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A film coating on a rough surface of an elastic body
Journal of Applied Mathematics and Mechanics 77 (2013) 79–90
Contents lists available at SciVerse ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
A film coating on a rough surface of an elastic body夽 M.A. Grekov ∗ , S.A. Kostyrko St Petersburg, Russia
a r t i c l e
i n f o
Article history: Received 2 October 2012
a b s t r a c t A solution of the plane problem of the theory of elasticity for a film–substrate composite is solved by a perturbation method for a substrate with a rough surface. An algorithm for calculating any approximation, which ultimately leads to the solution of the same Fredholm equation of the second kind, is given. Formulae for calculating the right-hand side of this equation, which depends on all the preceding approximations, are derived. An exact solution of the integral equation in the form of Fourier series, whose coefficients are expressed in quadratures, is given in the case of a substrate with a periodically curved surface. The stresses on the flat surface of the film and on the interfacial surface are found in a first approximation as functions of the form of bending of the surface, the mean thickness of the film and the ratio of Young’s moduli of the film and the substrate. It is shown, in particular, that the greatest stress concentration on the film surface occures on a protrusion of the softer substrate. ©2013 © 2013 Elsevier Ltd. All rights reserved.
The influence of the curvilinear shape of the surface of a film coating on the principal characteristics of the stress state of this surface and of the flat surface between the film and the substrate was investigated in Ref. 1. Bending of a film surface can occur in electronic and optoelectronic devices at the stage of growth and heat treatment of the film coating, which are accompanied by condensation and evaporation processes.2 Intense heating and large stresses3 transform the initially smooth film surface into a rough surface. In addition, the unstable state of the plane shape of the film surface and its morphological changes can be due not only to problems of a technological nature, but also to the unevenness of the substrate surface. In the present study small deviations of the interfacial surface from a plane shape are regarded as a defect of the surface, capable of degrading the functional properties of the devices by inducing the local growth of stresses not only on the joining surface itself, but also on the flat surface of the film. Under sufficiently large stresses, associated with mismatching of the parameters of the crystal lattices of the film and the substrate, as well as under the action of an external load, this can result in an even greater build up of dislocations at the interfacial surface,4 the generation and growth of cracks and ultimately in the complete cessation of the operation of the device. To analyse the non-uniform stress state of a film created by a curved interfacial surface, a perturbation method is used below together with the superposition method similar to the way in which the problem of a curvilinear crack located near a flat interfacial surface5,6 and the problem of a film coating with a curvilinear surface on a flat substrate surface1 were solved. Unlike the case of a weakly curved interfacial surface in an infinite body,7–9 here an important role is played by the mean thickness of the film and the difference between the elastic properties of the film and the substrate. Under the assumption that the curvature has a periodic form, the solution of the problem is found in each approximation in the form of Fourier series, whose coefficients are expressed in quadratures. 1. Statement of the problem Consider an elastic body, occupying the half-space x2 < f(x1 ), with a film coating of thickness h0 under the conditions of in-plane deformation or a plane stress state and with loads acting on the flat film surface and stresses at infinity. This enables us to formulate the corresponding plane problem of the theory of elasticity for the semi-infinite region of the complex variable z = x1 + ix2 , consisting of the semi-bounded region 1 , joined along the curvilinear interfacial boundary c to the band 2 (Fig. 1). We will assume that the shape of the interfacial surface differs only slightly from a flat surface and is specified in the x3 = const plane by the equation (1.1)
夽 Prikl. Mat. Mekh., Vol. 77, No. 1, pp. 113–128, 2013. ∗ Corresponding author. E-mail addresses: [email protected] (M.A. Grekov), [email protected] (S.A. Kostyrko). 0021-8928/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2013.04.010
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Fig. 1.
where f(x1 ) is a continuously differentiable periodic function with period , which satisfies the conditions
Thus, the amplitude of the deviation of points on the interfacial surface from the x2 = 0 plane is equal to A = . On the interfacial boundary c there are no discontinuities of the displacement vector u or of the stress (load) vector n , i.e., (1.2) Here
u1 and u2 are the components of the displacement vector along the x1 and x2 axes, and nn and nt are the normal and tangential loads on an area with the normal n (the unit vector n in equalities (1.2) is perpendicular to the line c ). The unit vectors n and t form a right-hand system of coordinates n,t. In the general case the loadqacts on the band boundary b : (1.3) Here
(1.4) whereP1 and P2 are components of the principal vector of the forces acting on a segment of the boundary b within one period. We will assume that the function q(x1 ) satisfies Hölder’s condition everywhere on b . The conditions at infinity are defined by the relations
(1.5) when x2 → –∞, where is the angle of rotation of a material particle.
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2. The superposition method As was done previously,1 in accordance with the superposition principle,10 we represent the solution of n (z), u(z) in the form of the sum of the solutions of two problems using the unified relation (␦k2 is the Kronecker delta) (2.1) in which the functions G(z, k ),Gb (z, 2 ),Gc (z, k ) are equal, respectively, to
k = (3 – 4k ) for in-plane deformation, k = (3–k )/(1 + k ) for a plane stress state, k is Poisson’s ratio, and k is the shear modulus of the medium k . The derivative d/dz is taken in the direction of the vector t. The quantities nb (z) and ub (z) are the load and the displacement in the uniform half-plane D = {z: Imz < h0 } with the elastic properties of the band 2 . On the straight boundary b of this region there are several unknown self-equalized loads p(x1 ) (2.2) which satisfy the periodicity condition
(2.3) and at infinity the stresses and the angle of rotation angle are equal to zero (Problem 1). The quantities cn (z) and uc (z) are the load and the displacement that occur in the two-component plane with the curvilinear interfacial boundary c , on which the load jump cn = cn + –cn – and the displacement jump uc = uc+ –uc– occur, under conditions (1.5) at infinity (Problem 2). Taking relations (2.1) into account, we find the expression for these jumps from the contact conditions between the film and the substrate (1.2) (2.4) When boundary conditions (1.3) and (2.2) are taken into account, the limit transition in relation (2.1) as z→zb ∈ b leads to the following equation, which must be satisfied by the unknown function p: (2.5) The original problem, is thus, reduced to solving Eq. (2.5), in which cn (zb ) is a functional that depends on the functionpand the point zb . Note that in the case of a straight interfacial boundary c , parallel to the x1 axis, cn (zb ) is a Fredholm integral operator, which acts on ¯ and Eq. (2.5) is a Fredholm equation of the second kind with continuous kernels.10,11 When the boundary the functionpand its conjugate p, c has an arbitrary form, the solution of Problem 2 reduces to a singular or hypersingular integral equation,10,12 and it is not possible to express the functions cn (z) and uc (z) in an explicit form in terms of the function p. In the case of a small deviation of the interfacial surface from a plane shape, the perturbation method can also be used to obtain the same explicit expression for the corresponding functions cn (z) and uc (z) for all approximations.8,9 Then, by analogy with the previously described approach,5,6 we find an explicit expression for the operator cn (zb ) in each approximation. 3. Solution of Problem 1 The load bn (z) and the displacement ub (z) in Problem 1 can be expressed10 by a single formula in terms of the complex function (), which is holomorphic in the complex plane = + i2 outside the straight line 2 = 0
(3.1) where ␣ is the angle between the direction of the area (the vector t) and the real axis of the complex variable , i.e., the 1 axis. Here and below the bar denotes the complex conjugate, and the prime denotes the derivative with respect to the argument. The function in relation (3.1) is defined by the equality10,13
(3.2)
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4. Solution of Problem 2 by the perturbation method The load cn (z) and the displacement uc (z) in Problem 2 for the combined deformation of the two uniform semi-bounded regions Dk (k = 1, 2, D1 = 1 ) with the curvilinear interfacial boundary c , which have the elastic properties of the respective media k , are defined by the relation8,9
(4.1) The functions k (z) are holomorphic in the region Dk , and the functions k (z) are holomorphic in Dk = {z:¯z ∈Dk } (k = 1, 2). Assuming in equality (4.1) that Im z→ ± ∞ for ␣ = 0 and ␣ = /2 and taking (1.5) into account, we arrive at the following relations, which specify the values of the functions k and k at infinity:
(4.2) where
(A and B are Dundurs parameters14 ). In relation (4.1) we take the limit as z → zc ∈ c and ␣ → ␣0 , where ␣0 is the angle between the positive direction of the tangent to the curve c at the point zc and the x1 axis. Then we obtain
(4.3) When conditions (2.4), expression (4.1) and the equality
are taken into account, relations (4.3) lead to two boundary conditions for finding the functions k (z) and k (z), which we write in the form of the single equality
(4.4) c = –bn (zc )
Equation (4.4) follows from the first of Eqs (4.3) when mk = k = 1 and and from the second when mk = k , k = –k and c = 2 1 2 (ub )’(zc ). The functions k (zc ) and c (¯zc ) are the limit values of the respective functions at z→zc . Bearing in mind the dependence of the solutions of both problems on the small parameter , we represent the complex potentials , k and k and the unknown function p in the form of the power series
(4.5)
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In turn, we expand the values of the functions m , km and km on c in corresponding Taylor series in the vicinity of the straight line x2 = 0, treating the real variable x1 as a parameter. We obtain
(4.6) Representing expansions (4.5) and (4.6) in Eq. (4.4) and equating the coefficients of the powers m (m = 0, 1, . . .), we arrive at the sequence of boundary conditions
(4.7) The functions Hm (x1 ) with m > 0 are defined in terms of known functions found in the preceding approximations, and the function Zm (x1 ) is an integral operator, which acts on the unknown function pm (x1 ), i.e.,
(4.8)
(4.9) The argument x1 in the function f, as well as on the right-hand side of the second equality in (4.8), has been omitted for brevity. In deriving of relations (4.8) and (4.9) we used the expansion
We introduce the functions
(4.10)
(4.11) which are holomorphic outside the straight line x2 = 0. Then, when relations (4.10) and (4.11) are taken into account, problem (4.7) reduces to the two Riemann–Hilbert boundary-value problems (4.12) (4.13)
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The solutions of problems (4.12) and (4.13) can be written in the form13 (4.14) where
(4.15)
kn and V kn are defined in terms of all the preceding solutions and are, thus, known functions in the m-th approximation. The functions ˙m m un are integral operators that act on the unknown function p and its conjugate. We introduce the notation The functions un and Vm m m
We find expressions for the complex potentials of the second problem in the m-th approximation km and km from relations (4.10) and (4.11)
(4.16) Thus, formulae (4.1), (4.8), (4.9), (4.14), (4.15) and (4.16) give the solution of the second problem in the m-th approximation, which is expressed in terms of the complex potentials of the first problem r (r = 0, 1,. . .,m), which, in turn, are expressed in terms of the functions pr according to formula (3.2) when the functions and p in them are replaced by r and pr . 5. The integral equation of m-th approximation To construct the integral equation we initially note that when expansions (4.5) are taken into account, relations (3.1) and (4.1) are written in the form
(5.1) b and Gc are defined by the same equalities (3.1) and (4.1) after the functions , and are replaced by the The functions Gm k k m corresponding functions m , km and km . By virtue of representations (5.1) we arrive in each approximation at an equation similar to Eq. (2.5)
(5.2) Here cnm is a Fredhom integral operator, which acts on all the functions pr (r = 0,1,. . .,m) and their conjugates, q0 = q, and qm = 0 for m > 0. c (z, ) when z = z = x + ih, ␣ = 0, and = 1. As a result, since c (z ) = Gc (z , We substitute expression (4.16) into the expression for Gm 2 1 2 b nm b m b 1) in this case, Eq. (5.2) takes the form
(5.3) where
(5.4)
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(5.5) Since h0 = / 0, the kernels of Eq. (5.3) K1 and K2 are continuous functions, which vanish at infinity. Thus, Eq. (5.3) is a Fredholm integral equation. In the zeroth approximation its right-hand side equals
When there are no external loads on the film surface, q(x1 ) = ∞ = ∞ = 0, and, as follows from physical arguments, the corresponding 22 12 homogeneous equation (5.3) has only the zeroth solution p0 (x1 ) = 0. We henceforth confine ourselves to the case when q(x1 ) = 0. When expressions (4.8) and (4.15) are taken into account, from equalities (3.2) and (4.16) we find the complex potentials in the zeroth approximation (5.6) They correspond to a uniform stress-strain state of a film and a substrate with a flat interfacial boundary, where 1∞ = s is the 11 longitudinal stress in the substrate, and 2∞ = 11
2 (1+1 ) 1 (1+2 )
= 0 is the longitudinal stress in the film.
6. The first approximation for q(x1 ) = 0 We substitute expressions (5.6) into equalities (4.8). Then (6.1) When relations (4.15) and (6.1) are taken into account, from equality (5.4) we find the right-hand side of Eq. (5.3) in a first approximation
(6.2) Here
(6.3) After the integral equation is solved, the stress-strain state of the film is specified by relations (3.1) and (4.1), and that of the substrate is specified by relations (4.1). In this relation, which was written for the potentials of the second problem in any approximation, the potentials of the first approximation are specified by the equalities
(6.4) where
It should be noted here that the preceding arguments, which include the derivation of integral equation (5.3), are valid in the general case when the shape of the interfacial surface and the loads acting on the surface of the film do not have the property of periodicity. In particular, when there is weak local bending of the interfacial surface, the periodicity conditions of the functions q and p (1.4) and (2.3) can be replaced by their decay conditions at infinity, and integration along the length of a period can be replaced by integration along the ∞ = ∞ = ∞ = 0. entire boundary b . In conditions (1.5) we should then take 22 12 Relations (2.1), (3.1), (4.1), (4.5), (4.8), (4.9), (4.14)–(4.16) and (5.3)–(5.5) do, in fact, give an algorithm for determining the stress-strain state of the film and the substrate for any degree of approximation. Initially, in the zeroth approximation the function p0 is found when integral equation (5.3) with the known right-hand side F0 is solved. Then, the expression for the function F1 is determined using formulae (4.8), (4.15), (4.16) and (5.4). After the function F1 is substituted into Eq. (5.3), its solution p1 is found. The subsequent approximations are constructed according to the same scheme.
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7. A periodically curved interfacial surface In most cases the rough surface of a substrate has a periodic type of bending or can be approximated by one. The periodicity of the shape of the interfacial surface and of the external loadqwith the same period is the reason why the function Fm defined by expression (5.4) is also periodic and can be represented in the form of a Fourier series with known coefficients
(7.1) In this case we obtain an analytical solution of Eq. (5.3) in the form of a Fourier series
(7.2) where
As an example we will assume that the shape of the interfacial surface is specified by the function
(7.3) The real quantity y plays of the role of the parameter that specifies the shape of the surface, and y ∈ (0, ∞). Figure 2 shows the contour of the interfacial surface within one period for y = 0.15, 0.5, and 2.0 (curves 1, 2 and 3, respectively) and ␥ = 0. The figure also presents the dependence of the radius of curvature R of curve (1.1) at the point x1 = 0 (the point of the minimum value of the radius R) on the parameteryfor = 0.05. When ␥ = 0, fairly narrow local depressions in the substrate can be described, and when ␥ = 1, similar protrusions can be described.
Fig. 2.
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Fig. 3.
Substituting expression (7.3) into equality (6.3), we obtain
(7.4)
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Fig. 4.
8. Results of the calculations We will henceforth confine ourselves to the first approximation and present some results that were obtained using formulae (6.2) and (7.1)–(7.4) in the case of a free film surface (q = 0). In this case only the longitudinal stress s acts in the substrate far from the interfacial boundary. Poisson’s ratios were assumed to be equal to 1 = 2 = 0.3. In this case f = rs , where r = E2 /E1 is the ratio of Young’s moduli of the film and the substrate. To obtain numerical results we take a portion of the Fourier series of the function f’(x1 ), whose length was determined to achieve agreement between the exact and approximate values of the minimum radius of curvature (at x1 = 0) with an error no greater than 0.1%. For y = 0.15 it was sufficient to take 22 terms of the Fourier series. It should be noted that the smaller the value of y, the larger the maximum value of the function and, consequently, the smaller the upper limit of the small parameter = A/ (Ais the maximum deviation of the interfacial surface from thex2 = 0 plane, and is the period of the curvilinear shape of this surface). Thus, wheny = 0.15, the condition |f’| < 1 leads to the inequality < 0.063. For this reason, all the results were obtained for = 0.05. The distribution of the ring stresses tt on the interfacial surface is presented on the left-hand side of Fig. 3 for h0 = 2A(+ tt is represented by the solid curves, and – tt is represented by the dashed curves), and the longitudinal stresses 11 on the film surface are presented on the right-hand side of Fig. 3 for h0 = 2A (the solid curves) and h0 = 10 A(the dashed curves). In view of the symmetry of the stresses about the x1 = 0 axis and the periodicity, they are shown for the regionx1 ≥ 0 within one period. The graphs in the upper part of Fig. 3 were constructed for the case of a substrate that is stiffer than the film (r = 1/3), and the graphs in the lower part were constructed for the opposite case (r = 3). In all cases the bottom of the depression atx1 = 0 is located in the stiffer material. Curves 1, 2 and 3 in Fig. 3, as well as in Fig. 4, correspond to the valuesy = 0.15, 0.5 and 2.0. Note that the ring stresses tt were calculated using the well-known estimate tt = 11 + O(2 ) (Ref 15). A comparison of the graphs in Fig. 3 shows that the stresses reach their highest values at the bottom of a depression in the stiffer material (– tt when r = 1/3 and + tt when r = 3). The corresponding stresses at the same point on the other side of the interfacial surface (+ tt when r = 1/3 and – tt when r = 3) are approximately as many times smaller as Young’s modulus of the softer material is smaller than that of the stiffer material. It follows from Fig. 3 (the right-hand side) that the bending of the interfacial surface results in an uneven distribution of the longitudinal stresses on the film surface and, in accordance with Hooke’s law, in an uneven distribution of the longitudinal and transverse strain on it. This, in turn, results in the formation of the relief on the flat surface of the film. The graphs in Fig. 3 show that the highest stresses on the film surface occur directly over a protrusion of the softer substrate (when r = 3). It is seen from Fig. 3 that when h0 = 10A, the stress 11 on the film surface depends weakly on the shape of the relief of the interfacial surface and approaches the values of f . It is noteworthy that when the film is sufficiently thin (h0 = 2A), a smoother depression causes higher stresses on the film surface than a more pointed depression. When h0 = 10A, the opposite picture is observed. This effect is clearly seen in Fig. 4, which shows how the stress intensity factor on the film surface K = 11 (0)/s depands on the ratio h0 /A when r = 3. The influence of the shape of the interfacial surface on the stress intensity factor K± = ± tt (0)/s on this surface is shown in Fig. 5 in the upper part for r = 1/3 and the two film thicknesses h0 = 2A and h0 = 10A and in the lower part for r = 3. It follows from the results that when the film stiffness is three times higher than that of the substrate, the corresponding values of K± for both thicknesses are nearly identical; therefore, in the lower part of Fig. 5 they are represented by a single. It is seen that the dependence of K± onyis most clearly displayed when y < 1, i.e., where the shape of the interfacial surface is sensitive to a variation of this parameter. A syincreases, the shape of the surface becomes closer to a cosine curve, and the stress intensity factors K± approach their asymptotic values. Note that when the
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Fig. 5. + , which for two film is sufficiently thick (h0 > 2, i.e.,h0 > 40A when A = 0.05), these values are close to the corresponding values of K∞ − + 16 semi-infinite regions with a curvilinear interfacial boundary and r = 3 are K∞ = 1.13 and K∞ = 3.42.
Acknowledgements This research was supported by the Russian Foundation for Basic Research (11–01–00230 and 12-08-31392) and St Petersburg State University (9.37.129.2011). References 1. 2. 3. 4. 5. 6.
Vikulina YuI, Grekov MA, Kostyrko SA. Model of a film coating with a weakly curved surface. Izv Ross Akad Nauk MTT 2010;(6):16–28. Freund LB. Evolution of waviness on the surface of a strained elastic solid due to stress-driven diffusion. Int J Solid Struct 1995;32(6/7):911–23. Gao H, Niz WD. Surface roughening of heteroepitaxial thin films. Ann Rev Mater Sci 1999;29:173–209. Andrews AM, Speck JS, Romanov AE, Bobeth M, Pompe W. Modeling cross-hatch surface morphology in growing mismatched layers. J Appl Phys 2002;91(4):1933–43. Grekov MA. A slightly curved crack near the joining surface of two different materials. Vestnik SPb Univ Ser 2008;1(1):93–101. Grekov MA, Morozov NF. Some modern methods in mechanics of cracks. In: Adamyan V, et al, editors. Modern Analysis and Applications. Ser. Operator Theory: Advances and Applications, 191. Basel: Birkhauser; 2009. p. 127–42. 7. Gao H. A boundary perturbation analysis for elastic inclusions and interfaces. Int J Solid Struct 1991;28(6):703–25.
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8. Grekov MA. The Perturbation method in the problem of the deformation of a two-component composite with a slightly curved interface. Vestnik SPb Univ Ser 2004;1(1):81–8. 9. Grekov MA. Two types of interface defects. J Appl Math Mech 2011;75(4):476–88. 10. Grekov MA. The Singular Plane Problem of the Theory of Elasticity. St Petersburg: Izd SPb Univ; 2001. 11. Grekov MA, Kostyrko SA. The Stress state of a thin coating under the action of a periodic system of concentrated surface forces. Vestnik SPb Univ Ser 2004;10(4):99–107. 12. Lin’kov AM. The Complex Method of Boundary Integral Equations of the Theory of Elasticity. St Petersburg: Nauka; 1999. 13. Muskhelishvili NI. Some Basic Problems of the Mathematical Theory of Elasticity. Leyden: Noordhoff; 1977. 14. Dundurs J. Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading. Trans ASME J Appl Mech 1969;36(3):650–2. 15. Gao H. Stress concentration at slightly undulating surfaces. J Mech Phys Solids 1991;39(4):443–58. 16. Grekov MA, Makarov SN. Stress concentration on a periodically curved interfacial surface. Vestnik SPb Univ Ser 2008;10(1):25–31.
Translated by P.S.
Contents lists available at SciVerse ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
A film coating on a rough surface of an elastic body夽 M.A. Grekov ∗ , S.A. Kostyrko St Petersburg, Russia
a r t i c l e
i n f o
Article history: Received 2 October 2012
a b s t r a c t A solution of the plane problem of the theory of elasticity for a film–substrate composite is solved by a perturbation method for a substrate with a rough surface. An algorithm for calculating any approximation, which ultimately leads to the solution of the same Fredholm equation of the second kind, is given. Formulae for calculating the right-hand side of this equation, which depends on all the preceding approximations, are derived. An exact solution of the integral equation in the form of Fourier series, whose coefficients are expressed in quadratures, is given in the case of a substrate with a periodically curved surface. The stresses on the flat surface of the film and on the interfacial surface are found in a first approximation as functions of the form of bending of the surface, the mean thickness of the film and the ratio of Young’s moduli of the film and the substrate. It is shown, in particular, that the greatest stress concentration on the film surface occures on a protrusion of the softer substrate. ©2013 © 2013 Elsevier Ltd. All rights reserved.
The influence of the curvilinear shape of the surface of a film coating on the principal characteristics of the stress state of this surface and of the flat surface between the film and the substrate was investigated in Ref. 1. Bending of a film surface can occur in electronic and optoelectronic devices at the stage of growth and heat treatment of the film coating, which are accompanied by condensation and evaporation processes.2 Intense heating and large stresses3 transform the initially smooth film surface into a rough surface. In addition, the unstable state of the plane shape of the film surface and its morphological changes can be due not only to problems of a technological nature, but also to the unevenness of the substrate surface. In the present study small deviations of the interfacial surface from a plane shape are regarded as a defect of the surface, capable of degrading the functional properties of the devices by inducing the local growth of stresses not only on the joining surface itself, but also on the flat surface of the film. Under sufficiently large stresses, associated with mismatching of the parameters of the crystal lattices of the film and the substrate, as well as under the action of an external load, this can result in an even greater build up of dislocations at the interfacial surface,4 the generation and growth of cracks and ultimately in the complete cessation of the operation of the device. To analyse the non-uniform stress state of a film created by a curved interfacial surface, a perturbation method is used below together with the superposition method similar to the way in which the problem of a curvilinear crack located near a flat interfacial surface5,6 and the problem of a film coating with a curvilinear surface on a flat substrate surface1 were solved. Unlike the case of a weakly curved interfacial surface in an infinite body,7–9 here an important role is played by the mean thickness of the film and the difference between the elastic properties of the film and the substrate. Under the assumption that the curvature has a periodic form, the solution of the problem is found in each approximation in the form of Fourier series, whose coefficients are expressed in quadratures. 1. Statement of the problem Consider an elastic body, occupying the half-space x2 < f(x1 ), with a film coating of thickness h0 under the conditions of in-plane deformation or a plane stress state and with loads acting on the flat film surface and stresses at infinity. This enables us to formulate the corresponding plane problem of the theory of elasticity for the semi-infinite region of the complex variable z = x1 + ix2 , consisting of the semi-bounded region 1 , joined along the curvilinear interfacial boundary c to the band 2 (Fig. 1). We will assume that the shape of the interfacial surface differs only slightly from a flat surface and is specified in the x3 = const plane by the equation (1.1)
夽 Prikl. Mat. Mekh., Vol. 77, No. 1, pp. 113–128, 2013. ∗ Corresponding author. E-mail addresses: [email protected] (M.A. Grekov), [email protected] (S.A. Kostyrko). 0021-8928/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2013.04.010
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Fig. 1.
where f(x1 ) is a continuously differentiable periodic function with period , which satisfies the conditions
Thus, the amplitude of the deviation of points on the interfacial surface from the x2 = 0 plane is equal to A = . On the interfacial boundary c there are no discontinuities of the displacement vector u or of the stress (load) vector n , i.e., (1.2) Here
u1 and u2 are the components of the displacement vector along the x1 and x2 axes, and nn and nt are the normal and tangential loads on an area with the normal n (the unit vector n in equalities (1.2) is perpendicular to the line c ). The unit vectors n and t form a right-hand system of coordinates n,t. In the general case the loadqacts on the band boundary b : (1.3) Here
(1.4) whereP1 and P2 are components of the principal vector of the forces acting on a segment of the boundary b within one period. We will assume that the function q(x1 ) satisfies Hölder’s condition everywhere on b . The conditions at infinity are defined by the relations
(1.5) when x2 → –∞, where is the angle of rotation of a material particle.
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2. The superposition method As was done previously,1 in accordance with the superposition principle,10 we represent the solution of n (z), u(z) in the form of the sum of the solutions of two problems using the unified relation (␦k2 is the Kronecker delta) (2.1) in which the functions G(z, k ),Gb (z, 2 ),Gc (z, k ) are equal, respectively, to
k = (3 – 4k ) for in-plane deformation, k = (3–k )/(1 + k ) for a plane stress state, k is Poisson’s ratio, and k is the shear modulus of the medium k . The derivative d/dz is taken in the direction of the vector t. The quantities nb (z) and ub (z) are the load and the displacement in the uniform half-plane D = {z: Imz < h0 } with the elastic properties of the band 2 . On the straight boundary b of this region there are several unknown self-equalized loads p(x1 ) (2.2) which satisfy the periodicity condition
(2.3) and at infinity the stresses and the angle of rotation angle are equal to zero (Problem 1). The quantities cn (z) and uc (z) are the load and the displacement that occur in the two-component plane with the curvilinear interfacial boundary c , on which the load jump cn = cn + –cn – and the displacement jump uc = uc+ –uc– occur, under conditions (1.5) at infinity (Problem 2). Taking relations (2.1) into account, we find the expression for these jumps from the contact conditions between the film and the substrate (1.2) (2.4) When boundary conditions (1.3) and (2.2) are taken into account, the limit transition in relation (2.1) as z→zb ∈ b leads to the following equation, which must be satisfied by the unknown function p: (2.5) The original problem, is thus, reduced to solving Eq. (2.5), in which cn (zb ) is a functional that depends on the functionpand the point zb . Note that in the case of a straight interfacial boundary c , parallel to the x1 axis, cn (zb ) is a Fredholm integral operator, which acts on ¯ and Eq. (2.5) is a Fredholm equation of the second kind with continuous kernels.10,11 When the boundary the functionpand its conjugate p, c has an arbitrary form, the solution of Problem 2 reduces to a singular or hypersingular integral equation,10,12 and it is not possible to express the functions cn (z) and uc (z) in an explicit form in terms of the function p. In the case of a small deviation of the interfacial surface from a plane shape, the perturbation method can also be used to obtain the same explicit expression for the corresponding functions cn (z) and uc (z) for all approximations.8,9 Then, by analogy with the previously described approach,5,6 we find an explicit expression for the operator cn (zb ) in each approximation. 3. Solution of Problem 1 The load bn (z) and the displacement ub (z) in Problem 1 can be expressed10 by a single formula in terms of the complex function (), which is holomorphic in the complex plane = + i2 outside the straight line 2 = 0
(3.1) where ␣ is the angle between the direction of the area (the vector t) and the real axis of the complex variable , i.e., the 1 axis. Here and below the bar denotes the complex conjugate, and the prime denotes the derivative with respect to the argument. The function in relation (3.1) is defined by the equality10,13
(3.2)
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4. Solution of Problem 2 by the perturbation method The load cn (z) and the displacement uc (z) in Problem 2 for the combined deformation of the two uniform semi-bounded regions Dk (k = 1, 2, D1 = 1 ) with the curvilinear interfacial boundary c , which have the elastic properties of the respective media k , are defined by the relation8,9
(4.1) The functions k (z) are holomorphic in the region Dk , and the functions k (z) are holomorphic in Dk = {z:¯z ∈Dk } (k = 1, 2). Assuming in equality (4.1) that Im z→ ± ∞ for ␣ = 0 and ␣ = /2 and taking (1.5) into account, we arrive at the following relations, which specify the values of the functions k and k at infinity:
(4.2) where
(A and B are Dundurs parameters14 ). In relation (4.1) we take the limit as z → zc ∈ c and ␣ → ␣0 , where ␣0 is the angle between the positive direction of the tangent to the curve c at the point zc and the x1 axis. Then we obtain
(4.3) When conditions (2.4), expression (4.1) and the equality
are taken into account, relations (4.3) lead to two boundary conditions for finding the functions k (z) and k (z), which we write in the form of the single equality
(4.4) c = –bn (zc )
Equation (4.4) follows from the first of Eqs (4.3) when mk = k = 1 and and from the second when mk = k , k = –k and c = 2 1 2 (ub )’(zc ). The functions k (zc ) and c (¯zc ) are the limit values of the respective functions at z→zc . Bearing in mind the dependence of the solutions of both problems on the small parameter , we represent the complex potentials , k and k and the unknown function p in the form of the power series
(4.5)
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In turn, we expand the values of the functions m , km and km on c in corresponding Taylor series in the vicinity of the straight line x2 = 0, treating the real variable x1 as a parameter. We obtain
(4.6) Representing expansions (4.5) and (4.6) in Eq. (4.4) and equating the coefficients of the powers m (m = 0, 1, . . .), we arrive at the sequence of boundary conditions
(4.7) The functions Hm (x1 ) with m > 0 are defined in terms of known functions found in the preceding approximations, and the function Zm (x1 ) is an integral operator, which acts on the unknown function pm (x1 ), i.e.,
(4.8)
(4.9) The argument x1 in the function f, as well as on the right-hand side of the second equality in (4.8), has been omitted for brevity. In deriving of relations (4.8) and (4.9) we used the expansion
We introduce the functions
(4.10)
(4.11) which are holomorphic outside the straight line x2 = 0. Then, when relations (4.10) and (4.11) are taken into account, problem (4.7) reduces to the two Riemann–Hilbert boundary-value problems (4.12) (4.13)
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The solutions of problems (4.12) and (4.13) can be written in the form13 (4.14) where
(4.15)
kn and V kn are defined in terms of all the preceding solutions and are, thus, known functions in the m-th approximation. The functions ˙m m un are integral operators that act on the unknown function p and its conjugate. We introduce the notation The functions un and Vm m m
We find expressions for the complex potentials of the second problem in the m-th approximation km and km from relations (4.10) and (4.11)
(4.16) Thus, formulae (4.1), (4.8), (4.9), (4.14), (4.15) and (4.16) give the solution of the second problem in the m-th approximation, which is expressed in terms of the complex potentials of the first problem r (r = 0, 1,. . .,m), which, in turn, are expressed in terms of the functions pr according to formula (3.2) when the functions and p in them are replaced by r and pr . 5. The integral equation of m-th approximation To construct the integral equation we initially note that when expansions (4.5) are taken into account, relations (3.1) and (4.1) are written in the form
(5.1) b and Gc are defined by the same equalities (3.1) and (4.1) after the functions , and are replaced by the The functions Gm k k m corresponding functions m , km and km . By virtue of representations (5.1) we arrive in each approximation at an equation similar to Eq. (2.5)
(5.2) Here cnm is a Fredhom integral operator, which acts on all the functions pr (r = 0,1,. . .,m) and their conjugates, q0 = q, and qm = 0 for m > 0. c (z, ) when z = z = x + ih, ␣ = 0, and = 1. As a result, since c (z ) = Gc (z , We substitute expression (4.16) into the expression for Gm 2 1 2 b nm b m b 1) in this case, Eq. (5.2) takes the form
(5.3) where
(5.4)
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(5.5) Since h0 = / 0, the kernels of Eq. (5.3) K1 and K2 are continuous functions, which vanish at infinity. Thus, Eq. (5.3) is a Fredholm integral equation. In the zeroth approximation its right-hand side equals
When there are no external loads on the film surface, q(x1 ) = ∞ = ∞ = 0, and, as follows from physical arguments, the corresponding 22 12 homogeneous equation (5.3) has only the zeroth solution p0 (x1 ) = 0. We henceforth confine ourselves to the case when q(x1 ) = 0. When expressions (4.8) and (4.15) are taken into account, from equalities (3.2) and (4.16) we find the complex potentials in the zeroth approximation (5.6) They correspond to a uniform stress-strain state of a film and a substrate with a flat interfacial boundary, where 1∞ = s is the 11 longitudinal stress in the substrate, and 2∞ = 11
2 (1+1 ) 1 (1+2 )
= 0 is the longitudinal stress in the film.
6. The first approximation for q(x1 ) = 0 We substitute expressions (5.6) into equalities (4.8). Then (6.1) When relations (4.15) and (6.1) are taken into account, from equality (5.4) we find the right-hand side of Eq. (5.3) in a first approximation
(6.2) Here
(6.3) After the integral equation is solved, the stress-strain state of the film is specified by relations (3.1) and (4.1), and that of the substrate is specified by relations (4.1). In this relation, which was written for the potentials of the second problem in any approximation, the potentials of the first approximation are specified by the equalities
(6.4) where
It should be noted here that the preceding arguments, which include the derivation of integral equation (5.3), are valid in the general case when the shape of the interfacial surface and the loads acting on the surface of the film do not have the property of periodicity. In particular, when there is weak local bending of the interfacial surface, the periodicity conditions of the functions q and p (1.4) and (2.3) can be replaced by their decay conditions at infinity, and integration along the length of a period can be replaced by integration along the ∞ = ∞ = ∞ = 0. entire boundary b . In conditions (1.5) we should then take 22 12 Relations (2.1), (3.1), (4.1), (4.5), (4.8), (4.9), (4.14)–(4.16) and (5.3)–(5.5) do, in fact, give an algorithm for determining the stress-strain state of the film and the substrate for any degree of approximation. Initially, in the zeroth approximation the function p0 is found when integral equation (5.3) with the known right-hand side F0 is solved. Then, the expression for the function F1 is determined using formulae (4.8), (4.15), (4.16) and (5.4). After the function F1 is substituted into Eq. (5.3), its solution p1 is found. The subsequent approximations are constructed according to the same scheme.
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7. A periodically curved interfacial surface In most cases the rough surface of a substrate has a periodic type of bending or can be approximated by one. The periodicity of the shape of the interfacial surface and of the external loadqwith the same period is the reason why the function Fm defined by expression (5.4) is also periodic and can be represented in the form of a Fourier series with known coefficients
(7.1) In this case we obtain an analytical solution of Eq. (5.3) in the form of a Fourier series
(7.2) where
As an example we will assume that the shape of the interfacial surface is specified by the function
(7.3) The real quantity y plays of the role of the parameter that specifies the shape of the surface, and y ∈ (0, ∞). Figure 2 shows the contour of the interfacial surface within one period for y = 0.15, 0.5, and 2.0 (curves 1, 2 and 3, respectively) and ␥ = 0. The figure also presents the dependence of the radius of curvature R of curve (1.1) at the point x1 = 0 (the point of the minimum value of the radius R) on the parameteryfor = 0.05. When ␥ = 0, fairly narrow local depressions in the substrate can be described, and when ␥ = 1, similar protrusions can be described.
Fig. 2.
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Fig. 3.
Substituting expression (7.3) into equality (6.3), we obtain
(7.4)
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Fig. 4.
8. Results of the calculations We will henceforth confine ourselves to the first approximation and present some results that were obtained using formulae (6.2) and (7.1)–(7.4) in the case of a free film surface (q = 0). In this case only the longitudinal stress s acts in the substrate far from the interfacial boundary. Poisson’s ratios were assumed to be equal to 1 = 2 = 0.3. In this case f = rs , where r = E2 /E1 is the ratio of Young’s moduli of the film and the substrate. To obtain numerical results we take a portion of the Fourier series of the function f’(x1 ), whose length was determined to achieve agreement between the exact and approximate values of the minimum radius of curvature (at x1 = 0) with an error no greater than 0.1%. For y = 0.15 it was sufficient to take 22 terms of the Fourier series. It should be noted that the smaller the value of y, the larger the maximum value of the function and, consequently, the smaller the upper limit of the small parameter = A/ (Ais the maximum deviation of the interfacial surface from thex2 = 0 plane, and is the period of the curvilinear shape of this surface). Thus, wheny = 0.15, the condition |f’| < 1 leads to the inequality < 0.063. For this reason, all the results were obtained for = 0.05. The distribution of the ring stresses tt on the interfacial surface is presented on the left-hand side of Fig. 3 for h0 = 2A(+ tt is represented by the solid curves, and – tt is represented by the dashed curves), and the longitudinal stresses 11 on the film surface are presented on the right-hand side of Fig. 3 for h0 = 2A (the solid curves) and h0 = 10 A(the dashed curves). In view of the symmetry of the stresses about the x1 = 0 axis and the periodicity, they are shown for the regionx1 ≥ 0 within one period. The graphs in the upper part of Fig. 3 were constructed for the case of a substrate that is stiffer than the film (r = 1/3), and the graphs in the lower part were constructed for the opposite case (r = 3). In all cases the bottom of the depression atx1 = 0 is located in the stiffer material. Curves 1, 2 and 3 in Fig. 3, as well as in Fig. 4, correspond to the valuesy = 0.15, 0.5 and 2.0. Note that the ring stresses tt were calculated using the well-known estimate tt = 11 + O(2 ) (Ref 15). A comparison of the graphs in Fig. 3 shows that the stresses reach their highest values at the bottom of a depression in the stiffer material (– tt when r = 1/3 and + tt when r = 3). The corresponding stresses at the same point on the other side of the interfacial surface (+ tt when r = 1/3 and – tt when r = 3) are approximately as many times smaller as Young’s modulus of the softer material is smaller than that of the stiffer material. It follows from Fig. 3 (the right-hand side) that the bending of the interfacial surface results in an uneven distribution of the longitudinal stresses on the film surface and, in accordance with Hooke’s law, in an uneven distribution of the longitudinal and transverse strain on it. This, in turn, results in the formation of the relief on the flat surface of the film. The graphs in Fig. 3 show that the highest stresses on the film surface occur directly over a protrusion of the softer substrate (when r = 3). It is seen from Fig. 3 that when h0 = 10A, the stress 11 on the film surface depends weakly on the shape of the relief of the interfacial surface and approaches the values of f . It is noteworthy that when the film is sufficiently thin (h0 = 2A), a smoother depression causes higher stresses on the film surface than a more pointed depression. When h0 = 10A, the opposite picture is observed. This effect is clearly seen in Fig. 4, which shows how the stress intensity factor on the film surface K = 11 (0)/s depands on the ratio h0 /A when r = 3. The influence of the shape of the interfacial surface on the stress intensity factor K± = ± tt (0)/s on this surface is shown in Fig. 5 in the upper part for r = 1/3 and the two film thicknesses h0 = 2A and h0 = 10A and in the lower part for r = 3. It follows from the results that when the film stiffness is three times higher than that of the substrate, the corresponding values of K± for both thicknesses are nearly identical; therefore, in the lower part of Fig. 5 they are represented by a single. It is seen that the dependence of K± onyis most clearly displayed when y < 1, i.e., where the shape of the interfacial surface is sensitive to a variation of this parameter. A syincreases, the shape of the surface becomes closer to a cosine curve, and the stress intensity factors K± approach their asymptotic values. Note that when the
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Fig. 5. + , which for two film is sufficiently thick (h0 > 2, i.e.,h0 > 40A when A = 0.05), these values are close to the corresponding values of K∞ − + 16 semi-infinite regions with a curvilinear interfacial boundary and r = 3 are K∞ = 1.13 and K∞ = 3.42.
Acknowledgements This research was supported by the Russian Foundation for Basic Research (11–01–00230 and 12-08-31392) and St Petersburg State University (9.37.129.2011). References 1. 2. 3. 4. 5. 6.
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Translated by P.S.