A new dislocation-like model for imperfect interfaces and their effect on load transfer

Composites Part A 29A (1998) 1057–1062 1359-835X/98/$ - see front matter q 1998 Published by Elsevier Science Ltd. All rights reserved

PII: S1359-835X(98)00010-4

A new dislocation-like model for imperfect interfaces and their effect on load transfer

H.Y. Yu Naval Research Laboratory, Washington, DC 20375-5343, USA

A dislocation-like model is proposed to describe the boundary conditions of an imperfect interface. In this new model, a thin layer of interphase material is introduced near the interface. In the limit of vanishing layer thickness, the interfacial tractions become continuous, but the displacements on either side of the interphase layer become discontinuous. The jump in the displacement at the interface is described by Somigliana dislocations. The variable discontinuity of displacement across the interface is assumed to be linearly proportional to the displacement at the interface of the constituent where the elastic singularity is. The result of applying this model is equivalent to introducing two effective interfacial moduli of rigidity. Using this model, the effect of imperfect interfaces on load transfer is studied. The Green’s function is obtained for two semi-infinite solids with a planar interface. The elastic fields due to defects such as inclusions and dislocations are also given. q 1998 Published by Elsevier Science Ltd. All rights reserved (Keywords: interfacial properties; dislocation-like model; elastic field; Green’s function; dislocations)

INTRODUCTION Interfaces are features in all materials other than infinite homogeneous solids or infinite single crystals. The properties of an imperfect interface are as important as the properties of the materials themselves. For a perfect interface, the two solids are either perfectly bonded together (both the tractions and displacements are continuous at the interface) or in smooth contact with each other (allowing free tangential slip at the interface). However, the interfaces are seldom perfect and the exact theoretical modeling is very difficult. Therefore, some simplified interfacial models have been introduced to simulate the actual behavior of imperfect interfaces. For example, Van der Merwe1,2 studied the strain energy of an interface by assuming that the interface is (1) a boundary caused by a difference in atomic spacing, (2) a twist boundary, or (3) a symmetrical tilt boundary. His calculations are based on the assumptions introduced by Peierls3 and Nabarro4 in dealing with a single dislocation. He noted5 that: A range of bond strength between the two crystals is likewise cared for in terms of an interfacial rigidity modulus m. Similarly, the three-phase model6–9 introduces an interphase or a mesophase with elastic constants different from those on either side of it to represent an imperfect interface. This layer has a given thickness. Continuity of tractions and displacements is assumed at both matrix–layer and layer–reinforcement interfaces. Another popular model is the linear spring-like model10–16 in which a thin layer of interphase material is introduced near the interface. In the limit of vanishing layer

thickness, the interfacial tractions become continuous, but the displacements at either side of the interface layer become discontinuous, the jump in displacement being linearly proportional to the interfacial traction. The boundary conditions at the planar interface x 3 ¼ 0 are j3i 9 ¼ j3i , uk 9 ¹ uk ¼ kT jk3 , u3 9 ¹ u3 ¼ kN j33

(1)

where i ¼ 1, 2, 3, k ¼ 1, 2, u i and j ij are, respectively, the displacement and stress in the half-space x 3 $ 0 (solid I), ui 9 and jij 9 are, respectively, the displacement and stress in the half-space x 3 # 0 (solid II), and k T and k N are the so-called spring constant-type material parameters in the tangential and the normal directions of the interface, respectively. It follows from eqn (1) that k T ¼ k N ¼ 0 corresponds to a perfectly bonded interface, while k T ¼ k N → ` represents a completely unbonded interface. Recently, a new dislocation-like model17 has been proposed to describe mathematically the effect of an imperfect interface on the load transfer. The boundary conditions for this new model are similar to the linear spring-like model except that the jump in displacement at the interface is assumed to be linearly proportional to the displacement at the interface of the constituent where the load is applied. By using this model, the maximum shear stresses calculated at the interface18 agree qualitatively with experimental measurements19. In experimental study19, the maximum shear stress along the interface in bimaterials due to a spherical inclusion with dilatational eigenstrain is measured by a photoelastic technique. The results showed that for a

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Dislocation-like model for imperfect interfaces: H. Y. Yu surface. Any values of either parameter between zero and 1 define an imperfect interface. In the following, the term ‘ideal interface’ means that the two solids are either perfectly bonded together or completely unbonded. The jumps in displacement at the interface can also be described by Somigliana’s dislocations. In the investigation of the solution of the elastic field due to an ellipsoidal inclusion in an infinite isotropic matrix when sliding takes place along the inclusion–matrix interface, Mura and Furuhashi20 proposed that the shear stress on the inclusion–matrix interface in the perfect bonding case be relaxed by the creation of Somigliana’s dislocations b i that equal the displacement jump at the interface, i.e. bi ¼ ui ¹ ui 9 ðx3 ¼ 0Þ

Figure 1

The boundary conditions in eqn (2) are then equivalent to the creation of Somigliana’s dislocations whose components are linearly proportional to the displacement u i in solid I at the interface, i.e.

Two joined elastic half-spaces with an imperfect interface

bk ¼ (1 ¹ hT )uk and b3 ¼ (1 ¹ hN )u3 perfectly bonded interface, the maximum shear stress at the interface decreases monotonically as the distance from the inclusion increases. For an imperfect interface, the maximum interfacial shear stress first decreases, then increases to some peak value before it gradually diminishes, i.e. it shows a minimum and a maximum. Even though the three-phase model and spring-like model have been used extensively, there is still no point force solution (Green’s function) available in the literature due to the mathematical difficulties. In this study, the dislocation-like model will be introduced first, then the Green’s function and the elastic field due to inclusions and dislocations in a bimaterial with a planar imperfect interface described by the dislocation-like model will be given.

As shown in Figure 1, two joined semi-infinite isotropic elastic solids consist of solid I (x 3 $ 0) with shear modulus m, and Poisson’s ratio n, and solid II (x 3 # 0) with shear modulus m9 and Poisson’s ratio n9. The dislocation-like model is similar to the linear spring-like model except that the jump in displacement at the interface is assumed to be linearly proportional to the displacement at the interface of the constituent where the load is applied. Therefore, when the cause of deformation is in solid I, the boundary conditions at the interface x 3 ¼ 0 are (2)

where i ¼ 1, 2, 3, k ¼ 1, 2, and h T and h N are two parameters that describe the bonding conditions in the tangential and the normal directions of the interface, respectively. It is seen that h T ¼ h N ¼ 1 implies vanishing of the displacement jumps and therefore the two solids are perfectly bonded together. At the other extreme, h T ¼ h N ¼ 0 implies that the two solids are completely unbonded and x 3 ¼ 0 is a free

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(4)

where k ¼ 1, 2.

GREEN’S FUNCTION The boundary condition at the interface x 3 ¼ 0 is given in eqn (2). The point force is acting at point (xp1 , xp2 , xp3 ) in solid I. To simplify the expression, Galerkin vectors G and G9 are used to represent the elastic fields at a point (x 1,x 2,x 3) in solid I and solid II, respectively, where21 u¼

1 [2(1 ¹ n)=2 G ¹ ==·G] and 2m

u9 ¼

1 [2(1 ¹ n9)=2 G9 ¹ ==·G9] 2m9

ð5Þ

where =, =· and = 2 are, respectively, the gradient, divergence and Laplacian operators. The stresses are

DISLOCATION-LIKE MODEL

j3i 9 ¼ j3i , uk 9 ¼ hT uk , u3 9 ¼ hN u3

(3)

jij ¼ lum, m þ m(ui, j þ uj, i )

(6)

for points in solid I and jij 9 ¼ l9um, m 9 þ m9(ui, j 9 þ uj, i 9)

(7)

for points in solid II where l and l9 are, respectively, the Lame´ constants for solid I and solid II. The usual subscript notation has been used (a repeated subscript indicates summation over the values 1, 2, 3, and subscripts preceded by a comma denote differentiation with respect to the Cartesian coordinates corresponding to those subscripts). The Green’s function is obtained by using the image method for obtaining the elastic fields due to nuclei of strain in either joined half-spaces with an ideal interface or two half-spaces in frictionless contact with each other22. The elastic solution for solid I corresponding to a point force S 0 at point (xp1 , xp2 , xp3 ) is composed of S 0 itself, the image of the point force a 0S 0, and other fictitious higher-order singularities (nuclei of strain) a jS j (j ¼ 1,2,3,…,n) at the image point (xp1 , xp2 , ¹ xp3 ), where S j denotes the type of nucleus of strain, and a j the corresponding effective strength of that

Dislocation-like model for imperfect interfaces: H. Y. Yu nucleus. The solution in region II is composed of the fictitious nuclei of strain A jS j (j ¼ 0,1,2,…,n) with effective strengths A j at point (xp1 , xp2 , xp3 ) in region I. The effective strengths a j and A j are then obtained for the different boundary conditions by solving a set of simultaneous linear algebraic equations (by using the software MATHEMATICA). It is found that, for S 0 in a bimaterial with an imperfect interface described by the dislocation-like model eqn (2), the set of S j is the same as that for the joined halfspaces with ideal interface problem22. It should be noted that, since the solutions are linear superpositions of different nuclei of strain, the supplementary conditions regarding equilibrium, compatibility and the vanishing of the displacement and stress fields at infinity are automatically satisfied. The solutions also exhibit the proper signularity for a nucleus of strain at the loading point because the image and fictitious nuclei of strain for the solution for region I are located in region II, and vice versa. The results are as follows, where, without loss of generality, a multiplying constant has been omitted in the Galerkin vectors g i for a point force in the x i-direction. (a) Point force parallel to the interface, i.e. in the x i-direction where i ¼ 1, 2. p gi ¼ (R1 þ a1 R2 þ a2 xp2 3 J2 þ a3 þ a4 x3 w2 )xi

¹ [a2 xp3 R2, i

¹ a5 (xi ¹ xpi )w2 ]x3

A2 ¼ 2(1 þ k)m(mN ¹ mT )D A3 ¼

(1 þ k)D {[(1 ¹ k)(1 ¹ k9)(mT ¹ m) ¹ 2(k ¹ k9)m]mmT m þ mT ¹(1¹k)m2 mN þ[(1 ¹ 2kþkk9)mþk(1 ¹ k9)mT ]mN mT }

A4 ¼ (1 þ k)D[(1 ¹ k)(2mT ¹ m)mN ¹ (1 þ k ¹ 2k9)mmT ] A5 ¼

(1 þ k)D {[(1 ¹ k)(1 ¹ k9)(mT þ mN ¹ m) (1 þ k9)(m þ mT ) ¹ 2(k ¹ k9)m]mmT ¹ (1 ¹ k)(1 þ k9)m2 mN þ 2k(1 ¹ k9)mN m2T }

D¼

1 (m þ kmT )(mN þ k9m) þ (m þ kmN )(mT þ k9m)

k ¼ 3 ¹ 4n, k9 ¼ 3 ¹ 4n9, mT ¼ hT m9, mN ¼ hN m9 (b) Single force normal to the interface, i.e. single force in the x 3-direction g3 ¼ (R1 þ b1 R2 þ b2 c2 þ b3 xp3 w2 þ b4 xp3 x3 J2 )x3 g3 9 ¼ (B1 R1 þ b2 c1 þ B3 xp3 w1 þ B4 xp3 x3 J1 )x3

ð8Þ

p gi 9 ¼ (A1 R1 þ A2 xp2 3 J1 þ A3 c1 þ A4 x3 w1 )xi

where b1 ¼ ¹ 1 þ (1 þ k)m[2k9m þ (1 þ k9)mT þ (1 ¹ k9)mN ]D

þ [A2 xp3 R1, i þ A5 (xi ¹ xpi )w1 ]x3

b2 ¼ ¹ where x j is the unit vector in the x j-direction,

m B ¼ (1 þ k)m[(1 ¹ k)k9m ¹ (1 ¹ k9)kmN ]D mT 2

b3 ¼ {(1 ¹ k9)m[(3 ¹ k)mT þ (1 ¹ 3k)mN ]

R2, ¼ (x1 ¹ xp1 )2 þ (x2 ¹ xp2 )2 þ z2,

¹ 4(1 ¹ k)(mN mT ¹ k9m2 )}D

1 J, ¼ ; w, ¼ log[R, þ ( ¹ 1), z, ] R,

b4 ¼ a 2

c, ¼ R, ¹ ( ¹ 1), z, w, ; z, ¼ x3 þ ( ¹ 1), xp3 ; , ¼ 1, 2

B1 ¼ (1 þ k)[(1 ¹ k)m(mN þ mT ) þ 2kmN (m þ mT )]D

a1 ¼

m ¹ mT m þ mT

þ (1 ¹ k)m(mN ¹ mT )}D B 4 ¼ A2

(1 þ k)mD {2(k ¹ k9)mN mT m þ mT þ (m ¹ mT )[(1 ¹ k9)hmT þ (1 ¹ k)mN i þ (1 ¹ k)k9m]}

a4 ¼ (1 þ k)(1 ¹ k9)m(mN ¹ mT )D a5 ¼

mD {(1¹ k9)[(1þk)mT ¹(1¹k)m]mT¹2(k¹k9)mN mT mþmT ¹ (1 ¹ k)[(1 ¹ k9)(m ¹ mT )mNþ 2k9m2 ]}

A1 ¼

2(1 þ k)mT (1 þ k9)(m þ mT )

ð11Þ

B3 ¼ (1 þ k){2[(1 ¹ k)mN ¹ (1 ¹ k9)m]mT

a2 ¼ 2D[2(mN mT ¹ k9m2 ) ¹ (1 ¹ k9)m(mN þ mT )] a3 ¼

(10)

ð9Þ

The displacements and stresses due to the point forces are obtained by substituting eqns (8) and (10) into eqns (5)–(7) which are functions of (x 1,x 2,x 3), (xp1 , xp2 , xp3 ), m, n, m9; n9, hT m9 and h Nm9. This means that making the assumption described by eqn (2) is equivalent to introducing two parameters h Tm9 and m Nm9 which will be called the ‘tangential interfacial modulus of rigidity’ and the ‘radial interfacial modulus of rigidity’, respectively. It should be noted that when h T ¼ h N ¼ h, eqns (8)–(11) are the Green’s functions for ideal interfaces when the shear modulus of solid II is hm922. This means that the effect of the imperfect

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Dislocation-like model for imperfect interfaces: H. Y. Yu interface on the load transfer is equivalent to lowering the modulus of rigidity of solid II by a factor of h. However, this is true only for the elastic displacement. To calculate the stresses by Hooke’s law, eqns (6) and (7), the original elastic moduli for solids I and II must be used. When h ¼ 1 or 0, eqns (8)–(11) are the Green’s functions for two solids perfectly bonded together or completely unbonded, respectively. The boundary condition eqn (2) does not include the sliding interface, i.e. two semi-infinite solids in frictionless contact with each other. For this sliding interface, the boundary conditions are j33 9 ¼ j 33, u3 9 ¼ h Nu 3, j3k 9 ¼ j 3k ¼ 0 (k ¼ 1, 2). The Green’s functions for this interface are obtained by substituting m9 by h Nm9 in the point force solutions given previously by Yu and Sanday22 for the ideal sliding interface. The Green’s functions for two semi-infinite solids with a planar imperfect interface given in eqns (8) and (10) consist of the same nuclei of strain as for those with a perfect interface22. The only difference between them is the coefficients a i, A i, b i and B i given in eqns (9) and (11) which are independent of the coordinates. The elastic fields due to defects in two semi-infinite solids with a perfect interface have been presented by using these Green’s functions. The defects are inclusions23–26, point defects27, dislocations28,29 and disclinations30. The results of these existing studies on defects are applicable directly to the present study simply by substituting the expressions for the coefficients a i, A i, b i and B i. In the following, the methods of obtaining the elastic solutions due to inclusions, dislocations and disclinations will only be outlined, and two simple examples will be presented.

at point (x 1,x 2,x 3) in solid I and solid II, respectively, where gjk ¼

gi ¼ R 1 x i

(15)

Substituting eqn (15) into eqns (12) and (5) gives Z 1 [eT R ¹ 4(1 ¹ n)eTik J1, k ¹ 2neTmm J1, i ]dQ ui ¼ 8p(1 ¹ n) Q jk 1, ijk (16) which is the solution given by Eshelby31. The Galerkin vectors due to a homogeneous inclusion of any shape and with eigenstrain eTij in a bimaterial with an imperfect interface is obtained by substituting eqns (8), (10) and (14) into eqns (12) and (13). The elastic displacements and stresses are obtained by substituting eqns (12) and (13) into eqns (5)–(7). Thus, as an illustrative example, if the eigenstrain of the inclusion is purely dilatational, i.e. eTij ¼ «dij

(17)

where d ij is the Kronecker delta and e is a constant, then eqns (12) and (13) become Z (1 ¹ n)m« g dQ (18) G¼ 4p(1 ¹ n)(1 ¹ 2n) Q nn and (1 þ n)m« 4p(1 ¹ n)(1 ¹ 2n)

Z Q

gnm 9dQ

(19)

respectively. By substituting eqns (8)–(10) into eqn (14), the Galerkin vectors for the sum of three double forces without moment are

INCLUSIONS A basic problem in elasticity is the transformation inclusion problem. In this problem, a subdomain Q (inclusion) of a domain (matrix) undergoes a spontaneous change of shape that, without the constrained of the domain, would be some prescribed stress-free transformation strain (misfit strain, eigenstrain), eTij . The constraint imposed by the matrix engenders an elastic field that is to be determined everywhere in the matrix and the inclusion. The problem considered here is that of the homogeneous inclusion, i.e. the inclusion has the same elastic constants as the matrix. By using Eshelby’s method31, Yu and Sanday23 showed that the Galerkin vectors due to a homogeneous inclusion with eigenstrain are Z   m n eTmm gnn dQ eTjk gjk þ G¼ 4p(1 ¹ n) Q 1 ¹ 2n

(12)

Z  Q

eTjk gjk 9 þ

 n eTmm gnn 9 dQ 1 ¹ 2n

gnn ¼ ¹ [(k ¹ 1)w1 þ c1 w2 þ c2 x3 J2 ]x3

(20)

gnn 9 ¼ ¹ (c1 9w1 þ c2 9x3 J1 )x3

(21)

and

where c1 ¼ b2 ¹ b3 ¹ (k ¹ 1)a1 þ a3 þ 2a5 , c2 ¼ ¹ (b1 þ a1 þ a2 þ a5 ) c1 9 ¼ B2 ¹ B3 þ (k9 ¹ 1)A1 ¹ A3 þ 2A5 , c2 9 ¼ B 1 ¹ A 2 ¹ A 1 þ A 5 Substituting eqns (9) and (11) into eqn (22) gives c1 ¼ {(k ¹ 2)[(m ¹ mT )(mN þ k9m) þ (m ¹ mN )(mT þ k9m)] þ (k þ 1)(mN ¹ mT )m}A

and

1060

(14)

g jk and gjk 9 are the Galerkin vectors for double force (when j ¼ k) and for double force with moment (when j Þ k). From eqns (8)–(11), the Galerkin vectors g i for the point force in the x i-direction in an infinite isotropic solid is

G9 ¼

m G9 ¼ 4p(1 ¹ n)

] ] g and gjk 9 ¼ p gk 9 ]xpj k ]xj

c2 ¼ 2[(m ¹ mT )(mN þ k9m) þ (m ¹ mN )(mT þ k9m)]A (13)

c1 9 ¼ 2(k þ 1)[mN (mN þ k9m) þ (mN ¹ mT )m]A

ð22Þ

Dislocation-like model for imperfect interfaces: H. Y. Yu c2 9 ¼ 2(k þ 1)(mT ¹ mN )m The following useful relationships between Galerkin vectors, which lead to the same elastic field, were used to obtain eqns (20) and (21) from eqns (8) and (10): =F ¼ 0 when =2 F ¼ 0 =R1 ¼ ¹ (1 ¹ k9)w1 x3 when x3 # 0 =R, ¼ ( ¹ 1), (1 ¹ k)w, x3 when x3 $ 0 {[(x1 ¹ xp1 )w, ], 1 þ [(x2 ¹ xp2 )w, ], 2 }x3 ¼ [2w, ¹ ( ¹ 1), R,, 3 ]x3 where , ¼ 1, 2. The Galerkin vectors in solid I and II for a center of dilatation are 1 1 gnn and gc 9 g 9 gc ¼ 1¹k 1 ¹ k nm

(25)

respectively. Substituting eqns (18) and (19) into eqn (5), the elastic displacements due to an inclusion with pure dilatational eigenstrain are ui ¼ ¹

(1 þ n)« {[(1 ¹ k)F1, i þ (c1 þ c2 )F2, i þ c2 (x3 F2, 3 ), i ] (1 ¹ k2 )p

¹ (1 þ k)c2 F2, 3 di3 }

ð26Þ

and the disclination problems. One is to formulate the defect theory with a continuous distribution of infinitesimal defect loop densities. This could be regarded as one of the simplest approaches, because the loop densities can be arbitrarily prescribed. Furthermore, any given defect can be built up from some loop distribution. Generally, the dislocation problem is a three-dimensional problem. However, if the surface of the cut is a half-plane, then the defect is a straight line dislocation or disclination and the problem becomes two-dimensional. Consider an infinitesimal Volterra dislocation loop at point (xp1 , xp2 , xp3 ), with the surface of the cut formally used to generate the dislocation loop being dS i. The displacement vector d i which generates the defect has components di ¼ bi þ «imn qm (xpn ¹ xn 0)

(30)

where b i is the Burger’s vector of the dislocation, q m is the Frank vector of the disclination, and « imn is the permutation symbol. The disclination axis passes through the point (x1 0, x2 0, x3 0). By using Volterra’s equation32,33, the Galerkin vectors at (x 1,x 2,x 3) for the infinitesimal dislocation loop is28–30  Z m 2n g d d dS G¼ g þ gkj þ (31) 8p(1 ¹ n) S jk 1 ¹ 2n nn jk j k for x 3 $ 0 and

and (1 þ n)m« {[(c1 9 ¹ c2 9)F1, i ¹ c2 9(x3 F1, 3 ), i ] ui 9 ¼ ¹ (1 ¹ k2 )pm9 þ ( þ k9)c2 9F1, 3 d3i } for x 3 $ 0 and x 3 # 0, respectively, where i ¼ 1, 2, 3. Z (28) F, ¼ J, dQ Q

is the Newtonian potential due to the inclusion Q with unit mass density when , ¼ 1 and it is the Newtonian potential due to the mirror image of the inclusion when , ¼ 2. The Newtonian potential is well known for many different inclusion shapes. The stresses in the matrix are obtained by substituting eqns (26) and (27) into eqns (6) and (7), respectively. The stress inside the inclusion is jQij ¼ jij ¹ jTij

(29)

jTij

where is the uniform stress derived from the eigenstrain eTij by using Hooke’s law and j ij is the stress in the matrix x 3 $ 0.

DISLOCATIONS There are two types of Volterra dislocations in the theory of elasticity. A dislocation is a Volterra dislocation if the two faces of the cut formally used to generate the dislocation displace parallel to each other by a translational vector known as the Burgers vector. A disclination is a Volterra dislocation if the two faces of the cut rotate with respect to each other by a rotational vector known as the Frank’s vector. There are several ways to deal with the dislocation

 Z m 2n g9 d d dS (32) G9 ¼ g9jk þ g9kj þ 8p(1 ¹ n) S 1 ¹ 2n nn jk j k

for x 3 # 0. The elastic field due to a dislocation (line or loop) with line vector x i and Burgers vector b i (whose sign is defined by the FS/RH convention) is obtained by letting q m ¼ 0 in eqn (30) and integrating eqns (31) and (32) over the surface of the cut that generates the dislocation. The surface is a closed surface for a dislocation loop and is a half-plane for a straight dislocation. For example, the cut surface of a straight screw dislocation is a half-plane bounded by a line parallel to b i, and the cut surface for a straight edge dislocation is a half-plane bounded by a line perpendicular to b i34. For straight edge dislocations, the opposite faces of the cut can move relative to one another in the direction either parallel or normal to the plane, therefore, one can choose different cut surfaces and different relative motions to create the same edge dislocation line35 and this provides the means to double check the results. The only task involved in obtaining the elastic solution is simply to find the appropriate potentials for these planes with unit mass density. One example will serve to illustrate this approach. For a screw dislocation at point (0,0,0) with Burgers vector d j ¼ b 2 in eqn (30), and dislocation line vector y ¼ (0,1,0), let the surface of cut that generates this screw dislocation be the positive x 1 –x 2 half-plane. Integrating eqns (31) and (32) (for j ¼ 2 and k ¼ 3) over dS 3 ¼ dx 1*dx 2* from x 1* ¼ 0 to ` and from x 2* ¼ ¹ ` to ` and substituting into eqn (5), gives   b2 m ¹ mT H ¹ H (33) u1 ¼ u 3 ¼ 0 u2 ¼ ¹ 4p 1, 3 m þ mT 2, 3

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Dislocation-like model for imperfect interfaces: H. Y. Yu and u2 9 ¼ ¹

mmT b2 H u 9 ¼ u3 9 ¼ 0 2pm9(m þ mT ) 1, 3 1

(34)

where Z` Z` z J, dxp2 ¼ 2z, tan ¹ 1 , ¹ x1 log(x21 þ z2, ) þ 2x1 H, ¼ dxp1 0 ¹` x1 (35) are the Newtonian potentials due to the positive x 1 –x 2 halfplane and its mirror image half-plane with unit surface mass density for , ¼ 1 and 2, respectively.

9. 10. 11. 12. 13. 14. 15.

SUMMARY A new model has been proposed to describe an imperfect interface. The boundary conditions are that the interfacial tractions are continuous but the interfacial displacements are discontinuous. The displacements at one side of the interface are linearly proportional to the displacements at the other side where the elastic singularity is located. The proportionality constants are h T for the tangential interfacial displacement and h N for the radial interfacial displacement. The results show that the effect of the imperfect interface is equivalent to introducing two effective interfacial moduli h Tm9 and h Nm9 in the calculation of the elastic displacements in both solids. The methods for obtaining the elastic solution due to defects have been presented and examples have been given for the inclusion and the dislocation problems.

16. 17.

18.

19.

20. 21. 22.

ACKNOWLEDGEMENTS

23.

The support of the Air Force Office of Scientific Research and the Office of Naval Research for this investigation is gratefully acknowledged.

24. 25. 26.

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