Adv. Studies Theor. Phys., Vol. 7, 2013, no. 9, 423 – 446 HIKARI Ltd, www.m-hikari.com

Elementary Mathematical Theory for Stress

Singularities at the Vertex of Plane Infinite Wedges

Yos Sompornjaroensuk 1,*, Wichitr Boonchareon 1,†, and Parichat Kongtong 2

1 Department of Civil Engineering, Faculty of Engineering Mahanakorn University of Technology, Bangkok, 10530, Thailand

* yos@mut.ac.th † valvevai48443@yahoo.com

2 Department of Automotive Engineering, Faculty of Engineering

Thai-Nichi Institute of Technology, Bangkok, 10250, Thailand Correspondence author: parichat@tni.ac.th

Copyright © 2013 Yos Sompornjaroensuk et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A particular singular problem in the class of plane linear elasticity theory that plays an important role in the development of linear elastic fracture mechanics (LEFM) is reviewed and mathematically studied. The specific problem is started with consideration of a planar wedge having stress-free on both radial edges but the remainder subjected to in-plane loading at the far-field. Employing the semi-inverse method together with the Fadle eigenfunction expansion technique, its solution can be written in terms of Airy stress function satisfying the two-dimensional biharmonic governing differential equation. Further analysis leads to a special case of crack problems, which is in the field of elastic fracture mechanics. Mathematics Subject Classification: 31A25, 74A45, 35Q74

Keywords: Airy stress functions, Biharmonic equation, Fracture mechanics, Singularities, Wedge.

424 Y. Sompornjaroensuk, W. Boonchareon and P. Kongtong 1 Introduction Because no real materials are capable of sustaining an infinite stress and if only consider no singularities due to the loadings, stress singularities are always associated with discontinuities in the geometry [15,19,54] or the change in type of boundary conditions [27,34,37,47,61,68,70]. It is important to remark on the work of England [30], who investigated the dependence of the singularities on the local geometry and on the types of boundary conditions and found out that the geometry has large influence on the magnitude of the singularity. Since stress singularities violate the basic assumptions behind mathematical solutions in linear elasticity theory [65] and behavior of a real material, hence, any situation having an infinite stress as predicted by the linear theory of elasticity will usually lead to yielding or some other kinds of failure in practice [23,53,57]. Pengfei et al.[49] used the complex stress functions in terms of power infinite series with the conformal mapping to study the orders of the stress singularity at the corner point of a diamond-shape rigid inclusion or hole in an infinite elastic plate under antiplane bending. At that period, Kohno and Ishikawa [40] also developed a new method to calculate the orders of stress singularity and stress intensities for the corner of a polygonal hole and a rigid polygonal inclusion under antiplane shear. Chue et al.[21] derived the characteristic equation governing the singularity orders of an anisotropic wedge for various boundary conditions from Lekhnitskii’s complex function method. The results showed that the singularity orders depend upon the wedge angle, boundary conditions, and material properties. Harding et al.[33] utilized the 3D finite element method and a standard regression technique for characterization of the notch singular modes. The influence of Poisson’s ratio, plate thickness and notch opening angle on the value of the notch stress intensity factor was comprehensively studied. Physically stress singularities correspond to regions of high stress in which plastic flow or even fracture of the material may occur, because the points having stress singularities are often the points of initiation of cracks. Generally, they can be found in many problems containing the initial cracks in the body [13,16,17,29,43], which are in the field of fracture mechanics [1,14,35,52]. Using the eigenfunction expansion method, Xie and Chaudhuri [73] obtained three-dimensional asymptotic stress fields in the vicinity of the front of a semi-infinite crack along a bimaterial interface of a plate that subjected to far-field extension and bending. Shahani and Adibnazari [55] applied the Mellin transform to determine the solution of governing differential equations for studying the antiplane shear deformation of perfectly bonded wedges as well as bonded wedges with an interface crack. In addition, Shahani [56] derived the analytical expressions by the conformal mapping techniques for mode III stress intensity factor of circular shafts with edge cracks, bonded half planes containing an interfacial edge crack, and bonded wedges with an interfacial edge crack. A semi-analytical boundary element method based on finite elements that called the scaled boundary finite element method was applied to fracture mechanics problems by Song and Wolf [58]. This method has been extended to analyze the in-plane singular stress fields at cracks and multi-material corners for evaluating

Elementary mathematical theory for stress singularities 425 orders of singularity and stress intensity factors [59]. Recently, Yoon and Chaudhuri [76] employed the eigenfunction expansion method for finding three-dimensional asymptotic displacement and stress fields in the vicinity of the front of a crack / anticrack type discontinuity weakening/reinforcing an infinite pie-shaped trimaterial plate that subjected to antiplane shear far field loading.

Figure 1 The state of stresses from the crack-tip

Based upon the linear theory of elasticity and together with confining the region of plastic flow to be very small and/or in the absence of plastic deformation, with these assumptions, the stress distributions and also the stress components near ahead of a crack-tip as seen in Figure 1 can be evaluated within the framework of linear elastic fracture mechanics (LEFM) as long as the plastic zone that occurred around the crack-tip is very small [23,43]. This is the case when the stress is low with respect to the yield stress of material strength. Furthermore, it can generally be noted that one of the important elastic parameters, which has been mostly used in fracture analysis is the stress intensity factor [7,29,40,43,56,59]. This factor significantly denotes the strengths of singularities and is mainly used to distinguish between different cracks that depended on different modes of failure as illustrated in Figure 2. Chaudhuri and Xie [17] derived the explicit expressions for three-dimensional singular stress fields in the neighborhood of the front of a semi-infinite crack. Additionally, the relationships between the strain energy release rate and the stress intensity factor was investigated. Im and Kim [36] used the eigenfunction expansion together with the M-integral for computing the local stress intensity factor of the singular near-tip field around the vertex of a generic composite wedge. Mohammed and Liechti [48] studied the effect of the corner angle on bimaterial corner stresses using the Betti’s law based reciprocal work contour integral theorem. The singularity and the stress intensity factors were found to depend on the angle. Similarly, Qian and Akisanya [51] applied a contour integral based on Betti’s reciprocal theorem in conjunction with the finite element method to evaluate the magnitude of the wedge corner stress intensities near the interface corner of a bi-material joint. Lee et al.[44] computed the stress intensities of 3D corner singularity for the tip of a transverse

 

x

y

z

zzzx

zyxx

yyyx

crack front r

yz

xyxz

426 Y. Sompornjaroensuk, W. Boonchareon and P. Kongtong crack terminating on the free surface in a laminated composite based on the two-state M-integral in conjunction with eigenfunction expansion. Kotousov [42] summarized theoretical and numerical studies and discussed some important features of three- dimensional singular solutions for sharp notches. Also the relationships between the intensities of the singular stress states corresponding to the 3D linear elastic solutions and the plate thickness were established in dimensionless form.

Figure 2 Three basic modes of failure in fracture mechanics

For certain failure modes in any isotropic linear elastic cracked body and subjected to applied external forces [9-12,31,38,60], it is possible to analytically derive the closed-form expressions for the stress fields in the vicinity of a crack dominated by the singularity of stress, depending on crack length, applied stress and configuration factor. Using separation of variables in the Airy stress function, Dempsey and Sinclair [25] examined and derived the singularities of the two- dimensional N-material composite wedge. In case of an isotropic wedge loaded by uniform antisymmetric shear tractions, the logarithmic stress singularity was found by Dempsey [26]. Furthermore, Dempsey and Sinclair [24] provided the analytical information on the stresses which can occur in the vicinity of the vertex of a two- dimensional bi-material wedge for a wide range of boundary and interface conditions. Stampouloglou and Theotokoglou [62] analyzed the problem of infinite isotropic wedge under shear and normal distributed loading along its external faces by the separation of variables method in which the stress and displacement fields were determined using the equilibrium conditions for forces and moments and the appropriate Airy stress function. The wedge loaded at its apex by a concentrated force was analyzed by Theotokoglou and Stampouloglou [64]. Consider the polar coordinates ( , )r with its origin placed at the center point along the line of a crack front in z - direction as illustrated previously in Figure 1,

the stress components ( )Nij corresponding to the three basic modes of failure that

presented in Figure 2 and , ,N I II III representing the mode numbers, can be given by,

( ) ( ) / 2 ( )

0

( ) ( )2

N N n nNij ij n ij

n

Kf A r g

r

, (1)

 

Elementary mathematical theory for stress singularities 427 where NK are the stress intensity factors ( )( , , ), ( )N

I II III ijK K K f and ( ) ( )nijg are

the known function of , and nA is a constant. It can be noted that the leading term

presented in (1) is proportional to the inverse-square-root stress singularity (1/ )r and approaches infinity when 0r , while the remaining higher order terms are depended only on the problem geometry. These terms are finite values for 0n and approach zero for 0n . Thus the higher order terms are negligible at location close to the crack-tip in comparison with the leading term. Hence, the stress near

the crack-tip varies with 1/ r and exhibits an 1/ r singularity, regardless of the configuration of the cracked body. However, it could be remembered that (1) can be used and is valid only as long as the plastic zone is very small [23,57]. Thus, it is limited to the region near the crack-tip in which the leading term is dominant and the other terms can be neglected [14,35]. Utilizing the concept as described above, the stress intensity factors can be determined from the stress fields or stress components ahead of a crack-tip as follows [1]:

( ) ( )

0lim ( )

2N NN

ij ijr

Kf

r

. (2)

It can also be concluded that (2) is only valid near the crack-tip, while the stresses far from the crack-tip are governed by the remote boundary conditions. In addition, (1) and (2) are restricted to the analysis of isotropic materials. For the problems of anisotropic materials, it can still be analyzed whereas the properties of material and the geometry boundaries have main influence on singularity behavior as expected [2,20-22,46]. It is also known that stress singularities can be encountered in the problems with material discontinuities or multimaterials. The strength of singularities depends on the materials that being used and the interfacial geometry of dissimilar materials [3,18,28,41,50,53,63,67]. For analyzing the stress singularities in three-dimensional problems, the solution methods are based on the three-dimensional theory of elasticity, which are found to be very difficult to solve because of the complexity of the problems [4,5,8,32,39,45,71-76]. Due to the significance of stress singularities arising in various fields of engineering applications, therefore, this kind of problems motivates many researchers to study the problems that involved the orders of stress singularities. Williams [68] first showed the general solution to a particular singular problem in the class of plane linear elasticity theory for an isotropic material that plays an important role in the early development of linear elastic fracture mechanics. The method used to solve the problem is the same to that of his previous work [69] for the plate-bending problems. Because of the analogy between the two classes of problems; namely, the plane elasticity problem [65] and the plate bending problem [66], it is able to convert the results for one class of problems to the results for another class of problems by using the same technique. In the class of plane linear elasticity problems, Williams [68] derived the characteristic equations for all three possible combinations

428 Y. Sompornjaroensuk, W. Boonchareon and P. Kongtong of free and clamped boundary conditions along the radial edges; namely, free-free, clamped-clamped, and free-clamped of isotropic wedges with apex angle 2 in extension. It has been shown that stress singularities occur at the corners having free and clamped boundary conditions if the angle 2 is larger than 63 and for

the other boundary conditions, stress singularities only happen when 2 180

(re-entrant corner). Because a problem of wedge having both free edges can be applied directly in the field of fracture mechanics (crack problems), therefore, only the case of free-free boundary conditions will be emphasized here to revisit in the present paper. In addition, the basic equations that significantly used for analyzing the problems in linear elastic fracture mechanics [35,52] will be given in this paper with details of their derivation. 2 Governing equation, stress field, and stress function Consider a problem of wedge of arbitrary apex angle 2 that has stress-free faces, but otherwise allows any system of forces on the remaining boundary as shown in Figure 3.

Figure 3 The coordinates defined in a wedge Using the semi-inverse method, Williams [68] solved this problem configuration by introducing the Airy stress function ( , )r in polar coordinates, which is easily be expressed in the same form of his previous work [69] using the method of variable separation in the form, ( , ) ( ) ( )r G r F , (3)

in which the functions of ( )F and ( )G r depend on and r , respectively, and( )G r is further assumed in the form as

1

1

( ) nn

n

G r c r

, (4)

 

x

y

Stress Free

Stress Free

Elementary mathematical theory for stress singularities 429 where nc is the unknown constant and n is the eigenvalue parameter (eigenfunction)

to be determined. For the plane linear elasticity problems [65], the governing equation is the two-dimensional biharmonic equation in terms of function ( , )r . Therefore, the differential equation can be found to be

4 2 2( , ) ( , ) 0r r , (5) with

2 2

22 2 2

1 1

r r r r

, (6)

where 2 is the two-dimensional Laplace operator. Substituting (3) for ( , )r into (5) and using (4), yields the following ordinary differential equation

2 2

2 22 2

( 1) ( 1) ( ) 0n n

d dF

d d

, (7)

which has a general solution of ( )F taken by

1 2 3 4( ) cos( 1) sin( 1) cos( 1) sin( 1)n n n nF C C C C , (8)

and 1C , 2C , 3C , and 4C are the arbitrary constants to be determined in the later

stage from the prescribed boundary conditions along straight edges. The stresses and the Airy stress function are generally known to be related as follows [37,65]:

2

2 2

1 1r r r r

, (9)

2

2r

, (10)

1

r r r

, (11)

where ,r are, respectively, defined to be the radial stress and tangential stress,

and r is the in-plane shearing stress.

Substitution of (3) into (9) through (11), leads to the expressions as follows:

1

1

[( 1) ( ) ( )]nr n

n

r F F

, (12)

1

1

[ ( 1) ( )]nn n

n

r F

, (13)

430 Y. Sompornjaroensuk, W. Boonchareon and P. Kongtong

1

1

[ ( )]nr n

n

r F

. (14)

The stress components presented in the above equations; (12), (13), and (14), they can be obtained from the boundary conditions on two radial edges. Thus, the boundary conditions of the stress-free faces of the wedge require the conditions listed below,

( , ) 0r , (15)

and ( , ) 0r r . (16)

Substitution of (13) and (14) into (15) and (16), respectively, the restrictions on ( )F are given by the requirements of

( ) ( ) 0F F , (17) and ( ) ( ) 0F F . (18)

It can be noted that (17) and (18) are provided for 0n by observing (13)

and (14). The next step is to find the characteristic equation of the problem. Applying the resulting boundary conditions given by (17) and (18) into the general form of function ( )F as previously presented in (8) leads to a system of four simultaneous equations in terms of the unknown constants as follows:

1 2 3 4cos( 1) sin( 1) cos( 1) sin( 1) 0n n n nC C C C , (19)

1 2 3 4cos( 1) sin( 1) cos( 1) sin( 1) 0n n n nC C C C , (20)

1 2( 1)sin( 1) ( 1)cos( 1)n n n nC C

3 4( 1)sin( 1) ( 1)cos( 1) 0n n n nC C , (21)

1 2( 1)sin( 1) ( 1)cos( 1)n n n nC C

3 4( 1)sin( 1) ( 1)cos( 1) 0n n n nC C . (22)

These four equations can obviously be separated into two independent sets of equations for the constants 1C , 3C and 2C , 4C , respectively, by the elementary

algebraic operations. Then, additions between (19) and (20), and (21) and (22), one obtains the results as

1 3cos( 1) cos( 1) 0n nC C , (23)

2 4( 1)cos( 1) ( 1)cos( 1) 0n n n nC C , (24)

and also subtractions between (19) and (20), and (21) and (22), yield the following

Elementary mathematical theory for stress singularities 431 results as 2 4sin( 1) sin( 1) 0n nC C , (25)

1 3( 1)sin( 1) ( 1)sin( 1) 0n n n nC C . (26)

It is more convenient to rewrite (23), (26) and (24), (25) in the matrix form as follows:

1

3

cos( 1) cos( 1) 0

( 1)sin( 1) ( 1)sin( 1) 0n n

n n n n

C

C

, (27)

2

4

sin( 1) sin( 1) 0

( 1)cos( 1) ( 1)cos( 1) 0n n

n n n n

C

C

. (28)

The meaningful solutions can be obtained only if the determinants of the coefficient matrices (augmented matrices) given in (27) and (28) are each equal to zero. Thus, the eigenvalues n are provided by the characteristic equations, which is

sin 2 sin 2n n . (29)

This equation ensures the non-trivial solutions of the problem of a wedge with free-free radial edge boundary conditions. For other combinations of edge boundary conditions, they can be obtained by the same procedure introduced here. However, it should be kept in mind that the choice of an Airy stress function used by Williams [68] as given in (3) is sufficient only if the induced stress and displacement behavior is of the power type. Dempsey [26] has pointed out that for the inhomogeneous boundary conditions of wedge loaded by uniform antisymmetric shear tractions, the Williams’ Airy stress function is not enough to handle the logarithmic stresses for the half-plane or crack geometry [24,25]. 3 Reduction to a problem of traction-free on edge crack Considering a special limiting case of a wedge with apex angle , that is a fully closed upon itself of wedge as shown in Figure 4. Therefore, (29) becomes

sin 2 0n , (30)

and the roots of (30) are immediately given by

2n

n ; 0, 1, 2, 3,...n (31)

However, some of these roots yield the physically unacceptable results for the crack problems and they have to be excluded, so that the detailed explanation for the possible nature of this problem field will be stated hereafter.

432 Y. Sompornjaroensuk, W. Boonchareon and P. Kongtong

Figure 4 A traction-free on edge crack For any 0n , the displacements are unbounded at the origin ( 0)r , then,n is positive value only. In the case of 0n , this yields an infinite amount of

strain energy stored in a finite volume, hence, 0n is also rejected. Therefore,

there will correspond a relationship between the constants 1nC and 3nC and 2nC and

4nC in (27) and (28), respectively. Consequently, the general solution consists of the

sum of all terms containing acceptable eigenvalues / 2n n and 0n . Moreover,

it is important to note that not all of the constants 1nC , 2nC , 3nC ,and 4nC are

independent [14,35]. Letting / 2n n and in (27) and (28), the results can

be satisfied for all values of n if,

3 1

4 2

2

2n n

n n

nC C

nC C

; 1,3,5,...n , (32)

and also,

3 1

4 2

2

2

n n

n n

C C

nC C

n

; 2, 4,6,...n (33)

Substitution of (32) and (33) into (8) by setting / 2n n , then, the Airy

stress function (3) becomes

1 / 21

1,3,5,...

2 2 2( , ) cos cos

2 2 2n

nn

n n nr r C

n

 

x

y

Elementary mathematical theory for stress singularities 433

2

2 2sin sin

2 2n

n nC

1 / 21

2,4,6,...

2 2cos cos

2 2n

nn

n nr C

2

2 2 2sin sin

2 2 2n

n n nC

n

. (34)

As it is easily seen in (34), the terms multiplied by 1nC will correspond to the

stress fields that are symmetric with respect to the crack plane (mode I in Figure 2-left), called the opening mode because of being even functions. Similarly, the terms involving 2nC will produce the antisymmetric stress fields (mode II in Figure

2-middle) with respect to 0 , sometimes called the inplane shearing mode, which are the odd functions. However, these two types of stress fields are mutually exclusive and, by using the principle of superposition, can be uncoupled and treated independently. Finally, the third mode in fracture mechanics (mode III in Figure 2-right) which is called the antiplane shearing mode, does not occur in the plane problems of elasticity. 4 Discussion of the solutions For instance as in the case of 2n , the symmetric part of the general solution for the Airy stress function that given in (34) is found to be

212( , ) (1 cos 2 )r r C , (35)

which can be recognized as the Airy stress function for a uniformly uniaxial stress field o in the x-direction [65] as demonstrated in Figure 5.

Figure 5 Uniform stress field o in infinite plate

 

x

y

r

oo

434 Y. Sompornjaroensuk, W. Boonchareon and P. Kongtong Application of stress boundary condition; x o into the stress function in

rectangular Cartesian coordinates for x-direction 2 2( / )x y and by noting that

siny r , then, yields the result as

124o C . (36)

The value of 12C is easily determined by considering the geometry and loading

of the body taken as a whole and cannot be a priori assigned. In order to derive the stress fields around the crack-tip, only the first term of symmetric part for the opening mode (mode I) in (34) corresponding to 1n , can be expressed by

3/ 2 211 12

1 3( , ) cos cos (1 cos 2 ) H.O.T.

2 3 2r r C r C

(37)

where H.O.T. denotes the higher order terms and then, substituting (37) into (9) and retaining explicitly only those terms that do not vanish as approach the crack-tip, which is

1112

35cos cos 2 (1 cos 2 ) H.O.T.

2 24r

CC

r

, (38)

and using (36) for 12C , therefore, (38) becomes

11 35cos cos (1 cos 2 ) H.O.T.

2 2 24o

r

C

r

(39)

Similarly, (10) and (11) can be written as

11 33cos cos (1 cos 2 ) H.O.T.

2 2 24oC

r

, (40)

11 3sin sin sin 2 H.O.T.

2 2 24o

r

C

r

(41)

For the in-plane shearing mode (mode II), by using the same procedure as described in mode I, the stresses are sought in the forms:

21 35sin 3sin H.O.T.

2 24r

C

r

, (42)

21 33sin 3sin H.O.T.

2 24

C

r

, (43)

21 3cos 3cos H.O.T.

2 24r

C

r

(44)

Elementary mathematical theory for stress singularities 435 It should be remarkable that (39) to (44) may be called the near-field equations or the modified near-field equations when the term of o is included [52]. These

equations describe the distribution of the stress field over some regions around the crack-tip. It is also seen that they all contain the same inverse-square-root singularity

(1/ )r multiplied with the known functions of , regardless of the geometry or details of loading [68].

Figure 6 Polar displacements at a wedgelike notch The radial and circumferential displacements ( , )ru u as demonstrated in Figure 6

can be obtained by integrating the strain-displacement relationships [65] which are represented by

rr

u

r

, (45)

1r uu

r r

, (46)

1 r

r

u uu

r r r r

, (47)

where r , , r are the polar strain components. Utilizing the generalized Hooke’s

law, the strains can further be expressed in terms of the stresses as follows:

r rE , (48)

rE , (49)

2(1 )r r r

EG

, (50)

that satisfied for plane stress problems, and

2 (1 )r rG , (51)

 

r

u ru

436 Y. Sompornjaroensuk, W. Boonchareon and P. Kongtong

2 (1 ) rG , (52)

r rG . (53)

for plane strain problems. The material properties E , G , and presented above are called the Young’s modulus, shear modulus, and Poisson’s ratio, respectively. These properties are related as / 2(1 )G E . After that, applying two sets of (48) to (50) and (51) to (53) into (45) to (47) the displacements are, for mode I:

1/ 211 3(2 1)cos cos ...

4 2 2r

Cu r

G

, (54)

1/ 211 3(2 1)sin sin ...

4 2 2

Cu r

G

, (55)

and for mode II:

1/ 221 3(2 1)sin 3sin ...

4 2 2r

Cu r

G

, (56)

1/ 221 3(2 1)cos 3cos ...

4 2 2

Cu r

G

, (57)

where the constant term is known as the Kolosov’s constant defined by [65]

3

1

for conditions of plane stress, (58a)

3 4 for conditions of plane strain. (58b)

Generally, (54) to (57) may be simply cast as follows:

1/ 2

1( )r n

u gr

, (59)

2/ 2

1( )

nu g

r , (60)

in which 1( )g , 2 ( )g are the known functions of . Note that if all roots 0n ,

the displacements are unbounded at 0r . For the case of 0n or 0n , this root

gives the stress and strain tensors ( , )ij ij of the type [1]

1

( )ij mr

, (61)

1

( )ij nr

, (62)

while ( )m and ( )n are the known functions in terms of only.

Elementary mathematical theory for stress singularities 437 The strain energy density oU of an infinitesimal element with an arbitrary

stress state for a linear elastic body is given by [65]

1

2o ij ijU , (63)

and also using (61) and (62), the strain energy becomes

2

1( ) ( )

2oU m nr

. (64)

Integrating (64) over a closed region surrounding the crack-tip, the total strain energy U stored within any circular area r R enclosing the crack front would

become unbounded as or [35],

2

0 o

R

o

r

U U rdrd

. (65)

Again if 0n , the general solution that given in (3) yields the physically

unacceptable results for the crack problem in real materials. Thus, the Airy stress function presented in (34) is the complete solution for problem of a single-ended sharp crack. For the last mode crack (mode III) which is the case of antiplane shearing mode, considering a body is bounded by two intersecting planes defining the solid within the angle 2 and by remote surfaces otherwise. The state of stress components is then shown in Figure 7. In this case, the states of stress and strain are independent of the coordinate z measured along the line of intersection. The certain stresses and displacements are known as [65]

0r r z ru u , (66)

and the nonzero stress components are rz , z , and also the nonzero displacement

is zu .

Figure 7 The cylindrical coordinates plane defined in antiplane stress components

 

r

rz

z x

yz

Free

Free

438 Y. Sompornjaroensuk, W. Boonchareon and P. Kongtong A closely realized problem is presented by considering the case of torsion, near shallow notch around the periphery [6,35,56] as indicated in Figure 8.

Figure 8 A torsion of near shallow notch around the periphery The equilibrium equation for antiplane state in polar coordinates is given by

( ) 0zrzr

r

, (67)

and the Hooke’s law for nonzero stress components are presented as follows:

zrz rz

uG G

r

, (68)

zz z

uGG

r

. (69)

The boundary conditions for the problem considered according to Figure 7 are enforced by the conditions:

0z or 0zu

; . (70)

Back substitution of (68) and (69) into the equilibrium equation of (67) leads to the potential equation for zu , which is a typical Laplace’s equation. Thus,

2 0zu , (71)

and further introducing the product form of solution zu as

1,2,3,...

( )nz

n

u r g

, (72)

where ( )g is the unknown function of only, and n is the eigenvalue to be

determined. Therefore, substituting (72) into (70) and (71), one obtains the new governing equations in terms of ( )g and n as follows

 

r

z

Elementary mathematical theory for stress singularities 439

2

22

0n

d gg

d

, (73)

0dg

d ; . (74)

The solution of (73) satisfying the antisymmetric displacement with respect to 0 is taken by

*( ) sinn ng D , (75)

where *nD is an arbitrary constant. The characteristic equation for n can be determined

by applying (75) into (74), which yields the result shown below,

cos 0n . (76)

If the sharp crack is of interest, setting in (76) gives the eigenvalues as

/ 2n n ; 1, 3, 5,...n , (77)

and the displacement and stress components involving the terms dominant at the crack-tip can be written from (68), (69), and (72) as in the following forms:

* 1/ 21 sin ...

2zu D r

, (78)

* 1/ 21 sin ...

2 2rz

GD r

, (79)

* 1/ 21 cos ...

2 2z

GD r

(80)

It can immediately be seen that the stresses are singular in the order of an inverse-square-root type at 0r . Because the stresses along the extended crack line and displacements along the crack sides are of particular importance, therefore,

it is more convenient to introduce the coefficients 11C , 21C , and *1D in the three

crack modes as follows [1,35]:

112

IKC

, (81)

212

IIKC

, (82)

*1

2IIIKD

G , (83)

in which the scalar quantities; , ,I II IIIK K K denote the strengths of singularities

440 Y. Sompornjaroensuk, W. Boonchareon and P. Kongtong corresponding to the crack modes I, II, III, respectively, and are called the stress intensity factors characterizing the dominant elastic fields near the crack-tip in the theory of linear elastic fracture mechanics. Furthermore, based on the superposition of the three modes it may be possible to describe the general case of failure mechanisms, because many problems, in general, are of the mixed mode type. However, the mode I is technically the most important. This is due to the fact that cracks which start in a single mode may become mixed mode later [1,14]. 5 Application in fracture mechanics In order to analyze the problems in fracture mechanics, it is useful to consider the crack embedded in the body as shown in Figure 9. The orthogonal coordinates

1 2 3( , , )x x x is used to define the quantities that involved in the analysis, which may

be the rectangular coordinates, cylindrical coordinates or the others corresponding to the problem formulation. Based on the linear elastic fracture mechanics, the stress components ( )ij and displacement components ( )iu that retained only the

dominant terms of interest of three crack modes are summarized in the following expressions below [35].

Figure 9 The coordinates 1 2 3( , , )x x x defined in a cracked body

For the opening mode (mode I); 2 0x ,

11 22 0 012

Ir

K

x

; 1 0x , (84)

12

1

2 2I

xu u K

G

; 1 0x . (85)

For the inplane shearing mode (mode II); 2 0x ,

21 012

IIr

K

x

; 1 0x , (86)

 

3x1x

2x

Elementary mathematical theory for stress singularities 441

11

1

2 2r II

xu u K

G

; 1 0x . (87)

For the antiplane shearing mode (mode III); 2 0x ,

23 012

IIIz

K

x

; 1 0x , (88)

13

2

2III

z

K xu u

G

; 1 0x , (89)

where iu represents the small displacements on the positive and negative faces (or

sides) with respect to that position considered and the constant is previously defined in (58). 6 Conclusion The principal aim of this paper is to present and describe the necessity of stress singularities considerations that can arise in the problems of plane linear elasticity theory. Thus, a particular singular problem of planar wedge with an apex angle 2 and having stress-free on two radial edges, but subjected to in-plane loading on the remaining boundary that have been analyzed by Williams [68] is reconsidered here. The details of derivation for obtaining the possible solutions of problem are clearly explained with some discussions on the results. Further analysis leads to a limiting case of problem with sharp edge crack in extension that is very important in the development of linear elastic fracture mechanics. However, in order to analyze the problems found in this class, it should be remembered that the obtained results are accurate as long as the plastic zone near the crack-tip is assumed to be very small and the actual stress due to the applied load are not larger than the yield stress of material. Acknowledgements The financial support from the Thai-Nichi Institute of Technology for the publication of this paper is gratefully acknowledged. Significantly, the authors would like to express their sincere appreciation to Asst. Prof. Dr.Sutja Boonyachut from the Department of Civil Engineering, Mahanakorn University of Technology for his careful proof reading the manuscript of this paper.

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Received: March 30, 2013