Crack tip plasticity of a half-infinite Dugdale crack embedded in an infinite space of one-dimensional hexagonal quasicrystal

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Crack tip plasticity of a half-infinite Dugdale crack embedded in an infinite space of one-dimensional hexagonal quasicrystal Peidong Li, Xiangyu Li ∗ , Guozheng Kang School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China

a r t i c l e

i n f o

Article history: Received 11 August 2014 Received in revised form 5 August 2015 Accepted 12 September 2015 Available online xxx Keywords: 1D hexagonal QC Half-infinite crack Dugdale model Plastic zone

a b s t r a c t The present paper is devoted to determining the crack tip plasticity of a half-infinite Dugdale crack embedded in an infinite space of one-dimensional hexagonal quasicrystal. A pair of equal but opposite line loadings is assumed to be exerted on the upper and lower crack lips. By applying the Dugdale hypothesis together with the elastic results for a half-infinite crack, the extent of the plastic zone in the crack front is estimated. The normal stress outside the enlarged crack and crack surface displacements are explicitly presented, via the principle of superposition. The validity of the present solutions is discussed analytically by examining the overall equilibrium of the half-space. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Quasicrystals (QCs), as a new structure of solid matter, were first discovered by Shechtman et al. [30]. Since then, QCs have become the focus of theoretical and experimental studies in the physics of condensed matter [8]. In the past three decades, a great progress has been made in understanding of the geometric structure and mechanical properties of QCs [6,8]. To date, QCs have been observed to have some desirable properties, such as low friction coefficient [5], low adhesion [25] and high wear resistance [2]. These desirable properties make QCs enjoy a high potential of practical applications to various engineerings. Recently, Kenzari et al. [18] find that QCs can be used as reinforced phases of polymer-matrix composites. Due to wide application prospects of QCs, researches of mechanical properties of QCs become extremely important and necessary. As a long-standing problem in solid mechanics, crack analysis has been extended to mechanics of QCs in the recent decade [9,11]. Up to now, a lot of research efforts to crack analyses for QCs have been made [11]. For example, with the help of crack theories in conventional linear elasticity fracture mechanics, Li et al. [22] derived an exact analytic solution for a uniformly pressured Griffith crack in decagonal QCs, by means of a general solution. Then, Peng and Fan [26–28], making use of integral transform techniques, investigated the problems of circular cracks in one-dimensional (1D) hexagonal QCs and two-dimensional (2D) decagonal QCs. Later, Fan

∗ Corresponding author. Tel.: +86 28 87634141; fax: +86 28 87600797. E-mail address: [email protected] (X. Li).

and Mai [9] gave a comprehensive review of the mathematical theory and methodology of elasticity of QCs and their applications to dislocations and cracks. For the anti-plane crack problems in 1D hexagonal QCs, some works [31,14,15] have been achieved under the framework of elasticity of QCs, to explore the effect of phason field on the deformations of the cracked materials. For the plane problems of an elliptic hole and a crack in three-dimensional (3D) QCs, Gao et al. [12,13] derived the explicit solutions for the phononphason coupled fields are obtained in closed forms, by a complex potential approach and the generalized Stroh formalism. Fan et al. [11] made a comprehensive review on the fracture theory of QCs concerning with linear, nonlinear and dynamic crack problems for QCs of various types. Recently, Li [19], using the potential theory method in conjunction with the general solutions, solved the mode I problems of three common planar cracks and presented the elastic fundamental fields as well as some important parameters in crack analysis. In the context of the classical theory of linear elasticity, the stresses at the crack tip are singular and exceed the yield stress of the material, which is an unrealistic behavior in practice [17,1]. Hence, the real crack problem should be taken into consideration under the framework of plasto-elasticity. To dispose the non-linear crack problem, Dugdale [7] proposed a simple model which considers a strip zone of concentrated plastic deformation in the crack front. In his model, Dugdale assumed that a constant cohesive stress, which is equal to the yield stress of the material, exists in the plastic zone. In fact, Dugdale model is very effective and has been verified by experiments [7,3]. Due to its simplicity and validity, the idea underlying the Dugdale model has been widely

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Please cite this article in press as: P. Li, et al., Crack tip plasticity of a half-infinite Dugdale crack embedded in an infinite space of one-dimensional hexagonal quasicrystal, Mech. Re. Commun. (2015), http://dx.doi.org/10.1016/j.mechrescom.2015.09.007

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adopted and developed to dispose a variety of crack or cracklike problems [3,29,33,4,24,16,21]. For instance, applying Dugdale’s approach, Maugis [24] successfully explained the transition of JKRDMT adhesive contact theory. Furthermore, Dugdale crack model is generalized to materials with more complicated yield surfaces, where the Tresca or von Mises yield criterion is required to satisfy in the crack tip plastic zone [3,16]. In recent years, some efforts on the Dugdale crack of QCs have been made by Fan and his coauthors [10,23,32], to investigate the size of plastic zone. To the best of authors’ knowledge, however, the problem of the half-infinite Dugdale crack embedded in an infinite space of 1D hexagonal QC has not been studied. The purpose of this paper is to study a half-infinite Dugdale crack in 1D hexagonal QC. The crack is assumed to be subjected to a pair of equal but opposite line phonon forces in the upper and lower crack surfaces. The Dugdale hypothesis in company with the elastic results for a half-infinite crack is applied to evaluate the extent of plasticity. With the aid of superposition principle, the normal stress outside the enlarged crack and crack surface displacements (CSDs) are explicitly presented. Numerical calculations are performed to examine the validity of the present solutions and to show the influence of some parameters on the distributions of some important physical quantities. Furthermore, the present solutions can be used to serve as benchmarks for computational fracture mechanics.

problem (MBVP) of the half-space z ≥ 0, with the following boundary conditions prescribed on the plane z = 0:

∀ (x, y) ∈ S :

⎫ ⎪ ⎬

0ı zm = Pm (y − ym ) ;

∀ (x, y) ∈ I − S : uzm = 0; ∀ (x, y) ∈ I :

z = zx + izy = 0, i =

√

−1,

⎪ ⎭

(2)

where  z1 ( z2 ) and uz1 (uz2 ) respectively denote the normal stress and the displacement in the z-direction in phonon (phason) field;  z is the complex phonon shear stress. Hereafter, the indices 1 and 2 indicate the physical quantities associated with the phonon and phason fields, respectively, unless otherwise stated. This MBVP has been investigated by [19], as a direct application of the fundamental solutions. For better usage, the normal stress on the crack plane, stress intensity factors (SIFs) and CSDs are recalled here. The generalized stress components on the intact region I − S read zm |z=0 = −

0 Pm  (ym − y)



ym , (y < 0) . −y

(3)

Defining the SIFs of mode I crack problem as km = lim



y→0−

−yzm (x, y, 0) ,

we arrive at 2. Preliminaries for a half-infinite crack Consider an infinite space of 1D hexagonal QC weakened by a half-infinite plane crack in parallel with the isotropic plane. In the Cartesian coordinate system (x, y, z), the atoms of 1D QCs are arranged periodically in the planes parallel to the xoy plane and quasiperiodically in the z-direction. For convenience, the cracked plane is assumed to be coincident with the plane z = 0 (denoted by I hereafter). For simplicity, the region of the crack is symbolized by S≡





(x, y, z) | − ∞ < x < +∞, y ≥ 0, z = 0 .

P0 km = − √m .  ym The CSDs in this case are of the form uz1 =

P10 g22 2 A

On the upper and lower crack lips are applied two pairs of equal but opposite line loads ±P10 (phonon) and ±P20 (phason) respectively along the lines y = ym (m = 1, 2 ; ym > 0, − ∞ < x < + ∞ , z = 0), as illustrated in Fig. 1. Therefore, the half-infinite crack problem is reduced to a plane strain problem and all the physical quantities would be independent of the variable x. In view of the symmetry with respect to the plane z = 0, the problem in question can be converted into a mixed boundary value

I (y; y1 ) −

P20 g12 2 A

I (y; y2 )

(5a)

and uz2 = −

(1)

(4)

P10 g21 2 A

I (y; y1 ) +

P20 g11 2 A

I (y; y2 ) ,

(5b)

where I (y; ym ) is defined by √ y + ym I (y; ym ) = ln √ √ (y > 0) . | y − ym | It is noted that the constants A and gij (i = 1, 2 ; j = 1, 2) in (5) are material parameters which are defined by [19] and presented in Appendix A. The constitutive laws of 1D hexagonal QC along with some auxiliary parameters are presented in Appendix A as well. 3. Half-infinite Dugdale crack Consider the aforementioned crack only subjected to a pair of equal but opposite phonon loads ±p0 exerted on the line y = y1 , namely, p01 (x, y) = −p0 ı (y − y1 ) , p02 (x, y) = 0.

Fig. 1. A schematic figure for a half-infinite crack subjected two pairs of line loads ±P10 (phonon) and ±P20 (phason) applied along the lines y = y1 and y = y2 , respectively.

(6)

It is necessary to note that phason loads are not considered in this case, on account of the weak effect of phason field on the deformations of the cracked material [20,19]. The same phason actions often appear as about 1% perturbations of the standard stress field. Therefore, phason loads are difficult to be controlled from the exterior, at the macroscopic scale [19]. In fact, the analytical approach for the case of phason loads can be readily extended from the present work, without any difficulty. Owing to the external source, plastic deformations occur at the crack front since the stresses in the neighborhood of the crack tip exceed the yield stress of the material constituting the body. Since the problem in question is a plane strain one, the crack front after deformation would be parallel to the original one. In

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together with the last two conditions in (7). Hereafter, the superscripts I and II indicate the physical quantities corresponding to the first and second sub-problems, respectively, without specification elsewhere. The problem in (8) have been solved in the last section with the parameters P10 , P20 , y and y1 replaced with −p0 , 0, y and y1 + d, respectively. Without any difficulties, we can get the following nonvanishing components of SIF,  zm |z=0 and CSDs as k1I =

p0





uIz1 = −

p0 g22 G1 y  2  A

with

other words, the plastic zone becomes an infinite strip symbolized by {(x, y, z)| − d < y ≤ 0, − ∞ < x < ∞ , z = 0}, as shown in Fig. 2. According to the Dugdale hypothesis [7], the normal stress  z1 in the infinite strip is equal to the yielding stress  Y of the constituent material under uniaxial tension along the z-direction. As a consequence, the original nonlinear plasto-elastic problem is transformed to a linear one which can be formulated by the following MBVP on the plane z = 0:

∀ (x, y) ∈

S:

⎫ ⎬

∀ (x, y) ∈ I :

⎪ ⎪ ⎭

z1 = 0,

p (x, y) =



and uIz2 = −



y1 + d|

x = x,

−d ≤ y ≤ 0

−p0 ı (y − y1 )

y>0

 II | z1 z=0

d

=− 0

Y  (y0 − y )



2 = − Y 

uIIz1

2 Y d;  Y g22  I2 y = 22 A



z = z.

Thus, the region

S occupied by the enlarged crack and the loads

p (x, y) are of the following forms  

S = (x , y , z ) ||x | < ∞, y ≥ 0, z = 0 , 

p =

Y ,

0 ≤ y ≤ d

−p0 ı (y − y1 − d) ,

y > d

I2 y

= y ln

I ∀ x  , y ∈

S : z1 = −p0 ı y − y1 − d , z2 = 0,

d2 + y1 d −



(8)

along with the last two conditions in (7). The second is specified by



∀ x ,y ∈

S: 



 II z1

=

Y , 0,

0 ≤ y ≤ d y > d

y > 0 .



y0 dy0 −y

d − arctan −y



 d −y



,



(11)

y <0 .

uIIz2 = −

and

Y g21  I2 y , 22 A

(12b)

y −

√ 2 d + d ln

y2



y d,



y + d √ 2 y − d

y > 0 .

p0





z2 = 0,

(9)

y1 + d

−

2 Y d 

which gives rise to 2 =0 4

(13)

with  = p0 / Y . It should be point out that the ratio  = p0 / Y has a unit of length. From (13), we can obtain that



,



The extent of plastic zone can be determined by applying the Dugdale hypothesis

.

0 = k1I + k1II =

(10c)

(12a)

(y + d)



+2

This makes the results in Section 2 directly applicable to the current MBVP (7). The MBVP (7) can be viewed as a superposition of the following two sub-problems. The first subproblem is formulated as



p0 g21 G1 y 2  A

where



y = y + d,

(10b)

k1II = −

Y ,





,

y < 0 ,

The solution to the second subproblem formulated by (9) can be also obtained from the solution in the last section by integrating the corresponding variables, with the help of concept of Green’s S function. For example, the normal stress in the intact region I −

can be delivered by

For convenience, we make the following transform 



y1 + d , −y

y + y1 + d

y −

|



Similarly, we can determine the SIF and CSDs, integrating (4) and (5). Since the integrals involved are basic and elementary, without details, we list the final results as

 

S = (x, y, z) ||x| < ∞, y ≥ −d, z = 0 



= ln

G1 y

(7)

where the enlarged half-infinite crack is symbolized by

and



z1 =

p (x, y) , z2 = 0; ⎪ ⎪

∀ (x, y) ∈ I −

S : uzm = 0;

(10a)

p0 =  (y1 + d − y )

I z1 |z=0

Fig. 2. Vertical cross section of an enlarged half-infinite plane crack and the normal stress distribution on the lower crack lip.

,

y1 + d

1 d = y1 2



1+

2 y12

−1



1 4

 p 2 0 Y y1

(14)

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Table 1 Material constants for a particular 1D hexagonal QC (cij , Ki and Ri are in GPa) [19]. Phonon elastic

c11 = 150 c33 = 90 K1 = 0.084 R1 = −1.68

0.20

c13 = 45

R3 = 1.20

0.15

D

Phason elastic Phonon-phason

c12 = 55 c44 = 50 K2 = 0.036 R2 = 1.20

0.22

0.10

where second equation holds if p0  Y y1 . The resultant normal stress and CSDs are determined by superposing the results in (10) associated with the first subproblem (8) and those specified by (11) and (12) pertinent to the second subproblem (9). 4. Numerical results and discussions

0.0 0.00

Let us discuss the validity of the present solutions by examining the overall equilibrium of the half-space z ≥ 0. Integrating the normal stress  z1 , which is the sum of (10b) and (11), with respect to y from negative infinity to zero, we arrive at





0

z1 |z=0 dy −∞



0

= −∞

I z1 |z=0 dy +

= p0 − Y d.

−∞

II z1 |z=0 dy

0.50

∙

0.75

1.00

Fig. 3. The dimensionless plastic zone D as a function of . ¯

(15)

0

z1 dy + Y d − p0 = 0.

(16)

−∞

Thus, the validity of the present solutions is analytically checked. Numerical calculations are performed to show the influence of some parameters. To fulfill our purpose, we consider a special 1D hexagonal QC with its elastic constants tabulated in Table 1 by referring to the previous work of Li [19]. For the sake of convenience, the following dimensionless quantities are introduced =

0.25

0

The detailed processes of (15) are displayed in Appendix B. As a result, we can find the following equilibrium condition of the halfspace z ≥ 0 is satisfied automatically,



0.05

z1 uz1  d y ,= , ¯ = ,D= and y¯ = , Y y1 y1 y1 y1

(17)

where the value of the yielding stress  Y is assumed to be 0.15 (unit: GPa). The extent of the dimensionless plastic zone D is shown in Fig. 3. It should be noted that the dimensionless coefficient ¯ can

be regarded as the dimensionless exerted load. From Fig. 3, we can see that the dimensionless plastic zone D rapidly increases with the dimensionless load . ¯ The distributions of the dimensionless normal stress  in the intact region I −

S are shown in Fig. 4. From Fig. 4(a), we know S , the normal stress  increases with both that, in the region I −

the dimensionless spatial coordinate y¯ and the dimensionless load . ¯ It is evident that, in the plastic zone −d ≤ y ≤ 0, the value of the dimensionless normal stress  is identical to 1, which is consistent with the Dugdale hypothesis [7]. Fig. 4(b) shows that the value of ¯ which is expected from  tends to be zero with the decrease of y, a physical point of view. Fig. 5 illustrates the variations of the dimensionless CSD  with ¯ Evidently, the dimensionless CSD  is singular at the acting y. position of line loading (y¯ = 1), which is not surprised and can be mathematically interpreted from (10c) and (12b). As expected, the values of CSDs are identical to zero at the tip of the enlarged crack and tend to vanish at infinity, which are respectively evidenced in Fig. 5(a) and Fig. 5(b). Further, Fig. 5 also indicates that  increases with . ¯

Fig. 4. Variations of the dimensionless normal stress  with y¯ for various magnitudes of coefficients , ¯ in linear scale (a) and logarithmic scale (b).

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Fig. 5. Variations of the dimensionless CSD  with y¯ for various magnitudes of coefficients , ¯ in linear scale (a) and logarithmic scale (b).

5. Conclusions

Appendix A.

In the present paper, we consider a half-infinite crack embedded in an infinite space of 1D hexagonal QC. The crack is assumed to be subjected to a pair of equal but opposite line loadings on the upper and lower crack surfaces. Dugdale crack model [7] is employed to linearize the nonlinear problem. With the help of the Dugdale hypothesis [7], the extent of the plastic zone in the crack front is explicitly obtained, based on the recent progress of crack analysis [19]. Moreover, the normal stress outside the enlarged crack and CSDs in the phonon and phason fields are also explicitly expressed in terms of elementary functions, via the principle of superposition. The validity of the present solutions is analytically verified examining the overall equilibrium of the half-space z ≥ 0. Numerical calculations have been performed to show the influence of some parameters. The present analytical results play important roles in fracture mechanics of QCs. The methods adopted in the present paper are not limited to plane problems of half-infinite cracks and can be extended to solve axisymmetric problems of planar cracks, for instance, penny-shaped and external circular cracks. Hence, a promising extension of the present study can be expected.

This section is devoted to defining some useful quantities which are involved in Section 2. The constitutive laws of 1D hexagonal QC read [8]

∂uy ∂ux ∂uz ∂wz + c12 + R1 , + c13 ∂x ∂y ∂z ∂z ∂uy ∂ux ∂uz ∂wz yy = c12 + c11 + c13 + R1 , ∂x ∂y ∂z ∂z ∂uy ∂ux ∂uz ∂wz zz = c13 + c33 + c13 + R2 , ∂x ∂y ∂z ∂z

xx = c11



yz = zy = c44

 zx = xz = c44

 xy = yx = c66

∂uy ∂uz + ∂z ∂y ∂uz ∂ux + ∂x ∂z ∂ux ∂uy + ∂x ∂y



+ R3

∂wz , ∂y

+ R3

∂wz , ∂x

 

(A.1a)

and

Acknowledgements

Hzz = R1

This work is supported by the National Natural Science Foundation of China (No: 11102171) and Program for New Century Excellent Talents in University of Ministry of Education of China (NCET-13-0973). The supports from Excellent Youth Foundation of Sichuan Scientific Committee (No.: 2015JQO008; China), State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics, China) (MCMS-0214G01) and the Alexander von Humboldt Foundation (Germany) are acknowledged as well.

Hzx = R3

∂uy ∂ux ∂uz ∂wz + R1 + R2 + K1 , ∂x ∂y ∂z ∂z



 Hzy = R3

∂uz ∂ux + ∂x ∂z ∂uy ∂uz + ∂z ∂y



+ K2

∂wz , ∂x

+ K2

∂wz ∂y



(A.1b)

where  ij (Hij ), ui (wz ) and cij (Ki ) are respectively stresses, displacements and elastic constants in the phonon (phason) field; Ri are elastic constants characterizing the phonon-phason coupling effect. For the sake of compactness, the notations  z1 =  zz ,  z2 = Hzz , uz1 = uz and uz2 = wz are applied in this paper. The eigen-values si (i = 1, 2, 3), which will be used later, are the eigen-roots with a positive real part of the following equation 4

as6 − bs + cs2 − d = 0

a=

c44



R22

− c33 K1 , d = c11 2





R32

− c44 K2 ,

where









2 − − K1 c11 c33 + c44 (c13 + c44 )2 + R2 [2c44 R3 + c11 R2 − 2 (c13 + c44 ) (R1 + R3 )] ,

b=

c33 −c44 K2 + (R1 + R3 )

c=

2 − c44 −c11 K1 + (R1 + R3 )2 − K2 c11 c33 + c44 (c13 + c44 )2 + R3 [2c11 R2 + c44 R3 − 2 (c13 + c44 ) (R1 + R3 )] .





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In order to specify the constants gmi and A in (5), we introduce the following constants ˛1j =

ˇ2j

and ˛2j =

ˇ1j sj

ˇ1j =

ˇ1j sj

,

− m1 ,

1

s→0

(A.3)

m1 = −K2 (c13 + c44 ) + R3 (R1 + R3 ) , (A.4)

2

m3 = −c11 K1 − c44 K2 + (R1 + R3 ) ,

s

−

arctan s s2



(A.5)

2j = R1 + R2 ˛1j sj + K1 ˛2j sj

d2m d3m

⎪ ⎭

=−

1 ⎢ ⎣ ˛11 2 ˛21

s3 13

˛12

˛13

˛22

˛23

⎤−1 ⎧ ⎫ ⎨ 0 ⎬ ⎥ ı1m , ⎦ ⎩ ⎭

(A.6)

ı2m

where m ranges from 1 to 2 and ıim is the Kronecker delta. The constants gmi and A in (5) are specified by gmi =

3 #

mk dki

and A = g11 g22 − g12 g21 .

(A.7)

k=1

Appendix B. This section is dedicated to evaluating the integrals involved in (15). I with respect to y from First, we consider the integral of z1 negative infinity to zero,



0

−∞

I z1 |z=0 dy

Setting t =



p0 = 



0

1 (y1 + d − y )

−∞

−y , we arrive at

0

−∞





+∞



y1 + d  dy . −y

(B.1)



y1 + d dt y1 + d + t 2 t 2p0 |+∞ = p0 . arctan = 0  y1 + d 2p0 = 

I z1 |z=0 dy

0

(B.2)

Then, the integral of the last part of  z1 is of the form



0

−∞

II z1 |z=0 dy

By setting s =





0

−∞

2Y =− 

II z1 |z=0 dy





0

−∞

d − arctan −y





d −y

dy .

−d/y , (B.3) can be simplified as

2Y d = 

$

+∞

sd 0

Integrating by parts, we have

1  s2

= 0,

(B.6)

(B.7)

 z1 |z=0 dy

−

0



0

= −∞

I z1 |z=0 dy +

= p0 − Y d.

0

−∞

II z1 |z=0 dy

(B.8)

References

1j = c13 + c33 ˛1j sj + R2 ˛2j sj ,

s2 12



0

−∞

The constants mk and dki are defined as

⎪ ⎩

'

|+∞ − arctan s|+∞ . 0 0

II z1 |z=0 dy = −Y d.

m4 = c11 R2 − c44 R1 − c13 (R1 + R3 ) .

s1 11



At this stage, from (B.2) and (B.7), we conclude that

m2 = −K1 (c13 + c44 ) + R2 (R1 + R3 ) ,

⎡

arctan s s2

0

−∞

and

s

−

the following results can be deduced



+ c44 R2 sj4 ,

with

⎧ ⎫ d ⎪ ⎨ 1m ⎪ ⎬

& 1

(B.5)

With the aid of lim

c11 R3 − m4 sj2

2Y d 

II z1 |z=0 dy =

(A.2)

ˇ2j = −c11 K2 − m3 sj2 − c44 K1 sj4 , ˇ3j =

0

−∞

ˇ3j

where m2 sj2



+∞

(arctan s) d

(B.3)

 1 %

. s2 (B.4)

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Please cite this article in press as: P. Li, et al., Crack tip plasticity of a half-infinite Dugdale crack embedded in an infinite space of one-dimensional hexagonal quasicrystal, Mech. Re. Commun. (2015), http://dx.doi.org/10.1016/j.mechrescom.2015.09.007