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Effect of interface plasticity on circular blisters
Scripta Materialia 113 (2016) 222–225
Contents lists available at ScienceDirect
Scripta Materialia journal homepage: www.elsevier.com/locate/scriptamat
Regular Article
Effect of interface plasticity on circular blisters Antoine Ruffini a,⁎, Alphonse Finel a, Jérôme Colin b, Julien Durinck b a b
Laboratoire d'Étude des Microstructures, CNRS-ONERA, B.P. 72, F92322 Châtillon Cedex, France Institut P', CNRS-Université de Poitiers-ENSMA, Département Physique et Mécanique des Matériaux, SP2MI-Téléport 2, F86962 Futuroscope-Chasseneuil cedex, France
a r t i c l e
i n f o
Article history: Received 3 September 2015 Received in revised form 12 October 2015 Accepted 23 October 2015 Available online xxxx Keywords: Buckling Dislocations Sliding Delamination Phase-field
a b s t r a c t Numerical simulations based on finite-strain elasticity and a phase-field model of dislocations reveal that dislocations are emitted at the crack-front delamination of a circular blister. It is shown that the phenomenon induces a sliding in the film-substrate interface that modifies the shape of the buckling structure. This phenomenon is theoretically quantified introducing an axisymmetric sliding into the Föppl and von-Kármán equations that describe the elastic behavior of the film. By extending the analytical investigations to the crack opening, it is shown that dislocation-induced sliding may stabilize the buckling-driven delamination of three-dimensional structures. © 2016 Elsevier B.V. All rights reserved.
Delamination and buckling constitute a damage phenomenon which is frequently observed in thin film materials [1–3]. Fundamentally, buckling results from the elastic behavior of a compressed thin film as described by the Föppl and von-Kármán (FvK) model [4], which explains a wide variety of experienced morphologies (circular blisters [5], telephone cord buckles [6], etc.). At the mesoscale, the understanding of the buckling-driven delamination process is based on the concepts of fracture mechanics considering an increase of the interface toughness with the shear modes [7,8]. Although some dissipative mechanisms, such as friction [9] and plasticity [10], have already been put forward to explain a part of this mode dependence, understanding the elementary microscopic mechanisms involved during decohesion remains nowadays an important challenge. In the last decade, several studies have been carried out to include other phenomena not considered in the aforementioned descriptions (compliant substrates [11], surface energy [12], pressure [13], etc.). The experimental observations concerning crystalline materials have also highlighted plasticity damage, like persistent folding at the base of structures, characterized subsequently by both analytical modeling and finite element simulations [14–17]. Concomitantly, atomistic simulations have been performed, notably revealing that the emission of interface dislocations modifies the shape of the straight-sided blisters [18] and affects their buckling-driven delamination [19]. A limitation of these studies is that they only consider two-dimensional situations where plasticity and fracturing are known to exhibit very different features than those observed in three-dimensional ones.
⁎ Corresponding author. E-mail address: antoine.ruffi[email protected] (A. Ruffini).
http://dx.doi.org/10.1016/j.scriptamat.2015.10.038 1359-6462/© 2016 Elsevier B.V. All rights reserved.
In this work, this question is addressed by investigating the effects of interface plasticity on circular blisters — the objective being to generalize such effects to more realistic three-dimensional structures. To do so, a new numerical model is used considering both cracks and individual dislocations within finite-strain elasticity formalism [20]. This letter is divided into two different parts. Simulation results showing dislocations nucleation at the crack-front of the structure and their propagation in the film-substrate interface are first reported. Then, their effects are characterized by introducing an axisymmetric sliding into the FvK equations. The results are discussed in view of the well-known features of buckling-driven delamination. The circular blisters are simulated using a code based on finite-strain elasticity and a new phase-field model of dislocations [20]. First, we briefly recall this model considering isotropic materials. The change in the free elastic energy is defined as: Z F el ¼
E ν ε2ij ðrÞ þ ε 2kk ðrÞ dr3 ; 2ð1 þ νÞ 1 − 2ν
ð1Þ
where E is the Young's modulus, v is the Poisson's ratio and εij is the Green-Lagrange strains reflecting the nonlinear geometry of the system. In this work, film, interface and substrate are considered with Young's moduli Ef, Eint and Es, respectively, taken such that Ef = 0.5 Es and Eint = 0.1 Es. For all of these materials, Poisson's ratio is the film's ratio, i.e. vf = 0.28. In this model, Eint also controls the energy required to break the interface (toughness). In thin-film materials where blistering occurs this value is generally low, justifying the choice of 0.1 Es [1]. Numerically, the strain/elastic field is divided into cubic elementary voxels of side d that contain the elastic information depending on whether the position r is related to the film, the interface or the
A. Ruffini et al. / Scripta Materialia 113 (2016) 222–225
substrate. These fields are computed via the displacements of the vertices of voxels which also constitute the nodes of the numerical grid. Within this framework, a crack resulting from the breaking of a given region is introduced by canceling the elastic coefficients related to the corresponding voxels. In this study, d stands for the interatomic distance but could also represent a higher length when the atomic resolution is not required. The model also involves dislocations corresponding to the six slipsystems {100}〈100〉 of a simple cubic material. They are introduced using the eigenstrain formalism [21] combined with a phase-field description formulated at finite-strain [20]. In the present work, dislocations have the following properties: their Burgers vector norm is b = d, the ratio between the core energy and the elastic energy is 0.12, the ~ p ¼ 0:011 and the core width is about 5 d, the Peierls stress is σ velocity is about V ≈ 0:06 b τ~ over the range of resolved shear stress τ~ ∈ ½0:01; 0:05. Here, tildes denote stresses given in the unit of the shear modulus μ = Es / (2(1 + νf)). A nucleation criterion based on the local geometry of voxels also allows the generation of dislocations at crack-front. To simulate the circular blisters, a 200 × 200 × 8 d3 system is built in the (x, y, z) Cartesian frame, which accounts for a thin film of thickness h = 6 d deposited on a substrate of thickness d (Fig. 1). They are separated from each other by an interface of thickness d initially free of dislocations (coherent). In the polar frame (r, θ, z), the film is delaminated from its substrate over a central circular region of diameter r = R0 = 50 d, performed by canceling the elastic coefficients of voxels constituting the interface in this region. The bottom nodes of the grid are only forced to move in their initial (r, θ) plane in order to mimic a noncompliant substrate. Free boundary conditions are set for all surfaces. Buckling is finally induced by applying a homogeneous in-plane compressive strain ε0 b 0. The purely elastic equilibrium configurations of the blister are first researched by minimizing the elastic energy of the system for different values of ε0. In this case, expected results are obtained confirming the classical behavior of the buckling structure (Fig. 3): the blister is formed beyond a critical strain εc = − 0.014 and conserves its circular shape during buckling with a maximal deflection varying as the square root of ε0. Other results concerning the elastic behavior are also reported in Ref. [20]. Then, plasticity is allowed, switching the phase-fields on. From εc to ε0 = −0.065, no dislocations are observed. At ε0 = −0.0675, eight dislocation-loops are nucleated at the crack-front in the (001) interface plane (Fig. 2a). In the (r, θ) polar frame, the two loops around θ ≈ 0° ! have the Burgers vector b ¼ −½100 , the two others at θ ≈ 180° ! have b ¼ ½100, and the four loops at θ ≈ ± 90° are characterized ! by b ¼ ∓½010, respectively. They occupy stable positions for the given strain, at a distance from the circular crack-front in the order of the Burgers vector. This process initiates an interface sliding mechanism of the film over the substrate. At ε0 = −0.07, the previous dislocations penetrate in the interface causing the merging of the loops with identical Burgers vectors (Fig. 2b). This results in four loops occupying stable
Fig. 1. Circular blister with 2R the diameter of the delaminated zone, δ the maximal deflection of the film, h its thickness and ε0 the homogeneous in-plane strain.
223
Fig. 2. Snapshots of the interface for different ε0 revealing dislocation loops (in red) around the circular crack-front of the blister (see color online). The black arrows indicate junctions (see text). Snapshot (f) shows a mapping of (e) with respect to the phase-field parameter η related to the (001)[100] slip-system. Assuming for instance that d = 0.5 nm, the size of the domain is 200 × 200 d2 = 100 nm2.
symmetrical positions in the vicinity of the crack-front. At ε0 = −0.08, the previous loops overlap partially causing an inward axisymmetric displacement of about d in the interface (Fig. 2c). From ε0 = −0.08 to −0.1025, the situation does not change significantly (Fig. 2d). The only change is that the four dislocation-loops penetrate further away in the interface as the compressive strain increases. They still occupy stable positions at a distance from the crack-front which roughly corresponds to the thickness of the film, i.e. h = 6 d. When ε0 = −0.105 is reached, four new dislocations are nucleated, identical to the first ones and spanning the same area (Fig. 2e). No equilibrium configurations are observed afterwards. To understand how the second dislocations interact with the first ones, the phase-field parameter η related to the (001)[100] slip-system is also displayed for the applied strain ε0 = − 0.105 (Fig. 2f). This field reflects the [100] glide along the interface plane and therefore, its discontinuities follow the dislocation line of the (001)[100] slip-system. By symmetry, the phase-field corresponding to the (001)[010] slip-system displays the same mapping, but is rotated by 90°. The analysis of Fig. 2e and f shows that these two slip-systems interact to form eight junctions (black arrows in Fig. 2e), each composed of an edge component of one slip-system and a screw component of the other. In the corresponding snapshot, film-substrate sliding consists of an inward isotropic displacement ≈2d in the interface. The simulations are finally stopped because
224
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radii of curvature close to the grid spacing are measured in the film which is likely to provoke numerical artifacts. In the present work, it is shown that interface plasticity can take place at the base of circular blisters causing an axisymmetric sliding in the film-substrate interface. As for straight-sided blisters, the first dislocations stopped in the vicinity of the crack-front, probably due to the shear stress generated by the blister in the interface, which can be canceled at a distance of about the thickness of the film [19,18]. With actual dislocation properties, this phenomenon is certainly accentuated by the high Peierls stress. Even for a three-dimensional structure, the sliding is found to increase the maximal deflection of the blister compared to its purely elastic behavior, a phenomenon which becomes obvious when the interface sliding is homogeneously axisymmetric (Fig. 3). Thus, the maximal deflection δ is about 2% higher when the first dislocations are nucleated. In the following, in order to characterize these results, an average sliding Δ located at the base of the blister is introduced in the Föppl von-Kármán (FvK) model which describes the elastic behavior of the film [4]. For this, we start from the two dimensionless equations controlling the mechanical equilibrium of the film in polar coordinate [7]: 6 1 − ν 2f 3 0 N ðr Þ þ ϕ2 ðrÞ ¼ 0; N″ðr Þ þ r r2 ϕ″ðr Þ þ
ϕ0 ðrÞ − r
1
r2
þ N ðr Þ − μ 2 ϕðr Þ ¼ 0;
N0 ð0Þ ¼ 0; N0 ð1Þ þ
ð4Þ
1 − ν f Nð1Þ ¼ −h R Δ E f = D f ;
ϕð0Þ ¼ ϕð1Þ ¼ 0:
ð2Þ
ð3Þ
NðrÞ ¼ R2 hðσ rr − σ 0 Þ = D f and the rotation ϕðrÞ ¼ w0 ðrÞ = h with w the out-of-plane displacement. In these equations, r ¼ r = R denotes the non-dimensional distance from the center of the film and μ2 = − R2hσ0 / Df the non-dimensional in-plane stress σ 0 ¼ 3
E f ð1 − ν f Þε 0 defined with the help of the bending stiffness D f ¼ E f h = ν 2f Þ.
The sliding is considered by introducing an inward displacement −Δ (with Δ N 0) at the crack-front of the blister. With u as the radial displacement, the following boundary conditions are thus considered, u(0) = 0, u(1) = −Δ,w(0) = δ and w(1) = 0, which are re-expressed as a function of N and ϕ. For stress N, Hooke's law is applied by noticing that the hoop stress σθθ is linked to the radial stressσ r r by formulaσ θθ ¼ ðrσ r r Þ0 atr ¼ 1,
ð6Þ
Nðr Þ ¼ sN1 ðr Þ þ s2 N2 ðr Þ þ O s3 ;
ð7Þ
ϕðr Þ ¼ sϕ1 ðr Þ þ O s3 ;
ð8Þ
μ 2 ¼ μ 2c þ sμ 21 þ s2 μ 22 þ O s3 ;
ð9Þ
with s = δ / h coming from the normalization w1(0) = h. Considering the boundary conditions given Eqs. (4)–(6), one obtains the following explicit solutions: ð10Þ
N 2 ðrÞ ¼ 3μ c 1 þ v f v f μ c r J 20 ðμ c Þ i h μ c r Þ −J 0 ðμ c r ÞJ 1 ðμ c r Þ þ μ c r J 21 ðμ c r Þ þ 1 − v f μ c r J 20 ð = r ð J 0 ðμ c Þ − 1Þ2 ; ð11Þ ϕ1 ðr Þ ¼ μ c J 1 ðμ c r Þ = ð J 0 ðμ c Þ − 1Þ;
ð12Þ
where J0 and J1 are the Bessel functions of the first kind of zero and first order, respectively, μc = 3.83171 is the first nontrivial zero of J1, μ12 = 12ΔR(1 + νf) / (hδ) and μ 22 = 3.6310(1 + νf) + 0.2231(1 − νf2). Considering Eq. (9) with s = δ / h and the explicit values of the series' terms, the asymptotic maximal deflection of the circular blister is finally found to be: 4R δ ¼ π
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Δ c1 ε c − ε 0 þ ; R
ð13Þ
with c1 = 2.2600 / (2.1086 − νf) and the critical buckling strain: εc ¼ −
Fig. 3. Maximal deflections of the circular blister δ versus the in-plane strain ε0. The critical buckling strain is εc given Eq. (14). The theoretical deflection is plotted using Eq. (13) and considering Δ = 0 without sliding and Δ ≈ b / 4 with sliding. Only equilibrium data have been displayed in this graph.
ð5Þ
As the FvK Eqs. (2) and (3) are not solvable in closed-form, a classical asymptotic analysis is conducted, reviving the work performed in Ref. [7]. In the present case, the solutions write:
N1 ðr Þ ¼ −12RΔ 1 þ ν f =ðhδÞ;
where the unknown quantities are the change in the radial in-plane stress
12 and E f ¼ E f = ð1 −
while it must simply be continuous and differentiable at r ¼ 0 [22]. This results in:
1:2235 1 þ νf
2 h : R
ð14Þ
When Δ = 0, the classical solutions of the purely elastic case are reproduced [7]. Eq. (13) confirms that the sliding Δ N 0 induces the increase of the maximal deflection of the blister δ. In Fig. 3, the theoretical deflection with and without sliding is plotted and compared to the values extracted from simulations considering R = (1 + ε0)R0. For this, it is noticed that the sliding of the analytical expressions is taken in the middle plane of the film while it is measured in the interface in the simulations. The value Δ ≈ b / 4 is thus pragmatically considered in the theoretical curve. Apart from the remaining discrepancies, which are due to both the asymptotic expansion of the solution and the weakly nonlinear regime assumed in the FvK model [23], it is found that the theoretical description reproduces the results of the simulations. Finally, the crack-opening caused by buckling-driven delamination is theoretically envisaged, considering a generalized situation where the sliding Δ reflects the continual and concomitant introduction of a large number of dislocations. To do this, the asymptotic energy release rate of the structure G is calculated, which is the quantity compared to the interface toughness Γ when it is to predict interface fracturing via the Griffith criterion [24].
A. Ruffini et al. / Scripta Materialia 113 (2016) 222–225
225
By definition, G = (−∂U / ∂R) / (2πR) where U is the free energy stored in the whole thin-film. In the buckled part, this energy is calculated into the FvK framework using the Δ-dependent solutions given in Eqs. (10)–(14) and following the procedure exposed in Refs. [7,24]. In the adherent part, it is assumed that the sliding Δ results from a frictionless axisymmetric linear decompression affecting a circular ring of constant outside diameter D N R. In this context, D delimitates a circular region around the blister beyond which the sliding relaxation does not occur. This length can thus be related to physical obstacles such as the distance between blisters, the distance between grain boundaries or the finite size of the system (here for instance D ≈ 4R0 which corresponds to the finite domain of the phase-field simulations). Under this assumption, the radial and hoop strains write εrr ¼ ε 0 þ Δ=ðD = 2 – RÞ and ε θθ ¼ ε0 − ðΔ = rÞðD = 2 – rÞ = ðD = 2 – RÞ, respectively. Using Hooke's law in plane strain, εrr and εθθ enable us to obtain the elastic energy stored in the adherent part of the film [1]. After calculations, the energy release rate is found to be: " " 2 # 2 # G0 εc G0 εc þ ε0 Δ Δ − 2 ; 1− þ g1 G ¼ c2 ε0 c2 R D = 2 − R ε0 ð15Þ with G0 ¼ hE f ð1 þ ν f Þε 20 the energy per unit area stored in the unbuckled film, c2 = 1 + 0.9021(1 − νf) and g1 = p2(4 ln p − p − 0.7726) / (8(1 + νf)(p − 2)) given with p ¼ D = R. When Δ = 0, the classical asymptotic expression of G is reproduced [7]. As in Ref. [19], the frictionless sliding is quantified considering that it results from the equilibrium of the radial stresses at the base of the circular blister. Using the previous results, one can show that Δ ¼ ðD = 2 − RÞ ðε c − ε 0 Þ = ð1 þ c3 pÞ with c3 = (0.4511 − 0.9511 νf) / (1 + νf). Introducing this expression in Eq. (15) finally yields: εc εc g − 1 ; 1 − 1 þ G ¼ 2 ε0 ε0 ð1 þ c3 pÞ2 2G0 g 1
ð16Þ
with g2 = (2c2g1 + (1 + 2c3)(1 + c3p)p) / (4c2g1). In Fig. 4, the typical evolution of G / Γ is plotted versus R for different values of strain ε0 considering the classical formula without sliding (Eq. (15) with Δ = 0) and the one with frictionless sliding affecting an initially adherent ring of diameter D N R (Eq. (16)). Since delamination proceeds when G / Γ = 1 for R = R0, it is shown that for a given applied strain ε0 = ε⁎, an infinite expansion is expected with the classical formulation (G / Γ N 1 when R increases) while a finite spreading is predicted when the sliding is considered (G / Γ → 0 when R increases). Usually, the paradox of the first description is removed considering a mode-dependent toughness increasing with shear (see [7] and references therein). Interestingly here, one such dependence is not required to explain the finite size of the observed blisters. As a consequence, interface plasticity occurring over a circular region delimited by D might explain a part of the mode-dependent toughness experienced in some systems. Of course, the analysis performed in this study is idealized and the physical mechanisms highlighted are certainly not sufficient to explain all the phenomenology of interface fracturing. Nonetheless, we believe that plasticity, as described in this work, realistically contributes to a significant part of the process and that some aforementioned results may be considered in mesoscopic descriptions [25]. In future works, effect of plasticity on the crack-front destabilization and the formation of complex patterns may be addressed using both
Fig. 4. Normalized energy release rate G / Γ versus radius R for different strains ε0. Gray dashed lines stand for the classical model in the purely elastic case (Eq. (15) with Δ = 0) while black lines stand for the current one including sliding (Eq. (16)). The present curves are obtained using the parameters of the simulations with D ¼ 4R0 . Rf denotes the final radius once delamination has proceeded.
simulations and theoretical investigations. To do so, the current simulations would have to be improved, considering the faced-centered cubic symmetry and anisotropic elasticity. Within an upscaling perspective, microstructures such as grain boundaries would also have to be introduced. References [1] L. Freund, S. Suresh, Thin Film Materials-Stress, Defects Formation and Surface Evolution, Cambridge Edition, Cambridge University Press, 2003. [2] M. Cordill, D. Bahr, N. Moody, W. Gerberich, Mater. Sci. Eng. A 443 (12) (2007) 150–155. [3] S.-J. Yu, X.-F. Xiao, M.-G. Chen, H. Zhou, J. Chen, P.-Z. Si, Z.-W. Jiao, Acta Mater. 64 (0) (2014) 41–53. [4] L.D. Landau, E.M. Lifshitz, Theory of Elasticity, Pergamon Edition, Vol. 7 of Course of theoretical physics, 1959. [5] A. Kuznetsov, M. Gleeson, F. Bijkerk, Thin Solid Films 545 (2013) 571–579. [6] A. Pundt, P. Pekarski, Scr. Mater. 48 (4) (2003) 419–423. [7] J.W. Hutchinson, Z. Suo, Adv. Appl. Mech. 29 (63) (1992) 63–191. [8] J.-Y. Faou, G. Parry, S. Grachev, E. Barthel, Phys. Rev. Lett. 108 (11) (2012) 116102. [9] B. Audoly, J. Mech. Phys. Solids 48 (11) (2000) 2315–2332. [10] Y. Wei, J.W. Hutchinson, Philos. Mag. 88 (30) (2010) 3841–3859. [11] K. Wu, J. Zhang, G. Liu, P. Zhang, P. Cheng, J. Li, G. Zhang, J. Sun, Acta Mater. 61 (20) (2013) 7889–7903. [12] T.Q. Lu, W.X. Zhang, T. Wang, Int. J. Eng. Sci. 49 (9) (2011) 967–975. [13] J. Colin, C. Coupeau, J. Durinck, A. Cimetière, J. Grilhé, Phys. Rev. E 89 (3) (2014) 032410. [14] M. Cordill, N. Moody, D. Bahr, Acta Mater. 53 (9) (2005) 2555–2562. [15] J. Colin, C. Coupeau, J. Grilhé, Phys. Rev. Lett. 99 (4) (2007) 046101. [16] R. Telyatnik, A. Osipov, S. Kukushkin, Phys. Solid State 57 (1) (2015) 162–172. [17] S. Hamade, J. Durinck, G. Parry, C. Coupeau, A. Cimetière, J. Grilhé, J. Colin, Phys. Rev. E 91 (1) (2015) 012410. [18] A. Ruffini, J. Durinck, J. Colin, C. Coupeau, J. Grilhé, Int. J. Solids Struct. 50 (2223) (2013) 3717–3722. [19] A. Ruffini, J. Durinck, J. Colin, C. Coupeau, J. Grilhé, Scr. Mater. 67 (2) (2012) 157–160. [20] A. Ruffini, A. Finel, Acta Mater. 92 (0) (2015) 197–208. [21] T. Mura, Micromechanics of defects in solids, vol. 3, Springer Netherlands, 1987. [22] A. El Doussouki, M. Guedda, M. Jazar, M. Benlahsen, Appl. Math. Comput. 219 (9) (2013) 4340–4345. [23] B. Audoly, Y. Pomeau, Elasticity and Geometry — From Hair Curls to the non-Linear Response of Shells, Oxford University Press Edition, New-York, 2010. [24] A. Evans, J. Hutchinson, Int. J. Solids Struct. 20 (5) (1984) 455–466. [25] K. Pan, Y. Ni, L. He, Phys. Rev. E 88 (6) (2013) 062405.
Contents lists available at ScienceDirect
Scripta Materialia journal homepage: www.elsevier.com/locate/scriptamat
Regular Article
Effect of interface plasticity on circular blisters Antoine Ruffini a,⁎, Alphonse Finel a, Jérôme Colin b, Julien Durinck b a b
Laboratoire d'Étude des Microstructures, CNRS-ONERA, B.P. 72, F92322 Châtillon Cedex, France Institut P', CNRS-Université de Poitiers-ENSMA, Département Physique et Mécanique des Matériaux, SP2MI-Téléport 2, F86962 Futuroscope-Chasseneuil cedex, France
a r t i c l e
i n f o
Article history: Received 3 September 2015 Received in revised form 12 October 2015 Accepted 23 October 2015 Available online xxxx Keywords: Buckling Dislocations Sliding Delamination Phase-field
a b s t r a c t Numerical simulations based on finite-strain elasticity and a phase-field model of dislocations reveal that dislocations are emitted at the crack-front delamination of a circular blister. It is shown that the phenomenon induces a sliding in the film-substrate interface that modifies the shape of the buckling structure. This phenomenon is theoretically quantified introducing an axisymmetric sliding into the Föppl and von-Kármán equations that describe the elastic behavior of the film. By extending the analytical investigations to the crack opening, it is shown that dislocation-induced sliding may stabilize the buckling-driven delamination of three-dimensional structures. © 2016 Elsevier B.V. All rights reserved.
Delamination and buckling constitute a damage phenomenon which is frequently observed in thin film materials [1–3]. Fundamentally, buckling results from the elastic behavior of a compressed thin film as described by the Föppl and von-Kármán (FvK) model [4], which explains a wide variety of experienced morphologies (circular blisters [5], telephone cord buckles [6], etc.). At the mesoscale, the understanding of the buckling-driven delamination process is based on the concepts of fracture mechanics considering an increase of the interface toughness with the shear modes [7,8]. Although some dissipative mechanisms, such as friction [9] and plasticity [10], have already been put forward to explain a part of this mode dependence, understanding the elementary microscopic mechanisms involved during decohesion remains nowadays an important challenge. In the last decade, several studies have been carried out to include other phenomena not considered in the aforementioned descriptions (compliant substrates [11], surface energy [12], pressure [13], etc.). The experimental observations concerning crystalline materials have also highlighted plasticity damage, like persistent folding at the base of structures, characterized subsequently by both analytical modeling and finite element simulations [14–17]. Concomitantly, atomistic simulations have been performed, notably revealing that the emission of interface dislocations modifies the shape of the straight-sided blisters [18] and affects their buckling-driven delamination [19]. A limitation of these studies is that they only consider two-dimensional situations where plasticity and fracturing are known to exhibit very different features than those observed in three-dimensional ones.
⁎ Corresponding author. E-mail address: antoine.ruffi[email protected] (A. Ruffini).
http://dx.doi.org/10.1016/j.scriptamat.2015.10.038 1359-6462/© 2016 Elsevier B.V. All rights reserved.
In this work, this question is addressed by investigating the effects of interface plasticity on circular blisters — the objective being to generalize such effects to more realistic three-dimensional structures. To do so, a new numerical model is used considering both cracks and individual dislocations within finite-strain elasticity formalism [20]. This letter is divided into two different parts. Simulation results showing dislocations nucleation at the crack-front of the structure and their propagation in the film-substrate interface are first reported. Then, their effects are characterized by introducing an axisymmetric sliding into the FvK equations. The results are discussed in view of the well-known features of buckling-driven delamination. The circular blisters are simulated using a code based on finite-strain elasticity and a new phase-field model of dislocations [20]. First, we briefly recall this model considering isotropic materials. The change in the free elastic energy is defined as: Z F el ¼
E ν ε2ij ðrÞ þ ε 2kk ðrÞ dr3 ; 2ð1 þ νÞ 1 − 2ν
ð1Þ
where E is the Young's modulus, v is the Poisson's ratio and εij is the Green-Lagrange strains reflecting the nonlinear geometry of the system. In this work, film, interface and substrate are considered with Young's moduli Ef, Eint and Es, respectively, taken such that Ef = 0.5 Es and Eint = 0.1 Es. For all of these materials, Poisson's ratio is the film's ratio, i.e. vf = 0.28. In this model, Eint also controls the energy required to break the interface (toughness). In thin-film materials where blistering occurs this value is generally low, justifying the choice of 0.1 Es [1]. Numerically, the strain/elastic field is divided into cubic elementary voxels of side d that contain the elastic information depending on whether the position r is related to the film, the interface or the
A. Ruffini et al. / Scripta Materialia 113 (2016) 222–225
substrate. These fields are computed via the displacements of the vertices of voxels which also constitute the nodes of the numerical grid. Within this framework, a crack resulting from the breaking of a given region is introduced by canceling the elastic coefficients related to the corresponding voxels. In this study, d stands for the interatomic distance but could also represent a higher length when the atomic resolution is not required. The model also involves dislocations corresponding to the six slipsystems {100}〈100〉 of a simple cubic material. They are introduced using the eigenstrain formalism [21] combined with a phase-field description formulated at finite-strain [20]. In the present work, dislocations have the following properties: their Burgers vector norm is b = d, the ratio between the core energy and the elastic energy is 0.12, the ~ p ¼ 0:011 and the core width is about 5 d, the Peierls stress is σ velocity is about V ≈ 0:06 b τ~ over the range of resolved shear stress τ~ ∈ ½0:01; 0:05. Here, tildes denote stresses given in the unit of the shear modulus μ = Es / (2(1 + νf)). A nucleation criterion based on the local geometry of voxels also allows the generation of dislocations at crack-front. To simulate the circular blisters, a 200 × 200 × 8 d3 system is built in the (x, y, z) Cartesian frame, which accounts for a thin film of thickness h = 6 d deposited on a substrate of thickness d (Fig. 1). They are separated from each other by an interface of thickness d initially free of dislocations (coherent). In the polar frame (r, θ, z), the film is delaminated from its substrate over a central circular region of diameter r = R0 = 50 d, performed by canceling the elastic coefficients of voxels constituting the interface in this region. The bottom nodes of the grid are only forced to move in their initial (r, θ) plane in order to mimic a noncompliant substrate. Free boundary conditions are set for all surfaces. Buckling is finally induced by applying a homogeneous in-plane compressive strain ε0 b 0. The purely elastic equilibrium configurations of the blister are first researched by minimizing the elastic energy of the system for different values of ε0. In this case, expected results are obtained confirming the classical behavior of the buckling structure (Fig. 3): the blister is formed beyond a critical strain εc = − 0.014 and conserves its circular shape during buckling with a maximal deflection varying as the square root of ε0. Other results concerning the elastic behavior are also reported in Ref. [20]. Then, plasticity is allowed, switching the phase-fields on. From εc to ε0 = −0.065, no dislocations are observed. At ε0 = −0.0675, eight dislocation-loops are nucleated at the crack-front in the (001) interface plane (Fig. 2a). In the (r, θ) polar frame, the two loops around θ ≈ 0° ! have the Burgers vector b ¼ −½100 , the two others at θ ≈ 180° ! have b ¼ ½100, and the four loops at θ ≈ ± 90° are characterized ! by b ¼ ∓½010, respectively. They occupy stable positions for the given strain, at a distance from the circular crack-front in the order of the Burgers vector. This process initiates an interface sliding mechanism of the film over the substrate. At ε0 = −0.07, the previous dislocations penetrate in the interface causing the merging of the loops with identical Burgers vectors (Fig. 2b). This results in four loops occupying stable
Fig. 1. Circular blister with 2R the diameter of the delaminated zone, δ the maximal deflection of the film, h its thickness and ε0 the homogeneous in-plane strain.
223
Fig. 2. Snapshots of the interface for different ε0 revealing dislocation loops (in red) around the circular crack-front of the blister (see color online). The black arrows indicate junctions (see text). Snapshot (f) shows a mapping of (e) with respect to the phase-field parameter η related to the (001)[100] slip-system. Assuming for instance that d = 0.5 nm, the size of the domain is 200 × 200 d2 = 100 nm2.
symmetrical positions in the vicinity of the crack-front. At ε0 = −0.08, the previous loops overlap partially causing an inward axisymmetric displacement of about d in the interface (Fig. 2c). From ε0 = −0.08 to −0.1025, the situation does not change significantly (Fig. 2d). The only change is that the four dislocation-loops penetrate further away in the interface as the compressive strain increases. They still occupy stable positions at a distance from the crack-front which roughly corresponds to the thickness of the film, i.e. h = 6 d. When ε0 = −0.105 is reached, four new dislocations are nucleated, identical to the first ones and spanning the same area (Fig. 2e). No equilibrium configurations are observed afterwards. To understand how the second dislocations interact with the first ones, the phase-field parameter η related to the (001)[100] slip-system is also displayed for the applied strain ε0 = − 0.105 (Fig. 2f). This field reflects the [100] glide along the interface plane and therefore, its discontinuities follow the dislocation line of the (001)[100] slip-system. By symmetry, the phase-field corresponding to the (001)[010] slip-system displays the same mapping, but is rotated by 90°. The analysis of Fig. 2e and f shows that these two slip-systems interact to form eight junctions (black arrows in Fig. 2e), each composed of an edge component of one slip-system and a screw component of the other. In the corresponding snapshot, film-substrate sliding consists of an inward isotropic displacement ≈2d in the interface. The simulations are finally stopped because
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radii of curvature close to the grid spacing are measured in the film which is likely to provoke numerical artifacts. In the present work, it is shown that interface plasticity can take place at the base of circular blisters causing an axisymmetric sliding in the film-substrate interface. As for straight-sided blisters, the first dislocations stopped in the vicinity of the crack-front, probably due to the shear stress generated by the blister in the interface, which can be canceled at a distance of about the thickness of the film [19,18]. With actual dislocation properties, this phenomenon is certainly accentuated by the high Peierls stress. Even for a three-dimensional structure, the sliding is found to increase the maximal deflection of the blister compared to its purely elastic behavior, a phenomenon which becomes obvious when the interface sliding is homogeneously axisymmetric (Fig. 3). Thus, the maximal deflection δ is about 2% higher when the first dislocations are nucleated. In the following, in order to characterize these results, an average sliding Δ located at the base of the blister is introduced in the Föppl von-Kármán (FvK) model which describes the elastic behavior of the film [4]. For this, we start from the two dimensionless equations controlling the mechanical equilibrium of the film in polar coordinate [7]: 6 1 − ν 2f 3 0 N ðr Þ þ ϕ2 ðrÞ ¼ 0; N″ðr Þ þ r r2 ϕ″ðr Þ þ
ϕ0 ðrÞ − r
1
r2
þ N ðr Þ − μ 2 ϕðr Þ ¼ 0;
N0 ð0Þ ¼ 0; N0 ð1Þ þ
ð4Þ
1 − ν f Nð1Þ ¼ −h R Δ E f = D f ;
ϕð0Þ ¼ ϕð1Þ ¼ 0:
ð2Þ
ð3Þ
NðrÞ ¼ R2 hðσ rr − σ 0 Þ = D f and the rotation ϕðrÞ ¼ w0 ðrÞ = h with w the out-of-plane displacement. In these equations, r ¼ r = R denotes the non-dimensional distance from the center of the film and μ2 = − R2hσ0 / Df the non-dimensional in-plane stress σ 0 ¼ 3
E f ð1 − ν f Þε 0 defined with the help of the bending stiffness D f ¼ E f h = ν 2f Þ.
The sliding is considered by introducing an inward displacement −Δ (with Δ N 0) at the crack-front of the blister. With u as the radial displacement, the following boundary conditions are thus considered, u(0) = 0, u(1) = −Δ,w(0) = δ and w(1) = 0, which are re-expressed as a function of N and ϕ. For stress N, Hooke's law is applied by noticing that the hoop stress σθθ is linked to the radial stressσ r r by formulaσ θθ ¼ ðrσ r r Þ0 atr ¼ 1,
ð6Þ
Nðr Þ ¼ sN1 ðr Þ þ s2 N2 ðr Þ þ O s3 ;
ð7Þ
ϕðr Þ ¼ sϕ1 ðr Þ þ O s3 ;
ð8Þ
μ 2 ¼ μ 2c þ sμ 21 þ s2 μ 22 þ O s3 ;
ð9Þ
with s = δ / h coming from the normalization w1(0) = h. Considering the boundary conditions given Eqs. (4)–(6), one obtains the following explicit solutions: ð10Þ
N 2 ðrÞ ¼ 3μ c 1 þ v f v f μ c r J 20 ðμ c Þ i h μ c r Þ −J 0 ðμ c r ÞJ 1 ðμ c r Þ þ μ c r J 21 ðμ c r Þ þ 1 − v f μ c r J 20 ð = r ð J 0 ðμ c Þ − 1Þ2 ; ð11Þ ϕ1 ðr Þ ¼ μ c J 1 ðμ c r Þ = ð J 0 ðμ c Þ − 1Þ;
ð12Þ
where J0 and J1 are the Bessel functions of the first kind of zero and first order, respectively, μc = 3.83171 is the first nontrivial zero of J1, μ12 = 12ΔR(1 + νf) / (hδ) and μ 22 = 3.6310(1 + νf) + 0.2231(1 − νf2). Considering Eq. (9) with s = δ / h and the explicit values of the series' terms, the asymptotic maximal deflection of the circular blister is finally found to be: 4R δ ¼ π
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Δ c1 ε c − ε 0 þ ; R
ð13Þ
with c1 = 2.2600 / (2.1086 − νf) and the critical buckling strain: εc ¼ −
Fig. 3. Maximal deflections of the circular blister δ versus the in-plane strain ε0. The critical buckling strain is εc given Eq. (14). The theoretical deflection is plotted using Eq. (13) and considering Δ = 0 without sliding and Δ ≈ b / 4 with sliding. Only equilibrium data have been displayed in this graph.
ð5Þ
As the FvK Eqs. (2) and (3) are not solvable in closed-form, a classical asymptotic analysis is conducted, reviving the work performed in Ref. [7]. In the present case, the solutions write:
N1 ðr Þ ¼ −12RΔ 1 þ ν f =ðhδÞ;
where the unknown quantities are the change in the radial in-plane stress
12 and E f ¼ E f = ð1 −
while it must simply be continuous and differentiable at r ¼ 0 [22]. This results in:
1:2235 1 þ νf
2 h : R
ð14Þ
When Δ = 0, the classical solutions of the purely elastic case are reproduced [7]. Eq. (13) confirms that the sliding Δ N 0 induces the increase of the maximal deflection of the blister δ. In Fig. 3, the theoretical deflection with and without sliding is plotted and compared to the values extracted from simulations considering R = (1 + ε0)R0. For this, it is noticed that the sliding of the analytical expressions is taken in the middle plane of the film while it is measured in the interface in the simulations. The value Δ ≈ b / 4 is thus pragmatically considered in the theoretical curve. Apart from the remaining discrepancies, which are due to both the asymptotic expansion of the solution and the weakly nonlinear regime assumed in the FvK model [23], it is found that the theoretical description reproduces the results of the simulations. Finally, the crack-opening caused by buckling-driven delamination is theoretically envisaged, considering a generalized situation where the sliding Δ reflects the continual and concomitant introduction of a large number of dislocations. To do this, the asymptotic energy release rate of the structure G is calculated, which is the quantity compared to the interface toughness Γ when it is to predict interface fracturing via the Griffith criterion [24].
A. Ruffini et al. / Scripta Materialia 113 (2016) 222–225
225
By definition, G = (−∂U / ∂R) / (2πR) where U is the free energy stored in the whole thin-film. In the buckled part, this energy is calculated into the FvK framework using the Δ-dependent solutions given in Eqs. (10)–(14) and following the procedure exposed in Refs. [7,24]. In the adherent part, it is assumed that the sliding Δ results from a frictionless axisymmetric linear decompression affecting a circular ring of constant outside diameter D N R. In this context, D delimitates a circular region around the blister beyond which the sliding relaxation does not occur. This length can thus be related to physical obstacles such as the distance between blisters, the distance between grain boundaries or the finite size of the system (here for instance D ≈ 4R0 which corresponds to the finite domain of the phase-field simulations). Under this assumption, the radial and hoop strains write εrr ¼ ε 0 þ Δ=ðD = 2 – RÞ and ε θθ ¼ ε0 − ðΔ = rÞðD = 2 – rÞ = ðD = 2 – RÞ, respectively. Using Hooke's law in plane strain, εrr and εθθ enable us to obtain the elastic energy stored in the adherent part of the film [1]. After calculations, the energy release rate is found to be: " " 2 # 2 # G0 εc G0 εc þ ε0 Δ Δ − 2 ; 1− þ g1 G ¼ c2 ε0 c2 R D = 2 − R ε0 ð15Þ with G0 ¼ hE f ð1 þ ν f Þε 20 the energy per unit area stored in the unbuckled film, c2 = 1 + 0.9021(1 − νf) and g1 = p2(4 ln p − p − 0.7726) / (8(1 + νf)(p − 2)) given with p ¼ D = R. When Δ = 0, the classical asymptotic expression of G is reproduced [7]. As in Ref. [19], the frictionless sliding is quantified considering that it results from the equilibrium of the radial stresses at the base of the circular blister. Using the previous results, one can show that Δ ¼ ðD = 2 − RÞ ðε c − ε 0 Þ = ð1 þ c3 pÞ with c3 = (0.4511 − 0.9511 νf) / (1 + νf). Introducing this expression in Eq. (15) finally yields: εc εc g − 1 ; 1 − 1 þ G ¼ 2 ε0 ε0 ð1 þ c3 pÞ2 2G0 g 1
ð16Þ
with g2 = (2c2g1 + (1 + 2c3)(1 + c3p)p) / (4c2g1). In Fig. 4, the typical evolution of G / Γ is plotted versus R for different values of strain ε0 considering the classical formula without sliding (Eq. (15) with Δ = 0) and the one with frictionless sliding affecting an initially adherent ring of diameter D N R (Eq. (16)). Since delamination proceeds when G / Γ = 1 for R = R0, it is shown that for a given applied strain ε0 = ε⁎, an infinite expansion is expected with the classical formulation (G / Γ N 1 when R increases) while a finite spreading is predicted when the sliding is considered (G / Γ → 0 when R increases). Usually, the paradox of the first description is removed considering a mode-dependent toughness increasing with shear (see [7] and references therein). Interestingly here, one such dependence is not required to explain the finite size of the observed blisters. As a consequence, interface plasticity occurring over a circular region delimited by D might explain a part of the mode-dependent toughness experienced in some systems. Of course, the analysis performed in this study is idealized and the physical mechanisms highlighted are certainly not sufficient to explain all the phenomenology of interface fracturing. Nonetheless, we believe that plasticity, as described in this work, realistically contributes to a significant part of the process and that some aforementioned results may be considered in mesoscopic descriptions [25]. In future works, effect of plasticity on the crack-front destabilization and the formation of complex patterns may be addressed using both
Fig. 4. Normalized energy release rate G / Γ versus radius R for different strains ε0. Gray dashed lines stand for the classical model in the purely elastic case (Eq. (15) with Δ = 0) while black lines stand for the current one including sliding (Eq. (16)). The present curves are obtained using the parameters of the simulations with D ¼ 4R0 . Rf denotes the final radius once delamination has proceeded.
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