Int J Thermophys (2012) 33:895–912DOI 10.1007/s10765-012-1190-4

Hyperbolic Heat Conduction in a CrackedThermoelastic Half-Plane Bonded to a Coating

Z. T. Chen · K. Q. Hu

Received: 3 November 2011 / Accepted: 22 March 2012 / Published online: 21 April 2012© Springer Science+Business Media, LLC 2012

Abstract In this paper, the transient temperature field around a thermally insulatedcrack in a substrate bonded to a coating is obtained using the hyperbolic heat conduc-tion model. Fourier and Laplace transforms are applied, and the thermal conductionproblem is reduced to solving a singular integral equation. Numerical results showthat the hyperbolic heat conduction parameters, the heat conductivity of the substrateand coating, and the geometric size of the composite have significant influence on thetransient temperature field. In the case of very small time scales, the results predictedby the hyperbolic model are more conservative than that by the parabolic model.

Keywords Coating · Fourier transforms · Hyperbolic heat conduction ·Laplace transform · Parabolic model · Singular integral equations · Substrate ·Transient temperature field

1 Introduction

High-rate heat transfer has become a major concern in modern industries especiallyin material processing, such as the application of pulsed laser heat and ultrasonicwaves. Recently, very strong substrate/coating interfaces have been obtained usingpulsed laser coating of bioceramic/metal nanomaterials on metal substrates by Zhangand Cheng [1]. Investigation of the temperature field is essential to calculating thermalstresses within the structures, which is necessary to understand the problem of thermaldamage, and accurate heat conduction analysis is of great importance for the materialand structural integrity.

Z. T. Chen (B) · K. Q. HuDepartment of Mechanical Engineering, University of New Brunswick, Fredericton, NB, Canadae-mail: ztchen@unb.ca

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896 Int J Thermophys (2012) 33:895–912

The Fourier heat conduction model, although it gives sufficient accuracy for manyengineering applications, implies an infinite thermal wave propagation speed and isineffective at the very small length and time scales associated with small-scale sys-tems [2,3]. For many technological applications that involve high thermal energy withextremely short times, the results predicted using the parabolic heat conduction modeldiffer significantly from the experimental results [4,5]. Examples include the transienttemperature field caused by pulsed laser heating of thin structures, and the measuredsurface temperature of a slab immediately after an intense thermal shock is 300 ◦Chigher than that obtained from the parabolic heat conduction model [6,7].

When relaxational effects are taken into account in the constitutive equation describ-ing the heat flux, as for instance in the Maxwell–Cattaneo equation, one has a hyper-bolic equation which implies a finite speed for heat transport [8]. Consideration ofthe hyperbolic heat conduction model becomes important if irreversible physical pro-cesses, such as crack or void initiation in a solid, are involved in the process of heattransport. In applications involving high rate heating where very extremely small timescales are important, it is appropriate to use the hyperbolic heat conduction model[9,10].

Inherent defects in materials such as dislocations and cracks may disturb the tem-perature distribution when thermal loading is applied to the material, and singularitiesmay be developed in the neighborhood of discontinuities. Some studies have beendevoted to studying the singular behavior of temperature gradients around a cracktip based on the classical Fourier heat conduction model [11–13]. Many researchershave paid attention to the crack problems under thermal loading in advanced materialsusing the Fourier heat conduction model and the stress intensity factors are obtained[14–21]. Qin [22] obtained the general solutions for thermo-piezoelectrics with var-ious holes under thermal loading. Gao and Noda [23] investigated thermal-inducedinterfacial cracking of magnetoelectroelastic materials.

A few investigations on crack problems in thermoelastic materials have been madeusing the hyperbolic heat conduction model. Among them are Manson and Rosakis[24], who derived a solution of the hyperbolic heat conduction equation for a travel-ing point heat source around a propagating crack tip, and measured the temperaturedistribution at the tip of a dynamically propagating crack experimentally. Tzou [25]investigated the near-tip behavior of the thermal field around a moving crack and stud-ied the effect of the crack velocity on thermal shock waves. Recently, Hu and Chen[26] studied the transient thermal stresses around a crack in a strip, and Chen and Hu[27] considered a crack in a half-plane under thermal impact loading.

To the author’s knowledge, the transient crack problem in a cracked substratebonded by a coating using the hyperbolic heat conduction model has not yet beenreported in the literature. Due to the pulsed laser causing rapid heating and coolingprocesses, the hyperbolic heat conduction model needs to be applied to investigatethe temperature fields in the coating and substrate. It is not unusual for structures tosuffer damage due to the applied thermal loading, and the local thermal stresses nearcracks can be elevated by the intensified temperature gradient, which may initiatecrack propagation or breakdown of the structure even under normal thermal condi-tions. An accurate analysis of the intensification of a temperature gradient near cracksis essential to predicting the failure behavior under thermal loading [28].

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)(0 tHT ⋅

cc

a

x

y

b

Fig. 1 Crack geometry and coordinates

In this paper, we analyze the transient temperature field around a crack in a sub-strate bonded by a coating under temperature impact loading using the hyperbolicheat conduction model. Considering that the substrate is much thicker than the coat-ing, the crack problem of a half-plane bonded to a coating strip is investigated. Fourierand Laplace transforms are employed to derive the singular integral equation aboutthe temperature field. The singular integral equation is solved numerically and theasymptotic fields around the crack tips are obtained. Laplace inversion is applied toget the temperature field in the time domain.

Dynamic temperature fields around the crack are obtained by solving singular inte-gral equations numerically using asymptotic analysis. The effect of the parameters ofthe hyperbolic heat conduction model and the geometric size of the composite on thetemperature disturbance is investigated. The results of the current problem lead to theexisting solution based on the parabolic heat conduction model when the relaxationtime vanishes. The results based on the hyperbolic heat conduction model show a muchhigher transient temperature disturbance in the very early stages of impact comparingto the parabolic model.

2 Basic Equations

Consider a thermoelastic half-plane containing a crack of length 2c parallel to theinterface between the half-plane and the coating, as shown in Fig. 1. The half-planeis initially at the uniform temperature zero, and the free surface of the coating aty = −(a +b) is suddenly heated to a temperature T0. The crack surfaces are assumedto be thermally insulated, which indicates no heat flux can go through the crack sur-faces. In this study, the effects of inertia and thermal–elastic coupling are neglectedwhich leads to an uncoupled and quasi-static problem.

Based on Fourier’s law, the temperature distribution in a body is governedby a parabolic partial differential equation, which predicts that the application of

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a thermal disturbance is instantaneously felt throughout the medium, and is referredto as an “infinite speed of propagation” of thermal disturbances. In the heat-trans-fer situations including extremely high temperature gradients, extremely large heatfluxes, or extremely short transient durations, the heat propagation speed is finite, andFourier’s law can be modified to a relation of the type [29],

h + τ∂h

∂t= −k · ∇T (1)

where T is the temperature change, k is the thermal conductivity of the material, and∇ is the spatial gradient operator. It is assumed that there is a phase-lag between theheat flux vector h and the temperature gradient ∇T , and τ is the so-called relaxationtime (a non-negative constant), or build-up period for the commencement of heat flowafter a temperature gradient has been imposed on the medium.

When this relation is incorporated in the local energy balance equation with avanishing heat source [30],

− ∇h = ρC∂T

∂t(2)

where ρ and C are the mass density and the specific heat capacity, respectively. Incor-porating Eq. 1 with Eq. 2 leads to the hyperbolic heat conduction equation for thesubstrate and the coating

di · �T (i) = ∂T (i)

∂t+ τi

∂2T (i)

∂t2 , (i = 1, 2) (3)

where � is Laplace’s differential operator, τi ’s are the relaxation times for the sub-strate and the coating, ki ’s are the thermal conductivities of the materials (i = 1, 2),and di = ki

ρi Ci(i = 1, 2) are the thermal diffusivities for the substrate and the coating.

It should be noted that the relaxation time for most engineering materials are of theorder of 10−14 s to 10−6 s, but experiments have shown that some non-homogeneousmaterials have relaxation times up to 10 s which are very important materials as ther-mal insulators [31–33]. From the literature, materials with thermal relaxation timeson the order of 1 s to 10 s are glass ballotini, sand, and some ion exchangers, amongothers. Investigation of heat conduction problems for these kinds of materials requiresthe application of the hyperbolic heat conduction theory, which can take into accountthe effect of the relaxation time.

For simplicity of the derivation, the following dimensionless variables are intro-duced:

(x, y, a, b

) = (x, y, a, b)/cT = T/T0, t = td0/c2 (4)

where T0 is the reference temperature and d0 is the reference thermal diffusivity (wecan choose either d1 or d2). The governing equations (Eq. 3) have the following form:

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∇2T (i) = ∂T (i)

∂t

d0

di+ τi d2

0

c2di

∂2T (i)

∂t2 , (i = 1, 2) (5)

In Eq. 5 and hereafter, the hat “−” of the dimensionless variables is omitted for simplic-ity. It is worth noting that Eq. 5 is dimensionless, although there are some dimensionalparameters in the equation which are related to material properties. In the present paper,the thermal relaxation time has an important effect on the thermal response based onthe hyperbolic heat conduction theory and the materials with thermal relaxation timeson the order of 1 s to 10 s are considered.

For a perfectly bonded, highly conducting interface, there is neither thermal resis-tance nor thermal inertia at the interface, and the temperature and heat flux are con-tinuous [34,35]. The hyperbolic heat equation (Eq. 5) is subjected to the followingboundary and initial conditions in dimensionless forms:

T (x,−(a + b)) = T0, (|x | < ∞, t > 0) (6)

T (x,−a+) = T (x,−a−), (|x | < ∞, t > 0) (7)

k1∂T (x,−a+)

∂y = k2∂T (x,−a−)

∂yor (|x | < ∞, t > 0)

h1 + τ1∂h1∂t = h2 + τ2

∂h2∂t

(8)

∂T (x, 0)

∂y= 0, (|x | < 1) (9)

T (x, 0+) = T (x, 0−), (|x | ≥ 1) (10-1)

∂T (x, 0+)

∂y= ∂T (x, 0−)

∂y, (|x | ≥ 1) (10-2)

It is assumed that both the initial temperature and the initial rate of temperature changein the composite are uniform, i.e.,

T = 0, (t = 0) (11-1)

∂T

∂t= 0, (t = 0) (11-2)

3 Temperature Field

Applying the Laplace transform to Eq. 5,

T (i)∗(x, y, p) = L(T (i)(x, y, t)

) =∞∫

0T (i)(x, y, t) exp(−pt) dt

T (i)(x, y, t) = L−1(

T (i)∗(x, y, p))

= 12π i

∫

BrT (i)∗(x, y, p) exp(pt) dp

(12)

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where Br stands for the Bromwich path of integration, and by considering the initialconditions (Eqs. 11-1 and 11-2), we have

∇2T (i)∗ = Ai pT (i)∗ + Bi p2T (i)∗ , (i = 1, 2) (13)

with Ai and Bi (i = 1, 2) defined as

Ai = d0

di, Bi = τi d2

0

c2di(i = 1, 2) (14)

The boundary conditions in the Laplace transform plane (p-plane) are

T ∗(x,−(a + b)) = T0/p, (|x | < ∞) (15)

T ∗(x,−a+) = T ∗(x,−a−), (|x | < ∞) (16)

k1∂T ∗(x,−a+)

∂y= k2

∂T ∗(x,−a−)

∂y, (|x | < ∞) (17)

∂T ∗(x, 0)

∂y= 0, (|x | < 1) (18)

T ∗(x, 0+) = T ∗(x, 0−), (|x | ≥ 1) (19-1)∂T (x, 0+)

∂y= ∂T ∗(x, 0−)

∂y, (|x | ≥ 1) (19-2)

The appropriate temperature field in the Laplace domain satisfying the boundarycondition and regularity condition can be expressed as

T (1)∗(x, y, p) =∞∫

−∞E1(ξ) exp(−r y) exp(−ixξ) dξ + W (y, p), (y ≥ 0) (20-1)

T (1)∗(x, y, p) =∞∫

−∞

[E2(ξ) exp(r y) + E3(ξ) exp(−r y)

]exp(−ixξ) dξ + W (y, p),

(−a ≤ y ≤ 0) (20-2)

T (2)∗(x, y, p) =∞∫

−∞

[D1(ξ) exp(ny) + D2(ξ) exp(−ny)

]exp(−ixξ) dξ + V (y, p),

(−(a + b) ≤ y ≤ −a) (20-3)

where Ei (ξ) (i = 1, 2, 3) and D j (ξ) ( j = 1, 2) are unknowns to be determined;the functions W (y, p), V (y, p), r , and n are defined in Appendix 1. Application ofEqs. 15–17 can lead to expressions of D2(ξ) and Ei (ξ) (i = 1, 2, 3) as the functionsof D2(ξ), as given in Appendix 1.

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Introduce the temperature density function as

φ(x) = ∂T (1)∗(x, 0+)

∂x− ∂T (1)∗(x, 0−)

∂x, (21)

It is clear from the boundary conditions (Eqs. 19-1, 19-2) that

1∫

−1

φ(t)dt = 0 (22-1)

and

φ(x) = 0, (|x | ≥ 1) (22-2)

Substituting Eqs. 20-1–20-3 into 21 and using a Fourier inverse transform, we have

D1(ξ) = 1

i4πλ1ξ

1∫

−1

φ(s) exp(isξ) ds (23)

Substituting Eqs. 20-1–20-3 into Eq. 18 and applying the relation (Eq. 23), we get asingular integral equation for φ(x) as follows:

1∫

−1

φ (t)

[1

t − x+ H (x, t)

]dt = 2πq f, (|x | < 1) (24)

where q and f are given in Appendix 1. The kernel function H(x, t) is defined as

H(x, t) =∞∫

0

{1 − r(λ1 − λ2)

λ1ξ

}sin [ξ(x − t)] dξ (25)

and λ1, λ2 are defined in Appendix 1.The integral equation (Eq. 24) under the singled-value condition in Eq. 22-1 has the

following form of solution [36]:

φ(x) = Φ(x)√1 − x2

, |x | < 1 (26)

where Φ(x) is bounded and continuous on the interval [−Φ(x)1, 1]. From the prop-erties of symmetry or from the condition (Eq. 22-1), it is seen that is an odd functionof x , i.e.,

Φ(−x) = −Φ(x) (27)

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902 Int J Thermophys (2012) 33:895–912

Following the numerical techniques of Erdogan [36], Eqs. 24 and 22-1 can be solvedat discrete points as

N∑

k=1

1

NΦ (tk)

[1

tk − xr+ H (xr , tk)

]= 2 f q, r = 1, 2, . . . , N − 1 (28-1)

N∑

k=1

π

NΦ (tk) = 0 (28-2)

tk = cos [(2k − 1)π/(2N )] , k = 1, 2, . . . , N . (28-3)

xr = cos(rπ/N ), r = 1, 2, . . . , N − 1 (28-4)

Once the function Φ(t) is obtained, the function D1(ξ) can be calculated using theChebyshev quadrature for integration as

D1(ξ) ∼= 1

4πξλ1

N∑

i=1

wiΦ (xi ) sin (ξ xi ) (29-1)

xi = cos

(2i − 1

2Nπ

), i = 1, 2, . . . , N (29-2)

wi = π/N (29-3)

The substitution of Eq. 29-1 into 20-1–20-3 can give the temperature in the p-plane,and the temperature in the time domain can be given by applying the Laplace inversetransform.

4 Temperature Gradients

A temperature gradient is a physical quantity that describes in which direction and atwhat rate the temperature changes the most rapidly around a particular location.

The temperature gradients in the Laplace domain can be obtained as

T (1)∗,y =

∞∫

−∞r(λ1 − λ2)D1(ξ) exp(−r y) exp (−iξ x)dξ − q f exp(−qy), (y ≥ 0)

(30-1)

T (1)∗,y =

∞∫

−∞r[λ1 exp(r y) − λ2 exp(−r y)

]D1(ξ) exp (−iξ x)dξ − q f exp(−qy),

(−a ≤ y ≤ 0) (30-2)

T (1)∗,x =

∞∫

−∞iξ(λ1 − λ2)D1(ξ) exp(−r y) exp (−iξ x)dξ, (y ≥ 0) (30-3)

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T (1)∗,x = −

∞∫

−∞iξ

[λ1 exp(r y) + λ2 exp(−r y)

]D1(ξ) exp (−iξ x)dξ,

(−a ≤ y ≤ 0) (30-4)

By considering the asymptotic nature of the integrands in Eqs. 30-1–30-4 for largevalues of the integration variable ξ and using the asymptotic formula [37],

1∫

−1

Fj (t)√1−t2 eiξ t dt =

√π

2|ξ |{

Fj (1) exp[i(ξ − πξ

4|ξ |)]

+Fj (−1) exp[−i

(ξ − πξ

4|ξ |)]

+ O(

1|ξ |

)},

(31-1)

∞∫

0xμ−1 exp (−sx)

{sincos

}(βx) dx = �(μ)

(s2+β2)μ/2

{sincos

} (μ tan−1

(βs

)),

s > 0, μ > 0(31-2)

the singular temperature gradients near the crack tip in the Laplace domain can beobtained as

T ∗,y (r, θ, p) = − Φ(1)

2√

2rcos

(θ

2

), (32-1)

T ∗,x (r, θ, p) = Φ(1)

2√

2rsin

(θ

2

), (32-2)

T ∗,r (r, θ, p) = − Φ(1)

2√

2rsin

(θ

2

), (32-3)

where (r, θ) are the polar coordinates measured from the crack tip defined by

r2 = (x − 1)2 + y2, tan(θ) = y/(x − 1) (33)

In front of the crack tip at θ = −π , the temperature gradient reaches a maximumvalue and the intensity factor of the temperature gradient (IFTG) at the crack tip canbe defined as [12]

K ∗T (p) = lim

r→02√

rT ∗,r (r, θ, p)

∣∣∣∣θ=−π

= Φ(1)√2

(34)

By applying the inverse Laplace transform to Eq. 34, the crack-tip temperaturegradients in the time domain can be obtained as

T,y (r, θ, t) = − KT (t)

2√

rcos

(θ

2

), (35-1)

T,x (r, θ, t) = KT (t)

2√

rsin

(θ

2

), (35-2)

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T,r (r, θ, t) = − KT (t)

2√

rsin

(θ

2

), (35-3)

where the IFTGs in the time domain, KT (t), is given by

KT (t) = L−1 (K ∗

T (p))

(36)

It can be seen from Eqs. 35-1–35-3 that the transient temperature gradients presenta r−1/2 singularity at the crack tip, which is in agreement with the corresponding staticthermal crack problem [11,12,28]. The dynamic effect is merely introduced by theIFTGs, which are time-dependent as shown in Eq. 36.

5 Numerical Results and Discussion

The temperature field in the time domain can be obtained from Eqs. 20-1–20-3 usingthe numerical inversion of the Laplace transform, as detailed in Miller and Guy [38]and Appendix 2. In the computation of the temperature field, the values of the parame-ters of the method are given as follows: N = 8 to 10, β = 0, 0.2 ≤ δ ≤ 0.3, as shownin Appendix 2.

By choosing the geometric size of the composite to be a/c = 2b/c = 1, the mate-rial parameters to be d2 = 2d1 = 2.0, k1 = 1.0, k2 = 0.5, τ1 = 1.0, τ2 = 0.4, andthe boundary condition T0 = 1.0, the temperature field can be obtained by solvingthe singular integral equation (Eq. 28-1–28-4) and substituting Eqs. 29-1–29-3 into20-1–20-3.

The steady temperature distribution in the cracked half-plane bonded to a coatingis shown in Fig. 2 as t → ∞. The disturbance of the crack on the temperature field canbe observed from the iso-temperature lines, and there is a temperature jump acrossthe crack faces. The appearance of the crack disturbed the temperature field in thecomposite, and there is a temperature jump across crack faces due to the insulationof the crack. The temperature on the lower crack face is higher than that on the uppercrack face.

The temperature distributions on crack faces and the extension line y = 0 at dif-ferent instants are shown in Fig. 3, where “Inf” denotes the long enough time, and“UF” and “LF” denote the values for the upper crack face and the lower crack face,respectively. Due to the assumption of the insulation of the crack faces, there is a tem-perature jump across crack faces. The highest temperature on crack faces appear at thecentral point on the lower crack face and the lowest temperature on the central pointof the upper crack face. As time increases, the temperature on crack faces increasesand eventually reaches a steady-state value.

The variation of temperature on the central point of the lower crack face with timeis shown in Fig. 4 for the hyperbolic and parabolic heat conduction models. The tem-perature approaches the steady-state values faster for the parabolic heat conductionmodel than for the hyperbolic heat conduction model. It can be seen that the peakvalue of temperature based on the hyperbolic heat conduction model in the very early

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-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5 0.6 0.6 0.6

0.65

0.65 0.65

0.65

0.7

0.7 0.7

0.7

0.75 0.75

0.75

0.8 0.8 0.8

0.85

0.85 0.85

0.850.9 0.9

0.9

0.9

0.9

0.9

0.95 0.95 0.951 1 1 1

1 1 11

x/c

y/c

Fig. 2 Temperature distribution in the cracked body when t → ∞

-3 -2 -1 0 1 2 3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x/c

Tem

pera

ture

t = 1, UFt = 1, LFt = 2, UFt = 2, LFt = 5, UFt = 5, LFt = Inf, UFt = Inf, LF

Fig. 3 Temperature distribution on crack faces and extension lines at different instants

stage is much higher than that based on the parabolic model, which is in agreementwith the experimental results [6,7].

The temperature distributions on the line x = 0 are shown in Fig. 5 when the tem-perature converges to a steady value (as t → ∞). In the figure, “Hyperbolic” denotesthe results correspond to the case τ1 = 1.0, τ2 = 0.4 and “Parabolic” denotes theresults correspond to the case τ1 = τ2 = 0.0. The shape of the temperature distribu-tion due to the impact temperature loading as predicted by the hyperbolic model is

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0 5 10 15 20 25 30 35 40-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time t

Tem

pera

ture

Hyperbolic

Parabolic

Fig. 4 The variation of temperature at the central point of lower crack face versus time

Fig. 5 Temperature on the line x = 0 when t → ∞

found to significantly differ from that predicted by the parabolic model. It can be seenthat the temperature jump for the hyperbolic model is larger than that of the parabolicmodel, and the maximum magnitude of temperature in the hyperbolic model is largerthan 1, which is the applied boundary condition on the coating. The maximum tem-perature point appears in the coating, and the computation based on hyperbolic heatconduction gives higher temperature values than those based on the parabolic heatconduction model. This is in agreement with the experimental results [6,7].

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Fig. 6 Intensity factors of temperature gradients vary with time

The dynamic IFTGs can be obtained from Eqs. 34 and 36 once the algebraic equa-tions (Eqs. 28-1–28-4) are solved and the numerical inverse of the Laplace transformis performed.

The variation of dynamic IFTGs versus time is shown in Fig. 6 for the hyperbolicheat conduction model and the parabolic model. For the hyperbolic heat conductionmodel, the IFTGs fluctuate and increase with time and reach peak values, and thenoscillate for some time to reach the steady value. The IFTGs for the parabolic modelincrease smoothly with time until the peak value is reached, then decrease graduallyto the steady value. The magnitudes of the IFTGs for the hyperbolic model are biggerthan those of the parabolic model, which shows the effect of the relaxation time onthe temperature field.

Figure 7 shows the dynamic IFTGs for different k2 values of the coating withk1 = 1.0 for the half-plane. It can be observed that the magnitudes of the IFTGsdecrease as the thermal conductivity of the coating k2 decreases. The limiting case ofk2 = 0.0 corresponds to the case that there is no heat conduction into the half-plane,which leads to the zero temperature field in the half-plane.

The effect of the geometric size b on the dynamic IFTGs is shown in Fig. 8 whenother parameters are kept unchanged. As the values of b increase, the dynamic IFTGsdecrease, which means that the temperature perturbation in the cracked half-plane canbe reduced by increasing the thickness of the coating. Figure 9 shows the dynamicIFTGs for different a values with the thickness of the coating being b/c = 1. It canbe observed that the magnitudes of the IFTGs decrease as a increases.

By setting the relaxation time of the half-plane (i.e., substrate) to be a constant(say 0), the effect of the relaxation time of the coating τ1 on the dynamic IFTGs isshown in Fig. 10. The magnitude of the dynamic IFTGs decreases as the relaxationtime decreases and the limiting case of τ1 = 0 corresponds to the parabolic heat

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908 Int J Thermophys (2012) 33:895–912

Fig. 7 Dynamic IFTGs for different k2 values when k1 = 1.0

Fig. 8 Dynamic IFTGs for different b values when a/c = 1.0

conduction model. It is clearly seen that there is much difference in the very earlystages of the thermal loading impact. In other words, the big difference is obvious inthe very small time scales, but as the time increases, the values of IFTGs for differentrelaxation times converge.

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Fig. 9 Dynamic IFTGs for different a values when b/c = 1.0

Fig. 10 Dynamic IFTGs for different τ1 values when τ2 = 0.0

6 Conclusions

The transient temperature distribution in a cracked half-plane bonded by a coatingunder temperature impact loading has been studied using the hyperbolic heat conduc-tion model. The crack lies parallel to the boundary of the substrate and coating andis assumed to be partially insulated. Fourier and Laplace transforms are applied tosolve the temperature field, and the mixed boundary problem is reduced to solving a

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910 Int J Thermophys (2012) 33:895–912

singular integral equation. An asymptotic analysis and inverse Laplace transform areused to obtain the temperature field in the time domain.

Numerical studies are conducted to analyze the effect of the hyperbolic heat con-duction on the transient temperature field compared to that of the parabolic model. Thegeometric size and the relaxation time parameter have strong effects on the transienttemperature field around the crack tip. The magnitudes of the IFTGs for the hyper-bolic model are bigger than that of the parabolic model, and it is observed that theresults predicted by the hyperbolic model should be used in thermal analysis in thecase of very small time scales, which will lead to more conservative design than forthe parabolic model. The thermal stresses around the crack in a half-plane bonded bya coating could be a future study based on the result of this paper.

Appendix 1

The functions W (y, p), V (y, p), r , and n in Eqs. 20-1–20-3 are defined as

W (y, p) = f exp(−qy) (37-1)

V (y, p) = C1 exp(my) + C2 exp(−my) (37-2)

q =√

A1 p + B1 p2, (37-3)

m =√

A2 p + B2 p2, (37-4)

r =√

ξ2 + A1 p + B1 p2, (37-5)

n =√

ξ2 + A2 p + B2 p2, (37-6)

Application of the boundary conditions (Eqs. 15–17) leads to the expression,

C1 = T0(k2m − k1q) exp [m(a − b)]

p[k2m(1 + exp(−2mb)) + k1q(1 − exp(−2mb))

] , (38-1)

C2 = T0

p exp [m(a + b)]− C1

exp [2m(a + b)]

= T0(k2m + k1q) exp [−m(a + b)]

p[k2m(1 + exp(−2mb)) + k1q(1 − exp(−2mb))

] , (38-2)

f = 2T0k2m exp [−mb − qa]

p[k2m(1 + exp(−2mb)) + k1q(1 − exp(−2mb))

] , (38-3)

D2(ξ) = − D1(ξ)

exp [2n(a + b)], (39-1)

E1(ξ) = [λ2(ξ) − λ1(ξ)] D1(ξ), (39-2)

E2(ξ) = λ1(ξ)D1(ξ), (39-3)

E3(ξ) = λ2(ξ)D1(ξ), (39-4)

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Int J Thermophys (2012) 33:895–912 911

λ1(ξ) = a1 + a2

2exp(ra), (40-1)

λ2(ξ) = a1 − a2

2exp(−ra), (40-2)

a1 = [1 − exp(−2nb)

]/exp(na), (40-3)

a2 = nk2[1 + exp(−2nb)

]/[rk1 exp(na)

], (40-4)

Appendix 2: Numerical Inversion of the Laplace Transform

Based on the numerical inversion of the Laplace transform given by Miller and Guy[38], the Laplace transform of the function f (t) is known as

F∗(p) =∞∫

0

f (t)e−pt dt (41)

which can be evaluated at N discrete points along the real positive p-axis in the Laplacedomain as

pk = (β + k + 1)δ, (k = 0, 1, 2, . . . , N , β > −1, δ > 0) (42)

The function f (t) can be approximated by

f (t) ≈N∑

n=0

Cn P(0,β)n

[2e−δ·t − 1

](43)

where P(0,β)n () is a Jacobi polynomial, and the coefficients f (t) can be determined

from the following set of equations:

k∑

n=0

k(k − 1) . . . [k − (n − 1)]

(β + k + 1)(β + k + 2) . . . (β + k + n + 1)Cn = δ · F∗(pk), n = 1, 2, . . .

C0

β + 1= δ · F∗(p0) (44)

The parameters β, δ, and N are selected such that f (t) can be best described withina particular period of time. If the parameter β is selected as 0, then the Jacobi polyno-mial becomes a Legendre polynomial, see [39].

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