs increases, new methodologies have to be developed to characterize FGMs, and also to analyze and design structural components made of these materials. * Corresponding author. Present address: School of Engineering and Design, Brunel University, Uxbridge, Middlesex UB8 3PH, UK. Tel.: +44 7950751674; fax: +44 1895 812556. E-mail addresses: [email protected] (S.A. Hosseini Kordkheili), [email protected] (R. Naghdabadi). 1 Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11365-9567, Tehran, Iran. Tel.: +98 21 6616 5546; fax: +98 2 6600 0021.

0263-8223/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.02.010

There have been some studies dealing with thermal stresses in the basic structural components of FGMs. Suresh and Mortensen [1] have provided an introduction to the fundamentals of FGMs. Noda [2] has presented an extensive review that covers a wide range of topics from thermoelastic to thermoinelastic problems. He has discussed the importance of temperature dependent properties on stresses and suggested that those properties of the material should be taken into account in order to perform more accurate analysis. Tanigawa [3] has compiled a comprehensive review on thermoelastic analysis of FGMs. Fukui and Yamanaka [4] have examined the effects of the gradation of components on the strength and deformation of thick-walled FG tubes under internal pressure in the case of plain strain. Fukui et al. [5] further extended their previous work by considering a thick-walled FG tube under uniform thermal loading. They also investigated the effect of graded components on residual stresses and estimated the optimum composition gradient generated by compressive circumferential stress at the inner surface. Using a perturbation approach, Obata and Noda [6] have investigated the thermal stresses in a FG hollow sphere and a FG hollow circular cylinder. Ootao and Tanigawa [7]

S.A. Hosseini Kordkheili, R. Naghdabadi / Composite Structures 79 (2007) 508–516

have conducted an approximate analysis of three-dimensional thermal stresses in a FG rectangular plate. Liew et al. [8] have presented an analysis of the thermomechanical behavior of hollow circular cylinders of FGM. They used the exponential volume fraction function and assumed the state of plain strain and developed an analytical model to deal with FG hollow circular cylinders that are subjected to the action of an arbitrary steady state or transient temperature field. Shahsiah and Eslami [9] have studied thermal buckling of FG cylindrical shells based on the first-order shear deformation shell theory (FSDST) and Sanders kinematic equations. They continued their study on thermal instability of FG cylindrical shells based on the FSDST and improved Donnell equations [10]. Ozturk and Tutuncu [11] have provided closed-form solutions for stresses and displacements in FG cylindrical vessels subjected to internal pressure alone using the infinitesimal theory of elasticity. They assumed material stiffness obeying a power law through the wall thickness with the Poisson’s ratio being constant. They approximated radially varying elastic modulus by E(r) = E0rb, where r is the radius of the cylinder. Since r is away from zero, by adjusting the constants E0 and b, it is possible to obtain physically meaningful results. Different values of the inhomogeneity constant b for a given geometry, renders FG cylinders with different base materials. However, the various b values demonstrate the effect of inhomogeneity on the stress distribution. Ootao and Tanigawa [12] have studied the theoretical treatment of a transient thermoelastic problem involving a FG thick strip due to a non-uniform heat supply in the width direction. They obtained the exact solution for the two-dimensional temperature change in a transient state, and thermal stresses of a simply supported strip under the state of plane strain. Naghdabadi and Hosseini Kordkheili [13] have derived a finite element formulation for the thermoelastic analysis of FG plates and shells. They assumed the power-aw distribution model for the composition of the constituent materials in shell thickness direction. Ruhi et al. [14] have presented an analytical thermoelasticity solution for thick-walled finitely long cylinders made of FGM. They reduced the governing partial differential equations to ordinary differential equations using Fourier expansion series and then used some virtual sub-domains to solution these ordinary differential equations. Durodola and Attia [15,16] have provided a finite element analysis for FG rotating disks using commercial software. They have modeled the disks as non-homogeneous orthotropic materials such as those obtained through non-uniform reinforcement of metal matrix by long fibers. Also, they have considered only three types of thermal loading cases with assigned temperature distributions. This work aims to investigate the relative influences of basic factors such as property gradation, centrifugal body loading and thermal loading on stresses and deformation in a FG rotating disk (Fig. 1). For this purpose, the equilibrium equations are derived based on the thermoelasticity theory. The radial domain is divided into some finite sub-

509

Fig. 1. Configuration of the thin-walled finitely radius FG rotating disk.

domains, in which the thermomechanical properties are assumed to be constant. Imposing the continuity conditions at the interface of the adjacent sub-domains, together with the global boundary conditions, a set of linear algebraic equations are derived. The non-linear heat transfer equation also is solved for thermal distribution through the radius by the proposed semi-analytical method. Solving the linear algebraic equations, the thermoelastic responses for a hollow or solid rotating axisymmetric FG disk are obtained and the normalized stress, strain and displacement components in the disk radius direction are presented. 2. Gradation relation There are some models for expressing the variation of material properties in FGMs in the literature. The most commonly used of these models is the power-aw distribution of the volume fraction. Based on this model, the material property gradation through the disk radius is represented in terms of the volume fraction by  n r  Ri pðrÞ ¼ ðpo  pi Þ þ p i ; Ri < r < Ro ð1Þ Ro  Ri where p(r) denotes a generic material property and po and pi denote the property of the outer and inner faces of the disk, respectively (e.g. elastic modulus) and n is a grading index that dictates the material variation profile through the thickness. Fig. 2 shows the distribution of the material 1 volume fraction through a cylinder thickness with Ro = 5Ri for various values of the grading index n. This study assumes that the elastic modulus E, thermal coefficient of expansion a, density q and heat conductivity k vary according to the gradation relations (1) and the Poisson’s ratio m is assumed to be constant. 3. Thermoelastic equations Consider a hollow axisymmetric FG disk with a constant and thin thickness, inner radius Ri and outer radius Ro, as shown in Fig. 1. This disk rotates at an angular

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where q(r)x2r2 is the body force per unit volume due to centrifugal force. Substituting Eqs. (2) into Eqs. (4) and then into Eq. (5), we obtain the Navier thermoelastic equation for a rotating FG disk as follows:     dEðrÞ dEðrÞ ð2  mÞ d2 u du  m EðrÞ u þ EðrÞ þ rEðrÞ 2 þ r dr dr r dr dr dðaðrÞ DT ðrÞ Þ ¼0 ð6Þ þ ð1  m2 ÞqðrÞ x2 r2  rð1 þ mÞ dr

1 0.9

Material #1 Volume Fraction

0.8

n=5.0 n=2.0

0.7

n=1.5

0.6 n=0.5

0.5

It is noted that in Navier equation (6), the displacement component and thermo mechanical properties are functions of r only (because of the axial symmetry and plane stress assumptions).

0.4 n=0.2

0.3 0.2

4. Boundary conditions

0.1 0 0.2

0.4

0.6

0.8

1

r/Ro Fig. 2. Material 1 volume fraction (V1) through the radius of the disk.

velocity x and is subjected to thermal loading DT(r) from a stress-free state. Since the disk is thin in thickness direction the plane stress condition is assumed. We also use the cylindrical coordinate system (r, h, z) and assume axial symmetry in geometry and loading. The considered FG disk is made of a metal-rich (material #1) surface at the inside and a ceramic-rich (material #2) surface at the outside, between these two surfaces material properties vary according to the gradation relation (1). The linear relations between the strain and the displacement components for described problem are er ¼

du ; dr

eh ¼

u r

ð2Þ

The following traction conditions on the inner and outer surfaces of the hallow rotating FG disk must be satisfied  rr ¼ 0 r ¼ Ri ð7Þ rr ¼ 0 r ¼ Ro In addition, for a solid rotating disk the boundary conditions are  u¼0 r¼0 ð8Þ rr ¼ 0 r ¼ Ro 5. Solution algorithms Providing an analytical solution for ODE (6) when E(r), a(r) and q(r) are expressed using relation (1) is difficult. Hence, in this study, a semi-analytical method is used to solve the problem. Using this method a solution for ODE (6) is got by dividing the radial domain into some finite sub-domains with thickness tk, as shown in Fig. 3. Evaluating coefficients of Eq. (6) at r = rk, mean radius of kth divi-

where u is the radial displacement. Also, the linear constitutive thermoelastic equations in the cylindrical coordinate system are used in the form of 1 ðrr  mrh Þ þ aðrÞ DT ðrÞ EðrÞ 1 eh ¼ ðrh  mrr Þ þ aðrÞ DT ðrÞ EðrÞ

er ¼

ð3Þ

kth division

or EðrÞ rr ¼ ððer þ meh Þ  ð1 þ mÞaðrÞ DT ðrÞ Þ ð1  m2 Þ EðrÞ rh ¼ ððmer þ eh Þ  ð1 þ mÞaðrÞ DT ðrÞ Þ ð1  m2 Þ

Ro

drr þ ðrr  rh Þ þ qðrÞ r2 x2 ¼ 0 dr

t k+1

th

t

(k-1) division t

ð4Þ

Due to the axisymmetry in the problem the shear stress is zero. The equilibrium equation for axisymmetric stresses in the present of body force is r

(k+1)th division

ð5Þ

k

k-1

rk

Ri Fig. 3. Dividing radial domain into some finite sub-domains.

S.A. Hosseini Kordkheili, R. Naghdabadi / Composite Structures 79 (2007) 508–516

sion, and using them instead of the variable coefficients in the original equation, a ODE’s with constant coefficients is obtained which is valid in kth sub-domain. That is   d2 d ck1 2 þ ck2 þ ck3 uk þ ck4 ¼ 0 ð9Þ dr dr where ck1 ¼ rk Eðrk Þ  dEðrÞ  ck2 ¼ rk þ Eðrk Þ dr r¼rk  dEðrÞ  ð2  mÞ ck3 ¼ m  Eðrk Þ rk dr  k

ð10Þ

r¼r

ck4

2

2

¼ ð1  m Þqðrk Þ ðxrk Þ  rk ð1 þ mÞ

 dðaðrÞ DT ðrÞ Þ  k dr r¼r

The temperature distribution DT(r) in Eq. (10), can be determined from the thermal analysis in the next section. Using the above technique, the ODE (6) with variable coefficients is changed into m ODEs with constant coefficients in which m is the number of virtual sub-domains. Therefore, the solution for Eq. (9) can be written in the form of uk ¼ X k1 expðkk1 rÞ þ X k2 expðkk2 rÞ 

ck4 ck3

ð11Þ

where kk1 ; kk2

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k k k ¼  c2  ck2 2ck1 2  4c3 c1

ð12Þ

and X k1 and X k2 are unknown constants for kth division. It is noted that the above solution is valid for k

rk 

k

t t 6 r 6 rk þ 2 2

ð13Þ

where rk and tk are the mean radius and thickness of the kth sub-domain, respectively. The unknowns X k1 and X k2 are determined by applying the necessary boundary conditions between each two adjacent sub-domains. For this purpose, the continuity of the radial displacement u as well as radial stress rr are imposed at the interfaces of the adjacent sub-domains. These continuity conditions at the interfaces are   uk r¼rk þtk ¼ ukþ1 r¼rkþ1 tkþ1 2 2   ð14Þ  kþ1 tkþ1 rkr r¼rk þtk ¼ rkþ1 r r¼r  2

2

The continuity conditions (14) together with the global boundary conditions (7) or (8) yield a set of linear algebraic equations in terms of X k1 and X k2 . Solving the resultant linear algebraic equations for X k1 and X k2 , the unknown coefficients of Eq. (11) are calculated. Then, the displacement component u is determined in each radial sub-domain. Increasing the number of subdivisions improves the accuracy of the results.

511

6. Thermal analysis If the prescribed surface temperature imposed on inner and outer sides of FG disk as DT ¼ DT i

at r ¼ Ri

ð15aÞ

DT ¼ DT o

at r ¼ Ro

ð15bÞ

and the variation of temperature is assumed to occur in the radius direction only, then solving the steady state heat transfer equation through the radius direction of the FG disk as   d2 DT ðrÞ dk ðrÞ dDT ðrÞ rk ðrÞ ¼0 ð16Þ þ k ðrÞ þ r dr2 dr dr carries out the thermal analysis. In Eq. (16) k(r) is the thermal conductivity coefficient. It is assumed that k(r) varies only along the disk radius according to Eq. (1) as follows  n r  Ri k ðrÞ ¼ ðk o  k i Þ þ k i ; Ri < r < Ro ð17Þ Ro  Ri A closed-form solution of Eq. (16) subject to the boundary conditions (15) as a function of the parameter n is not available. Hence, it is solved using the same proposed semianalytical method. Using this method DT ðrk Þ is achieved for each sub-domain in Fig. 3. For this purpose, the coefficients of Eq. (16) evaluate at r = rk, mean radius of kth division. The obtained ODE’s with constant coefficients which is valid only in kth sub-domain is   d2 d c1 2 þ c ð18Þ DT ðrk Þ ¼ 0 dr dr where c1 ¼ rk k ðrk Þ c2 ¼ k ðrk Þ þ rk

 dk ðrÞ  dr r¼rk

ð19Þ

Now we have m ODEs with constant coefficients. Hence, the exact solution for Eq. (18) can be written in the form of DT ðrk Þ ¼ X 1 þ X 2 expðrc2 =c1 Þ X k1

ð20Þ

X k2

where and are unknown constants for kth division. These unknowns are determined by applying the necessary boundary conditions between each two adjacent subdomains. The continuity of the value of temperature and the continuity of the heat flux are imposed at the interfaces of the adjacent sub-domains. These continuity conditions at the interfaces are   DT ðrÞ r¼rk þtk ¼ DT ðrÞ r¼rkþ1 tkþ1 2  2  ð21Þ dDT ðrÞ  dDT ðrÞ  ¼   dr r¼rk þtk dr r¼rkþ1 tkþ1 2

2

The continuity conditions (21) together with the global boundary conditions (15) yield a set of linear algebraic equations in terms of X k1 and X k2 . Solving the resultant linear algebraic equations result the unknown coefficients of

512

S.A. Hosseini Kordkheili, R. Naghdabadi / Composite Structures 79 (2007) 508–516

Eq. (20). Then, the DT value is determined in each radial sub-domain.

0.3

0.25

7. Numerical results 0.2

For verification of the presented Navier solution, a rotating FG disk with Ro = 5Ri is considered with nondimensionalized modulus of elasticity distribution as 

EðrÞ

r  Ri ¼ Ei þ ðEo  Ei Þ Ro  Ri

σr / ρm ω2 R2o

7.1. Verification of the solution

n

0.15

Present work Durodola & Attia (2000)

0.1

ð22Þ 0.05

where Ei and Eo are non-dimensional modulus of elasticity. Durodola and Attia [15,16] have used direct integration for governing equation and finite element method by using a commercial package for solution of this problem. A plot of the deformation and stress distribution using the presented Navier solution and finite element solution [15] are given in Figs. 4 and 5 for Ei ¼ 2:3, Eo ¼ 1:0 and n = 1.0. The results are plotted in a non-dimensional form same as [15]. According to Figs. 4 and 5 the results are comparable with those produced by more complicated finite element method.

0 0.2

0.3

0.4

0.5

0.7

0.8

0.9

1

Fig. 5. Comparing non-dimensionalized radial stress along the radius from present solution and finite element.

Ein ¼ EAl ¼ 70 GPa,

Eout ¼ ECer ¼ 151 GPa

6

ain ¼ aAl ¼ 23  10 =K, 3

qin ¼ qAl ¼ 2700 kg/m ; 7.2. Case studies

0.6

r/Ro

aout ¼ aCer ¼ 10  106 =K qout ¼ qCer ¼ 5700 kg/m

3

m ¼ 0:3

In order to show the abilities of the presented Navier solution for analyzing a FG rotating disk, two case studies are investigated. The analysis is conducted using aluminum as inner surface metal and zirconia as outer surface ceramic. According to Eq. (1) the following material properties are used in computing the numerical results

ð23Þ Two cases of loading are studied; uniform centrifugal force due to uniform rotating speed, x, and thermal loading, DT(r). In both cases for hollow disk Ro = 5Ri (Fig. 1) is considered.

0.17 0.16

0.16 0.15

0.15 Present work Durodola & Attia (2000)

0.14

ur ECer / ρCer ω2 Ro3

0.14

2

ur Em / ρm ω Ro3

0.13 0.12 0.11 0.1

0.13 0.12 0.11

Metal disk Ceramic disk FG disk, n=0.2 FG disk, n=0.5 FG disk, n=1.5 FG disk, n=2.0 FG disk, n=5.0

0.1 0.09

0.09 0.08

0.08 0.07 0.06 0.2

0.07 0.2 0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r/Ro Fig. 4. Comparing non-dimensionalized radial displacement along the radius from present solution and finite element.

0.4

0.6

0.8

1

r/Ro Fig. 6. Normalized radial displacements through the radius of the hollow disk due to uniform centrifugal force for different values of the grading index n.

S.A. Hosseini Kordkheili, R. Naghdabadi / Composite Structures 79 (2007) 508–516

7.2.1. Results for uniform centrifugal force In this study the results are presented in a non-dimensional form so that absolute values of properties and the loading speed are unimportant. Displacements, stresses and strains were normalized by dividing by the factors qCer x2 R3o =ECer , qCer x2 R2o and qCer x2 R2o =ECer respectively. Fig. 6 shows the non-dimensionalized radial displacement along the radius for different values of the grading index n. As it is expected, the radial displacement values for full metal (Al) disk are greater than those for full ceramic (zirconia) disk due to higher elasticity modulus of ceramic. Also, it is observed that the radial displacement

513

increases with the increase of the power-law index n from zero (homogenized aluminum disk) up to its maximum value for about n = 1.5 and then decreases to its minimum value for n = 1 (homogenized zirconia disk). Thus, for a given pair of materials, there is a particular volume fraction for which the radial displacement attains its maximum value under a given rotating speed. Figs. 7 and 8 show the variation of the normalized radial and angular strains along the radius for different values of the grading index n. As it is expected, eh is always positive whereas er gives also negative values for a prescribed centrifugal force. In addition, as it is observed in these figures,

0.2

0.25

0.15

0.2

σr / ρCer ω2 R 2o

εr ECer / ρCer ω2 R o2

0.1

0.05

0

Metal disk Ceramic disk FG disk, n=0.2 FG disk, n=0.5 FG disk, n=1.5 FG disk, n=2.0 FG disk, n=5.0

-0.05

0.15

0.1 Metal disk Ceramic disk FG disk, n=0.2 FG disk, n=0.5 FG disk, n=1.5 FG disk, n=2.0 FG disk, n=5.0

0.05

-0.1

-0.15

0.2

0.4

0.6

0.8

0 0.2

1

r/Ro

0.8

1

Fig. 9. Normalized radial stress through the radius of the hollow disk due to uniform centrifugal force for different values of the grading index n.

0.5

0.4

0.45

0.3

σθ / ρCer ω2 Ro2

0.35

Metal disk Ceramic disk FG disk, n=0.2 FG disk, n=0.5 FG disk, n=1.5 FG disk, n=2.0 FG disk, n=5.0

0.35

Metal disk Ceramic disk FG disk, n=0.2 FG disk, n=0.5 FG disk, n=1.5 FG disk, n=2.0 FG disk, n=5.0

0.4

εθ ECer / ρCer ω2 Ro2

0.6

r/Ro

Fig. 7. Normalized radial strain through the radius of the hollow disk due to uniform centrifugal force for different values of the grading index n.

0.3 0.25

0.25

0.2

0.15

0.2

0.1

0.15 0.1 0.2

0.4

0.4

0.6

0.8

1

r/Ro Fig. 8. Normalized angular strain through the radius of the hollow disk due to uniform centrifugal force for different values of the grading index n.

0.05 0.2

0.4

0.6

0.8

1

r/Ro Fig. 10. Normalized hoop stress through the radius of the hollow disk due to uniform centrifugal force for different values of the grading index n.

S.A. Hosseini Kordkheili, R. Naghdabadi / Composite Structures 79 (2007) 508–516

the extreme values for the strains er and eh occur at about n = 2.0 and n = 1.5, respectively. It is noted that these extreme values don’t occur in the homogenized state. Figs. 9 and 10 illustrate the through-the-radius variation of the normalized radial and hoop stresses for different values of the grading index n. As the figures show, these stresses take maximum and minimum values for full ceramic and metal disk, respectively, and for FG disks these values positioned between of them. Figs. 11–13 also show the non-dimensionalized radial displacement and stress values along the radius of a solid disk for different values of grading index n. From these fig-

0.25

0.2

σθ / ρCer ω2 Ro2

514

0.15

0.1

Metal disk Ceramic disk FG disk, n=0.2 FG disk, n=0.5 FG disk, n=1.5 FG disk, n=2.0 FG disk, n=5.0

0.05 0.16 0

0.14

0.4

r/Ro

0.6

0.8

1

Fig. 13. Normalized hoop stress through the radius of the solid disk due to uniform centrifugal force for different values of the grading index n.

0.12

ur ECer / ρCer ω2 R 3o

0.2

0.1

ures we can show that, the above discussions also are valid for a solid FG disk.

0.08 0.06

Metal disk Ceramic disk FG disk, n=0.2 FG disk, n=0.5 FG disk, n=1.5 FG disk, n=2.0 FG disk, n=5.0

0.04 0.02 0

0

0.2

0.4

r/Ro

0.6

0.8

1

Fig. 11. Normalized radial displacements through the radius of the solid disk due to uniform centrifugal force for different values of the grading index n.

7.2.2. Results for thermal loading In order to study the effects of thermal loading on FG disk behavior, a prescribed surface temperature (DT = DTo) imposed on outer sides of FG disk and for inner surface it is assumed DT = DTi = 0. Also, in this section the dividing factors aCerRoDTo and aCerDTo are used for doing non-dimensional of the displacements, stresses and strains, respectively. Fig. 14 illustrates the non-dimensionalized temperature distribution due to assumed imposed prescribed surface temperatures. Fig. 15 illustrates the

1

0.3

0.9 0.25

0.8 0.7

ΔT(r) / ΔTo

σr / ρCer ω2 R 2o

0.2

0.15

0.1

Metal disk Ceramic disk FG disk, n=0.2 FG disk, n=0.5 FG disk, n=1.5 FG disk, n=2.0 FG disk, n=5.0

0.05

0

0

0.2

0.4

0.6 0.5 0.4 0.3 0.2 0.1

0.6

0.8

1

r/Ro Fig. 12. Normalized radial stress through the radius of the solid disk due to uniform centrifugal force for different values of the grading index n.

0

0.2

0.4

0.6

0.8

1

r/Ro Fig. 14. Non-dimensionalized temperature distribution due to assumed imposed prescribed surface temperatures.

S.A. Hosseini Kordkheili, R. Naghdabadi / Composite Structures 79 (2007) 508–516 2 1.75

3.5 Metal disk Ceramic disk FG disk, n=0.2 FG disk, n=0.5 FG disk, n=1.5 FG disk, n=2.0 FG disk, n=5.0

Metal disk Ceramic disk FG disk, n=0.2 FG disk, n=0.5 FG disk, n=1.5 FG disk, n=2.0 FG disk, n=5.0

3

2.5

1.25

εθ / αCer ΔT

u/Ro αCer ΔTo

1.5

515

1 0.75

2

1.5

0.5 1

0.25 0 0.2

0.4

0.6

0.8

0.5 0.2

1

0.4

r/Ro

0.6

0.8

1

r/Ro

Fig. 15. Normalized radial displacements through the radius of the hollow disk due to thermal loading for different values of the grading index n.

Fig. 17. Normalized angular strain through the radius of the hollow disk due to thermal loading for different values of the grading index n.

non-dimensionalized radial displacements due to thermal loading along the radius of the disk. As shown in this figure, the radial displacements for the homogenized states and FG disks are positive. Also, it is noted that the minimum radial displacement occurs at ceramic disk. The through-the-radius variations of the normalized radial and angular strains for different values of the grading index n are presented in Figs. 16 and 17. From these two figures it is observed that the minimum value for er happens for FG disk with grading index about n = 0.2. But the minimum value for eh happens for full ceramic disk.

Figs. 18 and 19 show the variation of the normalized thermal radial and hoop stresses distribution through the radius of the disk due to the uniform thermal loading. From these two figures it is illustrated that, the values of thermal radial and hoop stresses are negative for the homogenized states (full metal or full ceramic) and for the selected values of the power-law index n. It is also noted that for different values of n, hoop stresses always attain their maximum values at the outer surface as shown in Fig. 19. From this study, it is observed that the response of the FG disk to thermal loading is complicated.

2.5 0.75 2

σr / αCer ΔT

ε r / α Cer ΔT

1.5

1

0.5 Metal disk Ceramic disk FG disk, n=0.2 FG disk, n=0.5 FG disk, n=1.5 FG disk, n=2.0 FG disk, n=5.0

0

-0.5

-1 0.2

0.4

0.6

0.8

Metal disk Ceramic disk FG disk, n=0.2 FG disk, n=0.5 FG disk, n=1.5 FG disk, n=2.0 FG disk, n=5.0

0.5

0.25

1

r/Ro Fig. 16. Normalized radial strain through the radius of the hollow disk due to thermal loading for different values of the grading index n.

0 0.2

0.4

0.6

0.8

1

r/Ro Fig. 18. Normalized radial stress through the radius of the hollow disk due to uniform thermal loading for different values of the grading index n.

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S.A. Hosseini Kordkheili, R. Naghdabadi / Composite Structures 79 (2007) 508–516

1.6 1.4 1.2

σθ / αCer ΔT

plays an important role in determining the thermomechanical responses of FG rotating disk as well as in optimal design of this structure.

Metal disk Ceramic disk FG disk, n=0.2 FG disk, n=0.5 FG disk, n=1.5 FG disk, n=2.0 FG disk, n=5.0

References

1 0.8 0.6 0.4 0.2 0

-0.2 0.2

0.4

0.6

0.8

1

r/Ro Fig. 19. Normalized hoop stress through the radius of the hollow disk due to thermal loading for different values of the grading index n.

8. Conclusions A semi-analytical solution for thermoelasticity equilibrium equations as well as for non-linear heat transfer equation of a thin axisymmetric rotating disk made of functionally graded materials is presented. The effects of the radial gradation of constitutive components on stress, strain and displacement components of the FG disk have been investigated for both centrifugal force and uniform thermal loadings. It is seen that for a given pair of materials, there is a particular volume fraction that extremes a specified mechanical response under a given thermomechanical loading. In addition, it is observed that for all values of the grading index n, the radial stress due to centrifugal force attains its maximum value at about r/Ro = 0.45. These maximum values are between the maximum values from homogenized disks. From the results, it is noted that the gradation of the constitutive components

[1] Suresh S, Mortensen A. Fundamentals of functionally graded materials. London, UK: IOM Communications Limited; 1998. [2] Noda N. Thermal stress in materials with temperature-dependent properties. Appl Mech Rev 1991;44:383–97. [3] Tanigawa Y. Some basic thermoplastic problems for nonhomogeneous structural materials. Appl Mech Rev 1995;48:377–89. [4] Fukui Y, Yamanaka N. Elastic analysis for thick-walled tubes of functionally graded material subjected to internal pressure. JSME Int J Ser I 1992;35:379–85. [5] Fukui Y, Yamanaka N, Wakashima K. The stresses and strains in a thick-walled tube for functionally graded material under uniform thermal loading. JSME Int J Ser A 1993;36(2):156–62. [6] Obata Y, Noda N. Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally gradient material. Thermal Stresses 1994;17:471–88. [7] Ootao Y, Tanigawa Y. Three-dimensional transient thermal stress of FG rectangular plate due to partial heating. Thermal Stresses 1999;22:35–55. [8] Liew KM, Kitipornachi S, Zhang XZ, Lim CW. Analysis of the thermal stress behavior of functionally graded hollow circular cylinders. Int J Solids Struct 2003;40:2355–80. [9] Shahsiah R, Eslami MR. Thermal buckling of functionally graded cylindrical shell. J Thermal Stresses 2003;26:277–94. [10] Shahsiah R, Eslami MR. Functionally graded cylindrical shell thermal instability based on improved Donnell equations. AIAA J 2003;41:1819–26. [11] Ozturk M, Tutuncu N. Exact solutions for stresses in FG pressure vessels. Composites, Part B 2001;32:683–6. [12] Ootao Y, Tanigawa Y. Transient thermoelastic problem of functionally graded thick strip due to nonuniform heat supply. Compos Struct 2004;63:139–49. [13] Naghdabadi R, Hosseini Kordkheili SA. A finite element formulation for analysis of functionally graded plates and shells. Arch Appl Mech 2005;74:375–86. [14] Ruhi M, Angoshtari A, Naghdabadi R. Thermoelastic analysis of thick-walled finite-length cylinders of functionally graded materials. J Thermal Stresses 2005;28:391–408. [15] Durodola JF, Attia O. Deformation and stresses in FG rotating disks. Compos Sci Technol 2000;60:987–95. [16] Durodola JF, Attia O. Property gradation for modification of response of rotating MMC disks. Mater Sci Technol 2000;16:919–24.