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Two types of interface defects
Journal of Applied Mathematics and Mechanics 75 (2011) 476–488
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Two types of interface defects夽 M.A. Grekov St Petersburg, Russia
a r t i c l e
i n f o
Article history: Received 24 June 2010
a b s t r a c t The solution of a plane problem in the theory of elasticity for a two-component body with an interface, a finite part of which is either weakly distorted or is a weakly curved crack is constructed using the perturbation method. In the first case, it is assumed that the discontinuities in the forces and displacements at the interface are known, and, in the second case, the non-equilibrium nature of the load in the crack is taken into account. General quadrature formulae are derived for the complex potentials, which enable any approximation to be obtained in terms of elementary functions in many important practical cases. An algorithm is indicated for calculating each approximation. Families of defects are studied, the form of which is determined by power functions. The effect of the amplitude of the distortion and the shape of the interface crack on the Cherepanov–Rice integral as well as the shape of the distorted part of the interface on the stress concentration is investigated in the first approximation. An analysis of the applicability of the oscillating solution for a distorted interface crack is carried out. The results of the calculations are shown in the form of graphical relations. © 2011 Elsevier Ltd. All rights reserved.
The insufficient strength of the joint between materials with different mechanical properties is the weak link in piecewise homogeneous media to which various laminated structures belong. One of the reasons for the reduction in the bonding strength is the existence of interface defects and, in particular, distortions of the interface. 1 At a specific scale level, the interface will, as a rule, not be flat. In a number of cases, the relief of the interface is formed by the formation of an oxidized coating 2 and plasma deposition. 3,4 Under certain conditions, an interface has a tendency to become non-planar due to its striving to attain a thermodynamically equilibrium state that ensures a minimum of the sum of the energy of deformation and the surface energy. 5,6 The pre-polished surfaces of silicon and germanium semiconductor crystals 7 demonstrate this tendency to stabilize the relief of the surface layer. On the one hand, roughness of the interface between media increases the bonding strength and, on the other hand, the distortion of the surface is the source of the non-uniformity in the stress-strain state that creates the prerequisites for the formation of cracks and the development of an exfoliation process. 1,8 The latter explains the appearance of a number of papers which analyze the local stress distribution at the distorted interface of materials. 1,5,9,10 In this paper, the analytical solutions of problems for two types of defects: a weak deviation of a continuous interface from a flat form and a weak distorted interface crack, are constructed in closed form. To do this, a unified approach is used based on the method of perturbations. Unlike in Ref. 5, in which an analytical solution of the first problem was obtained in the first approximation, in the present paper any approximation for the two problems is found in closed form. Although the solutions of the problems are presented for a single interface in an infinite body, by virtue of the smallness of the distortion amplitude it can be assumed that the interface is an element of a laminated structure. The problems are considered in a more general formulation than has been done previously. 11,12 It is assumed that the discontinuities in the forces and displacements at the interface are known in the first problem and that, in the second problem, the forces acting on the different surfaces of the crack, that are different from one another, are known. This extends the possibilities for using the solutions constructed in the treatment of closely related problems. 1. Formulation of the problem Consider an elastic two-component body which is in a state of plane deformation or a plane stress state. With the exception of a sufficiently small weakly distorted part, the interface of the given composite is plane. We therefore have a two-dimensional problem in
夽 Prikl. Mat. Mekh., Vol. 75, No. 4, pp. 678–697, 2011. E-mail address: [email protected] 0021-8928/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2011.09.012
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Fig. 1.
the theory of elasticity for the two-component plane of the complex variable z = x1 + ix2 constructed from the two semi-bounded domains:
The interface of this composite consists of a rectilinear part
and a weakly distorted part c (Fig. 1). We will assume that either the discontinuities in the forces and displacements on are known (Problem 1) or c is an interface crack and conditions of ideal bonding are satisfied on s (Problem 2). The boundary is considered as a small perturbation of the real x1 axis, which we shall call the unperturbed or basic boundary. The interface is defined by the equation
(1.1) The small parameter is equal to the ratio of the maximum deviation of the curve c from a rectilinear shape to the half-length of the interval [-l, l] on the unperturbed boundary. The function f (x1 ) specifies the shape of the curvilinear part c and is continuously differentiable. The boundary conditions for both problems are written in the following form. Problem 1.
The discontinuities in the forces n and the displacements u are specified at each point of (1.2)
Problem 2.
When there is an interface delamination crack c , we will assume that
f (±l) = 0
and that the conditions
(1.3) are satisfied. The following notation is used in equalities (1.2) and (1.3)
where u1 and u2 are the components of the displacement vector along the x1 and x2 axes, nn and nt are the normal and shear forces on an area with a normal n, and the unit vectors n and t form a right-handed system of coordinates (in relations (1.2) and (1.3) the direction
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of the vector t coincides with the direction of the tangent to ). We will assume that the functions n () and u() in relations (1.2) satisfy the Hölder condition almost everywhere on and that p() and q() satisfy the Hölder condition at all points of the curve c . In both problems, the stresses ij and the angle of rotation of a material particle are specified at infinity (1.4) Moreover, we will assume that, in relations (1.2),
(1.5) Note that Problem 1 with conditions (1.2) is ill-posed in the general case if this problem is considered as an independent problem. Thus, if it is specified that u = / 0 on some finite part of the interface and it is assumed that u≡0 on the remaining part of the boundary, then it is found that the shape of the opening in the part where the layers separate is one and the same for any forces acting in the interface including this part of it. In fact, the quantities u and n are interdependent and satisfy a singular or hypersingular integral equation. 13,14 The satisfaction of the conditions for the ideal bonding of two materials along the whole of the interface, that is, u()=0,∀∈, is one of the consistent formulations of Problem 1. In this case, the jump in the force (the stress vector) n () in the boundary can be arbitrary. Nevertheless, the quantities u and n can be considered as arbitrary up to a specific instant if it is kept in mind that Problem 1 that has been formulated is a component part in the method of superposition. In this case, conditions (1.2) have the right to exist since the connection between these quantities is laid down in this method. The effectiveness of this method when used in solving Problem 1 in the case of a rectilinear boundary is confirmed by many examples. 14,15 2. Basic relations According to results obtained earlier, 14 the force at a point z∈k in an area with normal n and the displacement at this point are expressed in terms of the Gursat–Kolosov potentials 16,17 k (z) and k (z),that are holomorphic in k , using the equalities
(2.1) where ␣ is the angle between the direction of the plane element (with vector t) and the x1 axis measured counterclockwise. k =(3−k )/(1+k ) for a plane stress state, k = 3−4k for plane deformation, and vk and k are Poisson’s ratio and the shear modulus of the medium k . Relation (2.1) is the basis for the use of the unified approach to solving many two-dimensional problems in the theory of elasticity by the perturbation method (for example, see the bibliography in Ref. 12 and, also, Refs 11 and 15). We now introduce the new functions ϒ k (z) (k = 1,2) that are holomorphic in the domains Dk = z : z¯ ∈ k with the boundary ¯ |x1 | < l L = Lc ∪ s , Lc = z : z = , (2.2) As a result, relation (2.1) is transformed to the form (2.3) Integration of this relation when k =-k leads to the expression for the displacements
(2.4) By virtue of conditions (1.2) and (1.3) and relation (2.3), the functions ϒ k (z), defined in the domain Dk by relation (2.2), are a continuous extension of the functions k (z) from k through those rectilinear parts of the interface s , on which n =0. The derivation of equality (2.3) is the first step on the way to constructing a solution by the perturbation method. Putting |z| → ∞ in equality (2.3) when ␣ = 0 and ␣ = /2 and taking relations (1.4) and (1.5) into account, we arrive at the following relations
(2.5)
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3. Solution of the problems In equality (2.3), we now take the limit as z→∈ assuming that ␣→␣0 , where ␣0 is the angle between the positive direction of the tangent to at the point and the x1 axis. Substituting the resulting relations into conditions (1.2), we arrive at the equalities
(3.1) If, however, c is an interface crack, it then follows from relations (1.3) and (2.3) that
(3.2) Bearing in mind the dependence of the solutions of both problems on the small parameter , we represent the values of the complex potentials k and ϒ k in the corresponding domains k and Dk in the form
(3.3) and the boundary values of the coefficients of the expansions in (3.5) on and L, as well as the functions n (), u(), p(), q() on , in the form of corresponding Taylor’s series in the neighbourhood of the straight line x2 = 0, treating the variable x1 as a parameter:
(3.4) Here,
and we mean by the function y each of the functions n , u, p and q. The question of the convergence of the power series in equalities (3.3) for each fixed point z when 1 is not in doubt even close to the singular points, that is, the vertices of the crack since, at the vertices, the coefficients of the expansions of km (z) and ϒ km (z) have one and the same singularity, which can be taken outside the limits of the summation. As far as the Taylor series (3.4) are concerned, the equality g = 0 at the points ±l enables us to assume that they converge everywhere subject to the condition of the infinite differentiability of the expanded functions at all the remaining points. Substituting expressions (2.3), taking account of equalities (3.3), into relations (3.1) and equating coefficients of m (m = 0, 1, . . .) we obtain, in the m-th approximation, two independent boundary conditions (3.5) Conditions (3.2) after the same operations reduce to the following
(3.6) The notation
has been introduced here. Moreover,
(3.7)
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It follows from relations (3.5) - (3.7) that the functions m and Vm are holomorphic outside the line x2 = 0 in Problem 1 and outside the interval |x1 | ≤ l in Problem 2. For each value of m, the functions Hmj (j = 1, 2, 3, 4) are expressed in terms of the limit values of the complex potentials k (x1 ), ϒ k (x1 ) (k = 1, 2, = 0, 1, . . ., m−1) on the line or in the interval according to the formulae
(3.8) where
For brevity, the argument x1 of the last equality on the right-hand side is omitted. It should be noted that, in formulae (3.8), n and u are quantities referring to Problem 1 and q is the jump in the stresses in the crack in Problem 2. According to these formulae, the function Hm (x1 ) when m > 0 is expressed in terms of the complex potentials of the preceding approximations. The representation
has been used in the deriving of relations (3.5) - (3.8). Following the well-known approach, 16 we find the solution of Problem 1 for the interface directly from boundary conditions (3.5)
(3.9) where
Note that the Cauchy-type integral in relations (3.9) converges by virtue of conditions (1.5) and the equality g(x1 ) = 0 when |x1 | ≥ l (see Ref. 16,17). In order to solve Problem 2, we multiply the first equality of (3.6) by −A(2 + 1 2 ), where A is one of the Dundurs parameters, 18 and add it to the second equality. The second equality is then transformed to the following (3.10) Here,
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By virtue of the holomorphicity of the functions m and Vm outside the interval |x1 | < l, from the last two boundary conditions (3.6) and condition (3.10), according to the well-known approach, 16 we find
(3.11) √
1
The branches of the multivalued functions in formula (3.11) are determined by the equalities 1 = 1, = 1, where is any complex number. The coefficient Bm is equal to the value of the left-hand side of equality (3.11) at infinity, and Am is equal to the coefficient of the second term of the expansion of this side of the equality at infinity, which is determined by the complex value F = F1 + iF2 of the principal vector of the forces F = (F1 , F2 ) applied to the surface of the interface crack in the second problem. The equality
holds for the quantity F. When account is taken of the expansion of the function q() in the Taylor series (3.4), it follows from the last equality that
(3.12) Bearing in mind expansion (3.12) and, also, relations (3.3) and (3.7), it can be
shown14
that
(3.13) Since
it follows from relations (3.9), (3.11) and (3.13) that
(3.14) Hence, the solutions of both problems are constructed for any approximation in quadratures. Each successive approximation is expressed in terms of all the preceding approximations. The complex potentials of the m-th approximation are found by solving system (3.7)
(3.15) In the case of Problem 1, the functions m and Vm are determined by formulae (3.8) and (3.9) and, for Problem 2, by (3.8), (3.11), (3.12) and (3.14). The stress-strain state of the composite is then found using formulae (2.3), (2.4), (3.3) and (3.15). It is important to note that the functions m and Vm in relations (3.9) and (3.121) can be obtained in many important practical cases in the form of algebraic expressions of elementary functions. This is possible in the case of a fairly smooth form of distortion and smooth boundary conditions on the surface of the crack, that is, when the functions g, p and q can be approximated by polynomials to a sufficient degree of accuracy.
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4. The zeroth approximation Either a continuous rectilinear interface or a rectilinear interface crack corresponds to the value m = 0. The general solution of Problem 1 has been constructed in this case. 14 Problem 2 has been studied by many authors in different formulations of whom Williams 19 was apparently the first. The solution of several versions of this problem can also be found in the monograph 14 and other papers in the bibliography given there. Problems of the propagation of a rectilinear interface crack have also been investigated. 20,21 When n = u≡0, a piecewise-homogeneous stress-strain state occurs in a medium with a continuous rectilinear boundary that, according to (2.5), is determined by the values of the complex potentials at infinity:
(4.1) Here, account has to be taken of the following relations, that follow from the first equality of (2.5)
(4.2) where
The coefficient B is identical to the second Dundurs parameter .18 The simplest version of the zeroth approximation of Problem 1 is the problem of the action of a concentrated force P=(P1 ,P2 ) at a certain point x10 of a rectilinear interface. In this case, it is obvious that
For null conditions at infinity, it then follows from relations (3.8) and (3.9) that
(4.3) The complex potentials k0 ϒ k0 (k = 1, 2) are then found using formulae (3.15). In the case of a rectilinear crack at the interface of two media, the solution of the corresponding problem is physically possible, generally speaking, only when there are normal forces on the surfaces of the crack and the stresses at infinity ∞ and k∞ . In these cases, the size 11 22 of the oscillation zone close to the crack tips is negligibly small, 14 and, on account of this, to solve Problem 2 we put Imp = Imq = ∞ =0 12 = ∞ = 0. Moreover, bearing in mind that the nature of the origin of the forces at the crack is associated with the and, for simplicity, ∞ 1 2 pressure of a gas or a liquid which has penetrated into the cavity, we shall henceforth assume that
With these assumptions, it follows from relations (3.11) that
(4.4) where
(4.5) Expression (4.4) and (4.5) conform with the oscillating solution obtained in a number of papers for a rectilinear interface crack when q0 = 0. In particular, under the action of only longitudinal stresses at infinity 1∞ , 2∞ and, also, when p0 = ∞ , such a crack does not open 11 11 22 and each half-plane is in a homogeneous stress-strain state. When 1 = 2 , 1 = 2 , and also when q0 = / 0, formulae (3.19), (4.4) and (4.5) give a solution of the problem of a crack in a homogeneous plane, which is identical to Muskhelishvili’s solution 16 if, in the latter solution, account is taken of the assumptions made above regarding the acting loads. The question of the opening of an interface crack when q0 = / 0 requires special discussion.
14
5. The first approximation The function H1 in relations (3.8) has the form
(5.1)
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and it follows from relations (3.8), (4.1) and (5.1) that, in the case of Problem 1 when u = n ≡ 0, ∞ = ∞ =0 1 2
(5.2) For Problem 2 when p = const and q = const
(5.3) where
It follows from relations (5.2) and (5.3) that the first approximation depends on the form of the defect. At the same time, if the load is determined exclusively by the stresses at infinity, then, in Problem 1, this approximation only depends on the derivative g (x1 ), that is, on the angle of inclination of the tangent to the interface, and, in Problem 2, further on the magnitude of the deviation of the crack g(x1 ) from a rectilinear shape. When n = / 0 in Problem 1, a dependence of the first approximation on the function g also holds as, for example, in the case when the zeroth approximation has the form (4.3). A curvilinear crack, located between two identical media and that is only opened by an internal pressure p0 <0 possesses an interesting property. In this case H1 (x1 )≡0, and this crack behaves in the same way as the corresponding rectilinear crack apart from the small quantity . Focussing our attention on the stress state of the interface , we write out the expressions for the stresses in this interface in the first approximation. From relations (2.3) and (3.3), it follows that
(5.4) whence the stresses on the other side of the interface can be obtained by replacing the subscript 2 by 1. We shall henceforth assume that the distortion of an interface itself and an interface crack is described in Eq. (1.1) by a function that is symmetrical about the origin of the coordinates x1 = 0: (5.5) The corresponding form of the defect is characterized by the fact that the whole of the interface is smooth and, when the parameter n is increased, the distorted part is progressively localized around the middle of the defect. An example of this defect when n = 3 is shown in Fig. 1. Although the function (5.5) is not a polynomial for any value of n, all the integrals in formulae (3.9) and (3.11) can be expressed in terms of elementary functions. A curvilinear interface without a crack. In the case of Problem 1 that takes account of relations (5.2), we find the integrals in equalities (3.9)
(5.6) where, using the properties of a Cauchy-type integral, 16
Here z1 = z/l, and Qn (z1 ) is a polynomial of degree n (n = 3, 4,...), the principal part in the expansion at infinity of the corresponding first term in the square brackets. Taking account of equalities (4.1) and ∞ = ∞ = 0, relations (5.4) take the simple form 1 2
(5.7)
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Fig. 2.
Using relations (3.9), (3.15), (5.6) and (5.7), the loads at any point of the interface can be calculated accurately up to o terms if the stresses , 2∞ , ∞ act at infinity. An analysis shows that, when ∞ = ∞ = 0 and 1 /2 < 1, the circumpherential stresses reach a maximum 1∞ 11 11 22 22 12 value at the point x1 = 0, that is, at the bottom of the hollow in the harder material 2 . The results of the calculations, for the case of plane deformation and the same Poisson’s ratios 1 = 2 = 0.3, are shown in Figs. 2 and 3 for n = 3, 9, 15. The dependences of the coefficient
on the ratio 1 /2 are shown in Fig. 2. It can be seen that the greater the difference between the shear moduli j of the materials (in the case of a Young’s modulus Ej ) the greater the coefficient of concentration of the circumpherential stresses K = 1 + T, which reaches its maximum value in the case when there is no medium 1 (1 /2 = 0). For a fixed value of , an increase in the parameter n leads to an increase in the concentration coefficient K, associated with a decrease in the radius of curvature at the point x1 = 0.
Fig. 3.
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When the circumpherential stresses tt are considerably greater than the longitudinal stresses 2∞ acting far from the defect, this can 11 give rise to cracks that are normal to the interface. Moreover, the distorted part of the interface is a concentrator not only of circumpherential stresses. Thus, if a stress ∞ acts far from the defect, then it is observed that the normal stress nn considerably exceeds ∞ in the distorted 22 22 part of the interface which can lead to delamination and the development of an interface crack. As an example, graphs of the change in the stresses nn (the solid curves) and tt (the dashed curves), obtained in the first approximation in the case of an isotropic stress 2∞ = ∞ = > 0 in the medium 2 when 1 /2 = 3, = 0.3, are shown in Fig. 3. 11 22 Note that, when 1 /2 > 1, the medium 1 is more rigid and the curve c for it has the shape of a protrusion. In this case, the vertex of the protrusion corresponds to the value x1 = 0 and the points on c close to the bottom of the protrusion become concentrators of the stresses tt as was revealed 22 in the case of longitudinal stretching for 2 /1 > 0. It can be seen from Fig. 3 that the points of a maximum in tt are shifted towards the middle of the protrusion as the parameter n increases, that is, in proportion to the localization extent of the distorted part of the interface. Calculations show that, in the case of longitudinal stretching, the positions of the maxima in tt near the protrusion are practically the same as in the case of bi-axial stretching in Fig. 3. However, the maximum values of the stresses tt are approximately 10% higher in the first case than in the second. A curvilinear interface. By analogy with Problem 1, the functions 1 and V1 can be obtained in closed form for any value of n > 2 in equality (5.5) by substituting expressions (5.1) and (5.3) into relations (3.11), taking account of relations (4.4) and (3.15) and the complex potentials (3.15) can be obtained in terms of them. We leave aside the unwieldy expressions for these functions and turn our attention to the stress state near the crack tips. Arguing in the same way as earlier 13 for a curvilinear crack in a homogeneous body, it can be shown that, in the case of the curvilinear interface cracks considered, the complex potentials and, consequently, the stresses for any values of and n > 2 have the same form of asymptotic formulae as in the case of a rectilinear interphase crack corresponding to the value = 0. This is due to the fact that the tangent to the cracks at the tip is the same in all cases. The crack geometry only has an effect on the coefficients in the first term of the asymptotic expansion of the stresses close to the crack tips K1 and K2 for which, in the case of the tip x1 = l, the equality 14,22
(5.8) holds. Substituting expression (2.3) into equality (5.8) when k =1, ␣=0, x1 >1 and taking account of relations (3.3) and (3.15), to a first approximation we obtain
(5.9) whence, on the basis of relations (3.11) and (4.4), we obtain (5.10)
(5.11) 14
for a rectilinear interface crack. If a rectilinear crack When q0 = 0, the quantity K0 is identical to the corresponding coefficient obtained ∞ ≥ p , that is, is located between two identical materials, then ␥ = A = 0. It follows from relations (5.11) in this case that K10 ≥ 0 when 22 0 the crack opens up regardless of the value of the quantity q0 . Requiring that K10 ≥ 0 also for a rectilinear interface crack, we arrive at the inequality (5.12) where =arctg 2␥. This inequality, like the coefficients K1 and K2 themselves in equality (5.10), makes sense if the coordinates are assumed to be dimensionless, having been divided by the half-length of the interface crack l, from the very start. It is then necessary to put l = 1 in relations (5.8) - (5.12). ≥ p0 since 2 ␥ ≤ ln 3 in the case of plane deformation and It can be shown that inequality (5.12) is satisfied when q0 = 0 if ∞ 22 0 ≤ j ≤ 1/2. In this case, it is well known that the oscillation zones are small and comparable with the zones of penetration of the surfaces of the crack. 14 If q0 = / 0, then, in the general case, it is only after an analysis of the opening of the crack and an estimation of the penetration zones that it is possible to establish whether there is a relation between inequality (5.12) and the condition of smallness of these zones or = p0 . It then follows from the equality (5.12) when not. However, this is obvious in two limit cases. Actually, for simplicity, suppose ∞ 22
that the condition q ≤ 0 must be satisfied. This means that the less rigid part of the surface of the crack is under a normal pressure 1 2 0 + + nn < 0 and a tensile normal force − nn = −nn > 0 is applied to the other, considerably more rigid part. Conversely, when 2 1 : q0 ≥ 0. It is clear that the penetration zones are negligibly small in both cases. They are identical to the analogous zones near a crack which − is located between elastic and absolutely rigid half-planes and which is opened by a constant normal force + nn < 0 (or nn < 0). This conclusion also obviously holds in the case of the weakly distorted crack described by function (5.5). As far as the oscillatory solution of the problem for the curvilinear cracks considered is concerned, a calculation of the jump in the displacement vector u in the first approximation for the most characteristic loads and loads realized in practice ± nn = −p0 < 0 and/or > 0, 1∞ > 0 when q0 = 0 showed that the size of the zone of penetration of the surface is of the same order as in the case of the ∞ 22 11
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Fig. 4.
corresponding rectilinear crack. In order to find the quantity u, it was sufficient to integrate the left-hand side of the penultimate equality of (3.2) using relations (2.3) and the expressions for the complex potentials obtained according to the scheme described above. In the case of an interface crack, the coefficients K1 and K2 do not have a definite physical meaning and they cannot therefore serve as characteristics of its limit state. The joining of two materials with elastic constants that satisfy the condition B = 0 is an exception. In the general case, the Cherepanov–Rice integral J, which is equal to the rate of release of the specific strain energy as the crack grows on both sides along the interface, is such a characteristic. 23,24 According to reference data 25
(5.13)
K12 + K22 and the integral J in the first approximation, obtained for the case of plane Some results of calculations of the quantity K = √ deformation and the values 1 = 2 = 0.3, p0 = q0 = 0, are shown in Figs. 4–6. Graphs of the dimensionless quantity k = K / ∞ l 22
against the ratio of the shear moduli of the materials 2 /1 in the case of uniaxial (∞ > 0, ∞ = 0, Fig. 4) and biaxial (1∞ = ∞ /2, 22 11 11 22 Fig. 5) stretching are shown in Figs. 4 and 5. The shape of the crack is determined by function (5.5) when n = 4 (the dashed curves) and when n = 16 (the dot-dash curves). Comparison of the corresponding curves for n = 4 and n = 16 enables us to reveal a general regularity that involves the fact that, as n increases, that is, in the case of a greater and greater localization of the distorted part of the crack around its middle, the value of the quantity k for a curvilinear crack differs to a lesser and lesser extent from the corresponding value for a rectilinear crack. It also follows from Figs. 4 and 5 that additional tension along the interface leads to a convergence of the graphs for a curvilinear crack and the corresponding graphs for a rectilinear crack. A characteristic feature of the relations presented is the fact that the maximum value of the quantity k is reached in the case when the materials are similar in their elastic properties, that is, 2 /1 ≈1. Graphs of the dimensionless integral J* = J/Jh against the ratio 2 /1 in the case of uniaxial tension when n = 4 are shown in Fig. 6. The integral (5.13) for a rectilinear crack located in a homogeneous elastic medium with constants vk and k under the action of stresses ∞ 22 is denoted by Jh , that is,
(5.14) The elastic constants of the medium 1 (in relation (5.14), the subscript k = 1) were fixed if 2 /1 < 1 when calculating the quantity J* and the elastic constants of the medium 2 (k = 2) were fixed if 2 /1 > 1. As a result, as can be seen in Fig. 6, a reduction in the stiffness of any component of the composite leads to an increase in J*. If the domains in which the ratio 2 /1 change places, we arrive at the fact that an increase in the stiffness of any component reduces the quantity J* . 25 The effect of the distortion of the crack on the integral J* is most > 0 when n = 4. Calculations showed that, when the parameter n is increased, the difference perceptible in the case of uniaxial tension ∞ 22
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Fig. 5.
Fig. 6.
between the integrals J* for a distorted crack and a rectilinear crack becomes smaller. In the case of the additional action of longitudinal tension ∞ > 0, the integral J* for a slightly distorted crack ( = 1/2) is practically the same as for a rectilinear crack even in the case when 11 n = 4. 6. Conclusion In combination, relations (2.2), (2.3), (3.3). (3.8), (3.9). (3.11), (3.14) and (3.15) give the solution of the problems formulated in closed form for any approximation. As an example, the symmetrical form of the distorted part of an interface (5.5), that smoothly changes into
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the plane part of the interface, has been considered. The arbitrary form of a local defect when the same smoothness is preserved can be approximated to a sufficient degree of accuracy by the function
where Qk (x1 ) is a k-th degree polynomial. In this case, any approximation in Problem 1, as well as for a sufficient degree of smoothness of the boundary conditions also in Problem 2, can be represented in terms of elementary functions. At the same time, as shown above, the first approximation already suffices to estimate the effect of the form and amplitude of the distortion of a defect on the stress concentration and limit state of the crack. When a more exact analysis is necessary, succeeding approximations can be taken into account using the algorithm described above. Acknowledgements This research was supported by the Russian Foundation for Basic Research (11.01-00230) and St Petersburg State University (9.0.165.2009, 9.37.129.2011). References 1. Gunnars J, Wikman B, Hogmark S. Effect of non-planar interfaces in layered materials subjected to residual stress. In: Gunnars J, editor. On Fracture of Layered Materials. Lulea: Lulea University of Technology; 1997. p. 1–33. 2. Liu YY, Nateso K. The adherence of nickel oxide on nickel during high-temperature oxidation Mater. Res Soc Symp Proc Adhesion in solids Reno 1988;119:213–56. 3. Steeper TJ, Varacalle DJ, Wilson GC, Idaho GA. Design of experimental study of plasma-sprayed alumina-titania coatings. J Thermal Spray Technol 1993;2(3):251–6. 4. Pukh VP, Baikova LG, Zvonareva, et al. The effect of a coating made of hydrogenated carbon on the strength and crack stability of glass in a high-strength state. Fiz Tverd Tela 2001;43(3):82–6. 5. Gao H. A boundary perturbation analysis for elastic inclusions and interfaces. Int J Solids and Struct 1991;28(6.):703–25. 6. Kung H, Chang H, Gibala R. Interfacial structures of MoSi2 −Mo5 Si3 eutectic alloys. Mat Res Soc Symp Proc Structure and Properties of Interfaces in Materials Boston 1992;238:599–604. 7. Betekhtin VI, Gorobei NN, Korsukov VE, Luk’yanenko AS, Obidov BA, Tomilin AN. Features of defect formation on the deformed (111) surface of silicon. Pis’ma v ZhTF 2002;28(21):29–35. 8. Evans AG, He MY, Hutchinson JW. Effect of non-planarity on the mixed mode fracture resistance of bimaterial interface. Acta Metall 1989;37(3):909–16. 9. Gunnars J, Alahelisten A. Thermal stresses in diamond coatings and their influence on coating wear and failure. Surf Coat Technol 1996;80:303–12. 10. Evans AG, He MY, Hutchinson JW. Effect of interface undulations on the thermal fatigue of thin films and scales on metal substrates. Acta Mater 1997;45(9):3543–54. 11. Grekov MA. The method of perturbations in the problem of the deformation of a two- component composite with a slightly distorted interface. Vestrik Sankt-Peterburg Univ Ser 1 Mat Mekh Astron 2004;1:81–8. 12. Grekov MA. The perturbation method applied to some problems in the theory of elasticity. In: Ivlev DD, Morozov NF, editors. Problems of the Mechanics of Deformable Solids and Rocks: Collection of Papers on the Occasion of the 75-th Birthday of Ye I. Shemyakin. Moscow: Fizmatlit; 2006. p. 188–98. 13. Lin’kov AM. The Complex Method of Boundary Integral Equations in the Theory of Elasticity. St Petersburg: Nauka; 1999. 14. Grekov MA. The Singular Plane Problem in the Theory of Elasticity. St Petersburg: Izd Univ; 2001. 15. Grekov MA, Morozov NF. Some modern methods in mechancs of cracks. In: Adamyan V, et al., editors. Modern Analysis and Application. Ser. Operator Theory: Advances and Applications, 191. Basel: Birkhauser; 2009. p. 127–42. 16. Muskhelishvili NI. Some Basic Problems in the Mathematical Theory of Elasticity. Leyden: Noordhoff; 1975. 17. Kolosov GV. The Use of Complex Diagrams and the Theory of Functions of a Complex Variable in the Theory of Elasticity. Leningrad–Moscow: ONTI; 1935. 18. Dundurs J. Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading. Trans ASME J Appl Mech 1969;36(3):650–2. 19. Williams ML. The stresses around a fault or crack in dissimalar media. Bull Seismol Soc Amner 1959;49(2):199–204. 20. Simonov IV. Prediction of arbitrary crack growth from the interface between two dissimialr elastic materials. Int J Fracture 1992;57(4.):349–63. 21. Simonov I, Osipenko K. Elastodynamic well-defined fields around a propagating interface open-crack edge. Int J Fracture 2002;116(4.):297–312. 22. Grekov MA, Makarov SN. Stress concentration near the slightly distorted part of the surface of an elastic body. Izv Ross Akad Nauk MTT 2004;6:53–61. 23. Rice JR, Sih GC. Plane problems of cracks in dissimilar media. Trans ASME Ser E J Appl Mech 1965;32(2):418–23. 24. Cherepanov GP. Mechanics of Brittle Fracture. New York: McGraw-Hill; 1979. 25. Stress Intensity Factors. In: Murakami, editor. Handbook. Oxford etc; 1987.
Translated by E.L.S.
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Two types of interface defects夽 M.A. Grekov St Petersburg, Russia
a r t i c l e
i n f o
Article history: Received 24 June 2010
a b s t r a c t The solution of a plane problem in the theory of elasticity for a two-component body with an interface, a finite part of which is either weakly distorted or is a weakly curved crack is constructed using the perturbation method. In the first case, it is assumed that the discontinuities in the forces and displacements at the interface are known, and, in the second case, the non-equilibrium nature of the load in the crack is taken into account. General quadrature formulae are derived for the complex potentials, which enable any approximation to be obtained in terms of elementary functions in many important practical cases. An algorithm is indicated for calculating each approximation. Families of defects are studied, the form of which is determined by power functions. The effect of the amplitude of the distortion and the shape of the interface crack on the Cherepanov–Rice integral as well as the shape of the distorted part of the interface on the stress concentration is investigated in the first approximation. An analysis of the applicability of the oscillating solution for a distorted interface crack is carried out. The results of the calculations are shown in the form of graphical relations. © 2011 Elsevier Ltd. All rights reserved.
The insufficient strength of the joint between materials with different mechanical properties is the weak link in piecewise homogeneous media to which various laminated structures belong. One of the reasons for the reduction in the bonding strength is the existence of interface defects and, in particular, distortions of the interface. 1 At a specific scale level, the interface will, as a rule, not be flat. In a number of cases, the relief of the interface is formed by the formation of an oxidized coating 2 and plasma deposition. 3,4 Under certain conditions, an interface has a tendency to become non-planar due to its striving to attain a thermodynamically equilibrium state that ensures a minimum of the sum of the energy of deformation and the surface energy. 5,6 The pre-polished surfaces of silicon and germanium semiconductor crystals 7 demonstrate this tendency to stabilize the relief of the surface layer. On the one hand, roughness of the interface between media increases the bonding strength and, on the other hand, the distortion of the surface is the source of the non-uniformity in the stress-strain state that creates the prerequisites for the formation of cracks and the development of an exfoliation process. 1,8 The latter explains the appearance of a number of papers which analyze the local stress distribution at the distorted interface of materials. 1,5,9,10 In this paper, the analytical solutions of problems for two types of defects: a weak deviation of a continuous interface from a flat form and a weak distorted interface crack, are constructed in closed form. To do this, a unified approach is used based on the method of perturbations. Unlike in Ref. 5, in which an analytical solution of the first problem was obtained in the first approximation, in the present paper any approximation for the two problems is found in closed form. Although the solutions of the problems are presented for a single interface in an infinite body, by virtue of the smallness of the distortion amplitude it can be assumed that the interface is an element of a laminated structure. The problems are considered in a more general formulation than has been done previously. 11,12 It is assumed that the discontinuities in the forces and displacements at the interface are known in the first problem and that, in the second problem, the forces acting on the different surfaces of the crack, that are different from one another, are known. This extends the possibilities for using the solutions constructed in the treatment of closely related problems. 1. Formulation of the problem Consider an elastic two-component body which is in a state of plane deformation or a plane stress state. With the exception of a sufficiently small weakly distorted part, the interface of the given composite is plane. We therefore have a two-dimensional problem in
夽 Prikl. Mat. Mekh., Vol. 75, No. 4, pp. 678–697, 2011. E-mail address: [email protected] 0021-8928/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2011.09.012
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Fig. 1.
the theory of elasticity for the two-component plane of the complex variable z = x1 + ix2 constructed from the two semi-bounded domains:
The interface of this composite consists of a rectilinear part
and a weakly distorted part c (Fig. 1). We will assume that either the discontinuities in the forces and displacements on are known (Problem 1) or c is an interface crack and conditions of ideal bonding are satisfied on s (Problem 2). The boundary is considered as a small perturbation of the real x1 axis, which we shall call the unperturbed or basic boundary. The interface is defined by the equation
(1.1) The small parameter is equal to the ratio of the maximum deviation of the curve c from a rectilinear shape to the half-length of the interval [-l, l] on the unperturbed boundary. The function f (x1 ) specifies the shape of the curvilinear part c and is continuously differentiable. The boundary conditions for both problems are written in the following form. Problem 1.
The discontinuities in the forces n and the displacements u are specified at each point of (1.2)
Problem 2.
When there is an interface delamination crack c , we will assume that
f (±l) = 0
and that the conditions
(1.3) are satisfied. The following notation is used in equalities (1.2) and (1.3)
where u1 and u2 are the components of the displacement vector along the x1 and x2 axes, nn and nt are the normal and shear forces on an area with a normal n, and the unit vectors n and t form a right-handed system of coordinates (in relations (1.2) and (1.3) the direction
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of the vector t coincides with the direction of the tangent to ). We will assume that the functions n () and u() in relations (1.2) satisfy the Hölder condition almost everywhere on and that p() and q() satisfy the Hölder condition at all points of the curve c . In both problems, the stresses ij and the angle of rotation of a material particle are specified at infinity (1.4) Moreover, we will assume that, in relations (1.2),
(1.5) Note that Problem 1 with conditions (1.2) is ill-posed in the general case if this problem is considered as an independent problem. Thus, if it is specified that u = / 0 on some finite part of the interface and it is assumed that u≡0 on the remaining part of the boundary, then it is found that the shape of the opening in the part where the layers separate is one and the same for any forces acting in the interface including this part of it. In fact, the quantities u and n are interdependent and satisfy a singular or hypersingular integral equation. 13,14 The satisfaction of the conditions for the ideal bonding of two materials along the whole of the interface, that is, u()=0,∀∈, is one of the consistent formulations of Problem 1. In this case, the jump in the force (the stress vector) n () in the boundary can be arbitrary. Nevertheless, the quantities u and n can be considered as arbitrary up to a specific instant if it is kept in mind that Problem 1 that has been formulated is a component part in the method of superposition. In this case, conditions (1.2) have the right to exist since the connection between these quantities is laid down in this method. The effectiveness of this method when used in solving Problem 1 in the case of a rectilinear boundary is confirmed by many examples. 14,15 2. Basic relations According to results obtained earlier, 14 the force at a point z∈k in an area with normal n and the displacement at this point are expressed in terms of the Gursat–Kolosov potentials 16,17 k (z) and k (z),that are holomorphic in k , using the equalities
(2.1) where ␣ is the angle between the direction of the plane element (with vector t) and the x1 axis measured counterclockwise. k =(3−k )/(1+k ) for a plane stress state, k = 3−4k for plane deformation, and vk and k are Poisson’s ratio and the shear modulus of the medium k . Relation (2.1) is the basis for the use of the unified approach to solving many two-dimensional problems in the theory of elasticity by the perturbation method (for example, see the bibliography in Ref. 12 and, also, Refs 11 and 15). We now introduce the new functions ϒ k (z) (k = 1,2) that are holomorphic in the domains Dk = z : z¯ ∈ k with the boundary ¯ |x1 | < l L = Lc ∪ s , Lc = z : z = , (2.2) As a result, relation (2.1) is transformed to the form (2.3) Integration of this relation when k =-k leads to the expression for the displacements
(2.4) By virtue of conditions (1.2) and (1.3) and relation (2.3), the functions ϒ k (z), defined in the domain Dk by relation (2.2), are a continuous extension of the functions k (z) from k through those rectilinear parts of the interface s , on which n =0. The derivation of equality (2.3) is the first step on the way to constructing a solution by the perturbation method. Putting |z| → ∞ in equality (2.3) when ␣ = 0 and ␣ = /2 and taking relations (1.4) and (1.5) into account, we arrive at the following relations
(2.5)
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3. Solution of the problems In equality (2.3), we now take the limit as z→∈ assuming that ␣→␣0 , where ␣0 is the angle between the positive direction of the tangent to at the point and the x1 axis. Substituting the resulting relations into conditions (1.2), we arrive at the equalities
(3.1) If, however, c is an interface crack, it then follows from relations (1.3) and (2.3) that
(3.2) Bearing in mind the dependence of the solutions of both problems on the small parameter , we represent the values of the complex potentials k and ϒ k in the corresponding domains k and Dk in the form
(3.3) and the boundary values of the coefficients of the expansions in (3.5) on and L, as well as the functions n (), u(), p(), q() on , in the form of corresponding Taylor’s series in the neighbourhood of the straight line x2 = 0, treating the variable x1 as a parameter:
(3.4) Here,
and we mean by the function y each of the functions n , u, p and q. The question of the convergence of the power series in equalities (3.3) for each fixed point z when 1 is not in doubt even close to the singular points, that is, the vertices of the crack since, at the vertices, the coefficients of the expansions of km (z) and ϒ km (z) have one and the same singularity, which can be taken outside the limits of the summation. As far as the Taylor series (3.4) are concerned, the equality g = 0 at the points ±l enables us to assume that they converge everywhere subject to the condition of the infinite differentiability of the expanded functions at all the remaining points. Substituting expressions (2.3), taking account of equalities (3.3), into relations (3.1) and equating coefficients of m (m = 0, 1, . . .) we obtain, in the m-th approximation, two independent boundary conditions (3.5) Conditions (3.2) after the same operations reduce to the following
(3.6) The notation
has been introduced here. Moreover,
(3.7)
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It follows from relations (3.5) - (3.7) that the functions m and Vm are holomorphic outside the line x2 = 0 in Problem 1 and outside the interval |x1 | ≤ l in Problem 2. For each value of m, the functions Hmj (j = 1, 2, 3, 4) are expressed in terms of the limit values of the complex potentials k (x1 ), ϒ k (x1 ) (k = 1, 2, = 0, 1, . . ., m−1) on the line or in the interval according to the formulae
(3.8) where
For brevity, the argument x1 of the last equality on the right-hand side is omitted. It should be noted that, in formulae (3.8), n and u are quantities referring to Problem 1 and q is the jump in the stresses in the crack in Problem 2. According to these formulae, the function Hm (x1 ) when m > 0 is expressed in terms of the complex potentials of the preceding approximations. The representation
has been used in the deriving of relations (3.5) - (3.8). Following the well-known approach, 16 we find the solution of Problem 1 for the interface directly from boundary conditions (3.5)
(3.9) where
Note that the Cauchy-type integral in relations (3.9) converges by virtue of conditions (1.5) and the equality g(x1 ) = 0 when |x1 | ≥ l (see Ref. 16,17). In order to solve Problem 2, we multiply the first equality of (3.6) by −A(2 + 1 2 ), where A is one of the Dundurs parameters, 18 and add it to the second equality. The second equality is then transformed to the following (3.10) Here,
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By virtue of the holomorphicity of the functions m and Vm outside the interval |x1 | < l, from the last two boundary conditions (3.6) and condition (3.10), according to the well-known approach, 16 we find
(3.11) √
1
The branches of the multivalued functions in formula (3.11) are determined by the equalities 1 = 1, = 1, where is any complex number. The coefficient Bm is equal to the value of the left-hand side of equality (3.11) at infinity, and Am is equal to the coefficient of the second term of the expansion of this side of the equality at infinity, which is determined by the complex value F = F1 + iF2 of the principal vector of the forces F = (F1 , F2 ) applied to the surface of the interface crack in the second problem. The equality
holds for the quantity F. When account is taken of the expansion of the function q() in the Taylor series (3.4), it follows from the last equality that
(3.12) Bearing in mind expansion (3.12) and, also, relations (3.3) and (3.7), it can be
shown14
that
(3.13) Since
it follows from relations (3.9), (3.11) and (3.13) that
(3.14) Hence, the solutions of both problems are constructed for any approximation in quadratures. Each successive approximation is expressed in terms of all the preceding approximations. The complex potentials of the m-th approximation are found by solving system (3.7)
(3.15) In the case of Problem 1, the functions m and Vm are determined by formulae (3.8) and (3.9) and, for Problem 2, by (3.8), (3.11), (3.12) and (3.14). The stress-strain state of the composite is then found using formulae (2.3), (2.4), (3.3) and (3.15). It is important to note that the functions m and Vm in relations (3.9) and (3.121) can be obtained in many important practical cases in the form of algebraic expressions of elementary functions. This is possible in the case of a fairly smooth form of distortion and smooth boundary conditions on the surface of the crack, that is, when the functions g, p and q can be approximated by polynomials to a sufficient degree of accuracy.
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4. The zeroth approximation Either a continuous rectilinear interface or a rectilinear interface crack corresponds to the value m = 0. The general solution of Problem 1 has been constructed in this case. 14 Problem 2 has been studied by many authors in different formulations of whom Williams 19 was apparently the first. The solution of several versions of this problem can also be found in the monograph 14 and other papers in the bibliography given there. Problems of the propagation of a rectilinear interface crack have also been investigated. 20,21 When n = u≡0, a piecewise-homogeneous stress-strain state occurs in a medium with a continuous rectilinear boundary that, according to (2.5), is determined by the values of the complex potentials at infinity:
(4.1) Here, account has to be taken of the following relations, that follow from the first equality of (2.5)
(4.2) where
The coefficient B is identical to the second Dundurs parameter .18 The simplest version of the zeroth approximation of Problem 1 is the problem of the action of a concentrated force P=(P1 ,P2 ) at a certain point x10 of a rectilinear interface. In this case, it is obvious that
For null conditions at infinity, it then follows from relations (3.8) and (3.9) that
(4.3) The complex potentials k0 ϒ k0 (k = 1, 2) are then found using formulae (3.15). In the case of a rectilinear crack at the interface of two media, the solution of the corresponding problem is physically possible, generally speaking, only when there are normal forces on the surfaces of the crack and the stresses at infinity ∞ and k∞ . In these cases, the size 11 22 of the oscillation zone close to the crack tips is negligibly small, 14 and, on account of this, to solve Problem 2 we put Imp = Imq = ∞ =0 12 = ∞ = 0. Moreover, bearing in mind that the nature of the origin of the forces at the crack is associated with the and, for simplicity, ∞ 1 2 pressure of a gas or a liquid which has penetrated into the cavity, we shall henceforth assume that
With these assumptions, it follows from relations (3.11) that
(4.4) where
(4.5) Expression (4.4) and (4.5) conform with the oscillating solution obtained in a number of papers for a rectilinear interface crack when q0 = 0. In particular, under the action of only longitudinal stresses at infinity 1∞ , 2∞ and, also, when p0 = ∞ , such a crack does not open 11 11 22 and each half-plane is in a homogeneous stress-strain state. When 1 = 2 , 1 = 2 , and also when q0 = / 0, formulae (3.19), (4.4) and (4.5) give a solution of the problem of a crack in a homogeneous plane, which is identical to Muskhelishvili’s solution 16 if, in the latter solution, account is taken of the assumptions made above regarding the acting loads. The question of the opening of an interface crack when q0 = / 0 requires special discussion.
14
5. The first approximation The function H1 in relations (3.8) has the form
(5.1)
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and it follows from relations (3.8), (4.1) and (5.1) that, in the case of Problem 1 when u = n ≡ 0, ∞ = ∞ =0 1 2
(5.2) For Problem 2 when p = const and q = const
(5.3) where
It follows from relations (5.2) and (5.3) that the first approximation depends on the form of the defect. At the same time, if the load is determined exclusively by the stresses at infinity, then, in Problem 1, this approximation only depends on the derivative g (x1 ), that is, on the angle of inclination of the tangent to the interface, and, in Problem 2, further on the magnitude of the deviation of the crack g(x1 ) from a rectilinear shape. When n = / 0 in Problem 1, a dependence of the first approximation on the function g also holds as, for example, in the case when the zeroth approximation has the form (4.3). A curvilinear crack, located between two identical media and that is only opened by an internal pressure p0 <0 possesses an interesting property. In this case H1 (x1 )≡0, and this crack behaves in the same way as the corresponding rectilinear crack apart from the small quantity . Focussing our attention on the stress state of the interface , we write out the expressions for the stresses in this interface in the first approximation. From relations (2.3) and (3.3), it follows that
(5.4) whence the stresses on the other side of the interface can be obtained by replacing the subscript 2 by 1. We shall henceforth assume that the distortion of an interface itself and an interface crack is described in Eq. (1.1) by a function that is symmetrical about the origin of the coordinates x1 = 0: (5.5) The corresponding form of the defect is characterized by the fact that the whole of the interface is smooth and, when the parameter n is increased, the distorted part is progressively localized around the middle of the defect. An example of this defect when n = 3 is shown in Fig. 1. Although the function (5.5) is not a polynomial for any value of n, all the integrals in formulae (3.9) and (3.11) can be expressed in terms of elementary functions. A curvilinear interface without a crack. In the case of Problem 1 that takes account of relations (5.2), we find the integrals in equalities (3.9)
(5.6) where, using the properties of a Cauchy-type integral, 16
Here z1 = z/l, and Qn (z1 ) is a polynomial of degree n (n = 3, 4,...), the principal part in the expansion at infinity of the corresponding first term in the square brackets. Taking account of equalities (4.1) and ∞ = ∞ = 0, relations (5.4) take the simple form 1 2
(5.7)
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Fig. 2.
Using relations (3.9), (3.15), (5.6) and (5.7), the loads at any point of the interface can be calculated accurately up to o terms if the stresses , 2∞ , ∞ act at infinity. An analysis shows that, when ∞ = ∞ = 0 and 1 /2 < 1, the circumpherential stresses reach a maximum 1∞ 11 11 22 22 12 value at the point x1 = 0, that is, at the bottom of the hollow in the harder material 2 . The results of the calculations, for the case of plane deformation and the same Poisson’s ratios 1 = 2 = 0.3, are shown in Figs. 2 and 3 for n = 3, 9, 15. The dependences of the coefficient
on the ratio 1 /2 are shown in Fig. 2. It can be seen that the greater the difference between the shear moduli j of the materials (in the case of a Young’s modulus Ej ) the greater the coefficient of concentration of the circumpherential stresses K = 1 + T, which reaches its maximum value in the case when there is no medium 1 (1 /2 = 0). For a fixed value of , an increase in the parameter n leads to an increase in the concentration coefficient K, associated with a decrease in the radius of curvature at the point x1 = 0.
Fig. 3.
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When the circumpherential stresses tt are considerably greater than the longitudinal stresses 2∞ acting far from the defect, this can 11 give rise to cracks that are normal to the interface. Moreover, the distorted part of the interface is a concentrator not only of circumpherential stresses. Thus, if a stress ∞ acts far from the defect, then it is observed that the normal stress nn considerably exceeds ∞ in the distorted 22 22 part of the interface which can lead to delamination and the development of an interface crack. As an example, graphs of the change in the stresses nn (the solid curves) and tt (the dashed curves), obtained in the first approximation in the case of an isotropic stress 2∞ = ∞ = > 0 in the medium 2 when 1 /2 = 3, = 0.3, are shown in Fig. 3. 11 22 Note that, when 1 /2 > 1, the medium 1 is more rigid and the curve c for it has the shape of a protrusion. In this case, the vertex of the protrusion corresponds to the value x1 = 0 and the points on c close to the bottom of the protrusion become concentrators of the stresses tt as was revealed 22 in the case of longitudinal stretching for 2 /1 > 0. It can be seen from Fig. 3 that the points of a maximum in tt are shifted towards the middle of the protrusion as the parameter n increases, that is, in proportion to the localization extent of the distorted part of the interface. Calculations show that, in the case of longitudinal stretching, the positions of the maxima in tt near the protrusion are practically the same as in the case of bi-axial stretching in Fig. 3. However, the maximum values of the stresses tt are approximately 10% higher in the first case than in the second. A curvilinear interface. By analogy with Problem 1, the functions 1 and V1 can be obtained in closed form for any value of n > 2 in equality (5.5) by substituting expressions (5.1) and (5.3) into relations (3.11), taking account of relations (4.4) and (3.15) and the complex potentials (3.15) can be obtained in terms of them. We leave aside the unwieldy expressions for these functions and turn our attention to the stress state near the crack tips. Arguing in the same way as earlier 13 for a curvilinear crack in a homogeneous body, it can be shown that, in the case of the curvilinear interface cracks considered, the complex potentials and, consequently, the stresses for any values of and n > 2 have the same form of asymptotic formulae as in the case of a rectilinear interphase crack corresponding to the value = 0. This is due to the fact that the tangent to the cracks at the tip is the same in all cases. The crack geometry only has an effect on the coefficients in the first term of the asymptotic expansion of the stresses close to the crack tips K1 and K2 for which, in the case of the tip x1 = l, the equality 14,22
(5.8) holds. Substituting expression (2.3) into equality (5.8) when k =1, ␣=0, x1 >1 and taking account of relations (3.3) and (3.15), to a first approximation we obtain
(5.9) whence, on the basis of relations (3.11) and (4.4), we obtain (5.10)
(5.11) 14
for a rectilinear interface crack. If a rectilinear crack When q0 = 0, the quantity K0 is identical to the corresponding coefficient obtained ∞ ≥ p , that is, is located between two identical materials, then ␥ = A = 0. It follows from relations (5.11) in this case that K10 ≥ 0 when 22 0 the crack opens up regardless of the value of the quantity q0 . Requiring that K10 ≥ 0 also for a rectilinear interface crack, we arrive at the inequality (5.12) where =arctg 2␥. This inequality, like the coefficients K1 and K2 themselves in equality (5.10), makes sense if the coordinates are assumed to be dimensionless, having been divided by the half-length of the interface crack l, from the very start. It is then necessary to put l = 1 in relations (5.8) - (5.12). ≥ p0 since 2 ␥ ≤ ln 3 in the case of plane deformation and It can be shown that inequality (5.12) is satisfied when q0 = 0 if ∞ 22 0 ≤ j ≤ 1/2. In this case, it is well known that the oscillation zones are small and comparable with the zones of penetration of the surfaces of the crack. 14 If q0 = / 0, then, in the general case, it is only after an analysis of the opening of the crack and an estimation of the penetration zones that it is possible to establish whether there is a relation between inequality (5.12) and the condition of smallness of these zones or = p0 . It then follows from the equality (5.12) when not. However, this is obvious in two limit cases. Actually, for simplicity, suppose ∞ 22
that the condition q ≤ 0 must be satisfied. This means that the less rigid part of the surface of the crack is under a normal pressure 1 2 0 + + nn < 0 and a tensile normal force − nn = −nn > 0 is applied to the other, considerably more rigid part. Conversely, when 2 1 : q0 ≥ 0. It is clear that the penetration zones are negligibly small in both cases. They are identical to the analogous zones near a crack which − is located between elastic and absolutely rigid half-planes and which is opened by a constant normal force + nn < 0 (or nn < 0). This conclusion also obviously holds in the case of the weakly distorted crack described by function (5.5). As far as the oscillatory solution of the problem for the curvilinear cracks considered is concerned, a calculation of the jump in the displacement vector u in the first approximation for the most characteristic loads and loads realized in practice ± nn = −p0 < 0 and/or > 0, 1∞ > 0 when q0 = 0 showed that the size of the zone of penetration of the surface is of the same order as in the case of the ∞ 22 11
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Fig. 4.
corresponding rectilinear crack. In order to find the quantity u, it was sufficient to integrate the left-hand side of the penultimate equality of (3.2) using relations (2.3) and the expressions for the complex potentials obtained according to the scheme described above. In the case of an interface crack, the coefficients K1 and K2 do not have a definite physical meaning and they cannot therefore serve as characteristics of its limit state. The joining of two materials with elastic constants that satisfy the condition B = 0 is an exception. In the general case, the Cherepanov–Rice integral J, which is equal to the rate of release of the specific strain energy as the crack grows on both sides along the interface, is such a characteristic. 23,24 According to reference data 25
(5.13)
K12 + K22 and the integral J in the first approximation, obtained for the case of plane Some results of calculations of the quantity K = √ deformation and the values 1 = 2 = 0.3, p0 = q0 = 0, are shown in Figs. 4–6. Graphs of the dimensionless quantity k = K / ∞ l 22
against the ratio of the shear moduli of the materials 2 /1 in the case of uniaxial (∞ > 0, ∞ = 0, Fig. 4) and biaxial (1∞ = ∞ /2, 22 11 11 22 Fig. 5) stretching are shown in Figs. 4 and 5. The shape of the crack is determined by function (5.5) when n = 4 (the dashed curves) and when n = 16 (the dot-dash curves). Comparison of the corresponding curves for n = 4 and n = 16 enables us to reveal a general regularity that involves the fact that, as n increases, that is, in the case of a greater and greater localization of the distorted part of the crack around its middle, the value of the quantity k for a curvilinear crack differs to a lesser and lesser extent from the corresponding value for a rectilinear crack. It also follows from Figs. 4 and 5 that additional tension along the interface leads to a convergence of the graphs for a curvilinear crack and the corresponding graphs for a rectilinear crack. A characteristic feature of the relations presented is the fact that the maximum value of the quantity k is reached in the case when the materials are similar in their elastic properties, that is, 2 /1 ≈1. Graphs of the dimensionless integral J* = J/Jh against the ratio 2 /1 in the case of uniaxial tension when n = 4 are shown in Fig. 6. The integral (5.13) for a rectilinear crack located in a homogeneous elastic medium with constants vk and k under the action of stresses ∞ 22 is denoted by Jh , that is,
(5.14) The elastic constants of the medium 1 (in relation (5.14), the subscript k = 1) were fixed if 2 /1 < 1 when calculating the quantity J* and the elastic constants of the medium 2 (k = 2) were fixed if 2 /1 > 1. As a result, as can be seen in Fig. 6, a reduction in the stiffness of any component of the composite leads to an increase in J*. If the domains in which the ratio 2 /1 change places, we arrive at the fact that an increase in the stiffness of any component reduces the quantity J* . 25 The effect of the distortion of the crack on the integral J* is most > 0 when n = 4. Calculations showed that, when the parameter n is increased, the difference perceptible in the case of uniaxial tension ∞ 22
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Fig. 5.
Fig. 6.
between the integrals J* for a distorted crack and a rectilinear crack becomes smaller. In the case of the additional action of longitudinal tension ∞ > 0, the integral J* for a slightly distorted crack ( = 1/2) is practically the same as for a rectilinear crack even in the case when 11 n = 4. 6. Conclusion In combination, relations (2.2), (2.3), (3.3). (3.8), (3.9). (3.11), (3.14) and (3.15) give the solution of the problems formulated in closed form for any approximation. As an example, the symmetrical form of the distorted part of an interface (5.5), that smoothly changes into
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the plane part of the interface, has been considered. The arbitrary form of a local defect when the same smoothness is preserved can be approximated to a sufficient degree of accuracy by the function
where Qk (x1 ) is a k-th degree polynomial. In this case, any approximation in Problem 1, as well as for a sufficient degree of smoothness of the boundary conditions also in Problem 2, can be represented in terms of elementary functions. At the same time, as shown above, the first approximation already suffices to estimate the effect of the form and amplitude of the distortion of a defect on the stress concentration and limit state of the crack. When a more exact analysis is necessary, succeeding approximations can be taken into account using the algorithm described above. Acknowledgements This research was supported by the Russian Foundation for Basic Research (11.01-00230) and St Petersburg State University (9.0.165.2009, 9.37.129.2011). References 1. Gunnars J, Wikman B, Hogmark S. Effect of non-planar interfaces in layered materials subjected to residual stress. In: Gunnars J, editor. On Fracture of Layered Materials. Lulea: Lulea University of Technology; 1997. p. 1–33. 2. Liu YY, Nateso K. The adherence of nickel oxide on nickel during high-temperature oxidation Mater. Res Soc Symp Proc Adhesion in solids Reno 1988;119:213–56. 3. Steeper TJ, Varacalle DJ, Wilson GC, Idaho GA. Design of experimental study of plasma-sprayed alumina-titania coatings. J Thermal Spray Technol 1993;2(3):251–6. 4. Pukh VP, Baikova LG, Zvonareva, et al. The effect of a coating made of hydrogenated carbon on the strength and crack stability of glass in a high-strength state. Fiz Tverd Tela 2001;43(3):82–6. 5. Gao H. A boundary perturbation analysis for elastic inclusions and interfaces. Int J Solids and Struct 1991;28(6.):703–25. 6. Kung H, Chang H, Gibala R. Interfacial structures of MoSi2 −Mo5 Si3 eutectic alloys. Mat Res Soc Symp Proc Structure and Properties of Interfaces in Materials Boston 1992;238:599–604. 7. Betekhtin VI, Gorobei NN, Korsukov VE, Luk’yanenko AS, Obidov BA, Tomilin AN. Features of defect formation on the deformed (111) surface of silicon. Pis’ma v ZhTF 2002;28(21):29–35. 8. Evans AG, He MY, Hutchinson JW. Effect of non-planarity on the mixed mode fracture resistance of bimaterial interface. Acta Metall 1989;37(3):909–16. 9. Gunnars J, Alahelisten A. Thermal stresses in diamond coatings and their influence on coating wear and failure. Surf Coat Technol 1996;80:303–12. 10. Evans AG, He MY, Hutchinson JW. Effect of interface undulations on the thermal fatigue of thin films and scales on metal substrates. Acta Mater 1997;45(9):3543–54. 11. Grekov MA. The method of perturbations in the problem of the deformation of a two- component composite with a slightly distorted interface. Vestrik Sankt-Peterburg Univ Ser 1 Mat Mekh Astron 2004;1:81–8. 12. Grekov MA. The perturbation method applied to some problems in the theory of elasticity. In: Ivlev DD, Morozov NF, editors. Problems of the Mechanics of Deformable Solids and Rocks: Collection of Papers on the Occasion of the 75-th Birthday of Ye I. Shemyakin. Moscow: Fizmatlit; 2006. p. 188–98. 13. Lin’kov AM. The Complex Method of Boundary Integral Equations in the Theory of Elasticity. St Petersburg: Nauka; 1999. 14. Grekov MA. The Singular Plane Problem in the Theory of Elasticity. St Petersburg: Izd Univ; 2001. 15. Grekov MA, Morozov NF. Some modern methods in mechancs of cracks. In: Adamyan V, et al., editors. Modern Analysis and Application. Ser. Operator Theory: Advances and Applications, 191. Basel: Birkhauser; 2009. p. 127–42. 16. Muskhelishvili NI. Some Basic Problems in the Mathematical Theory of Elasticity. Leyden: Noordhoff; 1975. 17. Kolosov GV. The Use of Complex Diagrams and the Theory of Functions of a Complex Variable in the Theory of Elasticity. Leningrad–Moscow: ONTI; 1935. 18. Dundurs J. Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading. Trans ASME J Appl Mech 1969;36(3):650–2. 19. Williams ML. The stresses around a fault or crack in dissimalar media. Bull Seismol Soc Amner 1959;49(2):199–204. 20. Simonov IV. Prediction of arbitrary crack growth from the interface between two dissimialr elastic materials. Int J Fracture 1992;57(4.):349–63. 21. Simonov I, Osipenko K. Elastodynamic well-defined fields around a propagating interface open-crack edge. Int J Fracture 2002;116(4.):297–312. 22. Grekov MA, Makarov SN. Stress concentration near the slightly distorted part of the surface of an elastic body. Izv Ross Akad Nauk MTT 2004;6:53–61. 23. Rice JR, Sih GC. Plane problems of cracks in dissimilar media. Trans ASME Ser E J Appl Mech 1965;32(2):418–23. 24. Cherepanov GP. Mechanics of Brittle Fracture. New York: McGraw-Hill; 1979. 25. Stress Intensity Factors. In: Murakami, editor. Handbook. Oxford etc; 1987.
Translated by E.L.S.