Heat Concentration around a Cylindrical Interface Crack in ...downloads.hindawi.com/journals/amp/2020/5849690.pdfstrip based on the memory-dependent heat conduction model. Fu et al.

Research ArticleHeat Concentration around a Cylindrical Interface Crack in aComposite Tube

J. W. Fu 1 and L. F. Qian 1,2

1Department of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China2Northwest Institute of Mechanical and Electrical Engineering, Xianyang, Shaanxi 712099, China

Correspondence should be addressed to L. F. Qian; [email protected]

Received 10 April 2020; Accepted 2 May 2020; Published 12 June 2020

Academic Editor: Marin Marin

Copyright © 2020 J. W. Fu and L. F. Qian. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Cracks always form at the interface of discrepant materials in composite structures, which influence thermal performances of thestructures under transient thermal loadings remarkably. The heat concentration around a cylindrical interface crack in a bilayeredcomposite tube has not been resolved in literature and thus is investigated in this paper based on the singular integral equationmethod. The time variable in the two-dimensional temperature governing equation, derived from the non-Fourier theory, iseliminated using the Laplace transformation technique and then solved exactly in the Laplacian domain by the employment of asuperposition method. The heat concentration degree caused by the interface crack is judged quantitatively with theemployment of heat flux intensity factor. After restoring the results in the time domain using a numerical Laplace inversiontechnique, the effects of thermal resistance of crack, liner material, and crack length on the results are analyzed with a numericalcase study. It is found that heat flux intensity factor is material-dependent, and steel is the best liner material among the threepotential materials used for sustaining transiently high temperature loadings.

1. Introduction

As a kind of star material, fiber reinforced resin matrix(FRRM) composite tubes with adhesive layers have beenextensively used in weight-sensitive engineering applications,such as automobile, aircrafts, and gun tube. FRRM compos-ites are used to reduce the weight of a structure, while theadhesive layer can usually protect the structure from theseverely thermal and mechanical loadings. However, due tothe manufacturing issue and the discrepancy of materialproperties between the FRRM composite and adhesive layer,their interface will sometimes debond and form an interfacecrack which affects the structure performance remarkably.The continuity of the heat flow in the structure under tran-sient thermal loadings is destroyed by the crack, which causesheat concentration around the crack tip. Various aspects ofcomposite structures have been studied, such as the constitu-tive equations of composite structures [1], the thermoelasticresponses under transient thermal loadings [2], and the wayto enhance the rigidity of composite plates [3].

Back in 1965, Sih [4] studied the thermal disturbanceproblem in an infinite region with lines of crack by adoptingthe complex variable method and found that heat flux proc-essed the inverse square root singularity around the crack tip.Tzou [5] confirmed the singularity behavior by introducingthe intensity factor of a temperature gradient. Heat conduc-tion of plates containing multiple insulated cracks underarbitrary Neumann thermal conditions was investigated byChen and Chang [6]. As to the interface crack, Chao andChang [7] obtained the exact temperature solutions for acomposite media made of dissimilar anisotropic materialswith a crack located at the interface. The similar problemwith the consideration of interstitial medium filled betweendissimilar anisotropic materials was investigated by Shiahand Shi [8]. Chiu et al. [9] calculated the temperature distribu-tion in an infinitely functionally graded (FG) plane containingan arbitrarily oriented crack. Zhou et al. [10] considered apartially insulated interface crack of a homogeneous ortho-tropic substrate coated by a graded orthotropic layer. And,an interface crack in a bilayered magnetoelectroelastic

HindawiAdvances in Mathematical PhysicsVolume 2020, Article ID 5849690, 13 pageshttps://doi.org/10.1155/2020/5849690

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https://doi.org/10.1155/2020/5849690

material under heat flow loadings was analyzed by Gao andNoda [11]. Wang et al. [12] discussed the effect of thermalresistance of a cohesive zone on the thermal fracture of modeII crack under static thermal loadings.

Recently, non-Fourier heat conduction theories havefound their applications on the thermal study of crackedstructures. These new theories were created to eliminate thedrawback of the classical Fourier one that the thermal wavepropagates at an infinitely large speed. Among these theories,the heat conduction model proposed by Cattaneo [13] andVernotte [14] (C-V model) has experimentally proved thatthis model could accurately predict the finite thermal propa-gation speed in processed meat and somematerials with non-homogeneous inner structures [15, 16]. Some other non-Fourier heat conduction models including the commonlyused dual-phase-lag (DPL) model were reviewed in Ref.[17]. The generalized thermoelastic theory, extended fromthe C-V model, has been used to study the thermomechani-cal coupling effects under various loading conditions, suchas pulsed laser heating [18]. Chen and Hu [19] got the dis-turbed temperature field of an infinite half-plane coated bya thin layer containing an insulated interface crack usingthe C-V model, while the DPL model was used to conductthe crack problem in a half-plane by Hu and Chen [20].The transient non-Fourier temperature and correspondingthermoelastic fields of a half-plane [21] or strip [22] with acrack parallel to the surface subjected to thermal shock wereinvestigated using the singular integral equation method. Xueet al. [23] revisited the thermoelastic problem of a crack in astrip based on the memory-dependent heat conductionmodel. Fu et al. [24] conducted the thermal analysis of asandwich structure with a cracked foam core based on thenon-Fourier heat conduction model. Cracks lead to a kindof mathematical problem termed as mixed-boundary-valueproblem; another similar mathematical problem termed asmixed-initial-boundary-value problem can be found in [25–27], in which thermoelastic responses of micropolar porousbodies are considered.

As to cylindrical structures, Guo et al. [28] obtained thetemperature field and stress intensity factor (SIF) for apenny-shaped crack using the DPL heat conduction theory.An insulated penny-shaped crack in an elastic half-spacewas handled by employing the fractional-order non-Fourierheat conduction model [29]. The transient thermoelasticfields of isotropic or transversely isotropic cylinders contain-ing a circumferential crack were investigated based on differ-ent kinds of non-Fourier models, and the correspondingthermal SIFs were calculated therein [30–34]. It should benoted that the circumferential crack, spreading along theradial direction, did not disturb the temperature field as thetemperature loading was applied on the lateral surfaces.

On the contrary, the cylindrical crack spreading along theaxial direction would disturb the heat flow in the radial direc-tion remarkably. However, the investigation on cylindricalcracks among most of the works published to date is limitedto mechanical loading conditions such as tension and torsion[35–39], and the work on its thermal analysis is rare. Thethermoelastic study of cylindrical cracks under static thermalloadings was conducted by Itou [40]. Recently, Fu et al. [41]

analyzed the disturbed temperature field of a FG cylindercontaining a cylindrical crack.

To the authors’ knowledge, the heat conduction in thecomposite tubes affected by the cylindrical interface crackhas not been resolved even using the classical Fourier the-ory. The current work is designed to study the heat concen-tration behavior of a cracked bilayered composite tube withthe adoption of non-Fourier heat conduction theory. TheC-V heat conduction model and the problem with corre-sponding boundary conditions are described in Section 2,and the solution procedure based on the singular integralequation is illustrated in Section 3 in detail. Section 4 givesthe expressions of temperature field and heat flux intensityfactor (HFIF). Moreover, the effects of crack face resistance,liner material, and crack length are analyzed in Section 5.Finally, the main conclusions of this paper are summarizedin Section 6.

2. Problem Formulation

A bilayered composite tube with an inner radius R0 and outerradius R2 is located in the cylindrical coordinate O − rφz, asillustrated in Figure 1. The sudden change of external andinternal environments of the tube will lead to a dynamic heatflow in the structure, which could be disturbed remarkably bythe cylindrical crack occupying the region −c < z < c at theinterface r = R1. The tube with uniform initial temperatureT0 is assumed infinitely long and will not deform under ther-mal loadings. The outer layer (2) is the fiber reinforced resinmatrix composite with fibers parallel to the axis and ran-domly distributed in the resin, while the material of the innerlayer (1) is a thermal protective liner and has significantlyhigher strength.

The heat flux in the radial and axial directions within theC-V model takes the form.

1 + τ ið Þ ∂∂t

� �q ið Þr r, z, tð Þ = −λ ið Þ

r∂T ið Þ r, z, tð Þ

∂r,

1 + τ ið Þ ∂∂t

� �q ið Þz r, z, tð Þ = −λ ið Þ

z∂T ið Þ r, z, tð Þ

∂z,

i = 1, 2,

ð1Þ

in which, qr and qz are the heat flux in the radial and axialdirection, respectively, t is the time, and T is the temperature.λr and λz are, respectively, the thermal conductivity in thetransverse plane and longitudinal direction for the

r

zo

(1)

(2)

2cR2

R1

R0𝜑

Figure 1: A composite tube with a cylindrical interface crack.

2 Advances in Mathematical Physics

transversely isotropic composite layer, and λr = λz holds forthe isotropic inner layer. Please note that the thermal proper-ties of fiber reinforced composite material will take its macro-scopically equivalent parameters. The phase lag of heat flux τ,which can be interpreted as the time lag needed to excite heatflow at a position when a temperature gradient loading isapplied on that position, is an intrinsic material propertywith the unit of time and varies from 10−14~102 s for differentmaterials. It can be easily seen from Equation (1) that τ = 0reduces the C-V model to the Fourier one.

Without considering the heat source, the energy conser-vation equation reads

−∂q ið Þ

r

∂r+q ið Þr

r+∂q ið Þ

z

∂z

!= ρ ið Þc ið Þ

P∂T ið Þ

∂t, ð2Þ

where cP and ρ are specific heat capacity and mass density,respectively.

The temperature governing equation can be obtained byeliminating the heat flux from Equation (2) and Equation(1) as

1 + τ ið Þ ∂∂t

� �∂T ið Þ

∂t= d ið Þ

r∂2

∂r2+

∂r∂r

!+ d ið Þ

z∂2

∂z2

" #T ið Þ, i = 1, 2,

ð3Þ

where d = λ/ρcP is the thermal diffusivity. Equation (3) alsoindicates that the speed of thermal wave in the radial direc-tion is CCV =

ffiffiffiffiffiffiffiffidr/τ

p.

The thermal boundary conditions for the cracked tubecan be written as

q 1ð Þr R0, z, tð Þ = −hi T

1ð Þ R0, z, tð Þ − Ta tð Þh i

, zj j <∞, ð4Þ

q 2ð Þr R2, z, tð Þ = ho T 2ð Þ R2, z, tð Þ − Tb tð Þ

h i, zj j <∞, ð5Þ

q 1ð Þr R1, z, tð Þ = q 2ð Þ

r R1, z, tð Þ, zj j <∞, ð6Þ

T 1ð Þ R1, z, tð Þ = T 2ð Þ R1, z, tð Þ, zj j ≥ c, ð7Þ

Vq 1ð Þr R1, z, tð Þ = T 1ð Þ R1, z, tð Þ − T 2ð Þ R1, z, tð Þ, zj j < c:

ð8ÞThe convective heat transfer conditions between surfaces

of the tube and its corresponding environments with temper-ature Ta and Tb are shown in Equations (4) and (5), where hiand ho are heat transfer coefficients. Equations (6) and (7)describe the thermal conditions at the interface. It can be eas-ily seen that the mixed boundary condition, given by Equa-tions (7) and (8), brings much mathematical complexity. Itshould be noted that a zero value of thermal resistance V rep-resents a perfectly conductive crack, while an insulated crackgenerates for an infinitely large value of V .

3. Solution of the Mixed-Boundary-Condition Problem

In this section, the Laplace transform and Fourier transformas well as the superposition method are adopted to solve theproblem. It should be mentioned that the singular integralequation method is originally proposed by Erdogan et al.[42] to study the mechanical fracture problems, and thispaper extends the method to investigate the mixed-boundary-condition problem in thermal fields. The mis-match of thermal properties of the material on the sides ofcrack improves the mathematical complexity and leads to amaterial-property-dependent HFIF.

In order to normalize the equations, the following nondi-mensional parameters are introduced:

t ′ = td 0ð Þ

c2,

τ ið Þ′ =τ ið Þd 0ð Þ

c2,

r′ = rc,

hi′=hic

λ 0ð Þ ,

T ið Þ′ =T ið Þ − T0

T0,

V ′ = Vλ 0ð Þ

c,

q ið Þ ′ = q ið Þc

λ 0ð ÞT0:

ð9Þ

It should be mentioned that other dimensional parame-ters with the same unit as those in Equation (9) could be nor-malized accordingly. The parameters with the superscript “0”are reference properties.

Substituting the nondimensional parameters in Equation(9) into Equation (3), the governing equation can be rewrit-ten as

∂2

∂r′2+

1r′

∂∂r′

+ υ ið Þ2 ∂2

∂z′2

!T ið Þ′ =

d 0ð Þ

drið Þ 1 + τ ið Þ′ ∂

∂t ′

� �∂T ið Þ′

∂t ′,

ð10Þ

where υðiÞ2 = dðiÞz /dðiÞr . When zero initial temperature changeand rate of temperature change are assumed, Laplace trans-formation is applied to Equation (10) to eliminate the timevariable t ′, as

∂2

∂r′2+

1r′

∂∂r′

+ υ ið Þ2 ∂2

∂z′2

!~T

ið Þ′ − k ið Þ1~T

ið Þ′ = 0, ð11Þ

in which, kðiÞ1 = sð1 + τðiÞ′sÞdð0Þ/drðiÞ and s is the Laplacevariable.

3Advances in Mathematical Physics

Equations (4)–(8) are treated with similar nondimensio-nalization and Laplace transformation operations, and theboundary conditions are rewritten as

~q 1ð Þr ′ R0′ , z′, s� �

= −hi′ ~T1ð Þ′

R0′ , z′, s� �

− ~Ta′ sð Þh i

, ð12Þ

~q 2ð Þr ′ R2′ , z′, s� �

= ho′ ~T2ð Þ′

R2′ , z′, s� �

− ~Tb′ sð Þh i

, ð13Þ

~q 1ð Þr ′ R1′ , z′, s� �

= ~q 2ð Þr ′ R1′ , z′, s� �

, ð14Þ

~T1ð Þ′

R1′ , z′, s� �

= ~T2ð Þ′

R1′ , z′, s� �

, z′�� ≥ 1, ð15Þ

V ′~q 1ð Þr ′ R1′ , z′, s� �

= ~T1ð Þ′

R1′ , z′, s� �

− ~T2ð Þ′

R1′ , z′, s� �

, z′�� < 1:

ð16ÞThe governing equation (11) under the boundaries

(12)–(16) can be solved using the superposition method. Thefirst problem (P1) can be described as inner and outer surfacesof an uncracked tube applied with inhomogeneous boundaryconditions. And the second problem (P2) is a cracked tubewithout thermal loadings applied on the surfaces. The mathe-matical expressions of P1 and P2 are, respectively, as follows:

P1:

d2

dr′2+

1r′

d

dr′

!~T

ið Þ1 ′ − k ið Þ

1~T

ið Þ1 ′ = 0,

~q 1ð Þr1 ′ R0′ , s� �

= −hi′ ~T1ð Þ1 ′ R0′ , s� �

− ~T ′a sð Þh i

,

~q 2ð Þr1 ′ R2′ , s� �

= ho′ ~T2ð Þ1 ′ R2′ , s� �

− ~T ′b sð Þh i

,

~q 1ð Þr1 ′ R1′ , s� �

= ~q 2ð Þr1 ′ R1′ , s� �

,

~T11ð Þ ′ R1′ , s� �

= ~T12ð Þ ′ R1′ , s� �

,

ð17ÞP2:

∂2

∂r′2+

1r′

∂∂r′

+ υ ið Þ2 ∂2

∂z′2

!~T

ið Þ2 ′ − k ið Þ

1~T

ið Þ2 ′ = 0, ð18Þ

~q 1ð Þr2 ′ R0′ , z′, s� �

= −hi′~T1ð Þ2 ′ R0′ , z′, s� �

, ð19Þ

~q 2ð Þr2 ′ R2′ , z′, s� �

= ho′~T2ð Þ2 ′ R2′ , z′, s� �

, ð20Þ

~q 1ð Þr2 ′ R1′ , z′, s� �

= ~q 2ð Þr2 ′ R1′ , z′, s� �

, ð21Þ~T

1ð Þ2 ′ R1′ , z′, s� �

= ~T2ð Þ2 ′ R1′ , z′, s� �

, z′�� ≥ 1, ð22Þ

V ′ ~q 1ð Þr1 ′ R1′ , s� �

+ ~q 1ð Þr2 ′ R1′ , z′, s� �h i

= ~T1ð Þ2 ′ R1′ , z′, s� �

− ~T2ð Þ2 ′ R1′ , z′, s� �

, z′�� < 1: ð23Þ

The actual results of temperature and heat flux are addi-

tion of the solution of P1 and P2 as ~TðiÞ′ = ~T

ðiÞ1 ′ + ~T

ðiÞ2 ′ and

~qðiÞr ′ = ~qðiÞr1 ′ + ~qðiÞr2 ′.

The solution of P1 can be easily obtained as

~Tið Þ1 ′ = S ið Þ

1 I0

ffiffiffiffiffiffik ið Þ1

qr′

� �+ S ið Þ

2 K0


qr′

� �, ð24Þ

~q ið Þr1 ′ = k ið Þ

2


qS ið Þ1 I1


qr′

� �− S ið Þ

2 K1


qr′

� �� ,

ð25Þ

where kðiÞ2 = −ðλðiÞ/λð0ÞÞð1/ð1 + τðiÞ′sÞÞ and InðÞ and KnðÞ rep-resent the nth-order modified Bessel functions of the first

kind and the second, respectively. The unknowns SðiÞ1 and

SðiÞ2 can be obtained from PS =Q, in which, S is defined as

S = S 1ð Þ1 S 1ð Þ

2 S 2ð Þ1 S 2ð Þ

2

h iT, ð26Þ

and the nonzero elements of 4 × 4 matrix P and 4 × 1 vectorQ are

P1,1 = k 1ð Þ2

ffiffiffiffiffiffiffik 1ð Þ1

qI1


qR0′

� �+ hi′I0


qR0′

� �,

P1,2 = −k 1ð Þ2


qK1


qR0′

� �+ hi′K0


qR0′

� �,

P2,3 = −k 2ð Þ2


qI1


qR2′

� �+ ho′I0


qR2′

� �,

P2,4 = k 2ð Þ2


qK1


qR2′

� �+ ho′K0


qR2′

� �,

P3,1 = k 1ð Þ2


qI1


qR1′

� �,

P3,2 = −k 1ð Þ2


qK1


qR1′

� �,

P3,3 = −k 2ð Þ2


qI1


qR1′

� �,

P3,4 = k 2ð Þ2


qK1


qR1′

� �,

P4,1 = −I0ffiffiffiffiffiffiffik 1ð Þ1

qR1′

� �,

P4,2 = −K0


qR1′

� �,

P4,3 = I0


qR1′

� �,

P4,4 = K0


qR1′

� �,

Q1 = hi′~T ′a sð Þ,Q2 = ho′ ~T ′b sð Þ:

ð27Þ


The application of the Fourier transformation,

�f ξð Þ =ð∞−∞

f z′� �

e−jξz ′dz′, ð28Þ

to Equation (18) leads to

∂2

∂r′2+

1r′

∂∂r′

− k ið Þ3

!�~T

ið Þ2 ′ = 0, ð29Þ

where kðiÞ3 = υðiÞ2ξ2 + kðiÞ1 and ξ is the Fourier transformationvariable. Thus, the temperature and heat flux in the double-transformed domain can be solved as

�~Tið Þ2 ′ = S ið Þ

3 I0


qr′

� �+ S ið Þ

4 K0


qr′

� �,

�~q ið Þ2 ′ = k ið Þ

2


qS ið Þ3 I1


qr′

� �− S ið Þ

4 K1


qr′

� �� :

ð30Þ

The corresponding temperature and the heat flux in theLaplace domain are then calculated using the Fourier inver-sion transform,

f z′� �

=12π

ð∞−∞

�f ξð Þejξz ′dξ, ð31Þ

as

~Tið Þ2 ′ = 1

2π

ð∞−∞

S ið Þ3 I0


qr′

� �+ S ið Þ

4 K0


qr′

� �� ejξz ′dξ, ð32Þ

~q ið Þ2 ′ = 1

2π

ð∞−∞

k ið Þ2


qS ið Þ3 I1


qr′

� �− S ið Þ

4 K1


qr′

� �� ejξz ′dξ,

ð33Þ

where the imaginary number j =ffiffiffiffiffiffi−1

p.

Using the boundary equations (18)–(21), the unknownsin Equation (32) are linked as

S 1ð Þ3 = −

Χ2Χ5Χ1

S 2ð Þ4 ,

S 1ð Þ4 =Χ5S

2ð Þ4 ,

S 2ð Þ3 = −

Χ4Χ3

S 2ð Þ4 ,

ð34Þ

in which

Χ1 = hi′I0ffiffiffiffiffiffiffik 1ð Þ3

qR0′

� �+ k 1ð Þ

2


qI1


qR0′

� �,

Χ2 = hi′K0


qR0′

� �− k 1ð Þ

2


qK1


qR0′

� �,

Χ3 = ho′I0ffiffiffiffiffiffiffik 2ð Þ3

qR2′

� �− k 2ð Þ

2


qI1


qR2′

� �,

Χ4 = ho′K0


qR2′

� �+ k 2ð Þ

2


qK1


qR2′

� �,

Χ5 =k 2ð Þ2


qI1


qR1′

� �Χ4/Χ3ð Þ + k 2ð Þ

2


qK1


qR1′

� �

k 1ð Þ2


qI1


qR1′

� �Χ2/Χ1ð Þ + k 1ð Þ

2


qK1


qR1′

� � :

ð35Þ

The last unknown coefficient Sð2Þ4 can be obtained usingthe mixed boundary conditions (22) and (23). A function,similar to dislocation density function in mechanical fields,is defined as

Θ z′, s� �

=∂∂z′

~T1ð Þ2 ′ R1′ , z′, s� �

− ~T2ð Þ2 ′ R1′ , z′, s� �h i

: ð36Þ

This gradient function is crucial for obtaining the two-dimensional temperature field caused by the crack with theadoption of the singular integral equation method.

Thus, from Equation (22), one could have

Θ z′, s� �

= 0, z′�� ≥ 1, ð37Þ

ð1−1Θ z′, s� �

dz′ = 0, z′�� < 1: ð38Þ

Equation (38), referred as the single-valuedness condi-tion, is obtained from the zero value of temperature jumpat the crack tips jz′j = 1. It also reflects the fact that the tem-perature jump starts from zero at z′ = −1, then increases to aspecific value within the crack, and finally decreases to zeroagain at the other tip z′ = 1. Substituting Equations (32)and (34) into Equation (36) results in

S 2ð Þ4 =

1jξΧ6

ð1−1Θ η, sð Þe−jξηdη, ð39Þ

in which

Χ6−Χ2Χ5Χ1

I0


qR1′

� �+Χ5K0


qR1′

� �

+Χ4Χ3

I0


qR1′

� �− K0


qR1′

� �: ð40Þ


Substituting Equations (25), (32), (33), (34) and (39) intoEquation (23) results in a singular integral equation,

ð1−1Θ η, sð Þ 1

z′ − η+ L z′, η� ��

dη =πH sð ÞΧ∞

, ð41Þ

in which

L z′, η� �

=ð∞0

Χ9 −Χ∞Χ∞

sin ξ z′ − η� �h i

dξ, ð42Þ

Χ9 = Χ5Χ8 −Χ2Χ5Χ7

Χ1+Χ4Χ3

I0


qR1′

� �− K0


qR1′

� �� /ξΧ6,

ð43Þ

Χ7 = I0


qR1′

� �−V ′k 1ð Þ

2


qI1


qR1′

� �, ð44Þ

Χ8 = K0


qR1′

� �+V ′k 1ð Þ

2


qK1


qR1′

� �, ð45Þ

Χ∞ = limξ→∞

Χ9 = −V ′

1/k 1ð Þ2 υ 1ð Þ

� �+ 1/k 2ð Þ

2 υ 2ð Þ� � , ð46Þ

H sð Þ =V ′k 1ð Þ2


qS 1ð Þ1 I1


qR1′

� �− S 1ð Þ

2 K1


qR1′

� �� :

ð47ÞThe singular integral equation (41) under the single-

valuedness condition (38) has the fundamental solution [42].

Θ η, sð Þ = f η, sð Þ 1 − η2 �−1/2

: ð48Þ

The Gauss-Jacobi integration formulas in [31] can beadopted to solve Equations (41) and (38) numerically as

〠N

μ=1

1Nf ημ, s� � 1

zω′ − ημ+ L zω′ , ημ, s� �" #

=H sð ÞΧ∞

, ð49Þ

〠N

μ=1f ημ, s� �

= 0, ð50Þ

in which

ημ = cos2μ − 12N

π

� �, μ = 1, 2,⋯,N , ð51Þ

zω′ = cosω

Nπ

� �, ω = 1, 2,⋯,N − 1: ð52Þ

After getting f ðη, sÞ, the four unknown coefficients in thegeneral solution of P2 given by Equation (2) can be calculatedusing Equations (34), (39) and (48).

4. Transient Temperature Field andHeat Concentration

Addition of Equation (24) and Equation (2) results in thetransient temperature

~Tið Þ ′ = 1

π

ð∞0

S ið Þ3 I0


qr′

� �+ S ið Þ

4 K0


qr′

� �� cos ξz′� �

dξ

+ S ið Þ1 I0


qr′

� �+ S ið Þ

2 K0


qr′

� �, i = 1, 2,

ð53Þ

in which, Sð1Þ3 , Sð1Þ4 , and Sð2Þ3 are expressions of Sð2Þ4 . And theGauss-Chebyshev integration equation can be adopted to

numerically calculate Sð2Þ4 from Equation (39),

S 2ð Þ4 =

−1ξΧ6

ð1−1Θ η, sð Þ sin ξηð Þdη = −1

ξΧ6〠N

μ=1

π

Nf ημ, s� �

sin ξημ

� �:

ð54Þ

Similar to the stress intensity factor, heat flux intensityfactor (HFIF) is introduced to judge the heat concentrationdegree around the crack tip quantitatively, as

Kq tð Þ = limz→c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 z − cð Þ

pq 2ð Þr R1, z, tð Þ: ð55Þ

Using the parameter Kq′ = Kqffiffic

p/ðλð0Þrr T0Þ, HFIF can be

nondimensionalized in the Laplace domain as

~K ′q sð Þ = limz ′→1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 z′ − 1� �r

~q′ 2ð Þr R1′ , z′, s� �

: ð56Þ

Substituting Equations (25) and (33) into Equation (56)leads to

~K ′q sð Þ =Y

λ2ð Þ6 f 1, sð Þ, ð57Þ

in which, f ð1, sÞ is calculated from f ðημ, sÞ using the interpo-lation method and

Y=

1

λ2ð Þ6 /λ 1ð Þ

6

� �1/υ 1ð Þ �

+ 1/υ 2ð Þ � : ð58Þ

It can be clearly seen from Equation (58) that the HFIF isdependent on thermal properties of the material on bothsides of the crack. Finally, the transient temperature fieldand HFIF in the time domain can be numerically trans-formed using the Laplace inversion technique given in [43].

5. Numerical Results

In this section, the solution procedure is verified first. Then,the effects of crack face resistance, liner material, and cracklength on the transient temperature field and HFIF of a


composite tube containing an interface crack are analyzed inthe following three subsections.

The composite tube with initial temperature T0 = 300Khas the size R0 = 0:006m, R1 = 0:008m and R2 = 0:01m. Byassuming the fully conductive inner and outer surfaces as hi= ho =∞, the non-Fourier heat conduction of the crackedtube with sudden temperature rise applied on the inner sur-face TaðtÞ = 330HðtÞK is investigated based on the C-Vmodel, in which, HðtÞ is the Heaviside step function. Theinner layer can be made of steel, silicon carbide ceramic, ortantalum, while the outer layer is carbon fiber reinforcedresin matrix composite. Table 1 lists properties of all thematerial. It should be mentioned that the accurate value ofphase lag of the carbon fiber reinforced composite cannotbe found in experimental tests, and it is assumed accordingto Ref. [41]. Phase lag of heat flux is in the order of picosec-onds for most of metals due to the phonon-electron couplingeffect. This parameter can be as large as 102 s [37] for nonho-mogeneous materials, as the thermal flow needs time to cir-cumvent inclusions or pass through the interface ofdifferent materials. The experimental evidence of non-Fourier heat conduction in some materials at room tempera-ture can also be found [44, 45].

The macroscopically equivalent material properties of thecomposite are calculated as [51]

λ 2ð Þr = λf +

1 − c11/ λm − λf

�+ c1/ 2λf

� ,λ 2ð Þz = c1λf + 1 − c1ð Þλm,

c 2ð ÞP = c1cPf + 1 − c1ð ÞcPm,

ρ 2ð Þ = c1ρf + 1 − c1ð Þρm,

ð59Þ

in which, the parameters with subscripts “f” and “m,” respec-tively, refer to those of fiber and matrix. Moreover, the fiberfraction c1 is taken to be 0:2 in this section. The structure pro-file described here is used in this whole section except Subsec-tion 5.1 designed for validation with reference.

5.1. Validation of the Solution Procedure. Before starting theanalysis of the parameters, the solution procedure presentedabove should be validated. As stated in Introduction, the C-Vheat conduction model is accurate to describe the tempera-ture distribution in processed meat and other materials mea-sured by especially designed experiments. However, thesekinds of experiment are not available for tubes with or with-

out cracks. The numerical calculation of the transient tem-perature field for cylindrical cracks using the Fourier ornon-Fourier models is also absent among the literature.Thus, the temperature result for uncracked tubes based onthe C-V model obtained by Babaei and Chen [52] is adoptedfor comparison. The same structure, material properties,thermal loading, and nondimensional parameters as thosein Ref. [52] for homogeneous tubes are employed, while thecylindrical crack in the present model spreads along the mid-dle plane R1 ′ = 0:8 and is assumed insulated. Figure 2 illus-trates the nondimensional temperature distribution alongthe radial direction at the instant t ′ = 0:126 for different axiallocations. The Saint-Venant principle indicates that the con-tinuous temperature field at a position far away from thecrack will not be affected, which means the same temperaturedistribution as that in uncracked tubes. This is verified bycomparing the temperature distribution at z′ = 10 for acracked tube with the results in [52] for an uncracked tube.The dash-dot line depicts the temperature in the cross sectionacross the crack face midpoints. As the crack is insulated,thermal energy will accumulate before the crack and a muchhigher temperature will then be obtained.

5.2. The Effect of Thermal Resistance. The effects of crackresistance V on the temperature field and HFIF are shownin Figures 3 and 4, respectively. An infinitely large V corre-sponding to an insulated crack is compared with three otherpartially insulated cracks with V = 1 × 10−3 km2sJ−1, V = 1× 10−4 km2sJ−1, and V = 1 × 10−5 km2sJ−1. The inner layeris made of steel, and a crack with a length of 0:006m affectsthe temperature field of the tube remarkably.

Figure 3(a) illustrates the temperature distribution alongthe interface for different values of V at the instant t = 3:184 swhen the transient temperature field approaches to the equi-librium state. It is shown that the temperature jump across

Table 1: Thermal properties [46].

Material ρ (kg/m3) cP (J/kgK) λ (W/mK) τ (s)

Carbon 1710 830 1900:8 [47]

Resin 1200 1050 8

Steel 7800 490 45 1:6 × 10−12 [48]Ceramic 3300 670 80 2:7 × 10−6 [49]Tantalum 16600 139 54:4 1:0 × 10−7 [50]

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1−0.5

0

0.5

1

1.5

2

Non-dimensional radial coordinate (r’)

Non

-dim

ensio

nal t

empe

ratu

re (T

’) R0 = 0.6 m, R2 = 1 mR1 = 0.8 m, c = 0.2 m𝜏’ = 0.35Insulated crackTb = 500 KT’ = 0.126

z’ = 10z’ = 0Ref. [52]

Figure 2: Validation of temperature distribution with Ref. [52].


the crack face exists for all the cases, while the temperature iscontinuous along other parts of the interface plane. Also, thetemperature at the inner crack face is higher than that in theouter crack face. With the reduction of V , the temperaturejump shrinks as well, and the perturbation effect of crackon the temperature fades away. Similar results were reportedby Chen and Hu [19] for layered structures as well. Further-more, the temperature histories at the crack face midpointsfor the insulated crack case and almost conducting crack case(V = 1 × 10−5 km2sJ−1) are shown in Figure 3(b). It can beseen that the temperature history performs in an obviouslywave-like manner and oscillates around a steady value.Figure 3(b) also depicts the fact that the temperature jump

across the insulated crack faces is much higher than thatacross the conducting crack faces all the while. The oscillat-ing behavior of transient HFIF is shown in Figure 4. Similarto the temperature results, the effect of V on the HFIF indi-cates that the heat concentration degree reduces with theraise of thermal conduction capability of the crack.

5.3. The Effect of Liner Material. Usage of thermally protec-tive liners is very helpful for the enhancement of the overallperformance of composite structure. In this subsection,three kinds of material are chosen to study their effects onthermal behaviors of composite tubes containing an

−6 −4 −2 0 2 4 6300

305

310

315

320

325

330

z (m)

Steel linerc = 0.003 mt = 3.184 s

T (K

)

×10−3

V = ∞V = 1e–3

V = 1e–4

V = 1e–5

(R1+,z)

(R1–,z)

(a)

0 1 2 3 4 5 6 7 8

t (s)

Steel linerc = 0.003 mz = 0

t (s)V = ∞V = 1e–5

(R1+,z)

(R1–,z)

300

305

310

315

320

325

330

T (K

)(b)

Figure 3: Effect of crack resistance on (a) temperature distribution along the interface plane and (b) temperature history at the crack facemidpoints.

0 1 2 3 4 5 6 70

1000

2000

3000

4000

5000

6000

7000

Steel linerc = 0.003 m

Kq (

Wm

−3/2

)

t (s)V = ∞V = 1e–3

V = 1e–4

V = 1e–5

Figure 4: Effect of crack resistance on the HFIF history.

−6 −4 −2 0 2 4 6300

305

310

315

320

325

330

335

340

z (m)

T (K

)

Insulated crackc = 0.003t = 3.184 s

×10−3

Steel linerSiC liner

Ta linerPure steel

(R1+,z)

(R1–,z)

Figure 5: Effect of liner material on the temperature distributionalong the interface plane.


insulated interface crack, while a tube with the same sizemade of pure steel is added for comparison. Please note thatthe phase lag of steel is so small that the non-Fourier effect isnegligible for the steel tube considered in this paper, whichhas been verified by comparing with the obtained resultswhen setting τ = 0 of steel manually. Thus, the results forsteel tube can generally represent thermal behaviors pre-dicted by the Fourier model. And, the crack length is takento be c = 0:003m.

The temperature distribution along the interface of thefour tubes at the instant t = 3:184 s is depicted in Figure 5,while Figure 6 shows the HFIF history. Besides the conclu-sions obtained from Figure 3(a), Figure 5 reveals that the

composite tube with steel liner has the lowest temperatureoff the crack compared with the other two tubes with sil-icon carbide and tantalum liner. However, their tempera-tures at the crack midpoints show a negligible difference.Although the temperature of the steel tube is lowest, ithas the highest weight as well as heat concentration degreeas shown in Figure 6. It can also been observed fromFigure 6 that HFIF of the steel tube increases rapidly tothe maximum value without oscillating as other tubes.This difference comes from the significant non-Fouriereffect of fiber reinforced resin matrix material. The oscilla-tion behavior of the heat flux and HFIF predicted by thenon-Fourier models is also described in [20, 53]. Similar

0 1 2 3 4 5 6 70

2000

4000

6000

8000

10000

12000

t (s)

Insulated crackc = 0.003 m

Steel linerCeramic liner

Ta linerPure steel

Kq (W

m−3

/2)

Figure 6: Effect of liner material on the HFIF history.

−6 −4 −2 0 2 4 6305

310

315

320

325

330

z (m)

Tem

pera

ture

(K)

Steel linerInsulated crackt = 3.184 s

×10−3

(R1+,z)

(R1–,z)

c = 0.003c = 0.0025

c = 0.002c = 0.0015

Figure 7: Effect of crack length on the temperature distributionalong the interface plane.

0 1 2 3 4 5 6 70

1000

2000

3000

4000

5000

6000

7000

t (s)

Steel linerInsulated crack

Kq (W

m−3

/2)

c = 0.003 mc = 0.0025 m

c = 0.002 mc = 0.0015 m

Figure 8: Effect of crack length on the HFIF history.

0.5

1

1.5

2

2.5

3

3.5


0 1 2 3 4 5 6 7t (s)

c = 0.003, z = 10c = 0.003, z = 0.0035

c = 0.002, z = 0.0035c = 0.0025, z = 0.0035

×105qr (

Wm

−2)

Figure 9: The effect of crack length on the history of heat flux atinterface.


to temperature, the tube with steel liner has the lowestHFIF among the three composite tubes.

5.4. The Effect of Crack Length. The effects of crack length onthe temperature field, heat flux, and HFIF are described inFigures 7–10. Four composite tubes with steel liner contain-ing different lengths of insulated interface crack, namely, c= 0:003m, 0:0025m, 0:002m, and 0:0015m, are utilized asexamples in this subsection. It can be figured out fromFigure 7, which shows the temperature distribution alongthe interface at the instant t = 3:184 s, that the shrink of cracklength reduces the temperature jump across the crack faces. Itcan be imagined that the temperature will be continuousacross the crack faces when the crack shortens to a size assmall as possible. Coincidentally, Figure 8 shows that theHFIF generally reduces as the crack length shrinks.

The histories of heat flux in the radial direction at thesame point (r = 0:008m, z = 0:0035m) for different cracklengths are shown in Figure 9, in which the result at the point(r = 0:008m, z = 10m) far away from the crack (c = 0:003m)is incorporated for comparison. The heat flux at the point z= 10m will not be affected by the crack and represents theresult for uncracked structure. It is seen that the heat fluxfrom the liner to composite material decreases with thereduction of crack length, as the point picked to plot theresult becomes away from the crack tip. The tiny discrepancybetween the black line and red line shows that the crack c =0:002m is too short to affect the heat flux at the point z =0:0035m.

In order to show the influence of crack on the tempera-ture field more clearly, Figure 10 draws the temperature con-tour around the crack for different crack lengths at the

z (m)

r (m

)

302.3862302.3862

302.3862

304.7122

304.7122

304.7122

307.0383

307.0383

307.0383309.3643

309.3643

309.3

643

309.3643311.6904

311.6904 311.6904

311.6904

314.0164

314.0164

314.0164316.3425

316.3425316.3425

318.6685318.6685

318.6685320.9946 320.9946 320.9946

323.3206323.3206

323.3206

325.6467

325.6467 325.6467

325.6467

327.9727

327.9

727 327.9727

327.9727

−6 −4 −2 0 2 4 66

6.5

7

7.5

8

8.5

9

9.5

10

Crack


×10−3

×10−3

(a)

r (m

)

6

6.5

7

7.5

8

8.5

9

9.5

10

z (m)

302.3951 302.3951 302.3951

304.7188304.7188

304.7188

307.0426307.0426

307.0426309.366 3

309.3663 309.3663311.69

311.69

311.6

9

311.69314.013 7

314.013 7 314.0

137

316.3375 316.3375

316.3375

318.6612 318.6612

318.6612320.9849 320.9849 320.9849

323.3086 323.3086

323.3086

325.6323

325.6323

325.6323

325.6323

327.9561

327.9561

327.9561

327.9561

−6 −4 −2 0 2 4 6


Crack

×10−3

×10−3

(b)

302.4049 302.4049 302.4049304.7257

304.7257304.7257

307.0466307.0466

307.0466309.3674

309.3674

309.3674311.6883

311.6883311.6883

311.6883314.0091

314.0091

314.0

091

314.0091316.33

316.33

316.33318.6508

318.6508318.6508

320.9716 320.9716 320.9716

323.2925

323.2925323.2925

325.6133

325.

6133

325.6133

325.6133

327.9342327.9342

327.9342

327.9342

Crack


z (m)−6 −4 −2 0 2 4 6

×10−3

r (m

)

6

6.5

7

7.5

8

8.5

9

9.5

10 ×10−3

(c)

302.4131 302.413 1 302.4131304.7308 304.7308 304.7308307.0485 307.048 5 307.0485309.3662

309.3662309.3662

311.6839311.6839

311.6839

314.0015314.0015

314.0015316.3192

316.3

192

316.3192318.6369

318.6369318.6369

32 0.9 546 320.9546 320.9546

323.2723

323.2 72 3323.2723

325.59

325.5

9

325.59

325.59

327.9077

327.9077327.9077Steel liner

Insulated crack

Crack

z (m)−6 −4 −2 0 2 4 6

×10−3

r (m

)

6

6.5

7

7.5

8

8.5

9

9.5

10 ×10−3

(d)

Figure 10: Effect of crack length on the 2D temperature distribution around the crack: (a) c = 0:003m, (b) c = 0:0025m, (c) c = 0:002m, and(d) c = 0:0015m.


instant when the maximum temperature happens. It is clearthat heat gathers around the crack tip and temperature jumpacross the crack faces exists, while the temperature fieldapproaches uniform at positions far away from the crack.Similar to the tendency shown in Figure 7, a shorter crackleads to a more homogeneous temperature field.

6. Conclusion

This paper studies the thermal concentration behavior of abilayered composite tube containing a cylindrical interfacecrack. The two-dimensional temperature field disturbed bythe crack is obtained using the singular integral equationmethod. By taking the crack as partially thermal insulated,the effects of crack resistance, liner material, and crack lengthon the temperature field and HFIF of the cracked tube areanalyzed numerically. The following main conclusions arearrived:

(1) Heat flux intensity factor serves as an effectiveparameter to describe the heat concentration degreecaused by the crack quantitatively;

(2) Steel is the best liner material among the three poten-tial materials used for protecting the barrel from thetransient temperature change of the inner environ-ment, as the composite tube with steel liner has thelowest temperature and HFIF results compared withthe other two liner materials;

(3) The insulating degree of crack affects the heat con-centration around the crack tip remarkably. The tem-perature jump across the crack faces is highest forfully insulated cracks, while the temperature jumpand corresponding HFIF decreases when the crackbecomes more conductive;

(4) When a small crack grows along the interface, aseverer temperature diagram with higher HFIFemerges. This indicates that the crack length in thetube should be controlled to a size as small as possibleduring the manufacturing process, and the crackdetection of the tube in service should be conductedregularly to avoid overheating around the crack tip

Data Availability

The data used to support the findings of this study areincluded within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (grant number 11702137), Natural Sci-ence Foundation of Jiangsu Province of China (grant numberBK20170816), and the Fundamental Research Funds for theCentral Universities (grant number 309171B8802).

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Heat Concentration around a Cylindrical Interface Crack in ...downloads.hindawi.com/journals/amp/2020/5849690.pdfstrip based on the memory-dependent heat conduction model. Fu et al.

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