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Size-Dependent of Interaction Between a Screw Dislocation and a Coated Nano-Inhomogeneity
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Procedia Engineering 00 (2011) 000–000 Procedia Engineering 14 (2011) 1629–1636
Procedia Engineering www.elsevier.com/locate/procedia
The Twelfth East Asia-Pacific Conference on Structural Engineering and Construction
Size-Dependent of Interaction Between a Screw Dislocation and a Coated Nano-Inhomogeneity D. X. LEIa and Z. Y. OUb School of Science, Lanzhou University of Technology, Lanzhou City, Gansu Province, China
Abstract
Effect of interface stress upon the interaction between a screw dislocation and a coated nanoinhomogeneity is investigated in the framework of surface elasticity. By using the complex variable function method, the stress fields in the inhomogeneous materials and the image force acting on the screw dislocation are derived analytically. The results indicate that, when the radius of the coated inhomogeneity is reduced to nanometers, the influence of interface stress on the motion of the dislocation near the inhomogeneity becomes significant and the image force acting on the screw dislocation depends on the coated inhomogeneity size, which differs from the classical solution. © 2011 Published by Elsevier Ltd. Keywords: Screw dislocation, coated inhomogeneity, image force, Complex variable function method.
1.
INTRODUCTION
Interactions between dislocations and inhomogeneities play an important role in the strength and stiffness of crystalline materials, and have attracted much attention since the pioneering work of Head (1953). For example, Dundurs (1967) calculated the displacement and stress fields of a screw dislocation near a circular elastic inhomogeneity. Sendeckyj (1970) extended this problem to the case of multiple inhomogeneities. Recently, the elastic interaction of screw dislocations with interphase boundaries of the circular inhomogeneities coated by one (Fan and Wang, 2003), two (Sudak, 2003) and many (Honein et al., 2006) intermediate layers has been studied. In their researches, the circular interface boundaries are assumed to be perfect (continuous displacement and traction across the interface) or imperfect. In the aforementioned studies, however, the effect of interface stress is not taken into account. The interface of solids is a special region with a very small thickness. Since the interface-to-volume ratio in the nano-scale domain is relatively high compared to that in the macro-scale domain, the interface stress
a b
Corresponding author: Email: [email protected] Presenter: Email: [email protected]
1877–7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.07.205
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2
plays a significant role on the size-dependent behavior of nanoscale elements or devices (Wong et al., 1997; Kulkarni and Zhou, 2006). Since there is no intrinsic length scale involved in the constitutive laws of the classical elastic theory, it can not predict the size-dependent behavior of nanosized structures and devices. Recently, the interaction between a screw dislocation and a circular nano- inhomogeneity incorporating interface stress has been considered by Fang’s research group (Liu and Fang, 2007). In their papers, interface stress is restricted at the interface between the inhomogeneity and the infinite matrix. It is well known that the three-phase (inclusion/interphase /matrix) model is of great practical and theoretical value in composites mechanics researches. In most fiber-reinforced composite materials, coated fibers are widely employed in the design stage to improve the bonding strength between fibers and the matrix. In the present paper, we study the problem of the elastic interaction between a screw dislocation and a coated nano-inhomogeneity incorporating interface stress. The interface stresses at the inner and outer interfaces of the annular coated layer are considered. By means of a complex variable technique, the explicit solutions of the complex potentials are obtained. The image forces exerted on the screw dislocations are derived. 2.
COATED INHOMOGENEITY WITH INTERFACE EFFECT
Let us consider an infinite isotropic medium containing a circular inhomogeneity and an annular coating layer under anti-plane problem, where R1 and R2 are the inner and outer radii of the coated annulus. L1 and L2 indicate the interfaces between the inhomogeneity and the coating layer, the coating layer and the matrix, respectively. A screw dislocation is located at the point z0 in the matrix. In anti-plane state, the displacement w, the shear stress components σ xz and σ yz of the materials can be expressed in terms of a single analytical function (complex potential function) with respect to the complex variable z=x+jy (j2=-1). Following Gurtin and Murdoch (1975), for the considered anti-plane problem, we obtain in the bulk (Muskhelishvili, 1975):
∂ 2 w( i ) ∂ 2 w( i ) + = 0 , τ rz( i ) = 2 μiε rz(i ) , τ θ(iz) = 2μiεθ( iz) 2 2 ∂x ∂y
(1)
and in the interfaces (Liu et al., 2003; Liu and Fang, 2007):
τ
( Lα ) θz
= 2 μ Lα ε
( Lα ) θz ,
1 ∂τ θ( Lzα ) ⎡⎣τ rz(α ) (t ) ⎤⎦ = Rα ∂θ
(2)
where let super- and subscript i to 1, 2, 3, which refers to the inhomogeneity, coating layer and the matrix (i ) (i ) (i ) (i ) region, respectively; μi are shear moduli, τ rz ( ε rz ) and τ θ z ( ε θ z ) are stress (strain) components in the polar coordinates system (r, θ ); the sub- and super-script Lα ( α =1, 2) denote the two interfaces, iθ respectively; μ Lα ( α =1, 2) are the elastic constants of the interfaces and t = R α e ( α =1, 2) denote (α ) the points on the circular arc interface Lα . In addition, ⎡⎣τ rz (t ) ⎤⎦ represents the jump of the interface (L ) stresses and points to the positive normal direction of inclusion and matrix. The interface strain ε θ zα ( α =1, 2) equals to the associated tangential strain in the abutting bulk materials. Referring to the work of Muskhelishvili (1975), in the bulk solid, the displacements w(i), shear (i ) (i ) stresses τ rz and τ θ z can be written in terms of the analytical functions Φi(z) as follows
w(i ) = ⎡⎣ Φ i ( z ) + Φ i ( z ) ⎤⎦ 2 , τ rz( i ) − jτ θ(iz) = ⎡⎣τ xz(i ) − jτ yz(i ) ⎤⎦ e jθ = μi e jθ ϕi ( z )
(3)
where ϕi ( z ) = Φ i ( z ) ，and the overbar represents the complex conjugate and the prime denotes the derivative with respect to the argument z. '
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For the present problem, the classical continuity conditions of displacements and the non-classical equilibrium conditions (2) on the interface can be given as (Liu and Fang, 2007)
μ L1 ∂τ θ(2)z (t ) w(1) (t ) = w(2) (t ) , τ rz(1) (t ) − τ rz(2) (t ) = , R1μ2 ∂θ
t = R1 ,
μ L ∂τ θ(2)z (t ) w (t ) = w (t ) , τ (t ) − τ (t ) = , R2 μ2 ∂θ (2)
(3)
(3) rz
(2) rz
(4)
t = R2 .
2
(5)
For the convenience of analysis, the following new analytically extensional functions are introduced in the corresponding regions according to the Schwarz symmetry principle (Muskhelishvili, 1975).
R12 ϕ1e ( z ) = − 2 ϕ1 ( R12 z ) , z > R1 , z
(6)
ϕ2 ei ( z ) = −
R12 ϕ2 ( R12 z ) , R12 R2 < z < R1 , 2 z
(7)
ϕ2 ee ( z ) = −
R22 ϕ2 ( R22 z ) , R2 < z < R22 R1 , z2
(8)
ϕ3 e ( z ) = −
R22 ϕ3 ( R22 z ) , z2
z < R2 ,
(9)
where ϕ• ( R z ) = ϕ• ( z ) . Substituting Eqs. (6)-(9) into Eqs. (4) and (5) and with the aid of Eq. (3), we have 2
[ϕ1 (t ) − ϕ2ei (t ) − ϕ3e (t )] = [ϕ2 (t ) − ϕ1e (t ) − ϕ3e (t )] I
[ϕ2 (t ) − ϕ1e (t ) − ϕ3e (t )]
C
(
C
t = R1 ,
,
(10)
= [ϕ3 (t ) − ϕ1e (t ) − ϕ2 ee (t ) ] , t = R2 , M
)
(
)
⎡ μ1ϕ1 (t ) + μ2 + μ L R1 ϕ2 ei (t ) + μ L R1 tϕ2' ei (t ) + μ3ϕ3e (t ) ⎤ 1 1 ⎣ ⎦
(
)
(
(11)
I
)
C
= ⎡⎣ μ1ϕ1e (t ) + μ2 − μ L1 R1 ϕ2 (t ) − μ L1 R1 tϕ2' (t ) + μ3ϕ3e (t ) ⎤⎦ , t = R1 ,
(
)
(
)
⎡ μ1ϕ1e (t ) + μ2 − μ L R2 ϕ2 (t ) − μ L R2 tϕ2' (t ) + μ3ϕ3e (t ) ⎤ 2 2 ⎣ ⎦
(
)
(
)
(12)
C
M
= ⎡⎣ μ1ϕ1e (t ) + μ2 + μ L2 R2 ϕ2 ee (t ) + μ L2 R2 tϕ2' ee (t ) + μ3ϕ3 (t ) ⎤⎦ , t = R2 .
(13)
where the superscript I, C and M refer to the boundary values of the physical quantity as z approaches the interfaces from the regions occupied by the inhomogeneity, the coating layer and matrix, respectively.
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3.
SOLUTION OF BOUNDARY VALUE PROBLEM
Let us analyze the singularities of the complex functions. If a screw dislocation with Burgers vector (0, 0, bz) is located inside the matrix, the complex potential ϕ3 ( z ) in the matrix can be taken the form (Smith, 1968) as
= ϕ3 ( z )
bz 1 + Γ + ϕ30 ( z ) 2π j z − z0
z > R2
(14)
where the first and second terms represent the complex potential for a screw dislocation in an infinite Γ τ xz∞ − jτ yz∞ μ3 is matrix in the absence of the inhomogeneity subjected to external shear loading; = ∞ ∞ identified with the longitudinal stresses τ xz , τ yz at infinity. The third term ϕ30 ( z ) is holomorphic in the infinite matrix and represents the disturbance of the complex potential due to the coated inclusion. The complex potential ϕ3e ( z ) can be evaluated by substituting Eq. (14) into Eq. (9) as
(
= ϕ3e ( z )
(
)
bz ⎛ 1 1 ⎞ R22 − Γ + ϕ3 e 0 ( z ) ⎜ ⎟− 2π j ⎝ z z − z * ⎠ z 2
)
z < R2
(15)
where ϕ3e 0 ( z ) is holomorphic in z < R2 . z = R2 z0 is assumed in the region z < R1 R2 . It will be seen that when the point falls in the other regions of z < R2 , the same resulting solution is reached. The potential function of the coated layer ϕ 2 ( z ) can be expanded into a Laurent series in the annular region *
2
2
ϕ= GN ( z ) + GP ( z ) R1 < z < R2 2 ( z)
(16)
where GN(z) and GP(z) represent the sum of the negative and positive power terms, respectively. ∞
∞
k =1
k =0
GN ( z ) = ∑ ak z − ( k +1) , GP ( z ) = ∑ bk z k R1 < z < R2
(17)
where ak and bk (k=0, 1, 2, …) are the complex constants. It is seen that GN(z) and GP(z) converge in the regions z > R1 and z < R2 , respectively. According to the work of Muskhelishvili (1975) and some complex calculations, thus the analytical functions ϕ1 ( z ) , ϕ 2 ( z ) and ϕ3 ( z ) can be obtained
= ϕ1 ( z )
∞ ∞ a 1 1 ( μ2 − μ3 − (k − 2)Λ1 ) 2kk z k −1 − ( Λ1 − Λ 2 ) kbk z k ∑ ∑ R1 μ1 + μ3 k 1 = μ1 + μ3 k 0 ∞ a bμ 2 μ3Γ 1 1 1 − + ( μ2 + μ3 + k Λ 2 ) 2kk z k −1 + z 3 ∑ μ1 + μ3 k =1 μ1 + μ3 π j z − z0 μ1 + μ3 R2
= ϕ2 ( z )
∞
∑a z
− ( k +1)
∞
+ ∑ bk z k
k k 1= k 0 =
= ϕ3 ( z )
(18)
∞ 1 ( μ1 − μ2 + (k + 1)Λ1 ) bk R12( k +1) z −( k + 2) ∑ μ1 + μ3 k =0
(19)
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∞ ∞ 1 1 2( k +1) −( k + 2 ) μ − μ + k + Λ b R z − ( 1) ( ) ∑ ∑ ( Λ − Λ 2 ) (k + 1)ak z −( k +1) 2 k 2 μ1 + μ3 k =0 1 2 μ1 + μ3 k =1 1
2 bz ( μ1 − μ3 ) ⎛ 1 bz 1 1 ⎞ ( μ1 − μ3 ) R2 + +Γ− Γ ⎜ − ⎟+ 2π j z − z0 2π j ( μ1 + μ3 ) ⎝ z z − z * ⎠ ( μ1 + μ3 ) z 2
(20)
μ L2 R2 . The coefficients ak and bk are represented as recursion where Λ1 =μ L1 R1 , Λ 2 = formulas ⎛ 4μ μ Γ ⎞ B0 ⎜ − E0 + 1 3 ⎟ μ1 + μ3 ⎠ − Bk Ek a1 = ⎝ , ak +1 = k ≥1 A1C0 + B0 D1 Ak +1Ck + Bk Dk +1
(21)
⎛ 4μ μ Γ ⎞ A1 ⎜ − E0 + 1 3 ⎟ μ1 + μ3 ⎠ − Ak +1 Ek k ≥1 bk = b0 = ⎝ Ak +1Ck + Bk Dk +1 A1C0 + B0 D1
(22)
( μ1 + μ2 + (k + 1)Λ1 ) , Bk = ( μ1 − μ2 + (k + 1)Λ1 ) R12( k +1) ( μ1 + μ2 − Λ1 − k Λ 2 )
where Ak +1=
C= k
⎡ ( μ1 − μ2 + (k + 1)Λ 2 ) ( μ1 − μ2 + (k + 1)Λ1 ) R12( k +1) ⎤ − ( μ1 − μ3 ) ⎢ − ⎥ ( μ1 + μ3 ) ( μ1 + μ3 ) R22( k +1) ⎣ ⎦ Dk +1 =−
(k + 2) ( μ1 − μ3 )( Λ1 − Λ 2 ) 1 − ( μ1 − μ2 − (k + 1)Λ 2 ) 2( k +1) 2( k +1) R2 ( μ1 + μ3 ) R2
⎡ ( μ1 − μ3 )2 bz 2 μ b ⎤ 1 ( μ − μ3 ) bz 1 Ek = −⎢ − 3 z ⎥ k +1 + 1 k +1 2π j ⎢⎣ 2π ( μ1 + μ3 ) j 2π j ⎥⎦ z0 ( z* ) Λ1 μ3 = μ L1 ( R1μ3 ) 1 and m2 = Λ 2 μ3 = μ L2 ( R2 μ3 ) 1 , that is, the effects of If m1 = interface is ignored, The solution of complex potentials ϕi ( z ) (i=1, 2, 3) agrees with the result of Liu et al. (2003). If we let μ1 = μ 2 and R1 = R2 , the solution will be reduced to the solution on the interaction between a screw dislocation and an inhomogeneity with the interface effect (Liu and Fang, 2007). 4.
STRESS FIELD AND IMAGE FORCE ON DISLOCATION
The image force acting on the dislocation plays an important role in understanding the mobility and socalled trapping mechanism of the dislocation. Referring to the work of Liu and Fang (2007) and supposing that the remote loads vanishes and the screw dislocation lies at the point x0 on the positive xaxis (z0 = x0>R2 is a real number). Due to ak and bk including j, in this case, fy = 0 and the component of 2 the normalized image force along the x-axis direction is defined as f x 0 = 2π R2 f x μ3bz . That is
(
)
5
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6
= f x 0 2π R2= f x ( μ3bz2 )
jb 2π R2 ∞ ( μ1 − μ2 + (k + 1)Λ1 ) k R12( k +1) z0−( k + 2) ∑ μ1 + μ3 k =0 bz
−
jb 2π R2 ∞ ( μ1 − μ2 + (k + 1)Λ 2 ) k R22( k +1) z0−( k + 2) ∑ μ1 + μ3 k =0 bz
−
R (μ − μ ) ⎛ 1 ja 2π R2 ∞ 1 ⎞ ( Λ1 − Λ 2 ) (k + 1) k z0− ( k +1) − 2 1 3 ⎜ − ⎟ ∑ bz μ1 + μ3 k =1 μ1 + μ3 ⎝ z0 z0 − z * ⎠
In the special case of
μ1 = μ2
and
(23)
Λ1 =Λ 2 =0 , Eq. (23) becomes
bz2 μ3 ( μ1 − μ3 ) R22 ⎛ 1 bz2 μ3 1 ⎞ f x0 = Im agef = − ⎜ − ⎟ 2π R2 2π ( μ1 + μ3 ) ⎝ z0 z0 − z * ⎠
(24)
which coincides with previous results for the circular inhomogeneity not accounting for interphase layer effect (Gong and Meguid, 1994) For the description of the effects of interfaces, one needs the constants m1 = μ L1 ( R1μ3 ) and m2 = μ L2 ( R2 μ3 ) . For simplicity, let Λ1 = Λ 2 . Here m1 and m2 characterize the effects of interfaces. For a macroscopic inhomogeneity with bigger value of R1, m1, m2<<1, the interface effect can be neglected. However, when the size of an inhomogeneity reduces to nanometers, m1, m2 become noticeable, thus the interface effect should be taken into account (Streitz et al., 1994; Miller and Shenoy, 2000; Ou et al., 2008). In addition, we define the relative shear modulus α = μ1 μ3 and β = μ 2 μ3 , the relative location of the dislocation r = x0 R2 and the relative coating thickness λ = R1 R2 . In Figs. 1, we illustrate the variation of the values of f x 0 with respect to the parameter r for the selected material constants, the interface parameters m1 and m2 , R1=15nm and coating thickness (λ=0.95). It is shown from Fig. 1 that, if the matrix is softer than the inhomogeneity, which is harder than the coating (β<α<1) and the interface characteristic constants is positive Λ1 =Λ 2 > 0 , the screw dislocation is first attracted then repelled by the coated inhomogeneity leading to an unstable equilibrium position near the inhomogeneity. It is remarkable different from the classical result, in which the soft inclusion will always attract the screw dislocation. It is seen that there exists significant local hardening at the interface due to the interface stress ( Λ1 =Λ 2 > 0 ) and the softest coating. As the above considerations, let Λ1 =Λ 2 for simplicity. The variation of the normalized force f x 0 with respect to the radius of a coated inclusion R2 is depicted in Figs. 2 for different values of α and β at r=1.05 with interface effects. It can be found that a positive value of Λ1 produces the repulsive force acting on the screw dislocation while a negative value of Λ1 produces the attractive force. The phenomenon cannot be predicted by classical elasticity without interface stress. This indicates that the local hardening and softening at the interface may exist due to the interface effect. In addition, the absolute values of image force increases with the decrease of the inclusion radius, and the size dependence becomes significant when the size of inhomogeneity is at nanoscale.
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-10
0.4
Λ1,2=10*10 m, α=0.5, β =0.8
0.2
Λ1,2=10*10 m, α=0.5, β =0.2
-10
0.0 -0.2 -0.4
-10
Λ1,2=-10*10 m, α=0.5, β =0.8
-0.6
-10
Λ1,2=-10*10 m, α=0.5, β =0.2
-0.8 -1.0 1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
r Fig. 1 The influence of moduli of a soft inhomogeneity and a soft coating on fx0 .
-2.4
−10
Λ1=Λ2=1∗10 (m),α=0.5,β=0.5 −10
Λ1=Λ2=−1∗10 (m),α=0.5,β=0.5
-2.7
Λ1=Λ2=0,α=0.5,β=0.5
fx0
-3.0 -3.3 -3.6 -3.9
-4.2 0
50
100
150
200
250
300
350
400
R2(nm) Fig. 2 The size effect of a soft inhomogeneity with soft coating layer on the fx0 at r=1.05.
5.
CONCLUSIONS
The elastic interaction between screw dislocations and a circular coated inhomogeneity incorporating interface stress is considered in the light of surface elasticity. The elastic fields and the imagine force are obtained by utilizing complex variable method. To illuminate the surface effects, the image forces acting on screw dislocations are discussed in details, which includes the mobility and the stability of the appointed screw dislocation in the matrix and the influence of the material elastic dissimilarity, the coating thickness and the interface stress as well as the relative position of the dislocation. The results show that, the size dependence becomes significant when the size of a coated inhomogeneity is at
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nanoscale, which is completely different from that in classic elasticity. Our results are helpful in the understanding of the motion mechanism of the dislocation and relevant physical phenomena in threephase nano-composite materials. 6.
ACKNOWLEDGMENTS
The NSFC (11062004) and Doctoral Fund of Lanzhou University of Technology are acknowledged. REFERENCES [1]
Dundurs J (1967). On the interaction of a screw dislocation with inhomogeneities. Recent Advances in Engineering Science 2, pp: 223–233.
[2]
Fan H and Wang GF (2003). Screw dislocation interacting with imperfect interface. Mech Mate. 35(10), pp: 943-953
[3]
Gong SX and Meguid SA (1994). A screw dislocation interacting with an elastic elliptical inhomogeneity. Int. J. Eng. Sci. 32, pp: 1221-1228.
[4]
Gurtin ME and Murdoch AI (1975). A continuum theory of elastic material surfaces. Arch. Rat. Mech. Anal. 57, pp: 291323.
[5] [6]
Head AK (1953). The interaction of dislocations and boundaries. Philosophical Magazine 44, pp: 92–94. Honein E, Rai H and Najjar MI (2006). The material force acting on a screw dislocation in the presence of a multi-layered circular inclusion. Int. J. Solids Struc. 43, pp: 2422–2440.
[7]
Kulkarni AJ and Zhou M (2006). Surface-effects-dominated thermal and mechanical responses. Acta Mech. Sin. 22, pp: 217–224.
[8]
Liu YW and Fang QH (2007). Size-dependent elastic interaction of a screw dislocation with a circular nano-inhomogeneity incorporating interface stress. Mater. Sci. Eng. A 464, pp: 117–123.
[9]
Liu YW, Jiang CP and Cheung YK (2003). A screw dislocation interacting with an interphase layer between a circular inclusion and the matrix. Int. J. Eng. Sci. 41, pp: 1883–1898.
[10]
Miller RE and Shenoy VB (2000). Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, pp: 139-147.
[11]
Muskhelishvili NL (1975). Some basic problems of mathematical theory of elasticity, Noordhoff, Leyden.
[12]
Ou ZY, Wang GF and Wang TJ (2008). Effect of residual surface tension on the stress concentration around a nanosized spheroidal cavity. Int. J. Eng. Sci.e 46, pp: 475–485.
[13]
Sendeckyj GP (1970). Screw dislocations near circular inclusions. Physica Status Solidi (a) 3, pp: 529–535.
[14]
Smith E (1968). The interaction between dislocations and inhomogeneities-1. Int. J. Eng. Sci. 6, pp: 129-143.
[15]
Streitz FH and Cammarata RCK (1994). Surface-Stress Effects on Elastic. Properties. I. Thin Metal Films. Phys. Rev. B. 49, 10699.
[16]
Sudak LJ (2003). Interaction between a screw dislocation and three-phase circular inhomogeneity with imperfect interface. Math. Mech. Solids 8, pp: 171–188.
[17]
Wang X, Pan E and Roy AK (2007). New phenomena concerning a screw dislocation interacting with two imperfect interfaces. J. Mech. Phys. Solids 55, pp. 2717–2734.
[18]
Wong E, Sheehan PE and Liebe CM (1997). Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science 277, pp. 1971–1975.
Procedia Engineering 00 (2011) 000–000 Procedia Engineering 14 (2011) 1629–1636
Procedia Engineering www.elsevier.com/locate/procedia
The Twelfth East Asia-Pacific Conference on Structural Engineering and Construction
Size-Dependent of Interaction Between a Screw Dislocation and a Coated Nano-Inhomogeneity D. X. LEIa and Z. Y. OUb School of Science, Lanzhou University of Technology, Lanzhou City, Gansu Province, China
Abstract
Effect of interface stress upon the interaction between a screw dislocation and a coated nanoinhomogeneity is investigated in the framework of surface elasticity. By using the complex variable function method, the stress fields in the inhomogeneous materials and the image force acting on the screw dislocation are derived analytically. The results indicate that, when the radius of the coated inhomogeneity is reduced to nanometers, the influence of interface stress on the motion of the dislocation near the inhomogeneity becomes significant and the image force acting on the screw dislocation depends on the coated inhomogeneity size, which differs from the classical solution. © 2011 Published by Elsevier Ltd. Keywords: Screw dislocation, coated inhomogeneity, image force, Complex variable function method.
1.
INTRODUCTION
Interactions between dislocations and inhomogeneities play an important role in the strength and stiffness of crystalline materials, and have attracted much attention since the pioneering work of Head (1953). For example, Dundurs (1967) calculated the displacement and stress fields of a screw dislocation near a circular elastic inhomogeneity. Sendeckyj (1970) extended this problem to the case of multiple inhomogeneities. Recently, the elastic interaction of screw dislocations with interphase boundaries of the circular inhomogeneities coated by one (Fan and Wang, 2003), two (Sudak, 2003) and many (Honein et al., 2006) intermediate layers has been studied. In their researches, the circular interface boundaries are assumed to be perfect (continuous displacement and traction across the interface) or imperfect. In the aforementioned studies, however, the effect of interface stress is not taken into account. The interface of solids is a special region with a very small thickness. Since the interface-to-volume ratio in the nano-scale domain is relatively high compared to that in the macro-scale domain, the interface stress
a b
Corresponding author: Email: [email protected] Presenter: Email: [email protected]
1877–7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.07.205
1630
D.X. LEI and Z.Y. OU / Procedia Engineering 14 (2011) 1629–1636 Author name / Procedia Engineering 00 (2011) 000–000
2
plays a significant role on the size-dependent behavior of nanoscale elements or devices (Wong et al., 1997; Kulkarni and Zhou, 2006). Since there is no intrinsic length scale involved in the constitutive laws of the classical elastic theory, it can not predict the size-dependent behavior of nanosized structures and devices. Recently, the interaction between a screw dislocation and a circular nano- inhomogeneity incorporating interface stress has been considered by Fang’s research group (Liu and Fang, 2007). In their papers, interface stress is restricted at the interface between the inhomogeneity and the infinite matrix. It is well known that the three-phase (inclusion/interphase /matrix) model is of great practical and theoretical value in composites mechanics researches. In most fiber-reinforced composite materials, coated fibers are widely employed in the design stage to improve the bonding strength between fibers and the matrix. In the present paper, we study the problem of the elastic interaction between a screw dislocation and a coated nano-inhomogeneity incorporating interface stress. The interface stresses at the inner and outer interfaces of the annular coated layer are considered. By means of a complex variable technique, the explicit solutions of the complex potentials are obtained. The image forces exerted on the screw dislocations are derived. 2.
COATED INHOMOGENEITY WITH INTERFACE EFFECT
Let us consider an infinite isotropic medium containing a circular inhomogeneity and an annular coating layer under anti-plane problem, where R1 and R2 are the inner and outer radii of the coated annulus. L1 and L2 indicate the interfaces between the inhomogeneity and the coating layer, the coating layer and the matrix, respectively. A screw dislocation is located at the point z0 in the matrix. In anti-plane state, the displacement w, the shear stress components σ xz and σ yz of the materials can be expressed in terms of a single analytical function (complex potential function) with respect to the complex variable z=x+jy (j2=-1). Following Gurtin and Murdoch (1975), for the considered anti-plane problem, we obtain in the bulk (Muskhelishvili, 1975):
∂ 2 w( i ) ∂ 2 w( i ) + = 0 , τ rz( i ) = 2 μiε rz(i ) , τ θ(iz) = 2μiεθ( iz) 2 2 ∂x ∂y
(1)
and in the interfaces (Liu et al., 2003; Liu and Fang, 2007):
τ
( Lα ) θz
= 2 μ Lα ε
( Lα ) θz ,
1 ∂τ θ( Lzα ) ⎡⎣τ rz(α ) (t ) ⎤⎦ = Rα ∂θ
(2)
where let super- and subscript i to 1, 2, 3, which refers to the inhomogeneity, coating layer and the matrix (i ) (i ) (i ) (i ) region, respectively; μi are shear moduli, τ rz ( ε rz ) and τ θ z ( ε θ z ) are stress (strain) components in the polar coordinates system (r, θ ); the sub- and super-script Lα ( α =1, 2) denote the two interfaces, iθ respectively; μ Lα ( α =1, 2) are the elastic constants of the interfaces and t = R α e ( α =1, 2) denote (α ) the points on the circular arc interface Lα . In addition, ⎡⎣τ rz (t ) ⎤⎦ represents the jump of the interface (L ) stresses and points to the positive normal direction of inclusion and matrix. The interface strain ε θ zα ( α =1, 2) equals to the associated tangential strain in the abutting bulk materials. Referring to the work of Muskhelishvili (1975), in the bulk solid, the displacements w(i), shear (i ) (i ) stresses τ rz and τ θ z can be written in terms of the analytical functions Φi(z) as follows
w(i ) = ⎡⎣ Φ i ( z ) + Φ i ( z ) ⎤⎦ 2 , τ rz( i ) − jτ θ(iz) = ⎡⎣τ xz(i ) − jτ yz(i ) ⎤⎦ e jθ = μi e jθ ϕi ( z )
(3)
where ϕi ( z ) = Φ i ( z ) ，and the overbar represents the complex conjugate and the prime denotes the derivative with respect to the argument z. '
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For the present problem, the classical continuity conditions of displacements and the non-classical equilibrium conditions (2) on the interface can be given as (Liu and Fang, 2007)
μ L1 ∂τ θ(2)z (t ) w(1) (t ) = w(2) (t ) , τ rz(1) (t ) − τ rz(2) (t ) = , R1μ2 ∂θ
t = R1 ,
μ L ∂τ θ(2)z (t ) w (t ) = w (t ) , τ (t ) − τ (t ) = , R2 μ2 ∂θ (2)
(3)
(3) rz
(2) rz
(4)
t = R2 .
2
(5)
For the convenience of analysis, the following new analytically extensional functions are introduced in the corresponding regions according to the Schwarz symmetry principle (Muskhelishvili, 1975).
R12 ϕ1e ( z ) = − 2 ϕ1 ( R12 z ) , z > R1 , z
(6)
ϕ2 ei ( z ) = −
R12 ϕ2 ( R12 z ) , R12 R2 < z < R1 , 2 z
(7)
ϕ2 ee ( z ) = −
R22 ϕ2 ( R22 z ) , R2 < z < R22 R1 , z2
(8)
ϕ3 e ( z ) = −
R22 ϕ3 ( R22 z ) , z2
z < R2 ,
(9)
where ϕ• ( R z ) = ϕ• ( z ) . Substituting Eqs. (6)-(9) into Eqs. (4) and (5) and with the aid of Eq. (3), we have 2
[ϕ1 (t ) − ϕ2ei (t ) − ϕ3e (t )] = [ϕ2 (t ) − ϕ1e (t ) − ϕ3e (t )] I
[ϕ2 (t ) − ϕ1e (t ) − ϕ3e (t )]
C
(
C
t = R1 ,
,
(10)
= [ϕ3 (t ) − ϕ1e (t ) − ϕ2 ee (t ) ] , t = R2 , M
)
(
)
⎡ μ1ϕ1 (t ) + μ2 + μ L R1 ϕ2 ei (t ) + μ L R1 tϕ2' ei (t ) + μ3ϕ3e (t ) ⎤ 1 1 ⎣ ⎦
(
)
(
(11)
I
)
C
= ⎡⎣ μ1ϕ1e (t ) + μ2 − μ L1 R1 ϕ2 (t ) − μ L1 R1 tϕ2' (t ) + μ3ϕ3e (t ) ⎤⎦ , t = R1 ,
(
)
(
)
⎡ μ1ϕ1e (t ) + μ2 − μ L R2 ϕ2 (t ) − μ L R2 tϕ2' (t ) + μ3ϕ3e (t ) ⎤ 2 2 ⎣ ⎦
(
)
(
)
(12)
C
M
= ⎡⎣ μ1ϕ1e (t ) + μ2 + μ L2 R2 ϕ2 ee (t ) + μ L2 R2 tϕ2' ee (t ) + μ3ϕ3 (t ) ⎤⎦ , t = R2 .
(13)
where the superscript I, C and M refer to the boundary values of the physical quantity as z approaches the interfaces from the regions occupied by the inhomogeneity, the coating layer and matrix, respectively.
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3.
SOLUTION OF BOUNDARY VALUE PROBLEM
Let us analyze the singularities of the complex functions. If a screw dislocation with Burgers vector (0, 0, bz) is located inside the matrix, the complex potential ϕ3 ( z ) in the matrix can be taken the form (Smith, 1968) as
= ϕ3 ( z )
bz 1 + Γ + ϕ30 ( z ) 2π j z − z0
z > R2
(14)
where the first and second terms represent the complex potential for a screw dislocation in an infinite Γ τ xz∞ − jτ yz∞ μ3 is matrix in the absence of the inhomogeneity subjected to external shear loading; = ∞ ∞ identified with the longitudinal stresses τ xz , τ yz at infinity. The third term ϕ30 ( z ) is holomorphic in the infinite matrix and represents the disturbance of the complex potential due to the coated inclusion. The complex potential ϕ3e ( z ) can be evaluated by substituting Eq. (14) into Eq. (9) as
(
= ϕ3e ( z )
(
)
bz ⎛ 1 1 ⎞ R22 − Γ + ϕ3 e 0 ( z ) ⎜ ⎟− 2π j ⎝ z z − z * ⎠ z 2
)
z < R2
(15)
where ϕ3e 0 ( z ) is holomorphic in z < R2 . z = R2 z0 is assumed in the region z < R1 R2 . It will be seen that when the point falls in the other regions of z < R2 , the same resulting solution is reached. The potential function of the coated layer ϕ 2 ( z ) can be expanded into a Laurent series in the annular region *
2
2
ϕ= GN ( z ) + GP ( z ) R1 < z < R2 2 ( z)
(16)
where GN(z) and GP(z) represent the sum of the negative and positive power terms, respectively. ∞
∞
k =1
k =0
GN ( z ) = ∑ ak z − ( k +1) , GP ( z ) = ∑ bk z k R1 < z < R2
(17)
where ak and bk (k=0, 1, 2, …) are the complex constants. It is seen that GN(z) and GP(z) converge in the regions z > R1 and z < R2 , respectively. According to the work of Muskhelishvili (1975) and some complex calculations, thus the analytical functions ϕ1 ( z ) , ϕ 2 ( z ) and ϕ3 ( z ) can be obtained
= ϕ1 ( z )
∞ ∞ a 1 1 ( μ2 − μ3 − (k − 2)Λ1 ) 2kk z k −1 − ( Λ1 − Λ 2 ) kbk z k ∑ ∑ R1 μ1 + μ3 k 1 = μ1 + μ3 k 0 ∞ a bμ 2 μ3Γ 1 1 1 − + ( μ2 + μ3 + k Λ 2 ) 2kk z k −1 + z 3 ∑ μ1 + μ3 k =1 μ1 + μ3 π j z − z0 μ1 + μ3 R2
= ϕ2 ( z )
∞
∑a z
− ( k +1)
∞
+ ∑ bk z k
k k 1= k 0 =
= ϕ3 ( z )
(18)
∞ 1 ( μ1 − μ2 + (k + 1)Λ1 ) bk R12( k +1) z −( k + 2) ∑ μ1 + μ3 k =0
(19)
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∞ ∞ 1 1 2( k +1) −( k + 2 ) μ − μ + k + Λ b R z − ( 1) ( ) ∑ ∑ ( Λ − Λ 2 ) (k + 1)ak z −( k +1) 2 k 2 μ1 + μ3 k =0 1 2 μ1 + μ3 k =1 1
2 bz ( μ1 − μ3 ) ⎛ 1 bz 1 1 ⎞ ( μ1 − μ3 ) R2 + +Γ− Γ ⎜ − ⎟+ 2π j z − z0 2π j ( μ1 + μ3 ) ⎝ z z − z * ⎠ ( μ1 + μ3 ) z 2
(20)
μ L2 R2 . The coefficients ak and bk are represented as recursion where Λ1 =μ L1 R1 , Λ 2 = formulas ⎛ 4μ μ Γ ⎞ B0 ⎜ − E0 + 1 3 ⎟ μ1 + μ3 ⎠ − Bk Ek a1 = ⎝ , ak +1 = k ≥1 A1C0 + B0 D1 Ak +1Ck + Bk Dk +1
(21)
⎛ 4μ μ Γ ⎞ A1 ⎜ − E0 + 1 3 ⎟ μ1 + μ3 ⎠ − Ak +1 Ek k ≥1 bk = b0 = ⎝ Ak +1Ck + Bk Dk +1 A1C0 + B0 D1
(22)
( μ1 + μ2 + (k + 1)Λ1 ) , Bk = ( μ1 − μ2 + (k + 1)Λ1 ) R12( k +1) ( μ1 + μ2 − Λ1 − k Λ 2 )
where Ak +1=
C= k
⎡ ( μ1 − μ2 + (k + 1)Λ 2 ) ( μ1 − μ2 + (k + 1)Λ1 ) R12( k +1) ⎤ − ( μ1 − μ3 ) ⎢ − ⎥ ( μ1 + μ3 ) ( μ1 + μ3 ) R22( k +1) ⎣ ⎦ Dk +1 =−
(k + 2) ( μ1 − μ3 )( Λ1 − Λ 2 ) 1 − ( μ1 − μ2 − (k + 1)Λ 2 ) 2( k +1) 2( k +1) R2 ( μ1 + μ3 ) R2
⎡ ( μ1 − μ3 )2 bz 2 μ b ⎤ 1 ( μ − μ3 ) bz 1 Ek = −⎢ − 3 z ⎥ k +1 + 1 k +1 2π j ⎢⎣ 2π ( μ1 + μ3 ) j 2π j ⎥⎦ z0 ( z* ) Λ1 μ3 = μ L1 ( R1μ3 ) 1 and m2 = Λ 2 μ3 = μ L2 ( R2 μ3 ) 1 , that is, the effects of If m1 = interface is ignored, The solution of complex potentials ϕi ( z ) (i=1, 2, 3) agrees with the result of Liu et al. (2003). If we let μ1 = μ 2 and R1 = R2 , the solution will be reduced to the solution on the interaction between a screw dislocation and an inhomogeneity with the interface effect (Liu and Fang, 2007). 4.
STRESS FIELD AND IMAGE FORCE ON DISLOCATION
The image force acting on the dislocation plays an important role in understanding the mobility and socalled trapping mechanism of the dislocation. Referring to the work of Liu and Fang (2007) and supposing that the remote loads vanishes and the screw dislocation lies at the point x0 on the positive xaxis (z0 = x0>R2 is a real number). Due to ak and bk including j, in this case, fy = 0 and the component of 2 the normalized image force along the x-axis direction is defined as f x 0 = 2π R2 f x μ3bz . That is
(
)
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= f x 0 2π R2= f x ( μ3bz2 )
jb 2π R2 ∞ ( μ1 − μ2 + (k + 1)Λ1 ) k R12( k +1) z0−( k + 2) ∑ μ1 + μ3 k =0 bz
−
jb 2π R2 ∞ ( μ1 − μ2 + (k + 1)Λ 2 ) k R22( k +1) z0−( k + 2) ∑ μ1 + μ3 k =0 bz
−
R (μ − μ ) ⎛ 1 ja 2π R2 ∞ 1 ⎞ ( Λ1 − Λ 2 ) (k + 1) k z0− ( k +1) − 2 1 3 ⎜ − ⎟ ∑ bz μ1 + μ3 k =1 μ1 + μ3 ⎝ z0 z0 − z * ⎠
In the special case of
μ1 = μ2
and
(23)
Λ1 =Λ 2 =0 , Eq. (23) becomes
bz2 μ3 ( μ1 − μ3 ) R22 ⎛ 1 bz2 μ3 1 ⎞ f x0 = Im agef = − ⎜ − ⎟ 2π R2 2π ( μ1 + μ3 ) ⎝ z0 z0 − z * ⎠
(24)
which coincides with previous results for the circular inhomogeneity not accounting for interphase layer effect (Gong and Meguid, 1994) For the description of the effects of interfaces, one needs the constants m1 = μ L1 ( R1μ3 ) and m2 = μ L2 ( R2 μ3 ) . For simplicity, let Λ1 = Λ 2 . Here m1 and m2 characterize the effects of interfaces. For a macroscopic inhomogeneity with bigger value of R1, m1, m2<<1, the interface effect can be neglected. However, when the size of an inhomogeneity reduces to nanometers, m1, m2 become noticeable, thus the interface effect should be taken into account (Streitz et al., 1994; Miller and Shenoy, 2000; Ou et al., 2008). In addition, we define the relative shear modulus α = μ1 μ3 and β = μ 2 μ3 , the relative location of the dislocation r = x0 R2 and the relative coating thickness λ = R1 R2 . In Figs. 1, we illustrate the variation of the values of f x 0 with respect to the parameter r for the selected material constants, the interface parameters m1 and m2 , R1=15nm and coating thickness (λ=0.95). It is shown from Fig. 1 that, if the matrix is softer than the inhomogeneity, which is harder than the coating (β<α<1) and the interface characteristic constants is positive Λ1 =Λ 2 > 0 , the screw dislocation is first attracted then repelled by the coated inhomogeneity leading to an unstable equilibrium position near the inhomogeneity. It is remarkable different from the classical result, in which the soft inclusion will always attract the screw dislocation. It is seen that there exists significant local hardening at the interface due to the interface stress ( Λ1 =Λ 2 > 0 ) and the softest coating. As the above considerations, let Λ1 =Λ 2 for simplicity. The variation of the normalized force f x 0 with respect to the radius of a coated inclusion R2 is depicted in Figs. 2 for different values of α and β at r=1.05 with interface effects. It can be found that a positive value of Λ1 produces the repulsive force acting on the screw dislocation while a negative value of Λ1 produces the attractive force. The phenomenon cannot be predicted by classical elasticity without interface stress. This indicates that the local hardening and softening at the interface may exist due to the interface effect. In addition, the absolute values of image force increases with the decrease of the inclusion radius, and the size dependence becomes significant when the size of inhomogeneity is at nanoscale.
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-10
0.4
Λ1,2=10*10 m, α=0.5, β =0.8
0.2
Λ1,2=10*10 m, α=0.5, β =0.2
-10
0.0 -0.2 -0.4
-10
Λ1,2=-10*10 m, α=0.5, β =0.8
-0.6
-10
Λ1,2=-10*10 m, α=0.5, β =0.2
-0.8 -1.0 1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
r Fig. 1 The influence of moduli of a soft inhomogeneity and a soft coating on fx0 .
-2.4
−10
Λ1=Λ2=1∗10 (m),α=0.5,β=0.5 −10
Λ1=Λ2=−1∗10 (m),α=0.5,β=0.5
-2.7
Λ1=Λ2=0,α=0.5,β=0.5
fx0
-3.0 -3.3 -3.6 -3.9
-4.2 0
50
100
150
200
250
300
350
400
R2(nm) Fig. 2 The size effect of a soft inhomogeneity with soft coating layer on the fx0 at r=1.05.
5.
CONCLUSIONS
The elastic interaction between screw dislocations and a circular coated inhomogeneity incorporating interface stress is considered in the light of surface elasticity. The elastic fields and the imagine force are obtained by utilizing complex variable method. To illuminate the surface effects, the image forces acting on screw dislocations are discussed in details, which includes the mobility and the stability of the appointed screw dislocation in the matrix and the influence of the material elastic dissimilarity, the coating thickness and the interface stress as well as the relative position of the dislocation. The results show that, the size dependence becomes significant when the size of a coated inhomogeneity is at
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nanoscale, which is completely different from that in classic elasticity. Our results are helpful in the understanding of the motion mechanism of the dislocation and relevant physical phenomena in threephase nano-composite materials. 6.
ACKNOWLEDGMENTS
The NSFC (11062004) and Doctoral Fund of Lanzhou University of Technology are acknowledged. REFERENCES [1]
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