The quasistatic approximation for a cracked interface between a layer and a substrate

The quasistatic approximation for a cracked interface between a layer and a substrate

Claudio Pecorari

Institute for Advanced Materials, European Commission, PO Box 2, 1755 ZG Petten, The Netherlands Piaras A. Kelly

Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland, New Zealand 共Received 7 January 1999; accepted 3 February 2000兲

Heuristic in nature, the quasistatic approximation 共QSA兲 describes the interaction of ultrasonic waves with imperfect interfaces by modeling the interfacial imperfection as distributions of springs and masses. The QSA does not provide any relationship between the interfacial stiffness constants and the micromechanics of the defects. The aims of this paper are threefold. First, a derivation from first principles of the QSA boundary conditions on a cracked interface is presented. Relationships linking the interfacial constants to the mechanical and geometrical properties of the distributed cracks are also obtained. Second, the stiffness dependence of a cracked interface between a layer and a substrate on the layer thickness is investigated. It is shown that the interfacial stiffnesses cannot be regarded as intrinsic properties of the interface, but they may also depend on the structural properties of the hosting system. Finally, the effect of the thickness dependence of the interfacial stiffnesses on the phase velocity of the lowest mode supported by the layered structure is investigated. © 2000 Acoustical Society of America.关S0001-4966共00兲01905-6兴

PACS numbers: 43.35.Cg, 43.35.Zc 关HEB兴

INTRODUCTION

For the last two decades the quasistatic approximation 共QSA兲 has been the approach most commonly used to de- scribe the interaction of ultrasonic waves with imperfect in- terfaces. The QSA is a low-frequency approximation and it can be used when the thickness of the interface and the ex- tension of the interfacial defects are much smaller than the wavelength of the wave used to inspect the interface. The most complete formulation of the QSA has been presented by Baik and Thompson,¹and models the real interfacial im- perfections as continuous, uniform distributions of springs along the interface plane 共see Fig. 1兲.

The mathematical formulation of the QSA is provided by the modified boundary conditions enforced at the inter- face plane. Following Baik and Thompson,¹the QSA bound- ary conditions can be written as follows,

1

2关␴33共x,z⫽0^⫹兲⫹␴33共x,z⫽0^⫺兲兴

⫽K_N关u₃共x,z⫽0^⫹兲⫺u₃共x,z⫽0^⫺兲兴, 共1a兲

1

2关␴31共x,z⫽0^⫹兲⫹␴31共x,z⫽0^⫺兲兴

⫽K_T关u₁共x,z⫽0^⫹兲⫺u₁共x,z⫽0^⫺兲兴, 共1b兲

␴33共x,z⫽0^⫹兲⫽␴33共x,z⫽0^⫺兲, 共1c兲

␴31共x,z⫽0^⫹兲⫽␴31共x,z⫽0^⫺兲. 共1d兲 In these equations K_N and K_T are the stiffness constants of the distributed springs, and relate the discontinuity of the displacement components to the corresponding components of the stress applied to the interface. The QSA boundary conditions are not derived from first principles. Rather, they are heuristic in nature.¹

The QSA does not provide any way to correlate the values of the spring stiffness constants K_N and K_T to the micromechanics of the defects. Baik and Thompson¹defined the spring constants as the ratio between the stress applied at

‘‘infinity,’’ ⌺N,T, and the extra displacement, ⌬N,T, mea- sured at a location far from the interface,

K_N,T⫽⌺N,T

⌬N,T

. 共2兲

The extra displacement,⌬N,T, is not zero when the interface contains imperfections that alter its elastic properties. A few simple cases, such as that of an interphase layer embedded between two infinite half-spaces, and those of periodic dis- tributions of one- and two-dimensional cracks, have been considered in detail,¹ and expressions for the spring con- stants have been presented in terms of the geometrical and mechanical properties of the distributed defects.

The QSA boundary conditions关Eqs.共1a兲and共1d兲兴have been widely applied to isolated interfaces¹ as well as to in- terfaces in layered media,^2,3or between fibers and a matrix⁴. In connection to systems featuring a characteristic length, such as the thickness of a layer, or the radius of a fiber, the legitimate question arises whether the system’s structure af- fects the elastic properties of the imperfect interface.

The objectives of this work are threefold. The first ob- jective is to present a derivation from first principles of the QSA boundary conditions for a randomly cracked interface.

In so doing, expressions for the spring constants that link the values of these quantities to the structural and microme- chanical properties of the crack distribution are obtained.

Second, the spring constants for a cracked interface between a layer and its substrate is evaluated numerically. To this end, the crack opening displacement 共COD兲 of an isolated

interfacial crack undergoing an external uniform load is ob- tained by numerically solving a system of integral equations for the dislocation densities associated with the components of the COD. The dislocation densities are equivalent to the crack surface displacement gradients. The introduction of the crack compliance tensor connecting the COD to the applied stress leads to the evaluation of the spring constants. The third objective is to assess the influence of the structural properties of the system on the interfacial stiffness and, con- sequently, on the dispersion of the modes supported by the layered structure.

I. THE QSA BOUNDARY CONDITIONS ON A CRACKED INTERFACE

In this section the cracks are assumed to be one- dimensional. However, the derivation presented here, as well as the relationship between the spring stiffness constants and the properties of the distributed imperfections, can be ex- tended to distributions of two-dimensional cracks in a straightforward manner.

Consider a random distribution of cracks at the interface between two media. Let z⫽0 be the interface plane, and u_i^⫹(x) and u_i^⫺(x) be the ith component of the displacement just above and just below the interface, respectively. The average displacement discontinuity at the interface can be written as

具^ui⫹共x兲⫺u_i^⫺共x兲典⫽具⌬u_i共x兲典⫽1

L

冕

L⌬u_i共x兲dx, i⫽x,z, 共3兲 where L is the representative length of the crack distribution.

The displacement discontinuity ⌬u_i(x) is zero except at the locations of the cracks, and there it is equal to the COD, b_i(x). Then, Eq.共3兲can be written as

具⌬u_i典⫽1

L _k

兺

_⫽^N₁

冕

a_k

b_i^k共x兲dx. 共4兲

In Eq. 共4兲, N is the number of cracks in L, and a_k is the length of the kth crack. For the sake of simplicity, let the distribution be uniform and the cracks identical to each other. Then Eq.共4兲becomes

具⌬u_i典⫽␯^a具^bi典^, 共5兲

where␯⫽N/L is the crack density, a is the crack length, and 具^bi典 is the average ith component of the COD. The crack compliance tensor, S_{i j}, that relates the average displacement components to the average stress,具␴i j典, applied to the crack faces, can now be introduced:⁵

具^bi典⫽aS_{i j}具␴jz典ⁿz, i, j⫽x,z. 共6兲

Note that here the z axis is assumed normal to the interface and, consequently, only the components of the stress tensor with indexes ‘‘jz’’ appear in this equation. The z-component of the unit vector n is equal to 1 and, therefore, it will be omitted hereafter. By introducing Eq. 共6兲 into Eq. 共5兲 the latter becomes

具^⌬ui典^⫽␯^a2Si j具␴jz典^{. 共7兲}

On the assumption that the cracks do not interact with each other, it can be shown numerically that the average normal 共tangential兲displacement component due to a uniform shear 共normal兲stress field is zero. Thus, only one term remains on the right-hand side of Eq.共7兲. By inverting Eq.共7兲, the QSA boundary condition for a cracked interface can be obtained,

具␴xz典^⫽Kxx具^⌬ux典^{, 共8a兲}

具␴zz典⫽K_zz具⌬u_z典^, 共8b兲

where

K_xx⫽K_T⫽ 1

␯^a2Sxx

. 共9a兲

K_zz⫽K_N⫽ 1

␯^a2Szz

. 共9b兲

Equations 共9a兲and共9b兲show that K_T and K_N are inversely proportional to the crack density, to the square of the crack length, and to the crack compliance. Therefore, both the geo- metrical and the micromechanical properties of the interfa- cial defects are included in the definition of the macroscopic interfacial properties. Note that the definition of K_T and K_N involves quantities that are intrinsic properties of the distri- bution, a and ␯, or that are derived from average values of the stress field and COD, S_xx and S_zz.

Disregarding the mutual interaction between neighbor- ing cracks of a planar distribution seems to be a reasonable simplification that is supported by the behavior of the total stress field in the neighborhood of a one-dimensional crack subjected to a static stress. In fact, as shown by Kachanov,⁵ although the total stress field is amplified in the plane con- taining the crack, the regions where the amplification effect occurs have an extent at most of the order of the crack length. Therefore, the range of validity of the independent crack approximation can be thought to reach values of the normalized crack density, a␯, up to 0.5. At larger crack den- sities crack interaction begins to occur, and off-diagonal terms in the boundary conditions are expected to play a sig- nificant role.

II. MICROMECHANICS OF A CRACKED INTERFACE A. Isolated interfacial crack

In this subsection the mechanical response to an external load of a crack located at the interface between a layer and a substrate is examined. To evaluate the COD of such a crack, a system of integral equations for the unknown dislocation densities is solved.^6–8 The problem is formulated in its full complexity, so that the crack closure near the crack tips due to the different elastic properties of the two media is cor- rectly described by the solution.⁶ However, to avoid large-

FIG. 1. Imperfect interface and its model according to the QSA.

scale crack closure during a compressive cycle,⁹ a tensile stress, T_o, is assumed to maintain the crack open all the times 共save for the small-scale crack-tip closure兲. The same system of integral equations is solved twice. The first time, the crack is subjected to the tensile stress T_oonly, while the second time, a normal, ⌬T, or a tangential, ⌬S, stress is superimposed onto T_o. The magnitude of the stresses ⌬T and⌬S are chosen to be one order of magnitude smaller than that of T_o. Once the dislocation densities共i.e., crack surface displacement gradients兲 are known, the components of the COD can be evaluated by integrating them over the crack extension. The difference between the two CODs obtained with and without the stress perturbation is eventually evalu- ated and used to obtain the effective average COD due to the stress wave. Finally, Eq.共6兲is used to calculate the compo- nents of the crack compliance tensor, S_{i j}.

Consider a system consisting of a layer of copper on a steel substrate, and let h be the thickness of the layer. Figure 2 illustrates the behavior of the normal and tangential com- ponent of the crack compliance as a function of the ratio h/a. The plots show that both components tend to infinity as the layer thickness decreases, while they approach the same limit as the layer approximates a half-space. It is worth not- ing that the normal compliance is always larger than the transverse compliance. This fact can be easily understood in terms of the amount of material surrounding the crack that opposes the crack deformation.

B. Cracked interface between dissimilar materials 1. Cracked interface between two identical half-spaces

Baik and Thompson¹ gave the solution for the normal spring constant for an interface consisting of a periodic array of one-dimensional cracks between two identical half-spaces.

In the limit a/⌳Ⰶ1, where a is the length of the crack and⌳ is the period of the distribution, they found

K⫽ 2

␲ E 1⫺␩²

1

␯^{a2, 共10兲}

where E and␩are the Young modulus and the Poisson ratio of the material, respectively, and ␯⫽⌳^⫺1. When the two

half-spaces are dissimilar, the closed form solution for the COD of an interface crack can be obtained using the well- known ‘‘open’’ interface crack model.⁷ This model, unlike the ‘‘closed’’ model used elsewhere in this study, does not assume crack closure near the crack-tips. The COD solution can be used to show that Eq. 共9兲 leads to the following result,⁸

K_N⫽K_T⫽⌫

冑

¹⫺␤^{2 1}

␯^a2Ic

. 共11兲

In Eq.共11兲,⌫is defined by the following expression,

⌫⫽ 2␮^(l)共1⫺␣兲

共1⫹␬^(l)兲共1⫺␤²兲, 共12兲

where ␮ is the shear modulus of medium, ⌲ is Kolosov’s constant, equal to 3 – 4␩ in plane strain, and the superscript 共l兲refers to the layer. The functions␣^and␤^{in Eq.}共12兲are the well-known Dundurs’ composite parameters,^7,10

␣⫽␮^(l)共␬^(s)⫹1兲⫺␮^(s)共␬^(l)⫹1兲

␮^(l)共␬^(s)⫹1兲⫹␮^(s)共␬^(l)⫹1兲,

共13兲

␤⫽␮^(l)共␬^(s)⫺1兲⫺␮^(s)共␬^(l)⫺1兲

␮^(l)共␬^(s)⫹1兲⫹␮^(s)共␬^(l)⫹1兲,

wherein the superscript 共s兲refers to the substrate. The quan- tity I_crepresents the following integral,

I_c⫽

冕

⫺1

⫹1

冑

¹⫺t²cos

冉

^␧ln

冏

^tt⫺⫹11

冏 冊

^{dt, 共14兲}

where⑀⫽(1/2␲^)ln兩(␤⫹1)/(␤⫺1)兩. The integral I_c is a func- tion of␤^only共see Fig. 3兲, and can be evaluated numerically.

However, a good approximation to I_c can be obtained by neglecting the cosine term, that is, by setting ␤⫽0 in which case I_c⫽␲/2. The maximum difference between the actual value of I_c and ␲/2 is obtained when ␤ is maximum, i.e.,

␤⫽0.5, and is equal to 0.044. When the materials are simi- lar,␤⫽0, I_c⫽␲^/2, ⌫⫽E/关4(1⫺␩²⁾兴, and Eq.共11兲reduces to the result of Baik and Thompson, Eq.共10兲. In the follow-

FIG. 2. Compliance of an isolated crack at the interface between a copper layer and a steel substrate versus the ratio of the layer thickness to the crack length, h/a.

FIG. 3. Plot of the integral Icversus the composite material parameter␤.

ing sections, the case of a crack between a layer and a sub- strate is considered. Equations 共11兲 and 共14兲 can be used when the thickness of the layer is infinite.

2. Cracked interface between a layer and a substrate Having assumed that the cracks do not interact with each other, and having evaluated the compliance tensor of an iso- lated crack as a function of the ratio h/a, it is now possible to investigate the effect of the geometrical properties of the layered structure on the interfacial stiffness.

Figure 4 presents plots of the normal and transverse in- terfacial stiffness constants versus the ratio h/a for three values of the crack length a, again for a copper layer on a steel substrate. The normalized crack density, a␯, is equal to 0.2. The plots show that the interface becomes more compli- ant as the layer thickness decreases, and, as expected, the larger the crack length, the more compliant the interface. As for the isolated crack, both spring constants approach the same limit as the thickness of the layer increases. Because of the inverse proportionality between the stiffness constants and the crack compliance, K_N is always smaller than the transverse spring constant. No physical interphase layer model of an imperfect interface could simulate such a prop- erty of a cracked interface.

Figure 5 shows plots of the interfacial constants versus the ratio h/a for three values of the layer thickness, h. The normalized crack density is again equal to 0.2. Here, the behavior of the spring constants is markedly different from that shown in Fig. 4. Such dependence of the spring con- stants is explained as follows. If h is constant, an increase of the ratio h/a is obtained by decreasing the crack length, a. In order to maintain the normalized crack density, a␯, constant, the crack density,␯ must be increased accordingly. Thus, as h/a increases the crack distribution changes its properties, becoming progressively a more dense distribution of smaller and smaller cracks. Figure 5, therefore, shows that among different interfaces with crack distributions having the same normalized crack density or, equivalently, the same cracked area, those having the smaller cracks are the stiffer. Results

similar to those presented in Figs. 2, 4, and 5 have also been obtained for a system consisting of a stiffening layer on a substrate.⁸

In an attempt to identify the nature of interfacial imper- fections, Nagy¹¹ considered the ratio between the reflection coefficients of longitudinal and shear waves, R_L and R_S, respectively, at normal incidence and at low frequencies, ␻^. He focused on interfaces between samples with identical ma- terial properties, and showed that the ratio r⫽R_L/R_S is pro- portional to the ratio between the transverse and normal in- terfacial constants,

r⫽lim

␻→0

R_L共␻兲 R_S共␻兲⫽K_T

K_L V_L

V_S. 共15兲

In Eq.共15兲, the symbols V_Land V_Sare the phase velocity of the longitudinal and shear waves, respectively. Nagy found that r is smaller than 1 for distributions of volumetric imper- fections, while it is larger than 1.5 for cracked interfaces.

The extension of Nagy’s analysis to interfaces between dif- ferent materials and to structures more complex than that of an isolated interface should be straightforward. Although this is not a primary objective of this work, the behavior of the ratio K_T/K_N is presented as a function of h/a in Fig. 6. The

FIG. 4. Interfacial stiffness constants versus h/a for three values of the crack length: a⫽100␮m, a⫽200␮m, and a⫽400␮m. The normalized crack density is constant, a␯⫽0.2.

FIG. 5. Interfacial stiffness constants versus h/a for three values of the layer thickness: h⫽50␮^{m, h}⫽100␮^{m, and h}⫽200␮m. The normalized crack density is constant, a␯⫽0.2.

FIG. 6. Stiffness ratio versus h/a. The normalized crack density, a␯^{, is} equal to 0.2.

plot shows that the ratio K_T/K_N is always greater than one, and progressively increases as the layer thickness decreases.

III. ULTRASONIC WAVES AND CRACKED INTERFACES

A. Cracked interface between two different half- spaces

Qu¹²presented a theoretical model to evaluate the nor- mal reflectivity of a cracked interface between two materials having different elastic properties. His approach was based on a differential self-consistent scheme and was used to ob- tain the reflection coefficient of a longitudinal wave as a function of the normalized crack length. The predictions of the self-consistent approach are reported in Fig. 7.¹³Qu used the following values for the phase velocities, V_L and V_T, modulus of rigidity,␮, and the Poisson ratio,␩, of the two materials:

Material 1: V_L¹⫽6300 m/s, V_T¹⫽3100 m/s,

␮1⫽26.1 GPa, ␩1⫽0.34, Material 2: V_L²⫽4600 m/s, V_T²⫽2300 m/s,

␮2⫽48.3 GPa, ␩2⫽0.33.

The incident wave propagates in the half-space 1, and the normalized crack density is a␯⫽0.3. Figure 7 also presents the values of the same reflection coefficient according to the QSA. The agreement between the two approaches is reason- able up to values of a/␭L⬃0.2, where␭L is the wavelength of the longitudinal incident wave. This relationship between the crack size and the wavelength of the interrogating wave will be used hereafter as an upper limit for the range of validity of the QSA.

B. Cracked interface between a layer and a substrate In several applications the nondestructive assessment of the interface bond between a layer and its substrate is per- formed by using the lowest mode supported by the system.

In this section the effect of a crack distribution on the phase velocity of this mode is briefly considered. In particular, the

investigation focuses on the relationship between the layer thickness, h, and the mode dispersion when the values of K_N and K_T depend on the ratio h/a.

Figure 8 shows the relative variation of the lowest mode’s phase velocity as a function of the frequency for a system consisting of a copper layer on a steel substrate. The normalized crack density is a␯⫽0.2, and the thickness of the layer is h⫽200␮m. Three values for the ratio h/a are cho- sen: 0.4, 0.65, and 0.9. They correspond to three interfaces with increasing values of the spring constants. The reference velocity is that of the same mode propagating along the sur- face of a system having a perfectly bonded interface. The interfacial constants used to generate these plots are those presented in Fig. 5. The plots stop at the frequency where the product a/␭SAW⬃0.2, where␭SAWis the wavelength of the propagating surface acoustic wave. The use of the QSA be- yond this limit leads to considerable errors in the evaluation of the phase velocity of the guided mode. Figure 8 shows that the QSA can describe interfaces with crack distributions that may cause relative phase velocity variations not greater than 10%.

Figure 9 reports the percentage error in the predicted phase velocity caused by the use of the spring constants of an isolated interface between two infinite half-spaces in place of those considered in Fig. 8. The layered system is that con- sidered in the previous figure. The error increases with the frequency and reaches the value of about 1% at the upper end of the range of validity of the QSA. Although a maxi- mum error of 1% may be acceptable in some experimental situation, careful consideration must be given to a notable exception that is provided by line-focus beam acoustic mi- croscopy of layered systems. This technique, in fact, allows measuring surface acoustic wave velocity with relative accu- racy lower that 0.1%.¹⁴Theoretical systematic errors greater than this limit may considerably affect the validity of any comparison between measured and theoretically predicted values of the phase velocity.

Finally, a system consisting of a stiffening NiT layer on

FIG. 7. Reflection coefficient from a cracked interface between two differ- ent media versus normalized crack dimension. The normalized crack density

is a␯⫽0.3. QSA: ———, self-consistent model: ---. FIG. 8. Relative variation of the phase velocity of the first mode propagat- ing along a copper layer on a steel substrate versus the ultrasonic frequency for three values of the ratio h/a: 0.4, 0.65, and 0.9. The layer thickness is h⫽200␮m. The normalized crack density is a␯⫽0.2.

a steel substrate is considered. The elastic properties of these materials are the following:¹⁵

NiT: ␳⫽5200 Kg/m³, V_T⫽6000 m/s, V_L⫽11021 m/s, steel: ␳⫽7932 Kg/m³, V_T⫽3260 m/s, V_L⫽5960 m/s.

Note that the ratio between the shear phase velocity of the layer and of the substrate is greater than 2. A well-known consequence of this fact¹⁶ is that there exists a cutoff fre- quency for the lowest mode supported by this system. At the cutoff frequency the velocity of the mode is equal to the velocity of a shear wave propagating in the substrate共hori- zontal line in Fig. 10兲. Figure 10 shows the three dispersion curves of the lowest mode for three interfaces. The first in- terface is characterized by a perfect bond, the second by a distribution of cracks with linear dimension a⬃150␮^{m, and} the third one by a distribution of cracks with length a⬃290

␮m. The normalized crack density is a␯⫽0.2, and the layer thickness is h⫽100␮m. The cut-off frequency for the lowest mode propagating along the perfect interface is f_cutoff⫽3.7

MHz. The most interesting feature of this figure is that the lowest mode propagating along an interface affected by some degree of damage continues to propagate even at fre- quencies higher than 3.7 MHz.

IV. SUMMARY

The QSA boundary conditions for a cracked interface were derived from first principles. The derivation leads to a definition of the interfacial stiffness constants in terms of local quantities that describe the geometrical and the micro- mechanical properties of the distributed cracks.

In the case of an interface between different materials, it has been shown that the QSA provides an accurate descrip- tion of the wave–interface interaction for ultrasonic frequen- cies such that a/␭⭐0.2, and for values of the normalized crack density smaller than 0.5. In this work, the stiffness constants have been obtained under the assumption that the distributed cracks do not interact with each other. The exten- sion of this modeling to include the interaction among first neighbors is conceptually straightforward. Multiple crack in- teraction is expected to introduce cross terms in the boundary conditions.

The spring constants of an interface between a layer and a substrate have been shown to depend, in general, on the layer thickness. Therefore, the interfacial constants can no longer be regarded as intrinsic properties of the interface, and determined only by the distributed imperfections. Neglecting their dependence of the structural properties of the hosting system may lead to relative errors on the predicted surface acoustic wave velocity of the order of 1%, which in some case may be not acceptable.

Finally, this investigation has shown that the first mode supported by a stiffening layer continues to propagate be- yond its characteristic cutoff frequency when the interface contains a crack distribution.

APPENDIX A: AN INTERFACIAL CRACK BETWEEN A LAYER AND A SUBSTRATE

The solution to the problem of a crack lying along the interface between a layer and a substrate may be solved by considering first the stress due to a single interfacial dis- placement discontinuity, or dislocation.⁷ The magnitude of the displacement discontinuity is called the Burgers vector, b. Let the resolved components of the Burgers vector be b_y in the direction normal to the interface共into the layer兲and b_x along the interface. The normal 共N兲 and shear 共S兲 stress ␴ arising at position x along the interface, due to an interfacial dislocation at position␰, may then be written as¹⁷

␲

⌫

再

^{␴␴NS共共xx兲兲}

冎

^⫽bhx

再

^{G˜G˜xNxS⫹⫹GGˆˆxNxS}

冎

^⫹bhy

再

^{G˜G˜y Ny S⫹⫹GGˆˆy Ny S}

冎

^,

共A1兲 where h is the thickness of the layer, and the parameter⌫is defined above. The influence functions G˜ in Eq. 共A1兲 give the stress due to a dislocation between two half-planes, and are

FIG. 9. Relative error versus ultrasonic frequency for the layered system of Fig. 7.

FIG. 10. Dispersion of the first mode supported by a stiffening NiT layer on a steel substrate versus ultrasonic frequency for a perfectly bonded interface 共———兲, an interface with the ratio h/a⫽0.4共---兲, and an interface with the ratio h/a⫽0.65共– – –兲. The thickness of the layer is h⫽100␮^{m, and the} normalized crack density is a␯⫽0.2.

G˜

xN⫽⫺␤␲␦共¯x⫺^¯␰兲, G˜

xS⫽1/共¯x⫺^¯␰兲,

共A2兲 G˜

y N⫽1/共¯x⫺^¯␰兲, G˜

y S⫽⫹␤␲␦共¯x⫺^¯␰兲,

where ␦ is the delta function and the overbar denotes nor- malization with respect to the layer thickness h, i.e., x¯

⫽x/h , etc. The influence functions Gˆ in Eq.共A1兲 describe the influence of the layer free-surface. They may be obtained using Fourier transform theory,⁸and are

Gˆ

xN⫽共1⫺␣兲

冕

0

⬁

F_xNe^⫺␭

⌬ cos关␭共¯x⫺^¯␰兲兴d␭,

Gˆ_xS⫽共1⫺␣兲

冕

0

⬁

F_xSe^⫺␭

⌬ sin关␭共¯x⫺^¯␰兲兴d␭,

共A3兲 Gˆ

y N⫽共1⫺␣兲

冕

0

⬁

F_{y N}e^⫺␭

⌬ sin关␭共¯x⫺^¯␰兲兴d␭,

Gˆ

y S⫽⫺Gˆ

xN, where

F_xN⫽2关共1⫹␤兲²␭²⫹␤兴e^⫹␭⫺␤共1⫹␣兲e^⫺␭, F_xS⫽⫺兵²共1⫹␤兲␭关共1⫹␤兲␭⫺共1⫺␤兲兴

⫹共1⫹␤²兲其^{e⫹␭⫹共}␣⫹␤²兲e^⫺␭, 共A4兲 F_{y N}⫽⫺兵²共1⫹␤兲␭关共1⫹␤兲␭⫹共1⫺␤兲兴

⫹共1⫹␤²兲其^{e⫹␭⫹共}␣⫹␤²兲e^⫺␭, and

⌬⫽共1⫺␤²兲e^2␭⫹共␣²⫺␤²兲e^⫺2␭

⫹4共1⫹␤兲共␤⫺␣兲␭²⫺2共␣⫺␤²兲. 共A5兲 Note that these equations correspond to Eqs.共1兲–共9兲of Ref.

18, but with the misprints in Ref. 18 corrected. The integrals in Eq.共A3兲may be evaluated numerically共replacing the up- per infinite limit of integration with a value of about 15 will provide convergence and accuracy兲.

Consider now an interface crack lying along 关⫺L,

⫹L兴. One assumes that the crack faces make contact near the crack-tips,⁶and that the crack is open along关⫺a,⫹b兴. Suppose also that the crack is loaded by a tension T_o⫹⌬T and a shear⌬S. The opening displacements of the crack are modeled as a continuous distribution of dislocations along its length. The boundary conditions along the crack are that the net shear traction at any point along the crack, due to the load and due to dislocations, must be zero共assuming that the crack-faces make frictionless contact in the contact-zones兲, and that the net normal traction must be zero along the open portion 关⫺a,⫹b兴. These conditions then lead, integrating Eq. 共A1兲, to the dual integral equations

⫺T_o⫹⌬T

⌫ ⫽⫺␤^Bx共x兲⫹ 1

␲^h

冕

⫺L

⫹L

B_x共␰兲关G˜_xN⫹Gˆ_xN兴d␰

⫹ 1

␲^h

冕

⫺a

⫹b

B_y共␰兲关G˜

y N⫹Gˆ

y N兴d␰^,

共A6兲

⫺⌬S

⌫ ⫽⫹␤^H共•兲B_y共x兲⫹ 1

␲^h

冕

⫺L

⫹L

B_x共␰兲关G˜

xS⫹Gˆ

xS兴d␰

⫹ 1

␲^h

冕

⫺a

⫹b

B_y共␰兲关G˜_{y S}⫹Gˆ_{y S}兴d␰^,

where H(•)⫽H(x⫹a)⫺H(x⫺b) is the Heaviside function, and B_x, B_yare the shear and normal crack-surface displace- ment gradients 共dislocation densities兲. Here, it is assumed that the contact zones are infinitesimal in length 共contact lengths ⬍10^⫺4L). It can be shown that this will be true provided (T_o⫹⌬T)/⌬S⬎1.13. The equations are solved by assuming that the shear densities B_xare square-root singular near ⫾L and that the normal densities B_y are so near ⫹a and⫺b 共even though the normal densities are known to be bounded there兲. The singularity in B_yis then driven to zero.

The contact lengths are unknown: an initial guess may be made of the position of a and b by using the bonded half- plane solution.¹⁹ The equations are solved in an iterative manner using the Gauss–Chebychev quadrature formulas 共see Ref. 7, Table 2.2, case I兲and Krenk’s interpolation for- mulas 共see Ref. 7, Table 2.4, case I兲. Full details are given elsewhere.⁸

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