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Thermal shock resistance of functionally gradient solid cylinders
Materials Science and Engineering A 418 (2006) 99–110
Thermal shock resistance of functionally gradient solid cylinders Jun Zhao ∗ , Xing Ai, Yanzheng Li, Yonghui Zhou School of Mechanical Engineering, Shandong University, Jinan 250061, PR China Accepted 16 November 2005
Abstract In this paper, an analysis of the transient thermo-mechanical behavior of a solid cylinder of functionally gradient material (FGM) under the convective boundary condition is presented theoretically. The analytical formula of the unsteady temperature distribution is derived by using the separation-of-variables method and hence the maximum thermal stress attained at the surface of the FGM solid cylinder as well as its time of occurrence can be calculated. Based on a local tensile stress criterion, the expression of critical temperature change Tc leading to the local tensile strength at the surface, which is designated as the thermal shock resistance parameter for FGM solid cylinder, is obtained. The effects of the radial distributions of thermo-physical properties on the thermal shock resistance of the FGM solid cylinder are investigated via numerical calculations in contrast to homogeneous solid cylinder, from which some suggestions on design of FGM solid cylinders with high thermal shock resistance are put forward. © 2005 Elsevier B.V. All rights reserved. Keywords: Functionally gradient materials; Thermal shock resistance; Critical temperature change; Solid cylinder; Unsteady thermal stresses
1. Introduction Functionally gradient materials (FGM) are new advanced composite materials intentionally designed so that they possess desirable properties for specific applications, especially for performance under thermal environment. In the theory of elasticity, FGMs are mostly treated as non-homogeneous materials with material constants varying continuously along one spatial direction. In the past two decades, a number of investigations have dealt with thermoelastic problems for the basic structural components of FGMs. Noda and co-workers [1–6] analyzed the one-dimensional steady state thermal stress problems of a FGM plate and other shapes, and proposed an analytical method of one-dimensional unsteady thermal stresses in a FGM plate by using the perturbation method. Araki et al. [7] and Sugano et al. [8,9] derived the analytical solution of the temperature distribution in a multilayered material under pulse or stepwise heating. Tanigawa et al. [10–12] used an iteration technique to analyze the one-dimensional transient thermal stresses of a FGM plate and a FGM cylinder using a laminated composite model, and discussed the temperature dependence of material properties, they [13] also analyzed the transient thermal stress of a non-homogeneous plate taking into account the relative heat transfer at boundary surfaces. Awaji et al. [14–16] also proposed techniques for analyzing one-dimensional steady state and transient temperature distribution in a FGM plate and a hollow FGM cylinder. By introducing the theory of laminated composites, Ootao and Tanigawa [17] treated the three-dimensional transient thermal stresses of functionally gradient rectangular plates due to partial heat supply, and analyzed the piezothermoelastic problem of a functionally gradient rectangular plate bonded to a piezoelectric plate [18]. Kim and Noda [19,20] researched the two-dimensional unsteady thermoelastic problems of functionally gradient infinite hollow cylinders by using a Green’s function approach. Jabbari et al. [21] derived analytical solutions for one-dimensional steady-state thermoelastic problems of functionally gradient circular hollow cylinders in the case of material models expressed as power functions of r, and treated the two-dimensional thermoelastic problems of the functionally gradient cylinder by using the Fourier transform [22]. Liew et al. [23] obtained analytical solutions of a functionally gradient circular hollow cylinder by a novel limiting process that employs the solutions of homogeneous circular hollow cylinders. ∗
Corresponding author. Tel.: +86 531 8839 3904; fax: +86 531 8839 3904. E-mail address: [email protected] (J. Zhao).
0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.11.019
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Fig. 1. Functionally gradient cylinder.
Wang et al. obtained the exact expression of dynamic thermal stress in a transversely isotropic sphere and consequently studied the thermal stress-focusing effect [24]. Their further investigations on the magnetothermodynamic stress and perturbation of magnetic vector in a conducting orthotropic thermoelastic cylinder subject to thermal shock [25], and the thermo-electro-elastic transient response in piezoelectric hollow structures subjected to arbitrary thermal shock, sudden mechanical load and electric excitation [26,27] provided valuable references to solve some transient coupled problems for non-homogeneous structures. To our knowledge, however, little work has been focused on the solution for the unsteady temperature field and unsteady thermal stress field of FGM solid cylinders under the convective boundary condition, and a strength-based fracture criterion for the thermal shock resistance evaluation of FGM solid cylinders has not been previously obtained probably because there is few application of this kind of FGMs currently. The present work is undertaken to propose a thermal shock resistance parameter for FGM solid cylinders. A FGM solid cylinder with composition and properties varying radially (Poisson’s ratio is constant) is studied, with its surface suddenly exposed to a convective medium of different temperature. The analytical formula of the unsteady temperature distribution is obtained by using the separation-of-variables method and hence the maximum thermal stress attained at the surface of the FGM solid cylinder as well as its time of occurrence can be calculated. Based on a local tensile stress criterion, the expression of critical temperature change Tc leading to the local tensile strength at the surface as the thermal shock resistance parameter for FGM solid cylinder is obtained. The effects of the distributions for thermo-physical properties on the thermal shock resistance of the FGM solid cylinders are discussed via numerical calculations in contrast to homogeneous solid cylinder. 2. Analytical methods 2.1. Unsteady temperature field As shown in Fig. 1, consider a long FGM solid cylinder with its radius being b. A cylindrical coordinate system (r, θ, z) is established for reference, with the z-axis lying on the axis of the cylinder. As we seek the unsteady thermal stress solutions of the FGM cylinder in the plane strain condition, the temperature is independent of z. It is assumed that initially it is at a uniform temperature T0 , and at time t = 0 the surface of the cylinder (at r = b) is suddenly exposed to a convective medium of temperature Ts . And the coefficient of heat transfer h is assumed to be a constant. The thermo-mechanical properties of the FGM solid cylinder are varied in the radial direction (Poisson’s ratio is assumed to be constant) continuously. Hence the temperature field and the thermal stress field are also functions of radial coordinate r. The one-dimensional unsteady heat conduction equation of non-homogeneous solid cylinders, the initial and boundary conditions are given as follows: ∂T 1 ∂ ∂T ρ(r)c(r) = rλ(r) (1) ∂t r ∂r ∂r T (r, 0) = T0 ∂T =0 ∂r r=0 ∂T −λ = h(T − Ts ) ∂r r=b
(2) (3a)
(3b)
where T is the temperature, t is the time, r is the coordinate variable in the radial direction, and c(r), ρ(r) and λ(r) are specific heat, specific gravity and thermal conductivity, respectively. For the purpose of simplicity, the following dimensionless variables are
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introduced:
t τ = κ(1) 2 b T (¯r , τ) − Ts T (¯r , τ) − Ts V (¯r , τ) = = T0 − T s T c(¯r ) c¯ (¯r ) = c(1) ρ(¯r ) ¯ r) = ρ(¯ ρ(1) λ(¯ r ) ¯ r) = λ(¯ λ(1) r¯ =
r b
(4)
where c(1), ρ(1), λ(1) and κ(1) are the specific heat, the specific gravity, the thermal conductivity and the thermal diffusivity of the surface (¯r = 1) of the cylinder. So Eqs. (1)–(3) have the following dimensionless form: ∂V 1 ∂ ∂V ¯ ¯ r )¯c(¯r ) ρ(¯ = r¯ λ(¯r ) (5) ∂τ r¯ ∂¯r ∂¯r V (¯r , 0) = 1 ∂V =0 ∂¯r r¯ =0 ∂V − = BV ∂¯r r¯ =1
(6) (7a) (7b)
where B = (h × b)/λ is the Biot number of the cylinder surface. To solve the dimensionless heat conduction Eq. (5), we use the separation-of-variables method by asking: V (¯r , τ) = W(¯r ) · U(τ)
(8)
Eq. (5) can be transferred into: d W ∂ ¯ r )] dW ¯ r) [¯r λ(¯ λ(¯ 2 . d¯r + ∂¯r = = −µ2 ¯ r )¯c(¯r ) W(¯r ) r¯ ρ(¯ ¯ r )¯c(¯r ) d¯r ρ(¯ U(τ) 2
dU dτ
(9)
where µ > 0 and µ2 is a constant. Eq. (9) can be decomposed into the following two differential equations: dU(τ) + µ2 U(τ) = 0 dτ
(10a)
and d2 W(¯r ) + d¯r 2
∂ ¯ ∂¯r [¯r λ(¯r )]
¯ r) r¯ λ(¯
·
¯ r )¯c(¯r ) ρ(¯ dW(¯r ) + µ2 · W(¯r ) = 0 ¯ r) d¯r λ(¯
(10b)
The solution to Eq. (10a) is: U(τ) = A exp(−µ2 τ)
(11)
where the coefficient A is a constant. To solve Eq. (10b), we assume the dimensionless thermal conductivity and the dimensionless thermal diffusivity to be described with a power law as: ¯ ¯ r ) = exp[cλ (1 − r¯ )] cλ = ln[λ(0)] λ(¯ ¯ r) 0 ≤ r¯ ≤ 1 (12) λ(¯ κ¯ (¯r ) = = exp[cκ (1 − r¯ )] cκ = ln[¯κ(0)] ¯ r) c¯ (¯r )ρ(¯ Eq. (10b) becomes: d2 W(¯r ) dW(¯r ) + p1 (¯r ) + p2 (¯r )W(¯r) = φ(¯r ) d¯r 2 d¯r
(13)
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where p1 (¯r) =
1 − cλ r¯ r¯
p2 (¯r) = µ2 exp[−cκ (1 − r¯ )] φ(¯r) = 0 We use the method of variable substitution to solve the second order differential Eq. (13) with variable coefficients by introducing: ξ=
∞ exp(cλ r¯ ) (cλ r¯ )n ln(cλ r¯ ) + d¯r = exp − p1 (¯r ) d¯r d¯r = r¯ n · n!
(14)
n=1
p2 (¯r ) F1 (¯r ) = 2 = µ2 r¯ exp(−cκ ) exp[(cκ − 2cλ )¯r ]
(15)
F2 (¯r ) = 0
(16)
dξ d¯r
Then Eq. (13) becomes: d2 W(ξ) + F1 (¯r )W(ξ) = F2 (¯r ) dξ 2
(17)
The solution to Eq. (17) is:
∞
c (cλ r¯ )n cκ − 2cλ κ W(¯r ) = D cos ξ F1 (¯r ) + G sin ξ F1 (¯r ) = D cos µ¯r ln(cλ r¯ ) + exp − exp r¯ n · n! 2 2 n=1
∞ c cκ − 2cλ (cλ r¯ )n κ exp + G sin µ¯r ln(cλ r¯ ) + exp − r¯ (18) 2 n · n! 2 n=1
By referring to Eq. (7a), we obtain: G = 0. By referring to Eq. (7b), the following transcendental equation can be obtained: ∞ n ∞ ∞ cκ + 2 − 2cλ cλ cλn cλn −cλ ln cλ + (19) ln cλ + µ · tan µe +1+ = Becλ 2 n! n · n! n · n! n=1
n=1
n=1
By referring to Eq. (6), we have: ∞
−cκ 1 (cλ r¯ )n (cκ −2cλ ) exp 2 exp r¯ d¯r 0 cos µ¯r ln(cλ r¯ ) + n·n! 2 D=
1 0
cos2 µ¯r ln(cλ r¯ ) +
n=1 ∞ n=1
(cλ r¯ )n n·n!
exp
−cκ 2
exp
(cκ −2cλ ) r¯ 2
(20) d¯r
Consequently the dimensionless transient temperature field for solid FGM cylinder can be expressed as:
∞ ∞
c (cλ r¯ )m cκ − 2cλ κ V (¯r , τ) = Dn cos µn r¯ ln(cλ r¯ ) + exp − exp r¯ exp −µ2n τ m · m! 2 2 n=1
(21)
m=1
where the discrete values µn and Dn are the roots of Eqs. (19) and (20), respectively. 2.2. Unsteady thermal stress field and the thermal shock resistance parameter for solid FGM cylinder The unsteady thermal stresses in the FGM solid cylinder that are caused by the axisymmetric temperature change are to be determined. We assume the Poisson’s ratio of the cylinder to be constant, while the Young’s modulus and the thermal expansion
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coefficient are described with a power law as:
E(¯r) = E(1) exp[cE (1 − r¯ )] cE = ln[E(0)/E(1)] α(¯r ) = α(1) exp[cα (1 − r¯ )] cα = ln[α(0)/α(1)] ν(¯r ) ≡ ν
0 ≤ r¯ ≤ 1
(22)
where E(1) and α(1) are the Young’s modulus and the thermal expansion coefficient of the surface layer (¯r = 1), ν is the Poisson’s ratio of the FGM cylinder. The following formulae are used to derive the expressions of stresses σ r , σ θ and σ z : ∂σr σ r − σθ + =0 ∂r r
(23a)
∂u ∂r u εθ = r εr =
(23b) (23c)
where u, σ and ε are the displacement, stress and strain, respectively. For a long FGM solid cylinder, εz = 0. By substituting Eqs. (23b) and (23c) into Hooke’s law, we obtain: E(r) ∂u(r, t) u(r, t) (24a) (1 − ν) +ν − (1 − ν)α(r)[T (r, t) − T0 ] σr (r, t) = (1 + ν)(1 − 2ν) ∂r r E(r) u(r, t) ∂u(r, t) (1 − ν) σθ (r, t) = +ν − (1 − ν)α(r)[T (r, t) − T0 ] (24b) (1 + ν)(1 − 2ν) r ∂r νE(r) ∂u(r, t) u(r, t) 1 − ν (24c) + − α(r)[T (r, t) − T0 ] σz (r, t) = ν[σr (r, t) + σθ (r, t)] − α(r)[T (r, t) − T0 ] = (1 + ν)(1 − 2ν) ∂r r ν The following displacement equation is obtained by substituting Eqs. (24a) and (24b) into Eq. (23a): 1 dE(r) 1 ∂u(r, t) ν r dE(r) 1 ∂2 u(r, t) + · + + · · − 1 2 u(r, t) ∂r 2 E(r) dr r ∂r 1 − ν E(r) dr r 1+ν ∂{α(r)[T (r, t) − T0 ]} 1 dE(r) = · α(r)[T (r, t) − T0 ] + 1 − ν E(r) dr ∂r
(25)
By substituting Eq. (22) into Eq. (25), we obtain: ∂2 u(¯r , τ) ∂u(¯r , τ) + q1 (¯r ) − q2 (¯r )u(¯r , τ) = q3 (¯r , τ) 2 ∂¯r ∂¯r where q1 (¯r ) =
1 − cE r¯
q2 (¯r) =
cE k1 r¯ + 1 r¯ 2
q3 (¯r , τ) = k2 b k1 =
ν 1−ν
k2 =
1+ν 1−ν
∂[α(¯r)T ∗ (¯r , τ)] − cE α(¯r )T ∗ (¯r , τ) ∂¯r
T ∗ (¯r , τ) = T (¯r , τ) − T0
(26)
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We also use the method of variable substitution to solve the second order differential Eq. (26) with variable coefficients by introducing: ω=
∞ 1 (cE r¯ )n exp − q1 (¯r ) d¯r d¯r = · exp(cE r¯ ) d¯r = ln |cE r¯ | + r¯ n · n!
(27)
n=1
q2 (¯r ) F3 (¯r ) = 2 = (cE k1 r¯ + 1) exp(−2cE r¯ )
(28)
dω d¯r
q3 (¯r , τ) F4 (¯r , τ) = 2 = k2 b¯r 2 dω d¯r
∂[α(¯r )T ∗ (¯r , τ)] − cE α(¯r )T ∗ (¯r , τ) ∂¯r
exp(−2cE r¯ )
(29)
Then Eq. (26) becomes: ∂2 u(ω, τ) − F3 (¯r )u(ω, τ) = F4 (¯r , τ) ∂ω2
(30)
The solution to Eq. (30) is: ∞ cE k1 r¯ + 1 (cE r¯ )n u(¯r , τ) = H exp exp(−cE r¯ ) ln |cE r¯ | + 2 n · n! n=1
∞
(cE r¯ )n cE k1 r¯ + 1 + S exp − exp(−cE r¯ ) ln |cE r¯ | + 2 n · n!
+
k2 b¯r 2 cE α(¯r )T ∗ (¯r , τ) −
n=1
cE k1 r¯ + 1
∂[α(¯r)T ∗ (¯r,τ)] ∂¯r
(31)
where the constant H and S can be determined by using the displacement at the center of the cylinder (u(0, τ) = 0), and hence ∂u (¯r , τ) /∂¯r can also be calculated consequently. We introduce the dimensionless thermal stresses for Eq. (24) by asking: ¯ r , τ) = σ(¯
σ(¯r , τ)(1 − ν) α(1)E(1)T
(32)
Then we obtain: σ¯ r (¯r , τ) =
¯ r ) ∂u(¯r , τ) ¯ r ) u(¯r , τ) k4 E(¯ k3 E(¯ ¯ r )α(¯ ¯ r )V ∗ (¯r , τ) · + · − k5 E(¯ bα(1)T ∂¯r bα(1)T r¯
(33a)
σ¯ θ (¯r , τ) =
¯ r ) u(¯r , τ) ¯ r ) ∂u(¯r , τ) k3 E(¯ k4 E(¯ ¯ r )α(¯ ¯ r )V ∗ (¯r , τ) · + · − k5 E(¯ bα(1)T r¯ bα(1)T ∂¯r
(33b)
σ¯ z (¯r , τ) =
¯ r ) ∂u(¯r , τ) ¯ r ) u(¯r , τ) k4 E(¯ k4 E(¯ ¯ r )α(¯ ¯ r )V ∗ (¯r , τ) · + · − k5 E(¯ bα(1)T ∂¯r bα(1)T r¯
(33c)
where k3 =
(1 − ν)2 (1 + ν)(1 − 2ν)
k4 =
ν(1 − ν) (1 + ν)(1 − 2ν)
k5 =
1−ν 1 − 2ν
V ∗ (¯r , τ) = 1 − V (¯r , τ)
J. Zhao et al. / Materials Science and Engineering A 418 (2006) 99–110
By substituting u(¯r , τ) and ∂u(¯r , τ)/∂¯r into Eq. (33), the following dimensionless thermal stresses can be obtained:
L1 c α L2 c 2 L3 σ¯ r (¯r , τ) = W(¯r ) − + α + [W1 (¯r ) − cα W2 (¯r ) + cα2 W3 (¯r )] exp[cα (1 − r¯ )] L L L ∞ L1 Pn (1) L2 Qn (1) L3 Rn (1) 2 W(¯r ) − + + + − W1 (¯r )Pn (¯r ) + W2 (¯r )Qn (¯r ) + W3 (¯r )Rn (¯r ) exp(−µ τ) L L L n=1
2 L1 c α L2 c α L3 2 σ¯ θ (¯r , τ) = M(¯r) − + + [M1 (¯r ) − cα M2 (¯r ) + cα M3 (¯r )] exp[cα (1 − r¯ )] L L L ∞ L1 Pn (1) L2 Qn (1) L3 Rn (1) M(¯r ) − + + + − M1 (¯r )Pn (¯r ) + M2 (¯r )Qn (¯r ) + M3 (¯r )Rn (¯r ) exp(−µ2 τ) L L L n=1
2 L1 c α L2 c α L3 2 σ¯ z (¯r , τ) = N(¯r ) − + + [N1 (¯r ) − cα N2 (¯r ) + cα N3 (¯r )] exp[cα (1 − r¯ )] L L L ∞ L1 Pn (1) L2 Qn (1) L3 Rn (1) 2 N(¯r ) − + + + − N1 (¯r )Pn (¯r ) + N2 (¯r )Qn (¯r ) + N3 (¯r )Rn (¯r ) exp(−µ τ) L L L
105
(34)
n=1
where L, L1 , L2 , L3 , Pn (¯r ), Pn (1), Qn (¯r ), Qn (1), Rn (¯r ), Rn (1), W(¯r ), W1 (¯r ), W2 (¯r ), W3 (¯r ), M(¯r ), M1 (¯r ), M2 (¯r ), M3 (¯r ), N(¯r ), N1 (¯r ), N2 (¯r ) and N3 (¯r ) are shown in Appendix A. Then the dimensionless thermal stresses at the surface (¯r = 1) can consequently be obtained as: σ¯ r (1, τ) = 0
L1 c α L2 cα2 L3 2 σ¯ θ (1, τ) = M(1) − + + [M1 (1) − cα M2 (1) + cα M3 (1)] L L L ∞ L1 Pn (1) L2 Qn (1) L3 Rn (1) 2 + M(1) − + + − M1 (1)Pn (1) + M2 (1)Qn (1) + M3 (1)Rn (1) exp(−µ τ) L L L (35) n=1
2 L1 c α L2 c L3 σ¯ z (1, τ) = N(1) − + α + [N1 (1) − cα N2 (1) + cα2 N3 (1)] L L L ∞ L1 Pn (1) L2 Qn (1) L3 Rn (1) 2 N(1) − + + + − N1 (1)Pn (1) + N2 (1)Qn (1) + N3 (1)Rn (1) exp(−µ τ) L L L n=1
where M(1), M1 (1), M2 (1) and M3 (1) are constants when r¯ = 1, which can be calculated by using the expressions of M(¯r), M1 (¯r ), M2 (¯r ) and M3 (¯r ) in Appendix A, respectively. N(1), N1 (1), N2 (1) and N3 (1) are constants when r¯ = 1, which can be calculated by using the expressions of N(¯r ), N1 (¯r ), N2 (¯r ) and N3 (¯r ) in Appendix A, respectively. The maximum tensile stress criterion for fracture initiation of the surface layer is: σmax (1, τc ) = σf (1)
(36)
where σ f (1) is the tensile strength of the surface layer, τ c is the dimensionless time for the surface layer to attain the maximum tensile stress, which can be obtained from the solution of the following equation: ¯ τ) dσ(1, =0 dτ
(37)
By referring to Eqs. (32) and (36), the critical temperature change for the surface layer of the cylinder to reach its tensile strength in cold shock (T0 > Ts , e.g. quenching) can be given by: Tc =
R(1) σ¯ max (1, τc )
(38)
where R(1) = σf (1)[1 − ν(1)]/[α(1)E(1)] [28] is the thermal shock resistance parameter of the surface layer. 3. Numerical results and discussion 3.1. Calculations of unsteady thermal stresses An Al2 O3 /TiC FGM solid cylinder is considered for the numerical calculations, which has the material properties of TiC at the surface (¯r = 1) and those of Al2 O3 at the center (¯r = 0). The composition varies continuously in the radial direction of the cylinder.
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Table 1 Data of the physical properties of Al2 O3 and TiC Materials
Specific gravity, ρ (g cm−3 )
Thermal conductivity (20 ◦ C), λ (W m−1 K−1 )
Specific heat (20 ◦ C), c (J g−1 K−1 )
Thermal expansion coefficient (20 ◦ C), α (10−6 K−1 )
Young’s modulus, E (GPa)
Poisson’s ratio, ν
␣-Al2 O3 TiC
3.99 4.93
30 20
0.77 0.56
8.5 7.6
380 440
0.19 0.19
Data of the physical properties of Al2 O3 and TiC are listed in Table 1. The Poisson’s ratio of Al2 O3 is designated as 0.19, which is equal to that of TiC considering the assumption of the present work (Eq. (22)). Beside the Al2 O3 /TiC FGM solid cylinder, a homogeneous solid cylinder made of pure TiC is also taken into consideration for comparison. The exponents cκ , cλ , cα and cE (referring to Eqs. (12) and (22)) are 0.2986, 0.4055, 0.1119 and −0.1466, respectively for the FGM solid cylinder, and are all zero for the homogeneous solid cylinder. Considering the computation capacity and computation time, only the first eight roots of Eq. (19) are used in the calculations of thermal stresses of the FGM cylinder at different Biot numbers. According to the calculation results, the value of the circumferential stress σ¯ θ (1, τ) of the FGM cylinder is always higher than that of the axial stress σ¯ z (1, τ) under a specific cold condition. Therefore we choose the circumferential stress to evaluate the thermal shock resistance of the FGM cylinder, and Eq. (38) becomes: Tc =
R(1) σ¯ θmax (1, τθc )
(39)
In case of the homogeneous solid cylinder, however, the circumferential stress is always equal to the axial stress. Data of the maximum dimensionless circumferential and axial stresses of the FGM cylinder (σ¯ θmax (1, τθc ) and (σ¯ zmax (1, τzc )) and the maximum dimensionless stresses σ¯ max (1, τc ) of the homogeneous cylinder and their times of occurrence at different Biot numbers are listed ¯ τ) of the in Table 2. The dimensionless circumferential stresses σ¯ θ (1, τ) of the FGM cylinder and dimensionless stresses σ(1, homogeneous cylinder at different values of Biot number in cold shock are plotted against dimensionless time τ in Fig. 2a–c, respectively. It can be seen clearly that the dimensionless circumferential stresses σ¯ θ (1, τ) of the FGM cylinder and the dimensionless stresses ¯ τ) of the homogeneous cylinder are all zero at the beginning of the cold shock (τ = 0). They increase with the dimensionless σ(1, time τ, reach their maximum values σ¯ θmax (1, τθc ) and σ¯ max (1, τc ), respectively, then decrease with further increase in time. At any given value of Biot number B (except B = ∝), the dimensionless time for the surface of the FGM cylinder to attain the maximum thermal stress is shorter than that of the homogeneous cylinder (i.e. τ θc < τ c ). The maximum dimensionless thermal stress σ¯ θmax (1, τθc ) at the surface of the FGM cylinder is lower than the maximum dimensionless thermal stress σ¯ max (1, τc ) at the surface of the homogeneous cylinder. Furthermore, the difference between τ c and τ θc decreases monotonously with the increase in Biot number (Table 2). Whereas, the difference between σ¯ max (1, τc ) and σ¯ θmax (1, τθc ) increases with the increase in Biot number (Table 2). ¯ τ) = 1 is achieved at the It is always the case that for the limiting case of an ideal cold shock (B = ∞), a maximum value of σ(1, surface at τ = 0 for all the two cylinders. 3.2. Discussions The maximum dimensionless thermal stress σ¯ θmax (1, τθc ) at the surface of the FGM cylinder and its dimensionless time of occurrence τ θc , as well as the critical temperature change Tc for the surface fracture are all dependent on the radial distributions of the thermo-physical parameters, and the compositional distribution of FGM cylinder along the radial direction. It can be concluded from the above calculation results that the critical temperature change Tc for the surface layer of the FGM cylinder (with positive cκ , cλ , cα and negative cE ) to reach its tensile strength σ f (1) in cold shock is higher than that of the homogeneous cylinder made of pure TiC (The surfaces of the FGM cylinder and the homogeneous cylinder have the same composition and hence the same tensile strength σ f so as to be comparable, see Eq. (36)). Therefore, some suggestions on the design Table 2 Data of the maximum dimensionless thermal stresses and their times of occurrence B
0.1 1 5 ∞
FGM cylinder
Homogeneous cylinder
τ θc
σ¯ θmax (1, τθc )
τ zc
σ¯ zmax (1, τzc )
τc
σ¯ max (1, τc )
0.2172 0.0964 0.0348 0
0.0196 0.1474 0.3732 1
0.2122 0.0933 0.0327 0
0.0186 0.1451 0.3683 1
0.2531 0.1093 0.0393 0
0.0231 0.1593 0.3913 1
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Fig. 2. Dimensionless surface stresses as a function of time τ. (a) B = 5, (b) B = 1 and (c) B = 0.1.
of FGM cylinders with high thermal shock resistance can be made as follows: Both the thermal expansion coefficient α and the thermal diffusivity κ (=λ/(cρ)) of the center should be higher than that of the surface, whereas the Young’s modulus E of the center should be lower than that of the surface. The previous work of the author [29] in the thermal shock resistance parameter for FGM plates with symmetrical structures supported the suggestions made above. Regardless of the technological difficulties in the fabrication of the FGM solid cylinders with continuously radial composition transition, the present work provides an approach to the design of potential FGM solid cylinders with high thermal shock resistance. 4. Conclusions The analytical formulae of unsteady temperature field and unsteady thermal stress field for a FGM solid cylinder under a convective boundary condition are derived by using the separation-of-variables method and the displacement method, respectively. A strength-based fracture criterion for thermal shock of FGM sold cylinder is put foreword and consequently the expression of critical temperature change Tc leading to the local tensile strength at the surface as the thermal shock resistance parameter for FGM sold cylinder is obtained. The effects of the distributions for thermo-physical properties on the thermal shock resistance of the FGM sold cylinder are analyzed via numerical calculations in contrast to homogeneous ceramics, from which some suggestions on design of FGM solid cylinders with high thermal shock resistance are made as follows: (1) Both the thermal expansion coefficient α and the thermal diffusivity κ (=λ/(cρ)) of the center should be higher than that of the surface. (2) The Young’s modulus E of the center should be lower than that of the surface.
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Acknowledgements This work was supported by the National Natural Science Foundation of China (50105011) and the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (200231) as well as the Natural Science Foundation of Shandong Province (Y2004F14). Appendix A The constants and functions in Eq. (34) are given by ∞ ∞ n n 2k c E k1 − c E − c E cE c E k1 + 1 cE 1 2 L = (cE k1 + 1) k3 exp(−cE ) + 1+ + k4 2 n · n! 2 n! n=1 n=1 ∞ n cE k1 + 1 cE ·exp exp(−cE ) ln |cE | + 2 n · n!
(A.1)
n=1
L1 = k5 (cE k1 + 1)2 − cE k2 k3 (cE k1 + 2) − cE k2 k4 (cE k1 + 1)
(A.2)
2 L2 = k2 k4 (cE k1 + 1) − k2 k3 (cE k1 + cE − cE k1 − 2)
(A.3)
L3 = k2 k3 (cE k1 + 1)
(A.4)
∞ (cλ r¯ )m
Pn (¯r ) = Dn exp[cα (1 − r¯ )] · cos µn r¯ ln |cλ r¯ | +
∞ cλm ln |cλ | + m · m!
Pn (1) = Dn cos µn
m=1
m · m!
c cκ − 2cλ κ exp − exp r¯ 2 2
(A.5)
exp(−cλ )
(A.6)
m=1
c cκ − 2cλ κ exp − exp r¯ exp[cα (1 − r¯ )] + µn exp[cα (1 − r¯ )] Qn (¯r ) = Dn cα cos µn r¯ ln |cλ r¯ | + m · m! 2 2 m=1
∞ ∞ c cκ − 2cλ cκ − 2cλ (cλ r¯ )m (cλ r¯ )m κ × exp − exp r¯ · r¯ + 1 ln |cλ r¯ | + +1+ 2 2 2 m! m · m! m=1 m=1
∞ c (cλ r¯ )m cκ − 2cλ κ × sin µn r¯ ln |cλ r¯ | + exp − exp r¯ (A.7) m · m! 2 2 ∞ (cλ r¯ )m
m=1
Qn (1) = Dn
∞ cλm ln |cλ | + m · m! m=1
cα cos µn
+1 +
∞ m cλ m=1
m!
e−cλ · sin µn
cκ + 2 − 2cλ + µn e 2 ∞ cλm ln |cλ | + e−cλ m · m!
−cλ
∞ cλm ln |cλ | + m · m!
m=1
(A.8)
m=1
∞ ∞ m m 2 c − 2c r ¯ ) r ¯ ) (c (c λ κ λ λ Rn (¯r) = Dn ecα (1−¯r) cα2 − µ2n e−cκ e(cκ −2cλ )¯r r¯ + 1 ln |cλ r¯ | + +1+ 2 m · m! m! × cos µn r¯ e ×
−cκ 2
e
cκ −2cλ r¯ 2
4cα + 2cλ − cκ 2
ln |cλ r¯ | +
∞ (cλ r¯ )m
m · m! m=1
cκ − 2cλ r¯ + 1 2
m=1
+ µn e
ln |cλ r¯ | +
−cκ 2
e
∞ (cλ r¯ )m m=1
m · m!
m=1
cκ −2cλ r¯ 2
+1+
∞ (cλ r¯ )m m=1
m!
−
cκ − 2cλ 2
J. Zhao et al. / Materials Science and Engineering A 418 (2006) 99–110
∞ (cλ r¯ )m
∞
1 (cλ r¯ )m cκ − 2cλ r¯ + 1 + 2 r¯ m · m! m! · r¯ m=1 m=1 ∞ cκ −2cλ −cκ (cλ r¯ )m r¯ 2 2 × sin µn r¯ e ln |cλ r¯ | + e m · m! ×
ln |cλ r¯ | +
−
−
∞ m=1
(cλ r¯ )m r¯ · (m − 1)!
109
(A.9)
m=1
∞ m 2 ∞ m c + 2 − 2c c c κ λ λ λ ln |cλ | + Rn (1) = Dn cα2 − µ2n e−2cλ +1+ 2 m! m · m!
m=1
m=1
∞ ∞ cλm cκ + 2 − 2cλ cλm −Cλ 4cα + 2cλ − cκ ·cos µn e ln |cλ | + + µn e ln |cλ | + m · m! 2 2 m · m! m=1 m=1 ∞ m ∞ ∞ m ∞ cλ cκ − 2cλ cλm cλ cλm cκ + 2 − 2cλ +1 + − ln |cλ | + 1+ − − m! 2 2 m · m! m! (m − 1)! m=1 m=1 m=1 m=1 ∞ cλm −cλ ·sin µn e ln |cλ | + (A.10) m · m! −cλ
m=1
∞ n 2 k r¯ c E k1 − c E − c E c k (c r ¯ + 1 r ¯ ) E 1 E 1 −c −c r ¯ r ¯ ¯ r ) exp (ln |cE r¯ | W(¯r ) = E(¯ e E ln |cE r¯ | + · k3 e E 2 n · n! 2 n=1 ∞ ∞ (cE r¯ )n cE k1 r¯ + 1 1 (cE r¯ )n k4 + + + + n · n! 2 r¯ n! · r¯ r¯
n=1
(A.11)
n=1
W1 (¯r ) =
¯ r) cE k2 k4 r¯ E(¯ ¯ r) cE k2 k3 (cE k1 r¯ 2 + 2¯r )E(¯ ¯ r) + − k5 E(¯ 2 cE k1 r¯ + 1 (cE k1 r¯ + 1)
(A.12)
W2 (¯r ) =
2 k r¯ 3 − c k r¯ 2 + c r¯ 2 − 2¯r )E(¯ ¯ r ) k2 k4 r¯ E(¯ ¯ r) k2 k3 (cE 1 E 1 E − 2 cE k1 r¯ + 1 (cE k1 r¯ + 1)
(A.13)
W3 (¯r ) = −
¯ r) k2 k3 r¯ 2 E(¯ cE k1 r¯ + 1
(A.14)
∞ ∞ n n 2 k r¯ c c k r ¯ + 1 (c r ¯ ) k − c − c (c r ¯ ) E 1 E E 1 E 1 E E −c r ¯ −c r ¯ ¯ r ) exp M(¯r ) = E(¯ e E ln |cE r¯ | + · k4 e E ln |cE r¯ | + 2 n · n! 2 n · n! n=1 n=1 ∞ cE k1 r¯ + 1 1 (cE r¯ )n k3 + + + (A.15) 2 r¯ n! · r¯ r¯ n=1
M1 (¯r ) =
¯ r ) cE k2 k3 r¯ E(¯ ¯ r) cE k2 k4 (cE k1 r¯ 2 + 2¯r )E(¯ ¯ r) − k5 E(¯ + 2 cE k1 r¯ + 1 (cE k1 r¯ + 1)
(A.16)
M2 (¯r ) =
2 k r¯ 3 − c k r¯ 2 + c r¯ 2 − 2¯r )E(¯ ¯ r) ¯ r ) k2 k3 r¯ E(¯ k2 k4 (cE 1 E 1 E − cE k1 r¯ + 1 (cE k1 r¯ + 1)2
(A.17)
M3 (¯r ) = −
¯ r) k2 k4 r¯ 2 E(¯ cE k1 r¯ + 1
(A.18)
∞ ∞ n n 2 k r¯ c c (c (c k r ¯ + 1 r ¯ ) k − c − c r ¯ ) E 1 E E 1 E E 1 E ¯ r ) exp N(¯r ) = E(¯ e−cE r¯ ln |cE r¯ | + · k4 e−cE r¯ ln |cE r¯ | + 2 n · n! 2 n · n! n=1 n=1 ∞ k4 cE k1 r¯ + 1 1 (cE r¯ )n + + (A.19) + 2 r¯ n! · r¯ r¯ n=1
J. Zhao et al. / Materials Science and Engineering A 418 (2006) 99–110
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N1 (¯r ) =
¯ r ) cE k2 k4 r¯ E(¯ ¯ r) cE k2 k4 (cE k1 r¯ 2 + 2¯r )E(¯ ¯ r) + − k5 E(¯ 2 cE k1 r¯ + 1 (cE k1 r¯ + 1)
(A.20)
N2 (¯r ) =
2 k r¯ 3 − c k r¯ 2 + c r¯ 2 − 2¯r )E(¯ ¯ (¯r ) ¯ r ) k2 k4 r¯ E k2 k4 (cE 1 E 1 E − cE k1 r¯ + 1 (cE k1 r¯ + 1)2
(A.21)
N3 (¯r ) = −
¯ r) k2 k4 r¯ 2 E(¯ cE k1 r¯ + 1
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
N. Noda, T. Tsuji, Trans. Jpn. Soc. Mech. Eng. 57A (533) (1991) 98–103 (in Japanese). N. Noda, T. Tsuji, Trans. Jpn. Soc. Mech. Eng. 57A (535) (1991) 625–631 (in Japanese). Y. Obata, N. Noda, T. Tsuji, Trans. Jpn. Soc. Mech. Eng. 58A (553) (1992) 1689–1695 (in Japanese). Y. Obata, N. Noda, J. Therm. Stresses 17 (553) (1994) 471–487. Y. Obata, N. Noda, Trans. Jpn. Soc. Mech. Eng. 59A (560) (1993) 1090–1096 (in Japanese). Y. Obata, N. Noda, Trans. Jpn. Soc. Mech. Eng. 59A (560) (1993) 1097–1103 (in Japanese). N. Araki, A. Makino, T. Ishiguro, J. Mihara, Int. J. Thermophys. 13 (3) (1992) 515–538. Y. Sugano, K. Sato, N. Kimura, N. Sumi, Trans. Jpn. Soc. Mech. Eng. 62A (595) (1996) 728–736 (in Japanese). Y. Sugano, K. Sato, N. Sumi, Trans. Jpn. Soc. Mech. Eng. 63A (606) (1997) 378–385 (in Japanese). Y. Tanigawa, T. Akai, T. Kasai, Trans. Jpn. Soc. Mech. Eng. 61A (583) (1995) 607–613 (in Japanese). Y. Tanigawa, M. Matsumoto, T. Akai, JSME Int. J. 40A (1) (1997) 84–93. Y. Tanigawa, N. Oka, T. Akai, R. Kawamura, JSME Int. J. 40A (2) (1997) 117–127. Y. Tanigawa, T. Akai, R. Kawamura, N. Oka, Trans. Jpn. Soc. Mech. Eng. 62A (593) (1996) 131–137 (in Japanese). H. Awaji, H. Takenaka, S. Honda, N. Nishikawa, J. Ceram. Soc. Jpn. 107 (9) (1999) 780–785. H. Awaji, H. Takenaka, S. Honda, N. Nishikawa, Trans. Jpn. Soc. Mech. Eng. 65A (639) (1999) 2318–2324 (in Japanese). H. Awaji, R. Sivakumar, J. Am. Ceram. Soc. 84 (5) (2001) 1059–1065. Y. Ootao, Y. Tanigawa, J. Therm. Stresses 22 (1999) 35–55. Y. Ootao, Y. Tanigawa, Int. J. Solids Struct. 37 (32) (2000) 4377–4401. K.S. Kim, N. Noda, Acta Mech. 156 (2002) 145–161. K.S. Kim, N. Noda, Arch. Appl. Mech. 72 (2–3) (2002) 127–137. M. Jabbari, S. Sohrabpour, M.R. Eslami, Int. J. Pres. Ves. Pip. 79 (7) (2002) 493–497. M. Jabbari, S. Sohrabpour, M.R. Eslami, ASME J. Appl. Mech. 70 (1) (2003) 111–118. K.M. Liew, S. Kitiporncai, X.Z. Zhang, C.W. Lim, Int. J. Solids Struct. 40 (10) (2003) 2355–2380. X. Wang, C. Wang, G. Lu, B.M. Zhou, J. Therm. Stresses 25 (1) (2002) 31–44. X. Wang, H.L. Dai, Int. J. Eng. Sci. 42 (5–6) (2004) 539–556. H.L. Dai, X. Wang, Eur. J. Mech. A: Solids 24 (2) (2005) 263–276. H.L. Dai, X. Wang, Int. J. Solids Struct. 42 (3–4) (2005) 1151–1171. W.D. Kingery, H.K. Bowen, D.R. Uhlmann, Introduction to Ceramics, 2nd ed., John Wiley & Sons Inc., New York, 1976. J. Zhao, X. Ai, J.X. Deng, Z.X. Wang, Mater. Sci. Eng. A 382 (1–2) (2004) 23–29.
(A.22)
Thermal shock resistance of functionally gradient solid cylinders Jun Zhao ∗ , Xing Ai, Yanzheng Li, Yonghui Zhou School of Mechanical Engineering, Shandong University, Jinan 250061, PR China Accepted 16 November 2005
Abstract In this paper, an analysis of the transient thermo-mechanical behavior of a solid cylinder of functionally gradient material (FGM) under the convective boundary condition is presented theoretically. The analytical formula of the unsteady temperature distribution is derived by using the separation-of-variables method and hence the maximum thermal stress attained at the surface of the FGM solid cylinder as well as its time of occurrence can be calculated. Based on a local tensile stress criterion, the expression of critical temperature change Tc leading to the local tensile strength at the surface, which is designated as the thermal shock resistance parameter for FGM solid cylinder, is obtained. The effects of the radial distributions of thermo-physical properties on the thermal shock resistance of the FGM solid cylinder are investigated via numerical calculations in contrast to homogeneous solid cylinder, from which some suggestions on design of FGM solid cylinders with high thermal shock resistance are put forward. © 2005 Elsevier B.V. All rights reserved. Keywords: Functionally gradient materials; Thermal shock resistance; Critical temperature change; Solid cylinder; Unsteady thermal stresses
1. Introduction Functionally gradient materials (FGM) are new advanced composite materials intentionally designed so that they possess desirable properties for specific applications, especially for performance under thermal environment. In the theory of elasticity, FGMs are mostly treated as non-homogeneous materials with material constants varying continuously along one spatial direction. In the past two decades, a number of investigations have dealt with thermoelastic problems for the basic structural components of FGMs. Noda and co-workers [1–6] analyzed the one-dimensional steady state thermal stress problems of a FGM plate and other shapes, and proposed an analytical method of one-dimensional unsteady thermal stresses in a FGM plate by using the perturbation method. Araki et al. [7] and Sugano et al. [8,9] derived the analytical solution of the temperature distribution in a multilayered material under pulse or stepwise heating. Tanigawa et al. [10–12] used an iteration technique to analyze the one-dimensional transient thermal stresses of a FGM plate and a FGM cylinder using a laminated composite model, and discussed the temperature dependence of material properties, they [13] also analyzed the transient thermal stress of a non-homogeneous plate taking into account the relative heat transfer at boundary surfaces. Awaji et al. [14–16] also proposed techniques for analyzing one-dimensional steady state and transient temperature distribution in a FGM plate and a hollow FGM cylinder. By introducing the theory of laminated composites, Ootao and Tanigawa [17] treated the three-dimensional transient thermal stresses of functionally gradient rectangular plates due to partial heat supply, and analyzed the piezothermoelastic problem of a functionally gradient rectangular plate bonded to a piezoelectric plate [18]. Kim and Noda [19,20] researched the two-dimensional unsteady thermoelastic problems of functionally gradient infinite hollow cylinders by using a Green’s function approach. Jabbari et al. [21] derived analytical solutions for one-dimensional steady-state thermoelastic problems of functionally gradient circular hollow cylinders in the case of material models expressed as power functions of r, and treated the two-dimensional thermoelastic problems of the functionally gradient cylinder by using the Fourier transform [22]. Liew et al. [23] obtained analytical solutions of a functionally gradient circular hollow cylinder by a novel limiting process that employs the solutions of homogeneous circular hollow cylinders. ∗
Corresponding author. Tel.: +86 531 8839 3904; fax: +86 531 8839 3904. E-mail address: [email protected] (J. Zhao).
0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.11.019
J. Zhao et al. / Materials Science and Engineering A 418 (2006) 99–110
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Fig. 1. Functionally gradient cylinder.
Wang et al. obtained the exact expression of dynamic thermal stress in a transversely isotropic sphere and consequently studied the thermal stress-focusing effect [24]. Their further investigations on the magnetothermodynamic stress and perturbation of magnetic vector in a conducting orthotropic thermoelastic cylinder subject to thermal shock [25], and the thermo-electro-elastic transient response in piezoelectric hollow structures subjected to arbitrary thermal shock, sudden mechanical load and electric excitation [26,27] provided valuable references to solve some transient coupled problems for non-homogeneous structures. To our knowledge, however, little work has been focused on the solution for the unsteady temperature field and unsteady thermal stress field of FGM solid cylinders under the convective boundary condition, and a strength-based fracture criterion for the thermal shock resistance evaluation of FGM solid cylinders has not been previously obtained probably because there is few application of this kind of FGMs currently. The present work is undertaken to propose a thermal shock resistance parameter for FGM solid cylinders. A FGM solid cylinder with composition and properties varying radially (Poisson’s ratio is constant) is studied, with its surface suddenly exposed to a convective medium of different temperature. The analytical formula of the unsteady temperature distribution is obtained by using the separation-of-variables method and hence the maximum thermal stress attained at the surface of the FGM solid cylinder as well as its time of occurrence can be calculated. Based on a local tensile stress criterion, the expression of critical temperature change Tc leading to the local tensile strength at the surface as the thermal shock resistance parameter for FGM solid cylinder is obtained. The effects of the distributions for thermo-physical properties on the thermal shock resistance of the FGM solid cylinders are discussed via numerical calculations in contrast to homogeneous solid cylinder. 2. Analytical methods 2.1. Unsteady temperature field As shown in Fig. 1, consider a long FGM solid cylinder with its radius being b. A cylindrical coordinate system (r, θ, z) is established for reference, with the z-axis lying on the axis of the cylinder. As we seek the unsteady thermal stress solutions of the FGM cylinder in the plane strain condition, the temperature is independent of z. It is assumed that initially it is at a uniform temperature T0 , and at time t = 0 the surface of the cylinder (at r = b) is suddenly exposed to a convective medium of temperature Ts . And the coefficient of heat transfer h is assumed to be a constant. The thermo-mechanical properties of the FGM solid cylinder are varied in the radial direction (Poisson’s ratio is assumed to be constant) continuously. Hence the temperature field and the thermal stress field are also functions of radial coordinate r. The one-dimensional unsteady heat conduction equation of non-homogeneous solid cylinders, the initial and boundary conditions are given as follows: ∂T 1 ∂ ∂T ρ(r)c(r) = rλ(r) (1) ∂t r ∂r ∂r T (r, 0) = T0 ∂T =0 ∂r r=0 ∂T −λ = h(T − Ts ) ∂r r=b
(2) (3a)
(3b)
where T is the temperature, t is the time, r is the coordinate variable in the radial direction, and c(r), ρ(r) and λ(r) are specific heat, specific gravity and thermal conductivity, respectively. For the purpose of simplicity, the following dimensionless variables are
J. Zhao et al. / Materials Science and Engineering A 418 (2006) 99–110
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introduced:
t τ = κ(1) 2 b T (¯r , τ) − Ts T (¯r , τ) − Ts V (¯r , τ) = = T0 − T s T c(¯r ) c¯ (¯r ) = c(1) ρ(¯r ) ¯ r) = ρ(¯ ρ(1) λ(¯ r ) ¯ r) = λ(¯ λ(1) r¯ =
r b
(4)
where c(1), ρ(1), λ(1) and κ(1) are the specific heat, the specific gravity, the thermal conductivity and the thermal diffusivity of the surface (¯r = 1) of the cylinder. So Eqs. (1)–(3) have the following dimensionless form: ∂V 1 ∂ ∂V ¯ ¯ r )¯c(¯r ) ρ(¯ = r¯ λ(¯r ) (5) ∂τ r¯ ∂¯r ∂¯r V (¯r , 0) = 1 ∂V =0 ∂¯r r¯ =0 ∂V − = BV ∂¯r r¯ =1
(6) (7a) (7b)
where B = (h × b)/λ is the Biot number of the cylinder surface. To solve the dimensionless heat conduction Eq. (5), we use the separation-of-variables method by asking: V (¯r , τ) = W(¯r ) · U(τ)
(8)
Eq. (5) can be transferred into: d W ∂ ¯ r )] dW ¯ r) [¯r λ(¯ λ(¯ 2 . d¯r + ∂¯r = = −µ2 ¯ r )¯c(¯r ) W(¯r ) r¯ ρ(¯ ¯ r )¯c(¯r ) d¯r ρ(¯ U(τ) 2
dU dτ
(9)
where µ > 0 and µ2 is a constant. Eq. (9) can be decomposed into the following two differential equations: dU(τ) + µ2 U(τ) = 0 dτ
(10a)
and d2 W(¯r ) + d¯r 2
∂ ¯ ∂¯r [¯r λ(¯r )]
¯ r) r¯ λ(¯
·
¯ r )¯c(¯r ) ρ(¯ dW(¯r ) + µ2 · W(¯r ) = 0 ¯ r) d¯r λ(¯
(10b)
The solution to Eq. (10a) is: U(τ) = A exp(−µ2 τ)
(11)
where the coefficient A is a constant. To solve Eq. (10b), we assume the dimensionless thermal conductivity and the dimensionless thermal diffusivity to be described with a power law as: ¯ ¯ r ) = exp[cλ (1 − r¯ )] cλ = ln[λ(0)] λ(¯ ¯ r) 0 ≤ r¯ ≤ 1 (12) λ(¯ κ¯ (¯r ) = = exp[cκ (1 − r¯ )] cκ = ln[¯κ(0)] ¯ r) c¯ (¯r )ρ(¯ Eq. (10b) becomes: d2 W(¯r ) dW(¯r ) + p1 (¯r ) + p2 (¯r )W(¯r) = φ(¯r ) d¯r 2 d¯r
(13)
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where p1 (¯r) =
1 − cλ r¯ r¯
p2 (¯r) = µ2 exp[−cκ (1 − r¯ )] φ(¯r) = 0 We use the method of variable substitution to solve the second order differential Eq. (13) with variable coefficients by introducing: ξ=
∞ exp(cλ r¯ ) (cλ r¯ )n ln(cλ r¯ ) + d¯r = exp − p1 (¯r ) d¯r d¯r = r¯ n · n!
(14)
n=1
p2 (¯r ) F1 (¯r ) = 2 = µ2 r¯ exp(−cκ ) exp[(cκ − 2cλ )¯r ]
(15)
F2 (¯r ) = 0
(16)
dξ d¯r
Then Eq. (13) becomes: d2 W(ξ) + F1 (¯r )W(ξ) = F2 (¯r ) dξ 2
(17)
The solution to Eq. (17) is:
∞
c (cλ r¯ )n cκ − 2cλ κ W(¯r ) = D cos ξ F1 (¯r ) + G sin ξ F1 (¯r ) = D cos µ¯r ln(cλ r¯ ) + exp − exp r¯ n · n! 2 2 n=1
∞ c cκ − 2cλ (cλ r¯ )n κ exp + G sin µ¯r ln(cλ r¯ ) + exp − r¯ (18) 2 n · n! 2 n=1
By referring to Eq. (7a), we obtain: G = 0. By referring to Eq. (7b), the following transcendental equation can be obtained: ∞ n ∞ ∞ cκ + 2 − 2cλ cλ cλn cλn −cλ ln cλ + (19) ln cλ + µ · tan µe +1+ = Becλ 2 n! n · n! n · n! n=1
n=1
n=1
By referring to Eq. (6), we have: ∞
−cκ 1 (cλ r¯ )n (cκ −2cλ ) exp 2 exp r¯ d¯r 0 cos µ¯r ln(cλ r¯ ) + n·n! 2 D=
1 0
cos2 µ¯r ln(cλ r¯ ) +
n=1 ∞ n=1
(cλ r¯ )n n·n!
exp
−cκ 2
exp
(cκ −2cλ ) r¯ 2
(20) d¯r
Consequently the dimensionless transient temperature field for solid FGM cylinder can be expressed as:
∞ ∞
c (cλ r¯ )m cκ − 2cλ κ V (¯r , τ) = Dn cos µn r¯ ln(cλ r¯ ) + exp − exp r¯ exp −µ2n τ m · m! 2 2 n=1
(21)
m=1
where the discrete values µn and Dn are the roots of Eqs. (19) and (20), respectively. 2.2. Unsteady thermal stress field and the thermal shock resistance parameter for solid FGM cylinder The unsteady thermal stresses in the FGM solid cylinder that are caused by the axisymmetric temperature change are to be determined. We assume the Poisson’s ratio of the cylinder to be constant, while the Young’s modulus and the thermal expansion
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coefficient are described with a power law as:
E(¯r) = E(1) exp[cE (1 − r¯ )] cE = ln[E(0)/E(1)] α(¯r ) = α(1) exp[cα (1 − r¯ )] cα = ln[α(0)/α(1)] ν(¯r ) ≡ ν
0 ≤ r¯ ≤ 1
(22)
where E(1) and α(1) are the Young’s modulus and the thermal expansion coefficient of the surface layer (¯r = 1), ν is the Poisson’s ratio of the FGM cylinder. The following formulae are used to derive the expressions of stresses σ r , σ θ and σ z : ∂σr σ r − σθ + =0 ∂r r
(23a)
∂u ∂r u εθ = r εr =
(23b) (23c)
where u, σ and ε are the displacement, stress and strain, respectively. For a long FGM solid cylinder, εz = 0. By substituting Eqs. (23b) and (23c) into Hooke’s law, we obtain: E(r) ∂u(r, t) u(r, t) (24a) (1 − ν) +ν − (1 − ν)α(r)[T (r, t) − T0 ] σr (r, t) = (1 + ν)(1 − 2ν) ∂r r E(r) u(r, t) ∂u(r, t) (1 − ν) σθ (r, t) = +ν − (1 − ν)α(r)[T (r, t) − T0 ] (24b) (1 + ν)(1 − 2ν) r ∂r νE(r) ∂u(r, t) u(r, t) 1 − ν (24c) + − α(r)[T (r, t) − T0 ] σz (r, t) = ν[σr (r, t) + σθ (r, t)] − α(r)[T (r, t) − T0 ] = (1 + ν)(1 − 2ν) ∂r r ν The following displacement equation is obtained by substituting Eqs. (24a) and (24b) into Eq. (23a): 1 dE(r) 1 ∂u(r, t) ν r dE(r) 1 ∂2 u(r, t) + · + + · · − 1 2 u(r, t) ∂r 2 E(r) dr r ∂r 1 − ν E(r) dr r 1+ν ∂{α(r)[T (r, t) − T0 ]} 1 dE(r) = · α(r)[T (r, t) − T0 ] + 1 − ν E(r) dr ∂r
(25)
By substituting Eq. (22) into Eq. (25), we obtain: ∂2 u(¯r , τ) ∂u(¯r , τ) + q1 (¯r ) − q2 (¯r )u(¯r , τ) = q3 (¯r , τ) 2 ∂¯r ∂¯r where q1 (¯r ) =
1 − cE r¯
q2 (¯r) =
cE k1 r¯ + 1 r¯ 2
q3 (¯r , τ) = k2 b k1 =
ν 1−ν
k2 =
1+ν 1−ν
∂[α(¯r)T ∗ (¯r , τ)] − cE α(¯r )T ∗ (¯r , τ) ∂¯r
T ∗ (¯r , τ) = T (¯r , τ) − T0
(26)
J. Zhao et al. / Materials Science and Engineering A 418 (2006) 99–110
104
We also use the method of variable substitution to solve the second order differential Eq. (26) with variable coefficients by introducing: ω=
∞ 1 (cE r¯ )n exp − q1 (¯r ) d¯r d¯r = · exp(cE r¯ ) d¯r = ln |cE r¯ | + r¯ n · n!
(27)
n=1
q2 (¯r ) F3 (¯r ) = 2 = (cE k1 r¯ + 1) exp(−2cE r¯ )
(28)
dω d¯r
q3 (¯r , τ) F4 (¯r , τ) = 2 = k2 b¯r 2 dω d¯r
∂[α(¯r )T ∗ (¯r , τ)] − cE α(¯r )T ∗ (¯r , τ) ∂¯r
exp(−2cE r¯ )
(29)
Then Eq. (26) becomes: ∂2 u(ω, τ) − F3 (¯r )u(ω, τ) = F4 (¯r , τ) ∂ω2
(30)
The solution to Eq. (30) is: ∞ cE k1 r¯ + 1 (cE r¯ )n u(¯r , τ) = H exp exp(−cE r¯ ) ln |cE r¯ | + 2 n · n! n=1
∞
(cE r¯ )n cE k1 r¯ + 1 + S exp − exp(−cE r¯ ) ln |cE r¯ | + 2 n · n!
+
k2 b¯r 2 cE α(¯r )T ∗ (¯r , τ) −
n=1
cE k1 r¯ + 1
∂[α(¯r)T ∗ (¯r,τ)] ∂¯r
(31)
where the constant H and S can be determined by using the displacement at the center of the cylinder (u(0, τ) = 0), and hence ∂u (¯r , τ) /∂¯r can also be calculated consequently. We introduce the dimensionless thermal stresses for Eq. (24) by asking: ¯ r , τ) = σ(¯
σ(¯r , τ)(1 − ν) α(1)E(1)T
(32)
Then we obtain: σ¯ r (¯r , τ) =
¯ r ) ∂u(¯r , τ) ¯ r ) u(¯r , τ) k4 E(¯ k3 E(¯ ¯ r )α(¯ ¯ r )V ∗ (¯r , τ) · + · − k5 E(¯ bα(1)T ∂¯r bα(1)T r¯
(33a)
σ¯ θ (¯r , τ) =
¯ r ) u(¯r , τ) ¯ r ) ∂u(¯r , τ) k3 E(¯ k4 E(¯ ¯ r )α(¯ ¯ r )V ∗ (¯r , τ) · + · − k5 E(¯ bα(1)T r¯ bα(1)T ∂¯r
(33b)
σ¯ z (¯r , τ) =
¯ r ) ∂u(¯r , τ) ¯ r ) u(¯r , τ) k4 E(¯ k4 E(¯ ¯ r )α(¯ ¯ r )V ∗ (¯r , τ) · + · − k5 E(¯ bα(1)T ∂¯r bα(1)T r¯
(33c)
where k3 =
(1 − ν)2 (1 + ν)(1 − 2ν)
k4 =
ν(1 − ν) (1 + ν)(1 − 2ν)
k5 =
1−ν 1 − 2ν
V ∗ (¯r , τ) = 1 − V (¯r , τ)
J. Zhao et al. / Materials Science and Engineering A 418 (2006) 99–110
By substituting u(¯r , τ) and ∂u(¯r , τ)/∂¯r into Eq. (33), the following dimensionless thermal stresses can be obtained:
L1 c α L2 c 2 L3 σ¯ r (¯r , τ) = W(¯r ) − + α + [W1 (¯r ) − cα W2 (¯r ) + cα2 W3 (¯r )] exp[cα (1 − r¯ )] L L L ∞ L1 Pn (1) L2 Qn (1) L3 Rn (1) 2 W(¯r ) − + + + − W1 (¯r )Pn (¯r ) + W2 (¯r )Qn (¯r ) + W3 (¯r )Rn (¯r ) exp(−µ τ) L L L n=1
2 L1 c α L2 c α L3 2 σ¯ θ (¯r , τ) = M(¯r) − + + [M1 (¯r ) − cα M2 (¯r ) + cα M3 (¯r )] exp[cα (1 − r¯ )] L L L ∞ L1 Pn (1) L2 Qn (1) L3 Rn (1) M(¯r ) − + + + − M1 (¯r )Pn (¯r ) + M2 (¯r )Qn (¯r ) + M3 (¯r )Rn (¯r ) exp(−µ2 τ) L L L n=1
2 L1 c α L2 c α L3 2 σ¯ z (¯r , τ) = N(¯r ) − + + [N1 (¯r ) − cα N2 (¯r ) + cα N3 (¯r )] exp[cα (1 − r¯ )] L L L ∞ L1 Pn (1) L2 Qn (1) L3 Rn (1) 2 N(¯r ) − + + + − N1 (¯r )Pn (¯r ) + N2 (¯r )Qn (¯r ) + N3 (¯r )Rn (¯r ) exp(−µ τ) L L L
105
(34)
n=1
where L, L1 , L2 , L3 , Pn (¯r ), Pn (1), Qn (¯r ), Qn (1), Rn (¯r ), Rn (1), W(¯r ), W1 (¯r ), W2 (¯r ), W3 (¯r ), M(¯r ), M1 (¯r ), M2 (¯r ), M3 (¯r ), N(¯r ), N1 (¯r ), N2 (¯r ) and N3 (¯r ) are shown in Appendix A. Then the dimensionless thermal stresses at the surface (¯r = 1) can consequently be obtained as: σ¯ r (1, τ) = 0
L1 c α L2 cα2 L3 2 σ¯ θ (1, τ) = M(1) − + + [M1 (1) − cα M2 (1) + cα M3 (1)] L L L ∞ L1 Pn (1) L2 Qn (1) L3 Rn (1) 2 + M(1) − + + − M1 (1)Pn (1) + M2 (1)Qn (1) + M3 (1)Rn (1) exp(−µ τ) L L L (35) n=1
2 L1 c α L2 c L3 σ¯ z (1, τ) = N(1) − + α + [N1 (1) − cα N2 (1) + cα2 N3 (1)] L L L ∞ L1 Pn (1) L2 Qn (1) L3 Rn (1) 2 N(1) − + + + − N1 (1)Pn (1) + N2 (1)Qn (1) + N3 (1)Rn (1) exp(−µ τ) L L L n=1
where M(1), M1 (1), M2 (1) and M3 (1) are constants when r¯ = 1, which can be calculated by using the expressions of M(¯r), M1 (¯r ), M2 (¯r ) and M3 (¯r ) in Appendix A, respectively. N(1), N1 (1), N2 (1) and N3 (1) are constants when r¯ = 1, which can be calculated by using the expressions of N(¯r ), N1 (¯r ), N2 (¯r ) and N3 (¯r ) in Appendix A, respectively. The maximum tensile stress criterion for fracture initiation of the surface layer is: σmax (1, τc ) = σf (1)
(36)
where σ f (1) is the tensile strength of the surface layer, τ c is the dimensionless time for the surface layer to attain the maximum tensile stress, which can be obtained from the solution of the following equation: ¯ τ) dσ(1, =0 dτ
(37)
By referring to Eqs. (32) and (36), the critical temperature change for the surface layer of the cylinder to reach its tensile strength in cold shock (T0 > Ts , e.g. quenching) can be given by: Tc =
R(1) σ¯ max (1, τc )
(38)
where R(1) = σf (1)[1 − ν(1)]/[α(1)E(1)] [28] is the thermal shock resistance parameter of the surface layer. 3. Numerical results and discussion 3.1. Calculations of unsteady thermal stresses An Al2 O3 /TiC FGM solid cylinder is considered for the numerical calculations, which has the material properties of TiC at the surface (¯r = 1) and those of Al2 O3 at the center (¯r = 0). The composition varies continuously in the radial direction of the cylinder.
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Table 1 Data of the physical properties of Al2 O3 and TiC Materials
Specific gravity, ρ (g cm−3 )
Thermal conductivity (20 ◦ C), λ (W m−1 K−1 )
Specific heat (20 ◦ C), c (J g−1 K−1 )
Thermal expansion coefficient (20 ◦ C), α (10−6 K−1 )
Young’s modulus, E (GPa)
Poisson’s ratio, ν
␣-Al2 O3 TiC
3.99 4.93
30 20
0.77 0.56
8.5 7.6
380 440
0.19 0.19
Data of the physical properties of Al2 O3 and TiC are listed in Table 1. The Poisson’s ratio of Al2 O3 is designated as 0.19, which is equal to that of TiC considering the assumption of the present work (Eq. (22)). Beside the Al2 O3 /TiC FGM solid cylinder, a homogeneous solid cylinder made of pure TiC is also taken into consideration for comparison. The exponents cκ , cλ , cα and cE (referring to Eqs. (12) and (22)) are 0.2986, 0.4055, 0.1119 and −0.1466, respectively for the FGM solid cylinder, and are all zero for the homogeneous solid cylinder. Considering the computation capacity and computation time, only the first eight roots of Eq. (19) are used in the calculations of thermal stresses of the FGM cylinder at different Biot numbers. According to the calculation results, the value of the circumferential stress σ¯ θ (1, τ) of the FGM cylinder is always higher than that of the axial stress σ¯ z (1, τ) under a specific cold condition. Therefore we choose the circumferential stress to evaluate the thermal shock resistance of the FGM cylinder, and Eq. (38) becomes: Tc =
R(1) σ¯ θmax (1, τθc )
(39)
In case of the homogeneous solid cylinder, however, the circumferential stress is always equal to the axial stress. Data of the maximum dimensionless circumferential and axial stresses of the FGM cylinder (σ¯ θmax (1, τθc ) and (σ¯ zmax (1, τzc )) and the maximum dimensionless stresses σ¯ max (1, τc ) of the homogeneous cylinder and their times of occurrence at different Biot numbers are listed ¯ τ) of the in Table 2. The dimensionless circumferential stresses σ¯ θ (1, τ) of the FGM cylinder and dimensionless stresses σ(1, homogeneous cylinder at different values of Biot number in cold shock are plotted against dimensionless time τ in Fig. 2a–c, respectively. It can be seen clearly that the dimensionless circumferential stresses σ¯ θ (1, τ) of the FGM cylinder and the dimensionless stresses ¯ τ) of the homogeneous cylinder are all zero at the beginning of the cold shock (τ = 0). They increase with the dimensionless σ(1, time τ, reach their maximum values σ¯ θmax (1, τθc ) and σ¯ max (1, τc ), respectively, then decrease with further increase in time. At any given value of Biot number B (except B = ∝), the dimensionless time for the surface of the FGM cylinder to attain the maximum thermal stress is shorter than that of the homogeneous cylinder (i.e. τ θc < τ c ). The maximum dimensionless thermal stress σ¯ θmax (1, τθc ) at the surface of the FGM cylinder is lower than the maximum dimensionless thermal stress σ¯ max (1, τc ) at the surface of the homogeneous cylinder. Furthermore, the difference between τ c and τ θc decreases monotonously with the increase in Biot number (Table 2). Whereas, the difference between σ¯ max (1, τc ) and σ¯ θmax (1, τθc ) increases with the increase in Biot number (Table 2). ¯ τ) = 1 is achieved at the It is always the case that for the limiting case of an ideal cold shock (B = ∞), a maximum value of σ(1, surface at τ = 0 for all the two cylinders. 3.2. Discussions The maximum dimensionless thermal stress σ¯ θmax (1, τθc ) at the surface of the FGM cylinder and its dimensionless time of occurrence τ θc , as well as the critical temperature change Tc for the surface fracture are all dependent on the radial distributions of the thermo-physical parameters, and the compositional distribution of FGM cylinder along the radial direction. It can be concluded from the above calculation results that the critical temperature change Tc for the surface layer of the FGM cylinder (with positive cκ , cλ , cα and negative cE ) to reach its tensile strength σ f (1) in cold shock is higher than that of the homogeneous cylinder made of pure TiC (The surfaces of the FGM cylinder and the homogeneous cylinder have the same composition and hence the same tensile strength σ f so as to be comparable, see Eq. (36)). Therefore, some suggestions on the design Table 2 Data of the maximum dimensionless thermal stresses and their times of occurrence B
0.1 1 5 ∞
FGM cylinder
Homogeneous cylinder
τ θc
σ¯ θmax (1, τθc )
τ zc
σ¯ zmax (1, τzc )
τc
σ¯ max (1, τc )
0.2172 0.0964 0.0348 0
0.0196 0.1474 0.3732 1
0.2122 0.0933 0.0327 0
0.0186 0.1451 0.3683 1
0.2531 0.1093 0.0393 0
0.0231 0.1593 0.3913 1
J. Zhao et al. / Materials Science and Engineering A 418 (2006) 99–110
107
Fig. 2. Dimensionless surface stresses as a function of time τ. (a) B = 5, (b) B = 1 and (c) B = 0.1.
of FGM cylinders with high thermal shock resistance can be made as follows: Both the thermal expansion coefficient α and the thermal diffusivity κ (=λ/(cρ)) of the center should be higher than that of the surface, whereas the Young’s modulus E of the center should be lower than that of the surface. The previous work of the author [29] in the thermal shock resistance parameter for FGM plates with symmetrical structures supported the suggestions made above. Regardless of the technological difficulties in the fabrication of the FGM solid cylinders with continuously radial composition transition, the present work provides an approach to the design of potential FGM solid cylinders with high thermal shock resistance. 4. Conclusions The analytical formulae of unsteady temperature field and unsteady thermal stress field for a FGM solid cylinder under a convective boundary condition are derived by using the separation-of-variables method and the displacement method, respectively. A strength-based fracture criterion for thermal shock of FGM sold cylinder is put foreword and consequently the expression of critical temperature change Tc leading to the local tensile strength at the surface as the thermal shock resistance parameter for FGM sold cylinder is obtained. The effects of the distributions for thermo-physical properties on the thermal shock resistance of the FGM sold cylinder are analyzed via numerical calculations in contrast to homogeneous ceramics, from which some suggestions on design of FGM solid cylinders with high thermal shock resistance are made as follows: (1) Both the thermal expansion coefficient α and the thermal diffusivity κ (=λ/(cρ)) of the center should be higher than that of the surface. (2) The Young’s modulus E of the center should be lower than that of the surface.
J. Zhao et al. / Materials Science and Engineering A 418 (2006) 99–110
108
Acknowledgements This work was supported by the National Natural Science Foundation of China (50105011) and the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (200231) as well as the Natural Science Foundation of Shandong Province (Y2004F14). Appendix A The constants and functions in Eq. (34) are given by ∞ ∞ n n 2k c E k1 − c E − c E cE c E k1 + 1 cE 1 2 L = (cE k1 + 1) k3 exp(−cE ) + 1+ + k4 2 n · n! 2 n! n=1 n=1 ∞ n cE k1 + 1 cE ·exp exp(−cE ) ln |cE | + 2 n · n!
(A.1)
n=1
L1 = k5 (cE k1 + 1)2 − cE k2 k3 (cE k1 + 2) − cE k2 k4 (cE k1 + 1)
(A.2)
2 L2 = k2 k4 (cE k1 + 1) − k2 k3 (cE k1 + cE − cE k1 − 2)
(A.3)
L3 = k2 k3 (cE k1 + 1)
(A.4)
∞ (cλ r¯ )m
Pn (¯r ) = Dn exp[cα (1 − r¯ )] · cos µn r¯ ln |cλ r¯ | +
∞ cλm ln |cλ | + m · m!
Pn (1) = Dn cos µn
m=1
m · m!
c cκ − 2cλ κ exp − exp r¯ 2 2
(A.5)
exp(−cλ )
(A.6)
m=1
c cκ − 2cλ κ exp − exp r¯ exp[cα (1 − r¯ )] + µn exp[cα (1 − r¯ )] Qn (¯r ) = Dn cα cos µn r¯ ln |cλ r¯ | + m · m! 2 2 m=1
∞ ∞ c cκ − 2cλ cκ − 2cλ (cλ r¯ )m (cλ r¯ )m κ × exp − exp r¯ · r¯ + 1 ln |cλ r¯ | + +1+ 2 2 2 m! m · m! m=1 m=1
∞ c (cλ r¯ )m cκ − 2cλ κ × sin µn r¯ ln |cλ r¯ | + exp − exp r¯ (A.7) m · m! 2 2 ∞ (cλ r¯ )m
m=1
Qn (1) = Dn
∞ cλm ln |cλ | + m · m! m=1
cα cos µn
+1 +
∞ m cλ m=1
m!
e−cλ · sin µn
cκ + 2 − 2cλ + µn e 2 ∞ cλm ln |cλ | + e−cλ m · m!
−cλ
∞ cλm ln |cλ | + m · m!
m=1
(A.8)
m=1
∞ ∞ m m 2 c − 2c r ¯ ) r ¯ ) (c (c λ κ λ λ Rn (¯r) = Dn ecα (1−¯r) cα2 − µ2n e−cκ e(cκ −2cλ )¯r r¯ + 1 ln |cλ r¯ | + +1+ 2 m · m! m! × cos µn r¯ e ×
−cκ 2
e
cκ −2cλ r¯ 2
4cα + 2cλ − cκ 2
ln |cλ r¯ | +
∞ (cλ r¯ )m
m · m! m=1
cκ − 2cλ r¯ + 1 2
m=1
+ µn e
ln |cλ r¯ | +
−cκ 2
e
∞ (cλ r¯ )m m=1
m · m!
m=1
cκ −2cλ r¯ 2
+1+
∞ (cλ r¯ )m m=1
m!
−
cκ − 2cλ 2
J. Zhao et al. / Materials Science and Engineering A 418 (2006) 99–110
∞ (cλ r¯ )m
∞
1 (cλ r¯ )m cκ − 2cλ r¯ + 1 + 2 r¯ m · m! m! · r¯ m=1 m=1 ∞ cκ −2cλ −cκ (cλ r¯ )m r¯ 2 2 × sin µn r¯ e ln |cλ r¯ | + e m · m! ×
ln |cλ r¯ | +
−
−
∞ m=1
(cλ r¯ )m r¯ · (m − 1)!
109
(A.9)
m=1
∞ m 2 ∞ m c + 2 − 2c c c κ λ λ λ ln |cλ | + Rn (1) = Dn cα2 − µ2n e−2cλ +1+ 2 m! m · m!
m=1
m=1
∞ ∞ cλm cκ + 2 − 2cλ cλm −Cλ 4cα + 2cλ − cκ ·cos µn e ln |cλ | + + µn e ln |cλ | + m · m! 2 2 m · m! m=1 m=1 ∞ m ∞ ∞ m ∞ cλ cκ − 2cλ cλm cλ cλm cκ + 2 − 2cλ +1 + − ln |cλ | + 1+ − − m! 2 2 m · m! m! (m − 1)! m=1 m=1 m=1 m=1 ∞ cλm −cλ ·sin µn e ln |cλ | + (A.10) m · m! −cλ
m=1
∞ n 2 k r¯ c E k1 − c E − c E c k (c r ¯ + 1 r ¯ ) E 1 E 1 −c −c r ¯ r ¯ ¯ r ) exp (ln |cE r¯ | W(¯r ) = E(¯ e E ln |cE r¯ | + · k3 e E 2 n · n! 2 n=1 ∞ ∞ (cE r¯ )n cE k1 r¯ + 1 1 (cE r¯ )n k4 + + + + n · n! 2 r¯ n! · r¯ r¯
n=1
(A.11)
n=1
W1 (¯r ) =
¯ r) cE k2 k4 r¯ E(¯ ¯ r) cE k2 k3 (cE k1 r¯ 2 + 2¯r )E(¯ ¯ r) + − k5 E(¯ 2 cE k1 r¯ + 1 (cE k1 r¯ + 1)
(A.12)
W2 (¯r ) =
2 k r¯ 3 − c k r¯ 2 + c r¯ 2 − 2¯r )E(¯ ¯ r ) k2 k4 r¯ E(¯ ¯ r) k2 k3 (cE 1 E 1 E − 2 cE k1 r¯ + 1 (cE k1 r¯ + 1)
(A.13)
W3 (¯r ) = −
¯ r) k2 k3 r¯ 2 E(¯ cE k1 r¯ + 1
(A.14)
∞ ∞ n n 2 k r¯ c c k r ¯ + 1 (c r ¯ ) k − c − c (c r ¯ ) E 1 E E 1 E 1 E E −c r ¯ −c r ¯ ¯ r ) exp M(¯r ) = E(¯ e E ln |cE r¯ | + · k4 e E ln |cE r¯ | + 2 n · n! 2 n · n! n=1 n=1 ∞ cE k1 r¯ + 1 1 (cE r¯ )n k3 + + + (A.15) 2 r¯ n! · r¯ r¯ n=1
M1 (¯r ) =
¯ r ) cE k2 k3 r¯ E(¯ ¯ r) cE k2 k4 (cE k1 r¯ 2 + 2¯r )E(¯ ¯ r) − k5 E(¯ + 2 cE k1 r¯ + 1 (cE k1 r¯ + 1)
(A.16)
M2 (¯r ) =
2 k r¯ 3 − c k r¯ 2 + c r¯ 2 − 2¯r )E(¯ ¯ r) ¯ r ) k2 k3 r¯ E(¯ k2 k4 (cE 1 E 1 E − cE k1 r¯ + 1 (cE k1 r¯ + 1)2
(A.17)
M3 (¯r ) = −
¯ r) k2 k4 r¯ 2 E(¯ cE k1 r¯ + 1
(A.18)
∞ ∞ n n 2 k r¯ c c (c (c k r ¯ + 1 r ¯ ) k − c − c r ¯ ) E 1 E E 1 E E 1 E ¯ r ) exp N(¯r ) = E(¯ e−cE r¯ ln |cE r¯ | + · k4 e−cE r¯ ln |cE r¯ | + 2 n · n! 2 n · n! n=1 n=1 ∞ k4 cE k1 r¯ + 1 1 (cE r¯ )n + + (A.19) + 2 r¯ n! · r¯ r¯ n=1
J. Zhao et al. / Materials Science and Engineering A 418 (2006) 99–110
110
N1 (¯r ) =
¯ r ) cE k2 k4 r¯ E(¯ ¯ r) cE k2 k4 (cE k1 r¯ 2 + 2¯r )E(¯ ¯ r) + − k5 E(¯ 2 cE k1 r¯ + 1 (cE k1 r¯ + 1)
(A.20)
N2 (¯r ) =
2 k r¯ 3 − c k r¯ 2 + c r¯ 2 − 2¯r )E(¯ ¯ (¯r ) ¯ r ) k2 k4 r¯ E k2 k4 (cE 1 E 1 E − cE k1 r¯ + 1 (cE k1 r¯ + 1)2
(A.21)
N3 (¯r ) = −
¯ r) k2 k4 r¯ 2 E(¯ cE k1 r¯ + 1
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