Fracture, Delamination, and Buckling of Elastic Thin Films on Compliant Substrates

FRACTURE, DELAMINATION, AND BUCKLING OF ELASTIC THIN FILMS ON COMPLIANT SUBSTRATES

Haixia Mei, Yaoyu Pang, Se Hyuk Im, Rui Huang Department of Aerospace Engineering and Engineering Mechanics University of Texas at Austin 1 University Station, C0600 Austin, Texas 78712 Phone: (512) 471-7558 Fax: (512) 471-5500 Email: [email protected]

ABSTRACT Greek symbols A series of studies have been conducted for mechanical fracture toughness (J/m2) behavior of elastic thin films on compliant substrates. Under first Dundurs parameter (dimensionless) tension, the film may fracture by growing channel cracks. The second Dundurs parameter (dimensionless) driving force for channel cracking (i.e., the energy release opening displacement (m) rate) increases significantly for compliant substrates. phase angle of mode mix (radian) Moreover, channel cracking may be accompanied by Poisson’s ratio (dimensionless) interfacial delamination. For a film on a relatively compliant 2 substrate, a critical interface toughness is predicted, which stress (N/m ) separates stable and unstable delamination. For a film on a relatively stiff substrate, however, a channel crack grows with Subscripts no delamination when the interface toughness is greater than a f film critical value. An effective energy release rate for the steady- i interface state growth of a channel crack is defined to account for the s substrate influence of interfacial delamination on both the fracture ss steady state driving force and the resistance, which can be significantly higher than the energy release rate assuming no delamination. Alternatively, when the film is under compression, it tends to INTRODUCTION buckle. Two buckling modes have been observed, one with interfacial delamination (i.e., buckle-delamination) and the Integrated structures with mechanically soft components have other without delamination (i.e., wrinkling). By comparing the recently been pursued over a wide range of novel applications, critical stresses for the onset of buckling, we give a criterion from high performance integrated circuits in microelectronics for the selection of the buckling modes, which depends on the [1] to unconventional organic electronics [2] and stretchable stiffness ratio between the film and the substrate as well as the electronics [3,4], along with the ubiquitous integration of hard interface defects. A general conclusion from these studies is and soft materials in biomedical systems [5]. In particular, the that, whether tension or compression, the interfacial properties integration of low dielectric constant (low k) materials in are critical in controlling the morphology and failure of elastic advanced interconnects of microelectronics has posed thin films on compliant substrates. significant challenges for reliability issues due to compromised mechanical properties. Two common failure KEY WORDS: fracture, delamination, buckling, interface modes have been reported, one for cohesive fracture [6] and the other for interfacial delamination [7]. The former pertains NOMENCLATURE to the brittleness of the low-k materials subjected to tension, and the latter manifests due to poor adhesion between low-k E elastic modulus (N/m2) and surrounding materials. Furthermore, a thin film layer G energy release rate (J/m2) under compression can buckle, which leads to failure due to K stress intensity factor (N m1/2) uneven surfaces, buckle-driven delamination and/or fracture W fracture energy (J/m2) [8,9]. These failure mechanisms are schematically illustrated b half width of buckle-delamination (m) in Figure 1. In this paper, the influence of interfacial d delamination width (m) delamination on fracture and buckling of elastic thin films is h thickness (m) investigated, with an emphasis on applications with compliant substrates.

762 CHANNEL CRACKING the film and the substrate or the buffer layers are assumed to remain perfectly bonded as the channel crack grows in the One common cohesive fracture mode for thin films under film (Figure 1a). Under this assumption, the energy release tension is channel cracking (Figure 1a). Previous studies have rate for the steady-state growth of a channel crack in a thin shown that the driving force (i.e., the energy release rate) for elastic film bonded to a thick elastic substrate is: the steady-state growth of a channel crack depends on the 2 h constraint effect of surrounding layers [10-12]. For a brittle f f , (1) Gss Z( , ) thin film on an elastic substrate, the driving force increases for E f increasingly compliant substrates. The effect of constraint can where is the tensile stress in the film, is the film be partly lost as the substrate deforms plastically [13] or f h f viscoelastically [14,15]. More recent studies have investigated 2 thickness, and E f E f 1 f is the plane strain modulus of the effects of stacked buffer layers [16,17] and patterned film the film with Young’s modulus E and Poisson’s ratio . The structures [6]. In most of these studies, the interfaces between f f dimensionless coefficient Z depends on the elastic mismatch between the film and the substrate, through the Dundurs’ parameters Gss Film E E f s , (2)

E f Es

h f f E (1 )(1 2 ) E (1 )(1 2 ) f f s s s f . (3) 2(1 f )(1 s ) E f Es

Substrate When the film and the substrate have identical elastic (a) moduli, we have 0 and Z 1.976 . The value of Z decreases slightly for a compliant film on a relatively stiff substrate ( E f Es and 0 ). A more compliant substrate Film ( 0 ), on the other hand, provides less constraint against film cracking. Thus, Z increases as increases. For very compliant substrates (e.g., low-k dielectrics, polymers, etc.), Z increases rapidly, with Z 30 for 0.99 [11,12]. The effect of is secondary and often ignored. hf In general, the steady-state energy release rate of channel dd cracking can be calculated from a two-dimensional (2D) Gd model [10-12] as follows:

1 h f Substrate G O (z)dz , (4) (b) ss 0 f 2hf

where (z) is the opening displacement of the crack surfaces far behind the channel front. Due to the constraint by the substrate, the crack opening does not change as the channel front advances and the energy release rate attains a steady state, independent of the channel length. Three-dimensional (c) substrate analyses have shown that the steady state is reached when the length of a channel crack exceeds two to three times the film thickness for a relatively stiff substrate [18], but the crack length to reach the steady state can be significantly longer for more compliant substrate materials [13]. The present study 2b focuses on the steady state. (d) substrate INTERFACIAL DELAMINATION FROM CHANNEL

Fig. 1: Schematic illustration of common failure mechanisms While some experimental observations clearly showed in thin film structures: (a) channel cracking; (b) concomitant channel cracks with no delamination [16,19], others observed channel cracking and interfacial delamination; (c) wrinkling; delamination of the interface [16,20]. There are two questions (d) buckle-delamination. to be answered: First, under what condition would the growth

763 of a channel crack be accompanied by interfacial where depends on the elastic mismatch and can be delamination? Second, how would the interfacial delamination determined by solving the equation [23] (if occurring) affect the fracture condition or reliability in 2 2 integrated thin film structures? To answer these questions, we cos 1 2 . (8) consider in this section interfacial delamination emanating 1 1 2 from the root of a channel crack (Fig. 1b). For a long, straight Here we discuss three scenarios at the short crack limit, channel crack, we assume a steady state far behind the channel which would eventually determine the condition for channel front, where the interfacial crack has a finite width, d. The cracking with or without interfacial delamination. First, when energy release rate for the interfacial crack can be written in a similar form as Eq. (1): 0 (no elastic mismatch), we have 0.5 . In this case, approaches a constant as . An analytical Zd d / h f 0 C d S 2 h G Z D ,, T f f , (5) solution [21] predicts that Z (0,0,0) 0.9878 , which d d D h T E d E f U f compares closely with the numerical results (Fig. 2). When 0 , we have 0.5 . Consequently, Z as d / h 0 . where Zd is a dimensionless function that can be determined d f from a two-dimensional plane strain problem [21]. In the As shown in Figure 2, for both 0 and 0 , the present study, a finite element model is constructed to interfacial energy release rate monotonically decreases as the calculate the interfacial energy release rate. The finite element delamination width increases. On the other hand, when 0 , package ABAQUS is employed. The method of J-integral is we have 0 0.5 , and thus, Zd 0 as d / h 0 . adopted for the calculation of the interfacial energy release f Interestingly, the numerical results in Figure 2 show that, rate. In all calculations, we set 1/3 such that f s instead of a monotonic variation with respect to the crack 4 , while the mismatch parameter is varied. length, the interfacial energy release rate oscillates between 1.2 the short and long crack limits for the cases with 0 . Such β α α=−0.99 an oscillation leads to local maxima of the interfacial energy 1.1 = /4 α=−0.6 release rate, which in some cases (e.g., 0.6 ) can be greater than the steady state value at the long crack limit. 1 α=0 α 0.9 =0.2 65 α=0.6

d β α 0.8 = /4 Z 60 0.7

0.6 55 0.5 (deg)

0.4 ψ 50 α=−0.99 0 1 2 3 α=−0.6 d/h f 45 α=0 Fig. 2: Normalized energy release rate of interfacial α=0.6 delamination from the root of a channel crack as a function of 40 the normalized delamination width for different elastic 0 1 2 3 4 5 mismatch parameters. d/h f As shown in Figure 2, the Z function has two limits. Fig. 3: Phase angle of the mode mix for interfacial d delamination as a function of the normalized delamination First, when (long crack limit), the interfacial crack d / hf width for different elastic mismatch parameters. The dashed reaches a steady state with the energy release rate line indicates of the steady-state phase angle (52°) for the case of zero elastic mismatch ( 0). 2 h d f f , (6) Gss 2E f A necessary condition for steady-state channel cracking with concomitant interfacial delamination is that the interfacial and thus Z 0.5 . The steady-state energy release rate for d crack arrests at a finite width. The delamination width can thus the interfacial crack is independent of the elastic mismatch. be determined by comparing the interfacial energy release rate On the other hand, when (short crack limit), the d / h f 0 in Eq. (5) to the interface toughness. In general, the interface interfacial energy release rate follows a power law [22]: toughness depends on the phase angel of mode mix [10], which in turn depends on the delamination width, as shown in C S1 2 Figure 3. Due to the oscillatory nature of the stress singularity D d T , (7) Zd ~ D T at the interfacial crack tip [24], a length scale has to be used to E hf U

764 define the phase angle. Here we take the film thickness as interface toughness, and film stress would be needed for hf the length scale, and define the mode angle as further validation of the predicted delamination map.

C Im Khi S Γ / Σ tan 1D f T , (9) i D i T E Re Khf U where K K iK is the complex stress intensity factor, and I II−A 1 2 0.99 1 C1 S lnD T . The real and imaginary parts of the complex 2 E1 U stress intensity factor are calculated by the interaction integral II−B 0.5 method in ABAQUS. Figure 3 shows that the phase angle quickly approaches a steady state, , . When the ss III−B III−A film and the substrate have identical elastic moduli ( 0 ), we have ss 0,0 52 . Considering the fact −1 0 1 that the variation of the phase angle with respect to the α = −0.89 α delamination width is relatively small and confined within a small range of short cracks ( ), we take the constant d h f Fig. 4. A map for interfacial delamination from the root of a steady-state phase angle in the subsequent discussions and channel crack: (I) no delamination, (II) stable delamination, assume that the interface toughness is independent of the and (III) unstable delamination, where A and B denote delamination width, i.e., . Then, the width of the delamination without and with an initiation barrier, i i ss respectively. interfacial delamination can be determined by requiring that

C d S E Z D s ,, T f i ss . (10) d D T i 2 CONCOMITANT CRACKING AND DELAMINATION E hf U f hf Depending on the elastic mismatch parameters and the With a stable delamination along each side of the channel normalized interface toughness ( ), the solution to Eq. (10) crack (Fig. 1b), the substrate constraint on the opening of the i channel crack is relaxed. Consequently, the steady-state predicts no delamination (ds 0 ), stable finite delamination energy release rate calculated from Eq. (4) becomes greater than Eq. (1). A dimensional consideration leads to (0 ds ), or unstable delamination (ds ). The result is summarized in Figure 4 as an interfacial delamination map. In 2 C d S h particular, for a stiff film on a relatively compliant substrate G* Z *D s ,, T f f , (11) ss D h T E ( 0 ), a stable delamination along the channel crack is E f U f predicted for 0.5 , whereas unstable delamination is i where Z * is a new dimensionless coefficient that depends on predicted for i 0.5. On the other hand, on a relatively stiff the width of interfacial delamination ( ) in addition to the d s h f substrate ( 0 ), the film cracks with no delamination in elastic mismatch parameters. Region I. The boundary between Region I and Region II-B is The same finite element model is employed to calculate determined from the finite element calculations, corresponding G* , by integrating the opening displacement along the surface the maximum interfacial energy release rate in the range ss of the channel crack as in Eq. (3). The stable delamination 0 0.89 (see Fig. 2 for 0.6 ). width, , is obtained as a function of the normalized In an experimental study by Tsui et al. [16], no interfacial ds / hf delamination was observed for channel cracking of a low k * interface toughness, i , by Eq. (10). Thus, the coefficient Z film directly deposited on a Si substrate, while a finite is plotted as a function of , as shown in Figure 5. When delamination was observed when a polymer buffer layer was i * * sandwiched between the film and the substrate. These 0 ,Z Z as i , and Z as i 0.5 ; in observations are consistent with the delamination map. In the between, Z * increases as decreases, because the interfacial former case, the elastic mismatch between the film and the i * substrate, 0.91 , thus no delamination when the delamination width increases. When 0 , Z Z for 0.9878 (i.e., no delamination). When 0 0.89 , Z * normalized interface toughness i 0.5 (i.e., Region I in Fig. i 4). With a polymer buffer layer, however, the elastic mismatch increases from Z to infinity within a narrow window of i , between the low k material and the polymer is, 0.4 . where stable delamination is predicted (Region II-B in Fig. 4). Although the polymer layer is relatively thin, it qualitatively For 0.89 , either Z * Z for no delamination or Z * changes the interfacial behavior from that for 0 (Region for unstable delamination. Apparently, with interfacial I) to that for 0 (Region II-A). More experimental delamination, the driving force for channel cracking can be evidences with different combinations of elastic mismatch, significantly higher than that assuming no delamination.

765 10 1.5 α=−0.6 α=−0.6 1.4 8 α=0 α=0 α=0.6 α=0.6 1.3 6 * /Z

Z 1.2 eff 4 Z 1.1

2 1

0 0.9 0.5 1 1.5 0.5 1 1.5 Interface toughness, Γ /Σ Interface toughness, Γ /Σ i i Fig. 5. Influence of the normalized interface toughness Fig. 6. Effective driving force for steady-state channel 2 cracking as a function of the normalized interface toughness ( f hf E f ) on the steady-state driving force for channel ( 2 h E ). cracking. f f f

While the interfacial delamination, if occurring, relaxes Using the effective energy release rate, the condition for the constraint on crack opening thus enhances the fracture the steady-state channel cracking is simply a comparison between eff and , the latter being a constant independent driving force, it also requires additional energy to fracture the Gss f interface as the channel crack advances. An energetic of the interface. Figure 6 plots the ratio, Z Z , , as a condition can thus be stated: if the increase in the energy eff function of for different elastic mismatch parameters. At release exceeds the fracture energy needed for delamination, i growth of the channel crack with interfacial delamination is the limit of high interface toughness ( i ),ds 0 and energetically favored; otherwise, the channel crack grows with Z Z , which recovers the case of channel cracking with no no delamination. Considering the interfacial fracture energy, a eff modified fracture condition for steady-state growth of a delamination. The effective driving force increases as the channel crack can be written as normalized interface toughness deceases. The present study predicts that channel cracking in an * , (12) Gss f Wd elastic thin film on a relatively compliant substrate is always accompanied by interfacial delamination, either stable or where is the cohesive fracture toughness of the film, and f unstable, depending on the interface toughness. This differs from the case for an elastic film on a relatively stiff substrate, Wd is the energy required to delaminate the interface accompanying per unit area growth of the channel crack. For in which channel cracks may grow without interfacial delamination (Region I in Fig. 4). This difference may have stable delamination of width d d at both sides of a channel s important implications for reliability of integrated structures. crack, the delamination energy is As an example, for interconnect structures in microelectronics,

d the low-k dielectrics is usually more compliant compared to 2 s ds . (13) Wd O i (a) da 2 i ss the surrounding materials [6]. Therefore, fracture of the low-k h 0 h f f dielectrics by channel cracking is typically not accompanied Equation (12) may not be convenient to apply directly, by interfacial delamination. However, when a more complaint since both sides of the equation (driving force and resistance, buffer layer is added adjacent to the low-k film, interfacial respectively) increase with the interfacial delamination. By delamination can occur concomitantly with channel cracking moving W to the left hand side and noting that the stable of the low-k film [16]. Moreover, a relatively stiff cap layer d (e.g., SiN) is often deposited on top of the low-k film [7]. delamination width is a function of the interface toughness, we Channel cracking of the cap layer on low-k could be define an effective driving force for the steady-state channel significantly enhanced by interfacial delamination. Flexible cracking electronics is another area of applications where compliant 2 h eff * f f , (14) substrates have to be used extensively along with Gss Gss Wd Zeff i , , E f mechanically stiffer films for the functional devices and with interconnects [3,4]. Here, interfacial delamination could play a critical role in the reliability assessment. As shown in a C d S d *D s T s . (15) previous study [25], the stretchability of metal thin-film Zeff Z D , , T 2 i E hf U hf interconnects on a compliant substrate can be dramatically reduced by interfacial delamination. For brittle thin films on compliant substrates, as considered in the present study,

766 interfacial delamination has a similar effect on the fracture and where b is the half width of the delamination (Fig. 1d). The thus deformability of the devices. buckling stress B0 is independent of the substrate properties but depends on the relative size of the interfacial delamination. Recent studies [35-37] have shown that the buckling WRINKLING AND BUCKLE-DELAMINATION stress can be significantly lower than that predicted by Eq. (17) when the elastic deformation of the substrate is Upon compression, buckling of thin films may lead to failure considered, especially for compliant substrates. Yu and of integrated structures interfacial delamination and fracture Hutchinson [37] derived an implicit expression for the critical [8,9]. Recently, understanding of buckle patterns has also led stress: to applications in metrology [26,27], stretchable interconnects C S h C 2 S B0 D B T f D a12 T , (18) [28,29], and optical gratings [30]. Previous studies on thin tanD T a22 12b D b / h a T film buckling have focused on one of the two buckling modes, B E B0 U E f 11 U buckle-delamination or wrinkling, as illustrated in Fig. 1(c) where a11, a22, and a12 are determined numerically, either by and (d). The former considers buckling of the film when it is direct finite element calculations or by solving an integral partly delaminated from the substrate [10], while the latter equation, as dimensionless spring constants at the edge of the assumes no delamination as the substrate deforms coherently buckled film, which depend on the ratio b/hf and the Dundurs’ with the film [31]. The characteristics of both buckling modes parameters. are often observable, with localized patterns for buckle- delamination (e.g., telephone cord blisters) and homogeneous patterns for wrinkling. One of the unanswered questions is: what determines the selection of either buckling mode for a given material system? For a thin elastic film bonded to a thick elastic substrate, the buckling instability is constrained by the substrate. Without delamination, buckling of the film (i.e., wrinkling) requires coherent deformation of the substrate, which is possible only when the substrate is relatively compliant. Based on an energetic analysis [31], the critical stress for wrinkling is

2 / 3 E C S f D 3Es T . (16) w D T 4 E E f U

When the compressive stress in the film, , the film f w buckles spontaneously, forming wrinkles throughout the film Fig. 7: Comparison between the critical stresses for wrinkling surface. A particular wrinkle wavelength is established to and buckle-delamination. The open symbols are numerical minimize the total elastic energy in the film and the substrate results from Eq. (18). The dashed lines indicate the limiting [31]. The effects of substrate and film thicknesses on the stresses for buckle-delamination. The vertical line represents the PS/PDMS system with . wrinkling stress have also been studied previously [31,32]. Es / E f 0.0005 For stiff substrates, buckling deformation of the film is highly constrained, leading to high critical stresses for A comparison between the critical stresses for wrinkling wrinkling. However, the substrate constraint may be locally and buckle-delamination is presented in Fig. 7. The stresses mitigated by interfacial defects that lead to partial are normalized by the plane-strain modulus of the film and delamination of the film. In this case, the delaminated portion plotted versus the substrate-film stiffness ratio. Using the log- of the film buckles, which in turn drives growth of log scales, the critical wrinkling stress in Eq. (16) is a straight delamination through interfacial fracture [10]. The co- line with a slope 2/3. The critical stress for buckling, ,B development of buckling and delamination leads to abundant B obtained from Eq. (3), decreases as the relative delamination blister patterns such as telephone-cord blisters. Compared to size b/h increases. For a constant delamination size, the wrinkling, the buckle-delamination patterns are typically buckling stress increases as the stiffness ratio increases, but at localized and sensitive to interfacial defects [33,34]. Early a slower rate compared to the increase of the wrinkling stress. studies of buckle-delamination often assumed a fixed-end The intersection of the two critical stresses defines a critical condition at the edge of delamination, which essentially stiffness ratio, R . When the relative substrate stiffness is neglected the effect of elastic deformation in the substrate. c Under such a condition, the critical stress for the onset of greater (i.e., Es / E f Rc ), the buckling stress is lower than the buckling is wrinkling stress, dictating that buckle-delamination occurs 2 C h S2 first as the compressive stress develops in the film. On the D f T , (17) B0 D T E f other hand, for more compliant substrates ( ), the 12 E b U Es / E f Rc wrinkling stress is lower and the film wrinkles. Therefore, a transition in the buckling mode is predicted quantitatively as

767 the stiffness ratio between the substrate and the film varies. SUMMARY

The critical value Rc as a function of the delamination size is plotted in Fig. 8. This plot represents a buckling mode This paper considers fracture and buckling of elastic thin selection map: if the stiffness ratio and the interfacial defect films under the influence of interfacial delamination. Three size render a point below the R curve, wrinkling is main conclusions are summarized as follows. c energetically favored; otherwise, buckle-delamination is Stable interfacial delamination along a channel crack favored. is predicted for certain combinations of film/substrate elastic mismatch, interface toughness, and film stress, as summarized in a delamination map (Fig. 4), together with conditions for no delamination and unstable delamination. buckle-delamination Interfacial delamination not only increases the fracture driving force for steady-state growth of the channel crack, but also adds to the fracture resistance by requiring additional energy for the interfacial fracture. An effective energy release rate for channel cracking is defined, which depends on wrinkling the interface toughness (Fig. 6) in addition to the elastic mismatch and can be considerably higher than the energy release rate assuming no delamination. Under compression, the film may buckle with or without delamination. The onset of the buckling and Fig. 8: The critical stiffness ratio as a function of the relative the selection of the buckling mode depend on the delamination size. elastic mismatch and the pre-existing interfacial defects, as summarized in a buckling map (Fig. 8). To validate the buckling mode selection criterion described above, an experiment was conducted, which observed both buckling modes as well as a mode transition Acknowledgments [38]. A polystyrene (PS) film (hf = 120 nm) was bonded to a 1 The authors are grateful for the financial support by National mm thick polydimethylsiloxane (PDMS) substrate. The Science Foundation under Grant No. 0547409. materials’ elastic moduli are estimated as E ~ 4 GPa and E f s References ~ 2 MPa, with a ratio . By annealing the Es / E f 0.0005 specimen at 120 °C, which is above the glass transition [1] P. S. Ho, G. Wang, M. Ding, J. H. Zhao, X. Dai, temperature of PS (Tg ~ 105 ºC), and then slowly cooling, a “Reliability issues for flip-chip packages,” Microelectronics compressive stress develops in the PS film due to differential Reliability, vol. 44, pp. 719-737, 2004. thermal expansion. Assuming an initially small defect at the [2] A. Dodabalapur, “Organic and polymer transistors for interface (say, b/hf = 5), as illustrated by the vertical line in electronics,” Materials Today, vol. 9, pp. 24-30, 2006. Fig. 7, the compressive stress in the PS film first reaches the [3] S. Wagner, S. P. Lacour, J. Jones, P. I. Hsu, J. C. Sturm, T. wrinkling stress, at which point wrinkling occurs Li, Z. Suo, “Electronic skin: architecture and components,” spontaneously over the film surface. As the temperature is Physica E, vol. 25, pp. 326-334, 2004. further decreased, the stress eventually reaches the critical [4] D. Y. Khang, H. Q. Jiang, Y. Huang, J. A. Rogers, “A buckling stress for the initial defect size, and buckle- stretchable form of single-crystal silicon for high-performance delamination occurs alongside the existing wrinkles. When the electronics on rubber substrate,” Science, vol. 311, pp. 208- material system was heated back to 120 °C from the room 212, 2006. temperature, the compressive stress in the film vanished and [5] H. Gao, “Application of fracture mechanics concepts to the elastic film recovered with no observable buckling or hierarchical biomechanics of bone and bone-like materials,” wrinkling. Next, the system was subjected to a second cooling. Int. J. Fracture 138: 101–137, 2006. Compared to the first cooling, the interfacial defect size has [6] X. H. Liu, M. W. Lane, T. M. Shaw, E. G. Liniger, R. R. increased due to the growth of buckle-delamination in the first Rosenberg, D. C. Edelstein, “Low-k BEOL mechanical cycle. An estimate of the delamination width gives b/hf ~ 20 modeling,” Proc. Advanced Metallization Conference, pp. for the second cycle, for which the buckling stress is lower 361-367, 2004. than the wrinkling stress as shown in Fig. 7. Consequently, [7] X. H. Liu, M. W. Lane, T. M. Shaw, E. Simonyi, buckle-delamination occurred first. Further cooling led to “Delamination in patterned films,” Int. J. Solids Struct., vol 44, growth of buckle-delamination and eventually wrinkling of the pp. 1706-1718, 2007. bonded region as well. [8] J. W. Hutchinson, M. D. Thouless, E. G. Liniger, “Growth and configurational stability of circular, buckling-driven film

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