Effect of heat conduction of penny-shaped crack interior on thermal stress intensity factors

International Journal of Heat and Mass Transfer 91 (2015) 127–134

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Effect of heat conduction of penny-shaped crack interior on thermal stress intensity factors Xian-Fang Li a,b, Kang Yong Lee b,⇑ a b

School of Civil Engineering, Central South University, Changsha 410075, PR China State Key Laboratory of Structural Analysis for Industrial Equipment and Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, PR China

a r t i c l e

i n f o

Article history: Received 6 June 2014 Received in revised form 3 July 2015 Accepted 22 July 2015

Keywords: Penny-shaped crack Heat flux Thermal stress intensity factor Hankel transform Partially insulated crack Heat conduction of crack

a b s t r a c t This paper analyzes the thermal stress induced by a penny-shaped crack in an elastic solid with uniform steady heat flux. Air inside an opening crack is taken as a thermally conducting medium and the crack is partially insulated. The Hankel transform technique is applied to convert the problem to a system of dual integral equations. Heat flux in the opening crack or temperature gradient across the crack depends on the crack opening displacement. Explicit expressions for the whole temperature change field and heat flux at any position in the cracked medium are given in terms of elementary functions. Thermal stresses and displacements are presented for a solid with a partially insulated crack under remote tensile loading and uniform heat flux. Stress intensity factors (SIFs) are determined. The mode-I SIFs depend only on external tensile loading, and are free of the material properties. The mode-II SIFs are related to both mechanical and thermal loading, in addition to the material properties. Numerical results for a cracked thermoelastic material are presented to show the influence of the thermal conductivity of air of the crack interior on the mode-II SIFs, and indicate that heat conduction of crack affects thermal SIFs. Insulated and isothermal cracks are two limiting cases of a partially insulated crack. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Thermally induced mechanical failure is one of the most typical failure and frequently encountered in electronic and photonic systems widely used in electronic devices. To promote structural reliability and lengthen service period of electronic devices, thermal stress analysis is a major concern in design and fabrication of electronic productions. Accordingly, the ability to understand the mechanical behavior of electronic structures induced by thermal effect is of quite importance. In particular, the presence of inclusion and defect in structures gives rise to thermal stress concentration and degrade the performance of structures [1,2]. Thermal stress analysis of an elastic medium with crack has received considerable attention. Along this line, a great number of investigations have been done. For example, Sih [3] applied a complex variable method to determine the singular behavior of thermal stress near a crack tip in a two-dimensional elastic medium. For a three-dimensional elastic solid, Olesiak and Sneddon [4] gave an analysis of a penny-shaped crack embedded in an isotropic homogeneous solid where heat flux or temperature change at the crack surface is given. This problem is symmetric with ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (K.Y. Lee). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.07.087 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

respect to the crack plane. On contrary, Florence and Goodier [5] analyzed an antisymmetric problem and determined thermal stresses disturbed by a penny-shaped crack when a simple uniform steady heat flux is given. Furthermore, thermal stresses in a semi-infinite elastic solid containing a penny-shaped crack has been studied by Srivastava and Palaiya [6]. For a transversely isotropic thermoelastic medium containing a penny-shaped crack, such symmetric and antisymmetric problems have been analyzed. Thermal stresses disturbed by a crack under heat flow has been determined [7]. When the temperature change at the crack surface is given, a fundamental solution has been obtained [8]. For an interface crack problem, Lee and Shul [9] determined thermal stress intensity factors for an interface crack under vertical uniform heat flow for a two-dimensional thermoelasticity problem. Similar three-dimensional thermoelastic problems for a penny-shaped crack lying at the interface of a bi-material with imperfect and perfect contact near the crack front have been dealt with and closed-form solutions have been derived, respectively [10,11]. For a bimaterial periodically-layered space, a thermoelastic problem related to an interface crack under heat flow has been solved [12]. Due to a distinct interface of two bonded dissimilar materials, thermal stresses in a functionally graded material instead of a bonded bi-material have been studied [13–15]. Considering the drawback of the classical Fourier law, Sherief

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and El-Maghraby [16] used a generalized thermoelasticity coupling theory to analyze the response of thermal stresses for a penny-shaped crack. Wang and Han [17] used non-Fourier heat conduction to study a crack problem in a finite solid. Hu and Chen [18] applied dual-phase-lag theory to investigate transient heat conduction in a cracked half-plane. For smart materials and structures with crack, the analysis of thermal stresses have been made. Shang et al. [19] handled a symmetric thermopiezoelectric problem related to a penny-shaped crack. Similarly, for uniform heat flux, thermal stress analysis corresponding to an antisymmetric problem has been coped with and the exact solution has been derived for a penny-shaped crack by Wang and Noda [20]. Further investigation for a magneto-electro-thermo-elastic material with a penny-shaped crack under uniform heat flow can be found in [21–23]. Ueda [24] treated a dynamic thermoelectroelastic response of a functionally graded piezoelectric strip with a penny-shaped crack. In the above-mentioned researches, the thermal boundary conditions at the crack surfaces are assumed either insulated or isothermal. As we know, due to the presence of crack, the temperature continuity cannot be met when crack is open. Meanwhile, air inside crack can transfer heat, but with thermal conductivity differing from that of the cracked medium. Accordingly, the above two cases are extremely ideal assumptions. Barber [25] considered a more realistic heat conductivity for a crack and solved a penny-shaped crack problem. The effect of the crack surface on thermal conductivity and thermal stress intensity factors were further analyzed by Hasselman [26] and Kuo [27]. Since air inside crack has lower heat conductivity than the cracked material, Lee and Park [28] analyzed a partially insulated interface crack under uniform heat flow. El-Borgi et al. [29] studied a partially insulated embedded crack in an infinite functionally graded medium under thermo-mechanical loading. Zhong and Lee [30] proposed a new thermal boundary condition at the crack surface, and solved a thermoelastic problem of a crack in a two-dimensional elastic medium. This paper further develops the partial insulation crack model and analyzes the effect of heat conduction of a penny-shaped crack on stress intensity factors in a three-dimensional thermoelastic material. By means of the Hankel transform technique, the problem is reduced to a system of coupled dual integral equations. Singular thermal stresses are derived and stress intensity factors are determined. The effects of the thermal conductivity of the air inside an opening crack on thermal stress intensity factors are presented graphically for a cracked solid. 2. Statement of the problem Consider a penny-shaped crack embedded in an infinite homogeneous isotropic thermoelastic solid which is subjected to simple uniform tensile loading and heat flux. The crack is located at the circular region in the xoy-plane, i.e. r 6 a; z ¼ 0, where ðr; u; zÞ is a system of cylindrical polar coordinates whose origin is at the crack center, as shown in Fig. 1. Here we assume that the elastic body under consideration is thermally conductive. With regard to the temperature change hðr; zÞ in the solid relative to the temperature of the solid in a state of zero stress and strain, it is required to satisfy Laplace’s equation

@ 2 h 1 @h @ 2 h þ þ ¼ 0; @r 2 r @r @z2

ð1Þ

in the steady state, where no thermal sources are assumed in the solid. Here an axisymmetric problem is considered for simplicity. In order to analyze thermal stresses in the cracked solid, in the absence of body forces, the equilibrium equations of stresses are

Fig. 1. Schematic of a penny-shaped crack embedded in a solid subjected to mechanical tension and heat flux.

@ rrr @ rrz rrr  rhh þ þ ¼ 0; @r @z r @ rrz @ rzz rrz þ þ ¼ 0; @r @z r

ð2Þ ð3Þ

where the stresses are related with the strains and temperature change by the following the constitutive relationships within the framework of the theory of linear thermoelasticity

@u Eah  ; @r 1  2m u Eah rhh ¼ ke þ 2l  ; r 1  2m @w Eah rzz ¼ ke þ 2l  ; @z 1    2m @u @w ; rrz ¼ l þ @z @r

rrr ¼ ke þ 2l

l being Lame’s constants, mE E k¼ ; l¼ ; 2ð1 þ mÞ ð1 þ mÞð1  2mÞ

ð4Þ ð5Þ ð6Þ ð7Þ

with k and

ð8Þ

and e being volume strain,

e¼

@u u @w þ þ : @r r @z

ð9Þ

where E is Young’s modulus, m Poisson’s ratio, a the coefficient of linear expansion, u and w are displacement components along the radial and axial directions, respectively, and rij stress tensor. In addition to the above differential equations, appropriate boundary conditions must be furnished. For elastic fields we have     rzz r;0þ ¼ rzz ðr; 0 Þ ¼ 0; rrz r;0þ ¼ rrz ðr;0 Þ ¼ 0; r < a; ð10Þ  þ  þ    rzz r;0 ¼ rzz ðr; 0 Þ; w r;0 ¼ wðr;0 Þ; u r;0þ ¼ uðr; 0 Þ r > a; ð11Þ   rrz r;0þ ¼ rrz ðr; 0 Þ; r P 0; ð12Þ

rzz ðr;zÞ ! r0 ; r2 þ z2 ! 1;

ð13Þ

where r0 is a prescribed constant. Notice that (10) implies that the crack surfaces are traction-free. Other boundary conditions indicate the continuity condition of elastic fields for crack-free part at z ¼ 0. As for the temperature change field, a general boundary condition usually reads

hðh  h0 Þ þ k

@h ¼0 @n

ð14Þ

where h is the heat transfer coefficient from the solid into external ambient at temperature h0 ; k is the heat conductivity, and n is the outward unit vector normal to the boundary of the solid. For an infinite cracked elastic solid, if heat flux far away from the crack is a constant, we can assume

qz ðr; zÞ ¼ q0 ;

r 2 þ z2 ! 1

ð15Þ

X.-F. Li, K.Y. Lee / International Journal of Heat and Mass Transfer 91 (2015) 127–134

where qz and q0 denote the heat flux and a prescribed heat flux value, respectively,

qz ðr; zÞ ¼ k

@h : @z

ð16Þ

For an opening crack full of air, if the crack is understood as an insulated medium, we have

  qz r; 0þ ¼ qz ðr; 0 Þ ¼ 0;

r < a:

we find that Eqs. (20) and (21) are automatically fulfilled. Consequently, the problem reduces to looking for two potential functions f and v through appropriate boundary conditions. Once they are determined, all physical quantities of interest to us are obtainable. In fact, the displacement components and temperature change are given by (25)–(27). In addition, elastic stresses and heat flux can be expressed in terms of f and v. They are

ð17Þ

In previous studies, an insulated crack is frequently assumed by many researchers such as [5,22,31], in particular for antisymmetric problem with prescribed uniform heat flux. In the present study, taking into account the fact that air inside crack has heat conductivity, although it is small enough as compared to those of most solids, it is therefore more reasonable to assume that

rrr

  qz r; 0þ ¼ qz ðr; 0 Þ ¼ qc ;

rzz

r
ð18Þ

instead of the zero heat flux boundary condition, where qc is governed by the following nonlinear relation of the temperature jump across the crack surfaces [30]

qc ¼ kc

  h r; 0þ  hðr; 0 Þ  þ ; w r; 0  wðr; 0 Þ

ð19Þ

kc being the heat conductivity of air inside the crack. 3. Solution of the problem In this section, we solve thermal stresses disturbed by a penny-shaped crack under uniform heat flux. Prior to presentation of the solution of the problem, let us first simplify the above equations. From the equilibrium equations (2) and (3) we have

u 1 @e 2ð1 þ mÞa @h þ  ¼ 0; r 2 1  2m @r 1  2m @r 1 @e 2ð1 þ mÞa @h r2 w þ  ¼ 0; 1  2m @z 1  2m @z

r2 u 

ð20Þ ð21Þ

where

r2 ¼

@2 1 @ @2 þ þ : @r 2 r @r @z2

ð22Þ

A usual treatment for such thermoelastic problems is finished via two steps. The first step is to determine the temperature change field, regardless of elastic fields, and after getting temperature change, the second step is to solve the stress response due to temperature change. Such a treatment is inconvenient in achieving concrete solution since in the second step determination of a desired solution is related to a system of nonhomogeneous partial differential equation subject to proper boundary conditions. In this paper, we present an approach for simultaneously determining temperature change field and thermal stresses according to given boundary conditions. To this end, similar to the well-known Love displacement potential function, let us introduce two new unknown potential functions f and v such that they satisfy biharmonic equation and harmonic equation, respectively,

r2 r2 f ¼ 0;

ð23Þ

2

r v ¼ 0:

@2f u¼ ; @r@z w ¼ 2ð1  mÞr2 f 

@ @z

ð28Þ



mr2 f 

" # @ @2f @2v 2 ¼ 2l ð1  mÞr f  2 þ 2lð1 þ mÞa ; @r @z @r@z

ð29Þ

ð30Þ

ð31Þ

qz ¼ k

@3v ; @z3

ð32Þ

qr ¼ k

@3v : @r@z2

ð33Þ

A straightforward check implies that the governing equations are all identically fulfilled. Additionally, if setting v ¼ 0, the well-known Love potential method is recovered for axisymmetric elastic problems, whereas if setting f ¼ 0, the above results reduce to the general solution of a pure temperature change field. As for a thermal stress problem related to the coupling of temperature and mechanical behavior, the above approach provides a method for simultaneously seeking temperature and elastic fields. It is pointed out that in the above derivation, the order of the governing equations is not increased. It infers that proper boundary conditions are enough to solve the problem in question. In order to obtain a desired solution, it is expedient to employ the Hankel transform technique. Performing the Hankel transform of the zeroth order to both sides of Eqs. (23) and (24) leads to two ordinary differential equations. After solving these equations and performing the Hankel transform one gets

Z

fðr; zÞ ¼

1

1

n2 Z 1 1

ðA1 þ 2mA2 þ znA2 Þenz J 0 ðnrÞdn;

ð34Þ

Benz J 0 ðnrÞdn;

ð35Þ

0

vðr; zÞ ¼

0

n2

for z P 0, where J n ðÞ is the nth order Bessel function of the first kind, A1 ; A2 and B are unknown functions in n to be determined through appropriate boundary conditions. Next we substitute (34) and (35) into (26) and (27), after some algebra yielding

u¼

Z

1

½A1  ð1  2mÞA2 þ znA2 enz J 1 ðnrÞdn;

ð36Þ

0

w¼

Z

1

½A1 þ 2ð1  mÞA2 þ znA2 enz J 0 ðnrÞdn 0

 2ð1 þ mÞa

ð25Þ @2f @v þ 2ð1 þ mÞa ; @z2 @z

2

 1 @f @2v  2lð1 þ mÞa 2 ; r @r @z " # @ @2f @2v ¼ 2l ð2  mÞr2 f  2 þ 2lð1 þ mÞa 2 ; @z @z @z

rhh ¼ 2l

rrz

! @2f @2v mr f  2  2lð1 þ mÞa 2 ; @r @z

@ ¼ 2l @z

ð24Þ

Then if choosing elastic displacement components and temperature change expressed in terms of f and v as follows

129

Z

1 0

Z

1

1 nz Be J 0 ðnrÞdn; n

Benz J 0 ðnrÞdn:

ð37Þ

ð26Þ

h¼

ð27Þ

Similarly, from (30)–(33) we obtain integral expressions for all of the stress components and heat flux. In particular, we have

0

ð38Þ

2

h¼

@ v ; @z2

130

X.-F. Li, K.Y. Lee / International Journal of Heat and Mass Transfer 91 (2015) 127–134

Z

1

where

nðA1 þ A2 þ znA2 Þezn J 0 ðnrÞdn Z 1 þ 2lð1 þ mÞa Benz J 0 ðnrÞdn; 0 Z 1 rrz ¼ 2l nðA1 þ znA2 Þezn J 1 ðnrÞdn 0 Z 1 þ 2lð1 þ mÞa Benz J 1 ðnrÞdn; 0 Z 1 nBenz J 0 ðnrÞdn; qz ¼ k Z0 1 qr ¼ k nBenz J 1 ðnrÞdn:

rzz ¼ 2l

0

ð39Þ

ð40Þ ð41Þ

ð1  2mÞð1 þ mÞa B ; n 2ð 1  m Þ B U 2 ¼ A1 þ A2 þ ð1 þ mÞa : n

U 1 ¼ A1 þ

Z

1

Z0 1

To obtain thermal stresses in z  0, an analogous procedure if taking functions

Z0 1

Z

fðr; zÞ ¼

1

n2 1 1

0

Z

vðr; zÞ ¼

1

n2

0

0 nz

ðC 1 þ 2mC 2  znC 2 Þe J 0 ðnrÞdn;

ð43Þ

nz

De J 0 ðnrÞdn;

ð44Þ

allows us to get the displacement components and temperature change below

u¼

Z 0

Z

ð45Þ

1

0

ð46Þ ð47Þ

0

Meanwhile, the above expressions are inserted into constitutive relationships to derive the stress components and heat flux. For instance, one has

rzz ¼ 2l

ð60Þ

U 2 J 0 ðnrÞdn ¼ 0;

r > a;

ð61Þ

BJ 0 ðnrÞdn ¼ 0;

Z

1

1 nz

nðC 1 þ C 2  znC 2 Þe J 0 ðnrÞdn Z 1 Denz J 0 ðnrÞdn; þ 2lð1 þ mÞa 0 Z 1 rrz ¼ 2l nðC 1  znC 2 Þenz J 1 ðnrÞdn 0 Z 1  2lð1 þ mÞa Denz J 1 ðnrÞdn; 0 Z 1 qz ¼ k nDenz J 0 ðnrÞdn; Z 10 nDenz J 1 ðnrÞdn: qr ¼ k 0

ð48Þ

ð49Þ

Z0 1

nU 2 J 0 ðnrÞdn ¼ 

k

Z

2ð1 þ mÞaB ; n

r
sffiffiffiffiffiffi qc  q0 2 3=2 a J3=2 ðnaÞ; B¼ k pn sffiffiffiffiffiffi r0 2 3=2 a J 3=2 ðnaÞ; U2 ¼  2l p n sffiffiffiffiffiffi ðqc  q0 Þð1 þ mÞa 2 5=2 U1 ¼  a J 5=2 ðnaÞ; 6kð1  mÞ pn

ð63Þ ð64Þ ð65Þ

ð66Þ ð67Þ ð68Þ

where in deriving the above solution, we have used some integrals of Bessel functions, listed in Appendix A. Recalling the known integral identities, listed in Appendix A, and considering that qc is not determined yet, from (66)–(68) one first gets

Mh ¼

Application of the boundary conditions rzz r; 0 ¼ rzz ðr; 0 Þ;  þ   rrz r; 0 ¼ rrz ðr; 0 Þ; qz r; 0þ ¼ qz ðr; 0 Þ for any r P 0 leads to

;

r
Thus Eqs. (60)–(65) form a system of coupled dual integral equations. Based on a multiplying-factor approach [32], solving the above system of coupled dual integral equations, we obtain the desired solution to be

ð51Þ þ

r0 2l

nBJ 0 ðnrÞdn ¼ qc  q0 ;

4ð1  mÞr0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2  r 2 ;

ð69Þ

4ðqc  q0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2  r2 : pk

ð70Þ

Mw ¼ 

ð62Þ

1

ð50Þ

0

r > a:

  ð1 þ mÞa B J ðnrÞdn ¼ 0; n U1 þ 2ð1  mÞ n 1

0

nz

Z

r > a;

0

½C 1 þ 2ð1  mÞC 2  znC 2 e J 0 ðnrÞdn Z 1 1 nz De J 0 ðnrÞdn; þ 2ð1 þ mÞa n 0 Z 1 h¼ Denz J 0 ðnrÞdn:

w¼

U 1 J 1 ðnrÞdn ¼ 0;

In addition, applying the remaining boundary conditions at the crack surface, we get

1

½C 1  ð1  2mÞC 2  znC 2 enz J 1 ðnrÞdn

ð59Þ

Making use of the continuity condition of the displacement components and temperature change for crack-free part at z ¼ 0, viz. Mu ¼ Mw ¼ Mh ¼ 0 for r > a, we have

ð42Þ

0

ð58Þ

pl

Hence qc can be gained by using (19) to be

lkc q0 : lkc þ ð1  mÞkr0

ð52Þ

qc ¼

C 1 ¼ A1 ;

ð53Þ

D ¼ B:

ð54Þ

From the above-obtained result, we find that the heat flux inside an opening crack qc is independent of the radial coordinate r, although the nonlinear relation at the right-hand side of Eq. (19) is related to r. However, qc nonlinearly depends on applied tensile stress r0 ðr0 > 0Þ. This is easily understood since if no external tensile loading is exerted, a crack with vanishing thickness does not open, which infers that heat flux does not change at all as if the crack is absent, viz. qc ¼ q0 follows from (71), as expected. In this case, uniform heat flux in the solid does not make a crack open. In other words, crack opening is only caused by applied tensile loading. This is different from a symmetric thermoelastic problem where uniform heat flux or temperature is solely prescribed on the

C 2 þ A2 ¼ 2A1 

Subtracting the displacement components and temperature at z ! 0 from their counterparts at z ! 0þ yields

  Mu ¼ u r; 0þ  uðr; 0 Þ ¼ 4ð1  mÞ

Z

1

U 1 J 1 ðnrÞdn;

ð55Þ

0

Z

1

  U 2 J 0 ðnrÞdn; Mw ¼ w r; 0þ  wðr; 0 Þ ¼ 4ð1  mÞ 0 Z 1  þ Mh ¼ h r; 0  hðr; 0 Þ ¼ 2 BJ0 ðnrÞdn; 0

ð56Þ ð57Þ

ð71Þ

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X.-F. Li, K.Y. Lee / International Journal of Heat and Mass Transfer 91 (2015) 127–134

crack surface, which causes a crack to open (e.g. [4]). Moreover, the present problem cannot be solved by superposition because of the nonlinearity characteristic of Eq. (71). Furthermore, owing to ð1  mÞkr0 > 0, a practical heat flux inside an opening crack is always less than applied uniform heat flux. With the above obtained results, the entire thermal stress field of a cracked solid can be determined. The displacement components and temperature can be derived by substituting (66)–(68) into (36)–(38). Making use of some known results involving infinite integrals of Bessel functions, which are listed in Appendix A, explicit expressions for the elastic displacements and temperature change are obtainable. For example, after superposing a temperature field corresponding to uniform heat flux without crack, the temperature change in the whole cracked solid is   qffiffiffiffiffiffiffiffiffiffiffiffiffiffi  q z 2ð1  mÞr0 q0 2 1 a h¼ 0 þ z sin  a2  l1 ; 8z 2 ð1; 1Þ; k p½lkc þ ð1  mÞkr0  l2

ð72Þ where the sign  takes  for z P 0, and þ for z < 0, respectively, and

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðr þ aÞ2 þ z2  ðr  aÞ2 þ z2 ; 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðr þ aÞ2 þ z2 þ ðr  aÞ2 þ z2 : l2 ¼ 2

l1 ¼

Note that limz!0

ð73Þ ð74Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 a2  l1 ¼ 0 as r > a and limz!0 a2  l1 ¼ a2  r 2

as r < a. It is readily found that the temperature change is continuos except for the crack region, and has a temperature difference across the crack. Moreover, this temperature difference is related to crack opening displacement. Once applied tensile stress disappears, crack is closed and a continuously varying temperature retrieves. In this case, with the aid of the integrals of Bessel functions (Appendix A), the heat flux vector in the whole solid can be evaluated as

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi3   2 a l2  a2 2ð1  mÞkr0 q0 4 1 l1 5;  2 qz ¼ q0  sin 2 p½lkc þ ð1  mÞkr0  r l2  l1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 al1 a2  l1 2ð1  mÞkr0 q0

: qr ¼  p½lkc þ ð1  mÞkr0  l2 l2  l2 2 1

rzz ðr; 0Þ ¼ (

rrz ðr; 0Þ ¼

ð75Þ

ð76Þ

At z ¼ 0, after a little manipulation the relative sliding of the radial displacement component is

ð77Þ

Obviously, we find that although crack opening displacement, given by (69), and crack relative sliding displacement (77) are both present, the mechanism of the occurrence of them is different. Crack opening displacement is caused by applied tensile stress, while crack relative sliding displacement Mu is produced by heat flux and tensile mechanical loading. From (77), Mu is apparently dependent on both r0 and q0 besides the material properties involved. In assessing the safety and integrity of a structure, of much significance is induced thermal stresses for the problem under consideration. Although full thermal stress field can be derived explicitly, here we omit lengthy expressions and only give the components of thermal stress rzz and rrz in z ¼ 0. After some calculations and making use of the known integrals, we obtain rzz and rrz to be

h

i 1 a 2r0 pffiffiffiffiffiffiffiffiffi a  sin ; 2 2 r p r a r0 ;

mÞaar0 q0  3pl½lð1þ kc þð1mÞkr0 

0;

r > a;

ð78Þ

r < a; a2 pffiffiffiffiffiffiffiffiffi ; r r 2 a2

r > a;

ð79Þ

r < a:

4. Numerical results and discussion In the foregoing section, expressions for the temperature change and the thermal stress responses have been obtained. This section is devoted to numerical calculation to show the influence of air as a conducting medium inside crack on the thermal stresses, in particular for the thermal stresses near the front of the penny-shaped crack. 4.1. Partial insulation coefficient Prior to the presentation of thermal stress distribution induced by a penny-shaped crack, let us first evaluate heat flux inside the crack. In other words, the crack is understood as a conducting medium, not a completely insulated medium since air of the crack interior has nonvanishing thermal conductivity, although it is low enough. On the other hand, the thermal conductivity of air kc ¼ 0:024 Wm1K1 is much less than that of most materials. Table 1 lists the material properties of several typical materials. Therefore, when heat flux passes a crack, the crack behaves unlike an insulated medium or a material itself. It is an intermediate between an insulation and the material itself. Consequently, based on the above result, we can define qc =q0 as the partial insulation coefficient g, which was taken as a known parameter in [28]. It is clear that in the present analysis g nonlinearly depends on applied mechanical loading and heat flux through the following formula

g¼

2

Z sffiffiffiffiffiffi 2ðqc  q0 Þð1 þ mÞa 1 2 5=2 a J 5=2 ðnaÞJ 1 ðnrÞdn Mu ¼ 3k pn 0   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 1  m2 ar0 q0 r a2  r 2 : ¼ 3p½lkc þ ð1  mÞkr0 

(

kc =k : kc =k þ 2ð1  m2 Þr0 =E

ð80Þ

Moreover, the partial insulation coefficient g reduces to zero if imposing the thermal conductivity of air to vanish, and the crack collapses to a completely insulated crack. In addition, if applied mechanical loading is absent, g ¼ 1 is implied and the crack surfaces in this case are isothermal. Once the crack is open, a realistic situation is that the crack is neither completely insulated nor isothermal at the crack surfaces. Therefore, partial insulation is more reasonable to simulate a realistic case. For the materials listed in Table 1, the ratio of k=kc ranges from 37.5 to 16250. Fig. 2 shows the variation of the partial insulation coefficient g against r0 =E for several different materials. From Fig. 2 one observes that for given applied loading, the partial insulation coefficient g may take different values, depending on chosen materials. This is easily interpreted from (80) because the crack is in close proximity to an isothermal crack for small enough applied loading r0 or equivalently r0 =E  kc =k, while the crack is in close proximity to an insulated crack for r0 =E  kc =k. Specially, from Fig. 2 it is viewed that for cracked glass, the crack is nearly isothermal crack if r0 =E < 103 , whereas for cracked copper, the crack may be approximately

Table 1 Properties of some materials.

Cu Al Pb Steel (stainless) Glass

k (Wm1 K1)

a (106 K1)

E (GPa)

m

k=kc

390 205 35 17 0.9

17 23.1 29 17.3 8.5

117 69 14 180 55

0.35 0.34 0.44 0.31 0.25

16250 8541.7 1458.3 708.33 37.5

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Fig. 2. Partial insulation coefficient g against

r0 =E for different materials.

Fig. 4. Comparison of temperature change field for an insulated and partially insulated penny-shaped crack in metal material aluminum, where solid lines correspond to a partially insulated crack, and dashed lines correspond to an insulated crack.

understood as an insulated crack if r0 =E > 104 . Fig. 3 displays the variation of the partial insulation coefficient g against kc =k if taking

r0 =E ¼ 104 ; 2 104 ; 5 104 . 4.2. Distribution of temperature change Now for a cracked thermoelastic solid, we compare the temperature change for an insulated or partially insulated crack. By superposing the temperature field corresponding to uniform heat flux, we get the entire temperature change as

2 6 2 h¼6 4  p 1 þ 2ð1Ekm2cÞkr0  

3   7 q0 z a  sin1  17 5 k l2

2

p 1 þ 2ð1Ekm2cÞkr0



q0 k

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2  l1

ð81Þ

where we have used the relation ar ¼ l1 l2 . Figs. 4 and 5 give a comparison of the temperature change in a cracked aluminum and lead solid, respectively, where solid lines correspond to the temperature distribution for a partially insulated crack, and dashed lines correspond to that for a completely insulated crack. In calculating temperature distribution, we have taken

r0 =E ¼ 104 . From Figs. 4

Fig. 5. Comparison of temperature change field for an insulated and partially insulated penny-shaped crack in metal material lead, where solid lines correspond to a partially insulated crack, and dashed lines correspond to an insulated crack.

and 5, we find that an opening crack indeed conducts heat through air inside the crack, and an insulation assumption is not suitable, in particular for materials with low thermal conductivity. For example, in Fig. 4, the temperature change in a solid made of aluminum with a penny-shaped crack subjected to uniform heat flux has an apparent difference when adopting the insulation and partial insulation assumptions. Furthermore, if aluminum is replaced by lead with lower thermal conductivity, a greater difference of the temperature distribution is viewed in Fig. 5 as compared to that for an insulated crack. In this case, temperature distribution is almost inconsistency with that without crack, i.e. the crack surfaces are isothermal. 4.3. Thermal stress intensity factors By inspection, not only the normal stress rzz but also shear stress rrz are singular near the boundary of the crack. However, the singularity of rzz results from applied tensile stress r0 , whereas the singularity of rrz originates from both heat flux and applied tensile stress. If we define the stress intensity factors (SIFs) near the crack front by

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pðr  aÞrzz ðr; 0Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K II ¼ limþ 2pðr  aÞrrz ðr; 0Þ;

K I ¼ limþ r!a

Fig. 3. Partial insulation coefficient g against kc =k with different values of

r0 =E.

r!a

ð82Þ ð83Þ

X.-F. Li, K.Y. Lee / International Journal of Heat and Mass Transfer 91 (2015) 127–134

133

we get

KI ¼

2r0

K II ¼ 

pffiffiffiffiffiffi pa

p

ð84Þ

;

lð1 þ mÞaar0 q0 3p½lkc þ ð1  mÞkr0 

pffiffiffiffiffiffi pa:

ð85Þ

It is clear that K I depends only on r0 , and does not depend on the material properties. On the contrary, for K II , not only applied mechanical loading but also thermal loading affect K II . In addition, K II is also related to the material properties such as l; m and k; kc . Since kc stands for the thermal conductivity of air inside an opening crack, the effect of air medium inside an opening crack has been taken into account. In particular, if setting kc ¼ 0, meaning that a crack is thermally insulated, we gain thermal SIFs as

K in II ¼ 

lð1 þ mÞaaq0 pffiffiffiffiffiffi pa 3pð1  mÞk

ð86Þ

identical to the classical result [5]. In addition, if setting r0 ¼ 0, one readily views K II ¼ 0. This is due to the fact that a crack is not open in the absence of tensile external loading and it is thermally conducting as if the crack is absent. Nonetheless, when a crack is open due to external loading, singular shear stress and crack relative sliding displacement can be induced by thermal effect. Furthermore, the thermal SIF K II linearly depends on the expansion coefficient a and heat flux q0 . Occurrence of K II is attributed to the fact that the temperature change is antisymmetric with respect to the crack plane, which gives rise to expansion of the material above and below the crack being not symmetric. Fig. 6 examines a dimensionless crack relative sliding displacement at the crack surface,  ¼ kMu=ð1 þ mÞaq0 a2 , for various values of b where b is defined Mu as a competition factor ðkc =kÞ=ðr0 =EÞ. Note that b ¼ 0 corresponds to an insulated crack, and b ! 1 corresponds to an isothermal crack. For a partially insulated crack, b takes a finite value. Obviously, at the crack center and crack front, there is no any relative sliding displacement, as expected. However, relative sliding displacement exists for other position in the penny-shaped crack: 0 < r < a; z ¼ 0. Consequently, shear stress is induced and moreover it is singular near the crack front. The ratio of thermal SIFs K II to K in II as a function of kc =k is presented in Fig. 7 for different values of r0 =E with Poisson’s ratio m ¼ 0:3. From Fig. 7, we find that the thermal SIF K II values are determined by applied tensile stress and the ratio of the thermal conductivity of crack interior medium to the material’s conductivity. Taking into account the thermal conductivity of the crack interior being a nonvanishing constant, even very low, kc =k

Fig. 7. Ratio of thermal SIFs K II =K in II versus kc =k for two different values of

r0 =E.

Fig. 8. Ratio of thermal SIFs K II =K in II as a function of ðr0 =EÞ=ðkc =kÞ.

is then fixed and also not equal to zero. The ratio of thermal SIFs K II to K in II against ðr0 =EÞ=ðkc =kÞ is demonstrated in Fig. 8. From Fig. 8, the thermal SIFs K II vanish if no external stress is exerted. Once tensile loading is applied, induced singular thermal stress takes place and nonvanishing thermal SIFs increases with r0 rising. Finally, when applied stress is sufficiently large or the ratio of thermal conductivity of the crack interior is much lower than that of the cracked solid, the thermal SIFs are very close to those corresponding to the insulation crack assumption. Otherwise, the thermal SIFs are severely lower than the counterpart results of the insulation crack. 5. Conclusions

Fig. 6. Dimensionless crack relative sliding displacement between two crack surfaces for different ratios of b.

Considering heat conduction of the crack interior, this paper developed a partial insulation crack model and analyzed singular thermal stresses induced by a penny-shaped crack embedded in an infinite homogeneous isotropic thermoelastic solid under remote tensile loading and heat flux. The Hankel transform technique was employed to reduce the mixed boundary value problem to a system of coupled dual integral equations. Explicit expressions for thermoelastic fields have been obtained. Some conclusions are drawn as follows.

134

X.-F. Li, K.Y. Lee / International Journal of Heat and Mass Transfer 91 (2015) 127–134

Partial insulation coefficient is nonlinearly dependent on applied mechanical loading and heat flux. It strongly affects singular thermal stresses near the crack front.

Without applied tensile loading, the partial insulation crack model reduces to an isothermal crack model. A closed crack does not result in singular thermal stresses when vertical heat flux is applied.

When tensile loading is applied, induced thermal stresses are overestimated for an insulated crack. The heat conduction of crack interior has an obvious effect on thermal stress intensity factors.

Thermal stress intensity factors depend both on applied mechanical loading and heat flux. Moreover, the material properties affect thermal stress intensity factors.

Penny-shaped insulated or isothermal cracks are two limiting cases of the present results. Conflict of interest None declared. Acknowledgements This work was supported by the Open Foundation of State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, P.R. China (No. GZ1308), the Faculty Research Foundation of Central South University, P.R. China (No. 2013JSJJ020). Appendix A In deriving explicit expressions for desired thermoelastic fields, the integrals of Bessel functions have been utilized

Z

pffiffiffi ecn J 1 ðnrÞJ 5=2 ðnaÞ ndn 0 " ! qffiffiffiffiffiffiffiffiffiffiffiffiffi#   1 2a2 2 1 l1 2 p ffiffiffiffiffiffi ffi  3 þ 2 2 l1 r 2  l1 3r sin ¼ r 2pa5=2 r l2  l1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Z 1 pffiffiffi 2l a2  l1 1

; ecn J 1 ðnrÞJ 3=2 ðnaÞ ndn ¼ pffiffiffiffiffiffiffi 2 2 0 2pa3=2 r l2  l1 Z 1 dn ecn J 1 ðnrÞJ 3=2 ðnaÞ pffiffiffi n 0  qffiffiffiffiffiffiffiffiffiffiffiffiffi   1 2 1 l1 2 ¼ pffiffiffiffiffiffiffi ; l1 r  l1 þ r 2 sin r 2pa3=2 r Z 1 pffiffiffi ecn J 0 ðnrÞJ 3=2 ðnaÞ ndn 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2   2 a l2  a2 2 l 1 1 4sin  2 2 5; ¼ pffiffiffiffiffiffiffi r 2pa3=2 l2  l1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi   Z 1 dn 2 2 1 l1 ; ecn J 0 ðnrÞJ 3=2 ðnaÞ pffiffiffi ¼ pffiffiffiffiffiffiffi a2  l1  c sin r n 2pa3=2 Z0 1 pffiffiffi ecn J 0 ðnrÞJ 3=2 ðnaÞn ndn 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi



3 2 2 2 2 2 2 a2  l1 l2 þ l1 2 a2  l1 2 5 ¼ pffiffiffiffiffiffiffi

2 43l1 þ 2 2 2 2 l2  l1 2pa3=2 l2  l1 1

for ReðcÞ > jImða  bÞj.

ðA1Þ

ðA2Þ

ðA3Þ

ðA4Þ ðA5Þ

ðA6Þ

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