Abstract—The generalized arbitrary cracked problem in an elastic plane is discussed in this paper. Comparing with the classical plane elastic crack model, in the discussion ， the problem solving become much easier through modifying some known conditions instead of adding any now known condition. The general exact analytic solution is given here based on this formulation though the problem is very complicated. Furthermore, the stress intensity factors IK , IIK of the problem are also given.

I. INTRODUCTION ENERALIZED curve cracks problem is one of the most important problems in plane elastic theory, which attracted many mathematicians and engineers. Since the

stress intensity factors could help to design new materials, the problem has great significance in both theory and application. Although great progress has been made in the region and a lot of effective solutions have been obtained on some special problems[1-2] during the past decades, the general method of solving a complicated curve cracks model has still not been set up. Because of the complex nature of the problem, researches in this area has been focusing on how to find the solution of the theory problem relate to the model, for instance, the existence and the uniqueness of solution. In fact, what engineers really care is the solution to the practical problem, which is just what we intent to find out here.

In the classical works [3] and all papers, the general method of dealing with the plane elastic crack problem based on the theory of elasticity. To be specific, we divide the problem into two parts according to the known conditions. One part is stress boundary value problem and another is displacement boundary value problem. It is the starting point and basis for further discussion.

It is well known, one of the most useful and effective method of solving the cracks problem is a complex variable function method [3-6]. Perhaps the purpose of works which can be read recently is just to prove the existence and the uniqueness of solution of the problem, so researchers always begin their study with using a special transform, for example, Sherman transform or the like to transfer the original problem to solve a regular singular integral equation or a integral

Manuscript received March 28, 2012. This work was supported by the

China Dalian Maritime University under Grant QN2012062. F. A. Ding Yun is with Mathematics Department of Dalian Maritime

University, 1 Linghai Road, Dalian, CO 116023 P.R.China (e-mail: dingyun@ dlmu.edu.cn).

S. B. Yang Xiaochun is with Mathematics Department of Dalian Maritime University, 1 Linghai Road, Dalian, CO 116023 P.R.China (phone:0411-84729207;fax:0411-84723070;e-mail:yangxiaochun@dlmu.edu.cn).

equation. After being transformed, the general equation was commonly written as [7-8]

1 2

( ) ( )( ) ( )2

( , ) ( ) ( , ) ( ) ( )L

L L

B tA t t di t

k t t d k t t d f t

ω τω τπ τ

τ ω τ τ ω τ

+−

+ + =

∫

∫ ∫,

(1.1) Where A t B t k t k t( ), ( ), ( , ), ( , ),1 2τ τ and f t( ) are known functions, and satisfy the Holder condition; L the sum of cracks; ω( )t the conjugation of ω( )t ; and A t B t( ) ( )± ≠ 0 .

As we all known, it is almost impossible for us to get the solution directly from solving the equation (1.1) because the equation is too complex. That is why the researchers give up obtaining the solution itself but to discuss the existence and the uniqueness of solution.

It seems that some known conditions are added here, but all of them are reasonable. Because these known conditions come from reality and they could be obtained in engineering. In fact, some of them were just neglected by us in the past. On this basis, the original problem is reduced to a well known solvable problem---Riemann-Hilbert boundary value problem and then a closed form of complex Airy function expression is obtained. The expression is exact and simple. Furthermore, the formulae of stress intensity factors K KI II, are given.

II. MATHEMATICAL MODEL OF A GENGRALIZED ELASTIC PLANE CRACKS PROBLEMS

Generally, the mathematical model that an infinite elastic plane exist arbitrary curve cracks is supposed as showing in Fig.1

There, Lj ( j=1, 2, …, p ) is the curve crack whose positive direction takes from aj to bj .

Discussion of Multi-cracks Problem

Yun Ding Xiaochun Yang

G

2012 Third International Conference on Intelligent Control and Information Processing July 15-17, 2012 - Dalian, China

978-1-4577-2143-4/12/$26.00 ©2012 IEEE 521

A..Plane stress problem In the situation, the known conditions are given by stress.

That is, X j ( t ) + iYj ( t ) ( ∈ H, j=1, 2, ..., p) is the known external stress acted on the crack L j , and it would be given as

( ) ( )

( ( ) ( )) ( ( ) ( ))j j j j

j j

n n n n j

X t iY t

X t X t i Y t Y t t L+ − + −

+

= + + + ∈,

(j=1, 2,…, p) , (2.1)

where X t iY t Hn nj j

± ±+ ∈( ) ( )( ) is the external stress that

respectively acted on both positive and negative side of Lj . Without losing of the generality, let the Xj+iYj=0 at very beginning ( j=1, 2 , …, p) , here

X iY X t iY t dsj j j jL j

+ = +∫ [ ( ) ( )] ,

( j=1,2,…,p ). In other words, the main external stress vector along the

cracks is supposed to a self-equilibrium system.

B. Plane strain problem The known conditions would be given by displacement in

the time. It is supposed that the displacement on the two sides of

cracks is known and belongs to H else. This means, if we note

)()( tivtu jj±± + ( j=1,2,…,p ) as displacement at the point

jLt ∈ in the positive and the negative side respectively,

)()( tivtu jj±± + , furthermore,

)](-)([)](-)([)( tvtvitututg j−+−+ += t L j∈ (

j=1,2,…,p ) , (2.2)

)]()([)]()([)( tvtvitututh j−+−+ +++= t L j∈

(j=1,2,…,p) are known here. For solving the problem of the two kinds of problems,

elasticity theory should be used here. According to the theory of plane elasticity [3], the stress function σ σ σxx xy yy, ,

and the displacement function u , v can be expressed by two complex Airy functions ( ), ( )z zφ ψ ( z x iy= + )

σ σ φσ σ σ φ ψ

μ κφ φ ψ

xx yy

yy xx xy

zz z z

u iv z z z z

+ = ′− + = ′′ + ′

+ = − ′ −

42

2

Re ( )[ ( ) ( )]

( ) ( ) ( ) ( )

,

(2.3) where φ ψ( ), ( )z z is analytic function of complex variable

z ; z z z, ( ) , ( )φ ψ are the complex conjugation of z z z, ( ) , ( )φ ψ respectively ,

′ = ′ =φ φ ψ ψ( ) ; ( ) ,z ddz

z ddz

and

κνν ν

=−− +

⎧⎨⎩

3 43 1

plane strainplane stress( ) ( )

where the parameters μ and ν are the shear modules and the Poisson’s ratio of the material respectively .

III. PRACTICAL ENGINEERING PROBLEM AND SOLVING

If we just consider how to use the least known conditions to obtain the solutions of the problem, in other words, solving the problem from the view of mathematics, the problem is solved in our past work (seeing refs. [9]), and results are obtained.

A. Stress problem In engineering, the cracks are always assumed as a straight

line segment but curve line indeed. The stress problem is discussed of two simple cases of tension and shear. That is, assuming the crack surface or two sides is subjected to a constant pressure p j or τ j on the two sides of cracks L j (j=1, 2, …, p ). The known conditions imply that the displacement of the two sides on cracks is equal to each other. Hence, the known condition in the stress problem, according to the complex variable function method, would be written as below boundary value conditions:

j( ) ( ) ( ) ( )

, 1, ,j

t t t t f t

t L j p

ϕ ϕ ψ± ± ± ±′+ + =

∈ =

(3.1)

[ ( ) ( )] [ ( ) ( )]

[ ( ) ( )] 0,, 1, ,j

t t t t t

t tt L j p

κ ϕ ϕ ϕ ϕ

ψ ψ

+ − + +

+ −

′ ′+ − +

− + =∈ =

(3.2) where

∫ +=± t

a jj dstiYtXtfj

)]()([)(j

ja

pajb

1b

1apbja

pajb

1b

1apb

Fig. 1. generalized crack model in an infinite elastic plane

522

j

t

ap d t= ∫ = 2 p t ± C j , (3.3)

C j is a known constant，it would be determined soon. By the equation (3.1) and (3.2), we have

2( ) ( ) ( )1

, 1, ,

j j

j

t t p t C

t L j p

ϕ ϕκ

+ −= − + ++

∈ =.

(3.4) Equation (3.4) is a constant coefficient regular

Riemann-Hilbert boundary value problem. Its solution can be written out easily [3, 6] ,

j2 ( )( )1 2 ( )( )

j

L

p t CX zz dti X t t z

φκ π +

+=

+ −∫ (3.5)

1 2

1 2

( ) ( )( ) ( )

( )( ) ( )

p

p

X z z a z a z a

z b z b z b

= − − − ⋅

− − − , (3.6)

∑=p

jLL1

C j is determined by

CX t

dtjLj

= ∫1( )

, ( , , , )j p= 1 2

, and the branch is taken as 1)(lim =−

∞→zXz p

z . (3.7)

Hence, the general expression of φ( )z is determined. With substituting φ( )z into equation (3.1), (3.2), and

by some simple algebraic and conjugate operations, the other complex Airy function ψ ( )z is obtained,

ψπ

( ) ( ) ( )( )( )

z X zi

H tX t t z

dtL

=−+∫2

, (3.8)

where

( ) 2 ( ) ( )

[ ( ) ( )]

jH t p t C t t

t t t

ϕ ϕ

ϕ ϕ

+ −

+ −

= + − −

′ ′− +.(3.9)

Thus, the general solution of the problem is given in analysis form.

B. Stains problem

Supposing that the displacement )()( tivtu jj±± + ( j = 1,

2, …, p ) is known and belongs to H. According to the complex variable function method, the known conditions would be written as below boundary value conditions:

( ) ( ) ( ) 2 [ ( ) ( )],

, 1, ,j j

j

t t t t u t iv t

t L j p

κϕ ϕ ψ μ± ± ± ± ±− − = +

∈ =

(3.10)

Because the external stress main vector along the cracks is supposed to a self-equilibrium system, the displacement of the two sides on a crack is often symmetrical. This fact implies that the condition

)()()( tttt +++ +′+ ψφφ

+ )()()( tttt −−− +′+ ψφφ = 0 (3.11)

is satisfied. Similar to the discussion in stress problem, we have

( ) ( ) ( )1

, 1, ,

j

j

t t h t

t L j p

μϕ ϕκ

+ −= − −+

∈ =.

(3.12) Equation (3.13) is still a constant coefficient regular

Riemann-Hilbert boundary value problem. Its solution can be written out like

dtzttX

thizXz

L∫ −+

= + ))(()(

2)(

1-)(

πκμφ ,

(3.13) Here, =)( th )( th j ( j = 1, 2, …, p ).

dtzttX

tGizXz

L∫ −

= + ))(()(

2)()(

πψ ,

(3.14) where

( ) ( ) ( ) ( ) ( )

[ ( ) ( )]

j jG t u t iv t t t

t t t

ϕ ϕ

ϕ ϕ

+ + + −

+ −

= + − −

′ ′− +,

(3.15) or

_ _( ) ( ) ( ) ( ) ( )

[ ( ) ( )]

j jG t u t iv t t t

t t t

ϕ ϕ

ϕ ϕ

+ −

+ −

= + − −

′ ′− +.

(3.16) The problem is solved too.

IV. STRESS ANALYSE In fracture mechanics, one of the most important

parameters is stress intensity factor. Its generalized definition is

)()(lim 21

11

zzziKKKzzIII φ′−=+=

→.

(4.1) Since stress intensity factor is just relative to (z)φ ′ ,

noted as (z)Φ , (z)Φ can be solved directly from below equation

2( ) ( )1

, 1, ,

j

j

t t p

t L j pκ

+ −Φ = −Φ ++

∈ =.

(4.2)

523

Because the complex Ariy function (z)Φ exists 1/2 order singularity at tip of every cracks L j, (4.2) need to be solved in h 0 class [5, 6] .

2 1 1 ( )1 2 ( ) ( )

(in pressure case)( )

2 1 1 ( )1 2 ( ) ( )

(in shear case)

j

L

j

L

p X t d ti X z t z

zX t d t

i X z t z

κ π

τκ π

+

+

⎧⋅ ⋅⎪ + −⎪

⎪⎪Φ = ⎨⎪ ⋅ ⋅⎪ + −⎪⎪⎩

∫

∫

(4.3) After some simple calculation [3,6], the stress intensity

factors can be written as

12lim( ) (z)

j

I II

jz z

K K iK

z z→

= +

= − Φ

12lim( )

jjz z

z z→

= −

p

1

p

1

( ) 1 1( )1 2

(in pressure case)

( ) 1 1( )1 2

(in shear case)

j

j j

j

j j

p X zi z a z b

X zi z a z b

κ π

τκ π

⎧⋅ ⋅ +⎪ + − −⎪

⎪⎪= ⎨⎪ ⋅ ⋅ +⎪ + − −⎪⎪⎩

∑

∑

,

(4.4) where z j is taken from a a a b b bp p1 2 1 2, , , , , , , . Expression (4.4) is the general calculation formula of

stress intensity factors of the problem.

Example When crack just is a single centre crack or it can be seen as

segment [-a, a], the stress intensity factor would be

I

II

2 ( 1)

2( 1)

aK K p

Ka

K K

πκ

πτ

κ

⎧= =⎪ +⎪= ⎨

⎪ = =⎪ +⎩

.

The results here are almost similar to the results [1-3]. It is easy to verify that the results are correct in the case that all cracks are collinear [7,10].

If someone want to study the strain problem and to get the stress intensity factor, our solving is still effective.

V. CONCLUSION AND DISCUSSION The paper provides a formula to determine stress intensity

factors K KI II, directly in the case of multi-cracks on an elastic plane.

The results discover a new fact that the stress intensity factor is relative to the physics parameter.

Formula (4.4) illustrates that the stress intensity factor in complicated situation is still determined by several factors which play an important role in the situation of a few straight segment cracks.

The method could be used to the multiple curve cracks case.

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Mech., 1965 [2] Isida M. Methods of Analysis and Solution of Crack Problems. ed. Sih G

C. Noordholf Int. Pub. 1973 [3] Muskhelishvili N I. Some Basic Problems of the Mathematical Theory of

Elasticity. 2nd. English ed. Noordholf, Groningen. 1963 [4] Gakhov F D. Boundary Value Problems. Pergamon, Oxford and

Addisonwesley, Reading Mass: 1966 [5] Muskhelishvili N I. Singular Integral equation. Shanghai Sci and Tech

Pub. Shanghai: 1962 (in Chinese) [6] Lu J K. Analytic Function Boundary Value Problem. Shanghai Sci and

Tech Pub. Shanghai: 1987(in Chinese) [7] Lu J K. Complex method on Plane Elasticity. Wuhan University Pub.

Wuhan: 1986(in Chinese) [8] X C Yang, L Wang, and T Y Fan. Analytic solution of an asymmetrical

crack in a strip, Chin. Phy.Lett., 15(2), 1998: 117-119 [9] X C Yang, T Y Fan, and S Q Liu. New formulation for plane elastic

generalized arbitrary cracks problem and its stress intensity factor, J. of Beijing Institute of Technology, Vol.16(4),1999 ,364-369

[10] Ioakimidis N I, Thecaris P C. A system of cuvilinear cracks in an elastic half-plane. Int J of Fract, 1979,15: 299~309

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