A screw dislocation interacting with a coated fiber

Mechanics of Materials 32 (2000) 485±494

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A screw dislocation interacting with a coated ®ber Z.M. Xiao *, B.J. Chen School of Mechanical & Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore Received 6 July 1999

Abstract A close-form analytical solution is obtained for the stress ®eld due to a screw dislocation near a coated ®ber inhomogeneity in isotropic material. The forces on dislocations are derived. The equilibrium positions of the dislocation are discussed in detail for various material constantsÕ combinations. It is found that when the coating layer is thick, the elastic property of the ®ber (inclusion) has no signi®cant in¯uence on the force of the dislocation, therefore the equilibrium and stability of the dislocation can be obtained similarly from the two-phase model adopted by Dunders (1967. Recent Advances in Engineering Science 2, 223±233). On the other hand when the coating layer is thin, if both the ®ber and the coating layer are ``softer'' (i.e., have lower modulus) than the matrix, the dislocation is always attracted by the ®ber, and if both the ®ber and the coating are ``harder'' (i.e., have higher modulus) than the matrix, the dislocation is always repelled by the ®ber. As a result, there are no equilibrium positions under these two conditions. While if the ®ber is harder and the coating is softer than the matrix, there is at least one unstable equilibrium position near the coating±matrix interface, if the ®ber is softer and the coating is harder than the matrix, there is at least one stable equilibrium position near the coating±matrix interface. Ó 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction The interaction of dislocations with inhomogeneities is an important topic in studying the strengthening and hardening mechanism of many materials. This is mainly due to the fact that the mobility of dislocation can be in¯uenced by various inhomogeneities in materials. In the absence of applied traction and residual stresses caused by mis®t, the interaction of dislocations with second-phase inclusions arises from two sources. One is the local disturbance in the atomic arrangements at the interfaces; the second is the mismatch in the elastic constants. When the dislocations are not extremely close to the interface and the size of inclusion is fairly large, the elastic interaction is expected to be predominant (Dunders, 1969). The ®rst investigation to assess the interaction of dislocations with large-scale inhomogeneities was made by Head (1953a,b), using the model of two connected semi-in®nite solids with di€erent elastic constants, and a straight dislocation running parallel to the interface. Under this circumstance, the dislocation is simply either repelled or attracted by the interface. Dunders and Mura (1964) have indicated, however, for the case of a ®nite inhomogeneity, an edge dislocation may have stable equilibrium positions near the

*

Corresponding author. Tel.: +65-799-4726; fax: +65-791-1859. E-mail address: [email protected] (Z.M. Xiao).

0167-6636/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 6 3 6 ( 0 0 ) 0 0 0 1 6 - 8

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interface even in the absence of volume mis®t between the matrix and the inclusion. This result suggests a so-called trapping mechanism of the motion of dislocations near inhomogeneities. Furthermore, the interaction is strongly a€ected not only by the mismatch between the shear moduli, but also by the di€erences in PoissonÕs ratios. While for a screw dislocation near a round inclusion in an in®nite solid, Dunders (1967) stated that a ``soft'' inclusion attracts while a ``hard'' one repels the screw dislocation. For the case of ®nite inclusion concentration, the interaction between an edge dislocation with surrounding inclusions was studied by Luo and Chen (1991a) with the aid of self-consistent model proposed by Christensen (1979) and Christensen and Lo (1979). In most ®ber-reinforced composite materials, control of the wetting, reaction and bonding of the matrix to ®bers is critical in achieving the desired property goals, therefore, coating on ®bers are often employed. In a ceramic matrix composite material, load transfer may be the secondary importance. While in metal matrix and polymer composite materials, because the mechanical properties are greatly enhanced by ecient load transfer from the matrix to the ®bers, the bond quality between the ®ber and the matrix is very important. The bond should be strong in order to obtain high quality composites. Coating on ®bers can improve the ®ber±matrix bonding quality. Articles on ®ber coating can be found in the literature (e.g. Mikata and Taya, 1985a,b; Walpole, 1978, to name a few.) In order to investigate the coating e€ect on bonding strength in composite materials with defects, it is of critical importance to have a comprehensive understanding on coated ®ber±defect interaction. The current study investigates the interaction between a screw dislocation and a circular inhomogeneity (the ®ber) with a coating layer. An analytical solution for the stress ®eld due to a screw dislocation near a coated ®ber is derived. The force on dislocation is calculated. Some other related results and discussion are given.

2. Formulation The physical problem to be investigated is shown in Fig. 1, where a screw dislocation b ˆ bz is located at point …e; 0†; e > b near a coated inclusion (the ®ber). The inner circular region r 6 a (phase ``1'') is the ®ber with radius a and elastic properties j1 ; l1 , the intermediate layer (phase ``2''), is the coating material with

Fig. 1. Geometry for a screw dislocation near a coated ®ber.

Z.M. Xiao, B.J. Chen / Mechanics of Materials 32 (2000) 485±494

487

elastic properties j2 ; l2 , occupying the annular region a 6 r 6 b, and the outside layer (phase ``3'') is the matrix material with elastic properties j3 ; l3 , occupying the in®nite region r P b. As the current case is an anti-plane elastic problem, the displacement components in each phase can be written as (Muskhelishvili, 1953): …i† u…i† x ˆ uy ˆ 0;

u…i† z ˆ ui …x; y†;

…1†

for i ˆ 1; 2; 3; where ui …x; y† are some unknown functions of x; y. The stress components related to the displacement components in Eq. (1) are thus obtained by r…i† zx ˆ li

oui ; ox

r…i† zy ˆ li

oui ; oy

…2†

and …i† …i† …i† r…i† zz ˆ rxx ˆ ryy ˆ rxy ˆ 0;

…3†

where li …i ˆ 1; 2; 3† are the shear modulus of the ®ber, coating layer and matrix, respectively. Assume there is no body force, the stress components given above must satisfy the equilibrium equations or…i† or…i† or…i† xy xx ‡ ‡ zx ˆ 0; ox oy oz …i† …i† or…i† or or xy yy zy ‡ ‡ ˆ 0; ox oy oz or…i† or…i† or…i† zy zx ‡ ‡ zz ˆ 0; ox oy oz

…4†

which lead to o2 ui o2 ui ‡ 2 ˆ 0: ox2 oy

…5†

In other words, ui …x; y† must be harmonic functions of the two variables x; y in the related regions. The traction components acting on the side surface are …i† Tz…i† ˆ r…i† zx cos …n; x† ‡ rzy cos…n; y† ˆ

oui ; on

…6†

and Tx…i† ˆ Ty…i† ˆ 0;

…7†

where n is the outward normal to the side surface. We now introduce the complex functions Ui …I† of the complex variable I ˆ x ‡ iy: Ui …I† ˆ ui ‡ iwi ;

…8†

where wi …x; y† are conjugate functions to ui …x; y†which satisfy oui owi ˆ ; ox oy

oui ow ˆ ÿ i: oy ox

…9†

Obviously, Ui …I† are holomorphic in the related regions. Hence, the displacement components and the stress components can be written in terms of the complex functions as

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u…i† z ˆ ReU…I†i ;

…10†

…i† 0 r…i† zy ‡ irzx ˆ ili Ui …I†:

…11†

and

In our problem, the holomorphic complex functions are given as 1 ÿ 0
bz X a ‡ ib0k Ik ; U1 …I† ˆ 2p kˆ0 k U2 …I† ˆ

…12†

1 ÿ 00
bz X ak ‡ ib00k Ik ; 2p kˆÿ1

U3 …I† ˆ ÿ

…13†

1 ÿ 000
k ibz bz X log …I ÿ e† ‡ a ‡ ib000 k I : 2p 2p kˆÿ1 k

…14†

The boundary conditions are: At r ˆ a;

uz…1† ˆ uz…2† ; Tz…1† ˆ Tz…2† ;

…15†

At r ˆ b;

uz…2† ˆ uz…3† ; Tz…2† ˆ Tz…3† :

…16†

Using (1) and (6), the boundary conditions are rewritten as ou1 l2 ou2 ˆ ; u1 jrˆa ˆ u2 jrˆa ; or rˆa l1 or rˆa ou2 l3 ou3 ˆ : u2 jrˆb ˆ u3 jrˆb ; or rˆb l2 or rˆb

…17†

When jIj < e, we have log …I ÿ e† ˆ log e ‡ ip ÿ

1 X Ik kˆ1

kek

:

…18†

Recalling Eq. (8), and writing I as I ˆ r… cos h ‡ i sin h†, by comparing the coecient of sin kh and cos kh in both sides of (17) and (18), the above boundary conditions lead to the following linear equations 0 00 00 000 which determine a00 , a000 , a0k , a00k , a00ÿk , a000 ÿk , bk , bk , bÿk , bÿk with k ˆ 1; 2; . . . a00 ˆ a000 ; a000 ˆ p;

a0k ˆ a00k ‡ a00ÿk =a2k ;

2k a00k ‡ a00ÿk =b2k ˆ a000 ÿk =b ;

a0k ˆ


l2 ÿ 00 ak ÿ a00ÿk =a2k ; l1

a00k

a00ÿk =b2k

ÿ

b0k ˆ b00k ÿ b00ÿk =a2k ;

b0k ˆ

l ˆ ÿ 3 a000 =b2k ; l2 ÿk

2k b00k ÿ b00ÿk =b2k ˆ ÿb000 ÿk =b ‡

…19a† 1 ; kek


l2 ÿ 00 bk ‡ b00ÿk =a2k ; l1 b00k

‡

b00ÿk =b2k

  l3 000 2k 1 ˆ b =b ‡ k : ke l2 ÿk

…19b† …19c† …19d†

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Solving Eqs. (19a)±(19d), we obtain a00 ˆ a000 ˆ p; b0k ˆ b00k ˆ

a0k ˆ a00k ˆ a00ÿk ˆ a000 ÿk ˆ 0;

…20a†

4 ; kek ‰…1 ‡ a2k †…1 ‡ …l1 =l3 †† ‡ …1 ÿ a2k †……l2 =l3 † ‡ …l1 =l2 ††Š

…20b†

2…1 ‡ …l1 =l2 †† ; ‡ …l1 =l3 †† ‡ …1 ÿ a2k †……l2 =l3 † ‡ …l1 =l2 ††Š

…20c†

kek ‰…1

‡

a2k †…1

b00ÿk ˆ

2a2k …1 ÿ …l1 =l2 †† ; kek ‰…1 ‡ a2k †…1 ‡ …l1 =l3 †† ‡ …1 ÿ a2k †……l2 =l3 † ‡ …l1 =l2 ††Š

…20d†

b000 ÿk ˆ

b2k ‰…1 ‡ a2k †……l1 =l3 † ÿ 1† ‡ …1 ÿ a2k †……l2 =l3 † ÿ …l1 =l2 ††Š ; kek ‰…1 ‡ a2k †…1 ‡ …l1 =l3 †† ‡ …1 ÿ a2k †……l2 =l3 † ‡ …l1 =l2 ††Š

…20e†

where a aˆ : b

…21†

3. Stress ®eld The stress components due to this screw dislocation in the ®ber, coating layer and matrix can be calculated using (11) from (12), (13), and (14), respectively, as 1 l bz X …1† ˆÿ 1 rkÿ1 k cos …k ÿ 1†h b0k ; …22a† rzy 2p kˆ1 …1† ˆÿ rzx

…2† rzy

…2† rzx

…3† rzy

…3† rzx

1 l1 b z X rkÿ1 k sin …k ÿ 1†h b0k ; 2p kˆ1

l bz ˆ 2 2p

"

# 1 1 X X cos …k ‡ 1†h 00 kÿ1 00 r k cos…k ÿ 1†h bk ‡ kbÿk ; ÿ rk‡1 kˆ1 kˆ1

…22b†

…23a†

" # 1 1 X l2 b Z X sin …k ‡ 1†h 00 kÿ1 00 ˆÿ r k sin …k ÿ 1†h bk ‡ kbÿk ; 2p kˆ1 rk‡1 kˆ1

…23b†

" # 1 X l3 bz xÿe cos…k ‡ 1†h 000 ˆ kbÿk ; ‡ rk‡1 2p …x ÿ e†2 ‡ y 2 kˆ1

…24a†

" # 1 X l3 bz ÿy sin …k ‡ 1†h 000 ˆ kbÿk : ÿ 2p …x ÿ e†2 ‡ y 2 kˆ1 rk‡1

…24b†

It is worth noting that from Eqs. (20a)±(20e), when the phase 1 and the phase 2 materials are the same, i.e., a ®ber without coating or l1 =l2 ˆ 1, the solutions are fully reduced to those of the two-phase model

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for a screw dislocation interacting with a circular inclusion given by Dunders (1967), as discussed in Section 1. 4. Force on dislocation When Q…x; y† is on the x-axis, from (24a) and (24b) we have " # 1 X l3 bz 1 k 000 …3† …3† b ; rzx ˆ 0: ‡ rzy …x; 0† ˆ 2p x ÿ e kˆ1 xk‡1 ÿk The strain energy is computed as the work required to reject the dislocation in the materials, thus, Z R 1 …3† rzy …x; 0† dx; W ˆ bz 2 e‡r0

…25†

…26†

where R is the distance corresponding to the material size and r0 is the core radius of the dislocation. One may take R ! 1 and r0 ! 0 for all terms in the integral that converge at those limits. Evaluating (26) by use of (25), we ®nd ! 1 l3 b2z R X Ak log ‡ ; …27† W ˆ 4p r0 kˆ1 kb2k where e bˆ ; b

Ak ˆ

…1 ‡ a2k †……l1 =l3 † ÿ 1† ‡ …1 ÿ a2k †……l2 =l3 † ÿ …l1 =l2 †† : …1 ‡ a2k †…1 ‡ …l1 =l3 †† ‡ …1 ÿ a2k †……l2 =l3 † ‡ …l1 =l2 ††

…28†

The ®rst term in (27) is the strain energy associated with a screw dislocation in a homogeneous material without the inclusion, and can be discarded if W is viewed as an interaction energy. It is clear that the force on dislocation is purely radial, and is de®ned as F ˆÿ

oW 1 oW ˆÿ : oe b ob

…29†

From (27), we have F ˆ

1 l3 b2z X Ak : 2pb kˆ1 b2k‡1

…30†

5. Examples and discussion From Eq. (30), it can be seen that the sign of F is determined solely by the coecients Ak . A positive value of F indicates that the coated ®ber repels the dislocation. In other words, Ak > 0 corresponds to repulsion, while Ak < 0 implies the dislocation is attracted by the coated ®ber. Let F ˆ 0, we have a2k ˆ ÿ

…1 ÿ …l2 =l3 ††…1 ‡ …l1 =l2 †† : …1 ÿ …l1 =l2 ††…1 ‡ …l2 =l3 ††

…31†

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491

Noting that 0 < a < 1, thus from Eqs. (30) and (31), for increasing b, we have the following sets of criteria: l1 l2 > 1 and < 1; …32a† …a† l3 l3 the dislocation is ®rst attracted, then repelled by the coated inclusion; l1 l2 < 1 and > 1; …b† l3 l3 the dislocation is ®rst repelled, then attracted by the coated inclusion; l1 l2 < 1 and < 1; …c† l3 l3 the dislocation is always attracted by the coated inclusion; l1 l2 > 1 and > 1; …d† l3 l3

…32b†

…32c†

…32d†

the dislocation is always repelled by the coated inclusion. De®ning the normalized force as F ˆ F

1 2pb X Ak ˆ : 2k‡1 2 l3 bz kˆ1 b

…33†

Figs. 2±5 show the variation of F  with respect to the dislocation location parameter b ˆ e=b for the aforesaid four cases with particular materials characterized by: (a) l1 =l3 > 1 and l2 =l3 < 1, (b) l1 =l3 < 1 and l2 =l3 > 1, (c) l1 =l3 < 1 and l2 =l3 < 1, (d) l1 =l3 > 1 and l2 =l3 > 1. For comparison, the corresponding results of the two-phase model (Dunders, 1967) are also displayed. As seen in Fig. 2, there is an unstable equilibrium position for the present three-phase case, while in Fig. 3, the equilibrium position is stable. These results are totally di€erent to the two-phase model case. For the cases shown in Figs. 4 and 5, there are no equilibrium positions for both the two-phase case and the present three-phase case. These results are consistent to the discussions in Eqs. (32a)±(32d). Also, from (28) we ®nd that if a2k is small, or b=a is large (the ®ber is thickly coated), the in¯uence of the ®ber phase on the equilibrium of the dislocation is ``shielded'' by the coating phase. In other words, the larger the value of b=a, the less in¯uence the ®ber

Fig. 2. Normalized force on dislocation bz with l1 : l2 : l3 ˆ 25 : 1 : 5 and b=a ˆ 1:05.

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Z.M. Xiao, B.J. Chen / Mechanics of Materials 32 (2000) 485±494

Fig. 3. Normalized force on dislocation bz with l1 : l2 : l3 ˆ 1 : 25 : 5 and b=a ˆ 1:05.

Fig. 4. Normalized force on dislocation bz with l1 : l2 : l3 ˆ 1 : 5 : 25 and b=a ˆ 1:05.

Fig. 5. Normalized force on dislocation bz with l1 : l2 : l3 ˆ 25 : 5 : 1 and b=a ˆ 1:05.

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493

Fig. 6. Normalized force on dislocation bz with l1 : l2 : l3 ˆ 25 : 1 : 5.

Fig. 7. Normalized force on dislocation bz with l1 : l2 : l3 ˆ 1 : 25 : 5.

phase has on the dislocation. Figs. 6 and 7 show these phenomena with respect to the following two kinds of material combinations: (a) l1 : l3 : l2 ˆ 25 : 5 : 1, (b) l2 : l3 : l1 ˆ 25 : 5 : 1. As shown in the ®gures, the equilibrium positions of the dislocation vary with b=a. When b=a is large, the equilibrium position disappears, and the result tends to the two-phase case. References Christensen, R.M., 1979. Mechanics of Composite Materials. Wiley, New York. Christensen, R.M., Lo, K.H., 1979. Solution for e€ective shear properties in three-phase sphere and cylinder models. Journal of Mechanics and Physics of Solids 27, 315±330. Dunders, J., Mura, T., 1964. Interaction between an edge dislocation and a circular inclusion. Journal of Mechanics and Physics of Solids 12, 177±189. Dunders, J., 1967. On the interaction of a screw dislocation with inhomogeneities. Recent Advances in Engineering Science 2, 223±233. Dunders, J., 1969. Elastic interaction of dislocations with inhomogeneities. Mathematical Theory of Dislocations. ASME, New York, pp. 70±115.

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Z.M. Xiao, B.J. Chen / Mechanics of Materials 32 (2000) 485±494

Head, A.K., 1953a. Edge dislocations in inhomogeneous media. Proceedings of the Physical Society of London 66, 793±801. Head, A.K., 1953b. The interaction of dislocations and boundaries. Philosophical Magazine 44, 92±94. Luo, H.A., Chen, Y., 1991a. An edge dislocation in a three-phase composite cylinder model. ASME Journal of Applied Mechanics 58, 75±86. Mikata, Y., Taya, M., 1985a. Stress ®eld in and around a coated short ®ber in in®nite matrix subjected to uniaxial and biaxial loading. ASME Journal of Applied Mechanics 52, 19±24. Mikata, Y., Taya, M., 1985b. Stress ®eld in a coated continuous ®ber composite subjected to thermal±mechanical loading. Journal of Composite Materials 19, 554±578. Muskhelishvili, N.I., 1953. Some Basic Problems of the Mathematical Theory of Elasticity. Noordho€, Groningen. Walpole, L.J., 1978. A coated inclusion in an elastic medium. Mathematical Proceedings of the Cambridge Philosophical Society 88, 495±505.