A 6 c 7 + A 7 c , BiJc^c5-c3+3A4A5c6)B^ ^ 3 = 0 , ( ^ 1 , 2 , 3 . : . ) , (2.32) dx +KMC2 +d2 + 3A 4 A 5 c \/ 3 F = - A 6 r 0 + 2 A 6 ( c 1 A 8 - c 4 ) ^ - 2 ( 2 A 5 A 6 + c , ) \/ \/ , V f = Q 2 ( A 6 c 7 + A 7 c , ) ' 2 ' ^ 3 = ^ . ( \" C ' + C 2 7 4 ^ 5 ^ ' ^ - = 0 , (\"=1,2,3...), (2.33) \/e, a, + KMC2 +d2+ 3 A 4 A 5 a 3 G , = - A 6 (- c}+c2-c3+ 3 A 4 A 5 c 6 )fl _ B] 3A 6 (- e,+c2 - c 4 - c5)ff y dx +KMC2 +d2 + 3A 4 A 5 c \/ 3 \/r, rf, + \/vMc 2 + rf2 + 3A 4 A 5 rf 3 G n + , =0, (w=l,2,3...), (2.34) where A 6 = ^ i \/ ^ M \u00bb A 7 = ( ^ i - 1 ) \/ 2 , A 8 = ( * r M - l ) \/ 2 , c , = l + 2A, , c 2 = A 6 ( l + 3A,), c3 = A 6 ( l - A , - A 6 ) \/ * r , , c 4 = A 2 ( A , + A 3 ) - l , c5 = A 6 [ l - A 2 ( A , + A 3 ) + 2A, + i r M A , ] , c 6 = A 6 ( A 6 - l ) \/ \/ c , , c 7 = l + A 4 A 5 , rf, =1 + 4A, +A 2 (A , + A 3 ) , d2 = A 6 [ l + A, + A 2 ( A , + A3) +KMA6]\/^ , rf3=A6(l + \/ c M A 6 ) \/ \/ c I . (2.35) The complete elastic field is explicitly given by Eqs. (2.1)-(2.6) together with Eqs. (2.23), (2.24) and (2.30)-(2.35). After some manipulation, the stresses along the interface are: In the matrix: 2\/4-.B, (5-35 3 )cos20 ( T e 0 = \u2014 + \" \u00bb 20 2 A + B, (-4A, - 5 + 35 3) cos 20 CTRR= L + - ! 3- , Ro Ro (-2 A, + B.+ 3B3) sin 20 O\"r0 = \u201e \u00bb U - i 0 ) ^0 In the inhomogeneity: _ 2F, | (12F3+G|)cos2.9 _ 2FX G,cos2r9 ^ i?0 \/?0 rr RQ R0 _^ ________ ( 2 3 7 ) Note that due to the surface\/interface stress effects, A , ( A 3 , A 5 ) * 0 and r\u00b0 * 0 , and the stress state depends on the size of the inhomogeneity R0 as the coefficients A \\, B\\, 5 3 , F\\, F), and G\\ in Eqs. (2.31)-(2.34) are nonlinear functions of R0. When R0 is quite larger than the intrinsic scales, these parameters A , , A 3 and A 5 are very closed to zero, and the surface effects will only depend on constant surface stress r \u00b0 . When R0 is comparable to the intrinsic length, A , , A 3 and A 5 are large and the surface effects would be important. When no surface\/interface stresses exist, i.e., A, = A 3 = A 5 = 0 and r\u00b0 = 0 , the above results reduce to the classical solution in which the elastic state is size-independent. The shear and radial stresses are equal on either side of the interface in the classical case and, A 6 ( l + O ( l - r f ) c r o s i n 2 0 2(1 + K M A 6 ) _ = A 6 ( l + A g ) ( l + flr)o-0-4\/v?' A 6 ( l + y M )(l-fl> 0 cos26> 2 ( A 6 + A 7 ) 2(l + * - M A 7 ) Hoop stress on the interface are different and given by, In the matrix: a = ( A 6 + 2 A 7 - A 6 A g ) ( l + <0o-0+4;\/,g* | (4 -3A 6 +\/c M A 6 ) ( l -^)cr 0 cos26? 2 ( A 6 + A 7 ) 2(1 + * M A 6 ) t K ' } In the inhomogeneity: _ = ( A 6 + A 6 A g ) ( l + rf)o-n-4\/vr* ( A6(\\ + KM){\\-d)oocos20 2(A 6 + A 7 ) 2{\\ + KMA,) 21 As can be seen from Eqs. (2.38)-(2.40), the stress state is independent of RQ in the classical solution. Note that the above solution reduces to the special case that the inhomogeneity is a hole by setting the shear modulus of the inhomogeneity \/\/, to be zero. 2.3 Numerical Results for Elastic Field Around a Circular Hole Selected numerical results are presented in this and next section based on the surface elastic constants obtained from past studies. Experiments have been performed to determine the surface stress which has an order of 1 N\/m [14,44,45,etc], but the surface elastic constants are difficult to be measured and no results are available at present. The embedded atom method was used by Miller and Shenoy [24] and Shenoy [25] to determine the surface elastic constants. Their results indicated that the surface elastic constants depend on the material type and the surface crystal orientation as shown in Table 2.1 for isotropic surface. Although the surface properties are generally anisotropic, it is assumed that isotropic case is sufficient to illustrate the main features of the size-dependent response. The plane strain case in which the surface effect is represented by the parameters A^ s and r\u00b0 is investigated in the numerical study without loss of any generality. While the following numerical results are not to provide very accurate values, the study is to show the main behavior with surface effects. In the calculations, unless specified otherwise, KS = \u00b1 1 0 N \/ m , (As + r \u00b0 ) = \u00b110N\/m and r\u00b0 is between - lN\/m and lN\/m. Table 2.1 Surface elastic constants. Units are N\/m. Surface r\u00b0 KS A s + r \u00b0 A l [100] -5.4251 3.4939 0.5689 -7.9253 4.0628 A l [111] -0.3760 6.8511 0.9108 5.1882 7.7619 N i [111] -0.6729 -1.8585 -0.1153 -3.0730 -1.9738 In this section, an infinite plane of aluminum containing a circular hole under far-field loading is considered. The bulk elastic constants for aluminum are: X_ = 58.17 GPa , MM =26.13 GPa [46]. 22 Based on the analysis presented in the previous section, hoop stress (plane strain) at 9 = 0 on hole surface is given by: Note that the first term is the classical elasticity result and the last two terms represent the surface stress effects and contain non-linear terms of R0. The first two terms are linear with respect to the loading magnitude while the third term is independent of external loading and linear with respect to the residual surface stress r \u00b0 . When d=\\, i.e., under radially symmetric loading, the result is the same as that obtained by Sharma and Ganti [32]. Following the classical definition, a stress concentration factor can be defined for a hole by normalizing hoop stress at 0 = 0 by the remote loading magnitude when T\u00b0 = 0. The effect of the surface elastic constant, Ks = 2\/us +AS - r \u00b0 , is first studied by setting r\u00b0 = 0. In this case based on Eq. (2.41), hoop stress concentration factor ( ) - ^ ) ^ i a J (3-1) am-xam =jXz) + Jw-[zf(z) + y,Xz)}i2ia, (3.2) where t is the unit tangent, and n is the outward unit normal at the interface which in complex form is e i a (where a is the angle between the normal direction n and the positive x\\-axis). Assume that there is perfect bonding at the inhomogeneity-matrix interface, then the displacements are continuous at the interface: k + l ^ L = (Un+\u2122, ),+(\"\u201e 0 n r > (3-3) where the last term is the displacement induced by the prescribed uniform dilatational eigenstrain e*, i.e., \u00a3\u2022*, =e*22 = s*, and: {un+iu,X =ze'e-'a, on T . (3.4) The surface traction on the inhomogeneity-matrix interface is discontinuous due to the surface stress effect. For isotropic bulk and interface, the field equations and constitutive relations based on the theory proposed by Gurtin and Murdoch [26] and Gurtin et al. [27] are given in Eqs. (2.8)-(2.11). In the (\u00ab, t, x 3) coordinates [X3 is the 34 direction perpendicular to the (n, r)-plane], the equilibrium equations (2.9) and (2.10) can be written as: On the surface\/interface r : In\/-direction: |x\u201eB, J + ^ - + - ^ - = 0 , (3.5) dt dx. In ^-direction: ] + ^ 2 L + ^ L = 0 , (3.6) dx3 dt Indirection: [ o r \u201e B J = ^ , (3.7) * o where R0 is the curvature radius. For plane problems, cr\u201eB3 = cr3, = af3 = 0 and the derivatives with respect to x3 are zero. Thus Eq. (3.6) is automatically satisfied. Eqs. (3.5) and (3.7) can be expressed in the following complex variable form: k . - ^ , J = 4 + i ^ - 0-8) The left-hand side of Eq. (3.8) can be obtained from Eq. (3.2). For the right-hand side, the surface normal stress in the tangential direction is: al =T\u00b0 + 2<7\/ s -r\u00b0K,+(A s +r\u00b0)(*3 3+*\u201e), on T . (3.9) Special attention is required when calculating the strain \u00a333 at the interface, because the strain at either side of the interface can be different even though the displacement is continuous as assumed. Thus the interface has associated with it two interface stresses. Here, the average of the two interface stresses is taken as this is consistent with the case of a spherical inhomogeneity in which the interface stress is continuous and only one interface stress or the average of the two is used in Eq. (3.8) [32]. In the case of a hole, there is only one interface stress, or more exactly, the surface stress, and only this stress appears in the right-hand side of Eq. (3.8). Tangential elastic stain at the surface en can be obtained from the following equations: etl+em=\u00a3u+e22=^\\z) + Vdj:|d;| = v ^ . (3.16) The complex potentials 0M(\u00a3), ^ , (\u00a3 ) , y M ( \u00a3 ) a n a \" V1O5) corresponding to the matrix and inhomogeneity are now expanded into the following Laurent series form.