Exact fundamental thermo-elastic solutions of a transversely isotropic elastic medium with a half infinite plane crack

International Journal of Mechanical Sciences 59 (2012) 83–94

Contents lists available at SciVerse ScienceDirect

International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

Exact fundamental thermo-elastic solutions of a transversely isotropic elastic medium with a half infinite plane crack X.Y. Li n School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, 610031, PR China

a r t i c l e i n f o

abstract

Article history: Received 30 October 2011 Received in revised form 20 December 2011 Accepted 20 March 2012 Available online 3 April 2012

This paper presents three-dimensional (3D) exact fundamental solutions of the thermoelastic field in a transversely isotropic elastic medium weakened by a half infinite plane crack subjected to a pair of point thermal loadings symmetrically acting on the crack surface. In view of the symmetric condition, the original problem is transformed into a mixed boundary value problem of a half space. By means of the general solutions conjugated with the generalized potential theory method, the problem is exactly solved and the corresponding Green’s functions are derived, for the first time. Complete and exact fundamental solutions are expressed in terms of elementary functions. The singular behavior of the crack tip is discussed and the stress intensity factor is given explicitly. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Fundamental solution Thermoelastic field Transverse isotropy Potential theory method Half infinite plane crack

1. Introduction As an important branch in solid mechanics, the study of thermoelastic problems has long been prevailing in the literature; for example, wave propagation [1,2], inclusion [3], contact problems [4–6], dislocation [7], heat transfer [8], and so on. In particular, the thermoelastic crack problem subjected to various types of external thermal loadings has been discussed extensively in the scientific community [9–17]. Most of the previous analytical works treated the axisymmetric [9–13] or two dimensional cases [14,15]. In the past crack analyses, integral transform method [10,12,13] or theory of dual/triple integral equations [11,16,17] were usually adopted. It is noted that there exist quite few reports concerning the non-axisymmetric problems within the framework of thermo-elasticity in literature. Kellogg [18] pointed that the potential theory method is a tool for studying problems governed by Laplace equations in several physic areas. This method is extended by Fabrikant [19,20] who creatively represented the reciprocal of the distance between two points in Euclidean space by an integral. Such a representation of the distance lends to various non-classic 3D elastic solutions for mixed boundary value problems arising in crack and punch problems, by using the potential theory along with the general solutions proposed by Elliott [21] for pure elasticity. The potential theory was further generalized by Chen and his coauthors for

n

Corresponding author. Tel.: þ86 28 87634181; fax: þ 86 28 87600797. E-mail address: [email protected]

0020-7403/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2012.03.007

crack and contact problems in multi-coupling disciplines [22], for instance, piezo-elasticity [23], thermo-elasticity [24], thermopiezo-elasticity [25], magneto-electro-thermo-elasticity [26], thermo-poro-elasticity [27], to name a few. In particular, the penny-shaped crack problem in Ref. [27] is axisymmetric, since the crack is subjected to pairs of identical mechanical, thermal and pressure loadings which are uniformly distributed on the upper and lower crack surfaces. From the mathematical point of view, the structures of governing equations for crack problems in piezoelasticity and magnetoelectroelasticity are identical to their counterparts in pure elasticity [24]. However, this is not so when the thermal effect is taken into count; certain new features appear, making some challenges to the potential theory method [24,28]. It is noted that most works (Refs. [22–27], among others) associated with crack problems mentioned above were dedicated to pennyshaped cracks. Half infinite crack has attracted a lot of attention from numerous scholars [29–34]. It is interesting to note that potential theory method was already employed to develop 3D non-axisymmetric analyses for half infinite plane cracks subjected to external loadings applied at an arbitrary point on the crack surfaces in elasticity [33], piezoelectricity [34] and piezo-thermo-elasticity [35]. However, there is no report yet in academic records, on the non-axisymmetric analyses within the framework of thermoelasticity for half infinite plane crack. This paper aims to make exact and complete 3D analyses for transversely isotropic media containing a half infinite plane crack subjected to temperature loads symmetrically applied on the

84

X.Y. Li / International Journal of Mechanical Sciences 59 (2012) 83–94

upper and lower crack surfaces. The original problem is turned into a mixed boundary value problem in view of the symmetry inherent to the problem. The transformed problem is solved by the generalized potential theory method conjugated with the general solutions. A new potential function is introduced to consider the thermal effect. The integral and integro-differential equations involved in the present study are, respectively similar to the governing equations for punch and crack problems in pure elasticity. The governing equations are solved by directly employing the results available in the literature. For a point temperature load, the corresponding Green’s functions along with their derivatives of various orders are derived. Exact and complete fundamental solutions are constructed in terms of elementary functions, for the first time. The singular behavior of the crack tip is examined and the stress intensity factor is given explicitly. For an arbitrary distributed thermal load, the stress intensity factor can be determined by an integral and an application of the current fundamental solutions has been presented for a particular plane strain problem. In the present study, the temperature field is derived via two different ways and a perfect agreement is achieved. In view of the merits mentioned above, the present solution can serve as a benchmark to various simplified analyses and numerical codes.

2. Steady state general solutions for thermoelasticity In Cartesian coordinate system Oxyz, the constitutive equation for transversely isotropic media with the isotropic plane perpendicular to the z-axis is described by the Duhamel–Neumann relations [12,24]   @u @v @w @v @w þ , sx ¼ c11 þ c12 þ c13 b1 T, tyz ¼ c44 @x @y @z @z @y   @u @v @w @u @w þ , sy ¼ c12 þ c11 þ c13 b1 T, tzx ¼ c44 @x @y @z @z @x     @u @v @w @u @v þ þ c33 b3 T, txy ¼ c66 þ , ð1Þ sz ¼ c13 @x @y @z @y @x where bi and cij are, respectively thermal moduli and elastic constants with the identity 2c66 ¼c11  c12 holding; u(v,w) and sij(si,tij) are the displacement and stress components, respectively; T is the temperature variation with T¼0 corresponding to the stress-free state. Substitution of the constitutive relations in Eq. (1) into the equilibrium equations (sji,j ¼0) gives rise to ! c11 þ c66 @2 c c @w D þ c44 2 U þ 11 66 L2 U þ ðc13 þ c44 ÞL b1 LT ¼ 0, 2 2 @z @z c44 D þ c33

! @2 c13 þ c44 @ @T ðLU þ LUÞb3 ¼ 0, wþ 2 @z @z @z2

ð2Þ

L ¼@/@xþi@/@y, D ¼ LL ¼ @2 =@x2 þ @2 =@y2 , U ¼ u þivði ¼ where p ffiffiffiffiffiffiffi 1Þ and the over bar represents the complex conjugate of the indicated quantity. The temperature field for the medium in a steady-state is governed by ! @2 k11 D þk33 2 T ¼ 0 ð3Þ @z where kij is the thermal conductivity coefficient. The general solutions to Eqs. (2) and (3) proposed by Chen et al. [24] are u¼

3 @c @c0 X j  , @y @x j¼1

v¼

3 @c @c0 X j  , @x @y j¼1

w¼

3 X j¼1

ai1

@cj , @zj

T¼

3 X

ai2

j¼1

@2 cj @z2j

,

ð4Þ

where cj(j ¼0,1,2,3) are quasi-harmonic functions ! @2 D þ 2 cj ¼ 0, ðj ¼ 0,1,2,3Þ; @zj

ð5Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c66 =c44 , zj ¼zsj and sj(j¼ 0,1,2,3) are material eigenvalues; s0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s3 ¼ k11 =k33 , s1 and s2 are the roots with a positive real part of the following algebraic equation a0 s4 b0 s2 þ c0 ¼ 0:

ð6Þ

It is noted that Eq. (4) is valid only in the case of distinct material eigenvalues. Introducing the following complex variables

s1 ¼ sx þ sy , s2 ¼ sx sy þ2itxy , tz ¼ txz þ ityz ;

ð7Þ

one can derive the thermoelastic field in the compact form as follows 0 1 3 3 X X @2 c cj þ ic0 A, sz ¼ gj1 2 j , U ¼ L@ @zj j¼1 j¼1 0 1 3 3 X X @2 cj 2@ s1 ¼ gj2 2 , s2 ¼ 2c66 L cj þ ic0 A, @zj j¼1 j¼1 0 1 3 X @cj @ c 0A : ð8Þ tz ¼ L@ gj3 is0 c44 @zj @z0 i¼j The constants aij, a0(b0,c0) and gij involved in Eqs. (4), (6) and (8) are given in Appendix A. Early researches [22] clearly revealed that the general solutions along with the potential theory method will definitely facilitate solving the mixed boundary value problems arising in the crack and contact analyses, especially the non-axisymmetric problems. Some non-classic 3D fundamental solutions were thus developed. To show the versatility of the potential theory method, an infinite body weakened by a half-infinite plane crack is considered in the next section.

3. Generalized potential method for half-infinite plane crack Consider an infinite transversely isotropic thermoelastic body containing a half-infinite plane crack whose surface is parallel to the plane of isotropy (see Fig. 1). The coordinate system is established such that the xoy plane is coincident with the crack surface, and the origin O locates at the edge of the crack. Two symbols S and S are introduced to denote the regions on the plane z¼0(denoted by I), respectively occupied by the crack and its complement and S {(x,y)90 ryoN,  Nox oN}, S [ S ¼ I and S \ S ¼ |, implying no intersection and separation. A pair of z y

N0(x0, y0) O

S N (x, y)

x O

(y0, 0+) (y0, 0─)

Fig. 1. Horizontal (a) and vertical (b) cross sections of a half infinite crack.

y

X.Y. Li / International Journal of Mechanical Sciences 59 (2012) 83–94

85

arbitrary thermal loads Y(x,y) is symmetrically applied on the upper and lower crack faces. Making use of the symmetric condition, one can transform the original problem to a mixed boundary value problem of the half-space z Z0 with the following boundary conditions on the plane z ¼0

Substituting Eq. (15) into Eq. (11)1 and changing the order of integration, one can arrive at Z þ1 Z þ1 g s3 H1 ðx,y,zÞ ¼ 12 2 dx0 P1 ðx,y,z; x0 ,y0 ÞYðx0 ,y0 Þdy0 , ð18Þ g 11 p 1 0

ðx,yÞ A S, sz ¼ 0, T ¼ Yðx,yÞ,

where P1(x,y,z; x0,y0) is a Green’s function with the expression Z þ1 Z þ1 pffiffiffiffiffiffiffiffi 2 yy0 1 dy0 : P1 ðx,y,z; x0 ,y0 Þ ¼ dx0 arctan ð19Þ R RR0 1 0

ðx,yÞ A S, w ¼ 0,

@T ¼ 0, @z

ðx,yÞ A I, tz ¼ 0:

ð9Þ

In order to generalize the potential method to thermoelasticity, it is assumed that

c0 ¼ 0, ci ðx,y,zÞ ¼

2 X

hij Hj ðx,y,zi Þ,ði ¼ 1,2,3Þ

ð10Þ

j¼1

where hij are constants to be determined, and R þ1 R þ1 H1 ðx,y,zÞ ¼ 1 dx0 0 oðxR00,y0 Þ dy0 ,   R þ1 R þ1 H2 ðx,y,zÞ ¼ 1 dx0 0 Wðx0 ,y0 Þ zln½R0 þ zR0 dy0 ,

p

ð11Þ

where the kernels of the potential of simple layer o and W are crack open displacement w(x,y,0 þ ) and temperature gradient qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @T=@z9z ¼ 0 þ , respectively; R0 ¼ ðxx0 Þ2 þ ðyy0 Þ2 þz2 is the distance between the points (x,y,z) and (x0,y0,0). Chen et al. [24] showed that the second and third boundary conditions in Eq. (9) for plane cracks with arbitrary shape are met if the constants hij satisfy 9 9 8 2 3 8 0 > h g13 g23 g33 1 > > = = < 1j > < d1j ,ðj ¼ 1,2Þ h2j ¼  1 6 a21 a31 7 ð12Þ 4 a11 5 > > > 2 p ; :h ; :d > a12 s1 a22 s2 a32 s3 2j 3j where dij is the Kronecker delta. The satisfaction of the first condition in Eq. (9) leads to the following two equations: Z þ1 Z þ1 Wðx0 ,y0 Þ dy0 ¼ 2ps3 Yðx,yÞ dx0 ð13Þ R 1 0 Z

þ1

Z

þ1

oðx0 ,y0 Þ

g 12 Yðx,yÞ  Yg ðx,yÞ ð14Þ R g 11 1 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 3 R ¼ ðxx0 Þ2 þ ðyy0 Þ2 and where g 1j ¼ ii ¼ ¼ 1 gi1 hij , ðj ¼ 1,2Þ, Yg(x,y) is seen as a generalized mechanical load. It is seen that the integral equation (13) is similar to the governing equation for punch problems while the integro-differential equation (14) similar to that for crack problem in pure elasticity. Due to the similarity, the solutions to Eqs. (13) and (14) can be obtained. This will be discussed in the next section.

D

dx0

It is very hard to express P1(x,y,z; x0,y0) in terms of elementary function. However, its derivatives necessary to construct the thermoelastic field can be obtained explicitly, and are listed in Appendix B. Inserting Eq. (16) into Eq. (11)2 results in Z Z þ1 2s3 þ 1 H2 ðx,y,zÞ ¼  dx0 P2 ðx,y,z; x0 ,y0 ÞYðx0 ,y0 Þdy0 : ð20Þ

dy0 ¼ 2ps3

where Kðx,y,z; x0 ,y0 Þ ¼

Z

y 0

dy1 d Ln ð2y1 Þ pffiffiffiffiffiffiffiffiffiffiffi yy1 dy1

Z

þ1 y1

dy0 pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Ln ðy0 ÞYðx0 ,y0 Þ y0 y1

ð22Þ

5. Exact and complete fundamental solutions In view of the structure of Eq. (10), in what follows, the thermoelastic field is divided into two parts, which correspond to the potential function H1 and H2, respectively. The physical quantity associated with H1 and H2 will be indicated by the superscripts (1) and (2), respectively. Assume the crack is subjected to a point temperature load Y ¼ Y0d(x  x0)d(y  y0) applied at the point (x0,y0,0 þ ). Here, the symbol Y0g ¼ 2ps3 g 12 Y0 =g 11 is introduced to denote the generalized concentrated force to facilitate the further discussion. By inserting Eq. (19) into Eqs. (4) and (8), the expressions for the first part can be derived

sð1Þ z ¼

3 2s3 g 12 Y0 X n h f ðz Þ, pg 11 j ¼ 1 j1 1 j

wð1Þ ¼

3 2s3 g 12 Y0 X g h f n ðz Þ , pg 11 j ¼ 1 j1 j1 3 j

sð1Þ 2 ¼

T ð1Þ ¼ ð17Þ

n

Note that the symbols h , t and l2 are defined in Eq. (B.2). Appendix B lists the derivatives of P2, some of which are completely new to literature. Once the derivatives of Pj(x,y,z; x0,y0) (j¼ 1,2) are available, one can readily construct the exact and complete fundamental solutions of thermoelastic field. This is the topic of the next section.

ð16Þ where Ln() is an operator with the following definition Z 1 þ 1 kf ðx0 ,yÞdx0 ,k4 0: Ln ðkÞf ðx,yÞ ¼ p 1 ðxx0 Þ2 þk2

2 !0:5 3  n  sffiffiffiffiffi @P2 1 h 2 4 1 it 5:  ¼  arctan Re pffiffiffiffi arctan n R0 y0 @z R0 2l2 it

n

U ð1Þ ¼ 

s Ln ðyÞ d Wðx,yÞ ¼ 3 p dy

0

It is evident that P2(x,y,z; x0,y0) is again a Green’s function, which is very difficult to express explicitly. Fortunately, P2 can be expressed by an integral, whose integrand is given by Fabrikant and Karapetian [33], i.e., Z P2 ðx,y,z; x0 ,y0 Þ ¼ Kðx,y,z; x0 ,y0 Þdz ð21Þ

4. Potential functions Comparing with the results in Ref. [33], one can derive the solutions to Eqs. (13) and (14) as Z þ1 Z þ1 pffiffiffiffiffiffiffiffi 2 yy0 g s Yðx0 ,y0 Þ dy0 , arctan oðx,yÞ ¼ 12 32 dx0 ð15Þ R R g 11 p 1 0

1

3 4c66 s3 g 12 Y0 X n hj1 f 4 ðzj Þ, pg 11 j¼1

3 2s3 g 12 Y0 X a h f n ðz Þ, pg 11 j ¼ 1 j1 j1 2 j

sð1Þ 1 ¼

3 2s3 g 12 Y0 X g h f n ðz Þ, pg 11 j ¼ 1 j2 j1 3 j

tð1Þ z ¼

3 2s3 g 12 Y0 X g h f n ðz Þ, pg 11 j ¼ 1 j3 j1 5 j

3 3 Y0 X 2s3 g 12 Y0 X aj2 hj1 f n3 ðzj Þ ¼  2g a h f n ðz Þ, pg 11 j ¼ 1 p j ¼ 1 j2 j1 3 j n

ð23Þ

where f j ðzÞðj ¼ 1,. . .,5Þ are closely related to the derivatives of the Green’s function P1 and are defined by Eq. (B.1)

86

X.Y. Li / International Journal of Mechanical Sciences 59 (2012) 83–94

Similarly, one can obtain the expressions of the second part as 3 2s3 Y X 0

U

ð2Þ

¼

p

sð2Þ z ¼

sð2Þ 2 ¼

3 2s3 Y X 0

n

hj2 g 1 ðzj Þ,

ð2Þ

w

¼

p

j¼1

3 2s3 Y0 X

p

gj1 hj2 g n3 ðzj Þ , sð2Þ 1 ¼

p

3 2s3 Y0 X

p

hj2 g n4 ðzj Þ,

tð2Þ z ¼

p

gj2 hj2 g n3 ðzj Þ,

j¼1

3 2s3 Y0 X

j¼1

gj3 hj2 g n5 ðzj Þ,

j¼1

Table 1 Material properties of hexagonal zinc [36]. Elastic constants (GPa) c11 162.8

c12 50.8

c13 36.2

Thermal moduli (Mpa/K)

b1

b3

1.798

1.3834

3 2s3 Y0 X

p

j¼1

Y0

p2

g n3 ðz3 Þ:

ð24Þ

c33 62.7

c44 38.5

The expressions of the corresponding functions g nj ðzÞðj ¼ 1,. . ., 5Þ are given by Eq. (B.2). The complete thermoelasic field can be obtained by superimposing the two parts in Eqs. (23) and (24). It should be pointed that the thermo-elastic field in Eqs. (23) and (24) seems similar to that in Ref. [24] for a penny-shaped crack. n However, the functions f j ðzÞ and g nj ðzÞðj ¼ 1,. . .,5Þ in Eqs. (23) and (24) are completely different from those involved in Ref. [24]. Moreover, there are two functions, which are new results to the literature. The singular behavior may be conveniently discussed, at the tip of the crack under the action of a point thermal load on the crack surface. In view of the property pffiffiffiffiffiffiffiffiffiffiffiffiffiffi n z-0, h =z- y0 =y, ðx,yÞ A S, ð25Þ one can readily deduce that

Thermal conductivity (W/mK) k11 ¼k33 124.0

aj2 hj2 g n3 ðzj Þ ¼

n

j¼1

j¼1

3 4c66 s3 Y0 X

aj1 hj2 g 2 ðzj Þ,

T ð2Þ ¼ 

sz 9 z ¼ 0 ¼ 

2s3 Y0 g 12

p

rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi y0 y þ , y0 y ðxx0 Þ2 þðyy0 Þ2 1

ðx,yÞ A S&ðx0 ,y0 Þ A S:

Fig. 2. The distribution of the dimensionless temperature G on various planes: Z ¼ 0 (a), Z ¼1 (b) and (c) and x ¼0 (d).

ð26Þ

X.Y. Li / International Journal of Mechanical Sciences 59 (2012) 83–94

6. Applications of the fundamental solutions

If the stress intensity factor (SIF) is defined as pffiffiffiffiffiffiffi K I ¼ lim½ ysz 9 z ¼ 0 

ð27Þ

y-0

it is obviously that KI ¼ 

2s3 g 12 Y0

pffiffiffiffiffi y0

p

ðxx0 Þ2 þ y20

:

ð28Þ

As far as the same half-infinite plane crack embedded in the pure elastic and thermoelastic media are concerned, the tip behavior is quite different. Specifically, KI for crack embedded in the pure elastic media depends only upon the location of the load [33], while KI is relative to both the load location and the material constants for crack embedded in the thermoelastic media. This observation is consistent with the results predicted by Chen et al. [24]. Furthermore, the crack surface displacement (CSD) can be also obtained from Eqs. (23) and (24) as

oðx,yÞ ¼ wðx,y,0Þ ¼

Y0 R

arctan

pffiffiffiffiffiffiffiffi 2 yy0 , R

87

ð29Þ

which can be retrieved directly from Eq. (15) on letting Y(x,y)¼

Y0d(x  x0)d(y  y0). The physical quantities specified by Eqs. (26), (28) and (29) play an important role in crack analyses. In the next section, a 2D crack problem is given as a direct application of the previous results.

Fig. 3. 3D curves (a) and contour (b) of the dimensionless temperature G in the rectangular region fðx, ZÞ910 r x r 10,10 r Z r 10g on the plane z ¼1.

For an arbitrarily distributed thermal loading Y(x,y), (x,y)AS, corresponding physical quantities can be determined by integrating the relative fundamental field variables given in Eqs. (23) and (24). To show the practical applications of the previous fundamental solutions, a linear constant heat source is considered to be applied at the line y¼y1 (y1 40). Consequently, the crack problem in question is reduced to a plane strain one. For simplicity, all the concerns are confined to these associated with the crack plane z¼0. In this case, the CSD, normal stress sz9 z ¼ 0 and SIF can be derived if the thermal loading is taken to be Y(x,y)¼ Y0 d(y y1)( N oxoN). For example, the normal stress can be expressed from Eq. (26) as

sz 9z ¼ 0 ¼ 

2s3 g 12

p

Z

1

dx0

Z 0

1

1

rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi y0 y þ dy0 : y0 y ðxx0 Þ þ ðyy0 Þ ð30Þ

Y0 dðy0 y1 Þ 2

2

By virtue of the property of the Dirac-delta function, the normal stress component in Eq. (30) can be figured out as rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi y1 y þ dx0 2 2 y1 p y 1 ðxx0 Þ þ ðyy1 Þ rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi 2s3 Y0 g 12 y1 y þ ¼ ð31Þ , ðy1 40,yo 0Þ: y1 y1 y y

sz 9z ¼ 0 ¼ 

2s3 Y0 g 12

Z

1

1

Fig. 4. 3D curves (a) and contour (b) of the dimensionless temperature G in the rectangular region fðx, ZÞ910 r x r 10,10 r Z r 10g on the plane z ¼ 2.

88

X.Y. Li / International Journal of Mechanical Sciences 59 (2012) 83–94

The SIF can be also determined by integrating Eq. (28) as KI ¼ 

2s3 g 12

p

Z

þ1

dx0

Z

þ1

ðxx0 Þ2 þ y20

0

1

pffiffiffiffiffi y0

Y0 dðy0 y1 Þdy0 ,

which can be also expressed explicitly as Z pffiffiffiffiffi y1 2s3 g 12 Y0 þ 1 2s3 g Y0 dx0 ¼  p12 KI ¼  ffiffiffiffiffi : 2 2 p y1 1 ðxx0 Þ þ y1

ð32Þ

ð33Þ

Note that the SIF in Eq. (33) can be retrieved from Eqs. (31) and (27) as well. From Eq. (29), the CSD can be also expressed by the following integral

oðx,yÞ ¼ wðx,y,0Þ ¼ arctan

g 12 s3 g 11 p2

pffiffiffiffiffiffiffiffi 2 yy0 dy0 : R

Z

þ1

1

dx0

Z 0

þ1

and Karapatian [33], as Z þ1 n Z y g s dt ½L ðy þ y0 2tÞYðx,y0 Þ pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi dy0 oðx,yÞ ¼ 12 3 g 11 p 0 yt u y0 t Z y Z þ1 n g s3 dt ½L ðy þ y0 2tÞY0 dðy0 y1 Þ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi dy0 : ¼ 12 g 11 p 0 yt u y0 t

ð35Þ

Taking advantage of the definition of the operator Ln in Eq. (17), one can derive that Z 1 1 ðy þ y0 2tÞY0 dðy0 y1 Þ Ln ðyþ y0 2tÞY0 dðy0 y1 Þ ¼ dx p 1 ðy þy0 2tÞ2 þ ðxx0 Þ2 0 ¼ Y0 dðy0 y1 Þ:

ð36Þ

Y0 dðy0 y1 Þ R ð34Þ

In contrast to the integrals in Eqs. (30) and (32), the integral in Eq. (34) cannot be directly obtained. However, Eq. (34) can be transformed into the following form by referring the results associated with potential theory method developed by Fabrikant

Fig. 5. The dimensionless CSD Oðx, ZÞ as functions of the dimensionless coordinates x (a) and Z (b).

Fig. 6. The distributions of dimensionless normal displacements W x ¼ W=y0 on various planes: z ¼ 1 (a), 2 (b) and 5 (c).

X.Y. Li / International Journal of Mechanical Sciences 59 (2012) 83–94

Substituting Eq. (36) into Eq. (35) and then using the property of the Dirac-delta function give rise to Z g s Y0 y dt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð37Þ oðx,yÞ ¼ 12 3 g 11 p ðytÞðy1 tÞ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi By means of Euler substitution ðytÞðy1 tÞ ¼ x0 t, Eq. (37) is reduced to ffi Z pffiffiffiffiffi yy1 g s Y0 dx0 oðx,yÞ ¼ 2 12 3 , ð38Þ g 11 p y þ y1 2x0 0

89

The solution to Eq. (45) is W0 ðx,yÞ ¼ 

Ln ðyÞ d 2p2 dy

Z

y 0

dy1 d Ln ð2y1 Þ pffiffiffiffiffiffiffiffiffiffiffi yy1 dy1

Z

þ1 y1

dy0 pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Ln ðy0 ÞYðx0 ,y0 Þ y0 y1

ð46Þ

where Ln() is the operator defined in Eq. (17).

which is equal to

oðx,yÞ ¼

g 12 s3 Y0 y þ y1 ln pffiffiffi pffiffiffiffiffi , g 11 p ð y y1 Þ2

ðy 4 0Þ:

ð39Þ

As expected, the CSD is singular on the line y¼y1. The results in Eqs. (31), (33) and (39) for this plane strain problem can work as benchmarks to clarify 2D analysis and numerical simulations.

7. Validation of the temperature In the present study, the uncoupled theory of thermoelasticity is employed. Hence, the presence of thermal loading will result in a variation of the elastic field (see Eq. 14), while the mechanical loading contributes nothing to the thermal field. In this sense, there seems to be a contradiction in Eq. (23): the generalized mechanical load would give arise to a change of temperature. However, such a contradiction is actually not true. Noticing the relations a12 ¼ a22 ¼0 and 2pa32s3h3j ¼  d2j (j¼1,2), one can immediately get that T ð1Þ ¼ 0,

T ¼ T ð2Þ ¼

Y0

p2

g n3 ðz3 Þ:

ð40Þ

This definitively removes the seeming contradiction. As pointed out in the previous studies [24,28], the temperature can be obtained independently and a prior. This provides a way to validate the temperature distribution predicted by Eq. (40). Actually, there are various methods such as Fourier transform to fulfill this purpose. At this stage, the potential theory method is again employed to show its simplicity. In fact, the temperature field under consideration can be formulated by the following mixed boundary value problem ! @2 ð41Þ Dþ 2 T ¼ 0 @z3 subjected to the boundary conditions ðx,yÞ A S :

@T ¼ 0; ðx,yÞ A S : @z3

T ¼ Yðx,yÞ

ð42Þ

Eq. (41) can be satisfied if the temperature is assumed to be Z þ1 0 Z þ1 W ðx0 ,y0 Þ dx0 dy0 : ð43Þ Tðx,y,zÞ ¼ R00 1 0 where

W0 ¼ 

1 @T , 2p @z3 z3 ¼ 0

R00 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxx0 Þ2 þðyy0 Þ2 þ z23

ð44Þ

It is noted that the first condition in Eq. (42) has been automatically met due to the property of simple layer potential. Satisfaction of the second condition in Eq. (42) requires Z þ1 Z þ1 W0 ðx0 ,y0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dy0 ¼ Yðx,yÞ: dx0 ð45Þ 1 0 ðxx0 Þ2 þðyy0 Þ2

Fig. 7. The contours of the normal displacements W z ¼ w=y0 on various planes: z ¼ 1 (a), 2 (b) and 5 (c).

90

X.Y. Li / International Journal of Mechanical Sciences 59 (2012) 83–94

Inserting Eq. (46) into Eq. (43) and following the procedures shown by Fabrikant and Karapetian [33], it is easy to obtain the temperature   n  Z þ1 Z þ1 1 z3 R00 h dx0 Yðx0 ,y0 Þdy0 Tðx,y,zÞ ¼ 2 n þ arctan 0 3 p 1 R00 ðR0 Þ h 0 ð47Þ On letting Y ¼ Y0d(x  x0)d(y y0), one can arrive at   n  Y0 z3 R0 h Y0 Tðx,y,zÞ ¼ 2 0 3 n0 þ arctan 0 ¼ 2 g 3 ðz3 Þ p ðR0 Þ h p R0

ð48Þ

which is exactly the result given in Eq. (40).

Wz ¼ X zy

w y0

¼

tzy c11

This section is devoted to numerically presenting the theoretical results derived in the previous sections. To this end, the temperature loads are assumed to be applied at the point (0,y0,0 7 ) and the magnitude of the loads is assumed to be Y0 ¼ y20 . For simplicity, the following dimensionless quantities are introduced ,

Z¼

y y0

,

z¼

z y0

,

G¼

T b1 c11

,

O¼

,

K 0I ¼

KI pffiffiffiffiffi c11 y0

s1 ¼ 1:20 þ 0:42i,

8. Numerical examples and discussions

x y0

VZ ¼

oðx,yÞ y0

¼

wðx,y,0Þ , y0

Fig. 8. The dimensionless displacement U x ¼ u=y0 as a function of the dimensionless coordinate x (a) and Z(b).

v , y0

Sz ¼

sz c11

,

X zx

¼

tzx c11

, ð49Þ

to denote the dimensionless coordinate, temperature, displacement, stress and stress intensity factor, respectively. In other words, the pffiffiffiffiffi quantities y0, c11/b1, c11 and c11 y0 are chosen as the reference scales of length, temperature, stress and SIF, respectively. The chosen material is hexagonal zinc [36], with material constants listed in Table 1. Consequently, the material parameters have the following numerical values

109 ,

x¼

u y0

Ux ¼

s2 ¼ 1:200:42i,

s3 ¼ 1,

g 11 ¼ 5:84

g 12 ¼ 8:46  104 :

Fig. 2 displays the distribution of the dimensionless temperature G, as functions of spatial coordinates. As expected, the temperature is symmetric with respect to the axis x ¼x/y0 ¼0, as shown in Fig. 2(a) and Fig. 2(b), for z ¼z/y0 ¼0 and z ¼1, respectively,. Variations of the temperature G, with the dimensionless Z on the plane z ¼1, and with z on the plane x ¼0, are, respectively shown in Fig. 2(c) and (d). It is seen from Fig. 1(c) that the temperature decrease with x, and from Fig. 2(d) that G generally decreases with both z ¼ z/y0 and Z ¼y/y0. Figs. 3 and 4 plot the 3D curves and corresponding contours of the dimensionless

Fig. 9. The dimensionless displacement V Z ¼ v=y0 as a function of the dimensionless coordinate x (a) and Z(b).

X.Y. Li / International Journal of Mechanical Sciences 59 (2012) 83–94

temperature in the same rectangular region {(x,Z)9  10 r x r10, 10 r Z r10} on the plane z ¼z/y0 ¼1 and 2, respectively. It is seen from Figs. 3 and 4 that the distributions of temperature are symmetric with respect to the x ¼x/y0 ¼ 0. Moreover, the locations with dimensionless coordinates (x,Z,z) ¼(0,1,z), where the maximum temperature on both horizontal planes occurs, are just below the point of the thermal loading. The crack surface displacement (CSD) w(x,y,0), (x,y)AS is an important physical parameter in crack analyses. The dimensionless CSDs O ¼ o(x,y)/y0 ¼w(x,y,0)/y0 as functions of dimensionless coordinates x and Z are plotted in Fig. 5(a) and (b), respectively. It is seen that O is singular at the point (x,Z,z)¼(0,1,0). For a specified value of Z, O decreases with x, as illustrated in Fig. 5(b). Variations of the dimensionless normal displacement Wz ¼ (w/y0)  106 on various horizontal planes z ¼1, 2 and 5 with the dimensionless coordinates x and Z are illustrated in Fig. 6. The corresponding contours of the normalized displacement are given in Fig. 7. Similar to the temperature distribution on the horizontal planes as shown in Figs. 3 and 4, the normal displacement is again observed to be symmetric to the axis x ¼0 and decreases with the dimensionless coordinate z. The maximum values of the dimensionless displacement Wz ¼(w/y0) on the horizontal plane z ¼1, 2 and 5 are, respectively 2.30  10  7, 9.20  10  9 and 2.51  10  9.

91

The dimensionless complex tangential displacements changing with the dimensionless coordinates x and Z on the horizontal cross section z ¼1 are shown in Figs. 8 and 9. As expected, the dimensionless displacements Ux and VZ in the x and y directions are anti-symmetric and symmetric with respect to the y axis, respectively, as shown in Fig. 8(a) and Fig. 9(a). The variations of the dimensionless normal stress Sz ¼ sz/c11 with the spatial coordinates are given in Fig. 10. As shown in Fig. 10(a), the magnitude of stress Sz(z ¼0) increases with the dimensionless coordinate Z ¼y/y0. The stress Sz ¼ sz =c11 near the crack edge (Z ¼  10  3) is at least ten times larger than those on the line Z ¼ 0.1. This is not surprising since the Sz is singular at the crack front, with a dimensionless stress intensity factor K 0I ¼

KI 3:31  107 : pffiffiffiffiffi ¼  2 c11 y0 1þx

ð50Þ

The distributions of the dimensionless normal stress on the plane z ¼ z=y0 ¼ 2 are shown in Fig. 10(b) and (c). It is interesting that the maximum stress occurs at the point (x,Z,z)¼(0,Z,2) and symmetry with respect to the line x ¼0 is again observed, as illustrated in Fig. 10(b). Furthermore, the normal stress on the plane z ¼ z=y0 ¼ 2 decrease with Z (see Fig. 10(c)). The distribution of Sz ¼ sz =c11 on the plane x ¼0 are shown in Fig. 10(d), which clearly shows that Sz is compressive for Z o0. As a

Fig. 10. The variations of the dimensionless normal stress Sz with the spatial coordinates. Data are for z ¼0 (a), z ¼ 2 (b) and (c) and x ¼0 (d).

92

X.Y. Li / International Journal of Mechanical Sciences 59 (2012) 83–94

posterior check of the first boundary condition in Eq. (9), Sz vanishes at the point (x,Z,z) ¼(0,0.5,0)AS. Furthermore, we can also expect that Sz tends to zero as z approaches to infinity. Fig. 11 depicts the 3D distribution of the dimensionless normal stress Sz on various cross sections z ¼1, 2 and 5. The maximum compressive stresses on these planes are, respectively equal to 52.09  10  8,  17.52  10  8 and 4.65  10  8. This means that Sz decreases very rapidly with the dimensionless coordinate z, which is consistent with the results in Fig. 10(d). The corresponding contours of the stress Sz on these planes are given in Fig. 12, where symmetry of the stress Sz with respect to the axis

Fig. 12. The contours of the dimensionless normal stress Sz in the rectangular region fðx, ZÞ910 r x r 10,10 r Z r 10g on various planes: z ¼1 (a), 2 (b) and 5 (c).

Fig. 11. The distributions of the dimensionless normal stress Sz in the rectangular region fðx, ZÞ910 r x r 10,10 r Z r 10g on various planes: z ¼ 1 (a), 2 (b) and 5 (c).

x¼0 is again observed and the maximum normal stress taking place at the neighborhood of the point (0,1,z) is seen. From a physical point of view, the shear stress components tzx and tzy should be anti-symmetric and symmetric with respect to the y axis, respectively. In addition, tzx should vanish in this case.

X.Y. Li / International Journal of Mechanical Sciences 59 (2012) 83–94

93

the general solutions will take other forms. The potential theory method generally works if appropriate potential functions are found and the succeeding procedure is similar to what have be shown in the present analysis. As an alternative, the thermoelastic field in the case of equal eigenvalues can be directly derive from the present results by means of L’Hospital rules, as suggested by Fabrikant and Karapetian [33].

Acknowledgment The work was supported by the National Natural Science Foundation of China (Nos. 11102171 and 11172250) and by the Fundamental Research Funds for the Central Universities (No.SWJTU11CX069). The author is indebted to Professors Ding H.J. and Chen W.Q. in Zhejiang University, who introduced him into this field.

Appendix A In this section, the constants involved in the general solutions developed by Chen et al. [24] are presented. The constants in Eq. (6) are a0 ¼ c33 c44 ,

b0 ¼ c11 c33 þ c244 ðc13 þ c44 Þ2 ,

c0 ¼ c11 c44 :

ðA:1Þ

The constants aij in Eq. (4) have the following forms

aj1 ¼

a2 sj b2 s3j a1 b1 s2j

,

aj2 ¼ d3j

a0 s43 b0 s23 þc0 , a1 b1 s23

ðj ¼ 1,2,3Þ

ðA:2Þ

where a1 ¼ b1 c44 ,

b1 ¼ b3 ðc13 þ c44 Þb1 c33 ,

a2 ¼ b3 c11 b1 ðc13 þ c44 Þ,

b2 ¼ b3 c44 :

ðA:3Þ

The constants gij in Eq. (8) are

gj1 ¼ c13 þ c33 sj aj1 b3 aj2 , gj2 ¼ 2½c11 c66 þ c13 sj aj1 b1 aj2 , Fig. 13. The contours of the dimensionless complex shear stress components: Szx ¼ tzx =c11 (a) and Szy ¼ tzy =c11 (b).

These conditions holds true, as shown in Fig. 13, where for the contours of the dimensionless shear stresses Szx ¼(tzx/c11)  108 and Szy ¼ (tzy/c11)  108 on the plane z ¼1 are illustrated. From s1 to s2 explicitly specified by Eqs. (23) and (24), one can numerically display the stress components sx, sy and txy without difficulties. For simplicity and saving space, they are not given here.

9. Concluding remarks This paper exactly investigates the thermoelastic field of a transversely isotropic medium with a half infinite plane crack subjected to external temperature load, on the basis of the 3D general solutions conjugated with the generalized potential theory method. For an arbitrarily located point thermal load, fundamental solutions are derived in terms of the elementary functions, for the first time, by using the new results of the potential theory method. The singular behavior at the edge of crack is examined for a point temperature load acting at the crack surface. Furthermore, the temperature field is obtained along two paths and an exact agreement is observed. It is noted that the present study makes sense only for transversely isotropic material, whose material eigenvalues sj(j ¼0,1,2,3) are distinct. For material with equal eigenvalues,

gj3 ¼ c44 ðaj1 sj Þ ¼ sj gj1 ,ðj ¼ 1,2,3Þ:

ðA:4Þ

Appendix B Here, the derivatives of the Green’s functions Pj(x,y,z; x0,y0) (j ¼1,2) are listed for the sake of completeness. These of P1(x,y,z; x0,y0) were given by Fabrikant and Karapetian [33] as 2 !0:5 3  n 2p 4 z h sn n 5, LP1 ¼ 2pf 1 ðzÞ ¼ arctan carctan n R0 t R0 2l1  n @P1 2p h n ¼ 2pf 2 ðzÞ ¼  arctan , @z R0 R0 " !#  n n n @2 P1 z h h l1 z2 n ¼ 2pf 3 ðzÞ ¼ 2p 3 arctan  , n n  2 n2 R0 @z2 R0 zðR20 þh Þ l1 l2 R0 2 !0:5   c i 2 sn arctan L2 P1 ¼ 2pf n4 ðzÞ ¼ 2p4 þ n t sn t 2l1 qffiffiffiffiffiffiffiffiffi n  n 2 y0 l1 zð3R20 z2 Þ h   arctan n 2 R0 t ðsn 2l1 Þsn t R30 !# n zh t 1  2 þ , n2 tR20 4ln22 4ln1 ln2 R0 þ h

94

X.Y. Li / International Journal of Mechanical Sciences 59 (2012) 83–94

2 !3  n n n 2 @ P1 t h h l z n 1 5,  ¼ 2pf 5 ðzÞ ¼ 2p4 3 arctan L þ

n n  n2 @z R0 2l1 2l2 R20 R0 R2 þ h 0

ðB:1Þ where pffiffiffiffiffiffiffiffiffiffiffiffiffi

t ¼ ðxx0 Þ þ i yy0 ,R0 ¼ tt þ z2 ,sn ¼ ðy þy0 Þiðxx0 Þ qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi h

0:5 i n n n : c ¼ 2y=sn ,h ¼ 2 y0 l2 ,l1,2 ¼ 0:5 y 8 y2 þ z2

ðB:2Þ

The derivatives of P2(x,y,z; x0,y0) read [33,35]     n 1 z h z 1 LP2 ¼ g n1 ðzÞ ¼ arctan  n þ pRe c R0 t R0 h !0:5  n 0:5 # n 1 s z 2l2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  arctan  arctan , n c it 2l1 2y0 it 2 !0:5 3  n  sffiffiffiffiffi @P2 1 h 2 4 1 it n 5, ¼ g 2 ðzÞ ¼  arctan Re pffiffiffiffi arctan  n R0 y0 @z R0 2l2 it   n  @2 P2 z R0 h ¼ g n3 ðzÞ ¼ 3 n þ arctan , 2 R0 @z Ro h " #   n n zð3R20 z2 Þ h z 2i h t LLP2 ¼ g n4 ðzÞ ¼ arctan   2 R0 it thn R20 ðR20 þhn2 Þ t R30    n 0:5 z 1 1 2l1 arctan  c þ þ n 2 n2 2 c sn h ðR0 þ h Þ t " #  n 0:5 n 3z 2l2 z it 2l2  2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arctan  2 n n  n it t t h it þ 2l2 it þ2l2 2iy0 t

p2

þ

ct

L

2

,

  n  @P2 t R0 h ¼ g n5 ðzÞ ¼ 3 n þarctan @z R0 Ro h 2 !0:5 3  n 0:5 1 4 2il2 t 5: arctan  n 1 n t h t 2il2

ðB:3Þ

References [1] Prasad R, Kumar R, Mukhopadhyay S. Propagation of harmonic plane waves under thermoelasticity with dual-phase-lags. Int J Eng Sci 2010;48:2028–43. [2] Chandrasekharaiah DS. Thermoelastic plane waves without energy dissipation. Mech Res Commun 1996;23:549–55. [3] Chao CK, Shen MH, Fung CK. On multiple circular inclusions in plane thermoelasticity. Int J Solids Struct 1997;34:1873–92. [4] Rahman M. The axisymmetric contact problem of thermoelasticity in the presence of an internal heat source. Int J Eng Sci 2003;41:1899–911. [5] Jang YH, Cho H, Barber JR. The thermoelastic Hertzian contact problem. Int J Solids Struct 2009;46:4073–8. [6] Kulchytsky-Zhyhailo R, Matysiak SJ, Perkowski DM. Plane contact problems with frictional heating for a vertically layered half-space. Int J Heat Mass Transfer 2011;54:1805–13. [7] Loveya FC, Condo AM, Torra V. A model for the interaction of martensitic transformation with dislocations in shape memory alloys. Int J Plast 2004;20:309–21.

[8] Eckert ERG, et al. Heat transfer-a review of 1995 literature. Int J Heat Mass Transfer 1995;31:1606–86. [9] Sih GC. On the singular character of thermal stress near a crack tip. J Appl Mech 1962;29:587–9. [10] Sneddon IN, Lowengrub M. Crack Problems in the Classical Theory of Elasticity. New York: John Wiley and Sons; 1969. [11] Kassir MK, Sih GC. Three-Dimensional Crack Problems (Mechanics of Fracture 2). Leyden: Noordhoff International Publishing; 1975. [12] Tsai YM. Thermal stress in a transversely isotropic medium containing a penny-shaped crack. J Appl Mech 1983;50:24–8. [13] Tsai YM. Transversely isotropic thermoelastic problem of uniform heat flow disturbed by a penny-shaped crack. J Therm Stresses 1983;6:379–89. [14] Liu L, Kardomateas GA. Thermal stress intensity factors for a crack in an anisotropic half plane. Int J Solids Struct 2005;42:5208–23. [15] Choi HJ. Thermoelastic problem of steady-state heat flows disturbed by a crack at an arbitrary angle to the graded interfacial zone in bonded materials. Int J Solids Struct 2011;48:893–909. [16] Tsai YM. Thermoelastic behavior of a transversely isotropic material containing a flat toroidal crack. J Therm Stresses 1998;21:881–95. [17] Tsai YM. Thermoelastic problem of uniform heat flow disturbed by a flat toroidal crack in a transversely isotropic medium. J Therm Stresses 2000;23:217–31. [18] Kellogg OD. Foundations of Potential Theory. New York: F. Ungar Publishing; 1929. [19] Fabrikant VI. Applications of Potential Theory in Mechanics: A Selection of New Results. The Netherlands: Kluwer Academic Publishers; 1989. [20] Fabrikant VI. Mixed Boundary value Problem of Potential Theory and their Applications in Engineering. The Netherlands: Kluwer Academic Publishers; 1991. [21] Elliott HA. Three-dimensional stress distributions in aeolotropic hexagonal crystals. Proc Cambridge Philos Soc 1948;44:522–33. [22] Chen WQ, Ding HJ. Potential theory method for 3D crack and contact problems of multi-field coupled media: a survey. J Zhejiang Univ Sci 2004;5:1009–21. [23] Chen WQ, Shioya T. Fundamental solution for a penny-shaped crack in a piezoelectric medium. J Mech Phys Solids 1999;47:1459–75. [24] Chen WQ, Ding HJ, Ling DS. Thermoelastic field of a transversely isotropic elastic medium containing a penny-shaped crack: exact fundamental solution. Int J Solids Struct 2004;41:69–83. [25] Chen WQ, Lim CW, Ding HJ. Point temperature solution for a penny-shaped crack in an infinite transversely isotropic thermo-piezo-elastic medium. Eng Anal Boundary Elem 2005;29:524–32. [26] Chen WQ, Lee KY, Ding HJ. General solution for transversely isotropic magneto-electro-thermo-elasticity and the potential theory method. Int J Eng Sci 2004;42:1361–79. [27] Li XY, Chen WQ, Wang HY. General steady state solutions for transversely isotropic thermoporoelastic media in three dimensions and its application. Euro J Mech A Solids 2010;29:317–26. [28] Ding HJ, Chen WQ, Zhang LC. Elasticity of Transversely Isotropic Materials. Dordrecht: Springer; 2006. [29] Gilman JJ. Cleavage Ductility and Tenacity in Crystals. New York: John Wiley & Sons; 1959. [30] Knauss WG. Stresses in an infinite strip containing a semi-infinite crack. J Appl Mech 1966;33:356–62. [31] Fan TY. Exact solutions of semi-infinite crack in a strip. Chin Phys Lett 1990;7:402–5. [32] Rubio-Gonzalez C, Mason JJ. Dynamic stress intensity factors at the tip of a uniformly loaded semi-infinite crack in an orthotropic material. J Mech Phys Solids 2000;48:899–925. [33] Fabrikant VI, Karapetian EN. Elementary exact method for solving boundaryvalue problems of potential theory with application to half-plane contact and crack problems. Quart J Mech Appl Math 1994;47:159–74. [34] Huang ZY, Bao RH, Bian ZG. The potential theory method for a half-plane crack and contact problems of piezoelectric materials. Compos Struct 2007;78:596–601. [35] Li XY. On the half-infinite crack problem in thermo-electro-elasticity. Mech Res Commun 2011;38:506–11. [36] Sharma JN, Sharma PK. Free vibration of homogeneous transversely isotropic cylindrical panel. J Thermal Stresses 2002;25:169–82.