Full-field analysis of a planar anisotropic
layered half-plane for concentrated
forces and edge dislocations
B y C h i e n-C h i n g M a a n d R u-L i L i nDepartment of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China
Received 6 March 2001; accepted 26 February 2002; published online 8 August 2002
The problem of a planar anisotropic elastic layered half-plane subjected to concen-trated forces and edge dislocations applied either in the layer or in the half-plane is analysed. One of the objectives of this study is to develop an effective analytical methodology to construct the exact full-field solution for this problem. By using the Lekhnitskii formalism for anisotropic elastic material with the Fourier-transformation technique, the explicit closed-form solutions for stresses in the layer and the half-plane are obtained. The solutions are suitable for loadings that are acting on the free surface or at the interface. The complete solutions for this problem consist only of the simplest solutions obtained from an infinite homogeneous medium with con-centrated forces and edge dislocations. The solutions include Green’s function for applied loadings in an infinite medium and an infinite number of image singularities that are induced to satisfy the boundary and interface conditions. It is shown that the physical meaning of the solution is the image method. The magnitudes and locations of image singularities are determined automatically from the mathematical method presented in this study. Numerical results for the full-field stress distribution in the layered half-plane medium subjected to concentrated forces or edge dislocations are discussed in detail.
Keywords: anisotropic elastic material; layered half-plane; Fourier transformation; image method
1. Introduction
Among the formalisms for two-dimensional linear anisotropic elasticity, the Lekhnit-skii formalism (LekhnitLekhnit-skii 1963) and the Stroh formalism (Stroh 1958) are the two most widely used methods. The Lekhnitskii formalism begins with the stresses and can be regarded as the generalization of the Airy stress function to anisotropic elas-ticity. The Stroh formalism can be traced to the work of Eshelby et al . (1953) and starts with the displacements. For a two-dimensional planar problem, the applied loadings are usually considered in two different types: concentrated forces and edge dislocations. In many applications, the elastic field due to loadings applied in the interior of the material is made up of dislocations, which are defects in the mate-rial. In order to understand the motion of a dislocation in an elastic material, one needs to calculate the stress distribution induced by the dislocation. For homoge-neous elastic material in an unbounded medium, the elastic field around dislocations Proc. R. Soc. Lond. A (2002) 458, 2369–2392
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or concentrated loadings is generally well known. In most engineering applications, the geometrical configurations all have finite boundaries. Thus the analysis of elastic fields for material with a finite boundary is necessary. Because of the complexity of the problem with a finite boundary, there are few analytical results available in the literature. On the other hand, the layered half-plane subjected to surface traction is applicable in a number of engineering fields. For example, the coating of a thin layer to protect soft matrices under contact and friction is a particular case of the problem. Furthermore, a wide range of electronic components are now manufactured by depositing semiconducting layers on supporting substrates. To develop our under-standing of the characteristics of these devices, a detailed knowledge of the behaviour of dislocations in thin-film structures is needed. In this study, the solutions of lay-ered half-plane media with a traction-free boundary and a perfectly bonded interface subjected to concentrated forces or edge dislocations in the interior of the material are presented. The Lekhnitskii formalism will be employed in conjunction with the Fourier-transform technique to obtain analytic explicit solutions.
Many research ventures start by addressing problems in a semi-infinite plane sub-jected to concentrated forces or dislocations, the simplest example of a medium with a boundary, for the first step of analysis. Head (1953) solved the problem for an isotropic half-plane. Ting & Barnett (1993) discussed the relation between the image force on a dislocation and a half-plane with a fixed boundary for an anisotropic material. Further work, leading to a bimaterial which consists of two per-fectly bound semi-infinite materials, has been done by Dundurs & Sendeckyj (1965) for isotropic material and by Barnett & Lothe (1974), Suo (1990) and Ting (1992) for the anisotropic case. The more advanced and complicated problem is supplied by a loading in an infinite strip or in an infinite layer bonded on a semi-infinite sub-strate. Buchwald (1964) showed that the solution of an isotropic infinite strip with Fourier integrals consisted of classic beam stretching and bending solutions from the singularities of the integrands. Nabarro & Kostlan (1978) and Moss & Hoover (1978) solved the problem of an edge dislocation in an isotropic strip with free sur-faces simultaneously. A similar problem of an orthotropic strip was completed by Chou (1963). Stagni & Lizzio (1986) obtained the elastic field of an isotropic infi-nite strip by means of the complex-variable method. They constructed the solution by using Muskhelishvili’s (1975) complex potentials with the aid of Laurent’s series expansions. Tullini & Savoia (1999) solved the interior problem of an orthotropic strip subjected to a given continuous distribution of normal and shear loads. Wu & Chiu (1995) and Chiu & Wu (1998) obtained the solution of an anisotropic infinite strip subjected to dislocations and concentrated forces from eigenfunction expan-sion by residue theory, respectively. The stress distribution of an edge dislocation in an isotropic layered half-plane was investigated by Weeks et al . (1968) and Lee & Dundurs (1973).
For anisotropic layered half-plane media, the problem becomes more complicated than that of their isotropic counterparts, due to the presence of many elastic con-stants. The goal of this study is to construct Green’s function for planar anisotropic layered half-plane media subjected to concentrated forces or edge dislocations in the interior. We will show that the solution obtained from the mathematical method pro-posed in this study has its own physical meaning, the method of images. The method of images is a technique that uses a simple fundamental solution in an infinite plane to construct the solution for other more complicated problems.
The use of the image method in solving two-dimensional isotropic problems deal-ing with screw dislocations is well known and has been used successfully for simple cases of multi-layered structures. It was found that an infinite number of image screw dislocations were required. The concept of semi-reflection and semi-transmission mir-rors were used to determine the magnitudes and locations of the image screw dis-locations without solving the boundary-value problem (Chou 1966; Kamat et al . 1987; ¨Ove¸co˘glu et al . 1987). The multiple-image problem for screw dislocations then becomes a combinatorial problem of counting reflections and transmissions for a given image path length. Basically, one can extend this methodology to include any number of layers, but as the number of layers increases, obtaining an explicit solution becomes extremely labourious and time consuming. Hence the work was not extended beyond the five-layer case (with three finite lengths) by Kamat et al . (1987) and ¨Ove¸co˘glu
et al . (1987). By using the Fourier-transform technique and a series expansion, an
effective analytical methodology was developed by Lin & Ma (2000) to construct explicit analytical solutions for an anisotropic multi-layered medium with n layers (the thickness in each layer is different), due to a screw dislocation in an arbitrary layer. The mathematical approach proposed by Lin & Ma (2000) is indeed the image method that provides an automatic determination for the magnitudes and locations of all the image screw dislocations. This is the most general case for the problem of screw dislocations in multi-layered media. For the isotropic edge-dislocation case, however, the locations of the images are independent of the material property and are the same as those for isotropic screw dislocations, but the image singularities on the image point are more complicated than the anti-plane one (Ma & Lin 2001). Due to the complexity and difficulty of the edge-dislocation problem, the solutions for a strip obtained by Stagni & Lizzio (1986) cannot be easily represented in terms of an infinite series, but must be expressed in terms of recursion formulae for the unknown quantities. On the other hand, for an anisotropic half-plane problem, the image point is not on the opposite side of the applied point. In general, the locations of image singularities for an anisotropic half-plane are different and depend on the anisotropic elastic constants (Ting 1996). In the planar anisotropic half-plane prob-lem, only five image points are required; four points for the in-plane problem and one point for the anti-plane problem. However, it is difficult to use the conventional image method to obtain the solution of the anisotropic layered half-plane problem directly.
In this study, an anisotropic layer perfectly bonded to an anisotropic dissimilar half-plane (as shown in figure 1) and subjected to in-plane concentrated forces and edge dislocations applied either in the thin layer or in the half-plane is investigated. The main objective is to develop an effective mathematical method to construct ana-lytical full-field solutions for this problem. The solution procedure follows Lekhnit-skii’s formalism and the Fourier transform. The analytical solutions for the stresses obtained in this study are exact and are expressed in an explicit closed form. The complete solutions consist only of the simplest solutions for an infinite homoge-neous medium with concentrated forces and edge dislocations. It can be shown that the physical meaning of the solution obtained in this study is the image method. The magnitudes and locations of image singularities will be determined automat-ically from the mathematical method presented in the study. For the special case when the thickness of the layer is infinitely large, the image singularities obtained by Ting (1992) for anisotropic bimaterials are recovered. The given formulae will
allow a thorough assessment of the significance of anisotropy in many areas of study. Numerical results for the full-field stress distribution based on the analytical solutions are presented.
2. Material constants and governing equations of a planar anisotropic material
The equilibrium equations for the elastostatic problem in the absence of body forces are
σij,j= 0, (2.1)
where the repeated indices imply summation and a subscript comma denotes differ-entiation. The generalized Hooke’s law for an anisotropic, homogeneous and linearly elastic solid is given by
σij= Cijklεkl, (2.2)
where εkl denotes the infinitesimal strain tensor and the Cijkl are the elasticity constants, which are assumed to be fully symmetric and positive-definite.
Introducing the contracted notation, the stress–strain law in (2.2) can be written in a matrix form as σxx σyy σzz σyz σxz σxy = C11 C12 C13 C14 C15 C16 C22 C23 C24 C25 C26 C33 C34 C35 C36 C44 C45 C46 sym. C55 C56 C66 εxx εyy εzz εyz εxz εxy (2.3) or εxx εyy εzz εyz εxz εxy = a11 a12 a13 a14 a15 a16 a22 a23 a24 a25 a26 a33 a34 a35 a36 a44 a45 a46 sym. a55 a56 a66 σxx σyy σzz σyz σxz σxy , (2.4)
where Cijand aij are the elastic stiffnesses and elastic compliances of an anisotropic material, respectively.
For a general anisotropic material, the in-plane and anti-plane deformations are coupled. When the compliance matrix (or stiffness matrix) of an anisotropic material possesses the form
[aij] = a11 a12 a13 0 0 a16 a22 a23 0 0 a26 a33 a34 a35 a36 a44 a45 0 sym. a55 0 a66 , (2.5)
then the in-plane and anti-plane deformation will be uncoupled. The problem associ-ated with the in-plane problem is known as the planar anisotropic problem. In fact,
monoclinic materials with the symmetry plane at z = 0 not only satisfy the condition of (2.5), but also require a34= a35 = 0.
For a two-dimensional planar anisotropic problem, the stress–strain relations are only dependent on six independent material constants and can be presented in the matrix form as follows. For plane-stress problems,
εεxxyy εxy = aa1112 aa1222 aa1626 a16 a26 a66 σσxxyy σxy (2.6)
and σzz = σxz= σyz= 0. For plane-strain problems, εεxxyy εxy = ss1112 ss1222 ss1626 s16 s26 s66 σσxxyy σxy (2.7)
and εzz = εxz = εyz = 0, where the sij are the reduced elastic compliances. The relationship between sij and aij is
sij = aij−
ai3aj3
a33
, i, j = 1, 2, 6. (2.8)
In many applications, the choice of the coordinate is very often dictated by the boundary conditions of the problem and hence may not coincide with the symme-try planes of the material. Therefore, transformation of the elastic compliances to a different coordinate system is necessary. For the special case in which the transfor-mation is a rotation, the z-axis with an angle θ, the elastic compliances in the new coordinate system (x, y, z) can be presented as
s= (T−1)TsT−1, (2.9) where T = cos 2θ sin2θ sin 2θ
sin2θ cos2θ − sin 2θ
− cos θ sin θ cos θ sin θ cos2θ− sin2
θ
.
For a two-dimensional problem, the in-plane stresses can be represented by the Airy stress function Φ(x, y) as follows:
σxx = ∂2Φ ∂y2, σyy= ∂2Φ ∂x2, σxy =− ∂2Φ ∂x∂y. (2.10)
In order to satisfy the compatibility equation, the governing equation of planar anisotropy in the absence of body forces can be obtained as
s22 ∂4Φ ∂x4 − 2s26 ∂4Φ ∂x3∂y+ (2s12+ s66) ∂4Φ ∂x2∂y2 − 2s16 ∂4Φ ∂x∂y3 + s11 ∂4Φ ∂y4 = 0 (2.11)
for plane strain and
a22 ∂4Φ ∂x4 − 2a26 ∂4Φ ∂x3∂y + (2a12+ a66) ∂4Φ ∂x2∂y2 − 2a16 ∂4Φ ∂x∂y3 + a11 ∂4Φ ∂y4 = 0 (2.12)
for plane stress. The two governing equations are similar, and the difference between (2.11) and (2.12) is the material constants.
3. Green’s function of an anisotropic infinite plane
Consider an anisotropic infinite space subjected to a line of uniformly distributed forces f per unit length applied on the z-axis and in the direction that is normal to the z-axis. In addition, there is a line dislocation at the z-axis, with Burgers’s vector b normal to the z-axis. This is a two-dimensional planar anisotropic problem. For the plane-strain problem, the governing equation is shown in (2.11). The expression for the field variables will be found by applying a Fourier transform over the spatial coordinate x with parameter ω. Take the Fourier-transform pairs of a function g(x, y), defined as ˜ g(ω, y) = ∞ −∞g(x, y)e −iωxdx, g(x, y) = 1 2π ∞ −∞g(ω, y)e˜ iωx dω, (3.1) and apply to the governing equation (2.11). Then (2.11) becomes an ordinary differ-ential equation, s22ω4+ 2s26iω3 d ˜Φ dy − (2s12+ s66)ω 2d2Φ˜ dy2 − 2s16iω d3Φ˜ dy3 + s11 d4Φ˜ dy4 = 0, (3.2)
with the general solutions ˜ Φ(ω, y) = 4 j=1 cjeςjωy, (3.3)
where cj are unknown constants to be determined and ςjare the roots of the following characteristic equation:
s22+ 2is26ς− (2s12+ s66)ς2− 2is16ς3+ s11ς4 = 0. (3.4)
There are four roots for ς, all of which are complex numbers. The magnitudes ςj are related to the eigenvalues of the elastic constant pj of the Lekhnitskii or Stroh formalism as
ςj = ipj, j = 1, 2, 3, 4,
where pk= αk+ iβk, pk+2= ¯pk, k = 1, 2, and αk and βk are real, with βk> 0. The general solutions for stresses and displacements of planar anisotropic material in the Fourier-transform domain are represented in a compact matrix form as
˜ σ ˜ u = Peiωp∗yc, (3.5) where ˜ σ = 1 ω2 ˜ σyy ˜ σxy , u =˜ 1 iω ˜ ux ˜ uy , P = −1 −1 −1 −1 p1 p2 p3 p4 ξ(p1) ξ(p2) ξ(p3) ξ(p4) η(p1) η(p2) η(p3) η(p4) , c = c1 c2 c3 c4
andeiωp∗y is the diagonal matrix, i.e.
in which ux and uy are displacement components in the x- and y-axis, respectively, and ξ(pj) = s11p2j + s12− s16pj, η(pj) = s12pj+ s22 pj − s26.
Consider the loadings f and b which are applied at (x, y) = (0, d) in an infinite plane. The jump conditions for applied forces and dislocations are
˜ σ ˜ u y=d+ − ˜ σ ˜ u y=d− = −f −(1/iω)b , (3.6) where f = fy fx and b = bx by .
Substituting (3.5) into (3.6), we get c+− c−= 1 ω2e −iωp∗dq, (3.7) where q = q1 q2 q3 q4 = P−1 −f b .
The vector q can be regarded as the strength of the applied loadings for anisotropic infinite plane rather than the force f and dislocation b.
After taking the inverse Fourier transform, Green’s functions of stresses and dis-placements for the infinite plane are
σyy σxy = 1 πRe − iq1 x + p1(y− d) − iq2 x + p2(y− d) ip1q1 x + p1(y− d) + ip2q2 x + p2(y− d) = 1 πRe 2 j=1 −iqj x + pj(y− d) 2 j=1 ipjqj x + pj(y− d) , ux uy =−1 πRe ξ(p3)q1ln(x + p1(y− d)) + ξ(p4)q2ln(x + p2(y− d)) η(p3)q1ln(x + p1(y− d)) + η(p4)q2ln(x + p2(y− d)) , (3.8)
where Re denotes the real part. The stresses display a singularity of order 1/r as the distance r from the applied loadings tends to zero. In the next section, we will show that Green’s functions for the layered half-plane derived in this study are composed of the fundamental solutions of an infinite plane as presented in (3.8).
1 2 x y f b d h half-plane layer
Figure 1. Configuration and coordinate system of an anisotropic layered half-plane.
4. Green’s function for an anisotropic layered half-plane
Consider a two-dimensional problem of an anisotropic layered half-plane subjected to a concentrated force and an edge dislocation, as shown in figure 1. Assume the thin layer with thickness h is occupied by material 1 and is perfectly bonded to the half-plane occupied by material 2. The x-axis is taken to be the surface of the thin layer. The thin layer occupies 0 < y < h, while the underlying half-plane (the substrate) occupies h < y <∞. A concentrated force f and an edge dislocation with Burgers’s vector b are applied at (x, y) = (0, d), d > 0, which may be located in the thin layer, the half-plane, the free surface or the interface of the layered half-plane. In each case, it is required to evaluate the stresses arising in both the thin layer and the half-plane. From the general solution presented in (3.5), the solutions in the Fourier-transform domain for materials 1 (thin layer) and 2 (half-plane) can be expressed as ˜ σ(1) ˜ u(1) = P(1)eiωp(1)∗ yc(1), (4.1) ˜ σ(2) ˜ u(2) = P(2)eiωp(2)∗ yc(2). (4.2)
The superscripts (1) and (2) on the field quantities are employed to label materials 1 and 2, respectively. The boundary conditions on the top surface of the thin layer are traction free, thus
˜
σ(1)(ω, 0) = 0. In terms of c(1)j from (4.1), we have
c(1)1 + c(1)2 + c(1)3 + c(1)4 = 0, (4.3)
p(1)1 c(1)1 + p(1)2 c(1)2 + p(1)3 c(1)3 + p(1)4 c(1)4 = 0. (4.4) The continuity conditions of stresses and displacements at the interface y = h are
˜ σ(1) ˜ u(1) = ˜ σ(2) ˜ u(2) at y = h. (4.5)
We rewrite (4.5) in terms of c(1) and c(2) as
P(1)eiωp(1)∗ hc(1)= P(2)eiωp(2)∗ hc(2). (4.6)
The bounded solution of stresses and displacements of the half-plane requires that
c(2)3 = c(2)4 = 0. (4.7)
The complete solutions for the applied loadings in the interior of thin layer and in the half-plane are different. Thus the problems are considered for two situations.
1+ x y f b 1− d
Figure 2. The jump conditions for applied loadings at the layer.
(a) The loadings are applied in the thin layer (0 d < h)
When the loadings are applied in the thin layer, the jump conditions for the thin layer are employed,
˜ σ(1) ˜ u(1) y=d+ − ˜ σ(1) ˜ u(1) y=d− = −f −(1/iω)b . (4.8)
The associated configuration of the jump condition is shown in figure 2. When we express (4.8) in terms of c(1), we have
(c(1))+− (c(1))−= 1
ω2e
−iωp(1)
∗ dq(1), (4.9)
where superscripts ‘+’ and ‘−’ indicate the undetermined constant c(1) in (4.1) for
0 y d and d y h, respectively, and q(1)= (P(1))−1 −f b .
With the aid of boundary conditions (4.3), (4.4) and (4.7), continuity conditions (4.6) and by omitting the algebraic derivation, the undetermined constants c(1)j and
c(2)k can be expressed explicitly and are summarized as
c(1)j = 2 k=1 bjkc(2)k eiω(p(2)k −p (1) j )h, (4.10) c(2)k = 1 ω2 (−1)k det| ∗ | 4 l=1 4 m=1 p−lmbm(3−k)ql(1)e−iω((p(1)m+p (2) k )h+p (1) l d), (4.11) where det| ∗ | = 3 j=1 4 k=j+1 p−kjBjke−iωp + jkh, Bjk = bj1bk2− bj2bk1, bij= 4 k=1 [Pik(1)]−1[Pkj(2)], p±ij= p(1)i ± p(1)j .
The explicit solutions of stress functions in the Fourier-transform domain for the layered half-plane are
˜ Φ(1)= 4 j=1 c(1)j eip(1)j ωy, (4.12) ˜ Φ(2)= 2 k=1 c(2)k eip(2)k ωy. (4.13)
Because the denominators in (4.10) and (4.11) are so complicated, it is not possible to obtain the inverse Fourier transform directly. In order to obtain the explicit analytical full-field solutions for the inverse Fourier transform, the denominator det| ∗ | in c(1)j and c(2)k should be reformulated. By examining the structure of det| ∗ |, the term with magnitude p−21B12e−ip
+
12ωhis taken out from det| ∗ |, so that
1 det| ∗ | = 1 p−21B12e−ip + 12ωh(1− det | · |) , (4.14) where det| · | = 1 − det| ∗ | p−21B12e−ip + 12ωh .
It can be shown that det| · | < 1 for ω > 0. By the expansion of (4.14) into a power series of det| · |, we obtain
1 p−21B12e−ip + 12ωh(1− det | · |) = e ip+12ωh p−21B12 ∞ n=0 (det| · |)n.
Because only five terms are included in det| · |, ∞n=0(det| · |)n can be expressed explicitly as eip+12ωh p−21B12 ∞ n=0 (det| · |)n=M e−iωgh, (4.15) where M = υ (p−21B12)n+1 (p−13B13)n1(p23−B23)n2(p−14B14)n3(p−24B24)n4(p−34B34)n5, (4.16) g = n1p+13+ n2p+23+ n3p+14+ n4p+24+ n5p+34− (n + 1)p+12, (4.17) = ∞ n=0 n n1=0 n−n1 n2=0 n−n1−n2 n3=0 n−n1−n2−n3 n4=0 , (4.18) n5 = n− n1− n2− n3− n4, (4.19) υ = n! n1!n2!n3!n4!n5! , (4.20)
and ‘!’ denotes factorial. By omitting the details of the derivation, the solutions for the stress functions in (4.12) and (4.13) can be expressed in the explicit forms as
follows: ˜ Φ(1)= 1 ω2 4 j=1 2 k=1 4 l=1 4 m=1 (−1)kbjkbm(3−k)p−lmq(1)l M eiω(p(1)j y−(p (1) j +p (1) m+g)h−p (1) l d), (4.21) ˜ Φ(2)= 1 ω2 2 k=1 4 l=1 4 m=1 (−1)kbm(3−k)p−lmql(1)M eiω(p(2)k y−(p (1) m+p (2) k +g)h−p (1) l d). (4.22) Now, the solutions for ˜Φ(1) and ˜Φ(2) are linear combinations of exponential
func-tions, i.e. MeiωG; each term represents Green’s function in the transform domain
for concentrated loadings in an infinite homogeneous medium. The term with G in the exponential functions indicates the location of the loading, andM represents the magnitude of the loading. The location of the loading depends on the thickness of the thin layer h, the position of applied loadings d and eigenvalues pj. The magnitude of
M only depends on material constants. Thus four terms in (4.21) are fundamental
solutions of an infinite plane subjected to concentrated forces and edge dislocations. All the remaining terms in (4.21) and all the terms in (4.22) are image singularities that are induced to satisfy the boundary and interface continuity conditions. Hence this method is referred to as the generalization method of images.
Since the solutions in the transformed domain expressed in (4.21) and (4.22) are exponential functions of ω, only the following inverse Fourier transformation is required, 1 2π ∞ −∞ eiω(py+H)eiωxdω = 1 πRe i x + py + H ,
where H is an arbitrary constant. Therefore, the inverse Fourier transformations for (4.21) and (4.22) can easily be obtained term by term. Finally, the explicit closed-form solutions of stresses for the layered half-plane are presented as follows,
σyy(j) σxy(j) σxx(j) = ψ (j)| α=0 ψ(j)| α=1 ψ(j)|α=2 , j = 1, 2, (4.23) where ψ(1)(x, y) = 1 π ∞ n=0 n n1=0 n−n1 n2=0 n−n1−n2 n3=0 n−n1−n2−n3 n4=0 4 j=1 4 l=1 4 m=1 Re i(−1)αBjmp−lmql(1)M (p(1)j )α x + p(1)j y− (p+jm+ g)h− p(1)l d (4.24 a) = 1 π 2 j=1 Re i(−1)α+1qj(1)(p(1)j )α x + p(1)j (y− d) (4.24 b) + 1 π 2 j=1 4 l=3 Re p− l(3−j) p−21 i(−1)j+αq(1) l (p (1) j )α x + (p(1)j y− p(1)l d) (4.24 c)
+ 1 π 4 j=1 4 l=1 4 m=1 (1− δ(j+m)3) Re i(−1)αB jmp−lmq (1) l M (p (1) j )α x + p(1)j y− (p+jm+ g)h− p(1)l d (4.24 d) + 1 π 4 j=1 4 l=1 4 m=1 Re i(−1)αBjmp−lmql(1)M (p(1)j )α x + p(1)j y− (p+jm+ g)h− p(1)l d (4.24 e) and ψ(2)(x, y) = 1 π ∞ n=0 n n1=0 n−n1 n2=0 n−n1−n2 n3=0 n−n1−n2−n3 n4=0 2 k=1 4 l=1 4 m=1 Re i(−1)k+αb m(3−k)p−mlq (1) l M (p (2) k ) α x + p(2)k (y− h) − (p(1)m + g)h− p(1)l d (4.25 a) = 1 π 2 k=1 2 m=1 Re 1 B12 i(−1)k+α+mb m(3−k)q3(1)−m(p(2)k )α x + p(2)k (y− h) − (p(1)m − p+12)h− p (1) 3−md (4.25 b) + 1 π 2 k=1 4 l=1 4 m=1 (1− δ(l+m)3) × Re i(−1)k+αb m(3−k)p−mlq (1) l M (p (2) k )α x + p(2)k (y− h) − (p(1)m + g)h− p(1)l d (4.25 c) + 1 π 2 k=1 4 l=1 4 m=1 Re i(−1)k+αb m(3−k)p−mlq (1) l M (p (2) k )α x + p(2)k (y− h) − (p(1)m + g)h− p(1)l d , (4.25 d) in which = ∞ n=1 n n1=0 n−n1 n2=0 n−n1−n2 n3=0 n−n1−n2−n3 n4=0
and δjk is Kronecker’s delta. It is noted that the whole-field solutions of stresses in the layered half-plane (ψ(1)(x, y) for the thin layer and ψ(2)(x, y) for the half-plane) can be explicitly expressed with compact forms, which are presented in (4.24 a) and (4.25 a). There are eight summation operators in ψ(1) and ψ(2), but only one summation operator has infinite summation numbers, i.e.∞n=0in ψ(1)and ψ(2). The infinite summation operator controls the convergence of the solution. Therefore, it is necessary to add all the terms in the infinite summation operator for the numerical calculation.
In order to explain the physical meaning of the terms in (4.24 a), the terms with
n = 0 in the solution are taken out and are expressed in (4.24 b)–(4.24 d). If the
thickness of the layer is infinitely large (h→ ∞), the layered half-plane solution will reduce to the half-plane solution, all the terms in (4.24 d) and (4.24 e) will be zero and only the terms in (4.24 b) and (4.24 c) will be left. The term in (4.24 b) with one summation is the same as that presented in (3.8) and is the solution for an infinite
plane with a concentrated force f and an edge dislocation b applied at the location (0, d). The terms in (4.24 c) with two summations are the image singularities that satisfy the free-surface condition of the top surface. The locations of these image singularities are determined by solving the equation x + p(1)j y− p(1)l d = 0 (j = 1, 2
and l = 3, 4) for x and y. There are four image singularities, which are located outside the layered plane and the results are identical to those obtained for the half-plane problem by Ting (1992). The image singularity at each image point consists of a concentrated force and an edge dislocation. It can be verified that all the terms in (4.24 d) and (4.24 e) are the multiple images of the free surface and interface of the layered half-plane. The locations of all the image singularities can be determined by solving the equation x + p(1)j y− (p+jm+ g)h− p(1)l d = 0 and are dependent on d, h and the material properties of the thin layer (i.e. material 1).
Similarly, the complete solutions for the half-plane are presented in (4.25 a), and are rewritten in an alternative form, as presented in (4.25 b)–(4.25 d) by taking n = 0 out of (4.25 a). The terms in (4.25 b) represent the solutions for the bimaterials that consist of two anisotropic half-planes bonded together along the interface. The solu-tion in (4.25 b) consists of the first four image singularities related to the interface. The locations of these image singularities can be determined by setting the denom-inator of (4.25 b) equal to zero; the results are dependent on d, h and the material properties for both materials.
(b) The loadings are applied in the half-plane (h d < ∞)
The general solutions in the Fourier-transform domain for the loadings applied in the half-plane are the same as those presented in (4.1) and (4.2) for materials 1 and 2, respectively. If the applied loadings are located in material 2, then the jump conditions are ˜ σ(2) ˜ u(2) y=d+ − ˜ σ(2) ˜ u(2) y=d− = −f −(1/iω)b . (4.26)
In terms of the unknown constants c(2), we have
(c(2))+− (c(2))−= 1 ω2e −iωp(2) ∗ dq(2), (4.27) where q(2)= (P(2))−1 −f b .
With the boundary conditions expressed in (4.3) and (4.4), and the continuity con-dition in (4.6), constants c(1)j and c(2)k are determined and are presented in a compact form as c(1)j = −1 ω2 4 k=3 bjkq(2)k eiω((p (2) k −p (1) j )h−p (2) k d) + 1 ω2det| ∗ | 2 k=1 4 l=3 3 m=1 4 r=m+1 (−1)l+s+tbjkp−stBmrkl ql(2) × eiω((p(2)l −p(1)j +p+mr)h−p (2) l d), (4.28)
c(2)k = 1 ω2q (2) k e−iωp (2) k + 1 ω2det| ∗ | 4 l=3 3 m=1 4 r=m+1 (−1)l+s+tp−stBmrkl ql(2)eiω((p(2)l +p + mr)h−p (2) l d), (4.29) where det| ∗ | = 3 m=1 4 r=m+1 (−1)s+tp−stB33mreiωp+mrh, s = 3− m +12(3− r)(4 − r) +12(m− 2)(m − 1), t = 10− m − r − s, bij= 4 k=1 [Pik(2)]−1[Pkj(1)], Bmrkl = bkmb(7−l)r− bkrb(7−l)m, p±ij= p(1)i ± p(1)j .
The solutions of stress functions in the transformed domain are ˜ Φ(1)= 4 j=1 c(1)j eip(1)j ωy, (4.30) ˜ Φ(2)= 2 k=1 c(2)k eip(2)k ωy. (4.31)
By using the method as discussed in§ 4 a, the denominator det |∗| in (4.28) and (4.29) can be reformulated by the series expansion as
1 det| ∗ | = 1 p−21B33 34eiωp + 34h(1− det | · |) = e −iωp+ 34h p−21B33 34 ∞ n=0 (det| · |)n =M eiωgh, (4.32) where M = υ (p−21B33 34)n+1 (p−31B2433)n1(p− 23B 33 14) n2(p− 34B 33 12) n3(p− 14B 33 23) n4(p− 42B 33 13) n5, g = n1p+24+ n2p+14+ n3p+12+ n4p+23+ n5p+13− (n + 1)p + 34, = ∞ n=0 n n1=0 n−n1 n2=0 n−n1−n2 n3=0 n−n1−n2−n3 n4=0 , n5= n− n1− n2− n3− n4, υ = n! n1!n2!n3!n4!n5! .
1 2 x y h 30º 135º η2 ξ2 η1 ξ1
Figure 3. The configuration for rotating the principal axis of orthotropic materials 1 and 2. The closed-form solutions of stresses in the layered half-plane are expressed explic-itly as follows, σyy(j) σxy(j) σxx(j) = ψ(j)| α=0 ψ(j)|α=1 ψ(j)| α=2 , j = 1, 2, (4.33) with ψ(1)(x, y) = 1 π 4 j=1 4 l=3 Re i(−1)αb jlq (2) l (p (1) j )α x + p(1)j (y− h) + p(2)l (h− d) + 1 π ∞ n=0 n n1=0 n−n1 n2=0 n−n1−n2 n3=0 n−n1−n2−n3 n4=0 4 j=1 2 k=1 4 l=3 3 m=1 4 r=m+1 Re i(−1)l+s+t+αbjkp−tsBmrkl M ql(2)(p(1)j )α x + p(1)j (y− h) + (p+mr+ g)h + p(2)l (h− d) (4.34 a) = 1 π 4 j=3 4 l=3 Re (bjl− (−1)lb j1B341l + bj2B2l34 B33 33 ) i(−1) αq(2) l (p (1) j )α x + p(1)j (y− h) + p(2)l (h− d) (4.34 b) + 1 π 2 j=1 4 l=3 Re i(−1)αb jlq (2) l (p (1) j )α x + p(1)j (y− h) + p(2)l (h− d) (4.34 c) + 1 π 4 j=1 2 k=1 4 l=3 3 m=1 λ r=m+1 Re i(−1)l+s+t+αb jkp−tsBmrkl M q (2) l (p (1) j )α x + p(1)j (y− h) + (p+mr+ g)h + p(2)l (h− d) (4.34 d) + 1 π 4 j=1 2 k=1 4 l=3 3 m=1 4 r=m+1 Re i(−1)l+s+t+αb jkp−tsBmrkl M q (2) l (p (1) j )α x + p(1)j (y− h) + (p+mr+ g)h + p(2)l (h− d) , (4.34 e)
where λ = 3, j = 3, 4, 4, j = 1, 2, and ψ(2)(x, y) = 1 π 2 k=1 Re i(−1)α+1q(2) k (p (2) k )α x + p(2)k (y− d) + 1 π ∞ n=0 n n1=0 n−n1 n2=0 n−n1−n2 n3=0 n−n1−n2−n3 n4=0 2 k=1 4 l=3 3 m=1 4 r=m+1 Re i(−1)l+s+t+αp−tsBmrkl M ql(2)(p(2)k )α x + p(2)k (y− h) + (p+mr+ g)h + p(2)l (h− d) . (4.35) The terms in (4.34 b) represent the four image singularities related to the interface. The terms in (4.35) with one summation represent Green’s function of material 2 in an infinite plane with a concentrated force f and an edge dislocation b applied at the location (0, d).
The analytical solutions of stresses for loadings applied either in the layer or in the half-plane can be expressed in compact forms. Each term in the complete solution represents Green’s functions of simplest solutions for concentrated forces and edge dislocations in an infinite plane. To conclude, the physical meaning of the solutions is, indeed, the image method. The mathematical derivation in this study provides an automatic determination for the magnitude and locations of all the image singu-larities.
If the loadings are applied on the free surface (d = 0) of material 1 or at the interface (d = h), the solutions can be obtained directly from that derived in § 4 a or§ 4 b. When the loadings are applied on the free surface of material 1, the complete solutions can be obtained from (4.24 a) and (4.25 a) by setting d = 0. However, if the loadings are applied at the interface, the solutions can be constructed from (4.34 a) and (4.35) by setting d = h.
5. Numerical results for stresses
In this section, the full-field stress distributions of a layered half-plane subjected to concentrated forces or edge dislocations applied at different locations are investigated. The material of T300/934 graphite/epoxy is employed for numerical calculations. The engineering constants of the material in its principal coordinate are (Gibson 1994)
E11= 131 GPa, E22 = 10.3 GPa, G12 = 6.9 GPa, ν12= 0.22,
where E11and E22 are Young’s moduli, G12 is the shear modulus and v12 is Poisson’s
ratio. The corresponding elastic compliance constants are ss1112 ss1222 ss1626 s16 s26 s66 = −1.687.63 −1.6897.1 00 0 0 144.5 × 10−3 (GPa−1).
x/h 3 2 1 0 y /h 2 1 0 −1 −2 −1.2 −1.0 −0.8 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 −0.02 −0.003 0.001 0.001 0.02 −0.1 −0.2 −0.1 −0.2 −0.3 0.1 0.2 0.3 0.5 1.0 −0.1 −0.5 −1.0 0.0 3 2 1 0 y /h 0.0 (a) (b)
Figure 4. (a) Full-field distribution of σyyfor the layered half-plane subjected to a surface
con-centrated vertical force fy(σyyh/fy). (b) Full-field distribution of σxy for the layered half-plane
subjected to a surface concentrated vertical force fy(σxyh/fy).
In order to model the planar anisotropic of the material, the principal axis of thin layer and half-plane are rotated16π and34π, respectively, with respect to the x, y
coor-dinate of the layered half-plane. The associated geometrical configuration is shown in figure 3, where (ξ1, η1) and (ξ2, η2) indicate the principal axis of materials 1 and 2,
respectively. The elastic compliance constants of materials 1 and 2 can be obtained by using (2.9) and are expressed as follows,
ss1112 ss1222 ss1626 s16 s26 s66 = −8.59 81.63 30.7636.91 −8.59 46.71 46.71 30.76 117.3 × 10−3 (GPa−1) for material 1, and
ss1112 ss1222 ss1626 s16 s26 s66 = −10.8961.57 −10.89 −44.7361.57 −44.73 −44.73 −44.73 108.1 × 10−3 (GPa−1)
−0.05 −0.03 0.0 −0.03 −0.05 −0.1 0.3 0.0 −0.03 −0.1 −0.3 0.3 0.15 0.1 −0.03 0.0 0.03 0.05 0.05 0.03 0.0 0.030.05 0.1 −0.15 −0.1 0.4 −0.05 −0.1 −0.15 −0.25 −0.4 −0.05 −0.05 0.0 0.2 0.1 −0.6 x/h 3 2 1 0 y /h 2 1 0 −1 −2 3 2 1 0 y /h (a) (b)
Figure 5. (a) Full-field distribution of σyyfor the layered half-plane subjected to a concentrated
horizontal force fx in the layer (σyyh/fx). (b) Full-field distribution of σxy for the layered
half-plane subjected to a concentrated horizontal force fx in the layer (σxyh/fx).
for material 2. A computational program for the numerical calculation of the full-field configuration of stresses is constructed by using the explicit formulation of the solutions presented in the previous section. To generate the full-field stress contours, the series solution is truncated if the numerical calculation is to within an accuracy of 0.01%. In the stress-distribution contours, short dashed lines and solid lines are used to indicate negative and positive values, respectively.
Parts (a) and (b) of figure 4 show the contours of stress components σyy and σxy in the layered half-plane due to a surface vertical concentrated force with magni-tude fy applied at (x, y) = (0, 0), respectively. Both σyy and σxy are continuous at the interface and the inclination of the contour is due to the anisotropy of the material.
Figures 5 and 6 are stress contours, achieved by applying a horizontal concentrated force fx and an edge dislocation with vertical Burgers’s vector by in the thin layer
0.0 0.0 0.02 0.05 0.02 0.0 −0.02 −0.05 0.02 0.05 −0.02 −0.05 −0.1 −0.4 −1.0 0.1 0.4 1.0 1.2 0.0 0.0 −0.05 −0.1 −0.2 0.0 0.05 0.0 0.1 −0.05 −0.1 −0.2 −0.5 0.05 0.1 0.2 0.5 −1.2 x/h 3 2 1 0 y /h 2 1 0 −1 −2 3 2 1 0 y /h (a) (b)
Figure 6. (a) Full-field distribution of σyy for the layered half-plane subjected to a dislocation
with Burgers’s vector byin the layer (σyys66h/by). (b) Full-field distribution of σxyfor the layered
half-plane subjected to a dislocation with Burgers’s vector by in the layer (σxys66h/by).
5h 4h h x y 4h 8h −4h −8h layer half-plane
applied loading point image points of layer image points of half-plane
−0.05 0.0 0.05 −0.05 −0.05 −0.1 −0.3 0.0 0.1 0.3 0.6 1.0 −0.05 −0.1 −0.3 −0.6 −1.0 0.1 0.3 0.6 1.0 0.1 0.0 −0.05 0.0 0.05 0.1 0.3 0.05 −0.1 −0.3 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.20.4 0.6 1.1 0.0 −0.1 0.2 0.4 0.6 1.1 0.1 0.0 −0.1 −0.2 −0.4 −1.1 0.0 0.1 0.2 0.4 0.6 −0.6 x/h 3 2 1 0 y /h 2 1 0 −1 −2 3 2 1 0 y /h (a) (b)
Figure 8. (a) Full-field distribution of σyy for the layered half-plane subjected to a
disloca-tion with Burgers’s vector bx at the interface (σyys66h/bx). (b) Full-field distribution of σxy
for the layered half-plane subjected to a dislocation with Burgers’s vector bx at the interface
(σxys66h/bx).
at (x, y) = (0, 0.5h), respectively. The stresses σyy and σxy are continuous at the interface and are zero on the free surface. Figure 7 shows the image points of the layer (circles, 137 points) and the half-plane (triangles, 113 points), within−5h y 5h, for applied loadings at (0, 0.5h) in the layer, as indicated in figures 5 and 6. We can see that the image points for the layer spread out all over the plane, except at the layer. However, the image points for the half-plane are occupied only in the region
y h. Usually, thousands of image points are used to obtain the result to within an
accuracy of 0.01%.
Figure 8 shows the stress contours of a layered half-plane subjected to an inter-face edge dislocation bx at (x, y) = (0, h). Figures 9 and 10 are stress distributions of a layered half-plane subjected to a vertical concentrated force fy and an edge dislocation bx applied at (x, y) = (0, 1.3h) in the half-plane, respectively.
0.2 0.0 −0.01 −0.05 −0.1 −0.2 −1.0 1.0 0.01 0.05 0.1 0.3 0.2 0.1 0.05 0.01 0.0 0.4 −0.07 −0.05 −0.05 −0.01 −0.1 −0.4 0.8 0.2 0.3 0.4 0.1 0.05 0.0 0.0 −0.01 0.05 0.1 x/h 3 2 1 0 y /h 2 1 0 −1 −2 3 2 1 0 y /h (a) (b)
Figure 9. (a) Full-field distribution of σyyfor the layered half-plane subjected to a concentrated
vertical force fy in the half-plane (σyyh/fyy). (b) Full-field distribution of σxy for the layered
half-plane subjected to a concentrated vertical force fy in the half-plane (σxyh/fy). 6. Concluding remarks
This study presents analytical investigations of an anisotropic layered half-plane sub-jected to concentrated forces and edge dislocations. By using the Fourier-transform technique and a series expansion in the transformed domain, analytical full-field solutions for stresses are obtained in an explicit closed form. The complete solution in series form consists of simple solutions for an infinite homogeneous medium with concentrated loadings. The analytical solution includes two parts: Green’s function for applied loadings and an infinite number of image singularities that are induced to satisfy the boundary and interface conditions. The physical meanings of the solu-tion can be regarded as the image method for the planar anisotropic problem. The mathematical method introduced in this study provides an automatic determination for the locations and magnitudes of all the image singularities. A computational pro-gram for calculating full-field stress distributions of the layered half-plane is easily
−0.3 −0.05 0.0 0.05 0.1 0.2 0.3 0.5 1.0 0.20.1 0.05 0.0 −0.05 −0.1 −0.2 0.05 0.1 0.0 −0.5 −1.0 −0.05 −0.05 −0.2 0.2 0.0 0.1 0.1 0.0 0.3 0.5 1.0 −1.0 −0.5 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.1 −0.1 0.0 x/h 3 2 1 0 y /h 2 1 0 −1 −2 3 2 1 0 y /h (a) (b)
Figure 10. (a) Full-field distribution of σyyfor the layered half-plane subjected to a dislocation
with Burgers’s vector bx in the half-plane (σyys66h/bx). (b) Full-field distribution of σxy for
the layered half-plane subjected to a dislocation with Burgers’s vector bx in the half-plane
(σxys66h/bx).
constructed from the explicit formulation of the solutions. Detailed numerical results of full-field stress distributions are provided in this study for applying concentrated forces and/or edge dislocations on the boundary or in the interior of the layered half-plane.
The study of dislocations in multi-layered structures is important as a means of understanding the mechanical properties of composites and thin films. The motion of a dislocation is related to the image force or the Peach–Koehler (Peach & Koehler 1950) force exerted on the dislocation. The image force that exerts on the dislocation is the negative of the gradient of the interaction energy that depends on the location of the dislocation. The force on the dislocation is most conveniently obtained from the Peach–Koehler formula. A method for evaluating the image forces on disloca-tions in the thin film is required to investigate the problem. The soludisloca-tions available in the literature have been obtained for relatively simple problems, such as isotropic
screw or edge dislocations, most of which used multiple-image analysis. However, this method is not applicable for the problem of edge dislocations in a multi-layered anisotropic medium. The explicit form of the image force acting on the edge dislo-cation in an anisotropic thin-film structure can be easily obtained from the results provided in this study. The image force on the edge dislocation can be calculated as the summation of force due to the stress fields of all the image singularities in the thin-film solution. A detailed investigation of the micro-mechanical behaviour for the interactions of edge dislocations in an anisotropic thin-film structure will be given in a subsequent paper.
The authors thank the National Science Council, Republic of China, and gratefully acknowledge grant no. NSC 89-2212-E002-018 to the National Taiwan University.
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