Engineering Fracture Mechanics

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1 Engineering Fracture Mechanics 80 (0) 60 7 Contents lists available at ScienceDirect Engineering Fracture Mechanics journal homepage: Mode III interfacial crack in the presence of couple-stress elastic materials A. Piccolroaz a,, G. Mishuris a, E. Radi b a Institute of Mathematical and Physical Sciences, Aberystwyth University, Ceredigion SY3 3BZ, Wales, UK b Dipartimento di Scienze e Metodi dell Ingegneria, Università di Modena e Reggio Emilia, Viale Amendola, I-4 Reggio Emilia, Italy article info abstract Keywords: Interface fracture Couple stress elasticity Asymptotic analysis Stress singularity In this paper we are concerned with the problem of a crack lying at the interface between dissimilar materials with microstructure undergoing antiplane deformations. The micropolar behavior of the materials is described by the theory of couple-stress elasticity developed by Koiter. This constitutive model includes the characteristic lengths in bending and torsion and thus it is able to account for the underlying microstructure of the two materials. We perform an asymptotic analysis to investigate the behavior of the solution near the crack tip. It turns out that the stress singularity at the crack tip is strongly influenced by the microstructural parameters and it may or may not show oscillatory behavior depending on the ratio between the characteristic lengths. Ó 0 Elsevier Ltd. All rights reserved.. Introduction Nowadays, bimaterials are efficiently and widely used in many advanced engineering applications, such as layered composite structures, electronic packaging and thin film coatings. For the prediction of failure of these structures and the assessment of acceptable stress level under the condition experienced during service, it becomes essential to estimate the magnitude and distribution of the interfacial stress and strain fields along the interface and mainly near the tip of interface cracks, which may arise and extend under general loading conditions. In particular, antiplane shear loading condition may frequently occur in the life span of composite structures, both alone or accompanied by plane deformation. Within the classical LEFM theory, the crack tip fields for an interface crack under antiplane strain are similar to the Mode III crack tip fields in a homogeneous medium [,]. In both cases, indeed, the shear stresses on the crack plane are the same in the upper and lower bodies, whereas the out-of-plane displacement is zero on the uncracked region of the crack plane. Thus, it is possible to combine the lower and the upper bodies to obtain equilibrium, without changing displacements or stresses in the two halves. Stresses exhibit Mode III symmetry, but displacements do not and thus crack sliding profiles are not symmetric. Due to the lack of a length scale, the classical theory of elasticity is not able to characterize the constitutive behavior of brittle materials at the micron scale. This lack is expected to be particularly significant for the analysis of the stress and deformation fields very near the crack tip. For a proper investigations of the crack tip fields at the micron scale it becomes necessary to adopt enhanced constitutive models, which account for the presence of microstructure. A way of doing that consists in the inclusion of one or more characteristic lengths, typically of the same order of the compositional grain size, generally few microns, for many advanced materials. The indeterminate theory of couple-stress elasticity (CSE) developed by Koiter [3] involves the material characteristic lengths in bending and torsion. It is sufficiently accurate to simulate the Corresponding author. Tel.: ; fax: address: roaz@ing.unitn.it (A. Piccolroaz) /$ - see front matter Ó 0 Elsevier Ltd. All rights reserved. doi:0.06/j.engfracmech

2 A. Piccolroaz et al. / Engineering Fracture Mechanics 80 (0) Nomenclature G ± m ± l b l t l ± shear modulus (+/ stands for upper/lower half-plane) Poisson s ratio material characteristic length in bending material characteristic length in torsion material characteristic length (Koiter s notation) g ± ratio between the characteristic lengths in bending and torsion (Koiter s notation, <g ± 6 ) u displacement field w out-of-plane displacement u rotation vector strain tensor v curvature tensor t nonsymmetric stress tensor r symmetric part of the stress tensor s skew-symmetric part of the stress tensor l couple-stress tensor p reduced tractions vector q reduced couple-stress tractions vector k order of stress singularity behavior of materials at the micron scale as well as the size effects occurring at distances to the crack tip comparable to characteristic lengths, but it is also simple enough to allow the achievement of closed-form solutions. Although the presence of the microstructure is expected to modify the interface crack tip field with respect to the classical solution of the LEFM, no analytical investigations have been so far performed about the problem of an antiplane crack along the interface between micropolar and classical elastic materials (the only related work regards a crack terminating perpendicular to the interface [4]). Most of related studies available in literature instead concern the problem of an interface crack under plane deformations, e.g., Itou [5] examined the effect of couple-stresses on the strain-energy release rate for an interface crack loaded by an internal pressure, neglecting somehow the oscillatory behavior of the crack tip fields. In order to provide an experimental basis for studying the interfacial behavior of a bimaterial specimen under shear loading, Kang et al. [6] applied a method which combines moiré interferometry with phase shift and image processing to measuring the interfacial displacement and strain fields within the interfacial region. Their experimental results show that there is a boundary layer characteristic with a peak value of shear strain and high gradient of rotation angle in the interfacial region. Their study also shows that similar results can be analytically predicted by means of the couple-stress theory, considering the additional freedom of the rotation angle effect. Hutapea et al. [7] investigated the micro-stress generated along a fiber/matrix interface under generalized plane deformation, which are expected to dominate the failure initiation process in composite laminate. They showed that the micropolar theory is able to capture the interface micro-stress accurately. A small number of interface crack problems have been investigated by using the strain gradient theory of plasticity [8,9]. In particular, Hao and Liu [8] analyzed the crack propagation in bimaterial systems showing that high stress triaxiality always occurs on the softer material, which may promote ductile damage and facilitate crack growth. Chen and Wang [9] explored the interface crack tip fields at micron scales under plane strain conditions. Their numerical investigations show that the singularity of stresses in the strain gradient theory slightly exceeds or equals to the square-root singularity independently of the material hardening exponents. Askes and Gitman [0] showed numerically that in gradient elasticity no singular behavior is found for a crack terminating perpendicular to the interface. The analysis of singular stress concentration in homogeneous micropolar elastic solids shows that several pathological predictions of classical elasticity in singular stress concentration problems are altered, mitigated, or possibly eliminated when couple-stresses are taken into account []. In particular, the problem of a Mode III crack in a homogeneous materials modeled by the couple-stress elastic theory was first analyzed by Zhang et al. [] and later by Geogiadis [3] by considering a single characteristic length. The results obtained therein indicate that the skew-symmetric stress components have r 3/ singularity near the crack tip, where r is the distance to the crack tip. Although this singularity is much stronger than the conventional square-root singularity, it does not violate the boundness of strain-energy surrounding the crack tip and leads to a finite energy release rate. Their asymptotic analysis also provides a negative out-of-plane displacement ahead of the crack tip. This unphysical result is due to the exclusion of the lowest order terms for the displacement and symmetric stress components, which do not contribute to the energy release rate. The effects of both characteristic lengths in bending and torsion and a complete investigation of the crack tip fields under Mode III loading condition in homogeneous CSE materials have been properly addressed in a recent work by Radi [4]. The roles of both characteristic lengths are therein examined in detail and their influence on the crack tip is analytically explored by using Fourier transform and the Wiener Hopf method. The asymptotic and full-field analyses show that the symmetric stress is finite at the crack tip, whereas the skew-symmetric stress is negative and strongly singular. Ahead of the crack tip

3 6 A. Piccolroaz et al. / Engineering Fracture Mechanics 80 (0) 60 7 within a zone smaller than the characteristic length in torsion, both the total shear stress and reduced tractions occur with the opposite sign with respect to the classical LEFM solution, as predicted by the asymptotic analysis. However, the zone of dominance of the asymptotic fields has limited physical relevance and becomes vanishing small for a characteristic length in torsion of zero. In this limit, the full-field solution recovers the classical K III field with square-root stress singularity. Outside this zone, the total shear stress exhibits a positive maximum, thus providing more realistic predictions on the tractions level ahead of the crack tip than the singular LEFM solution. A sharp crack profile is also observed. It may denote that the crack becomes stiffer, thus revealing that the presence of microstructures may shield the crack tip from fracture. In the present work, the effects of strain rotation gradients on a stationary antiplane crack along the interface between two different couple-stress elastic materials are analytically investigated by performing an asymptotic analysis of the crack tip fields. The special problem of a crack along the interface between a couple-stress elastic solid and a classical elastic medium is also addressed in Section 3. The results of the present asymptotic analysis are expected to hold in a small zone near to the crack tip, whose extent may vary with the size of the characteristic lengths, and provide valuable information for performing a full-field analysis of the interface crack problem, e.g., by using the Wiener Hopf method, which will be the subject of further investigations.. Crack at the interface between couple-stress elastic materials We consider a bimaterial plane made of two dissimilar materials, joined along a perfect interface. The two materials are assumed to have an underlying microstructure, described by the material characteristic lengths in bending and in torsion, denoted by l b and l t, respectively. The elastic moduli are denoted by G ± (shear modulus) and m ± (Poisson s ratio). A semi-infinite plane crack is placed along the interface, and a Cartesian reference system is assumed centered at the crack tip, see Fig.. The fundamentals of the Couple Stress (CS) elasticity theory Koiter [3] can be found in several text books and research papers [5,6,4]. It is recalled here that the main characteristic of this theory is that the rotation vector u is not independent of the displacement vector u, but it is subject to the condition u ¼ curl u: ðþ Consequently, all the kinematical quantities can be derived from the displacement field. In particular, for antiplane shear deformations, the following kinematical relations between the out-of-plane displacement w, rotation vector u, strain tensor e and curvature tensor v are derived u ; ; e 3 ; e 3 v ¼ v w ; v w ; w An infinitesimal surface element transmits a force and a couple vector, which give rise to a nonsymmetric stress tensor, t, and a couple-stress tensor, l. The nonsymmetric stress tensor can be decomposed into a symmetric part r and a skew-symmetric part s, such that t = r + s. The isotropic constitutive equations are given by r ¼ mgðtreþ m I þ Ge; l ¼ Gl ðv T þ gvþ; ð3þ where I is the identity tensor, l and g the CS parameters introduced by Koiter [3], with <g6, the superscript T denotes transposition. The material parameters l and g characterize the microstructure of the material and can be expressed in terms of the material characteristic lengths in bending and in torsion as follows: p l b ¼ l= ffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ; lt ¼ l þ g: ð4þ For antiplane shear deformations, the nonzero stress and couple-stress components are ðþ x G+ l + η + r θ G l η x Fig.. A crack at the interface between dissimilar couple-stress materials.

4 r 3 ; r 3 ; l ¼ l ¼ Gl ð þ w A. Piccolroaz et al. / Engineering Fracture Mechanics 80 (0) 60 ; l ¼ Gl w ; l ¼ w w In the absence of body forces and body couples, the equations of equilibrium read ð5þ! : ð6þ div t T ¼ 0; div l T þðse Þe þðse Þe þðse 3 Þe 3 ¼ 0; ð7þ where {e,e,e 3 } is an orthonormal basis. For antiplane shear deformations, the nonzero skew-symmetric stress components, derived from (6) and (7), are s 3 ¼ ; s 3 ¼ ; ð8þ where D stands for the laplacian operator. In the CS theory, due to the internal constraint () between rotations and displacements, the Neumann boundary conditions are prescribed in terms of the so called reduced force tractions vector p and couple tractions vector q defined as p ¼ t T n þ rl nn n; q ¼ lt n l nn n; ð9þ respectively, where n denotes the outward unit normal and l nn = n ln. Additionally, if the external surface of the body is not smooth but piecewise smooth, the boundary conditions include the following equation along each edge Q ¼ ðlþ nn l nn Þ; ð0þ where Q is a prescribed line load tangential to the edge [3], and superscripts + and stand for the values on the surface at each side of the edge. It then appears that the condition (0) becomes essential in the case of bodies with non-regular boundaries, such as cusps, wedges and cracks. A substitution of (5) and (8) into (7) gives the following governing equation for the displacements w ± in the two halfplanes: Dw l D w ¼ 0; ðþ where D denotes the bilaplacian operator. We assume that the crack faces are traction-free, so that the following boundary conditions apply for x =0 ± and x <0: ( " #) p 3 :¼ ð w w ( ) ðþ q :¼ G g ¼ Assuming also that no tangential line load Q is applied along the crack edge, we enforce that l þ l ¼ G þl þ ð þ g þ w þ þ ð þ g w ðx ;x Þ¼ð0 ;0 þ Þ ðx ;x Þ¼ð0 ;0 Þ ð3þ The formulation is completed by the transmission conditions for ideal interface, which imply continuity of the displacements, rotations, reduced stress and couple-stress components for x = 0 and x > 0: swt ¼ 0; t ¼ ( " #) s ð þ w @x t ¼ 0; ð4þ s Gl ( w w t ¼ 0; where the notation sft stands for the jump of the function f across the interface: sft = f + f.

5 64 A. Piccolroaz et al. / Engineering Fracture Mechanics 80 (0) Asymptotic analysis and singularity of stresses Assuming a polar reference system centered at the crack tip, we search for the main asymptotic term of the solution as r? 0 in the standard form as follows w ðr; hþ ¼r k F ðh; kþ: ð5þ We are interested in finding the leading term of the asymptotic solution corresponding to finite elastic energy. This requires that k P 3/ [4]. It is noted that the values k = 0 and k = are also admissible, as long as the respective terms in (5) correspond to a rigid body motion (constant displacement) and a uniform deformation (linear displacement), respectively. Moreover, the expression (5) can be used to find more terms in the asymptotic solution in the form P i rk i F ðh; k i Þ, provided that jk i k j j <, "i j. If more terms are required with exponents differing by or more than, then a two-terms asymptotic procedure should be used instead, as explained in Section 3. Keeping into account only the leading term as r? 0, the governing Eq. () yields the following ODE for the unknown functions F ± F 0000 þ ðk k þ ÞF 00 þ k ðk Þ F ¼ 0: ð6þ We first investigate the simplest cases k = 0 and k =, for which Eq. (6) admits the solutions F ðh; 0Þ ¼B þ B h þ B 3 sin h þ B 4 cos h; ð7þ F ðh; Þ ¼ðB þ B hþ sin h þðb 3 þ B 4 hþ cos h; ð8þ respectively. Taking into account all boundary and transmission conditions, one can conclude that for k = 0 and k =, Eqs. (7) and (8) take, as expected, the forms F ðh; 0Þ ¼b 0 ; F ðh; Þ ¼b sin h þ b cos h; ð9þ respectively. One can also use the representation (5) to find the solution for the case k =, since the term corresponding to k = 0 is not involved in the analysis (as it vanishes after differentiation). Thus, for k =, Eq. (6) admits the solution F ðh; Þ ¼B þ B h þ B 3 sin h þ B 4 cos h; ð0þ which, taking into account all boundary and transmission conditions, reduces to F ðh; Þ ¼b ðcos h þ g sin hþþb sin h: ðþ For all other cases, Eq. (6) admits the following solution: F ðhþ ¼B sinðkhþþb cosðkhþþb 3 sin½ðk ÞhŠþB 4 cos½ðk ÞhŠ: ðþ By imposing the boundary and transmission conditions, we obtain a 8 8 homogeneous algebraic system, whose characteristic equation is sin ðpkþ½cosðpkþþjš ¼ 0; ð3þ where j = C/D > 0 for any g ± > and C ¼ G þ l4 þ ð5 g þg Þð3 g þ Þ ð þg þ Þ þ G l4 ð5 g þ þg þ Þð3 g Þ ð þg Þ þ G þ l þ G l ð3 þg þg g þ þg Þð3 g þ Þð þg þ Þð3 g Þð þg Þ; n D ¼ð3 g þ Þð þg þ Þð3 g Þð þg Þ G þ l4 þ ð3 g þ Þð þg þ ÞþG l4 ð3 g Þð þg ÞþG þl þ G l ð5 g g þg o þ þg Þ : ð4þ The first term in Eq. (3) leads to the conclusion that k = 3 (the cases k = 0,, have been investigated above), while the second term may exhibit singular behavior depending on the value of the parameter j. Ifj >, then the solution of Eq. (3) is complex and the singularity shows oscillatory behavior in the vicinity of the crack tip. Otherwise, the solution is real and there are no oscillations. More precisely, since j is strictly positive, the following three cases may occur: (i) 0 < j < : the first admissible value of the exponent is in the interval 3/ < k < 7/4 (simple root). (ii) j = : the first admissible value of the exponent is k = 3/ (double root). (iii) j > : the first admissible value of the exponent is k = 3/ ± ic (simple root), where c ¼ p logðj þ p j ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ: ð5þ

6 A. Piccolroaz et al. / Engineering Fracture Mechanics 80 (0) In the case of a homogeneous material, G + = G, l + = l, g + = g, the ratio j is equal to and thus the first admissible value for the exponent is 3/ (this case has been analysed in Radi [4]). Some other special cases are investigated in the next section... Some particular and special cases To decrease number of parameters, let us first consider the case where g + = g = g. Then the ratio j reduces to G þ j l4 þ þ G l4 5 g þ g þ Gþ l þ G l ð3 gþð þ gþ ¼ ðg þ l4 þ þ G l4 Þð3 gþð þ gþþg þl þ G l ð5 g þ g Þ ¼ ða þ b Þc þ ab a þ b þ abc ; ð6þ where we use the notations a ¼ G þ l þ, b ¼ G l and 5 g þ g cðgþ ¼ ð3 gþð þ gþ : Since c P for any admissible value of g ( <g 6 ), it is easy to show that, for dissimilar materials, j is always greater than and equal to if and only if c = (or equivalently g = ), see Fig.. In the limiting case g = the asymptotic solution in the vicinity of the crack tip is given by w ðr;hþ¼b 0 þ rðb sinh þ b coshþþr 3= l b G 3 3sin h sin3h þ b 4 cos h cos3h ð7þ þ r ðb 5 þ b 6 sinhþþoðr 5= Þ; r! 0; where the constants b i are amplitude factors depending on far-field loading and specimen geometry. It is noted that logarithmic terms are excluded, since geometric and algebraic multiplicities of the double root coincide [7]. Correspondingly, the asymptotics of symmetric stress, couple-stress and skew-symmetric stress as r? 0 are r 3 ¼ b G r = l h sin 3b 3 sin h þ b 4 cos h þ rg ðb 5 cos h þ b 6 sin hþþoðr 3= Þ; r 3 ¼ b G þ r = l sin h ð 3b 3 sin h þ b 4 ð3 þ cos hþþþrg ðb 5 sin h þ b 6 cos hþþoðr 3= Þ; ð8þ l ¼ r = l þ l sin h ½ 3b 3ðsin h þ sin hþþb 4 ð þ cos h þ cos hþšþ4b 6 G l þ Oðr= Þ; l ¼ l ¼ r = l þ l b 3 cos h ðsin h sin hþþb 4 cos h ð cos h þ cos hþ þ Oðr = Þ; ð9þ s 3 ¼ r 3= l þ l ð3b 3 sin 3h þ b 4 cos 3h ÞþOðr = Þ; s 3 ¼ r 3= l þ l ð3b 3 cos 3h b 4 sin 3h ÞþOðr = Þ: Applying now the condition (3), we obtain l þ ðh ¼ pþ l ðh ¼ pþ ¼4b 4l þ l pffiffi þ 4b 6 ðg l r G þl þ ÞþOðr= Þ; r! 0: ð30þ The limit, as r? 0, of (30) should equal the tangential line load Q applied to the crack edge. It then appears that b 4 is always zero and b 6 does not vanish only if Q is different from zero. It is also noted that the constant b 3 plays the role of stress intensity factor. Fig.. Plot of the ratio j given by (6) as a function of g for a/b =,3,6,0.

7 66 A. Piccolroaz et al. / Engineering Fracture Mechanics 80 (0) 60 7 The asymptotics for the crack opening, swt = w + (h = p) w - (h = p), and for the skew-symmetric stress ahead of the crack tip, s 3ðh ¼ 0Þ, asr? 0, are given by swt ¼ 4b 3 G þ l þ þ G l G þ G r 3= þ Oðr 5= Þ; ð3þ s 3 ðh ¼ 0Þ ¼ 3 b 3l þ l r 3= þ Oðr = Þ: ð3þ Therefore, in the vicinity of the crack tip, the skew-symmetric stress at h = 0 displays a sign opposite to that of the crack opening, in contrast to the classical result of LEFM. A similar effect has been found by Radi [4] in the case of a crack in an homogeneous CS material. In the opposite case (g + = g = g < ), the solution always exhibits oscillatory behavior near the crack tip. Moreover, this behavior is quite different from that we encounter in the case of classical materials. In the classic case, the region near the crack tip where the oscillatory behavior appears is very small, while in the considered case this zone can be quite pronounced or its size can even tend to infinity if g?. Taking into account that the asymptotic analysis given here is valid only in a small neighborhood of the crack tip where the micropolar theory controls the behavior of the solution, such situation has limited physical meaning. Let us now assume that the parameters g + and g are different and one of them, say g -, tends to the limiting value g?, while the other is separated from, so that g + > +, where is a small positive parameter. In this case, one can easily check that j? for any fixed value of g +. This again corresponds to the case where the exponent is a complex number with the imaginary part becoming infinite, c?, and therefore it has no physical relevance. It is noted that the case g =, g + cannot be recovered from the limiting case discussed above, and thus it will be discussed separately in Section 3.3. Another case of interest is when one of the multipliers involved in the parameter j, say G l, diminishes, G l! 0, then j! 5 g þ g ð3 g Þð þ g Þ ¼ cðg Þ P : ð33þ Once again, this solution has physical relevance if and only if g =. The assumption G l? 0 takes place if one assumes that l? 0, or, in other words, when the material in the lower half-plane reduces to a classical elastic material. This suggests considering the problem of a crack at the interface between a micropolar material (occupying the upper half-plane) and a classical one (occupying the lower half- plane). However, it is not possible to recover from (8) the solution for a classical elastic material, due to the singular perturbation character of the Eq. () as l ±? 0. For this reason, the problem of a crack lying at the interface between couple-stress elastic and classical elastic materials is addressed separately in the next section..3. Energy release rate In this section, the energy release rate is evaluated for the asymptotic representation (8), valid in the case g + = g =,by means of the J-integral argument. The conservation law for couple-stress elasticity [8] implies that Z J ¼ Wn t T l T ds ¼ C for every closed contour C, provided that there is no singularity within C. In(34), n is an outward unit normal on C and W denotes the strain-energy density W ¼ G þ Gl ðv v þ gv v T Þ: We define C ¼ C [ C [ C cr [ C in (see Fig. 3), so that X ðj J cr J J in Þ¼0; ð35þ ð36þ x Γ + Γ + cr Γ cr Γ + R R Γ Γ + in Γ in Γ x Fig. 3. Path of integration for the evaluation of the energy release rate.

8 A. Piccolroaz et al. / Engineering Fracture Mechanics 80 (0) with evident meaning of the symbols. Then we evaluate each term according to the representation (8), thus obtaining in which J ¼ J þ þ J ¼ b b 4 l þ l pffiffiffiffiffi þ P þ OðR = Þ; ð37þ R J ¼ J þ þ J ¼ b b 4 l þ l pffiffiffiffiffi þ P þ OðR = Þ; ð38þ R J cr ¼b b 4 l þ l pffiffiffiffiffi b b 4 l þ l pffiffiffiffiffi þ OðR = Þ; ð39þ R R J in ¼ 3b b 3 l þ l pffiffiffiffiffi þ 3b b 3 l þ l pffiffiffiffiffi þ OðR = Þ; ð40þ R R P ¼ X l l4 G ð9pb 3 þ pb 4 b 3b 4 Þ: ð4þ Note that the integrals along the interface cancel out from (36), whereas the integrals along the crack faces give J cr Jþ cr ¼ FðR Þ FðR Þ; ð4þ where FðRÞ ¼ b b 4 l þ l pffiffiffi þ OðR = Þ: ð43þ R It is possible now to introduce a path-independent parameter as follows: J ¼ lim R!0 ðj þ FðR ÞÞ ¼ lim R!0 ðj þ FðR ÞÞ ¼ P: In consideration of the condition (3), b 4 vanishes and the path-independent parameter J gives the energy release rate (derived through direct energy balance considerations by Atkinson and Leppington [9] and Eshelby [0])! J ¼ 9 l pl þ l þ l þ b 3 G þ G : ð45þ In the case of homogeneous body (G + = G = G, l + = l = l), our formula coincides with earlier results by Radi [4]: J ¼ 9pl6 G b 3 : ð46þ It is noted here that particular attention should be paid when using the J-integral in CS materials. The contribution of the integrals along the crack faces is not zero in general, since the force tractions t T n and couple tractions l T n are not vanishing as in the classical case. ð44þ 3. Crack at the interface between couple-stress elastic and classical elastic materials In this section we consider the problem of a crack lying at the interface between a micropolar material (occupying the upper half-plane) and a classical elastic one (occupying the lower half-plane). Then the governing equations are Dw þ l D w þ ¼ 0; Dw ¼ 0: ð47þ The traction-free boundary conditions along the crack faces are: p þ 3 ¼ 0 and qþ ¼ 0 for h ¼ p; r 3 ¼ 0 for h ¼ p: ð48þ Along the ideal interface the continuity of displacement and force tractions needs to be enforced, so that we have the following transmission conditions: w þ ¼ w ; p þ 3 ¼ r 3 for h ¼ 0: ð49þ However, since the orders of the two governing equations are different, an additional transmission condition is needed. This additional condition can be chosen in two different ways. At the boundary of the micropolar material one can prescribe the value of the reduced couple traction q þ, or, alternatively, the value of the rotation uþ (note that uþ ¼ u follows immediately from (49) ). We analyse these two cases separately in the next subsections.

9 68 A. Piccolroaz et al. / Engineering Fracture Mechanics 80 (0) Couple transmission conditions The additional transmission condition in this case takes the form q þ ¼ 0 for h ¼ 0: ð50þ Bearing in mind that, in this case, we need to match the solutions of governing equations having different orders, we use here a two-terms asymptotic analysis, so that a solution is searched for in the form w þ ðr; hþ ¼r k F ðhþþr kþ F ðhþþoðr kþ4 Þ; r! 0; ð5þ w ðr; hþ ¼r k H ðhþþr kþ H ðhþþoðr kþ4 Þ; r! 0: ð5þ Substituting (5) in (47) and collecting like powers of r, we obtain D ðr k F Þ¼0; D ðr kþ F Þ¼ l Dðrk F Þ: ð53þ For k 0,,, the system (53) admits the solution F ¼ B ðþ sinðkhþþbðþ F ¼ B ðþ þ BðÞ 3 4kl cosðkhþþbðþ 3 sin½ðk ÞhŠþBðÞ 4 cos½ðk ÞhŠ; sin½ðk þ ÞhŠþBðÞ cos½ðk þ ÞhŠþBðÞ 3 sinðkhþþbðþ 4 cosðkhþ ðþ B4 sin½ðk ÞhŠþ 4kl cos½ðk ÞhŠ; ð54þ Substituting (5) in (47) and collecting like powers of r, we obtain Dðr k H Þ¼0; Dðr kþ H Þ¼0: ð55þ For k 0,,, the system (55) admits the solution H ¼ A ðþ sin kh þ AðÞ cos kh; H ¼ A ðþ sin½ðk þ ÞhŠþAðÞ cos½ðk þ ÞhŠ: ð56þ Substituting the two-terms asymptotics (5), with F, and H, given by (54) and (56) respectively, into the boundary and transmission conditions, we obtain a homogeneous algebraic system, whose characteristic equation is ð þ gþ cos ðpkþ sin 4 ðpkþ ¼0; ð57þ so that the exponent k admits the values k = k/, where k =,3,5. Note that for k = / the first term in (5) for the classical elastic material corresponds to bounded elastic energy, while the first term in (5) for the micropolar material corresponds to infinite elastic energy. However, from the analysis of all boundary and transmission conditions, it is found that F (h) 0 in this case, so that the energetic requirements are fulfilled. For the special cases k = 0,, the analysis is straightforward and the final asymptotic representation of the solution taking into account all terms k = k/, k = 0,,,3, is given by w ðr; hþ ¼a 0 þ r = a sin h þ ra cos h þ r 3= a 3 sin 3h þ r a 4 cos h þ r 5= a 5 sin 5h þ Oðr3 Þ; r! 0; w þ ðr; hþ ¼a 0 þ rða cos h þ b sin hþþr þ g a 4 þ g cos h þ b sin h r 5= a G 3 5g 5h sin 3G þ l ð3 gþ 5ð þ gþ þ sin h þ Oðr 3 Þ; r! 0; ð58þ The asymptotics of stress in the lower half-plane, occupied by the classic elastic material, as r? 0 are given by r 3 ¼ r = a G sin h þ a G þ 3 r= a 3 G sin h þ ra 4G cos h þ 5 r3= a 5 G sin 3h þ Oðr Þ; r 3 ¼ r = a G cos h þ 3 r= a 3 G cos h ra 4G sin h þ 5 r3= a 5 G cos 3h þ Oðr Þ: ð59þ The asymptotics of symmetric stress, couple-stress and skew-symmetric stress in the upper half-plane, occupied by the micropolar material, as r? 0 are given by

10 A. Piccolroaz et al. / Engineering Fracture Mechanics 80 (0) r þ 3 ¼ a G þ þ rg þ ða 4 cos h þ b sin hþ 3g þð5 3gÞcos h 3 r3= a G sin h l ð3 gþð þ gþ þ Oðr Þ; r þ 3 ¼ b G þ þ rg þ ða 4 g sin h þ b cos hþ 3 r3= a G þ 5g þð 7gÞ cos h l ð3 gþð þ gþ l þ ¼ b G þ l ð þ gþ r = a G 3g þð þ gþ cos h 3 g l þ ¼ r= a G ð þ gþ sin h 3 g cos h þ OðrÞ; l þ ¼ a 4G þ l ð g Þþr = a G 3 5g ð þ gþ cos h 3 g cos h þ OðrÞ; sin h þ OðrÞ; cos h þ Oðr Þ; ð60þ s þ 3 ¼ r = a G 3 g sin h þ OðÞ; s þ 3 ¼ r = a G 3 g cos h þ OðÞ: The condition (3) becomes now l þ ðr ¼ 0; h ¼ pþ ¼0, which requires b to vanish, except for the case where a tangential line load is applied along the crack edge. It is now possible to analyse the jump of the rotation component u across the interface and it is found that there is a mismatch between the micropolar material and the elastic one given by su t ¼ 4 a r = þ b 3 4 a 3r = þ b r þ Oðr 3= Þ; r! 0: ð6þ It is concluded that in the classical elastic material the solution shows a square-root singularity, as in the classic case. Moreover, in the micropolar material, the symmetric stress and the couple-stress are bounded, and only the skew-symmetric stress shows a square-root singularity. The energy release rate, computed in the same manner as in Section.3, is given by J ¼ p 8 G a ; ð6þ which shows that the constant a plays the role of stress intensity factor. 3.. Rotation transmission conditions The problem is defined by the Eq. (47) with the boundary and transmission conditions (48) and (49). The additional transmission condition used here is the continuity of the rotation vector, namely u þ ¼ u for h ¼ 0: ð63þ We search for a solution again in the form (5) and (5). For k 0,,, the characteristic equation now takes the form ð þ gþ sin ðpkþ½ þ g ð 3 þ gþ cos pkš ¼ 0; so that the exponent k admits real values, namely integer positive numbers and k ¼ p arccos g þ g 3 þ k; where k is a non negative integer. In the particular case g =, the complete asymptotics of the solution up to the forth order is given by w ðr; hþ ¼a 0 þ ra cos h þ r a cos h þ r 3 a 3 cos 3h þ r 4 a 4 cos 4h þ Oðr 5 Þ; r! 0; w þ ðr; hþ ¼a 0 þ ra cos h þ r a þ r 5= b cos 5h cos h þ r 3 3 a 3 cos h cos 3h þ r 7= b cos 7h 3h cos a þ r 4 4l a 4 cos 4h þ a 4 a cos h þ a þ Oðr 9= Þ; r! 0: ð66þ 6l 8l The asymptotics of stress in the classic elastic material occupying the lower half-plane as r? 0 become ð64þ ð65þ r 3 ¼ a G þ ra G cos h þ 3r a 3 G cos h þ 4r 3 a 4 G cos 3h þ Oðr 4 Þ; r 3 ¼ ra G sin h 3r a 3 G sin h 4r 3 a 4 G sin 3h þ Oðr 4 Þ; ð67þ

11 70 A. Piccolroaz et al. / Engineering Fracture Mechanics 80 (0) 60 7 Finally, the asymptotics of symmetric stress, couple-stress and skew-symmetric stress in the micropolar material occupying the upper half-plane as r? 0 become þ 4r 3 a 4 G þ r þ 3 ¼ a G þ þ ra G þ cos h 3r 3= b G þ sin h sin h þ 3r a 3 G þ 0r 5= b G þ sin h cos h 3h þ cos cos hð cos hþþoðr 7= Þ; r þ 3 ¼ ra G þ sin h r3= b G þ 3 sin h 3h þ 7 sin þ 3r a 3 G þ sin h þ r5= b G þ 5 sin h 5h 9 sin þ 3l r3 G þ sin h½a ða 8a 4 l Þ cos hšþoðr 7= Þ; l þ ¼ 3 r= b G þ l 5 sin h 3h þ sin þ ra 3 G þ l sin h 5 r3= b G þ l 3 sin h 3h þ 7 sin þ 4r a 4 G þ l sin h þ Oðr 5= Þ; l þ ¼ lþ ¼ 3 r= b G þ l 5 cos h 3h cos þ 5 r3= b G þ l 3 cos h 3h 7 cos þ 4r G þ sin hða a 4 l ÞþOðr 5= Þ; s þ 3 ¼ 3 r = b G þ l cos h 6a 3G þ l þ 5 r= b G þ l cos h 4ra 4G þ l cos h þ Oðr 3= Þ; s þ 3 ¼ 3 r = b G þ l sin h 5 r= b G þ l sin h 4rG þða 6a 4 l Þ sin h þ Oðr 3= Þ: In this case, the reduced couple traction q along the interface has the form: q þ ðr; h ¼ 0Þ ¼ 6r= b G þ l 0r 3= b G þ l þ Oðr 5= Þ; r! 0: ð69þ Of course, it is possible to construct the asymptotic solution for arbitrary jgj <. However, the general form for arbitrary g is rather complicated. As an additional example, we provide the result for g = /3 and up to the first four terms (k = 0, /3, /3, ): w ðr; hþ ¼a 0 þ ra cos h þ r a cos h þ r 7=3 a 3 sin 7h 3 p ffiffiffi cos 7h þ r 8=3 a 4 sin 8h p ffiffiffi 3 w þ ðr; hþ ¼a 0 þ ra cos h þ r a 3 þ cos h 3 0 7h 0 þr 7=3 a 3 sin p ffiffiffi cos 7h sin h 3 þ p 7 3 ffiffiffi cos h 3 3 8h þr 8=3 a 4 sin 9 3 þ p 3 ffiffiffi cos 8h h sin 9 3 p 8 3 ffiffiffi cos h þ Oðr 3 Þ; r! 0: 3 3 cos 8h 3 ð68þ þ Oðr 3 Þ; r! 0; It is found that the behavior of the displacement and stress fields is similar for any <g 6. In particular, stresses are always bounded in the classical elastic material, while in the micropolar material singular behavior appears only in the skewsymmetric stress (with different level of singularity depending on the value of g). Moreover, the energy release rate is always zero for any <g 6. This shows that this type of transmission conditions does not allow for the propagation of the crack along the interface, and thus has limited physical meaning. ð70þ 3.3. The special case g = and g + In the particular case of g = and g +, the governing Eq. () and the traction-free crack face conditions () for the material in the lower half-plane are satisfied by the classical solution defined by the field equation Dw ¼ 0; ð7þ together with the following ¼ 0 for x < 0; x ¼ 0: ð7þ Moreover, in this case the couple-stress and skew-symmetric stress fields in the lower half-plane identically vanish, so that the transmission conditions along the interface x >0,x = 0 become w þ ¼ w ; p þ 3 ¼ r 3 ; ð73þ together with one of the following additional conditions q þ ¼ 0; uþ ¼ u ; ð74þ which correspond to the cases investigated in Sections 3. and 3., respectively.

12 A. Piccolroaz et al. / Engineering Fracture Mechanics 80 (0) Discussion and conclusions In the present work, the effects of strain rotation gradients on a stationary Mode III crack along the interface between dissimilar couple stress elastic materials have been analytically investigated by performing an asymptotic analysis of the crack tip fields. It is shown that solutions without oscillations appear in the following two cases: when the two materials are the same (homogeneous material) and when the two materials are dissimilar but g + = g =. In the latter case, the solution displays the same r 3/ singularity (appearing in the skew-symmetric stress components) as for the problem of a crack in an homogeneous CS material. In other cases, the solution exhibits oscillatory behavior in the vicinity of the crack tip, with the overlapping zone becoming increasingly large as the ratio g between the characteristic lengths in one of the materials approaches the value. The energy release rate has been calculated by means of the conservation J-integral. It is shown that contributions of the integrals along the crack faces have to be retained and the additional boundary condition along the crack edge [3] is essential to guarantee that the generalized J-integral [8] remains bounded. This additional boundary condition has always been omitted in the earlier literature because so far only the symmetrical problem in homogeneous materials was discussed. The boundary condition along the crack edge breaks the symmetry and becomes fundamental for interface problems. The special problem of a crack along the interface between couple-stress and classical elastic materials has also been addressed. Two types of transmission conditions have been considered: couple and rotation transmission conditions. In the former case, it is assumed that the couple-stress traction is continuous, and thus vanishes, at the interface. In the latter, it is assumed instead that the rotations are continuous at the interface. It turns out that the solutions are quite different in the two cases and it is not possible to satisfy simultaneously both type of transmission conditions, so that a mismatch is always present at the interface, resulting either in a non-balanced couple-stress or a discontinuity in the micro-rotations. It is shown also that the special case g = and g + reduces to the problem of a crack at the interface between classical and couple-stress elastic materials. Acknowledgements The paper has been completed during the Marie Curie Fellowship of A.P. at Aberystwyth University supported by the European Union Seventh Framework Programme under contract number PIEF-GA E.R. gratefully aknowledges financial support from the Cassa di Risparmio di Modena in the framework of the International Research Project Modelling of crack propagation in complex materials. References [] Willis JR. 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