DSpace at VNU: Interaction between a cracked hole and a line crack under uniform heat flux

DSpace at VNU: Interaction between a cracked hole and a line crack under uniform heat flux tài liệu, giáo án, bài giảng...

DOI 10.1007/s10704-004-7138-3 © Springer 2005

Interaction between a cracked hole and a line crack under uniform heat flux

PHAM CHI VINH1, NORIO HASEBE2,∗, XIAN-FENG WANG2 and

TAKAHIRO SAITO2

1Faculty of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam

2Department of Civil Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555 Japan

∗Author for correspondence (E-mail: hasebe@kozo4.ace.nitech.ac.jp; Fax: +81-52-735-5482)

Received 21 July 2004; accepted in revised form 2 December 2004

Abstract This article deals with the interaction between a cracked hole and a line crack under

uniform heat flux Using the principle of superposition, the original problem is converted into three particular cracked hole problems: the first one is the problem of the hole with an edge crack under uniform heat flux, the second and third ones are the problems of the hole under distributed tem-perature and edge dislocations, respectively, along the line crack surface Singular integral equations satisfying adiabatic and traction free conditions on the crack surface are obtained for the solution of the second and third problems The solution of the first problem, as well as the fundamental solutions

of the second and third, is obtained by the complex variable method along with the rational mapping function approach Stress intensity factors (SIFs) at all three crack tips are calculated Interestingly, the results show that the interaction between the cracked hole and the line crack under uniform heat flux can lead to the vanishing of the SIFs at the hole edge crack tip The fact has never been seen for the case of a cracked hole and a line crack under remote uniform tension.

Key words: Cracked hole, dislocation, heat flux, interaction, integral equation, mapping function, stress

intensity factor, thermal stress.

1 Introduction

Due to the stress concentration effect, cracks are likely to initiate at a hole bound-ary under the action of monotonous or fatigue loading A number of papers deal-ing with the hole edge crack problem are available (Bowie, 1956; Tweed and Rooke, 1973; Hasebe and Ueda, 1980; Schijve, 1983; Hasebe et al., 1988, 1994a, b; Zhang and Hasebe, 1993; Chao and Lee, 1996; Hasebe and Chen, 1996) Among them, the rational mapping function approach is significant to solve the hole edge crack prob-lem (Hasebe and Ueda, 1980; Hasebe et al., 1988, 1994a, b; Hasebe and Chen, 1996)

Up to now, there seem to be only a few papers concerning the interaction between the crack emanating from a hole and another independent crack Hasebe et al (1994a) solved a second mixed boundary value problem analytically and as an exam-ple of the solution, the interaction of a square hole with an edge crack and a line crack was investigated for a geometrically symmetric case Hasebe and Chen (1996) treated a cracked circular hole and a crack when the remote stresses were applied at

368 P.C Vinh et al.

d

f -a

e

2b

y

temperature dislocation

edge dislocation

A

B

C

y

x

y

x 0

y

x 0

C

A

B

0

I

γπ(1−γ)π

I

B

A I

A

B

no crack

C

y

x

A temperature dislocation

y

x 0

C

zo

zo edge dislocation

Problem A

Problem B

Problem C

Problem E (a)

(b)

Figure 1 Superposition of problems.

infinity The interactions of a hole and a rigid inclusion, respectively, with a crack were considered by Hasebe et al., (2003a) The problem of a crack initiating from a rigid inclusion interacting with a line crack was also solved by Hasebe et al (2003b)

It is known that when steady heat flow is disturbed by the presence of crack, there is a high local intensification of temperature gradient at the crack tips and large thermal stresses arise around them Thermal disturbances of this kind may, in some cases, cause crack propagation resulting in serious damage to structural components Consequently, the study of the behavior of thermal stresses near the crack tips is of importance in fracture mechanics

The aim of this paper is to investigate the interaction between a cracked hole and

a line crack under uniform heat flux The hole edge and the faces of the cracks are assumed to be adiabatic and traction free The problem to be considered is shown in Figure 1a It is clear that the original problem A can be reduced to three following problems by the principle of superposition:

Problem B: Problem of the hole with an edge crack under uniform heat flux

(Fig-ure 1b)

Problem C: Problem of the hole with an edge crack under distributed temperature

dislocation along the line crack surface (Figure 1c)

Problem D: Problem of the cracked hole under distributed edge dislocation along

the line crack surface (Figure 1d)

In order to solve these problems we apply the complex function method along with the rational mapping function approach By means of the principle of superposition,

problem C (D) can be reduced to a problem named problem E (F), in which the cracked hole subjected to a temperature point dislocation (an edge point dislocation) placed at some point on the crack line AB After problem E (F) is solved, the solu-tion of problem C (D) can be obtained by integrating the Green’s funcsolu-tions of prob-lem E (F) along the line AB By summation of the solutions of probprob-lems B, C and

D, we can ascertain problem A’s solution, in which there are two unknown functions: the distribution density of temperature dislocation along AB (coming from prob-lem C’s solution) and the distribution density of edge dislocation along AB (com-ing from problem D’s solution) These functions are determined from the s(com-ingular integral equations which are derived from the adiabatic and traction-free conditions along the faces of the line crack AB Finally, by numerical integration of the singular integral equation, stress intensity factors (SIFs) at crack tips are obtained It is inter-esting to find that the interaction between the cracked hole and the line crack under uniform heat flux can lead to the vanishing of the SIFs at the hole edge crack tip The fact has never been seen for the case of a cracked hole and a line crack under remote uniform tension

2 Rational mapping function

As mentioned above, the rational mapping function technique is used to deal with the cracked hole with arbitrary shapes in an infinite plane In the computation, we will consider a cracked elliptical hole and a cracked square hole as examples Here, with-out loss of generality, the general form of rational mapping function is given (Hasebe and Ueda, 1980; Hasebe et al., 2003b):

z = ω(ς) = E0ς +

N

k=1

Ek

which maps the exterior of the cracked hole in the z-plane onto the exterior of the unit circle in the ς-plane as shown in Figure 2 Here E−1, E0, Ek, ςk(k = 1, 2, 3, , N) are

complex constants and |ς k | < 1 for all k = 1, 2, 3, , N It should be noted that other configurations can be readily tackled using this general formulation Taking E0= 0 and

Ek = 0 (k = 1, 2, 3, , N) in (1), the mapping function is reduced to the one for a circle Likewise, when E0= (a + b)/2 and E1= (b − a)/2, ς1= 0 and E k = 0 (k = 2, 3, , N) are taken, the mapping function is for ellipse with semi-axes on the x and y-axes being a and

b, respectively.

3 Basic formulae

3.1 Temperature field

In the ς-plane, the temperature function θ(ς, ς) for the two-dimensional steady-state

thermoelasticity satisfies the Laplace equation Thus, the temperature function can be

expressed as the real part of an analytic function Y (ς), which gives temperature and

heat flux (Hasebe et al., 1988; Han and Hasebe, 2001) as:

370 P.C Vinh et al.

Figure 2 Rational mapping functions for a cracked square hole and a cracked elliptical hole.

qx − iq y = −k



Y(ς)

ω(ς)



qρ − iq θ= ςω(ς)

where q x and q y denote the heat flux components in the x- and y-axes, respectively,

qρ and q θ represent these in the orthogonal curvilinear coordinates generated by

ω(ς), and k signifies the thermal conductivity of the material Also the heat flux

boundary condition is given as follows:

−k[Y (σ ) − Y (σ)] = 2i



where σ and q n denote a value of ς and the normal heat flux component on the

boundary, respectively, and the integration is carried out along the boundary From

(5), the adiabatic condition (q n= 0) along the boundary is:

3.2 Thermal stress field

Employing complex functions φ(ς) and ψ(ς), the stresses in the elastic body are

(Muskhelishvili, 1963):

σx + σ y= 4Re



φ(ς)

ω(ς)



σy − σ x + 2iτ xy= 2



ω(ς)



φ(ς)

ω(ς)



+ ψ(ς)

 

σθ + σ ρ = σ x + σ y , (9)

σθ − σ ρ + 2iτ ρθ= ς2ω(ς)

The stress boundary condition is written as:

φ(σ ) + ω(σ )

ω(σ ) φ

(σ ) + ψ(σ ) = i



where p x and p y denote external force components applied to the boundary in the

x- and y-directions, respectively.

Using the complex stress functions φ(ς), ψ(ς) and the temperature function Y (ς),

the displacement expression can be put into the following form:

κφ(ς) − ω(ς)

ω(ς) φ

(ς) − ψ(ς) + 2Gα



where G is the shear modulus, κ and α are: κ = 3 − 4v, α= (1 + v)α for plane strain and κ = (3 − v)/(1 + v), α=α for generalized plane stress; v and α are Poisson’s ratio

and the coefficient of thermal expansion, respectively

4 Solution of problem B

4.1 Temperature field

Consider the heat conduction problem shown in Figure 1b, in which q is the inten-sity of the uniform heat flux; δ is the angle between the direction of the heat flux and the x-axis Herein the hole edge and crack faces are assumed to be adiabatic The temperature function YB(ς) of problem B can be broken down into two parts:

where the first function denotes the one induced from the uniform heat flux; the

sec-ond one denotes the complementary part From (3), function Y1B(ς) can be obtained:

Y1B(ς) = − q

k e

Substituting (13) and (14) into (6) yields:

Y2B(σ ) − Y2B(σ ) = q

k e

−iδ ω(σ ) − q

k e

Multiplying (15) by the factor dσ/[2πi(σ − ς)] and carrying out the Cauchy integra-tion along the unit circle, we obtain Y2B(ς) and finally (Hasebe et al., 1988):

YB(ς) = − q

k



e −iδ E0ς + e iδ E¯0

ς



372 P.C Vinh et al.

4.2 Thermal stress field

The stress function for the thermoelastic problem can be split into two parts: a

non-holomorphic part [φ1B(ς), ψ1B(ς)] and a holomorphic part [φ2B(ς), ψ2B(ς)]:

φB(ς) = φ1B(ς) + φ2B(ς),

The last term of the left-hand side of (12) denotes the thermal displacement This integration contains logarithmic term which represents dislocation in the

thermo-elasticity To remove it, we consider the following stress functions φ1B(ς), ψ1B(ς)

(Florence and Goodier, 1960):

where the constants AB and BB are determined using the conditions that the stress and the displacement components around the hole are single-valued Substituting (17) and (18) into (11), and using the stress single valuedness condition, we have

Next, substitute (17) and (18) into (12) Multiple values of logarithmic terms must

be cancelled due to the single-valuedness of displacement Consequently, the constant

AB is determined as:

AB=αqGR

2k E0

 N

k=1

Ek e −iδ+ ¯E0e iδ

where R = (1 + v)/(1 − v) for plane strain and R = (1 + v) for generalized plane stress.

Now we consider the boundary condition to derive the holomorphic functions

φ2B(ς) and ψ2B(ς) The hole edge and crack faces are assumed traction free without

loss of generality, i.e., p x = p y= 0 Substituting (17) into (11) yields

φ2B(σ ) + ω(σ )

ω(ς) φ

 2B(σ ) + ψ2B(σ ) = −φ1B(σ ) − ω(σ )

ω(σ ) φ

 1B(σ ) − ψ1B(σ ). (21)

Multiplying (21) by the factor dσ/[2πi(σ − ς)] and carrying out the Cauchy integra-tion along the unit circle, we obtain φ2B(ς) as:

φ2B(ς) =

N

k=1

¯

AkBk

ς − ςk+ ¯AB

N

k=1

Bkςk

in which B k ≡ E k/ω(ς k) with ς k≡ 1/ ¯ς k and A k = φ

2B(ς k) Here the real and

imagi-nary values of A k are determined by the simultaneous equations of 2N derived by differentiating (22) and substituting ς = ς k

Thus, the stress function φB(ς) is (Hasebe et al., 1988):

φB(ς) = ABlog ς +

N

k=1

¯

Ak Bk

ς − ςk + ¯AB

N

k=1

Bk ςk

The stress function ψB(ς) can be derived by analytic continuation along the

free-traction boundary of the unit circle Indeed, by introducing the following function:

φB(ς) = − ¯ω ω(ς)(1/ς)

¯

φB(1/ς) − ¯ ψB(1/ς), ς ∈ S−= {ς : |ς| < 1} (24)

from (11) with regarding p x = p y= 0 on the unit circle and (24), we have

This means the function φB(ς), ς ∈ S− is a continuation of the function φB(ς), ς ∈

S+={ς : |ς| > 1} from the outside of the unit circle to its inside From (25) it follows:

ψB(ς) = − ¯ φB(1/ς) − ¯ω(1/ς)

ω(ς) φ



5 Solution of problem C

Consider an infinite plane with a cracked hole subjected to distributed tempera-ture dislocations along the line crack surface, as shown in Figure 1c The hole edge and crack faces are assumed to be traction-free and adiabatic As previously said, by means of the principle of superposition, problem C can be reduced to

a problem E, in which the cracked hole subjected to a temperature point

dislo-cation placed at some point z0(= ω(ς0), ς0 is a point in the ζ -plane correspond-ing to z0) on the line AB To solve problem E, we need to find the Green’s function for the temperature field, as well as the stress field of the cracked hole under a temperature point dislocation After problem E is solved, the solution

of problem C can be obtained by integrating the Green’s function along the line AB

It is not difficult to verify that, for the temperature field, the Green’s function of problem E for a couple temperature dislocation is (Hasebe and Han, 2001):

YE(ς) = 

2πk

e iϑ

ω(ς0)(ς − ς0)− e −iϑ ς

2

p

ω(ς0)(ς − ςp)

where k is thermal conductivity of the material, β is the angle between the line AB and the x-axis,  denotes the magnitude of the couple temperature dislocation, and

ς0= ω−1(z0), ςp ≡ 1/ ¯ς0

Next, for the stress field, the stress functions φE(ς) and ψE(ς) for problem E can

be expressed in the following form:

φE(ς) = φ1E(ς) + φ2E(ς),

where φ2E(ς) and ψ2E(ς) are holomorphic in S+; φ1E(ς) and ψ1E(ς) are expressed as

φ1E(ς) = − 2π C log(ς − ς0) + AElog ς,

ψ1E(ς) = − 2π C log(ς − ς0) + 2π C ω(ς0)

ω(ς )(ς−ς ) + BElog ς

(29)

374 P.C Vinh et al.

with C = k(κ+1) 2Gα e iϑ and AElog ς and BElog ς are functions to cancel the dislocation

of both stress and displacement around the hole Substituting (28) and (29) into (11), and moving around the unit circle once, the requirement that the stress are single val-ued around the hole gives

Substituting (28), (29) and (30) into the displacement expression (12), after mov-ing around the unit circle, the displacements must recover their initial values This

requirement fixes the constant AE in the following expression:

AE= C

2π

N

k=1

Ek

ω(ς0)(ς0− ς k)2+ ¯C

2π

E0ς p2

On account of (28), the traction-free boundary condition (11) is of the form:

φ2E(σ ) + ω(σ )

ω(ς) φ

 2E(σ ) + ψ2E(σ ) = −φ1E(σ ) − ω(σ )

ω(σ ) φ

 1E(σ ) − ψ1E(σ ). (32)

For obtaining φ2E(ς), we substitute (29) into (32), multiply both sides of the resulting

equation by the factor dσ/[2πi(σ − ς)], (ς ∈ S+), and carry out the Cauchy

integra-tion along the unit circle Using the following:

(i) The function (ω(ς)/ ¯ ω(1/ς)) ¯ φ2E (1/ς) is holomorphic in S− except the points

ςk (k = 1, , N), which are poles with the principal parts: E k

ς k −ς φ

 2E(ς k)

ω(ς k)

(ii) The function (ω(ς)/ ¯ ω(1/ς)) ¯ φ1E (1/ς) is holomorphic in S− except the points

ςk (k = 1, , N) and ςp, which are poles with the principal parts, respectively, being:

Ek

ω(ς k)(ς − ςk)

 ¯C

2π

ςpςk

ςp − ς k− ¯AEςk

 and ¯C 2π

ω(ςp)

ω(ς0)

ς p2

and taking φ2E(∞) = 2π C log(−ς0), we obtain the function φ2E(ς) as:

φ2E(ς) = C

2π log

 1

ς − ς0

 +

N

k=1

Ek

ω(ς k)(ς − ςk)



φ2E (ς k) + ¯ AEςk− ¯C

2π

ςpςk

ςp − ς k



+ ¯C

2π

ω(ς0) − ω(ςp) ς2

p

It is noted that the real and the imaginary values of φ2E (ς k) in (34) are determined by

solving the simultaneous equations of 2N derived by differentiating (34) and substi-tuting ς = ς k Thus, the function φE(ς) is:

φE(ς) = C

2π log

 1

ς − ς0

 +

N

k=1

Ek

ω(ς k)(ς − ςk )



φ2E (ς k) + ¯ AEςk− ¯C

2π

ςpςk

ςp − ς k



+ ¯C

2π

ω(ς0) − ω(ςp) ς p2

ω(ς0)(ς − ςp ) − C

2π log(ς − ς0) + AElog ς. (35)

Similar to Section 4, the stress function ψE(ς) can be derived directly by analytic

continuation along the traction-free boundary as:

ψE(ς) = − ¯ φE(1/ς) − ¯ω(1/ς)

ω(ς) φ



As previously stated, by means of the principle of the superposition, after problem E

is solved, the solution of problem C is obtained by using the Green’s functions YE(ς),

and φE(ς) and ψE(ς), respectively, along the line AB (see Equations (45) and (48)).

6 Solution of problem D

Problem D is shown in Figure 1d, in which an infinite plane having a hole with

an edge crack is under distributed edge dislocations along the line crack AB The hole edge and crack surfaces are traction-free Naturally, by means of the principle

of superposition, problem D can be reduced to a problem F, in which an edge point

dislocation with magnitude D is placed at a point z0 (= ω(ς0)) on the line AB

Simi-lar to the previous sections, the Green’s function of problem F: φF(ς) and ψF(ς) can

be found in the form:

φF(ς) = φ1F(ς) + φ2F(ς),

where φ2F(ς) and ψ2F(ς) are functions holomorphic in S+ and (Hasebe and Chen, 1996):

φ1F(ς) = − D

2π log(ς − ς0),

(38)

ψ1F(ς) = − D¯

2π log(ς − ς0) +

D

2π

ω(ς0)

ω(ς0)(ς − ς0) .

From (37), the traction-free boundary condition (11) for problem F is of the form:

φ2F(σ ) + ω(σ )

ω(ς) φ

 2F(σ ) + ψ2F(σ ) = −φ1F(σ ) − ω(σ )

ω(σ ) φ

 1F(σ ) − ψ1F(σ ). (39)

Multiplying both sides of (39) by the factor dσ/[2πi(σ − ς)], (ς ∈ S+), and carrying

out the Cauchy integration along the boundary yield the result of φ2F(ς) Similar to

the procedures used in the section 5, we obtain:

φ2F(ς) = D

2π log

 1

ς − ¯ς0

 +

N

k=1

Ek

ω(ς k)(ς − ςk ) φ

 2F(ς k) + D¯

2π

1

¯ς0− ¯ς

k

+D¯

2π

ω(ς0) − ω(ςp) ς p2

It should be noted that the real and imaginary parts of φ2F (ς k) in (40) are

deter-mined by solving the simultaneous equations of 2N derived by differentiating (40)

376 P.C Vinh et al.

and substituting ς = ς k Taking into account (37), (38) and (40), the function φF(ς)

is obtained

φF(ς) =

N

k=1

Ek

ω(ς k)(ς − ςk ) φ

 2F(ς k) + D¯

2π

1

¯ς0− ¯ς

k

+D¯

2π

ω(ς0) − ω(ςp) ς p2

ω(ς0)(ς − ςp) − D

2π log(ς − ς0) +

D

2π log

 1

ς − ¯ς0



The stress function ψF(ς) is derived directly by analytic continuation along the

traction-free boundary of the unit circle as:

ψF(ς) = − ¯ φF(1/ς) − ¯ω(1/ς)

ω(ς) φ



By means of the principle of superposition, the solution of problem D can be

obtained by using φF(ς) and ψF(ς), respectively, along the line AB (see Equation

(48))

7 Solution of problem A: Integral equations

The original problem A can be reduced to the superposition of three subproblems

B, C and D which have been studied in Sections 4–6, respectively Thus, from these results, the solution of problem A is derived as follows:

Temperature function:

where YB(ς) is defined by (17) and YC(ς) is obtained by integrating YE(ς) with

corresponding density along the line AB

Stress functions:

φ(ς) = φB(ς) + φC(ς) + φD(ς),

where φB(ς) and ψB(ς) are defined by (23), (26); φC(ς) and ψC(ς), and φD(ς) and

ψD(ς) are obtained by integrating the Green’s functions φE(ς) and ψF(ς), and φF(ς)

and ψF(ς), respectively, along the line AB.

It should be noted that in (43) and (44), there are still two unknown functions

(t) and D(t) Therefore, in order to calculate the solution of problem A, we need

to find equations (singular integral equations) for the unknown functions: the

distri-bution density of the temperature dislocation γ (t)dt = d(t) and the density of edge dislocation D(t) From the adiabatic condition along the crack faces expressed by (6) and (43), the following singular integral equation for γ (t) is derived:

Im





b



−b

γ (t)YE(s, t)dt −

b



−b

γ (t)YE(s0, t)]dt



 =−Im [YB(s) − YB(s0)] , |s| ≤ b, (45)

where s0 is an arbitrary point (standard point) on the crack face AB, Y E is expressed

by (27) with  = 1 Function γ (t) can be further reduced to the following form for a

crack problem:

an edge crack is under distributed edge dislocations along the line crack AB The hole edge and crack surfaces are traction-free Naturally, by means...

where φB(ς) and ψB(ς) are defined by (23), (26); φC(ς) and ψC(ς), and φD(ς) and< /i>

ψD(ς)...

ψD(ς) are obtained by integrating the Green’s functions φE(ς) and ψF(ς), and φF(ς)

and ψF(ς),