The interaction of an edge dislocation with an inhomogeneity of arbitrary shape in an applied stress field

Mechanics Research Communications 48 (2013) 19–23

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The interaction of an edge dislocation with an inhomogeneity of arbitrary shape in an applied stress field Cuili Zhang a , Shu Li b , Zhonghua Li a,∗ a b

Department of Engineering Mechanics, Shanghai Jiaotong University, 200240 Shanghai, PR China Institute of Forming Technology & Equipment, Shanghai Jiaotong University, 200240 Shanghai, PR China

a r t i c l e

i n f o

Article history: Received 30 June 2012 Received in revised form 14 November 2012 Available online 10 December 2012 Keywords: Edge dislocation Inhomogeneity Interaction

a b s t r a c t A general, approximate solution is presented for an edge dislocation interacting with an inhomogeneity of arbitrary shape under combined dislocation and applied stress fields. The solution shows that the contributions of the dislocation stress field and the applied stress field to the interaction follow a simple superposition principle. The dislocation stress field has a short range effect, while the applied stress field has a long range effect. As special cases, explicit solutions for some common inhomogeneity shapes are obtained for the interaction induced by the applied stress field. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction It is well-known that the mobility of dislocations plays an important role in the understanding of the macro strain hardening behavior and micro damage mechanisms of crystalline materials, and this mobility is strongly influenced by the interaction between dislocation and inhomogeneity in the materials. Therefore, the study of interaction between dislocation and inhomogeneity has received a great deal of attention during the last few decades. Most of the studies were traditionally based on solution of appropriate boundary value problems in the linear theory of elasticity. However, due to the complexity of the elastic boundary and interfacial problems only a handful of analytical solutions can be found for highly idealized inhomogeneity shapes, such as circular (Dundurs and Mura, 1964; Dundurs and Gangadharan, 1967; Wang and Pan, 2010, 2011; Fang and Liu, 2006a,b) and elliptical inhomogeneities (Stagni and Lizzio, 1983; Santare and Keer, 1986; Gong and Meguid, 1994; Qaissaunee and Santare, 1995), and a surface layer (Weeks et al., 1968). Eshelby inclusion theory (Eshelby, 1956, 1961) provides an alternative method to solve the problem. According to this theory, if a homogenous inclusion (the inclusion has the same elastic behavior as the matrix) undergoes a stress-free transformation strain in an applied stress field, the interaction force between the homogenous inclusion and the applied stress field can be determined from the work done by the stress field during the transformation. For

∗ Corresponding author. Tel.: +86 021 64352492; fax: +86 021 34206334. E-mail address: [email protected] (Z. Li). 0093-6413/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechrescom.2012.11.009

an inhomogeneity, it may be transformed to a homogenous one with an equivalent transformation strain on the base of the Eshelby equivalent inclusion theory (Eshelby, 1961; Withers et al., 1989; Li et al., 2011; Zhou et al., 2011). Consequently, the interactions between the stress field and the inhomogeneity can be evaluated by the same method. Based on this approach some approximate solutions for the interaction of an inhomogeneity of arbitrary shape with dislocation have been obtained (Li and Shi, 2002; Shi and Li, 2003). However, these solutions are limited to the inhomogeneity interacting with dislocation stress field only. It is evident that the interaction between dislocation and inhomogeneity in an applied stress field is of more practical importance because the macro strain hardening behavior and micro damage mechanisms of the materials have to be known just under such conditions. In this study, on the basis of Eshelby inhomogeneity theory, a general, approximate analytical solution is developed for an edge dislocation interacting with an inhomogeneity of arbitrary shape under combined dislocation and remotely applied stress fields. The solution shows that the contributions of the dislocation stress field and the applied stress field to the interaction follow a simple superposition principle. The dislocation stress field becomes dominant to the interaction only when the distance between dislocation and inhomogeneity approaches to nanometer scale, i.e., the dislocation stress field has only a short range effect, while the applied stress field has a long range effect. In view of the fact that the interaction between dislocation and inhomogeneity induced by the dislocation stress field has been extensively studied, in the present paper we focus our attention on the interaction produced by the remotely applied stress field.

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C. Zhang et al. / Mechanics Research Communications 48 (2013) 19–23

2. Model and formulation

where

Fig. 1 shows the physical problem to be studied. A straight edge dislocation is located at the origin of a Cartesian coordinate system (o − xyz), and the dislocation line coincides with the z-axis. Consider a two-dimensional (2D) inhomogeneity of arbitrary shape (domain ˝) embedded in an infinite matrix and subjected to the dislocation stress field ijd and remotely applied stress field  ij . This 2D implementation conveys the essence of 3D problem, and represents an inhomogeneity extending throughout the thickness, consistently with experimental observations in thin metal film. According to the Eshelby theory (Eshelby, 1961), the inhomogeneity will undergo an equivalent transformation strain eT induced by combined action of the dislocation stress field and the applied stress field. Now consider a differential element dA within the inhomogeneity. The transformation strain in dA can be expressed by Eshelby (1961) and Withers et al. (1989) eT = [(Ci − Cm )S + Cm ]−1 (Cm − Ci )e,

(1)

where S is the Eshelby tensor, dependent solely upon the shape of the inhomogeneity and the Poisson’s ratio  of the matrix material. Ci and Cm are the elastic tensors of the inhomogeneity and the matrix material, respectively. e is the combined strain field of the edge dislocation field and the applied strain field in the absence of the inhomogeneity. For plane strain condition, the non-zero components of the combined strain e are given by e11

11 − (11 + 22 ) b(1 − 2)sin  = − 2m 4r(1 − )

e22

22 − (11 + 22 ) b(1 − 2)sin  = − 2m 4r(1 − )

e12

12 b cos  = + m 4r(1 − )

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

i m

(4)

i is the shear module of the inhomogeneity. Combining Eqs. (1) and (3), it gives eT = Le,

(5)

where L = [(˛ − 1)S + I]−1 (1 − ˛),

,

(2)

(6)

I is the unit tensor. Thus, the tensor L relates the equivalent transformation strain eT in the inhomogeneity to the combined strain e without going into the details of the form of the Ci and Cm tensors. For a differential element with circular section inside the inhomogeneity, the non-zero components of the Eshelby tensor are given by Mura (1987) S1111 = S2222 =

5 − 4 , 8(1 − )

S1122 = S2211 =

S1133 = S2233 =

 , 2(1 − )

S1212 =

S1313 = S2323 =

1 2

4 − 1 8(1 − )

3 − 4 4(1 − )

(1 − ˛)(1 − )(3 − 4 + 5˛ − 4˛) (1 + ˛ − 2)(1 + 3˛ − 4˛)

L1122 = L2211 = −

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

Substituting Eq. (7) into Eq. (6) yields L1111 = L2222 =

(1 − ˛)2 (1 − )(1 − 4) (1 + ˛ − 2)(1 + 3˛ − 4˛)

(3)

.

(7)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ L3333 = (1 − ˛) L1133 = L2233 ⎪ ⎪ ⎪ ⎪ ⎪ 4(1 − ˛)(1 − ) 2(1 − ˛) ⎪ ⎪ ⎭ , L1313 = L2323 = L1212 = (1 − ˛)2  = , (1 + ˛ − 2)

in which b is the Burger’s vector of the dislocations, m ,  are the shear modulus and Poisson’s ratio of the matrix material. The first and second terms in Eq. (2) are, respectively, the applied strains and dislocation strains. As shown in Eq. (1), the equivalent transformation strain eT is not zero for an inhomogeneity (Ci = / Cm ). For simplicity, it is assumed that the inhomogeneity and the matrix material are isotropic and Poisson’s ratio of the inhomogeneity is the same as the matrix material. Then we have Ci = ˛Cm ,

˛=

,

(8)

1+˛

(1 + 3˛ − 4˛)

And other components of the L tensor are zero. Combining Eqs. (2), (5) and (8), the non-zero components of the transformation strain in dA are given by

⎫ ⎬

T =L e11 1111 e11 + L1122 e22 ⎪ T =L e22 2211 e11 + L2222 e22 T e12

= L1212 e12

⎪ ⎭

,

(9)

for plane strain. The elastic interaction energy per unit length in z-direction of dislocation on the differential element is given by Mura (1994) dUint = −ijd eijT dA.

(10)

The non-zero components of ijd are d = d = − 11 22 d = 12

⎫

m b sin  ⎪ ⎬ 2r(1 − )

⎪ ⎭

m b cos  2r(1 − )

.

Then we have dUint =

C1 C2 d + 11 + 22 )sin  − (211 r r

(11)

1 2



d + 12 cos , 12

(12)

where

Fig. 1. An edge dislocation interacts with an inhomogeneity of arbitrary shape in an applied stress field.

C1 =

b(1 − ˛)(1 − 2) , 2(1 + ˛ − 2)

C2 =

2b(1 − ˛) , (1 + 3˛ − 4˛)

(13)

C. Zhang et al. / Mechanics Research Communications 48 (2013) 19–23

From (12) we gain the force acting on the dislocation unit length

21

w

y

∂ C1 d + 11 + 22 )sin  dFr = − (dUint ) = 2 (411 ∂r r C2 d − 2 (12 + 12 )cos . r

y

r

(14)

m b2 (1 − ˛) 2 r 3



r0





dFr cos dA,

(b)

(a) A small circular inhomogeneity of radius R is centered at (r, ). Combining Eq. (17) and (15) the glide force can be approximated by

(15)

 C1 (11 + 22 )sin  cos  − C2 12 cos2  , 2 ˇ

(18)

where ˇ = r/R.

˝



dFr sin dA.

(16)

˝

The integrations are carried out over the whole domain ˝ occupied by the inhomogeneity. A positive (negative) value of F is corresponding to repulsion (attraction). Noting that the dislocation stress field in Eq. (14) is proportional to m b/r. It is well-known that for most metal materials the Burgers vector b is in the order of 10−10 m and the shear modulus m is about 105 MPa. Therefore, as compared with the applied stress field the role of the dislocation stress field may become dominant only when the distance between dislocation and inhomogeneity approaches to nanometer scale. Eq. (14) explicitly indicates that when r is larger than submicrometer scale the contribution of the dislocation stress field to the interaction is negligible as compared with the applied stress field. Thus, the dislocation stress field has only a short range effect, while the applied stress field has a long range effect. In view of the fact that the contributions of the dislocation stress field and the applied stress field to the interaction follow a simple superposition principle (see Eq. (14)), and that the interaction between dislocation and inhomogeneity induced by dislocation stress field has been extensively studied (seen Li and Shi (2002) and references cited therein), of particular importance in the present study is the interaction induced by the applied stress field. Thus, Eq. (14) can be rewritten as dFr =

x

l

Fig. 2. An edge dislocation interacts with a lamellar inhomogeneity (l  w, r0 ): (a) the lamellar inhomogeneity perpendicular to the x-axis and centered at (r0 , 0) and (b) the lamellar inhomogeneity lies on the x-axis.

Fglide ≈

Fc lim b =

r0

l

(a)

(14-1)

This is just the Eq. (13) of reference (Shi and Li, 2003). From this equation an explicit solution was obtained for a small circular inclusion (Shi and Li, 2003), which is consistent with the classical solution obtained by Dundurs and Mura (1964). The total forces on the dislocation along x and y-directions are, respectively, the glide and climb forces Fglide =

x

1 (1 − )(1 + 3˛ − 4˛)

2(1 − 4) sin2  , (1 + 3˛ − 4˛)(1 + ˛ − 2)

+

w

q

In the case of free applied stress, Eq. (14) reduces to dFr = −

l

1 C1 (11 + 22 )sin  − C2 12 cos  , 2 r

(17)

which is controlled by the applied stress field only. Besides, since the mobility of an edge dislocation depends on the glide force, more attention will be paid to the glide force in the following discussion.

(b) A lamellar inclusion of length 2l and width w (l  w) perpendicular to x axis and its center is located at (r0 , 0) as shown in Fig. 2a. By use of dA = wrd/ cos  in Eq. (15) we have Fglide =

2wb(˛ − 1) r0 (1 + 3˛ − 4˛)



0 +



1 sin 20 12 , 2

(19)

where  0 = arctg l/r0 . When l/r0  1, one obtains Fglide =

wb(˛ − 1) 12 . r0 (1 + 3˛ − 4˛)

(20)

(c) A lamellar inclusion of length l and width w(l  w) lies on the x-axis, Fig. 2(b). The distance of the near dislocation end of the inclusion to the dislocation is r0 . By use of dA = wdr and sin  = 0, cos  = 1 in Eq. (17), we have Fglide =

2bw(1 − ˛) (1 + 3˛ − 4˛)

 1 r0 + l

−

1 r0

 12 ,

(21)

when l/r0  1, it gives Fglide =

2wb(˛ − 1) 12 . r0 (1 + 3˛ − 4˛)

(22)

The sign of the interaction force depends on the relative hardness of the inhomogeneity and matrix. As shown in Eqs. (19)–(22), a hard inhomogeneity (˛ > 1) is repelled to a dislocation whereas a soft one (˛ < 1) is attracted, acting as a barrier and sink, respectively. Two extremes when the inhomogeneity is very hard (rigid, ˛→ ∞) and very soft (void or crack, ˛ = 0) give an upper bound on the repulsion and the attraction, respectively. In addition, it is evident that if there is a periodical array of lamellar inhomogeneities in vertical and horizontal direction in Fig. 2, as appeared in fiber-reinforced materials, the interaction force can be obtained by integrating Eq. (15) over the domain occupied by all inhomogeneities.

3. Some special cases 4. Numerical example and discussion Spherical particle and fiber are the most common reinforcements in many engineering structure materials. Explicit expressions for the interaction of edge dislocation with such inhomogeneity shapes under remotely applied stress field can be obtained.

In the case of an inhomogeneity of irregular shape or the inhomogeneity located in a non-uniformly applied stress field, simple numerical calculations are required. As an example, the inserted figure in Fig. 3 shows an edge dislocation located at mode I crack-tip

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C. Zhang et al. / Mechanics Research Communications 48 (2013) 19–23

0.15

alpfa=2.0 alpfa=5.0 alpfa=0.2 alpfa=0.5

0.10

r 0.8

0.00 -0.05

KI

-0.15

-3

-2

-1

0

1

θ

x 2

-0.4 3

interacts with a small circular inhomogeneity of radius R centered at (r, ). Since the circular inhomogeneity is small, Eq. (18) can be used to predict the glide force acting on the dislocation. Substituting the well-known mode I crack-tip stress field

KI 22 = √ 2r

1 − sin

 1 + sin

 3 sin 2 2  3 sin 2 2



cos

cos

 2  2

KI  3  12 = √ cos sin cos 2 2 2 2r into Eq. (18) yields Fglide =

b(1 − ˛)KI √ 4ˇ2 2r



⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

,

(24)

 1 − 2 cos 1 + ˛ − 2 2

2 3 − cos cos  1 + 3˛ − 4˛ 2



sin 2.

a = 0.2

Eq. (15)

-0.8

√ Fig. 3. The normalized glide force Fglide (in unit of bKI /(4ˇ2 2r)) as a function of  for an edge dislocation located at mode-I crack tip interacting with an small circular inhomogeneity of ˛ = 2, 5, − 2, − 5 at fixed  = 0.3.



σ 12

0.0

r

q (rad)

KI 11 = √ 2r

x

Fglide

2R

y

2l

θ

r0

a = 5

0.4

-0.10

-0.20

w

1.2

0.05

Fglide

σ 12

y

(25)

Fig. 3√ shows the normalized glide force (in unit of bKI /(4ˇ2 2r)) as a function of  for ˛ = 2, 5, − 2, − 5 at fixed  = 0.3. The glide force fluctuates as the inhomogeneity changes its location angle , and is anti-symmetrical with the crack plane. Depending on its location, a hard (or soft) inhomogeneity may produce either repulsion or attraction to the dislocation. It is therefore concluded that the overall interaction force is negligible when the inhomogeneities in composites are uniformly distributed around the crack tip. Note that the direct action of the applied stress field on the dislocation is not considered in the present study. The contribution of the applied stress field to interaction comes from the transformation strain induced by the applied stress field only. Hence, Eq. (25) is not equivalent to the problem on the interaction of dislocation with inhomogeneity and crack. As another example, the inserted figure in Fig. 4 shows an edge dislocation located in a plane shearing stress field interacting with a rectangular inhomogeneity of area w × 2l = r02 centered at (r0 , 0), where r0 is a constant for given area of the inhomogeneity. Substi(17) and (13) into Eq. (15) and using the relationship of tuting Eqs. √ r0 = 2l/ ˇ, w = l/ˇ we have  1  √2/ˇ+1/2ˇ  2 x 2b12 (˛ − 1) Fglide = dy √ dx 2 2 (1 + 3˛ − 4˛) −1 x +y 2/ˇ−1/2ˇ

-1.2

Eq. (19) 0

1

2

3

4

5

b (= l / w) Fig. 4. The interaction force Fglide (in unit of b 12 ) as a function of ˇ for an edge dislocation located in a plane shearing stress filed interacting with a rectangular inhomogeneity of ˛ = 5, 0.2 at fixed  = 0.3.

Numerically integrating this equation we get the Fglide forces (in unit of b 12 ) as a function of ˇ for ˛ = 5, 0.2, corresponding to that an inhomogeneity changes its shape but keeping its area constant. The integrating values (red lines) are compared with the predictions of the approximate Eq. (19) for the case of ˇ  1 (blue lines). Strong shape and module ratio dependences of the interaction are shown. As shown in Fig. 4, when ˇ > 1.5, the approximate Eq. (19) gives good prediction for the numerical solution. It should be noted that the present solution is approximated for an inhomogeneity of arbitrary shape, because we determine the transformation strain in the inhomogeneity by Eshelby inclusion theory, which is mathematically rigorous only for an infinite matrix containing a single ellipsoidal inclusion. In the past, in order to utilize the Eshelby approach in more realistic situations there has been considerable activity in extending it to various problems: the interaction of two ellipsoidal inclusions (Moschobidis and Mura, 1975), the short fiber reinforced composites (Withers et al., 1989), the stress field inside a non ellipsoidal inclusion (Johnson et al., 1980), an arbitrarily shaped polygonal inclusion in anisotropic piezoelectric full-and half-planes (Pan, 2004). In particular, the Eshelby technique has been used calculating the stress intensity factor induced at a crack-tip by the transformation of one particle (McMeeking and Evans, 1982) and by an inclusion of arbitrary shape embedded in crack-tip field (Li and Chen, 2002; Li and Yang, 2004). The extended application of the Eshelby approach has fairly good accuracy as validated by a number of numerical examples (Li and Chen, 2002; Li and Yang, 2004) and by experimental results (Ippolito et al., 2005, 2006), as well as in comparison with available classical solutions (Li and Shi, 2002; Shi and Li, 2003). Recently, Li et al. (2011) further demonstrated, from the principle of stress equivalence, that the Eshelby theory is applicable to a variety of inhomogeneities, such as pore (or crack), gas bubble, shear band and plastically deformed zone, and developed a general, approximate continuum theory for interaction between dislocations and inhomogeneity of any shape and properties. As shown in the present study, on the basis of the general theory some explicit approximate solutions for practical problem can be obtained which are very useful for engineering application.

C. Zhang et al. / Mechanics Research Communications 48 (2013) 19–23

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