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On the interaction between a dislocation and a circular inhomogeneity with imperfect interface in antiplane shear
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 30 (2003) 53–59 www.elsevier.com/locate/mechrescom
On the interaction between a dislocation and a circular inhomogeneity with imperfect interface in antiplane shear L.J. Sudak Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Alberta, Canada T2N-1N4 Received 7 February 2002
Abstract The solution of appropriate elasticity problems involving the interaction between inclusions and dislocations plays a fundamental role in many practical and theoretical applications, namely, it increases the understanding of material defects thereby providing valuable insight into the mechanical behavior of composite materials. Although the problem of a three-phase circular inclusion interacting with a dislocation in antiplane shear has been presented [Xiao and Chen, Mech. Mater. 32 (2000) 485], the analysis is limited to the classical perfect bonding condition. The current paper considers the solution for a homogeneous circular inclusion interacting with a dislocation under thermal loadings in antiplane shear. The bonding along the inhomogeneity–matrix interface is considered to be imperfect with the assumption that the interface imperfections are constant. It is found that when the inhomogeneity is soft, regardless of the level of interface imperfection, the inhomogeneity will always attract the dislocation. As a result, no equilibrium positions are available. Alternatively, when the inhomogeneity is hard, an unstable equilibrium position is found which depends on the imperfect interface condition and the shear moduli ratio l2 =l1 . Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Dislocation; Imperfect interface; Antiplane shear; Inclusion
1. Introduction Problems involving the elastic interaction between dislocations and inhomogeneities has received a considerable amount of attention in the mechanics of solids literature (see, for example, Dundurs and Mura, 1964; Dundurs and Sendeckyj, 1965; Dundurs, 1967; Stagni and Lizzio, 1983; Warren, 1983; Luo and Chen, 1991; Qaissaunee and Santare, 1995; Stagni, 1999; Xiao and Chen, 2000). In almost all cases, the classical situation where the displacements and surface tractions are continuous across the material interface has been assumed––the so-called perfect bonding condition. However, the perfect bonding model is inadequate for the accurate representation of many practical cases in which it is of interest to examine the
E-mail address: [email protected] (L.J. Sudak). 0093-6413/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 3 - 6 4 1 3 ( 0 2 ) 0 0 3 5 2 - X
54
L.J. Sudak / Mechanics Research Communications 30 (2003) 53–59
effects of interface damage (e.g. debonding, sliding and/or cracking across an interface) in composite materials. Recently, Xiao and Chen (2000) investigated the effects of a coated fiber interacting with a dislocation in antiplane shear under the limited condition of perfect bonding and no external loadings. Recognizing the existence of a coating layer (or interphase layer) implies that the composite must be regarded as a threephase assemblage. Such a consideration requires the complete knowledge of the physical properties of the interphase, information which is extremely difficult if not impossible to obtain because of the diminutive nature of the interphase and the spatial variation of the material properties. Moreover, since load transfer between fiber and matrix depends on, and is controlled by, the degree of contact along the material interface, the presence of interfacial damage (such as damage arising from imperfect adhesions, microcracks and voids) cannot be ignored. It is this realistic fact which has been neglected in the analysis presented in Xiao and Chen (2000). To address this shortcoming, the concept of an imperfect interface or an imperfect bonding condition has been adopted. The significance of interface imperfections, in any micro-mechanical analysis, is paramount in understanding the mechanical behavior of composite materials. One of the most widely used mechanical models of an imperfect bonding condition is based on the assumption that tractions are continuous but the displacements are discontinuous across the material interface. Specifically, the displacement jumps are proportional (in terms of Ôspring-factor typeÕ interface parameters) to their respective tractions (see Sudak et al., 1999). The advantage of using the imperfect bonding model over the three-phase perfect bonding description of Xiao and Chen (2000) are: the thickness of the interphase layer and its elastic constant are combined into one interface parameter (only for the elastic antiplane case), can simulate intermediate states of bonding (i.e. from perfect bonding to complete debonding), the model can be utilized when the interphase layer cannot be identified or defined, can take into account various degrees of damage within the interphase layer, and it is more mathematically tractable. The only drawback of the imperfect bonding model is with respect to the notion of material overlapping. A detailed discussion of this condition is given in Hashin (1991). In this paper, the interaction between a screw dislocation and a circular inhomogeneity undergoing uniform eigenstrains is investigated. Unlike the work of Xiao and Chen (2000), the bonding condition along the inhomogeneity–matrix interface is assumed to be imperfect with the assumption that the interface imperfections are constant. Closed-form expressions are derived for the elastic fields inside and outside the inhomogeneity, respectively, and the force on the dislocation is also given. Results illustrating the equilibrium positions of the dislocation, under various material property combinations and bonding conditions, are discussed. 2. Formulation Consider a domain in R2 , infinite in extent, containing a single internal elastic inhomogeneity with elastic properties different from the surrounding matrix. The linearly elastic materials occupying the matrix and the inhomogeneity are assumed to be homogeneous and isotropic with associated shear moduli l1 and l2 , respectively. In addition, a screw dislocation with Burgers vector b ¼ bz is located at the point ðp; 0Þ, p > R near the inhomogeneity. The matrix is represented by the domain D1 and the inhomogeneity occupies a circular region D2 with center at the origin and radius R. The inhomogeneity–matrix interface will be denoted by the curve C. In what follows, subscripts and superscripts 1 and 2 will refer to the regions D1 and D2 , respectively, and uðx; yÞ will denote the elastic (antiplane) deformation at the point ðx; yÞ. It is prescribed that the circular inhomogeneity is imperfectly bonded to the matrix along C by the Ôspring-layer typeÕ interface condition. The boundary value problem which describes the elastic (antiplane) deformation of a circular inclusion with imperfect interface condition on C is given by (Ru and Schiavone, 1997)
L.J. Sudak / Mechanics Research Communications 30 (2003) 53–59
r 2 u1 ¼ 0
in D1 ; ou2 þ au0 ðx; yÞ a½u1 u2 ¼ l2 on
on C;
r2 u2 ¼ 0 in D2 ; ou1 ou2 ¼ l2 on C; l1 on on
55
) ð1Þ
where n is the outward normal to C, a is the homogeneous imperfect interface parameter and u0 ðx; yÞ represents the additional displacement induced by the uniform (stress-free) eigenstrains ðe0xz ; e0yz Þ prescribed within the inclusion. These eigenstrains are proportional to the change in temperature and the difference between the thermal expansion coefficients of the matrix and the inclusion. In addition, note that if a ¼ 0, condition (1) reduces to the case of a traction-free interface, while if a ¼ 1, condition (1) corresponds to a perfectly bonded interface. Denote uj ðzÞ ¼ uj ðx; yÞ þ ivj ðx; yÞ ðj ¼ 1; 2Þ;
ð2Þ
where z ¼ x þ iy and vj ðx; yÞ are the harmonic functions conjugate to uj ðx; yÞ which satisfy the Cauchy– Reimann equations. Then, it is clear that vj ðx; yÞ are single valued and uniquely determined within an integration constant and the corresponding complex potentials u1 ðzÞ and u2 ðzÞ are analytic within D1 and D2 , respectively. Thus, the displacement components and the stress components can be written in terms of the complex functions uj ðzÞ as 2uj ðzÞ ¼ uj ðzÞ þ uj ðzÞ;
rxz iryz ¼ lj u0j ðzÞ;
z 2 Dj
ðj ¼ 1; 2Þ:
ð3Þ
Furthermore, note that 2
ou2 ¼ u02 ðzÞeinðzÞ þ u02 ðzÞeinðzÞ ; on
z 2 C;
ð4Þ
where einðzÞ represents (in complex form) the outward normal to C at z. Hence, the boundary value problem given in (1) can now be reformulated in terms of the functions uj ðzÞ. ^i dx ^j, the traction continuity condition Utilizing the Cauchy–Reimann equations and noting that n ¼ dy ds ds can be written in the following form, l1 v1 ¼ l2 v2
ð5Þ
on C:
Then, using conditions (4) and (5), the displacement discontinuity condition can be rewritten in terms of u1 and u2 as i l h ð6Þ u1 ðzÞ ¼ 2 u02 ðzÞeinðzÞ þ u02 ðzÞeinðzÞ þ du2 ðzÞ þ ð1 dÞu2 ðzÞ þ u0 ðzÞ; z 2 C; 2a where d¼
l1 þ l2 1 > ; 2 2l1
u0 ðzÞ ¼ zðe0xz ie0yz Þ þ zðe0xz þ ie0yz Þ:
ð7Þ
To solve the problem of a screw dislocation interacting with a circular inhomogeneity with imperfect interface let us define two analytic functions. The analytic complex potential within the matrix can be represented as the sum of two functions, namely u1 ðzÞ ¼ i
1 bz bz X logðz pÞ þ Ak zk ; 2p 2p k¼1
z 2 D1 ;
ð8Þ
where the first term represents the complex potential for the screw dislocation in an infinite homogeneous material and the second term represents the disturbance of the complex potential due to the presence of the inhomogeneity. Similarly, the analytic complex potential within the inhomogeneity can be expressed as
56
L.J. Sudak / Mechanics Research Communications 30 (2003) 53–59
u2 ðzÞ ¼
1 bz X B k zk ; 2p k¼0
z 2 D2 ;
ð9Þ
where Ak and Bk are undetermined complex coefficients. Substituting (8) and (9) into the interface condition (7), and noting that for a circular inclusion ReinðzÞ ¼ z ðz 2 CÞ we obtain, by comparing the coefficients for the powers of z: log p ; Im½B0 ¼ Re½B0 ¼ p; 1 2d 2p 1 2p 0 þ e e0xz þ i bz p bz yz B1 ¼ ; l dþ 2 2aR
2p 2 2p 1 l2 2 0 0 R ð2d 1Þexz þ i ð2d 1Þeyz þ d 1 R ð10Þ b bz p 2aR ; A1 ¼ z l dþ 2 2aR 2 3 2 3 l2 d 1 k 6 7 1 6 2aR 7 7 Bk ¼ i 6 Ak ¼ iR2k 4 5; k ¼ 2; 3; 4; . . . l2 4 5; kl k 2 d þ k kp k kp d þ 2aR 2aR It should be noted that when e0xz ¼ e0yz ¼ 0 (no eigenstrains) and when the parameter a is infinite (i.e. corresponding perfect bonding condition) the solutions given in (10) are identical to those obtained by Xiao and Chen (2000) for the two-phase system. In addition, in view of (8) and (9), the elastic fields can be calculated via Eq. (3). 3. Discussion To examine the effects of the imperfect interface on the stability of the dislocation let us adopt the procedure outlined in Xiao and Chen (2000), namely, by evaluating what the force on the dislocation is one can determine if the dislocation is attracted or repelled by the inhomogeneity and whether the dislocation is stable or not. In this paper, we limit ourselves to the study of the conditions when the force on the dislocation can move only in the radial direction. Then, as in Xiao and Chen (2000), let us define the normalized force as 1 X Xk F ¼ ; ð11Þ 2kþ1 b k¼1 where it can be shown that b¼
p > 1; R
l2 l 1 k 2 l aR Xk ¼ 1 : l2 l2 þ1 þk l1 aR
ð12Þ
Then, it can be seen from Eq. (11) that the sign of F is determined solely by the coefficients Xk . A positive value of F indicates that the inhomogeneity repels the dislocation. In other words, Xk > 0 corresponds to a repulsion of the dislocation by the inhomogeneity, while Xk < 0 corresponds to an attraction of the dislocation by the inhomogeneity. In addition, let us introduce a convenient non-dimensional parameter M ¼ aR=l2 , into Eq. (12). Here, M characterizes the effectiveness of bonding (degree of interface
L.J. Sudak / Mechanics Research Communications 30 (2003) 53–59
57
imperfection) at the interface in transferring load between the matrix and the inhomogeneity. Physically, a very small value of M corresponds to a traction-free condition while a very large value of M corresponds to perfect bonding condition. Figs. 1–3 show the variation of the normalized force with respect to the dislocation position parameter (b ¼ p=R) for various values of M and material property combinations (i.e. l2 =l1 ). As well, for comparison,
0
-1
-2 Μ = 0.01, µ2/µ1 = 0.1
F*
-3
Μ = 1, µ2/µ1 = 0.1 Μ = 1Ε+9, µ2/µ1 = 0.1
-4
-5
-6
-7
-8 1
1.1
1.2
1.3
1.4
1.5
1.6
β = p/R
1.7
1.8
1.9
2
2.1
2.2
Fig. 1. The effect of the dislocation position parameter on the normalized force for various bonding conditions when the inclusion is soft.
8 6 4 2
F*
0 -2
Μ = 0.01, µ2/µ1 = 5 Μ = 0.1, µ2/µ1 = 5
-4
Μ = 1, µ2/µ1 = 5 Μ = 3, µ2/µ1 = 5
-6
Μ = 5, µ2/µ1 = 5
-8
Μ = 1Ε+9, µ2/µ1 = 5
-10 1
1.2
1.4
1.6
1.8
2
2.2
β = p/R Fig. 2. The effect of the dislocation position parameter on the normalized force for various bonding conditions when the inclusion is hard.
58
L.J. Sudak / Mechanics Research Communications 30 (2003) 53–59 6
4
2
F*
0
Μ = 1; µ2/µ1 = 0.1
-2
Μ = 1, µ2/µ1 = 1 Μ = 1, µ2/µ1 = 3
-4
Μ = 1, µ2/µ1 = 5 Μ = 1, µ2/µ1 = 10
-6
Μ = 1, µ2/µ1 = 25
-8 1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
β = p/R Fig. 3. The effect of the dislocation position parameter on the normalized force for various combinations of shear moduli ratio.
the corresponding perfect bonding condition is also illustrated. The results have been generated by truncating the series, given in (11), at k ¼ 15. As seen in Fig. 1, there is no equilibrium position for either the perfect or imperfect bonding condition, respectively, when the ratio l2 =l1 < 1 (soft inhomogeneity). Moreover, the results illustrate that no matter what the level of interface imperfection, a soft inhomogeneity will always attract the dislocation. This result is in agreement to that obtained by Dundurs (1967) for the perfect bonding condition. In contrast, when the inhomogeneity is hard (i.e. l2 =l1 > 1), for example when l2 =l1 ¼ 5 (see Fig. 2), and M is taken to be small (e.g. M ¼ 0:1) the inhomogeneity will attract the dislocation and no equilibrium positions are available. Alternatively, when M is large (e.g. M ¼ 3) the inhomogeneity will repel the dislocation and again no equilibrium positions are available. However, when M ¼ 1 the dislocation is first attracted then repelled by the inhomogeneity. In this case, an equilibrium position is found between b ¼ 1:05 and 1.1. Since oF =ob > 0 the equilibrium position must be unstable. Fig. 3 reinforces the previous results in that the stability of the dislocation not only depends on the imperfect bonding condition but also on the material property combinations (i.e. l2 =l1 ).
4. Conclusions The interaction between dislocations and inhomogeneities is an important topic in analyzing the physical characteristics of many materials. In the analysis of such problems, the basic assumption that has been made is that the inhomogeneity is perfectly bonded to the surrounding matrix. However, this idealized condition effectively ignores the presence of interfacial damage between the inhomogeneity and the matrix. Thus, the concept of an imperfect interface has been adopted in order to take into account the various degrees of damage that exist along a material boundary. In this paper, the interaction between a screw dislocation and a circular inhomogeneity with homogeneous imperfect interface undergoing uniform eigenstrains is investigated. Using complex variable techniques, closed-form expressions have been obtained for the elastic stress fields both inside and outside the inhomogeneity, respectively. In addition, the force on the dislocation is also given. It has been shown that
L.J. Sudak / Mechanics Research Communications 30 (2003) 53–59
59
depending on interface conditions and whether the inclusion is soft or hard will determine whether the inhomogeneity attracts or repels the dislocation.
References Dundurs, J., 1967. On the interaction of a screw dislocation with inhomogeneities. Recent Adv. Eng. Sci. 2, 223–233. Dundurs, J., Mura, T.J., 1964. Interaction between an edge dislocation and a circular inclusion. Mech. Phys. Solids 12, 177–189. Dundurs, J., Sendeckyj, G.P., 1965. Edge dislocations inside a circular inclusion. J. Mech. Phys. Solids 13, 141–147. Hashin, Z., 1991. The spherical inclusion with imperfect interface. J. Appl. Mech. (ASME) 58, 444–449. Luo, H.A., Chen, Y., 1991. An edge dislocation in a three-phase composite cylinder model. J. Appl. Mech. (ASME) 58, 75–86. Qaissaunee, M.T., Santare, M.H., 1995. Edge dislocation interacting with an elliptical inclusion surrounded by an interfacial zone. Quart. J. Mech. Appl. Math. 48, 465–482. Ru, C.Q., Schiavone, P., 1997. A circular inclusion with circumferentially inhomogeneous interface in antiplane shear. Proc. R. Soc. (London) A 453, 1–22. Stagni, L., 1999. The effect of the interface on the interaction of an interior edge dislocation with an elliptical inhomogeneity. Z. Angew. Math. Phys. 50, 327–337. Stagni, L., Lizzio, R., 1983. Shape effects in the interaction between an edge dislocation and an elliptic inclusion. J. Appl. Phys. A 30, 217–221. Sudak, L.J., Ru, C.Q., Schiavone, P., Mioduchowski, A., 1999. A circular inclusion with inhomogeneously imperfect interface in plane elasticity. J. Elasticity 55, 19–41. Warren, W.E., 1983. The edge dislocation inside an elliptical inclusion. Mech. Mater. 2, 319–330. Xiao, Z.M., Chen, B.J., 2000. A screw dislocation interacting with a coated fiber. Mech. Mater 32, 485–494.
Mechanics Research Communications 30 (2003) 53–59 www.elsevier.com/locate/mechrescom
On the interaction between a dislocation and a circular inhomogeneity with imperfect interface in antiplane shear L.J. Sudak Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Alberta, Canada T2N-1N4 Received 7 February 2002
Abstract The solution of appropriate elasticity problems involving the interaction between inclusions and dislocations plays a fundamental role in many practical and theoretical applications, namely, it increases the understanding of material defects thereby providing valuable insight into the mechanical behavior of composite materials. Although the problem of a three-phase circular inclusion interacting with a dislocation in antiplane shear has been presented [Xiao and Chen, Mech. Mater. 32 (2000) 485], the analysis is limited to the classical perfect bonding condition. The current paper considers the solution for a homogeneous circular inclusion interacting with a dislocation under thermal loadings in antiplane shear. The bonding along the inhomogeneity–matrix interface is considered to be imperfect with the assumption that the interface imperfections are constant. It is found that when the inhomogeneity is soft, regardless of the level of interface imperfection, the inhomogeneity will always attract the dislocation. As a result, no equilibrium positions are available. Alternatively, when the inhomogeneity is hard, an unstable equilibrium position is found which depends on the imperfect interface condition and the shear moduli ratio l2 =l1 . Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Dislocation; Imperfect interface; Antiplane shear; Inclusion
1. Introduction Problems involving the elastic interaction between dislocations and inhomogeneities has received a considerable amount of attention in the mechanics of solids literature (see, for example, Dundurs and Mura, 1964; Dundurs and Sendeckyj, 1965; Dundurs, 1967; Stagni and Lizzio, 1983; Warren, 1983; Luo and Chen, 1991; Qaissaunee and Santare, 1995; Stagni, 1999; Xiao and Chen, 2000). In almost all cases, the classical situation where the displacements and surface tractions are continuous across the material interface has been assumed––the so-called perfect bonding condition. However, the perfect bonding model is inadequate for the accurate representation of many practical cases in which it is of interest to examine the
E-mail address: [email protected] (L.J. Sudak). 0093-6413/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 3 - 6 4 1 3 ( 0 2 ) 0 0 3 5 2 - X
54
L.J. Sudak / Mechanics Research Communications 30 (2003) 53–59
effects of interface damage (e.g. debonding, sliding and/or cracking across an interface) in composite materials. Recently, Xiao and Chen (2000) investigated the effects of a coated fiber interacting with a dislocation in antiplane shear under the limited condition of perfect bonding and no external loadings. Recognizing the existence of a coating layer (or interphase layer) implies that the composite must be regarded as a threephase assemblage. Such a consideration requires the complete knowledge of the physical properties of the interphase, information which is extremely difficult if not impossible to obtain because of the diminutive nature of the interphase and the spatial variation of the material properties. Moreover, since load transfer between fiber and matrix depends on, and is controlled by, the degree of contact along the material interface, the presence of interfacial damage (such as damage arising from imperfect adhesions, microcracks and voids) cannot be ignored. It is this realistic fact which has been neglected in the analysis presented in Xiao and Chen (2000). To address this shortcoming, the concept of an imperfect interface or an imperfect bonding condition has been adopted. The significance of interface imperfections, in any micro-mechanical analysis, is paramount in understanding the mechanical behavior of composite materials. One of the most widely used mechanical models of an imperfect bonding condition is based on the assumption that tractions are continuous but the displacements are discontinuous across the material interface. Specifically, the displacement jumps are proportional (in terms of Ôspring-factor typeÕ interface parameters) to their respective tractions (see Sudak et al., 1999). The advantage of using the imperfect bonding model over the three-phase perfect bonding description of Xiao and Chen (2000) are: the thickness of the interphase layer and its elastic constant are combined into one interface parameter (only for the elastic antiplane case), can simulate intermediate states of bonding (i.e. from perfect bonding to complete debonding), the model can be utilized when the interphase layer cannot be identified or defined, can take into account various degrees of damage within the interphase layer, and it is more mathematically tractable. The only drawback of the imperfect bonding model is with respect to the notion of material overlapping. A detailed discussion of this condition is given in Hashin (1991). In this paper, the interaction between a screw dislocation and a circular inhomogeneity undergoing uniform eigenstrains is investigated. Unlike the work of Xiao and Chen (2000), the bonding condition along the inhomogeneity–matrix interface is assumed to be imperfect with the assumption that the interface imperfections are constant. Closed-form expressions are derived for the elastic fields inside and outside the inhomogeneity, respectively, and the force on the dislocation is also given. Results illustrating the equilibrium positions of the dislocation, under various material property combinations and bonding conditions, are discussed. 2. Formulation Consider a domain in R2 , infinite in extent, containing a single internal elastic inhomogeneity with elastic properties different from the surrounding matrix. The linearly elastic materials occupying the matrix and the inhomogeneity are assumed to be homogeneous and isotropic with associated shear moduli l1 and l2 , respectively. In addition, a screw dislocation with Burgers vector b ¼ bz is located at the point ðp; 0Þ, p > R near the inhomogeneity. The matrix is represented by the domain D1 and the inhomogeneity occupies a circular region D2 with center at the origin and radius R. The inhomogeneity–matrix interface will be denoted by the curve C. In what follows, subscripts and superscripts 1 and 2 will refer to the regions D1 and D2 , respectively, and uðx; yÞ will denote the elastic (antiplane) deformation at the point ðx; yÞ. It is prescribed that the circular inhomogeneity is imperfectly bonded to the matrix along C by the Ôspring-layer typeÕ interface condition. The boundary value problem which describes the elastic (antiplane) deformation of a circular inclusion with imperfect interface condition on C is given by (Ru and Schiavone, 1997)
L.J. Sudak / Mechanics Research Communications 30 (2003) 53–59
r 2 u1 ¼ 0
in D1 ; ou2 þ au0 ðx; yÞ a½u1 u2 ¼ l2 on
on C;
r2 u2 ¼ 0 in D2 ; ou1 ou2 ¼ l2 on C; l1 on on
55
) ð1Þ
where n is the outward normal to C, a is the homogeneous imperfect interface parameter and u0 ðx; yÞ represents the additional displacement induced by the uniform (stress-free) eigenstrains ðe0xz ; e0yz Þ prescribed within the inclusion. These eigenstrains are proportional to the change in temperature and the difference between the thermal expansion coefficients of the matrix and the inclusion. In addition, note that if a ¼ 0, condition (1) reduces to the case of a traction-free interface, while if a ¼ 1, condition (1) corresponds to a perfectly bonded interface. Denote uj ðzÞ ¼ uj ðx; yÞ þ ivj ðx; yÞ ðj ¼ 1; 2Þ;
ð2Þ
where z ¼ x þ iy and vj ðx; yÞ are the harmonic functions conjugate to uj ðx; yÞ which satisfy the Cauchy– Reimann equations. Then, it is clear that vj ðx; yÞ are single valued and uniquely determined within an integration constant and the corresponding complex potentials u1 ðzÞ and u2 ðzÞ are analytic within D1 and D2 , respectively. Thus, the displacement components and the stress components can be written in terms of the complex functions uj ðzÞ as 2uj ðzÞ ¼ uj ðzÞ þ uj ðzÞ;
rxz iryz ¼ lj u0j ðzÞ;
z 2 Dj
ðj ¼ 1; 2Þ:
ð3Þ
Furthermore, note that 2
ou2 ¼ u02 ðzÞeinðzÞ þ u02 ðzÞeinðzÞ ; on
z 2 C;
ð4Þ
where einðzÞ represents (in complex form) the outward normal to C at z. Hence, the boundary value problem given in (1) can now be reformulated in terms of the functions uj ðzÞ. ^i dx ^j, the traction continuity condition Utilizing the Cauchy–Reimann equations and noting that n ¼ dy ds ds can be written in the following form, l1 v1 ¼ l2 v2
ð5Þ
on C:
Then, using conditions (4) and (5), the displacement discontinuity condition can be rewritten in terms of u1 and u2 as i l h ð6Þ u1 ðzÞ ¼ 2 u02 ðzÞeinðzÞ þ u02 ðzÞeinðzÞ þ du2 ðzÞ þ ð1 dÞu2 ðzÞ þ u0 ðzÞ; z 2 C; 2a where d¼
l1 þ l2 1 > ; 2 2l1
u0 ðzÞ ¼ zðe0xz ie0yz Þ þ zðe0xz þ ie0yz Þ:
ð7Þ
To solve the problem of a screw dislocation interacting with a circular inhomogeneity with imperfect interface let us define two analytic functions. The analytic complex potential within the matrix can be represented as the sum of two functions, namely u1 ðzÞ ¼ i
1 bz bz X logðz pÞ þ Ak zk ; 2p 2p k¼1
z 2 D1 ;
ð8Þ
where the first term represents the complex potential for the screw dislocation in an infinite homogeneous material and the second term represents the disturbance of the complex potential due to the presence of the inhomogeneity. Similarly, the analytic complex potential within the inhomogeneity can be expressed as
56
L.J. Sudak / Mechanics Research Communications 30 (2003) 53–59
u2 ðzÞ ¼
1 bz X B k zk ; 2p k¼0
z 2 D2 ;
ð9Þ
where Ak and Bk are undetermined complex coefficients. Substituting (8) and (9) into the interface condition (7), and noting that for a circular inclusion ReinðzÞ ¼ z ðz 2 CÞ we obtain, by comparing the coefficients for the powers of z: log p ; Im½B0 ¼ Re½B0 ¼ p; 1 2d 2p 1 2p 0 þ e e0xz þ i bz p bz yz B1 ¼ ; l dþ 2 2aR
2p 2 2p 1 l2 2 0 0 R ð2d 1Þexz þ i ð2d 1Þeyz þ d 1 R ð10Þ b bz p 2aR ; A1 ¼ z l dþ 2 2aR 2 3 2 3 l2 d 1 k 6 7 1 6 2aR 7 7 Bk ¼ i 6 Ak ¼ iR2k 4 5; k ¼ 2; 3; 4; . . . l2 4 5; kl k 2 d þ k kp k kp d þ 2aR 2aR It should be noted that when e0xz ¼ e0yz ¼ 0 (no eigenstrains) and when the parameter a is infinite (i.e. corresponding perfect bonding condition) the solutions given in (10) are identical to those obtained by Xiao and Chen (2000) for the two-phase system. In addition, in view of (8) and (9), the elastic fields can be calculated via Eq. (3). 3. Discussion To examine the effects of the imperfect interface on the stability of the dislocation let us adopt the procedure outlined in Xiao and Chen (2000), namely, by evaluating what the force on the dislocation is one can determine if the dislocation is attracted or repelled by the inhomogeneity and whether the dislocation is stable or not. In this paper, we limit ourselves to the study of the conditions when the force on the dislocation can move only in the radial direction. Then, as in Xiao and Chen (2000), let us define the normalized force as 1 X Xk F ¼ ; ð11Þ 2kþ1 b k¼1 where it can be shown that b¼
p > 1; R
l2 l 1 k 2 l aR Xk ¼ 1 : l2 l2 þ1 þk l1 aR
ð12Þ
Then, it can be seen from Eq. (11) that the sign of F is determined solely by the coefficients Xk . A positive value of F indicates that the inhomogeneity repels the dislocation. In other words, Xk > 0 corresponds to a repulsion of the dislocation by the inhomogeneity, while Xk < 0 corresponds to an attraction of the dislocation by the inhomogeneity. In addition, let us introduce a convenient non-dimensional parameter M ¼ aR=l2 , into Eq. (12). Here, M characterizes the effectiveness of bonding (degree of interface
L.J. Sudak / Mechanics Research Communications 30 (2003) 53–59
57
imperfection) at the interface in transferring load between the matrix and the inhomogeneity. Physically, a very small value of M corresponds to a traction-free condition while a very large value of M corresponds to perfect bonding condition. Figs. 1–3 show the variation of the normalized force with respect to the dislocation position parameter (b ¼ p=R) for various values of M and material property combinations (i.e. l2 =l1 ). As well, for comparison,
0
-1
-2 Μ = 0.01, µ2/µ1 = 0.1
F*
-3
Μ = 1, µ2/µ1 = 0.1 Μ = 1Ε+9, µ2/µ1 = 0.1
-4
-5
-6
-7
-8 1
1.1
1.2
1.3
1.4
1.5
1.6
β = p/R
1.7
1.8
1.9
2
2.1
2.2
Fig. 1. The effect of the dislocation position parameter on the normalized force for various bonding conditions when the inclusion is soft.
8 6 4 2
F*
0 -2
Μ = 0.01, µ2/µ1 = 5 Μ = 0.1, µ2/µ1 = 5
-4
Μ = 1, µ2/µ1 = 5 Μ = 3, µ2/µ1 = 5
-6
Μ = 5, µ2/µ1 = 5
-8
Μ = 1Ε+9, µ2/µ1 = 5
-10 1
1.2
1.4
1.6
1.8
2
2.2
β = p/R Fig. 2. The effect of the dislocation position parameter on the normalized force for various bonding conditions when the inclusion is hard.
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L.J. Sudak / Mechanics Research Communications 30 (2003) 53–59 6
4
2
F*
0
Μ = 1; µ2/µ1 = 0.1
-2
Μ = 1, µ2/µ1 = 1 Μ = 1, µ2/µ1 = 3
-4
Μ = 1, µ2/µ1 = 5 Μ = 1, µ2/µ1 = 10
-6
Μ = 1, µ2/µ1 = 25
-8 1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
β = p/R Fig. 3. The effect of the dislocation position parameter on the normalized force for various combinations of shear moduli ratio.
the corresponding perfect bonding condition is also illustrated. The results have been generated by truncating the series, given in (11), at k ¼ 15. As seen in Fig. 1, there is no equilibrium position for either the perfect or imperfect bonding condition, respectively, when the ratio l2 =l1 < 1 (soft inhomogeneity). Moreover, the results illustrate that no matter what the level of interface imperfection, a soft inhomogeneity will always attract the dislocation. This result is in agreement to that obtained by Dundurs (1967) for the perfect bonding condition. In contrast, when the inhomogeneity is hard (i.e. l2 =l1 > 1), for example when l2 =l1 ¼ 5 (see Fig. 2), and M is taken to be small (e.g. M ¼ 0:1) the inhomogeneity will attract the dislocation and no equilibrium positions are available. Alternatively, when M is large (e.g. M ¼ 3) the inhomogeneity will repel the dislocation and again no equilibrium positions are available. However, when M ¼ 1 the dislocation is first attracted then repelled by the inhomogeneity. In this case, an equilibrium position is found between b ¼ 1:05 and 1.1. Since oF =ob > 0 the equilibrium position must be unstable. Fig. 3 reinforces the previous results in that the stability of the dislocation not only depends on the imperfect bonding condition but also on the material property combinations (i.e. l2 =l1 ).
4. Conclusions The interaction between dislocations and inhomogeneities is an important topic in analyzing the physical characteristics of many materials. In the analysis of such problems, the basic assumption that has been made is that the inhomogeneity is perfectly bonded to the surrounding matrix. However, this idealized condition effectively ignores the presence of interfacial damage between the inhomogeneity and the matrix. Thus, the concept of an imperfect interface has been adopted in order to take into account the various degrees of damage that exist along a material boundary. In this paper, the interaction between a screw dislocation and a circular inhomogeneity with homogeneous imperfect interface undergoing uniform eigenstrains is investigated. Using complex variable techniques, closed-form expressions have been obtained for the elastic stress fields both inside and outside the inhomogeneity, respectively. In addition, the force on the dislocation is also given. It has been shown that
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depending on interface conditions and whether the inclusion is soft or hard will determine whether the inhomogeneity attracts or repels the dislocation.
References Dundurs, J., 1967. On the interaction of a screw dislocation with inhomogeneities. Recent Adv. Eng. Sci. 2, 223–233. Dundurs, J., Mura, T.J., 1964. Interaction between an edge dislocation and a circular inclusion. Mech. Phys. Solids 12, 177–189. Dundurs, J., Sendeckyj, G.P., 1965. Edge dislocations inside a circular inclusion. J. Mech. Phys. Solids 13, 141–147. Hashin, Z., 1991. The spherical inclusion with imperfect interface. J. Appl. Mech. (ASME) 58, 444–449. Luo, H.A., Chen, Y., 1991. An edge dislocation in a three-phase composite cylinder model. J. Appl. Mech. (ASME) 58, 75–86. Qaissaunee, M.T., Santare, M.H., 1995. Edge dislocation interacting with an elliptical inclusion surrounded by an interfacial zone. Quart. J. Mech. Appl. Math. 48, 465–482. Ru, C.Q., Schiavone, P., 1997. A circular inclusion with circumferentially inhomogeneous interface in antiplane shear. Proc. R. Soc. (London) A 453, 1–22. Stagni, L., 1999. The effect of the interface on the interaction of an interior edge dislocation with an elliptical inhomogeneity. Z. Angew. Math. Phys. 50, 327–337. Stagni, L., Lizzio, R., 1983. Shape effects in the interaction between an edge dislocation and an elliptic inclusion. J. Appl. Phys. A 30, 217–221. Sudak, L.J., Ru, C.Q., Schiavone, P., Mioduchowski, A., 1999. A circular inclusion with inhomogeneously imperfect interface in plane elasticity. J. Elasticity 55, 19–41. Warren, W.E., 1983. The edge dislocation inside an elliptical inclusion. Mech. Mater. 2, 319–330. Xiao, Z.M., Chen, B.J., 2000. A screw dislocation interacting with a coated fiber. Mech. Mater 32, 485–494.