Stress analysis of functionally graded discs under mechanical and thermal loads

Indian Journal of Engineering & Materials Sciences Vol. 8, April 0, pp. -8 Stress analysis of functionally graded discs under mechanical and thermal loads Hasan Çallioğlu, Metin Sayer* & Ersin Demir Department of Mechanical Engineering, Pamukkale University, 000, Denizli, Turkey Received September 00; accepted March 0 The analytical study deals with stress analysis of functionally graded rotating annular discs subjected to internal pressure and various temperature distributions, such as uniform T 0 (reference temperature), linearly increasing T b (outer surface temperature) and decreasing T a (inner surface temperature) in radial direction. Infinitesimal deformation theory of elasticity and for graded parameters power law functions are used in the solution procedure. The results show that the tangential stresses increase at the inner surface and decrease at the outer surface with increasing T 0 and T b temperatures. If T a is increased, they decrease at the inner surface and rise at the outer surface. The radial stresses reduce gradually along the radial section with increasing T 0 and T a temperatures, whereas they rise along the radial section when T b is increased. The radial displacement values in the discs subjected to uniform temperature T 0 are higher compared to the discs subjected to other kind of temperature distributions. The radial displacement has higher value at the outer surface than that of the inner surface for all the temperature distributions, except for radial displacement in the reference temperature case. The stresses and displacement values in a disc under uniform temperature are compared with the values given in the literature and they are found to be consistent with each other. Keywords: Functionally graded disc, Rotating disc, Thermal stress, Internal pressure The fiber-reinforced composites have been extensively used as structural materials in many aerospace and automotive applications. The reinforcement in these composites is generally distributed uniformly. Functionally graded materials (FGMs) are being used as interfacial zone to improve the bonding strength of layered composites, to reduce the residual and thermal stresses in bonded dissimilar materials and as wear resistant layers in machine and engine components,. One of the advantages of FGMs over laminates is that, due to continuous material property variation, there is no stress build-up at sharp material boundaries thus eliminating potential structural integrity issues such as delamination. Therefore, FGMs have been the subject of intense researches and attracted considerable attention in recent years. The rotating discs have extensively been used in many mechanics and engineering applications such as in steam and gas turbine rotors, turbo generators, internal combustion engines, casting ship propellers, turbojet engines, reciprocating and centrifugal compressors 3. A disc in a turbine rotor can be made of FGM with ceramic-rich at the outer surface and *Corresponding author (E-mail: msayer@pau.edu.tr) metal-rich at the inner surface. While the heat resistant property of the ceramic at the outer surface prevents heat from being transferred, the metal at the inner surface helps carry the stress for the transmission of torque from the disc to the shaft. A free-free condition can be applied for the disc connected to the shaft by means of splines where small axial movement is allowed. And then, an internal pressure from shaft to the disc can be emerged by increasing deformation in the internal surface of the disc. An analytical solution for the stress analysis in a homogeneous-isotropic disc under centrifugal force, pressure or thermal loads can be found in literature 4. Çallıoğlu 5,6 analyzed the stresses in rotating rectilinearly or polar orthotropic discs subjected to various temperature distributions. Sayman and Arman 7 carried out an elastic-plastic stress analysis in a thermoplastic composite disc reinforced by steel fibers curvilinearly under a steady state temperature distribution. Singh and Ray 8 investigated creep in an orthotropic aluminum-silicon carbide composite rotating disc by using Hill s anisotropic yield criteria. In that study, the results obtained have also been compared with the results obtained by using von Mises yield criterion for the composites.

INDIAN J. ENG. MATER. SCI., APRIL 0 Since the mathematical problems arising are complicated, much of the work on FGMs has been carried out numerically. Nevertheless, the mechanical and mathematical modeling of FGMs is currently an active research area. Horgan and Chan 9,0 investigated the stress response in rotating discs and pressurized hollow cylinder or disc made of Functionally graded (FG) isotropic linearly elastic materials. Durodola and Attia investigated deformation and stresses in FG rotating discs. They compared two methods, finite element method (ABAQUS) and direct numerical integration of governing differential equations, with each other. Chen et al. presented three-dimensional analytical solution for a rotating disc composed of transversely isotropic FGMs. A significant amount of these studies has been done in order to see the effects of the FGMs on the isotropic discs. Mohammadi and Dryden 3 examined the role of nonhomogeneous stiffness on the thermoelastic stress field in a FG curved beam. In all the studies with FGMs, it has only been considered a body with Young s modulus varying radially. Gupta et al. 4 investigated the creep behavior of a rotating disc made of isotropic composite containing varying amounts of silicon carbide in the radial direction in the presence of a thermal gradient, also in the radial direction. You et al. 5 investigated the stresses on the FG rotating circular discs under uniform temperature. Çallıoğlu 6 studied the stress analysis of the rotating hollow discs made of FGMs under internal and external pressures. Bayat et al. 7 presented thermoelastic solutions in a rotating FG disc with variable thickness under a steady temperature field. Bayat et al. 3 presented the elastic solutions for axisymmetric rotating discs which made of FGM with variable thickness. They investigated the effects of material grading index and geometry of the disc on the stresses and displacements. Kordkheili and Naghdabadi 8 presented a semi-analytical thermoelasticity solution for hollow and solid rotating axisymmetric discs made of FGMs. The static analysis of FGM plates subjected to transverse mechanical loadings was studied by Carrera et al. 9. In that study the unified formulation and principle of virtual displacements were employed to obtain both closed-form and finite element solutions. Brischetto and Carrera 0 investigated FGM plates subjected to a transverse mechanical load. They extended the unified formulation (UF) and the Reissner s mixed variational theorem (RMVT) to FGMs and proposed a model. The thermo-mechanical analysis of a simply supported FG shell was studied by Cinefra et al.. They considered the refined shell theories to account for grading material variation in the thickness direction. In this study, the closed-form solutions for stresses and displacements in FG annular discs rotating at a constant angular velocity and subjected to uniform, linearly decreasing and linearly increasing temperature distributions in radial direction are obtained by using the infinitesimal deformation theory of elasticity. The effect of internal pressure is also considered. The material is assumed to be an isotropic material with constant Poisson s ratio ν and varying elasticity modulus E, density ρ and thermal expansion coefficient α through the radial direction. For this purpose, the power-law function is used. The validity of the proposed theory is preliminarily verified by comparing the obtained results with the available literature. Stress Analysis Due to the fact that a rotating thin disc is an axisymmetric problem, its equilibrium equation is 4 d r r r dr σ r + σ r σ θ + ρ ( ) ω = 0 () where σ r, σ θ, ω and ρ(r) are, respectively, radial stress, tangential stress, angular velocity and radially varying material mass density, which is used instead of mass density (ρ) of the isotropic, homogeneous materials. r is the radial distance (a r b). a and b are inner and outer radii of the disc, respectively as shown in Fig.. The solution can be efficiently handled by using a special stress function that automatically satisfies the Fig. A functionally graded rotating disc under various temperature profiles

ÇALLIOĞLU et al.: STRESS ANALYSIS OF FUNCTIONALLY GRADED DISCS 3 equilibrium Eq. (). A particular stress-stress function relation is given by F df σ r =, σ θ = + ρ( r) ω r () r dr where F=F(r) is the stress function. The governing equation for the stress function is determined from the compatibility condition. For this axisymmetric case, the displacement field is of the form u=u r =u r (r) and u θ =0. Therefore, the strain field is given by 4 du u ε r =, ε θ =, γ rθ = 0 (3) dr r where ε r, ε θ and u are the strains in radial and tangential directions, and displacement component in the radial direction, respectively. If u is eliminated from these equations, a simple compatibility condition can be developed as 4 dε ε = ε + r θ (4) dr r θ Using Hooke s law for plane stress case, the strains are given by ε r = ( σr νσ θ ) + α( r) T ( r) E ( r) ε θ = ( σθ ν σ r ) + α( r) T ( r) E ( r) (5) where E, α and T are, elasticity modulus, thermal expansion coefficient and temperature change, respectively, and it is assumed that material properties (E, ρ, α) and temperature (T) vary through the radial section. Poisson s ratio ν can be considered as a constant because of its variation has much less practical significance than that of the other material properties. Substitution of Eqs () and (5) in the compatibility relation (4) generates the following governing equation: E ( r) E ( r) r F + r F r + F ν r = E( r) E( r) 3 E ( r) 4 ρ( r) ω r 3 + ν r ρ ( r) ω r E( r) E( r) r ( r) T ( r) E( r) r ( r) T ( r) α α (6) The superscript ( ) represents derivative with respect to r. Let us assume for the sake of argument that r E( r) = E b r ρ ( r) = ρ b n n r α ( r) = α b n3 r a T ( r) = Ta T0 + ( Tb Ta ) b a (7) where n, n and n 3 are dimensionless arbitrary constants (gradient parameters). T a, T b and T 0 are inner, outer and reference surface temperatures, respectively. Introducing these variables in Eq. (6), differential equation reduces to ( ) ( ) r F + r F n + F ν n = ( 3 n n ) n ρ ω + ν + n b + 3 E α n3 Tb Ta n + n3 + T n a T0 a r + n + b 3 b a E α ( + n3) Tb Ta n n + r n + n b 3 b a r + 3+ The stress function F can be written as n + m n m n + 3+ n + 3 n + n3 + F = C r + C r + Ar + B r (8) + C r (9) where m = n 4ν n + 4, C and C are the integration constants and A, B and C terms are: ( 3 n n ) ρ ω + ν + A = n b ( n + 6n n n 3n + ν n + 8) E α n B = 3 n + n3 b ( n3 + n3 + nn 3 + n + ν n ) Tb T T a a T0 a b a

4 INDIAN J. ENG. MATER. SCI., APRIL 0 E α ( + n3 ) C = n + n3 b ( n3 + 4n3 + nn 3 + n + ν n + 3) Tb Ta b a (0) The stress components can be obtained from the stress function in Eq. (9) as, n + m n m n+ σ r = C r + C r + A r + B r + C r n + n 3+ n + n3 n + m n m + C r C r n m n m σ θ = + n + 3 + ( n + 3) Ar + ( n + n + ) B r n + n3+ ( n n3 ) C r ( r) r n + n3 + + + + ρ ω () The boundary conditions that are the disc is subjected to internal pressure P i at the inner surface and free at the outer surface, give C and C integration constants: n + m+ n + m+ D b D a C = m m b a n + m+ n + m+ m m D b a D a b C = m m b a where n + n n3 n n3 D = Pi A a B a + C a + + n + n n3 n n3 D = Ab B b + C b + + () Radial displacement component Using the small deformation theory of elasticity, radial displacement can be determined as 6, r u = ( σθ ν σ r ) + r α ( r) T ( r) (3) E ( r) Results and Discussion In this paper, a stress analysis is carried out on FG rotating annular discs under internal pressure and various temperatures by using an analytical solution including small deformation of theory of elasticity. The inner and outer radii of the discs are a=00 mm and b=500 mm, respectively. Mechanical properties of the discs, such as elasticity modulus, density and thermal expansion coefficient, and temperature applied are assumed to be varying along the radial direction. In order to compare the results with the values reported in the literature 5, the material coefficients are taken to be E=50 GPa, ν = 0. 3, ρ=5600 kg/m 3 and α= 3 0-6 / ο C. Gradient parameters are n =-0.594, n =-0.4873 and n 3 =0.536 for Disc (D) and n =0.594, n =0.4873 and n 3 =- 0.536 for Disc (D). The results are presented for angular velocity ω = 650 rad / s 5. It is considered that D is a disc that is made of FGM with ceramicrich at the inner surface, whereas D is a disc that is made of FGM with metal-rich at the inner surface. Temperature change is set to T=0, 300 and 600 C for uniform case. For linearly decreasing case inner surface temperature change is 300 or 600 C and linearly decreases to 0 C at the outer surface along the radius, and for linearly increasing case the situation is opposite. If the room temperature will be taken into account, it should be added to the temperature T. Internal pressure in both discs is considered as 0 MPa. The radial stress values in D under uniform temperature 50 C are compared with the values given in the literature 5 and are given in Table. They are found to be consistent with each other. Nevertheless, the maximum radial stress is 5.895 MPa at r = 30 mm and zero at the inner and outer surfaces while it is 5.03 MPa at r = 300 mm, - 3.9e-05 at inner surface and zero at the outer surface in Ref. 5 Elasticity modulus, density and thermal expansion coefficient variations are given as normalized values along the radial direction of the discs in order to demonstrate the effects of FGMs on the discs. For E, ρ and α, the following formal normalized variables are used: Table Comparison of the radial stress values in Disc under uniform temperature 50 C with values of Ref. 5 r, mm 00 50 300 350 400 450 500 σ r, MPa Ref. 5-3.9e-05.94 5.03 38.5308 05.47 57.969 0.0000 Present 0.0000 5.443 5.880 36.9004 0.083 53.90 0.0000

ÇALLIOĞLU et al.: STRESS ANALYSIS OF FUNCTIONALLY GRADED DISCS 5 E ( r) E =, E ρ( r) α( r) ρ =, α = (4) ρ α In the following figures, the stresses and displacements in D and D are compared with each other. Figure illustrates the variations of the normalized elasticity modulus, density and temperature expansion coefficient along the radial direction of D and D. As can be seen from this figure, the elasticity modulus, density and temperature expansion coefficient are equal to those of the isotropic, homogeneous disc at the outer surface. Through the radius of D, thermal expansion coefficient value decreases whereas elasticity modulus and density values increase gradually. For D, this situation is completely opposite according to D. The normalized thermal expansion coefficient value decreases to about 0.6 times of an isotropic, homogeneous disc when the normalized elasticity modulus and density values for D increase about.6 times of an isotropic, homogeneous disc. These values for D are approximately the opposite of those of D. Performance of FGM discs under an internal pressure The variations of the stresses and displacements are depicted in Fig. 3 for rotating D and D with internal pressure of 0 MPa. Line and dotted line in the figures represent the rotating D and D with internal pressure at T 0 = 0 C, respectively. It can be seen that the radial stresses at the inner surface are equal to internal pressure. If the stresses at D and D are compared with each other, it can be seen from Fig. 3a that the stresses at D are higher than those of D. Therefore, the selection of D material properties can be recommended in order to carry more loads for a disc under these situations. As can be seen in Fig. 3b, when radial displacements in the inner edge are higher for both discs subjected to an internal pressure, they decrease gradually through the radius. Radial displacements in D are higher than the ones in D, and they are parallel to the each other. Performance of FGM discs under constant temperature The variations of tangential and radial stresses in both discs subjected to uniform temperature (T=T a =T b =T 0 ) are shown in Fig. 4. The discs rotate with a constant angular speed when there are no temperature effects, T 0 = 0 C. Maximum radial stress value is obtained as 07.093 MPa at r = 305 mm for D, and tangential stress at the inner surface (70.985 MPa) is higher than that of the outer surface (63.4458 MPa). It decreases gradually along the radial direction, as seen in Fig. 4a. With Fig. Variations of the normalized elasticity modulus, density and temperature expansion coefficient along the radial direction of (a) D and (b) D Fig. 3 Variations of (a) tangential and radial stresses and (b) radial displacements at D and D with internal pressure

6 INDIAN J. ENG. MATER. SCI., APRIL 0 increasing temperature, radial stress at the inner section rises. On the other hand, tangential stress reduces at the outer surface while it increases at the inner surface. The maximum radial stress value is 7.885 MPa at r = 39 mm for D when the tangential stress at the inner surface (35.637 MPa) is higher than that of the outer surface (00.8488 MPa). It can be seen from Fig. 4b that radial stress at the inner section decreases. Tangential stress in the outer surface increases while it decreases in the inner surface with increasing temperature. Figure 5 shows the variations of radial displacements in both discs under uniform temperature. As T 0 = 0 C, the radial displacement values at both discs decrease gradually from the inner surface through the outer surface whereas by increasing temperature they increase more and more. In addition, radial displacement values in D are higher than the ones of D, as seen from these figures. Performance of FGM discs under thermal gradient Figure 6 illustrates the variations of radial stresses at both discs under linearly increasing and decreasing temperatures. As T=T a =T b =T 0 =0 C, the discs rotate with a constant angular velocity alone. As there is T a and T b =T 0 =0 C, the temperature T decreases linearly along the radial direction whereas the temperature increases linearly when there is T b and T a = T 0 = 0 C. Fig. 5 Variations of radial displacements at (a) D and (b) D under uniform temperature Fig. 4 Variations of tangential and radial stresses at (a) D and (b) D under uniform temperature Fig. 6 Variations of radial stresses at (a) D and (b) D under linearly increasing and decreasing temperatures

ÇALLIOĞLU et al.: STRESS ANALYSIS OF FUNCTIONALLY GRADED DISCS 7 Fig. 7 Variations of tangential stresses at (a) D and (b) D under linearly increasing and decreasing temperatures By increasing temperature T a, the radial stresses decrease gradually according to the rotating disc alone while they increase more and more with increasing temperature T b for both discs. Although the absolute radial stress values in D with increasing temperature T a are higher than those of D, it can be seen from these figures that the radial stress values in D with increasing temperature T b are higher than those of D. Figure 7 shows the variations of tangential stresses at both discs subjected to linearly increasing and decreasing temperatures. With increasing temperature T a, the tangential stresses at the inner edge decrease gradually and they increase at the outer edge according to the rotating disc alone, for D. By increasing temperature T b, they rise more and more at the inner surface and decrease at the outer surface. For D, when temperature T a are increased the tangential stresses at the inner edge decrease gradually and they increase at the outer edge. By increasing temperature T b, they rise at the inner surface and decrease at the outer surface. The absolute tangential stress values at both inner and outer edges of D by increasing temperature T a are higher than those of D, except for the stresses in the rotating disc alone. Unlike this situation, it is found that the absolute tangential stress values in D are higher than those of D with increasing temperature T b. Fig. 8 Variations of radial displacements at (a) D and (b) D under linearly increasing and decreasing temperatures The variations of radial displacements at both discs under linearly increasing and decreasing temperatures are depicted in Fig. 8. As T a =T b =T 0 =0 C, the radial displacement values at both discs decrease gradually from the inner surface through the outer surface, as can also be seen from Fig. 5. By increasing temperatures T a and T b, the radial displacement values at the outer surface are higher than those at the inner surface of both discs. Radial displacement values in D are higher than those of D for all temperatures. Conclusions The following conclusions can be derived from the thermal stress analysis of the FG discs: (i) As there is an internal pressure in the rotating discs, the radial stresses are equal to internal pressure at the inner surface and zero at the outer surface. The radial stresses at D are higher than those of D. As for the tangential stresses, the values of D are also higher than those of D. (ii) Therefore, the selection of D material properties can be recommended in order to carry more loads for a disc under these situations. (iii) The tangential stresses are found to be the highest at the inner surface but the lowest at the

8 INDIAN J. ENG. MATER. SCI., APRIL 0 outer surface for both discs. They increase at the inner surface whereas decrease at the outer surface with increasing T 0 and T b temperatures. If T a is increased, they decrease at the inner surface and rise at the outer surface, for both discs. (iv) With increasing T 0 and T a temperatures, the radial stresses decrease gradually along the radial section. They rise along the radial section when T b is increased. (v) The magnitudes of the tangential stresses are higher than those of the radial stresses. (vi) The radial displacement values in both discs subjected to uniform temperature T 0 are higher than those of discs that are subjected to linearly increasing T b and decreasing T a temperatures. Acknowledgements The authors would like to thank Pamukkale University Scientific Research Council for supporting this study under Project Contract No. 00FBE096. References Pindera M J, Aboudi J, Glaeser A M & Arnold S M, Compos Pt B-Eng, 8 (997). Erdoğan F, Compos Eng, 5 (995) 753-770. 3 Bayat M, Saleem M, Sahari B B, Hamouda A M S & Mahdi E, Mech Res Commun, 35(5) (008) 83-309. 4 Timoshenko S & Goodier J N, Theory of Elasticity (McGraw- Hill, California), 95. 5 Çallıoğlu H, J Reinf Plast Comp, 3 (004) 859-867. 6 Çallıoğlu H, J Thermoplast Compos Mater, 0 (007) 357-369. 7 Sayman O & Arman Y, J Reinf Plast Comp, 5(6) (006) 709-7. 8 Singh S B & Ray S, Mech Mater, 34 (00) 363-37. 9 Horgan C O & Chan A M, J Elasticity, 55() (999) 43-59. 0 Horgan C O & Chan A M, J Elasticity, 55(3) (999) 9-30. Durodola J F & Attia O, Compos Sci Technol, 60 (000) 987-995. Chen J, Ding H & Chen W, Arch Appl Mech, 77 (007) 4-5. 3 Mohammadi M & Dryden J R, J Therm Stresses, 3 (008) 587 598. 4 Gupta V K, Singh S B, Chandrawat H N & Ray S, J Eng Mater & Technol, 7() (005) 97-05. 5 You L H, You X Y, Zhang J J & Li J, Math Physik ZAMP, 58 (007) 068-084. 6 Çallıoğlu H, Adv Compos Lett, 7(5) (008) 47-53. 7 Bayat M, Saleem M, Sahari B B, Hamouda A M S & Mahdi E, Int J Pres Ves Pip, 86(6) (009) 357 37. 8 Hosseini Kordkheili S A & Naghdabadi R, Compos Struct, 79 (007) 508-56. 9 Carrera E, Brischetto S & Robaldo A, AIAA J, 46() (008) 94-03. 0 Brischetto S & Carrera E, Comput Struct, 88(3-4) (00) 474-483. Cinefra M, Carrera E, Brischetto S & Belouettar S, J Therm Stresses, 33(0) (00) 94-963. Noda N, Hetnarski R B & Tanigawa Y, Thermal Stresses (Lastran Co., NY), 000.