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Stress intensity factors for fibrous composite with a crack embedded in an infinite matrix under a remote uniform load
Accepted Manuscript Stress intensity factors for fibrous composite with a crack embedded in an infinite matrix under a remote uniform load C.K. Chao, F.M. Chen, T.H. Lin PII: DOI: Reference:
S0013-7944(17)30308-9 http://dx.doi.org/10.1016/j.engfracmech.2017.04.045 EFM 5519
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
24 March 2017 24 April 2017 25 April 2017
Please cite this article as: Chao, C.K., Chen, F.M., Lin, T.H., Stress intensity factors for fibrous composite with a crack embedded in an infinite matrix under a remote uniform load, Engineering Fracture Mechanics (2017), doi: http://dx.doi.org/10.1016/j.engfracmech.2017.04.045
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Stress intensity factors for fibrous composite with a crack embedded in an infinite matrix under a remote uniform load
C.K. Chaoa, F.M. Chenb and T.H. Lina a
Department of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan
b
Department of Mechanical Engineering, Nan Kai University of Technology, Nantou County, Taiwan
Abstract
The interaction between a crack and two circular elastic inclusions for a fibrous tri-material composite under a remote uniform load is investigated. Based on the method of analytical continuation combined with the alternation technique, the solutions to the cracked composite are derived. A rapidly convergent series solution for the stress field, either in the matrix or the fibers, is obtained in an elegant form. The stress intensity factors are obtained numerically in terms of the dislocation density functions of the logarithmic singular integral equations. The merit of the present approach is the formulation of weakly singular integration that the integrals involved in the weakly singular allow easy calculations in the singular integral equation which can be treated to solve the problem even when a crack is closer to the interface. The stress intensity factors as a function of the dimensionless crack length for various material properties and geometric parameters are shown in graphic form. It is shown that the fibers act to restrain or accelerate crack growth depending on the shear moduli
between the fibers and the matrix. With this understanding, an optimum design can thus be conceptualized for a fibrous composite system. The present proposed method can be further extended to deal with the corresponding crack problem associated with any number of inclusions. Keywords: Two circular elastic inclusions; stress intensity factors; analytical continuation
1. Introduction
The use of advanced composite materials has become increasingly popular in engineering applications. One type of fibrous composite consisting of parallel fibers embedded in a matrix has grown to be an important class of composites [1-4]. Even though composites offer excellent performance, a small quantity of crack-like defects can cause a remarkable effect on the composite strength [5-6]. It is also known that the low-fiber and high-fiber concentration ranges are dominated by different failure mechanisms [7]. For a low-fiber concentration system, cracks are generally first observed in the matrix material where the fibers act to restrain crack growth [8]. For a high-fiber concentration system, cracks appear to develop and grow in regions where fibers are closest together following the fiber-to-matrix and fiber-tofiber interfaces. In order to analyze and predict crack propagation in a cracked composite, it is of great importance to have a better understanding of the stress field in the vicinity of the crack tip. The stress intensity factors (SIF) introduced by Irwin [9] play an important role in linear fracture mechanics. This fracture parameter may be used to study the problem in the
framework of microcracking for composite materials, fiber/matrix crack initiation and propagation [10-16]. The Singular Integral Equation (SIE) has established itself as one of the most extensively used tools in the evaluation of SIF for plane crack problems. The fundamental solution of a dislocation is required as a Green's function in this approach. By placing a continuous distribution of dislocations along a prospective crack site and applying a superposition technique, which is basically the summation of two solutions, a system of SIE’s can be formulated. The first solution is obtained for the given external loads and the given medium without the crack, and the second solution is obtained for the same medium having a crack. Once established, the system of SIE’s is then solved numerically to calculate the SIF. Integral transforms [17] and Muskhelishvili's complex potential [18] are two major approaches to obtain the SIE’s. The integral transform method [17] requires some inverse transform procedures which are somewhat cumbersome. Consequently, the conventional approach to stress analysis for a multi-media problem requires a system of simultaneous equations with a large number of unknown parameters to be solved. In addition, when selecting the methods of analyzing a cracked composite system, we must also consider that the solutions must be forced to satisfy the continuity conditions of multiple constituent interfaces. In order to overcome these difficulties, application of complex potential in conjunction with the analytical continuation technique [19-20] has been successfully applied [21-22]. The advantage of this method is that the analytical continuation technique allows the interface continuity conditions of multi-layered composites to be easily dealt with. Furthermore, the derived series solution is in terms of the material parameters instead of the geometries of cracks and interfaces, making the series solution rapidly convergent, even for cracks or dislocations located relatively closer to interfaces. Thus, compared to the method using
integral transforms, the method of complex potentials in conjunction with the analytical continuation technique [18] provides the simplest method to analyze plane crack problems, particularly in problems involving multi-layered materials. This method has been successfully applied to many examples of cracks or singularities interacting with multicomponent media, such as a point force with an isotropic tri-material [22], pairs of point heat sources and sinks interacting with isotropic [23] and viscoelastic [24] tri-materials, cylindrically layered media with either an arbitrarily located singularity [25] or a point heat source [26], an edge dislocation interacting with a coated elliptic inclusion [27], and a coated circular inclusion under uniaxial tension [28]. The present study examines a fiber-reinforced composite with a low-fiber concentration, whereby considering two elastic circular regions in a cross-sectional model of a tri-material. These circular regions could be considered anything from a spherical particle or round fiber reinforcement, to an inclusion, or even a hole. These two regions may be comprised of differing materials as well. The analysis will emphasize the effects of the two elastic circular inclusions or fibers embedded in the cracked composite, while under a remote uniform load applied perpendicular to the fibers. Matrix cracking will be the dominant failure mechanism [4]. A material characterization can be achieved by the determination of the stress intensit y factors from the point of view of linear elastic fracture mechanics. A new integral equation with logarithmic singular kernels [29] is used in this work. The development of weakly singular (logarithmic singular) equations allows easier calculations in obtaining the unknown constants. Linear interpolation formulae with undetermined coefficients are applied to approximate the distribution of dislocations along the crack, except in the vicinity of the crack tip where the dislocation distribution preserves a square-root singularity. Once the undetermined dislocation coefficients are solved, the stress intensity factors can be obtained.
The layout of the present work is as follows. Following this brief introduction in Section 1, the basic principle concerning complex potential functions is briefly given in Section 2. The derivation of Green’s function for an edge dislocation interacting with two circular elastic inclusions is provided in Section 3. The weakly singular integral equations are established in Section 4. Several numerical examples are discussed in Section 5. Finally Section 6 states the conclusions of the study. For the sake of brevity, we will use the term “inclusion” to refer to the solid phases other than the matrix.
2. Problem formulation
Consider two circular elastic inclusions interacting with a crack of length 2c located in an infinite matrix under a remote uniform tension T along the y-axis schematically illustrated in Fig. 1. The center of the left inclusion is placed at the origin in the rectangular Cartesian coordinate system while the center of the right inclusion is placed at the point (d , 0) in the rectangular Cartesian coordinate system. Let a denote the left inclusion of radius r1 , b denote the matrix, and c denote the right inclusion of radius r2 , respectively. The boundaries between two circular inclusions and a matrix are two circles 1 and 2 respectively, which are assumed to be perfect, i.e. both tractions and displacements are continuous across the two interfaces. Based on the complex variable theory for a twodimensional plane elasticity, the component of the displacement ux + iuy and the resultant forces -Fy + iFx can be described by two complex functions ( z ) and ( z ) , each of which is analytic in its argument z = x + i y, as [18] 2G(ux iu y ) ( z ) z ' ( z ) ( z )
(1)
Fy iFx ( z ) z ' ( z ) ( z)
(2)
where G is the shear modulus, 3 4 for plane strain, and (3 ) / (1 ) for plane stress, with being the Poisson’s ratio. The prime notation (‘) is designated as the derivative with respect to the associated argument, and a superimposed bar represents the complex conjugate. To solve this problem, we use the conformal mapping function m( ) [30] as
z m( )
x2 x1 z x1 1 , m ( z) 1 z x2
(3)
where x1 and x2, as shown in Fig. 1, are the points relatively symmetrical to the circumferences 1 and 2 . They satisfy the following relations:
x1 x2 r12 , (d x1 )(d x2 ) r22
(4)
From Eq. (4), we can obtain
x1 br1 , x2 r1 / b
(5)
where
b
1 2 2 2 d r1 r2 (d 2 r12 r22 )2 4r12 d 2 2r1 d
(6)
As illustrated in Fig. 2, the above mapping function m( ) can map the two circular interfaces 1 ,
formed by the left inclusion and the matrix, and 2 , formed by the right inclusion and the
matrix in the z-plane, onto two concentric circles with radii 1 x1 / x2
and
2 (d x1 ) /(d x2 ) respectively in the -plane. The left inclusion a is mapped onto a circular region 1 , the matrix b is mapped onto an annulus 1 2 , and the point at z is mapped to the point 1 , the right inclusion c is mapped onto the region 2 . With this transformation, Eqs. (1-2) can be rewritten as
2G(u x iu y ) ( )
Fy iFx ( )
m( ) m ( ) '
m( ) m ( ) '
' ( ) ( )
' ( ) ( )
(7)
(8)
For a region bounded by a circle, say c , we introduce an auxiliary stress function ( ) such that m(
c2
)
' ( ) ' ( ) ( ) m ( )
(9)
It should be noted that unlike the standard Muskhelishvili complex functions ( ) and ( ) , the function ( ) is dependent on the radius of the circular interface.
For the single-valued conditions of the traction force and the displacement, the stress functions
( ) and ( ) must take the following form
an ( ) n 1 ( ) 0 ( ) n ( ) bn ( ) n 1 n 1 cn ( ) n 1
Sa Sb
(10)
Sc
Sa an ( ) n 1 ( ) 0 ( ) n ( ) bn ( ) Sb n 1 n 1 Sc cn ( ) n 1
(11)
For the problem associated with a tri-material composite, the complex potential functions are found to depend on non-dimensional parameters as follows
ab
Ga b Gb a Ga Gb a
(12)
ab
Ga Gb Ga b Gb
(13)
3. Green’s functions for a tri-material composite
3.1 Homogeneous solution The complex potential for an infinite homogeneous medium subjected to a remote uniform load T can be trivially given as [18]
0 ( )
T x2 x1 4 1
(14)
0 ( )
T x2 x1 2 1
(15)
For the plane elasticity problem, it is assumed that the distributed edge dislocation with the density b(s)=b1(s)+ib2(s) is placed along the prospective crack segment in an infinite plane. The appropriate complex potentials for a crack embedded within a homogeneous medium will take the following form [29] ( 0 )( x1 x2 ) ds ( 1)( 0 1)
0 ( ) Q log 2c
( 0 )( x1 x2 ) ( 1)( 0 1)( x2 0 x1 ) ds ds Q ( 0 )( x1 x2 )( 0 1) ( 1)( 0 1) 2c
0 ( ) Q log 2c
where
(16)
(17)
Q( s )
G b1 ( s) ib2 ( s) i 1
The solution of an edge dislocation in a circularly cylindrical layered media as shown in Fig. 2 is expressed in terms of the corresponding homogeneous problem subjected to the same loading. Based on the analytical continuation theorem that is alternately applied across two different interfaces in the complex plane, the series form solution will be provided in the following sections.
3.2 Edge dislocation in an infinite matrix Referring to Fig. 2, we consider the case of an edge dislocation located in an annulus Sb . Based on the method of analytical continuation associated with the alternation technique, the additional terms an ( z ), bn ( z ), cn ( z ), n1 ( z ), and an ( z ), bn ( z ), cn ( z ), n1 ( z ) appearing in Eqs. (10) and (11) can finally be obtained as (see Appendix A.)
a1 ( ) (1 ab )0 ( ) ba
( 12 1) 2 Ca1 ( 1)
(18)
12 ( 12 1)2 ( 12 1) 2 1 ( ) ab 0 ( ) 2 C0 (1 ba ) 2 Ca1 1 1 ( 1) ( 1) 1 ( ) ab 0 (
(19)
12 ( 2 1)2 ) ab 1 C0 ( 1)
a1 ( ) (1 ab )0 ( ) (1 ab )
( 12 1) 2
2 ( 1 1)
(20)
C0
( 12 1) 2
2 ( 1 1)
Ca1
(21)
2 ( 2 1)2 (C0* C1* ) 2 b1 ( ) cb 0* ( 2 ) 1* ( 2 ) cb 2 ( 1) c1 ( ) (1 cb ) 0* ( ) 1* ( ) (1 cb )(C0* C1* )
( 22 1)2 (
c1 ( ) (1 cb ) 0 ( ) 1 ( )
b1 ( ) cb 0 (
(22)
22 1)
( 22 1) 2 Cc1 (
22 1)
bc ( 22 1)2 Cc1 1
( 12 1)2 Ca 2 ( 1)
( 2 1)2 Cb*1 12 ) ab 1 ( 1)
a 2 ( ) (1 ab ) ( )
( 12 1)2 Ca 2
12 ( 1)
(27)
(28)
(1 ab )
( 12 1)2 Cb*1
12 ( 1)
* 22 bn ( ) cb n 1 ( ) h( )Cn*1
cn ( ) (1 cb ) n*1 ( ) h(
(25)
(26)
( 12 1)2 Ca 2 ( 12 1)2 Cb*1 12 2 ( ) ab b1 ( ) (1 ba ) 12 12 ( 1) ( 1)
* b1
(23)
(24)
22 2 ( 2 1)2 (C0* C1* ) (1 bc )( 22 1) 2 Cc1 ) 1 ( 2 ) 2 22 2 ( 1) ( 2 1)
a 2 ( ) (1 ab )b1 ( ) ba
2 ( ) ab b*1 (
22 * 2 )Cn 1 h( 2 )Cc ( n 1)
(29)
(30)
(31)
cn ( ) (1 cb )n1 ( ) bc h( )Cc ( n1)
bn ( ) cb n 1 (
(32)
22 2 2 ) h( 2 )Cn*1 (1 bc )h( 2 )Cc ( n 1)
(33)
a ( n1) ( ) (1 ab )bn ( ) bau( )Ca ( n1)
(34)
12 12 12 * n1 ( ) ab bn ( ) (1 ba )u ( )Ca ( n 1) u( )Cbn
(35)
12 * ) u ( )Cbn
(36)
* n1 ( ) ab bn (
* a ( n1) ( ) (1 ab ) bn ( ) u (
(37)
12 * 2 )Cbn u ( 1 )Ca ( n 1)
where C0 0' ( 12 ) and Ca1 a' 1 ( 12 ) , C0* 0' ( 22 ) , C1* 1' ( 22 ) , Cc1 c'1 ( 22 ) , ' * ' Ca 2 2' ( 12 ) , Cb*1 b' 1 ( 12 ) , Cn n' ( 12 ) , Cn* n' ( 22 ) , Can an ( 12 ) , Can an ( 22 ) ,
Cbn ( ) , ' bn
2 2
C ( ) , * bn
' bn
2 1
Ccn ( ) ,
( 22 1) 2 , (n=2,3,4……). h( ) ( 1)
4. Singular integral equations
' cn
2 2
( 12 1) 2 C ( ) , u ( ) ( 1) * cn
' cn
2 1
and
Let a crack be located in an annulus of a circularly cylindrical layered media. The corresponding complex potentials are given by substituting Eqs. (16) and (17) into Eqs. (10) and (11) for Sb, respectively such as
Sb
(38)
( ) 0 ( ) 1 ( ) b1 ( ) Sb
(39)
( ) 0 ( ) 1 ( ) b1 ( ) and
where
( 2 1) 1 2 1 0 Q log ( 1 0 )( x1 x2 ) 1 ( ) ab Q 2 2 ( 1 )( 0 1) 2c ( 1 0 )( 1) Q
(40)
( 12 )( 0 )( 0 1) ds ( 12 0 )( 0 1)( 1)
( 2 )( x x ) ( 12 1) 0 1 1 0 1 2 1 ( ) Q ab log Q 2 2 ( 1 )( 12 0 ) ( )( 1) 2c 1 0 ( 12 1) 1 ab 0 1 1 2 Q ba 0 Q 2 ( 1 )( ba 1) 1 0 12 0
ds
(41)
2 2 ( 0 1) ( 1 1) 2 (1 ab ) ( 0 1) ba Q Q 2 ( 12 22 )( ba 1) ( 12 0 ) ( ) 1 0
b1 ( ) ab 2c
Q Q
( 12 1) 22 ( 0 1) 12 ( 12 22 )( 22 ) 2 ( 0 1) ab Q ( 1)( 22 12 ) 2 ( 22 12 )( 12 0 )
12 ( 12 22 )( 22 ) 2 ( 0 1) 2 ab ( 12 0 0 ) ( 1)( 22 12 )( 12 0 )( 12 22 0 ) 2 ( 0 1)
( 1) 2 ( 1) ( 2 1) 2 2 Q 2 1 2 0 2 ab2 Q 2 0 2 ( 1) ( 2 1)( 2 0 ) ( 1 2 )( 1 2 0 ) ( x1 x2 )( 22 0 12 ) 12 ( 0 1) 2 ab ( 12 0 0 ) Q Q ab log 2 2 ( 12 0 )( 12 22 0 ) 2 ( 0 1) ( 2 1 )( 0 1)
(42)
( x1 x2 )( 22 0 ) Q Q log ds 2 2 ( 1)( 0 1)( 2 0 0 ) ( 2 )( 0 1) ( 22 )( 0 )( 0 1)
( 22 1) 2 (1 cb ) ( 0 1) 12 ab ( 0 1) Q Q ( 22 )( bc 1) ( 22 1)( 22 0 ) ( 12 22 )( 12 22 0 ) 2c
b1 ( ) Q
12 ab ( 0 1) 2 ( 12 0 0 ) ( 1) Q 2 bc 0 2 2 2 2 2 ( 0 1)( 1 0 )( 1 2 0 ) ( 2 1)( 2 0 )
Q
bc 12 ( 0 1) ab bc 12 ( 0 1) 2 ab ( 12 0 0 ) Q ( 12 22 )( 12 22 0 ) ( 12 0 )( 12 22 0 ) 2 ( 0 1)
( 0 1) 12 ( 0 1) ab ( 22 1) 2 Q 2 Q ( 22 ) ( 22 1)( 22 0 ) ( 1 22 )( 12 22 0 ) 2 ( 1) 2 ab ( 12 0 0 ) Q 1 0 2 ( 0 1)( 1 0 )( 12 22 0 ) 2 cb ab ( 12 1)( 0 1) ( 22 12 )( 0 1)( 22 0 ) Q Q ( 22 ) ( 12 0 ) ( 12 22 0 )( 0 1) ( x x )( 2 2 ) Q cb ab log 1 2 2 2 2 0 1 ( 2 1 )( 0 1) ( x x )( 2 ) 0 Q log 1 2 2 ds 2 ( 2 )( 0 1)
From Eq. (9) we can obtain
(43)
( ) 0 ( ) 1 ( ) b1 ( )
Sb
(44)
The resultant force across the crack surface can be obtained by substituting two complex solutions ( z ) and ω(z) from Eqs. (38) and (44) into Eq. (2). Applying the principle of superposition leads to the following integral equation with logarithmic singular kernels
K ( , ,
2c
0
, 0 )Q(s)ds Kcon ( , , 0 , 0 )Q(s)ds C1 +iC2 Fy0 +iFx0
(45)
2c
where
K ( , , 0 , 0 ) H ( , , 0 , 0 ) L1 ( , , 0 , 0 ) L2 ( , , 0 , 0 )
(46)
Kcon ( , , 0 , 0 ) H con ( , , 0 , 0 ) L1con ( , , 0 , 0 ) L2con ( , , 0 , 0 )
(47)
and H、Hcon、L1、L1con、L2、L2con can be obtained from the following equations
HQ H con Q 0 ( )
L1Q L1con Q 1 ( )
m( )
0' ( ) 0 ( )
(48)
1' ( ) 1 ( )
(49)
m' ( ) m( ) m ( )
L2Q L2con Q b1 ( )
'
m( ) m' ( )
b' 1 ( ) b1 ( )
(50)
in which H, Hcon, L1, L1con , L2 and L2con are the coefficients of Q and Q from Eqs. (48)-(50). Meanwhile the resultant force Fy 0 ( 0 ) iFx 0 ( 0 ) , appearing on the right hand side of Eq. (45), corresponding to the unflawed media can be obtained by substituting the homogeneous solution from Eqs. (14) and (15) into Eq. (2) as
Fy 0 ( 0 ) iFx 0 ( 0 ) 0 ( )
m( )
0' ( ) 0 ( )
m ( ) m( ) ' 1 ( ) ' 1 ( ) 1 ( ) m ( ) m( ) ' b1 ( ) ' b1 ( ) b1 ( ) m ( ) '
(51)
where
1 ( )
T (3x2 12 2 x1 ) ab 4 ( 12 )
(52)
1 ( )
2 T 2 x1 ab ba x2 1 ab ba 1 ab 2 ba ab ba 2 4 1 1
(53)
2 2 2 T x2 2 2 3 x1 2 2 3 b1 ( ) cb 4 ( 22 )( 1)
x1 x2 14 24 2 12 22 1 2 ab ab 24 ab ( 12 22 ) 2 ( 1)
(54)
2 x1 x2 12 ( 12 22 )( 22 ) 2 ab 1 2 ( 1 22 )3 ( 1)
2 2 12 2 x1 2 ab ba ( ba 1) x2 12 ab ( ba 1) 22 ab 2 ba ab ba 2 2
x 2 x ab 2 x1 22 x2 22 3 12 T b1 ( ) cb 2 22 1 4 22 12 2
x1 x2 14 24 2 12 22 1 2 ab ab 24 ab ( 12 22 ) 2 ( 1)
x1 x2 1 cb 14 24 2 12 22 1 2 ab ab 24 ab 1 bc ( 12 22 ) 2 ( 22 ) bc 1 In addition, the single-valued condition of the dislocation density must be satisfied, i.e.
(55)
[b (s) ib (s)]ds 0 1
(56)
2
2c
The dislocation density function can be found by separating Eq. (45) into real and imaginary parts, respectively given as
Re K ( , ,
0
2c
, 0 ) Kcon ( , , 0 , 0 )b1 ( s) ds
Im K ( , , 0 , 0 ) + Kcon ( , , 0 , 0 )b2 ( s) ds +C1 Fy0 ( 0 )
(57)
2c
and
Im K ( , ,
2c
0
, 0 ) Kcon ( , , 0 , 0 )b1 ( s) ds
Re K ( , , 0 , 0 ) Kcon ( , , 0 , 0 )b2 ( s) ds +C2 = Fx0 ( 0 )
(58)
2c
Eqs. (51), (57) and (58) together with the subsidiary condition in Eq. (44) will be solved numerically for the dislocation coefficients of bi(s).
4.1 Stress intensity factors In order to perform the numerical calculation, the crack is divided into N line segments as indicated in Fig. 3. Then the linear interpolation formulae with the undetermined coefficients are applied to approximate the dislocation distribution along the elements, except in the vicinity of the crack tip where the dislocation distribution preserves a square-root singularity. The interpolation formulae in local coordinate's sj (1 ≤ j ≤ N) are defined as [28] 2d1 bi ( s1 ) bi ,1 1 bi ,2 d1 s1
for the left tip
(59)
2d N bi ( sN ) bi ,N 1 1 bi ,N d N sN
bi ( s j ) bi , j
d j sj 2d j
bi , j 1
d j sj 2d j
for the right tip
(60)
for intermediate segments
(61)
where (i = 0, 1, 2), dj (1 ≤ j ≤ N) are the half length of each line segment, and bi,j (1 ≤ j ≤ N+1) are the unknown coefficients. If the above formulae are used, the singular integral equation together with the subsidiary condition can be carried out to yield N+2 algebraic equations for N+2 unknown coefficients. For the crack tip segment, the following exact logarithmic integrations are used [28] 2d
I1
0
2d 1 log( r )dr 2d (log(2d ) 3) r
(62)
2d
I2
log(r )dr 2d (log(2d ) 1)
(63)
0
2d
I3
0
2d 1 log(2d r )dr 2d log(2d ) 3 4log(2d ) r
(64)
Meanwhile for the intermediate segment from P 2P3 until PN-1PN, the following exact logarithmic integration can be used 2d
I4
0
2d r log(r )dr d log(2d ) 1.5 2d
(65)
2d
I5
r
2d log(r )dr d log(2d ) 0.5
(66)
0
Otherwise, the following approximate Gauss-Chebyshev integration rule is used
d
d
G( s )ds
d
M
G( s M m 1
m
)sin(
2m 1 2m 1 ) with sm d cos( ) ; m = 1, 2, …., M M 2M
(67)
After having the coefficients b1,1 , b2,1 , b1, N 1 , b2, N 1 , the mode-I and mode-II stress intensity factors for the in-plane line crack problems with inclined angle λ can be obtained as [29] K A K IA iK IIA e i 2 2 lim sb( s) 2 2 2d1 e i b1,1 ib2,1 3
3
1/2
s0
3
K B K IB iK IIB e i 2 2 lim sl
(68)
3
a s b( s) 2 2 2d N 1 1/2 e i b1,N 1 ib2,N 1
(69)
where b( s) b1 ( s) ib2 ( s) and the angle of inclination λ is the angle between the direction of a uniform load and the x-axis.
5. Numerical examples
The single inclusion problem with a crack located in matrix is first considered to examine the accuracy of the present approach. Comparisons between the present calculated results with Gc / Gb 1, Ga / Gb 0.5 and the corresponding available results obtained by Lam et al. [31] for different crack locations are given in Fig. 4. From the above numerical examination, we see that the present approach provides reliable results compared with available results. Next, we consider a crack between a circular inclusion and a hole under a
remote uniform load (see Fig. 5). The calculated values of the normalized stress intensity factors closely compare to the available results from Dong and Lee [32] as shown in Fig. 6. The calculated results of the normalized stress intensity factors for different number of crack segments are shown in Table 1 and Table 2. It is seen that the calculated results of the normalized stress intensity factors with N 15 yield an excellent agreement with published values. It is expected that the error percentage will continue to decrease if we increase the number of line segments. We like to emphasize that the computer running time for N=15 takes only a little longer than that for N=10. Note that additional accuracy could be achieved (see Table 3) if the near-tip segments are made smaller by placing d1 d15 d / 9 and d 2 d14 4d / 9 where d is the half length of the intermediate segments as shown in Fig. 3.
In the following discussion, we assume that r1 r2 a , a b c 0.3 , and c / a 1/ 2 ur interest will be focused on the mode-1 stress intensity factors where a crack
is perpendicular to the direction of applied load which presents the worst scenario case of crack propagation. The present proposed methodology can be also applied to the mixed mode fracture problem for the case with the inclined crack where the mode-I and mode-II stress intensity factors are obtained from Eqs. (68) and (69). When the crack is located by the distance h to the left side of the left-hand counterpart of the two inclusions, the normalized mode-I stress intensity factors at tip-A versus the dimensionless location of a crack for different values of Ga / Gb with Gc / Gb 2 , d/ a = 4 are shown in Fig. 7. It is seen that the softer materials may always enhance the stress intensity factors while the harder materials may always reduce the stress intensity factors when a crack approaches the interface 1 .This trend is more pronounced at tip-B which is closer to the interface 1 as shown in Fig. 8. It is clearly seen that the normalized mode-I stress intensity factors at tip-A and tip-B have similar trends even though the material properties Gc / Gb vary as shown in Figs. 9-10. It can be
concluded that the material adjacent to the crack tip dominates the effects on the variation of stress intensity factors in a tri-material composite. When a crack is placed between two circular inclusions at distance h away from the left-hand inclusion with Gc / Gb = 2 and d / a 5 , the normalized mode-I stress intensity factors at the crack tip approaching the
nearest inclusion, tip-A, would increase (or decrease) if the nearest inclusion becomes softer (or stiffer) as indicated in Fig. 11. This trend is less pronounced at tip-B which is farther from the interface 1 as shown in Fig. 12. It is understood that the normalized mode-I stress intensity factors at both tip-A and tip-B are equal to each other when h / a 1 due to the symmetric geometry. When the distance between the two circular inclusions decreases to d / a 4 , the normalized mode-I stress intensity factors at tip-A have a similar trend
compared to the previous case d / a 5 when a crack is approaching the left-hand inclusion as indicated in Fig. 13 and Fig. 15. However, when the crack is approaching the right-hand inclusion, the normalized mode-I stress intensity factors at tip-B would increase (or decrease) if the left inclusion becomes softer (or stiffer) regardless of the value of Ga / Gb as shown in Fig. 14 and Fig. 16. Therefore, we can conclude again that the material adjacent to the crack tip dominates the effects on the variation of stress intensity factors in a tri-material composite. In order to demonstrate the convergence of the series solution presented in this study, the contribution of the stress intensity factors for the first three terms of a serious solution is 88.11%, 10.56%, and 1.33%, respectively. The contribution accounts for the ratio of each term to the summation of the first three terms of a series solution. Since the last term lends less than 2% of the contribution, we conclude that the series solution is rapidly convergent. The calculated results based on our proposed method yield an excellent agreement with the results based on the finite element method which shows that our present proposed method is efficient and general.
6. Concluding remarks
A solution of a crack interacting with two circular inclusions for a tri-material fibrous composite is presented in this study. The solution is based on the method of analytical continuation in conjunction with the alternation technique. This method provides a reliable result compared with published values of two-phase composite case. All the stress intensity factors presented here, including the crack located near the interface, have been obtained by dividing 15 equally spaced segments (N=15) along the crack surface with the summation of a series solution up to the first three terms or more depending on the distance between the crack and the interface. It is worthy to note that the convergence of the series solution is still assured even when the crack tip approaches the boundary of the inclusion. This is because that
our
series
solution
is
expressed
in
terms
of
the
material
parameters
ab (or bc ) and ab (or bc ) instead of the geometric parameter h / a . Some numerical examples have been calculated to investigate the effects of the material properties and geometric configurations on the mode-I stress intensity factors. The interaction between a crack and a tri-material fibrous composite indicates that the material adjacent to the crack tip plays a more significant influence on the stress intensity factors. It also shows that a reinforcement of increasing stiffness, such as a particle or fiber, may result in a decrease of local stress intensification in the crack tips. Compared with published values of two-phase composite case, the presented method has provided reliable results. The calculated results based on our proposed method yield an excellent agreement with the results based on the finite element method which shows that our present proposed method is efficient and general. The present proposed method can be further extended to model the interaction between a
multi-layered media with different types of crack such as arc cracks, kink cracks and branch cracks. The results obtained in this study may provide some guidance for minimizing the SIF by carefully selecting the material and geometry. The problem of a crack contained wholly within the reinforcement remains as another subject to be solved in the future.
Acknowledgment
The research was supported by the Ministry of Science and Technology of Taiwan under grant MOST 103-2221-E-011-023-MY3.
Nomenclature
( z ), ( z)
Complex potential functions for in-plane elasticity
( z )
Auxiliary stress function
G
Shear modulus
Poisson’s ratio
,
Non-dimensional parameters
z
Location of view point in the physical domain
z0
Location of singular point in the physical domain
Location of view point in the transformed domain
0
Location of singular point in the transformed domain
r
Radius of circular inclusion in the physical
domain
Radius of circular inclusion in the transformed domain
a
The first circular inclusion in the physical domain
b
Matrix in the the physical domain
c
The second circular inclusion in the physical domain
Sa
The first circular inclusion in the transformed domain
Sb
Matrix in the transformed domain
Sc
The second circular inclusion in the transformed domain
1
Interface between
a
and
b
2
Interface between
c
and
b
L1
Interface between
Sa
and
Sb
L2
Interface between
Sc
and
Sb
x, y
Coordinates in the physical domain
,
Coordinates in the transformed domain
c
Half-length of a crack
b1 ib2
Density function of edge dislocations
bi , j
Unknown coefficient of dislocation distribution
d
Half-length of crack segment
h K I , K II
Distance between the interface and the crack tip Mode-1 and Mode-11 stress intensity factors
N
Number of crack segments
Y iX
Resultant forces
Remote uniform tensile load
Appendix A. Derivations of Eqs. (18-37) We start to regard Sb and Sc composed of the same material b and region S a of material
a . 1 ( ) and 1 ( ) holomorphic in Sb Sc (except at z 0 ), a1 ( ) and a1 ( ) holomorphic in S a are introduced to satisfy the continuity of traction and displacement across
L1 that
a1 ( ) a1 ( ) 0 ( ) 0 ( ) 1 ( ) 1 ( )
L1
(A1)
1 1 aa1 ( ) a1 ( ) b0 ( ) 0 ( ) b1 ( ) 1 ( ) Gb Ga
L1
(A2)
12 m( ) ' 0 ( ) 0 ( ) . where 0 ( ) ' m ( ) By standard analytical continuation arguments, it follows 1 ( ) 0 (
12 2 ( 2 1)2 ( 2 1) 2 ) a1 ( 1 ) 1 C0 1 Ca1 0 ( 1) ( 1)
1
(A3)
a1 ( ) 0 ( ) 1 (
b Gb
a Ga
1 ( )
a1 ( )
12 ( 2 1)2 ( 2 1)2 ) 1 C0 1 Ca1 0 ( 1) ( 1)
1
2 2 ( 2 1)2 C0 ( 12 1) 2 Ca1 1 1 0 ( 1 ) a1 ( 1 ) 1 0 Gb Ga ( 1) Gb ( 1) Ga
b Gb
0 ( )
2 ( 2 1)2 C0 ( 12 1)2 Ca1 1 1 ( 1 ) 1 0 Gb ( 1) Gb ( 1) Ga
(A4)
1
1
(A5)
(A6)
Uncoupling Eqs. (A3) - (A6), we obtain a1 ( ) (1 ab )0 ( ) ba
( 12 1) 2 Ca1 ( 1)
(A7)
12 ( 12 1)2 ( 12 1) 2 1 ( ) ab 0 ( ) 2 C0 (1 ba ) 2 Ca1 1 1 ( 1) ( 1) 1 ( ) ab 0 (
(A8)
12 ( 2 1)2 ) ab 1 C0 ( 1)
a1 ( ) (1 ab )0 ( ) (1 ab )
( 12 1) 2
2 ( 1 1)
(A9)
C0
( 12 1) 2
2 ( 1 1)
Ca1
(A10)
where C0 0' ( 12 ) and Ca1 a' 1 ( 12 ) .
Since this result is based on the assumption that region Sc is made up of material b , it cannot satisfy the continuity condition at the interface L2 which lies between material b and
c . In the second step, we assume region Sb and S a be made up of the same material b and region Sc of material c . Additional terms b1 ( ) and b1 ( ) holomorphic in Sb Sa (except at 0 ), c1 ( ) and c1 ( ) holomorphic in Sc are introduced to satisfy the continuity conditions across L2 that
1 ( ) 1* ( ) 0 ( ) 0* ( ) b1 ( ) b1 ( ) c1 ( ) c1 ( ) L2 1 b0 ( ) 0* ( ) b1 ( ) 1* ( ) bb1 ( ) b1 ( ) Gb 1 cc1 ( ) c1 ( ) L2 Gc
m( where 0* ( ) 0 ( )
(A11)
(A12)
22 2 2 2 ) m( 1 ) m ( 2 ) m( 1 ) ' ' 0 ( ) and 1* ( ) 1 ( ) 1 ( ) . ' ' m ( ) m ( )
Based on the method of analytical continuation, we have
b1 ( ) 0* (
22 2 2 ( 2 1) 2 C0* ) 1* ( 2 ) c1 ( 2 ) 2 ( 1)
( 2 1) 2 C1* ( 22 1) 2 Cc1 2 0 ( 1) ( 1)
c1 ( ) 0 ( ) 1 ( ) b1 (
2
22 ( 22 1) 2 C0* ) ( 1)
( 2 1) 2 C1* ( 22 1) 2 Cc1 2 0 ( 1) ( 1)
(A13)
(A14)
2
b 22 ( 22 1)2 C0* ( 22 1)2 C1* 1 * 12 1 * 12 1 b1 ( ) 0 ( ) 1 ( ) c1 ( ) Gb Gb Gb Gc Gb ( 1) Gb ( 1) (A15)
( 22 1)2 Cc1 0 Gc ( 1)
2
c b b 22 ( 22 1)2 C0* ( 22 1)2 C1* 1 c1 ( ) 0 ( ) 1 ( ) b1 ( ) Gc Gb Gb Gb Gb ( 1) Gb ( 1) (A16)
( 1) Cc1 0 Gc ( 1) 2 2
2
2
Solve Eqs. (A13) - (A16) to obtain
2 ( 2 1)2 (C0* C1* ) 2 b1 ( ) cb 0* ( 2 ) 1* ( 2 ) cb 2 ( 1) c1 ( ) (1 cb ) 0* ( ) 1* ( ) (1 cb )(C0* C1* )
( 22 1)2 (
c1 ( ) (1 cb ) 0 ( ) 1 ( )
b1 ( ) cb 0 (
(A17)
22 1)
( 22 1) 2 Cc1 (
22 1)
bc ( 22 1)2 Cc1 1
(A18)
(A19)
22 2 ( 2 1)2 (C0* C1* ) (1 bc )( 22 1) 2 Cc1 ) 1 ( 2 ) 2 22 2 ( 1) ( 2 1)
(A20)
where C0* 0' ( 22 ) ， C1* 1' ( 22 ) ， Cc1 c'1 ( 22 ) .
Since this result is based on the assumption that region Sb is made up of material b, it cannot satisfy the continuity condition across L1 which lies between material a and b . In the third step, we again regard region Sb and Sc composed of the same material b and region S a of
material
a
.
Similar
to
the
previous
approach,
additional
terms
a 2 ( ), a 2 ( ), 2 ( ), 2 ( ) , which are introduced to satisfy the continuity conditions at the interface L1 , can be obtained as
( 12 1)2 a 2 ( ) (1 ab )b1 ( ) ba Ca 2 ( 1)
2 ( ) ab b1 (
( 2 1)2 Ca 2 ( 12 1)2 Cb*1 12 ) (1 ba ) 1 2 1 2 ( 1) ( 1 1)
(A21)
(A22)
2 ( ) ab b*1 (
( 2 1)2 Cb*1 12 ) ab 1 ( 1)
a 2 ( ) (1 ab )b*1 ( )
( 12 1)2 Ca 2 (
12 1)
(A23)
(1 ab )
( 12 1)2 Cb*1 (
12 1)
(A24)
where Ca 2 2' ( 12 ) and Cb*1 b' 1 ( 12 ) .
In the fourth step, regions Sa and Sb are assumed to make up with material b again. Repetitions of second and third step, the analytical continuation method is alternatively applied to two interfaces to obtain the additional terms an ( z ), bn ( z ), cn ( z ), n 1 ( z) and
an ( z), bn ( z), cn ( z), n1( z) (n 2,3,4,...) . The stress functions can be finally obtained as
bn ( ) cb n*1 (
22 ) h( )Cn*1
cn ( ) (1 cb ) n*1 ( ) h(
22 * 2 )Cn 1 h( 2 )Cc ( n 1)
(A25)
(A26)
cn ( ) (1 cb )n1 ( ) bc h( )Cc ( n1)
(A27)
22 22 * 22 bn ( ) cb n 1 ( ) h( )Cn 1 (1 bc )h( )Cc ( n 1)
(A28)
a ( n1) ( ) (1 ab )bn ( ) bau( )Ca ( n1)
(A29)
n1 ( ) ab bn (
12 2 2 * ) (1 ba )u ( 1 )Ca ( n 1) u ( 1 )Cbn
(A30)
* n1 ( ) ab bn (
12 * ) u ( )Cbn
* 2 * 12 a ( n1) ( ) (1 ab ) bn ( ) u ( 1 )Cbn u ( )C a ( n 1)
(A31)
(A32)
* ' * where Cn n' ( 12 ) , Cn* n' ( 22 ) , Can an' ( 12 ) , Can an ( 22 ) , Cbn bn' ( 22 ) , Cbn bn' ( 12 ) , * Ccn cn' ( 22 ) , Ccn cn' ( 12 ) , u ( )
( 12 1) 2 ( 2 1) 2 and h( ) 2 , (n=2,3,4……). ( 1) ( 1)
References [1] Piggott M. Load bearing fibre composites. Kluwer Academic Publishers; 2002.
[2] Schiavone P. Neutrality of the elliptic inhomogeneity in the case of non-uniform loading. Int. J. Eng. Sci 2003;41(18):2081-2090. [3] Mahboob M, Schiavone P. Designing a neutral elliptic inhomogeneity in the case of a general non-uniform loading. Appl. Math. Letters 2005;18(11):1312-1318. [4] Benveniste Y, Miloh T. Soft neutral elastic inhomogeneities with membrane-type interface conditions. J. Elas 2007;88(2);87-111. [5] Mantic V. Interface crack onset at a circular cylindrical inclusion under a remote transverse tension. Application of a coupled stress and energy criterion. Int J Solids Struct 2009;46(6):1287-1304. [6] Galiotis C, Young RJ, Yeung PHJ, Batchelder DN. The study of model polydiacetylene/epoxy composites: Part 1 the axial strain in the fibre. J Mater Sci 1984;9(11):3640-3648. [7] Kelly A, Tyson WR. Tensile properties of fiber-reinforced metals: copper/tungsten and copper/molybdenum. J Mech Phys Solids 1965;13(1):39-50. [8] Hull D, Clyne T. An introduction to composite materials. Cambridge University Press, Cambridge, UK; 1996. [9] Irwin GR. Analysis of stresses and strains near the end of a crack traversing a plate. ASME J Appl Mech 1957;4(3):167-170. [10] Greco F, Leonette L, Nevone Blasi P. Non-linear macroscopic response of fiberreinforced composite materials due to initiation and propagation of interface cracks. Engng Fract Mech 2012;80(1):92-113. [11] Greco F, Leonette L, Lonetti P. A two-scale failure analysis of composite materials in presence of fiber/matrix crack initiation and propagation. Composite Structures 2013;95(1):582-597. [12] Greco F, Leonette L, Nevone Blasi P. Adaptive multiscale modeling of fiber-reinforced
composite materials subjected to transverse microcracking. Composite Structures 2014;113(1):249-263. [13] Greco F, Leonette L, Lonetti P, Nevone Blasi P. Crack propagation analysis in composite materials by using moving mesh and multiscale techniques. Computers and Structures 2015;153(1):201-216. [14] Dal Corso F, Shahzad S, Bigoni D. Isotoxal star-shaped polygonal voids and rigid inclusions in nonuniform antiplane shear fields: Part II: Singularities, annihilation and invisibility. Int J Solids Struct 2016;85-86:76-88. [15] Shahzad S, Dal Corso F, Bigoni D. Hypocycloidal inclusions in nonuniform out-ofplane elasticity: stress singularity vs stress reduction. J. Elas 2017;126(2):215-229. [16] Chao CK, Wikarta A. Solutions of a crack interacting with a three-phase composite in plane elasticity. Appl Math Model 2016;40(3):2454-2472. [17] Erdogan F, Gupta G. The stress analysis of multi-layered composites with a flaw. Int J Solids Struct 1971;7(1):pp. 39-61. [18] England AH. Complex variable methods in elasticity, Wiley, London; 1971. [19] Suo Z. Singularities interacting with interfaces and cracks. Int J Solids Struct 1989;25(10):1133-1142. [20] Xiao ZM, Chen BJ. On the interaction between an edge dislocation and a coated inclusion. Int J Solids Struct 2001;38(15):2533-4842. [21] Chao CK, Kao B. A thin cracked layer bonded to an elastic half-space under an antiplane concentrated load. Int J Frac 1997;83(3):223-241. [22] Choi ST, Earmme YY. Elastic study on singularities interacting with interfaces using alternating technique: Part 2 Isotropic trimaterial. Int J Solids Struct 2001;39(5):11991211. [23] Chao CK, Chen FM. Thermal stresses in an isotropic trimaterial interacted with a pair of
point heat source and heat sink. Int J Solids Struct 2004;41(22):6233-6247. [24] Porter GA, Chuang CT, Chao CK, Chang RC. Thermal stresses in a viscoelastic trimaterial with a combination of a point heat source and a point heat sink. J Eng Math 2008;61(2):99-109. [25] Chao CK, Chen FM, Shen MH. Circularly cylindrical layered media in plane elasticity. Int J Solids Struct 2006;43(16):4739-4756. [26] Chao CK, Chen FM, Shen MH. Green's functions for a point heat source in circularly cylindrical layered media. J Therm Stresses 2006;29(9):809-847. [27] Chen FM, Chao CK, Chen CK. Interaction of an edge dislocation in a coated elliptic inclusion. Int J Solids Struct 2011;48(10):1451-1465. [28] Pak YE, Mishra D, Yoo SH. Closed-form solution for a coated circular inclusion under uniaxial tension. Acta Mech 2012;223(5):937-951. [29] Chen YZ, Cheung YK. New integral equation approach for the crack problem in elastic half-plane. Int J Frac 1990;46(1):57-69. [30] Lars VA. Conformal invariants: Topics in geometric function theory, McGraw-Hill Book Co, New York; 1973. [31] Lam KY, Ong PP, Wude N. Interaction between a circular inclusion and a symmetrically branched crack. Theore Appl Frac Mech 1998;28(3):197-211. [32] Dong CY, Lee KY. A new integral equation formulation of two-dimensional inclusion– crack problems. Int J Solids Struct 2005;42(18):5010-5020.
Table 1. Comparisons between the calculated and published values of KI at tip-A
h/a 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Ga/Gb =0.5 Lam et al. [31] 1.297 1.206 1.155 1.121 1.098 1.081 1.068 1.058
N=5 1.4253 1.3217 1.2614 1.2206 1.1911 1.1689 1.1517 1.1381
N=10 1.2934 1.1976 1.1476 1.1176 1.0975 1.0886 1.0894 1.0645
N=15 1.2983 1.2096 1.1585 1.1244 1.0983 1.0858 1.0754 1.0634
Table 2. Comparisons between the calculated and published values of KI at tip-A Ga/Gb
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Gc=0 Dong and Lee [32] 3 1.8 1.43 1.25 1.125 1.05 1 0.97 0.94 0.93 0.92
N=5
N=10
N=15
2.6091 2.2983 2.1085 2.0134 1.9567 1.9191 1.8924 1.8724 1.8569 1.8445 1.8344
2.9824 1.8989 1.5246 1.3355 1.2209 1.1440 1.0886 1.0468 1.0141 0.9878 0.9662
2.9872 1.8108 1.4386 1.2588 1.1329 1.0574 1.0068 0.9768 0.9466 0.9374 0.9273
Table 3. Comparisons between equally spaced segments and non-equally spaced segments of the crack discretization at tip-A Ga/Gb
Gc=0 Dong and Lee [32]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
3 1.8 1.43 1.25 1.125 1.05 1 0.97 0.94 0.93 0.92
Gc=0 Non-Equally spaced segments (N=15) 2.9891 1.8076 1.4332 1.2536 1.1289 1.0535 1.0032 0.9742 0.9438 0.9357 0.9261
Gc=0 Equally spaced segments (N=15) 2.9872 1.8108 1.4386 1.2588 1.1329 1.0574 1.0068 0.9768 0.9466 0.9374 0.9273
Figure 1: Interaction of a crack with two circular inclusions embedded in a homogeneous isotropic material under a remote uniform load
Figure 2: The problem in the plane
PN-1
P3 P1
s1 P2
P4
PN+1
SN PN
2dN
2d1
Figure 3: Division and nodal distribution of a crack
Figure 4: A comparison between the calculated and published values of the normalized mode-I stress intensity factors at tip-A for a crack located in an infinite matrix
Figure 5: A crack located in an infinite matrix between a circular inclusion and a hole under a remote uniform load
Figure 6: A comparison between the calculated and published values of the normalized mode-I stress intensity factors at tip-A for a crack located in an infinite matrix
Figure 7: Normalized mode-I stress intensity factors at tip-A versus dimensionless location of crack for different values of Ga / Gb with Ga / Gb =2, d/ a = 4
Figure 8: Normalized mode-I stress intensity factors at tip-B versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 2, d/ a = 4
Figure 9: Normalized mode-I stress intensity factors at tip-A versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 0.5, d/ a = 4
Figure 10: Normalized mode-I stress intensity factors at tip-B versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 0.5, d/ a = 4
Figure 21: Normalized mode-I stress intensity factors at tip-A versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 2, d/a= 5
Figure 32: Normalized mode-I stress intensity factors at tip-B versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 2, d/a = 5
Figure 13: Normalized mode-I stress intensity factors at tip-A versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 2, d/a= 4
Figure 14: Normalized mode-I stress intensity factors at tip-B versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 2, d/a= 4
Figure 15: Normalized mode-I stress intensity factors at tip-A versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 0.5, d/a = 4
Figure 16: Normalized mode-I stress intensity factors at tip-B versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 0.5, d/a= 4
Highlights
． A logarithmic singular integral equation is proposed.
． The analytical continuation technique and the alternation method are used.
． The derived series solution is rapidly convergent.
． The softer material may give enhancement effect on SIF.
． The stiffer material may give retardation effect on SIF.
S0013-7944(17)30308-9 http://dx.doi.org/10.1016/j.engfracmech.2017.04.045 EFM 5519
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
24 March 2017 24 April 2017 25 April 2017
Please cite this article as: Chao, C.K., Chen, F.M., Lin, T.H., Stress intensity factors for fibrous composite with a crack embedded in an infinite matrix under a remote uniform load, Engineering Fracture Mechanics (2017), doi: http://dx.doi.org/10.1016/j.engfracmech.2017.04.045
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Stress intensity factors for fibrous composite with a crack embedded in an infinite matrix under a remote uniform load
C.K. Chaoa, F.M. Chenb and T.H. Lina a
Department of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan
b
Department of Mechanical Engineering, Nan Kai University of Technology, Nantou County, Taiwan
Abstract
The interaction between a crack and two circular elastic inclusions for a fibrous tri-material composite under a remote uniform load is investigated. Based on the method of analytical continuation combined with the alternation technique, the solutions to the cracked composite are derived. A rapidly convergent series solution for the stress field, either in the matrix or the fibers, is obtained in an elegant form. The stress intensity factors are obtained numerically in terms of the dislocation density functions of the logarithmic singular integral equations. The merit of the present approach is the formulation of weakly singular integration that the integrals involved in the weakly singular allow easy calculations in the singular integral equation which can be treated to solve the problem even when a crack is closer to the interface. The stress intensity factors as a function of the dimensionless crack length for various material properties and geometric parameters are shown in graphic form. It is shown that the fibers act to restrain or accelerate crack growth depending on the shear moduli
between the fibers and the matrix. With this understanding, an optimum design can thus be conceptualized for a fibrous composite system. The present proposed method can be further extended to deal with the corresponding crack problem associated with any number of inclusions. Keywords: Two circular elastic inclusions; stress intensity factors; analytical continuation
1. Introduction
The use of advanced composite materials has become increasingly popular in engineering applications. One type of fibrous composite consisting of parallel fibers embedded in a matrix has grown to be an important class of composites [1-4]. Even though composites offer excellent performance, a small quantity of crack-like defects can cause a remarkable effect on the composite strength [5-6]. It is also known that the low-fiber and high-fiber concentration ranges are dominated by different failure mechanisms [7]. For a low-fiber concentration system, cracks are generally first observed in the matrix material where the fibers act to restrain crack growth [8]. For a high-fiber concentration system, cracks appear to develop and grow in regions where fibers are closest together following the fiber-to-matrix and fiber-tofiber interfaces. In order to analyze and predict crack propagation in a cracked composite, it is of great importance to have a better understanding of the stress field in the vicinity of the crack tip. The stress intensity factors (SIF) introduced by Irwin [9] play an important role in linear fracture mechanics. This fracture parameter may be used to study the problem in the
framework of microcracking for composite materials, fiber/matrix crack initiation and propagation [10-16]. The Singular Integral Equation (SIE) has established itself as one of the most extensively used tools in the evaluation of SIF for plane crack problems. The fundamental solution of a dislocation is required as a Green's function in this approach. By placing a continuous distribution of dislocations along a prospective crack site and applying a superposition technique, which is basically the summation of two solutions, a system of SIE’s can be formulated. The first solution is obtained for the given external loads and the given medium without the crack, and the second solution is obtained for the same medium having a crack. Once established, the system of SIE’s is then solved numerically to calculate the SIF. Integral transforms [17] and Muskhelishvili's complex potential [18] are two major approaches to obtain the SIE’s. The integral transform method [17] requires some inverse transform procedures which are somewhat cumbersome. Consequently, the conventional approach to stress analysis for a multi-media problem requires a system of simultaneous equations with a large number of unknown parameters to be solved. In addition, when selecting the methods of analyzing a cracked composite system, we must also consider that the solutions must be forced to satisfy the continuity conditions of multiple constituent interfaces. In order to overcome these difficulties, application of complex potential in conjunction with the analytical continuation technique [19-20] has been successfully applied [21-22]. The advantage of this method is that the analytical continuation technique allows the interface continuity conditions of multi-layered composites to be easily dealt with. Furthermore, the derived series solution is in terms of the material parameters instead of the geometries of cracks and interfaces, making the series solution rapidly convergent, even for cracks or dislocations located relatively closer to interfaces. Thus, compared to the method using
integral transforms, the method of complex potentials in conjunction with the analytical continuation technique [18] provides the simplest method to analyze plane crack problems, particularly in problems involving multi-layered materials. This method has been successfully applied to many examples of cracks or singularities interacting with multicomponent media, such as a point force with an isotropic tri-material [22], pairs of point heat sources and sinks interacting with isotropic [23] and viscoelastic [24] tri-materials, cylindrically layered media with either an arbitrarily located singularity [25] or a point heat source [26], an edge dislocation interacting with a coated elliptic inclusion [27], and a coated circular inclusion under uniaxial tension [28]. The present study examines a fiber-reinforced composite with a low-fiber concentration, whereby considering two elastic circular regions in a cross-sectional model of a tri-material. These circular regions could be considered anything from a spherical particle or round fiber reinforcement, to an inclusion, or even a hole. These two regions may be comprised of differing materials as well. The analysis will emphasize the effects of the two elastic circular inclusions or fibers embedded in the cracked composite, while under a remote uniform load applied perpendicular to the fibers. Matrix cracking will be the dominant failure mechanism [4]. A material characterization can be achieved by the determination of the stress intensit y factors from the point of view of linear elastic fracture mechanics. A new integral equation with logarithmic singular kernels [29] is used in this work. The development of weakly singular (logarithmic singular) equations allows easier calculations in obtaining the unknown constants. Linear interpolation formulae with undetermined coefficients are applied to approximate the distribution of dislocations along the crack, except in the vicinity of the crack tip where the dislocation distribution preserves a square-root singularity. Once the undetermined dislocation coefficients are solved, the stress intensity factors can be obtained.
The layout of the present work is as follows. Following this brief introduction in Section 1, the basic principle concerning complex potential functions is briefly given in Section 2. The derivation of Green’s function for an edge dislocation interacting with two circular elastic inclusions is provided in Section 3. The weakly singular integral equations are established in Section 4. Several numerical examples are discussed in Section 5. Finally Section 6 states the conclusions of the study. For the sake of brevity, we will use the term “inclusion” to refer to the solid phases other than the matrix.
2. Problem formulation
Consider two circular elastic inclusions interacting with a crack of length 2c located in an infinite matrix under a remote uniform tension T along the y-axis schematically illustrated in Fig. 1. The center of the left inclusion is placed at the origin in the rectangular Cartesian coordinate system while the center of the right inclusion is placed at the point (d , 0) in the rectangular Cartesian coordinate system. Let a denote the left inclusion of radius r1 , b denote the matrix, and c denote the right inclusion of radius r2 , respectively. The boundaries between two circular inclusions and a matrix are two circles 1 and 2 respectively, which are assumed to be perfect, i.e. both tractions and displacements are continuous across the two interfaces. Based on the complex variable theory for a twodimensional plane elasticity, the component of the displacement ux + iuy and the resultant forces -Fy + iFx can be described by two complex functions ( z ) and ( z ) , each of which is analytic in its argument z = x + i y, as [18] 2G(ux iu y ) ( z ) z ' ( z ) ( z )
(1)
Fy iFx ( z ) z ' ( z ) ( z)
(2)
where G is the shear modulus, 3 4 for plane strain, and (3 ) / (1 ) for plane stress, with being the Poisson’s ratio. The prime notation (‘) is designated as the derivative with respect to the associated argument, and a superimposed bar represents the complex conjugate. To solve this problem, we use the conformal mapping function m( ) [30] as
z m( )
x2 x1 z x1 1 , m ( z) 1 z x2
(3)
where x1 and x2, as shown in Fig. 1, are the points relatively symmetrical to the circumferences 1 and 2 . They satisfy the following relations:
x1 x2 r12 , (d x1 )(d x2 ) r22
(4)
From Eq. (4), we can obtain
x1 br1 , x2 r1 / b
(5)
where
b
1 2 2 2 d r1 r2 (d 2 r12 r22 )2 4r12 d 2 2r1 d
(6)
As illustrated in Fig. 2, the above mapping function m( ) can map the two circular interfaces 1 ,
formed by the left inclusion and the matrix, and 2 , formed by the right inclusion and the
matrix in the z-plane, onto two concentric circles with radii 1 x1 / x2
and
2 (d x1 ) /(d x2 ) respectively in the -plane. The left inclusion a is mapped onto a circular region 1 , the matrix b is mapped onto an annulus 1 2 , and the point at z is mapped to the point 1 , the right inclusion c is mapped onto the region 2 . With this transformation, Eqs. (1-2) can be rewritten as
2G(u x iu y ) ( )
Fy iFx ( )
m( ) m ( ) '
m( ) m ( ) '
' ( ) ( )
' ( ) ( )
(7)
(8)
For a region bounded by a circle, say c , we introduce an auxiliary stress function ( ) such that m(
c2
)
' ( ) ' ( ) ( ) m ( )
(9)
It should be noted that unlike the standard Muskhelishvili complex functions ( ) and ( ) , the function ( ) is dependent on the radius of the circular interface.
For the single-valued conditions of the traction force and the displacement, the stress functions
( ) and ( ) must take the following form
an ( ) n 1 ( ) 0 ( ) n ( ) bn ( ) n 1 n 1 cn ( ) n 1
Sa Sb
(10)
Sc
Sa an ( ) n 1 ( ) 0 ( ) n ( ) bn ( ) Sb n 1 n 1 Sc cn ( ) n 1
(11)
For the problem associated with a tri-material composite, the complex potential functions are found to depend on non-dimensional parameters as follows
ab
Ga b Gb a Ga Gb a
(12)
ab
Ga Gb Ga b Gb
(13)
3. Green’s functions for a tri-material composite
3.1 Homogeneous solution The complex potential for an infinite homogeneous medium subjected to a remote uniform load T can be trivially given as [18]
0 ( )
T x2 x1 4 1
(14)
0 ( )
T x2 x1 2 1
(15)
For the plane elasticity problem, it is assumed that the distributed edge dislocation with the density b(s)=b1(s)+ib2(s) is placed along the prospective crack segment in an infinite plane. The appropriate complex potentials for a crack embedded within a homogeneous medium will take the following form [29] ( 0 )( x1 x2 ) ds ( 1)( 0 1)
0 ( ) Q log 2c
( 0 )( x1 x2 ) ( 1)( 0 1)( x2 0 x1 ) ds ds Q ( 0 )( x1 x2 )( 0 1) ( 1)( 0 1) 2c
0 ( ) Q log 2c
where
(16)
(17)
Q( s )
G b1 ( s) ib2 ( s) i 1
The solution of an edge dislocation in a circularly cylindrical layered media as shown in Fig. 2 is expressed in terms of the corresponding homogeneous problem subjected to the same loading. Based on the analytical continuation theorem that is alternately applied across two different interfaces in the complex plane, the series form solution will be provided in the following sections.
3.2 Edge dislocation in an infinite matrix Referring to Fig. 2, we consider the case of an edge dislocation located in an annulus Sb . Based on the method of analytical continuation associated with the alternation technique, the additional terms an ( z ), bn ( z ), cn ( z ), n1 ( z ), and an ( z ), bn ( z ), cn ( z ), n1 ( z ) appearing in Eqs. (10) and (11) can finally be obtained as (see Appendix A.)
a1 ( ) (1 ab )0 ( ) ba
( 12 1) 2 Ca1 ( 1)
(18)
12 ( 12 1)2 ( 12 1) 2 1 ( ) ab 0 ( ) 2 C0 (1 ba ) 2 Ca1 1 1 ( 1) ( 1) 1 ( ) ab 0 (
(19)
12 ( 2 1)2 ) ab 1 C0 ( 1)
a1 ( ) (1 ab )0 ( ) (1 ab )
( 12 1) 2
2 ( 1 1)
(20)
C0
( 12 1) 2
2 ( 1 1)
Ca1
(21)
2 ( 2 1)2 (C0* C1* ) 2 b1 ( ) cb 0* ( 2 ) 1* ( 2 ) cb 2 ( 1) c1 ( ) (1 cb ) 0* ( ) 1* ( ) (1 cb )(C0* C1* )
( 22 1)2 (
c1 ( ) (1 cb ) 0 ( ) 1 ( )
b1 ( ) cb 0 (
(22)
22 1)
( 22 1) 2 Cc1 (
22 1)
bc ( 22 1)2 Cc1 1
( 12 1)2 Ca 2 ( 1)
( 2 1)2 Cb*1 12 ) ab 1 ( 1)
a 2 ( ) (1 ab ) ( )
( 12 1)2 Ca 2
12 ( 1)
(27)
(28)
(1 ab )
( 12 1)2 Cb*1
12 ( 1)
* 22 bn ( ) cb n 1 ( ) h( )Cn*1
cn ( ) (1 cb ) n*1 ( ) h(
(25)
(26)
( 12 1)2 Ca 2 ( 12 1)2 Cb*1 12 2 ( ) ab b1 ( ) (1 ba ) 12 12 ( 1) ( 1)
* b1
(23)
(24)
22 2 ( 2 1)2 (C0* C1* ) (1 bc )( 22 1) 2 Cc1 ) 1 ( 2 ) 2 22 2 ( 1) ( 2 1)
a 2 ( ) (1 ab )b1 ( ) ba
2 ( ) ab b*1 (
22 * 2 )Cn 1 h( 2 )Cc ( n 1)
(29)
(30)
(31)
cn ( ) (1 cb )n1 ( ) bc h( )Cc ( n1)
bn ( ) cb n 1 (
(32)
22 2 2 ) h( 2 )Cn*1 (1 bc )h( 2 )Cc ( n 1)
(33)
a ( n1) ( ) (1 ab )bn ( ) bau( )Ca ( n1)
(34)
12 12 12 * n1 ( ) ab bn ( ) (1 ba )u ( )Ca ( n 1) u( )Cbn
(35)
12 * ) u ( )Cbn
(36)
* n1 ( ) ab bn (
* a ( n1) ( ) (1 ab ) bn ( ) u (
(37)
12 * 2 )Cbn u ( 1 )Ca ( n 1)
where C0 0' ( 12 ) and Ca1 a' 1 ( 12 ) , C0* 0' ( 22 ) , C1* 1' ( 22 ) , Cc1 c'1 ( 22 ) , ' * ' Ca 2 2' ( 12 ) , Cb*1 b' 1 ( 12 ) , Cn n' ( 12 ) , Cn* n' ( 22 ) , Can an ( 12 ) , Can an ( 22 ) ,
Cbn ( ) , ' bn
2 2
C ( ) , * bn
' bn
2 1
Ccn ( ) ,
( 22 1) 2 , (n=2,3,4……). h( ) ( 1)
4. Singular integral equations
' cn
2 2
( 12 1) 2 C ( ) , u ( ) ( 1) * cn
' cn
2 1
and
Let a crack be located in an annulus of a circularly cylindrical layered media. The corresponding complex potentials are given by substituting Eqs. (16) and (17) into Eqs. (10) and (11) for Sb, respectively such as
Sb
(38)
( ) 0 ( ) 1 ( ) b1 ( ) Sb
(39)
( ) 0 ( ) 1 ( ) b1 ( ) and
where
( 2 1) 1 2 1 0 Q log ( 1 0 )( x1 x2 ) 1 ( ) ab Q 2 2 ( 1 )( 0 1) 2c ( 1 0 )( 1) Q
(40)
( 12 )( 0 )( 0 1) ds ( 12 0 )( 0 1)( 1)
( 2 )( x x ) ( 12 1) 0 1 1 0 1 2 1 ( ) Q ab log Q 2 2 ( 1 )( 12 0 ) ( )( 1) 2c 1 0 ( 12 1) 1 ab 0 1 1 2 Q ba 0 Q 2 ( 1 )( ba 1) 1 0 12 0
ds
(41)
2 2 ( 0 1) ( 1 1) 2 (1 ab ) ( 0 1) ba Q Q 2 ( 12 22 )( ba 1) ( 12 0 ) ( ) 1 0
b1 ( ) ab 2c
Q Q
( 12 1) 22 ( 0 1) 12 ( 12 22 )( 22 ) 2 ( 0 1) ab Q ( 1)( 22 12 ) 2 ( 22 12 )( 12 0 )
12 ( 12 22 )( 22 ) 2 ( 0 1) 2 ab ( 12 0 0 ) ( 1)( 22 12 )( 12 0 )( 12 22 0 ) 2 ( 0 1)
( 1) 2 ( 1) ( 2 1) 2 2 Q 2 1 2 0 2 ab2 Q 2 0 2 ( 1) ( 2 1)( 2 0 ) ( 1 2 )( 1 2 0 ) ( x1 x2 )( 22 0 12 ) 12 ( 0 1) 2 ab ( 12 0 0 ) Q Q ab log 2 2 ( 12 0 )( 12 22 0 ) 2 ( 0 1) ( 2 1 )( 0 1)
(42)
( x1 x2 )( 22 0 ) Q Q log ds 2 2 ( 1)( 0 1)( 2 0 0 ) ( 2 )( 0 1) ( 22 )( 0 )( 0 1)
( 22 1) 2 (1 cb ) ( 0 1) 12 ab ( 0 1) Q Q ( 22 )( bc 1) ( 22 1)( 22 0 ) ( 12 22 )( 12 22 0 ) 2c
b1 ( ) Q
12 ab ( 0 1) 2 ( 12 0 0 ) ( 1) Q 2 bc 0 2 2 2 2 2 ( 0 1)( 1 0 )( 1 2 0 ) ( 2 1)( 2 0 )
Q
bc 12 ( 0 1) ab bc 12 ( 0 1) 2 ab ( 12 0 0 ) Q ( 12 22 )( 12 22 0 ) ( 12 0 )( 12 22 0 ) 2 ( 0 1)
( 0 1) 12 ( 0 1) ab ( 22 1) 2 Q 2 Q ( 22 ) ( 22 1)( 22 0 ) ( 1 22 )( 12 22 0 ) 2 ( 1) 2 ab ( 12 0 0 ) Q 1 0 2 ( 0 1)( 1 0 )( 12 22 0 ) 2 cb ab ( 12 1)( 0 1) ( 22 12 )( 0 1)( 22 0 ) Q Q ( 22 ) ( 12 0 ) ( 12 22 0 )( 0 1) ( x x )( 2 2 ) Q cb ab log 1 2 2 2 2 0 1 ( 2 1 )( 0 1) ( x x )( 2 ) 0 Q log 1 2 2 ds 2 ( 2 )( 0 1)
From Eq. (9) we can obtain
(43)
( ) 0 ( ) 1 ( ) b1 ( )
Sb
(44)
The resultant force across the crack surface can be obtained by substituting two complex solutions ( z ) and ω(z) from Eqs. (38) and (44) into Eq. (2). Applying the principle of superposition leads to the following integral equation with logarithmic singular kernels
K ( , ,
2c
0
, 0 )Q(s)ds Kcon ( , , 0 , 0 )Q(s)ds C1 +iC2 Fy0 +iFx0
(45)
2c
where
K ( , , 0 , 0 ) H ( , , 0 , 0 ) L1 ( , , 0 , 0 ) L2 ( , , 0 , 0 )
(46)
Kcon ( , , 0 , 0 ) H con ( , , 0 , 0 ) L1con ( , , 0 , 0 ) L2con ( , , 0 , 0 )
(47)
and H、Hcon、L1、L1con、L2、L2con can be obtained from the following equations
HQ H con Q 0 ( )
L1Q L1con Q 1 ( )
m( )
0' ( ) 0 ( )
(48)
1' ( ) 1 ( )
(49)
m' ( ) m( ) m ( )
L2Q L2con Q b1 ( )
'
m( ) m' ( )
b' 1 ( ) b1 ( )
(50)
in which H, Hcon, L1, L1con , L2 and L2con are the coefficients of Q and Q from Eqs. (48)-(50). Meanwhile the resultant force Fy 0 ( 0 ) iFx 0 ( 0 ) , appearing on the right hand side of Eq. (45), corresponding to the unflawed media can be obtained by substituting the homogeneous solution from Eqs. (14) and (15) into Eq. (2) as
Fy 0 ( 0 ) iFx 0 ( 0 ) 0 ( )
m( )
0' ( ) 0 ( )
m ( ) m( ) ' 1 ( ) ' 1 ( ) 1 ( ) m ( ) m( ) ' b1 ( ) ' b1 ( ) b1 ( ) m ( ) '
(51)
where
1 ( )
T (3x2 12 2 x1 ) ab 4 ( 12 )
(52)
1 ( )
2 T 2 x1 ab ba x2 1 ab ba 1 ab 2 ba ab ba 2 4 1 1
(53)
2 2 2 T x2 2 2 3 x1 2 2 3 b1 ( ) cb 4 ( 22 )( 1)
x1 x2 14 24 2 12 22 1 2 ab ab 24 ab ( 12 22 ) 2 ( 1)
(54)
2 x1 x2 12 ( 12 22 )( 22 ) 2 ab 1 2 ( 1 22 )3 ( 1)
2 2 12 2 x1 2 ab ba ( ba 1) x2 12 ab ( ba 1) 22 ab 2 ba ab ba 2 2
x 2 x ab 2 x1 22 x2 22 3 12 T b1 ( ) cb 2 22 1 4 22 12 2
x1 x2 14 24 2 12 22 1 2 ab ab 24 ab ( 12 22 ) 2 ( 1)
x1 x2 1 cb 14 24 2 12 22 1 2 ab ab 24 ab 1 bc ( 12 22 ) 2 ( 22 ) bc 1 In addition, the single-valued condition of the dislocation density must be satisfied, i.e.
(55)
[b (s) ib (s)]ds 0 1
(56)
2
2c
The dislocation density function can be found by separating Eq. (45) into real and imaginary parts, respectively given as
Re K ( , ,
0
2c
, 0 ) Kcon ( , , 0 , 0 )b1 ( s) ds
Im K ( , , 0 , 0 ) + Kcon ( , , 0 , 0 )b2 ( s) ds +C1 Fy0 ( 0 )
(57)
2c
and
Im K ( , ,
2c
0
, 0 ) Kcon ( , , 0 , 0 )b1 ( s) ds
Re K ( , , 0 , 0 ) Kcon ( , , 0 , 0 )b2 ( s) ds +C2 = Fx0 ( 0 )
(58)
2c
Eqs. (51), (57) and (58) together with the subsidiary condition in Eq. (44) will be solved numerically for the dislocation coefficients of bi(s).
4.1 Stress intensity factors In order to perform the numerical calculation, the crack is divided into N line segments as indicated in Fig. 3. Then the linear interpolation formulae with the undetermined coefficients are applied to approximate the dislocation distribution along the elements, except in the vicinity of the crack tip where the dislocation distribution preserves a square-root singularity. The interpolation formulae in local coordinate's sj (1 ≤ j ≤ N) are defined as [28] 2d1 bi ( s1 ) bi ,1 1 bi ,2 d1 s1
for the left tip
(59)
2d N bi ( sN ) bi ,N 1 1 bi ,N d N sN
bi ( s j ) bi , j
d j sj 2d j
bi , j 1
d j sj 2d j
for the right tip
(60)
for intermediate segments
(61)
where (i = 0, 1, 2), dj (1 ≤ j ≤ N) are the half length of each line segment, and bi,j (1 ≤ j ≤ N+1) are the unknown coefficients. If the above formulae are used, the singular integral equation together with the subsidiary condition can be carried out to yield N+2 algebraic equations for N+2 unknown coefficients. For the crack tip segment, the following exact logarithmic integrations are used [28] 2d
I1
0
2d 1 log( r )dr 2d (log(2d ) 3) r
(62)
2d
I2
log(r )dr 2d (log(2d ) 1)
(63)
0
2d
I3
0
2d 1 log(2d r )dr 2d log(2d ) 3 4log(2d ) r
(64)
Meanwhile for the intermediate segment from P 2P3 until PN-1PN, the following exact logarithmic integration can be used 2d
I4
0
2d r log(r )dr d log(2d ) 1.5 2d
(65)
2d
I5
r
2d log(r )dr d log(2d ) 0.5
(66)
0
Otherwise, the following approximate Gauss-Chebyshev integration rule is used
d
d
G( s )ds
d
M
G( s M m 1
m
)sin(
2m 1 2m 1 ) with sm d cos( ) ; m = 1, 2, …., M M 2M
(67)
After having the coefficients b1,1 , b2,1 , b1, N 1 , b2, N 1 , the mode-I and mode-II stress intensity factors for the in-plane line crack problems with inclined angle λ can be obtained as [29] K A K IA iK IIA e i 2 2 lim sb( s) 2 2 2d1 e i b1,1 ib2,1 3
3
1/2
s0
3
K B K IB iK IIB e i 2 2 lim sl
(68)
3
a s b( s) 2 2 2d N 1 1/2 e i b1,N 1 ib2,N 1
(69)
where b( s) b1 ( s) ib2 ( s) and the angle of inclination λ is the angle between the direction of a uniform load and the x-axis.
5. Numerical examples
The single inclusion problem with a crack located in matrix is first considered to examine the accuracy of the present approach. Comparisons between the present calculated results with Gc / Gb 1, Ga / Gb 0.5 and the corresponding available results obtained by Lam et al. [31] for different crack locations are given in Fig. 4. From the above numerical examination, we see that the present approach provides reliable results compared with available results. Next, we consider a crack between a circular inclusion and a hole under a
remote uniform load (see Fig. 5). The calculated values of the normalized stress intensity factors closely compare to the available results from Dong and Lee [32] as shown in Fig. 6. The calculated results of the normalized stress intensity factors for different number of crack segments are shown in Table 1 and Table 2. It is seen that the calculated results of the normalized stress intensity factors with N 15 yield an excellent agreement with published values. It is expected that the error percentage will continue to decrease if we increase the number of line segments. We like to emphasize that the computer running time for N=15 takes only a little longer than that for N=10. Note that additional accuracy could be achieved (see Table 3) if the near-tip segments are made smaller by placing d1 d15 d / 9 and d 2 d14 4d / 9 where d is the half length of the intermediate segments as shown in Fig. 3.
In the following discussion, we assume that r1 r2 a , a b c 0.3 , and c / a 1/ 2 ur interest will be focused on the mode-1 stress intensity factors where a crack
is perpendicular to the direction of applied load which presents the worst scenario case of crack propagation. The present proposed methodology can be also applied to the mixed mode fracture problem for the case with the inclined crack where the mode-I and mode-II stress intensity factors are obtained from Eqs. (68) and (69). When the crack is located by the distance h to the left side of the left-hand counterpart of the two inclusions, the normalized mode-I stress intensity factors at tip-A versus the dimensionless location of a crack for different values of Ga / Gb with Gc / Gb 2 , d/ a = 4 are shown in Fig. 7. It is seen that the softer materials may always enhance the stress intensity factors while the harder materials may always reduce the stress intensity factors when a crack approaches the interface 1 .This trend is more pronounced at tip-B which is closer to the interface 1 as shown in Fig. 8. It is clearly seen that the normalized mode-I stress intensity factors at tip-A and tip-B have similar trends even though the material properties Gc / Gb vary as shown in Figs. 9-10. It can be
concluded that the material adjacent to the crack tip dominates the effects on the variation of stress intensity factors in a tri-material composite. When a crack is placed between two circular inclusions at distance h away from the left-hand inclusion with Gc / Gb = 2 and d / a 5 , the normalized mode-I stress intensity factors at the crack tip approaching the
nearest inclusion, tip-A, would increase (or decrease) if the nearest inclusion becomes softer (or stiffer) as indicated in Fig. 11. This trend is less pronounced at tip-B which is farther from the interface 1 as shown in Fig. 12. It is understood that the normalized mode-I stress intensity factors at both tip-A and tip-B are equal to each other when h / a 1 due to the symmetric geometry. When the distance between the two circular inclusions decreases to d / a 4 , the normalized mode-I stress intensity factors at tip-A have a similar trend
compared to the previous case d / a 5 when a crack is approaching the left-hand inclusion as indicated in Fig. 13 and Fig. 15. However, when the crack is approaching the right-hand inclusion, the normalized mode-I stress intensity factors at tip-B would increase (or decrease) if the left inclusion becomes softer (or stiffer) regardless of the value of Ga / Gb as shown in Fig. 14 and Fig. 16. Therefore, we can conclude again that the material adjacent to the crack tip dominates the effects on the variation of stress intensity factors in a tri-material composite. In order to demonstrate the convergence of the series solution presented in this study, the contribution of the stress intensity factors for the first three terms of a serious solution is 88.11%, 10.56%, and 1.33%, respectively. The contribution accounts for the ratio of each term to the summation of the first three terms of a series solution. Since the last term lends less than 2% of the contribution, we conclude that the series solution is rapidly convergent. The calculated results based on our proposed method yield an excellent agreement with the results based on the finite element method which shows that our present proposed method is efficient and general.
6. Concluding remarks
A solution of a crack interacting with two circular inclusions for a tri-material fibrous composite is presented in this study. The solution is based on the method of analytical continuation in conjunction with the alternation technique. This method provides a reliable result compared with published values of two-phase composite case. All the stress intensity factors presented here, including the crack located near the interface, have been obtained by dividing 15 equally spaced segments (N=15) along the crack surface with the summation of a series solution up to the first three terms or more depending on the distance between the crack and the interface. It is worthy to note that the convergence of the series solution is still assured even when the crack tip approaches the boundary of the inclusion. This is because that
our
series
solution
is
expressed
in
terms
of
the
material
parameters
ab (or bc ) and ab (or bc ) instead of the geometric parameter h / a . Some numerical examples have been calculated to investigate the effects of the material properties and geometric configurations on the mode-I stress intensity factors. The interaction between a crack and a tri-material fibrous composite indicates that the material adjacent to the crack tip plays a more significant influence on the stress intensity factors. It also shows that a reinforcement of increasing stiffness, such as a particle or fiber, may result in a decrease of local stress intensification in the crack tips. Compared with published values of two-phase composite case, the presented method has provided reliable results. The calculated results based on our proposed method yield an excellent agreement with the results based on the finite element method which shows that our present proposed method is efficient and general. The present proposed method can be further extended to model the interaction between a
multi-layered media with different types of crack such as arc cracks, kink cracks and branch cracks. The results obtained in this study may provide some guidance for minimizing the SIF by carefully selecting the material and geometry. The problem of a crack contained wholly within the reinforcement remains as another subject to be solved in the future.
Acknowledgment
The research was supported by the Ministry of Science and Technology of Taiwan under grant MOST 103-2221-E-011-023-MY3.
Nomenclature
( z ), ( z)
Complex potential functions for in-plane elasticity
( z )
Auxiliary stress function
G
Shear modulus
Poisson’s ratio
,
Non-dimensional parameters
z
Location of view point in the physical domain
z0
Location of singular point in the physical domain
Location of view point in the transformed domain
0
Location of singular point in the transformed domain
r
Radius of circular inclusion in the physical
domain
Radius of circular inclusion in the transformed domain
a
The first circular inclusion in the physical domain
b
Matrix in the the physical domain
c
The second circular inclusion in the physical domain
Sa
The first circular inclusion in the transformed domain
Sb
Matrix in the transformed domain
Sc
The second circular inclusion in the transformed domain
1
Interface between
a
and
b
2
Interface between
c
and
b
L1
Interface between
Sa
and
Sb
L2
Interface between
Sc
and
Sb
x, y
Coordinates in the physical domain
,
Coordinates in the transformed domain
c
Half-length of a crack
b1 ib2
Density function of edge dislocations
bi , j
Unknown coefficient of dislocation distribution
d
Half-length of crack segment
h K I , K II
Distance between the interface and the crack tip Mode-1 and Mode-11 stress intensity factors
N
Number of crack segments
Y iX
Resultant forces
Remote uniform tensile load
Appendix A. Derivations of Eqs. (18-37) We start to regard Sb and Sc composed of the same material b and region S a of material
a . 1 ( ) and 1 ( ) holomorphic in Sb Sc (except at z 0 ), a1 ( ) and a1 ( ) holomorphic in S a are introduced to satisfy the continuity of traction and displacement across
L1 that
a1 ( ) a1 ( ) 0 ( ) 0 ( ) 1 ( ) 1 ( )
L1
(A1)
1 1 aa1 ( ) a1 ( ) b0 ( ) 0 ( ) b1 ( ) 1 ( ) Gb Ga
L1
(A2)
12 m( ) ' 0 ( ) 0 ( ) . where 0 ( ) ' m ( ) By standard analytical continuation arguments, it follows 1 ( ) 0 (
12 2 ( 2 1)2 ( 2 1) 2 ) a1 ( 1 ) 1 C0 1 Ca1 0 ( 1) ( 1)
1
(A3)
a1 ( ) 0 ( ) 1 (
b Gb
a Ga
1 ( )
a1 ( )
12 ( 2 1)2 ( 2 1)2 ) 1 C0 1 Ca1 0 ( 1) ( 1)
1
2 2 ( 2 1)2 C0 ( 12 1) 2 Ca1 1 1 0 ( 1 ) a1 ( 1 ) 1 0 Gb Ga ( 1) Gb ( 1) Ga
b Gb
0 ( )
2 ( 2 1)2 C0 ( 12 1)2 Ca1 1 1 ( 1 ) 1 0 Gb ( 1) Gb ( 1) Ga
(A4)
1
1
(A5)
(A6)
Uncoupling Eqs. (A3) - (A6), we obtain a1 ( ) (1 ab )0 ( ) ba
( 12 1) 2 Ca1 ( 1)
(A7)
12 ( 12 1)2 ( 12 1) 2 1 ( ) ab 0 ( ) 2 C0 (1 ba ) 2 Ca1 1 1 ( 1) ( 1) 1 ( ) ab 0 (
(A8)
12 ( 2 1)2 ) ab 1 C0 ( 1)
a1 ( ) (1 ab )0 ( ) (1 ab )
( 12 1) 2
2 ( 1 1)
(A9)
C0
( 12 1) 2
2 ( 1 1)
Ca1
(A10)
where C0 0' ( 12 ) and Ca1 a' 1 ( 12 ) .
Since this result is based on the assumption that region Sc is made up of material b , it cannot satisfy the continuity condition at the interface L2 which lies between material b and
c . In the second step, we assume region Sb and S a be made up of the same material b and region Sc of material c . Additional terms b1 ( ) and b1 ( ) holomorphic in Sb Sa (except at 0 ), c1 ( ) and c1 ( ) holomorphic in Sc are introduced to satisfy the continuity conditions across L2 that
1 ( ) 1* ( ) 0 ( ) 0* ( ) b1 ( ) b1 ( ) c1 ( ) c1 ( ) L2 1 b0 ( ) 0* ( ) b1 ( ) 1* ( ) bb1 ( ) b1 ( ) Gb 1 cc1 ( ) c1 ( ) L2 Gc
m( where 0* ( ) 0 ( )
(A11)
(A12)
22 2 2 2 ) m( 1 ) m ( 2 ) m( 1 ) ' ' 0 ( ) and 1* ( ) 1 ( ) 1 ( ) . ' ' m ( ) m ( )
Based on the method of analytical continuation, we have
b1 ( ) 0* (
22 2 2 ( 2 1) 2 C0* ) 1* ( 2 ) c1 ( 2 ) 2 ( 1)
( 2 1) 2 C1* ( 22 1) 2 Cc1 2 0 ( 1) ( 1)
c1 ( ) 0 ( ) 1 ( ) b1 (
2
22 ( 22 1) 2 C0* ) ( 1)
( 2 1) 2 C1* ( 22 1) 2 Cc1 2 0 ( 1) ( 1)
(A13)
(A14)
2
b 22 ( 22 1)2 C0* ( 22 1)2 C1* 1 * 12 1 * 12 1 b1 ( ) 0 ( ) 1 ( ) c1 ( ) Gb Gb Gb Gc Gb ( 1) Gb ( 1) (A15)
( 22 1)2 Cc1 0 Gc ( 1)
2
c b b 22 ( 22 1)2 C0* ( 22 1)2 C1* 1 c1 ( ) 0 ( ) 1 ( ) b1 ( ) Gc Gb Gb Gb Gb ( 1) Gb ( 1) (A16)
( 1) Cc1 0 Gc ( 1) 2 2
2
2
Solve Eqs. (A13) - (A16) to obtain
2 ( 2 1)2 (C0* C1* ) 2 b1 ( ) cb 0* ( 2 ) 1* ( 2 ) cb 2 ( 1) c1 ( ) (1 cb ) 0* ( ) 1* ( ) (1 cb )(C0* C1* )
( 22 1)2 (
c1 ( ) (1 cb ) 0 ( ) 1 ( )
b1 ( ) cb 0 (
(A17)
22 1)
( 22 1) 2 Cc1 (
22 1)
bc ( 22 1)2 Cc1 1
(A18)
(A19)
22 2 ( 2 1)2 (C0* C1* ) (1 bc )( 22 1) 2 Cc1 ) 1 ( 2 ) 2 22 2 ( 1) ( 2 1)
(A20)
where C0* 0' ( 22 ) ， C1* 1' ( 22 ) ， Cc1 c'1 ( 22 ) .
Since this result is based on the assumption that region Sb is made up of material b, it cannot satisfy the continuity condition across L1 which lies between material a and b . In the third step, we again regard region Sb and Sc composed of the same material b and region S a of
material
a
.
Similar
to
the
previous
approach,
additional
terms
a 2 ( ), a 2 ( ), 2 ( ), 2 ( ) , which are introduced to satisfy the continuity conditions at the interface L1 , can be obtained as
( 12 1)2 a 2 ( ) (1 ab )b1 ( ) ba Ca 2 ( 1)
2 ( ) ab b1 (
( 2 1)2 Ca 2 ( 12 1)2 Cb*1 12 ) (1 ba ) 1 2 1 2 ( 1) ( 1 1)
(A21)
(A22)
2 ( ) ab b*1 (
( 2 1)2 Cb*1 12 ) ab 1 ( 1)
a 2 ( ) (1 ab )b*1 ( )
( 12 1)2 Ca 2 (
12 1)
(A23)
(1 ab )
( 12 1)2 Cb*1 (
12 1)
(A24)
where Ca 2 2' ( 12 ) and Cb*1 b' 1 ( 12 ) .
In the fourth step, regions Sa and Sb are assumed to make up with material b again. Repetitions of second and third step, the analytical continuation method is alternatively applied to two interfaces to obtain the additional terms an ( z ), bn ( z ), cn ( z ), n 1 ( z) and
an ( z), bn ( z), cn ( z), n1( z) (n 2,3,4,...) . The stress functions can be finally obtained as
bn ( ) cb n*1 (
22 ) h( )Cn*1
cn ( ) (1 cb ) n*1 ( ) h(
22 * 2 )Cn 1 h( 2 )Cc ( n 1)
(A25)
(A26)
cn ( ) (1 cb )n1 ( ) bc h( )Cc ( n1)
(A27)
22 22 * 22 bn ( ) cb n 1 ( ) h( )Cn 1 (1 bc )h( )Cc ( n 1)
(A28)
a ( n1) ( ) (1 ab )bn ( ) bau( )Ca ( n1)
(A29)
n1 ( ) ab bn (
12 2 2 * ) (1 ba )u ( 1 )Ca ( n 1) u ( 1 )Cbn
(A30)
* n1 ( ) ab bn (
12 * ) u ( )Cbn
* 2 * 12 a ( n1) ( ) (1 ab ) bn ( ) u ( 1 )Cbn u ( )C a ( n 1)
(A31)
(A32)
* ' * where Cn n' ( 12 ) , Cn* n' ( 22 ) , Can an' ( 12 ) , Can an ( 22 ) , Cbn bn' ( 22 ) , Cbn bn' ( 12 ) , * Ccn cn' ( 22 ) , Ccn cn' ( 12 ) , u ( )
( 12 1) 2 ( 2 1) 2 and h( ) 2 , (n=2,3,4……). ( 1) ( 1)
References [1] Piggott M. Load bearing fibre composites. Kluwer Academic Publishers; 2002.
[2] Schiavone P. Neutrality of the elliptic inhomogeneity in the case of non-uniform loading. Int. J. Eng. Sci 2003;41(18):2081-2090. [3] Mahboob M, Schiavone P. Designing a neutral elliptic inhomogeneity in the case of a general non-uniform loading. Appl. Math. Letters 2005;18(11):1312-1318. [4] Benveniste Y, Miloh T. Soft neutral elastic inhomogeneities with membrane-type interface conditions. J. Elas 2007;88(2);87-111. [5] Mantic V. Interface crack onset at a circular cylindrical inclusion under a remote transverse tension. Application of a coupled stress and energy criterion. Int J Solids Struct 2009;46(6):1287-1304. [6] Galiotis C, Young RJ, Yeung PHJ, Batchelder DN. The study of model polydiacetylene/epoxy composites: Part 1 the axial strain in the fibre. J Mater Sci 1984;9(11):3640-3648. [7] Kelly A, Tyson WR. Tensile properties of fiber-reinforced metals: copper/tungsten and copper/molybdenum. J Mech Phys Solids 1965;13(1):39-50. [8] Hull D, Clyne T. An introduction to composite materials. Cambridge University Press, Cambridge, UK; 1996. [9] Irwin GR. Analysis of stresses and strains near the end of a crack traversing a plate. ASME J Appl Mech 1957;4(3):167-170. [10] Greco F, Leonette L, Nevone Blasi P. Non-linear macroscopic response of fiberreinforced composite materials due to initiation and propagation of interface cracks. Engng Fract Mech 2012;80(1):92-113. [11] Greco F, Leonette L, Lonetti P. A two-scale failure analysis of composite materials in presence of fiber/matrix crack initiation and propagation. Composite Structures 2013;95(1):582-597. [12] Greco F, Leonette L, Nevone Blasi P. Adaptive multiscale modeling of fiber-reinforced
composite materials subjected to transverse microcracking. Composite Structures 2014;113(1):249-263. [13] Greco F, Leonette L, Lonetti P, Nevone Blasi P. Crack propagation analysis in composite materials by using moving mesh and multiscale techniques. Computers and Structures 2015;153(1):201-216. [14] Dal Corso F, Shahzad S, Bigoni D. Isotoxal star-shaped polygonal voids and rigid inclusions in nonuniform antiplane shear fields: Part II: Singularities, annihilation and invisibility. Int J Solids Struct 2016;85-86:76-88. [15] Shahzad S, Dal Corso F, Bigoni D. Hypocycloidal inclusions in nonuniform out-ofplane elasticity: stress singularity vs stress reduction. J. Elas 2017;126(2):215-229. [16] Chao CK, Wikarta A. Solutions of a crack interacting with a three-phase composite in plane elasticity. Appl Math Model 2016;40(3):2454-2472. [17] Erdogan F, Gupta G. The stress analysis of multi-layered composites with a flaw. Int J Solids Struct 1971;7(1):pp. 39-61. [18] England AH. Complex variable methods in elasticity, Wiley, London; 1971. [19] Suo Z. Singularities interacting with interfaces and cracks. Int J Solids Struct 1989;25(10):1133-1142. [20] Xiao ZM, Chen BJ. On the interaction between an edge dislocation and a coated inclusion. Int J Solids Struct 2001;38(15):2533-4842. [21] Chao CK, Kao B. A thin cracked layer bonded to an elastic half-space under an antiplane concentrated load. Int J Frac 1997;83(3):223-241. [22] Choi ST, Earmme YY. Elastic study on singularities interacting with interfaces using alternating technique: Part 2 Isotropic trimaterial. Int J Solids Struct 2001;39(5):11991211. [23] Chao CK, Chen FM. Thermal stresses in an isotropic trimaterial interacted with a pair of
point heat source and heat sink. Int J Solids Struct 2004;41(22):6233-6247. [24] Porter GA, Chuang CT, Chao CK, Chang RC. Thermal stresses in a viscoelastic trimaterial with a combination of a point heat source and a point heat sink. J Eng Math 2008;61(2):99-109. [25] Chao CK, Chen FM, Shen MH. Circularly cylindrical layered media in plane elasticity. Int J Solids Struct 2006;43(16):4739-4756. [26] Chao CK, Chen FM, Shen MH. Green's functions for a point heat source in circularly cylindrical layered media. J Therm Stresses 2006;29(9):809-847. [27] Chen FM, Chao CK, Chen CK. Interaction of an edge dislocation in a coated elliptic inclusion. Int J Solids Struct 2011;48(10):1451-1465. [28] Pak YE, Mishra D, Yoo SH. Closed-form solution for a coated circular inclusion under uniaxial tension. Acta Mech 2012;223(5):937-951. [29] Chen YZ, Cheung YK. New integral equation approach for the crack problem in elastic half-plane. Int J Frac 1990;46(1):57-69. [30] Lars VA. Conformal invariants: Topics in geometric function theory, McGraw-Hill Book Co, New York; 1973. [31] Lam KY, Ong PP, Wude N. Interaction between a circular inclusion and a symmetrically branched crack. Theore Appl Frac Mech 1998;28(3):197-211. [32] Dong CY, Lee KY. A new integral equation formulation of two-dimensional inclusion– crack problems. Int J Solids Struct 2005;42(18):5010-5020.
Table 1. Comparisons between the calculated and published values of KI at tip-A
h/a 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Ga/Gb =0.5 Lam et al. [31] 1.297 1.206 1.155 1.121 1.098 1.081 1.068 1.058
N=5 1.4253 1.3217 1.2614 1.2206 1.1911 1.1689 1.1517 1.1381
N=10 1.2934 1.1976 1.1476 1.1176 1.0975 1.0886 1.0894 1.0645
N=15 1.2983 1.2096 1.1585 1.1244 1.0983 1.0858 1.0754 1.0634
Table 2. Comparisons between the calculated and published values of KI at tip-A Ga/Gb
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Gc=0 Dong and Lee [32] 3 1.8 1.43 1.25 1.125 1.05 1 0.97 0.94 0.93 0.92
N=5
N=10
N=15
2.6091 2.2983 2.1085 2.0134 1.9567 1.9191 1.8924 1.8724 1.8569 1.8445 1.8344
2.9824 1.8989 1.5246 1.3355 1.2209 1.1440 1.0886 1.0468 1.0141 0.9878 0.9662
2.9872 1.8108 1.4386 1.2588 1.1329 1.0574 1.0068 0.9768 0.9466 0.9374 0.9273
Table 3. Comparisons between equally spaced segments and non-equally spaced segments of the crack discretization at tip-A Ga/Gb
Gc=0 Dong and Lee [32]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
3 1.8 1.43 1.25 1.125 1.05 1 0.97 0.94 0.93 0.92
Gc=0 Non-Equally spaced segments (N=15) 2.9891 1.8076 1.4332 1.2536 1.1289 1.0535 1.0032 0.9742 0.9438 0.9357 0.9261
Gc=0 Equally spaced segments (N=15) 2.9872 1.8108 1.4386 1.2588 1.1329 1.0574 1.0068 0.9768 0.9466 0.9374 0.9273
Figure 1: Interaction of a crack with two circular inclusions embedded in a homogeneous isotropic material under a remote uniform load
Figure 2: The problem in the plane
PN-1
P3 P1
s1 P2
P4
PN+1
SN PN
2dN
2d1
Figure 3: Division and nodal distribution of a crack
Figure 4: A comparison between the calculated and published values of the normalized mode-I stress intensity factors at tip-A for a crack located in an infinite matrix
Figure 5: A crack located in an infinite matrix between a circular inclusion and a hole under a remote uniform load
Figure 6: A comparison between the calculated and published values of the normalized mode-I stress intensity factors at tip-A for a crack located in an infinite matrix
Figure 7: Normalized mode-I stress intensity factors at tip-A versus dimensionless location of crack for different values of Ga / Gb with Ga / Gb =2, d/ a = 4
Figure 8: Normalized mode-I stress intensity factors at tip-B versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 2, d/ a = 4
Figure 9: Normalized mode-I stress intensity factors at tip-A versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 0.5, d/ a = 4
Figure 10: Normalized mode-I stress intensity factors at tip-B versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 0.5, d/ a = 4
Figure 21: Normalized mode-I stress intensity factors at tip-A versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 2, d/a= 5
Figure 32: Normalized mode-I stress intensity factors at tip-B versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 2, d/a = 5
Figure 13: Normalized mode-I stress intensity factors at tip-A versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 2, d/a= 4
Figure 14: Normalized mode-I stress intensity factors at tip-B versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 2, d/a= 4
Figure 15: Normalized mode-I stress intensity factors at tip-A versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 0.5, d/a = 4
Figure 16: Normalized mode-I stress intensity factors at tip-B versus dimensionless location of crack for different values of Ga / Gb with Gc / Gb = 0.5, d/a= 4
Highlights
． A logarithmic singular integral equation is proposed.
． The analytical continuation technique and the alternation method are used.
． The derived series solution is rapidly convergent.
． The softer material may give enhancement effect on SIF.
． The stiffer material may give retardation effect on SIF.