Interaction Between an Edge Dislocation and a Crack

Xu Wang School of Mechanical and Power Engineering, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, China e-mail: [email protected]

Peter Schiavone1 Department of Mechanical Engineering, University of Alberta, 4-9 Mechanical Engineering Building, Edmonton, AB T6G 2G8, Canada e-mail: [email protected]

Interaction Between an Edge Dislocation and a Crack With Surface Elasticity We undertake an analytical study of the interaction of an edge dislocation with a finite crack whose faces are assumed to have separate surface elasticity. The surface elasticity on the faces of the crack is described by a version of the continuum-based surface/interface theory of Gurtin and Murdoch. By using the Green’s function method, we obtain a complete exact solution by reducing the problem to three Cauchy singular integrodifferential equations of the first-order, which are solved by means of Chebyshev polynomials and a collocation method. The correctness of the solution is rigorously verified by comparison with existing analytical solutions. Our analysis shows that the stresses and the image force acting on the edge dislocation are size-dependent and that the stresses exhibit both the logarithmic and square root singularities at the crack tips when the surface tension is neglected. [DOI: 10.1115/1.4029472] Keywords: surface elasticity, crack, edge dislocation, plane strain deformation, singular integrodifferential equation

1

Introduction

2

The elastic solution to the interaction problem of a dislocation near a crack is fundamental to the development of a general theory of crack shielding and antishielding by dislocations in a solid (see, for example, Refs. [1–6]). These classical studies are confined to the case in which the surface effects on the crack faces are ignored. When the size of a crack is measured at the nanoscale, surface stresses, tension, and energies come into play [7]. One of the most celebrated continuum-based theories of surface/interface mechanics is the model proposed by Gurtin et al. [8–10]. This model was most recently clarified and further developed by Ru [11]. A curvature-dependent surface elasticity model first proposed by Steigmann and Ogden [12] was further developed by Chhapadia et al. [13] to explain some atomistic level phenomena (e.g., the asymmetry in the elastic response of nanostructures under bending versus tension). The effects of surface stress, which is assumed to be independent of surface strain, on cracks were studied in Refs. [14–16]. The Gurtin–Murdoch surface elasticity model has been adopted in the analysis of a crack in isotropic homogenous materials and bimaterials under uniform or nonuniform remote antiplane and in-plane stresses [17–23]. In this research, the Gurtin–Murdoch surface model is incorporated into the analysis of an edge dislocation interacting with a finite crack. In our discussion, the residual surface tension is ignored due to the fact that its contribution is minimal [19,21]. The boundary value problem is formulated using a continuous distribution of both line dislocations and line forces on the crack, and is finally reduced to three first-order Cauchy singular integrodifferential equations, which can be numerically solved using a collocation method. Our analysis indicates that the size-dependent stress field at the crack tips exhibits both weak logarithmic and strong square root singularities. This observation is consistent with that made in Kim et al. [22].

Bulk and Surface Elasticity

2.1 The Bulk Elasticity. In a fixed rectangular coordinate system xi ði; j; k ¼ 1; 2; 3Þ, let ui , rij , and eij be, respectively, the displacement, stress, and strain components in an isotropic elastic bulk material. The equilibrium equation and stress–strain law are given by rij;j ¼ 0;

rij ¼ 2leij þ kekk dij ;

1 eij ¼ ðui;j þ uj;i Þ 2

(1)

where k and l are the Lame constants and dij is the Kronecker delta. For plane strain deformations of an isotropic elastic material, the nontrivial stresses, displacements, and stress functions (u1 ; u2 ) can be expressed in terms of two analytic functions /(z) and wðzÞ of the complex variable z ¼ x1 þ ix2 as [24] h i r11 þ r22 ¼ 2 /0 ðzÞ þ /0 ðzÞ r22  r11 þ 2ir12 ¼ 2½z/00 ðzÞ þ w0 ðzÞ 2lðu1 þ iu2 Þ ¼ j/ðzÞ  z/0 ðzÞ  wðzÞ h i u1 þ iu2 ¼ i /ðzÞ þ z/0 ðzÞ þ wðzÞ

(2)

where j ¼ ðk þ 3lÞ=ðk þ lÞ ¼ 3  4 with ð0    1=2Þ being the Poisson’s ratio. In addition, the stresses are related to the stress functions through [25] r11 ¼ u1;2 ;

r12 ¼ u1;1

r21 ¼ u2;2 ;

r22 ¼ u2;1

(3)

2.2 The Surface Elasticity. The equilibrium conditions on the surface incorporating interface/surface elasticity can be expressed as [8–11] 1 Corresponding author. Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received December 16, 2014; final manuscript received December 22, 2014; accepted manuscript posted December 31, 2014; published online January 7, 2015. Assoc. Editor: Pradeep Sharma.

Journal of Applied Mechanics

½raj nj ea  þ rsab;b ea ¼ 0; ½rij ni nj  ¼

C 2015 by ASME Copyright V

rsab jab ;

ðtangential directionÞ

ðnormal directionÞ

(4)

FEBRUARY 2015, Vol. 82 / 021006-1

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 10/04/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use

where a, b ¼ 1,3; ni is the unit normal vector to the surface, ½ denotes the jump of the quantities across the surface, rsab is the surface stress tensor, and jab is the curvature tensor of the surface. In addition, the constitutive equations on the isotropic surface are given by rsab ¼ r0 dab þ 2ðls  r0 Þesab þ ðks þ r0 Þescc dab þ r0 rs u

(5)

/ðzÞ ¼

1 2pðj þ 1Þ

ða

f2l½b2 ðnÞ  ib1 ðnÞ  ½f1 ðnÞ þ if2 ðnÞg

a

 lnðz  nÞdn þ wðzÞ ¼

1 2pðj þ 1Þ

ða

lðby  ibx Þ lnðz  z0 Þ pðj þ 1Þ

f2l½b2 ðnÞ þ ib1 ðnÞ þ j½f1 ðnÞ  if2 ðnÞg

a

 lnðz  nÞdn ða 1 2ln½b2 ðnÞ  ib1 ðnÞ  n½f1 ðnÞ þ if2 ðnÞ dn  2pðj þ 1Þ a zn

where esab is the surface strain tensor, r0 is the surface tension, ks and ls are the two surface Lame parameters, and rs is the surface gradient. In the present study, the residual surface tension r0 is taken to be zero due to the fact that its contribution to the system is minimal [21].

þ

lðby þ ibx Þ lðby  ibx Þ z0 lnðz  z0 Þ  pðj þ 1Þ pðj þ 1Þ z  z0

3 An Edge Dislocation Near a Crack With Surface Elasticity

(11)

Consider the plane strain deformations of a linearly elastic and homogeneous isotropic solid containing an isolated finite crack, the cross section of which occupies the region ½a; a of the real axis. Tractions on the crack faces are zero, i.e., r12 ¼ r22 ¼ 0 on a < x1 < a and x2 ¼ 60. An edge dislocation with Burgers vector bx þ iby is located at z ¼ z0 , and there exists no other external loading. Let the upper (x2 > 0) and lower (x2 < 0) half-planes be designated the “þ” and “” sides of the crack, respectively. It follows from Eq. (4) that the boundary conditions on the crack faces can be written as rs11;1 þ ðr12 Þþ  ðr12 Þ ¼ 0 ðr22 Þþ  ðr22 Þ ¼ 0

on the upper crack face

(6a)

rs11;1 þ ðr12 Þþ  ðr12 Þ ¼ 0 ðr22 Þþ  ðr22 Þ ¼ 0

on the lower crack face

(6b)

where ðr12 Þ ; ðr22 Þ in Eq. (6a) and ðr12 Þþ ; ðr22 Þþ in Eq. (6b) are zero. By assuming a coherent interface (esab ¼ eab ) and r0 ¼ 0, Eq. (5) can then reduce to rs11 ¼ ðks þ 2ls Þu1;1

(7)

Consequently, it follows from Eqs. (6) and (7) that the surface conditions on the crack faces can be given as follows: ðr12 Þþ ¼ ðks þ 2ls Þuþ 1;11 ðr22 Þþ ¼ 0

on the upper crack face

(8)

ðr12 Þ ¼ ðks þ 2ls Þu 1;11 ðr22 Þ ¼ 0

on the lower crack face

(9)

Through satisfaction of the boundary conditions in Eq. (10), we can finally obtain the following hypersingular integrodifferential equations: ð 1 a b1 ðnÞ ðj þ 1Þðks þ 2ls Þ 0 jþ1 0 b1 ðx1 Þ þ r dn ¼  p a n  x1 4l 2l 12 f2 ðx1 Þ ¼ 0 (12) ð b2 ðnÞ j  1 a f1 ðnÞ dn  dn þ 2r022 ¼ 0 pðj þ 1Þ a n  x1 a n  x1 ð ðks þ 2ls Þðj  1Þ a b2 ðnÞ dn f1 ðx1 Þ ¼  2 pðj þ 1Þ a ðn  x1 Þ ð ðks þ 2ls Þj a f1 ðnÞ dn þ 2ðks þ 2ls Þu01;11 þ plðj þ 1Þ a ðn  x1 Þ2 (13) 

4l pðj þ 1Þ

ða

where ( )   bx þ iby 2lbx 1 2lImfz0 g  Im ¼ Re pðj þ 1Þ x1  z0 pðj þ 1Þ ðx1  z0 Þ2 ( )   2lby bx þ iby 1 2lImfz0 g 0 Re  Re r22 ¼ x1  z0 pðj þ 1Þ pðj þ 1Þ ðx1  z0 Þ2     by  ibx by þ ibx j 1  Re Re u01;1 ¼ 2pðj þ 1Þ 2pðj þ 1Þ x1  z0 x1  z0 ( ) bx þ iby Imfz0 g þ Re pðj þ 1Þ ðx1  z0 Þ2 ( ) ( ) by  ibx by þ ibx j 1 0 Re Re þ u1;11 ¼  2pðj þ 1Þ 2pðj þ 1Þ ðx1  z0 Þ2 ðx1  z0 Þ2 ( ) 2Imfz0 g bx þ iby Re  pðj þ 1Þ ðx1  z0 Þ3 r012

which are equivalent to

(14) By making use of Eq. (13)1, Eq. (13)2 can be equivalently written in the form

 ðr12 Þþ þ ðr12 Þ ¼ ðks þ 2ls Þðuþ 1;11  u1;11 Þ  ðr12 Þþ  ðr12 Þ ¼ ðks þ 2ls Þðuþ 1;11 þ u1;11 Þ þ

(10)



ðr22 Þ ¼ ðr22 Þ ¼ 0 The formulation of the problem can be achieved by considering a distribution of glide and climb edge dislocations with densities b1 ðx1 Þ and b2 ðx1 Þ, and horizontal and vertical line forces with densities f1 ðx1 Þ and f2 ðx1 Þ on the crack. Thus, the two analytic functions /ðzÞ and wðzÞ take the following form: 021006-2 / Vol. 82, FEBRUARY 2015

1 p

ða

f1 ðnÞ 4l dn  s n  x ðj þ 1Þðk þ 2ls Þ 1 a 2ðj  1Þ 0 8l 0 r  u ¼ j þ 1 22 j þ 1 1;1

ð x1

f1 ðnÞdn

a

(15)

In addition, the following conditions can be derived using Eq. (11) Transactions of the ASME

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 10/04/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use

 Du1 ¼ uþ 1  u1 ¼ 

ð x1 a

 b1 ðnÞdn; Du2 ¼ uþ 2  u2 ¼ 

ð x1

b2 ðnÞdn

a

 þ  rþ 12  r12 ¼ f1 ðx1 Þ; r22  r22 ¼ f2 ðx1 Þ  0;  a < x1 < a

(16) Thus, the single valuedness of the displacements and balance of force for a contour surrounding the crack surface require that ða ða ða b1 ðnÞdn ¼ 0; b2 ðnÞdn ¼ 0; f1 ðnÞdn ¼ 0 (17) a

a

a

Now the original boundary value problem has been reduced to three Cauchy singular integrodifferential equations in Eqs. (12)1, (13)1, and (15) together with the three auxiliary conditions in Eq. (17). Due to the assumption of the residual surface tension being zero (r0 ¼ 0), we have f2 ðx1 Þ  0 on the crack.

4

Solution to the Singular Integrodifferential Equations

Set x ¼ x1 =a and t ¼ n=a in Eqs. (12)1, (13)1, (15), and (17). For convenience, we write b1 ðxÞ ¼ b1 ðaxÞ, b2 ðxÞ ¼ b2 ðaxÞ, and f1 ðxÞ ¼ f1 ðaxÞ. Consequently, Eqs. (12)1, (13)1, (15), and (17) can be expressed into the following normalized form:   ð1 ^ 1 b1 ðtÞ dt ¼ pSe b^01 ðxÞ þ cos wRe x  ^z0 1 t  x ( ) iw e  Imf^z0 gIm ;  1 < x < 1 (18) ðx  ^z0 Þ2 ð ð 1 1 f^1 ðtÞ 1 x ^ dt  f1 ðtÞdt p 1 t  x Se 1 ( )   cos w 1 Imf^z0 g eiw ¼  Im Re ; 1 > > > = cos w < 1

Im ¼ > > ip p > :cos ; þ ^z0 > N 8 9 > > > > > > < = Imf^z0 g eiw Re 

 2 ; i ¼ 1; 2; …; N  1 > > p > > cos ip þ ^z0 > > : ; N   N X d0 cos w 1 Im mdm ¼   pþ p 1  ^z0 Se m¼1 ( ) Imf^z0 g eiw Re  ; d0 ¼ 0 p ð1  ^z0 Þ2 N X

(pffiffiffiffiffiffiffiffiffiffiffiffiffi)!  1 ^z20  1 b^1 ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi cos w 1 þ Re x  ^z0 p 1  x2 ( ) eiw ð^z0 x  1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi  Imf^z0 gIm ðx  ^z0 Þ2 ^z20  1 (pffiffiffiffiffiffiffiffiffiffiffiffiffi)!  1 ^z20  1 ^ b2 ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi sin w 1 þ Re x  ^z0 p 1  x2 ( ) eiw ð^z0 x  1Þ p ffiffiffiffiffiffiffiffiffiffiffiffi ffi ; f^1 ðxÞ ¼ 0  Imf^z0 gRe ðx  ^z0 Þ2 ^z20  1

By letting Se ! 0 in Eqs. (19), (23), and (31), the result in Eq. (37) can be recovered. On the other extreme, when Se ! 1, the boundary conditions on the crack surfaces become  þ  rþ 22 ¼ r22 ¼ u1;1 ¼ u1;1 ¼ 0

(38)

In this case, the exact solution can be finally derived as lðby  ibx Þ 1 pðj þ 1Þ z  z0 " # pffiffiffiffiffiffiffiffiffiffiffiffiffiffi lðby  ibx Þ 1 1 z20  a2 ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pðj þ 1Þ z  z0 z2  a2 z2  a2 ðz  z0 Þ

/0 ðzÞ 

" # pffiffiffiffiffiffiffiffiffiffiffiffiffiffi lðby þ ibx Þ 1 z20  a2 1   pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pðj þ 1Þ z  z0 z2  a2 ðz  z0 Þ z2  a2

(35) 021006-4 / Vol. 82, FEBRUARY 2015

(37)

(39) Transactions of the ASME

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 10/04/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use

lðby þ ibx Þ 1 2lðbx þ iby Þ Imfz0 g þ pðj þ 1Þ z  z0 pðj þ 1Þ ðz  z0 Þ2 " # pffiffiffiffiffiffiffiffiffiffiffiffiffiffi lðby þ ibx Þ z20  a2 1 1  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pðj þ 1Þ z  z0 z2  a2 ðz  z0 Þ z2  a2 " # pffiffiffiffiffiffiffiffiffiffiffiffiffiffi lðby  ibx Þ 1 1 z20  a2   pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pðj þ 1Þ z  z0 z2  a2 z2  a2 ðz  z0 Þ

w0 ðzÞ þ /0 ðzÞ þ z/00 ðzÞ 



r11

r22

ffi pffiffiffiffiffiffiffiffiffiffiffiffiffi z20  a2

lImfz0 gðbx þ iby Þ 1 1  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðj þ 1Þ ðz  z0 Þ2 z2  a2 ðz  z0 Þ2

 lImfz0 gðbx  iby Þ z0 ffi þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 pðj þ 1Þ z0  a ðz  z0 Þ " !# pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 z0 z20  a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi   þ ðz  z0 Þ2 z20  a2 ðz  z0 Þ z2  a2 ðz  z0 Þ2 þ

ða 2l b1 ðnÞ dn þ r012 pðj þ 1Þ a n  x1 ða ða 2l b2 ðnÞ jþ3 f1 ðnÞ ¼ dn þ dn pðj þ 1Þ a n  x1 2pðj þ 1Þ a n  x1   by  ibx 4l  r022 Re þ pðj þ 1Þ z  z0 ða ða 2l b2 ðnÞ j1 f1 ðnÞ ¼ dn  dn þ r022 pðj þ 1Þ a n  x1 2pðj þ 1Þ a n  x1 x1 < a or x > a (43)

r12 ¼ 

ðr12 Þ ðr22 Þþ

(40)

ðr11 Þþ It is further derived from Eqs. (39) and (40) that b^1 ðxÞ ¼ 0

(pffiffiffiffiffiffiffiffiffiffiffiffiffi)! 1 ^z20  1 sin w 1 þ Re C B C x  ^z0 B C B C B (pffiffiffiffiffiffiffiffiffiffiffiffiffi) C B 2 1 ^z0  1 C B ðj  1Þ cos w Im b^2 ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi B þ C C jþ1 x  ^z0 p 1  x2 B C B B ( )C C B 2Imf^z g iw e ð^z0 x  1Þ A @ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi  Re 2 jþ1 ðx  ^z0 Þ ^z20  1 " (pffiffiffiffiffiffiffiffiffiffiffiffiffi) 1 ^z20  1 ^ ffiffiffiffiffiffiffiffiffiffiffiffi ffi p f1 ðxÞ ¼  cos wIm x  ^z0 p 1  x2 ( )# eiw ð^z0 x  1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi þ Imf^z0 gRe (41) ðx  ^z0 Þ2 ^z20  1 0

It is easily verified that b^2 ðxÞ and f^1 ðxÞ in Eq. (41) satisfy the relationship in Eq. (23). Remember that Eq. (23) is valid for any value of Se . In addition, f^1 ðxÞ obtained from Eq. (19) for Se ! 1 just recovers the corresponding result in Eq. (41). Thus, the correctness of the solution is analytically verified for the two extreme cases Se ¼ 0 and Se ! 1.

6

The Size-Dependent Stress Field

The stress field can be obtained by substituting the following expressions of /0 ðzÞ and w0 ðzÞ into Eq. (2): ða 1 2l½b2 ðnÞ  ib1 ðnÞ  f1 ðnÞ /0 ðzÞ ¼ dn 2pðj þ 1Þ a zn lðby  ibx Þ 1 pðj þ 1Þ z  z0 ða 1 2l½b2 ðnÞ þ ib1 ðnÞ þ jf1 ðnÞ dn w0 ðzÞ ¼ 2pðj þ 1Þ a zn ða 1 2ln½b2 ðnÞ  ib1 ðnÞ  nf1 ðnÞ þ dn 2pðj þ 1Þ a ðz  nÞ2 þ

þ

lðby þ ibx Þ 1 lðby  ibx Þ z0 þ pðj þ 1Þ z  z0 pðj þ 1Þ ðz  z0 Þ2

(42)

In particular, the stresses are distributed along the real axis as follows: Journal of Applied Mechanics

ks þ 2ls 0 1 b1 ðx1 Þ  f1 ðx1 Þ 2 2 ks þ 2ls 0 1 ¼ b1 ðx1 Þ þ f1 ðx1 Þ 2 2 ¼ ðr22 Þ ¼ 0 ða 4lb1 ðx1 Þ 2 2lb2 ðnÞ  f1 ðnÞ þ ¼ dn jþ1 pðj þ 1Þ a x1  n   by  ibx 4l Re þ pðj þ 1Þ x1  z0 ða 4lb1 ðx1 Þ 2 2lb2 ðnÞ  f1 ðnÞ þ ¼ dn jþ1 pðj þ 1Þ a x1  n   by  ibx 4l ; a < x1 < a Re þ pðj þ 1Þ x1  z0

ðr12 Þþ ¼

ðr11 Þ

(44)

In view of the fact that b2 ðx1 Þ and f1 ðx1 Þ exhibit the square root singularity at the crack tips whereas b1 ðx1 Þ is finite at the crack tips, it is simply deduced from Eq. (42) that the stress components exhibit both the square root and logarithmic singularities at the crack tips. This observation is in agreement with that by Kim et al. [22]. It is observed from Eq. (44) that on the real axis, r12 exhibits the logarithmic singularity outside the crack and exhibits the square root singularity on the crack faces; r22 exhibits the square root singularity outside the crack and is zero on the crack faces; r11 exhibits the square root singularity outside the crack and on the crack faces. Once the two densities b1 ðxÞ and f1 ðxÞ are determined (b2 ðxÞ can be obtained by using Eq. (23)), the stress field can then be arrived at by using the two analytic functions /0 ðzÞ and w0 ðzÞ given by Eq. (42). In addition, the image force acting on the edge dislocation can be arrived at by using the Peach–Koehler formula [26] once the stress field is known. For example, the image force acting on a glide edge dislocation located on the real axis (w ¼ 0 and Imf^z0 g ¼ 0) can be derived as F1 ¼

2lb2 paðj þ 1Þ

ð1 ^ b1 ðtÞ dt; z0  t 1 ^

F2 ¼ 0

(45)

where F1 and F2 are, respectively, the force components along the x1 and x2 directions. Since the surface parameter Se is controlled by the crack size (see Eq. (22)), the induced stress field and image force acting on the edge dislocation are then both dependent on the crack size. The above results indicate that the interaction problem is completely solved once b1 ðxÞ and f1 ðxÞ have been determined. In Sec. 7, b1 ðxÞ and f1 ðxÞ will be numerically determined.

7

Numerical Results

We illustrate in Figs. 1–3 the distributions of b1 ðxÞ, Ðx Du1 ¼ a 1 b1 ðtÞdt and f1 ðxÞ for five different values of Se ¼ 0:001; 0:1; 0:5; 1; 1000 with w ¼ 0 and ^z0 ¼ i=2. b1 ðxÞ for Se ¼ 0:001 in Fig. 1 has been very close to the exact solution in FEBRUARY 2015, Vol. 82 / 021006-5

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 10/04/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Fig. 1 The distribution of b1 ðx Þ for different values of Se with w 5 0 and z^0 5 i=2

Fig. 4 The distribution of b1 ðx Þ for different values of Se with w 5 p=2 and z^0 5 i=2

Fig. 2 The distribution of Du1 for different values of Se with w 5 0 and z^0 5 i=2

Fig. 5 The distribution of Du1 for different values of Se with w 5 p=2 and z^0 5 i=2

Fig. 3 The distribution of f1 ðx Þ for different values of Se with w 5 0 and z^0 5 i=2

Eq. (37)1 for Se ¼ 0; f1 ðxÞ for Se ¼ 1000 in Fig. 3 is also very close to the exact solution in Eq. (41)3 for Se ! 1. b1 ð1Þ ¼ b1 ð1Þ for Se ¼ 0:1; 0:5; 1; 1000 in Fig. 1 are apparently finitely valued. As illustrated in Fig. 2, both the magnitude of Du1 at a fixed value of x and the crack-tip opening angle are decreasing functions of Se . This observation is in agreement with that by 021006-6 / Vol. 82, FEBRUARY 2015

Antipov and Schiavone [27] for a mode III finite crack in a strip. It is observed from Fig. 3 that a nonzero value of Se will cause f1 ðxÞ 6¼ 0. Even when Se ¼ 0:001  0, f1 ðxÞ at the points extremely very close to x ¼ 61 are apparently nonzero. When w ¼ 0 (i.e., by ¼ 0) and the dislocation is on the x2 -axis, b1 ðxÞ is an even function of x, whilst Du1 and f1 ðxÞ are odd functions of x. Illustrated in Figs. 4–6 are the distributions of b1 ðxÞ, Du1 and f1 ðxÞ for five different values of Se ¼ 0:001; 0:1; 0:5; 1; 1000 with w ¼ p=2 and ^z0 ¼ i=2. Again, b1 ðxÞ for Se ¼ 0:001 in Fig. 4 is approaching the exact solution in Eq. (37)1 for Se ¼ 0; f1 ðxÞ for Se ¼ 1000 in Fig. 6 is approaching the exact solution in Eq. (41)3 for Se ! 1. b1 ð1Þ ¼ b1 ð1Þ for Se ¼ 0:1; 0:5; 1; 1000 in Fig. 4 are apparently finitely valued. Both the magnitude of Du1 at a fixed value of x and the crack-tip opening angle are decreasing functions of Se , as shown in Fig. 5. The result in Fig. 5 also shows that a nonzero value of Se will induce f1 ðxÞ 6¼ 0. When w ¼ p=2 (i.e., bx ¼ 0) and the dislocation is on the x2 -axis, b1 ðxÞ is an odd function of x, while Du1 and f1 ðxÞ are even functions of x. We illustrate in Fig. 7 the variation of Du1 for five different values of ^z0 ¼ 1:0001; 1:05; 1:2; 2; 10 with w ¼ 0 and Se ¼ 1. As the glide edge dislocation (by ¼ 0) on the positive real axis moves closer to the right crack tip, the magnitude of the induced Du1 enlarges and meanwhile the crack-tip opening angles at the two crack tips also increase, with the opening angle at the right crack tip increasing more rapidly. The result in Fig. 7 suggests that the surface elasticity can effectively suppresses the magnitude of Du1 , especially when the dislocation is very close to the crack tip. Transactions of the ASME

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 10/04/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use

effects decrease the magnitude of the attractive force, as observed in Fig. 8. The image force for Se ¼ 0:001 in Fig. 8 matches quite well the following exact solution for Se ¼ 0 obtained by Zhang and Li [4] F1 ¼ 

2lb2 1 h pffiffiffiffiffiffiffiffiffiffiffiffiffii ; paðj þ 1Þ ð^z2  1Þ ^z0 þ sgnð^z0 Þ ^z2  1 0

j^z0 j > 1

0

(46) Thus, the correctness of the solution has also been numerically verified in this section.

8

Fig. 6 The distribution of f1 ðx Þ for different values of Se with w 5 p=2 and z^0 5 i=2

Conclusions

We analytically examine the contribution of Gurtin–Murdoch surface elasticity to the elastic interaction between an edge dislocation and a finite crack. We obtain a full-field analytical solution via three Cauchy singular integrodifferential equations in Eqs. (18)–(20), which are solved numerically by using a modified collocation method. The correctness of the obtained solution is carefully verified by comparison with existing classical solutions for the two extreme cases Se ¼ 0 and Se ! 1, both analytically (in Sec. 5) and numerically (in Sec. 7). The method can be used to analyze other pointwise singularities, such as a line force, concentrated moment, and a circular Eshelby’s inclusion, interacting with a finite crack with surface elasticity. The corresponding interaction problem in the absence of the surface elasticity has been discussed by Suo [28].

Acknowledgment This work was supported by the National Natural Science Foundation of China (Grant No. 11272121) and through a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.

References Fig. 7 The variation of Du1 for different values of z^0 with w 5 0 and Se 5 1

Fig. 8 The image force on a glide edge dislocation located on the positive real axis for different values of Se

Finally, Fig. 8 shows the image force acting on a glide edge dislocation located on the positive real axis outside the crack for four different values of Se ¼ 0:001; 0:1; 0:5; 1. The size-dependency of the image force can be clearly seen from Fig. 8. The surface Journal of Applied Mechanics

[1] Majumdar, B. S., and Burns, S. J., 1981, “Crack Tip Shielding—An Elastic Theory of Dislocations and Dislocation Arrays Near a Sharp Crack,” Acta Metall., 29(4), pp. 579–588. [2] Lin, I. H., and Thomson, R., 1986, “Cleavage, Dislocation Emission and Shielding for Cracks Under General Loading,” Acta Metall., 34(2), pp. 187–206. [3] Zhang, T. Y., and Li, J. C. M., 1991, “Image Forces and Shielding Effects of a Screw Dislocation Near a Finite-Length Crack,” Mater. Sci. Eng. A, 142(1), pp. 35–39. [4] Zhang, T. Y., and Li, J. C. M., 1991, “Image Forces and Shielding Effects of an Edge Dislocation Near a Finite Length Crack,” Acta Metall., 39(11), pp. 2739–2744. [5] Lee, K. Y., Lee, W. G., and Pak, Y. E., 2000, “Interaction Between a SemiInfinite Crack and a Screw Dislocation in a Piezoelectric Material,” ASME J. Appl. Mech., 67(1), pp. 165–170. [6] Kwon, J. H., and Lee, K. Y., 2002, “Electromechanical Effects of a Screw Dislocation Around a Finite Crack in a Piezoelectric Material,” ASME J. Appl. Mech., 69(1), pp. 55–62. [7] Sharma, P., and Ganti, S., 2004, “Size-Dependent Eshelby’s Tensor for Embedded Nano-Inclusions Incorporating Surface/Interface Energies,” ASME J. Appl. Mech., 71(5), pp. 663–671. [8] Gurtin, M. E., and Murdoch, A. I., 1975, “A Continuum Theory of Elastic Material Surfaces,” Arch. Ration. Mech. Anal., 57(4), pp. 291–323. [9] Gurtin, M. E., and Murdoch, A. I., 1978, “Surface Stress in Solids,” Int. J. Solids Struct., 14(6), pp. 431–440. [10] Gurtin, M. E., Weissmuller, J., and Larche, F., 1998, “A General Theory of Curved Deformable Interface in Solids at Equilibrium,” Philos. Mag. A, 78(5), pp. 1093–1109. [11] Ru, C. Q., 2010, “Simple Geometrical Explanation of Gurtin–Murdoch Model of Surface Elasticity With Clarification of Its Related Versions,” Sci. China, 53(3), pp. 536–544. [12] Steigmann, D. J., and Ogden, R. W., 1999, “Elastic Surface Substrate Interactions,” Proc. R. Soc. London, Ser. A, 455 (1982), pp. 437–474. [13] Chhapadia, P., Mohammadia, P., and Sharma, P., 2011, “Curvature-Dependent Surface Energy and Implications for Nanostructures,” J. Mech. Phys. Solids, 59(10), pp. 2103–2115. [14] Wu, C. H., 1999, “The Effect of Surface Stress on the Configurational Equilibrium of Voids and Cracks,” J. Mech. Phys. Solids, 47(12), pp. 2469–2492. [15] Wu, C. H., and Wang, M. L., 2000, “The Effect of Crack-Tip Point Loads on Fracture,” J. Mech. Phys. Solids, 48(11), pp. 2283–2296.

FEBRUARY 2015, Vol. 82 / 021006-7

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 10/04/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use

[16] Wu, C. H., and Wang, M. L., 2001, “Configurational Equilibrium of Circular-Arc Cracks With Surface Stress,” Int. J. Solids Struct., 38(24–25), pp. 4279–4292. [17] Kim, C. I., Schiavone, P., and Ru, C. Q., 2010, “The Effects of Surface Elasticity on an Elastic Solid With Mode-III Crack: Complete Solution,” ASME J. Appl. Mech., 77(2), p. 021011. [18] Kim, C. I., Schiavone, P., and Ru, C. Q., 2010, “Analysis of a Mode III Crack in the Presence of Surface Elasticity and a Prescribed Non-Uniform Surface Traction,” Z. Angew. Math. Phys., 61(3), pp. 555–564. [19] Kim, C. I., Schiavone, P., and Ru, C. Q., 2011, “Analysis of Plane-Strain Crack Problems (Mode I and Mode II) in the Presence of Surface Elasticity,” J. Elasticity, 104(1–2), pp. 397–420. [20] Kim, C. I., Schiavone, P., and Ru, C. Q., 2011, “The Effects of Surface Elasticity on Mode-III Interface Crack,” Arch. Mech., 63(3), pp. 267–286. [21] Kim, C. I., Schiavone, P., and Ru, C. Q., 2011, “Effect of Surface Elasticity on an Interface Crack in Plane Deformations,” Proc. R. Soc. London, Ser. A, 467(2174), pp. 3530–3549.

021006-8 / Vol. 82, FEBRUARY 2015

[22] Kim, C. I., Ru, C. Q., and Schiavone, P., 2013, “A Clarification of the Role of Crack-Tip Conditions in Linear Elasticity With Surface Effects,” Math. Mech. Solids, 18(1), pp. 59–66. [23] Wang, X., 2014, “A Mode III Arc Shaped Crack With Surface Elasticity,” Z. Angew. Math. Phys. (in press). [24] Muskhelishvili, N. I., 1953, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen, The Netherlands. [25] Ting, T. C. T., 1996, Anisotropic Elasticity—Theory and Applications, Oxford University Press, New York. [26] Dundurs, J., 1969, “Elastic Interaction of Dislocations With Inhomogeneities,” Mathematical Theory of Dislocations, T. Mura, ed., ASME, New York, pp. 70–115. [27] Antipov, Y. A., and Schiavone, P., 2011, “Integro-Differential Equation for a Finite Crack in a Strip With Surface Effects,” Q. J. Mech. Appl. Math., 64(1), pp. 87–106. [28] Suo, Z., 1989, “Singularities Interacting With Interfaces and Cracks,” Int. J. Solids Struct., 25(10), pp. 1133–1142.

Transactions of the ASME

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 10/04/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use