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ISSN 0894-9166

EFFECT OF FRACTIONAL ORDER PARAMETER ON THERMOELASTIC BEHAVIORS IN INFINITE ELASTIC MEDIUM WITH A CYLINDRICAL CAVITY⋆⋆ Yingze Wang⋆

Dong Liu

Qian Wang

(Department of Energy and Power Engineering, Jiangsu University, Zhenjiang 212013, China) Received 12 September 2013, revision received 27 February 2014

ABSTRACT The thermal shock problems involved with fractional order generalized theory is studied by an analytical method. The asymptotic solutions for thermal responses induced by transient thermal shock are derived by means of the limit theorem of Laplace transform. An infinite solid with a cylindrical cavity subjected to a thermal shock at its inner boundary is studied. The propagation of thermal wave and thermal elastic wave, as well as the distributions of displacement, temperature and stresses are obtained from these asymptotic solutions. The investigation on the effect of fractional order parameter on the propagation of two waves is also conducted.

KEY WORDS generalized thermoelasticity, fractional order theory, asymptotic solutions, thermal shock

I. INTRODUCTION The classical coupled theory of thermoelasticity[1] , apart from its mechanical constitutive equation, utilizes a thermal constitutive equation which accommodates infinite speeds of propagation of heat waves. This is not compatible with physical observations and some experimental results[2, 3] . As for conventional heat conduction process, the duration of heat effect is long enough to satisfy the quasi equilibrium hypothesis, which leads to the consistent predictions of classical theory with experiments. However, when analyzing the thermal behaviors involving extremely short duration or high heat flux, such as laser processing or rapid solidification of metal, the classical theory can’t give an accurate prediction since the heat wave has a finite speed of propagation. In order to overcome the shortcoming of classical coupled theory, some modified theories, which are usually referred to as the generalized theories of thermoelasticity[4–8] , permit a finite speed of thermal signal referred as the second sound. In these generalized theories, the Fourier’s law is modified from different perspective to obtain a wave-type heat conduction equation, which can accurately describe the second sound effect. Recently, due to the extremely successful application of the fractional calculus to modify many existing models of physical processes in the area of mechanics of solids[9] , it has also been employed in the area of thermoelasticity theory by some researchers. Povestnko[10] has constructed a quasi-static uncoupled thermoelasticity model based on the heat conduction equation with fractional order time derivatives. Youssef[11] and Sherief et al.[12] have proposed a generalized theory of thermoelasticity in the context of a Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (No. 11102073), the National Science Foundation for Post-doctoral Scientists of China (No. 2012M511207), the Research Foundation of Advanced Talents of Jiangsu University (No. 10JDG055) and the Priority Academic Program Development of Jiangsu Higher Education Institutions. ⋆

⋆⋆

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new consideration of the heat conduction equation with fractional-order time derivatives, respectively. The uniqueness of the solution has also been proved in the same work. Then some investigations involved with these new generalized theories of thermoelasticity have been conducted[13–15] . In these investigations, the general thermoelastic phenomenon and the effect of fractional order parameter on thermal behaviors have been obtained. Due to the complexity of governing equations of these generalized theories, the integral transform technique and the numerical inversion are usually used to solve the governing equations in these investigations. Consequently, the truncation error and discretization error produced in numerical inversion would reduce the precision of prediction, the wavelike behaviors of heat propagation especially the jumps in the locations of wavefront can’t be revealed accurately[16]. These would be unbeneficial to reveal the effect of each generalized characteristic factors such as relaxation time, thermal coupling coefficient and fractional order parameter on thermal behaviors. Taking the transient characteristics of generalized thermoelastic problems into account, an asymptotic analysis method[17–19] has been introduced to solve these problems in the present work. The asymptotic solutions of governing equations can be obtained to predict the wave behaviors of heat propagation[18, 19], which is important to reveal the effect of each characteristic factors on thermal behaviors. In this paper, a generalized thermoelastic problem involving fractional order theory of thermoelasticity is investigated by this asymptotic analysis. The asymptotic solutions of the problem for an infinite elastic medium with a cylindrical cavity are solved. With these asymptotic solutions, the propagation of the thermal elastic wave and thermal wave, as well as the distributions of displacement, temperature and stresses are obtained. By illustration and comparison of results for different values of fractional order parameter, their effect on thermal behaviors has been studied.

II. FORMULATION OF THE PROBLEMS Due to the fractional order generalized theory of thermoelasticity[11] , the heat conduction equation for a homogenous and isotropic medium can be expressed as (1) kI α−1 θ,ii = ρcp θ˙ + τ0 θ¨ + T0 β (γ˙ ii + τ0 γ¨ii )

where I α is the fractional integral operator and can be defined as follows: Z t 1 α−1 I α f (t) = (t − s) f (s) ds (0 < α ≤ 2) Γ (α) 0

(2)

in which Γ (α) is the gamma function and 0 < α < 1, α = 1, 1 < α ≤ 2,

for weak conductivity for normal conductivity for strong conductivity

The equation of motion can be expressed as ρ¨ ui = σij,j + ρfi

(3)

The constitutive equation takes the form σij = λγii δij + 2µγij − βθδij

(4)

In the preceding equations, k is the thermal conductivity, cp is the specific heat at constant strain, ρ is the density, β = αT (3λ + 2µ) is the material constant, αT is the coefficient of linear thermal expansion, λ and µ are Lame’s constants, τ0 is the relaxation time constant, θ = T − T0 is the increment of the dynamic temperature, T is the absolute temperature, T0 is the reference temperature, ui are the components of the displacement, fi are the components of the body force per unit mass, σij are the components of stress tensor, γij = (ui,j + uj,i ) /2 are the components of strain tensor, δij is the Kronecker delta. Meanwhile, the superscript dot (·) denotes the derivative respect to the time, and the subscript comma (,) denotes the derivative respect to the coordinates. Now, we consider a homogenous isotropic elastic medium with a cylindrical cavity. The displacement components in the cylindrical coordinate (r, φ, z) have the following form: ur = u (r, t) ,

uφ = uz = 0

(5)

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The strain components can be written as γrr =

∂ur , ∂r

γφφ =

ur , r

γzz = γrz = γrφ = γφz = 0

(6)

Substituting Eq.(6) into Eqs.(4), the nonzero stress components can be expressed as ∂ur ur + λ − βθ ∂r r ∂ur ur σφφ = λ + (λ + 2µ) − βθ r ∂r ur ∂ur + − βθ σzz = λ ∂r r

σrr = (λ + 2µ)

(7) (8) (9)

Correspondingly, the equation of motion (3) without body force can be expressed as follows: σrr,r +

1 (σrr − σθθ ) = ρ¨ ur r

(10)

Substituting these stress components (7)-(9) into Eqs.(1) and (10), we have ∂θ ∂ ∂ur ∂2θ ∂2 ur kI α−1 △2 θ = ρcp + τ0 2 + T0 β + τ0 + ∂t ∂t ∂t ∂t ∂r r 2 ∂ ur ∂θ ρ 2 = (λ + 2µ) ∇2 ur − β ∂t ∂r

(11) (12)

where ∇2 =

∂ 1 ∂ (r) , ∂r r ∂r

△2 =

1 ∂ r ∂r

∂ r ∂r

Equations (7)-(12) are the general equations involving fractional order theory of thermoelasticity for the infinite elastic medium with a cylindrical cavity. For simplicity, some non-dimensional variables are introduced as follows: r∗ = aCL r,

t∗ = aCL2 t,

τ0∗ = aCL2 τ0 ,

u∗r = aCL

λ + 2µ ur , βT0

θ∗ =

θ , T0

∗ σij =

1 σij βT0

Substituting these non-dimensional variables into governing equations (11) and (12) and dropping the asterisks for convenience, we get ∂ 2θ ∂ ∂2 ∂ur ur ∂θ + τ0 2 + ϑ + τ0 + (13) I α−1 △2 θ = ∂t ∂t ∂t ∂t ∂r r ∂θ ∂ 2 ur = ∇2 ur − (14) 2 ∂t ∂r p where a = ρcp /k is the thermal viscosity, CL = (λ + 2µ)/ρ is the speed of thermal elastic wave, ϑ = β 2 T0 /[ρCL (λ + 2µ)] is the thermal coupling constant. Correspondingly, the non-dimensional components of stresses can be expressed as ∂ur ur + kv −θ ∂r r ∂ur ur σφφ = kv + −θ r ∂r ∂ur ur σzz = kv −θ + ∂r r

σrr =

where kv = λ/(λ + 2µ).

(15) (16) (17)

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III. ASYMPTOTIC SOLUTIONS OF THE PROBLEM Taking the Laplace transform, which is defined as follows: Z ∞ L [t] = f¯ (s) = e−st f (t) dt 0

for the both side of Eqs.(13)-(17), we get 1

s

△2 θ¯ = s + τ0 s2 θ¯ + ϑ s + τ0 s2 α−1

dθ¯ dr d¯ ur u ¯r ¯ σ ¯rr = + kv −θ dr r u¯r ¯ d¯ ur σ ¯φφ = kv + −θ dr r u ¯r d¯ ur + − θ¯ σ ¯zz = kv dr r

d¯ ur u ¯r + dt r

∇2 u¯r = s2 u¯r +

(18) (19) (20) (21) (22)

where the homogenous initial conditions and the following rule for the Laplace transform of the RiemanLiouville fractional integral[10] are used to obtain above equations: L [I n f (t)] =

1 L [f (t)] sn

(n > 0)

Eliminating terms θ¯ and u ¯r separately by combining with Eqs.(17) and (18) renders: ∇4 u¯r − s2 + χ s + τ0 s2 + χϑ s + τ0 s2 ∇2 u ¯r + χs2 s + τ0 s2 u¯r = 0 △4 θ¯ − s2 + χ s + τ0 s2 + χϑ s + τ0 s2 △2 θ¯ + χs2 s + τ0 s2 θ¯ = 0

(23)

(24) (25)

where χ = sα−1 , a parameter depending on fractional order parameter α, describes the same conductivity cases that described by parameter α by different values as follows: 0 < χ < 1, χ = 1, χ > 1,

for weak conductivity for normal conductivity for strong conductivity

This means the effect of fractional order parameter α on thermal behaviors can be obtained by the comparison of prediction for different values of parameter χ. The general solutions of equations (24) and (25) can be written as u ¯r = θ¯ =

2 X

[Ai (s) I1 (Ri r) + Bi (s) K1 (Ri r)]

i=1 2 X

[Ci (s) I0 (Ri r) + Di (s) K0 (Ri r)]

(26) (27)

i=1

where Ri2 (i = 1, 2) are the roots of following characteristic equation: R4 − s2 + χ (1 + ϑ) s + τ0 s2 R2 + χs2 s + τ0 s2 = 0

(28)

I0 (x), I1 (x), K0 (x) and K1 (x) are the zero- and first-order modified Bessel functions of the first kind and second kind, respectively, Ai (s), Bi (s), Ci (s) and Di (s) are coefficients depending on s and determined by given boundary conditions. Substituting the general solutions (26) and (27) into Eqs.(18) or (19), one can obtain Ci (s) =

Ri2 − s2 Ai (s) , Ri

Di (s) = −

Ri2 − s2 Bi (s) Ri

(29)

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Hence, the solution (27) can be rewritten as 2 2 X Ri − s2 ¯ θ= [Ai (s) I0 (Ri r) − Bi (s) K0 (Ri r)] Ri i=1

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(30)

Now we consider a cylindrical cavity with non-dimensional radius r = 1, whose inner boundary plane (r = 1) is subjected to a thermal shock and is traction free. Thus, we have θ (1, t) = θ0 H (t) ,

σrr (1, t) = 0

(31)

where θ0 is a non-dimensional constant, H (t) is the Heavside unit function. Meanwhile, when r → ∞, ur and θ satisfy the following relation: {ur (r, t) , θ (r, t)} → 0

(t > 0)

(32)

Due to the limit properties of modified Bessel functions, the function Iv (x) (v = 0, 1) in general solutions (26) and (27) or (30) should be abandoned to satisfy above constraint condition (32). Thus, we can obtain u ¯r = B1 (s) K1 (R1 r) + B2 (s) K1 (R2 r) R 2 − s2 R 2 − s2 θ¯ = − 1 B1 (s) K0 (R1 r) − 2 B2 (s) K0 (R2 r) R1 R2

(33) (34)

Using Laplace transform as we defined before to boundary condition (31), we have θ0 θ¯ (1, s) = , s Combining with Eqs.(20), (33) and (34) results in B1 (s) =

S1∗ , B∗

σ ¯rr (1, s) = 0

(35)

S2∗ B∗

(36)

B2 (s) =

where s2 s2 θ0 θ0 ∗ kw K1 (R2 ) − K0 (R2 ) , S2 = kw K1 (R1 ) − K0 (R1 ) , kw = kv − 1 =− s R1 R2 s 2 2 2 2 2 2 s R2 − s s R1 − s ∗ K0 (R1 ) kw K1 (R2 ) − K0 (R2 ) − K0 (R2 ) kw K1 (R1 ) − K0 (R1 ) B = R1 R2 R2 R1 S1∗

Substituting Eq.(36) into Eqs.(33) and (34), then combining with the stress component expressions (20)-(22), the general solutions of ur , θ and σii (i = 1, 2, 3) in transform domain can be obtained. By taking the inverse Laplace transform to these solutions, the corresponding solutions in time domain can also be obtained. However, as for the complex expressions of the terms Ri and Bi (s) (i = 1, 2) contained in these transform solutions, it is practically impossible to construct the exact solutions in a closed form in time domain by inverse Laplace transform. Hence, an asymptotic analysis method[17–19] is introduced to solve the present problem. In accordance with a limit theorem of Laplace transform, if the time is a lower limit value, the transform parameter is a upper limit. Then some approximations for roots Ri (i = 1, 2) can be expressed as[18, 19] R1,2 = k1,2 s + m1,2

(37)

where

√ √ 1 + χτ0 + χϑτ0 ± a1 1/2 χ (1 + ϑ) ± b1 / a1 k1,2 = , m1,2 = 2 4k1,2 a1 = (1 + χτ0 + χϑτ0 )2 − 4χτ0 , b1 = χ2 τ0 (1 + ϑ)2 + χ (ϑ − 1) The coefficients Bi (s) (i = 1, 2) can also be expressed as r θ0 1 2R2 B1 (s) = − 2 kw − s exp (R2 ) s (ps + q) k2 r π θ0 1 2R1 B2 (s) = 2 kw − s exp (R1 ) s (ps + q) k1 π

(38)

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where

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k4 − k3 p= , k1 k2

q = kw

k4 k3 − k1 k2

,

k3,4

−1 + χτ0 + χϑτ0 ± = 2

2015

√

a1

In above derivation, the following asymptotic formulas for modified Bessel functions Kv (x) are used: r 1 π exp (−x) x ≥ v 2 − (39) Kv (x) → 2x 4

Inserting these approximations into the transform solutions, one can obtain the forms that are convenient to inverse Laplace transform. Using the standard results of Laplace transform, one can derive the asymptotic solutions of ur , θ and σii (i = 1, 2, 3) in time domain as follows: θ0 −1/2 r exp [− (r − 1) m1 ] [t − (r − 1) k1 ] H (t − (r − 1) k1 ) k2 p θ0 −1/2 − r exp [− (r − 1) m2 ] [t − (r − 1) k2 ] H (t − (r − 1) k2 ) (40) k1 p k3 θ0 −1/2 q θ= r exp [− (r − 1) m1 ] −1 + k2 kw + [t − (r − 1) k1 ] H (t − (r − 1) k1 ) k1 k2 p p k4 θ0 −1/2 q [t − (r − 1) k2 ] H (t − (r − 1) k2 ) (41) − r exp [− (r − 1) m2 ] −1 + k1 kw + k1 k2 p p θ0 −1/2 k1 q σrr = − r exp [− (r − 1) m1 ] 1 − kw + k2 + [t − (r − 1) k1 ] k1 k2 p r p θ0 −1/2 k2 q ·H (t − (r − 1) k1 ) + r exp [− (r − 1) m2 ] 1 − kw + k1 + k1 k2 p r p · [t − (r − 1) k2 ]} · H (t − (r − 1) k2 ) (42) θ0 −1/2 1 1 m1 1 q σφφ = r exp[−(r − 1)m1 ] − kw k1 + + kw k2 kw k1 + − − + k2 p k1 k1 k2 rk2 p θ0 −1/2 1 · [t − (r − 1) k1 ]} H (t − (r − 1) k1 ) − r exp[−(r − 1)m2 ] − kw k2 + k1 p k2 1 m2 1 q + kw k1 kw k2 + − − + · [t − (r − 1) k2 ] H (t − (r − 1) k2 ) (43) k2 k1 rk1 p 1 1 m1 q θ0 −1/2 r exp[−(r − 1)m1 ] − kw k1 + + kw k2 kw k1 + − + σzz = k2 p k1 k1 k2 p θ0 −1/2 1 · [t − (r − 1) k1 ]} H (t − (r − 1) k1 ) − r exp[−(r − 1)m2 ] − kw k2 + k1 p k2 1 m2 q + kw k1 kw k2 + − + · [t − (r − 1) k2 ] H (t − (r − 1) k2 ) (44) k2 k1 p ur =

where the parameter χ is regarded as a constant that can take different values to indicate different cases for heat conductivity defined before.

IV. RESULTS AND DISCUSSION 4.1. The Effect of Fractional Order Parameter on Propagation of Waves Since Heaviside unit functions in these asymptotic solutions predict the occurrence of waves, we can obtain two waves generated in elastic medium by the action of thermal shock, one is a thermal wave associated with the propagation in elastic medium of heat, and another is a thermal elastic wave associated with the thermal deformation induced by temperature. Due to the properties of Heaviside unit functions, the speeds and locations of wavefront for these two waves can be obtained as v1,2 =

1 , k1,2

ξ1,2 = 1 +

t k1,2

(45)

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Combining with the expressions of k1,2 , we can observe that both the speeds and locations of wavefront for these two waves are dependent of the parameters τ0 , ϑ0 and χ, which means the fractional order parameter also has an effect, such as the delay effect and coupling effect, on propagation of these two waves. This is different from the prediction made by previous investigation using other generalized theories[18, 19] . If τ0 → 0, which corresponds to Fourier heat conduction v1 → 1 and v2 → 0, we can make v1 represent the speed of thermal elastic wave and v2 represent the speed of thermal wave. Furthermore, it is worth noting that the terms χ and τ0 are together with the form χτ0 in Eqs.(45), which means the parameter χ can be regarded as the influence factor of relaxation time τ0 . Figure 1 shows the distributions of the speeds for two waves versus relaxation time τ0 with different values of parameter χ, where χ = 0.5, 1.0, 1.5 corresponds to the weak, normal and strong conductivity, respectively. As for the propagation of these two waves, both speeds are decreasing as χ increases, which is the same as the influence of τ0 . However, the magnitude of decrease of speed of thermal elastic wave (v1 ) is increasing as τ0 increases but decreasing for the speed of thermal wave (v2 ), which means the effect of fractional parameter χ on delay effect is different for these two waves.

Fig. 1. Distribution of non-dimensional speeds of thermal elastic wave and thermal wave versus τ0 at different χ.

4.2. The Eeffect of Fractional Order Parameter on Distributions of Thermal Response For better illustration, some non-dimensional constants are chosen for numerical evaluations as follows[11, 18]: τ0 = 1.5, ϑ = 0.2, θ0 = 1, kv = 0.5 Figures 2-4 display the distributions of displacement ur , temperature θ, and radial stress σrr for a wide range (1 ≤ r ≤ 2.5) and given non-dimensional time t = 0.5 with different values of parameter χ. As for these predictions, the general phenomenon for generalized thermoelastic problem that the delay between the heat flux and temperature gradient, and the jumps in locations of wavefront for temperature and stresses are also obtained. These results are the same as the previous predictions made by other generalized theories[18, 19] . The difference is that the influence on each field is added with the effect of fractional order parameter χ. For different values of parameter χ, the time of each field begins to establish, the peak values of jumps for temperature and stresses and its intervals between two jumps are changed. With the decreases of χ, the time of establishment is earlier, the intervals between two jumps are shorter, and the peak values of stresses are larger, which means the conduction with weaker heat conductivity is closer to Fourier heat conduction. Furthermore, it is worth noting that the locations of wavefront for thermal wave are nearly the same at χ ≥ 1, which leads to the same effect domains of boundary thermal shock for normal and strong conductivity. Taking account into the effect of parameter χ on propagation of thermal elastic wave and thermal wave, one can ignore the influence of the parameter χ on the speed of thermal wave at τ0 > 1, which leads to these distributions. For better illustration, the distributions of each field at τ0 = 0.5 for different values of parameter χ are displayed in Figs.5-7. As for the propagation of thermal wave, the speed has a significant change for different values of parameter χ at τ0 = 0.5, which leads to the changed effect domain of boundary thermal shock for different locations of wavefront for thermal wave. Consequently, we can obtain that the effect of parameter χ on distributions of each field is different for different values of τ0 .

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Fig. 2. Distribution of normalized displacement in radius direction (τ0 = 1.5).

Fig. 3. Distribution of normalized temperature in radius direction (τ0 = 1.5).

Fig. 4. Distribution of normalized radial stress in radial direction (τ0 = 1.5).

Fig. 5. Distribution of normalized displacement in radial direction (τ0 = 0.5).

Fig. 6. Distribution of normalized temperature in radial direction (τ0 = 0.5).

Fig. 7. Distribution of normalized radial stress in radial direction (τ0 = 0.5).

V. CONCLUSIONS In this paper, a thermoelasic problem involving with fractional order theory of thermoelasticity has been studied by an analytical method. The effect of fractional order parameter on thermal behaviors is discussed and conclusions can be drawn as follows: (1) As a generalized theory of thermoelasticity, the delay phenomenon of distribution of each field and the jumps at the locations of wavefront are predicted by fractional order theory of thermoelasticity. (2) The fractional order parameter has a significant effect on the propagation of thermal elastic wave and thermal wave induced by boundary thermal shock, which can be regarded as the influence factor of relaxation time. (3) The fractional order parameter also has a significant effect on distributions of displacement, temperature and stresses. The time of each field begins to establish, the peak values of jumps and the

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intervals of two jumps are changed for different values of fractional order parameter. As the fractional order parameter decreases, the conductivity is weakening and is closer to Fourier heat conduction. (4) For different values of relaxation time, the effect of fractional order parameter on propagation of thermal elastic wave and thermal wave is different, which also leads to the different effect on distributions of each filed. This means the value of relaxation time should be considered when utilizing fractional order theory to investigate the thermal behaviors.

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