H c of the corner stress intensity. The estimated pulloff

Transcription

1 Cohesive detahment of an elasti pillar from a dissimilar substrate N. A. Flek 1,, S. N. Khaderi 1, R. M. MMeeking,3,4, and E. Arzt,5 1 Cambridge University Engineering Dept., Trumpington St., Cambridge, CB 1PZ, UK INM-Leibniz Institute for New Materials, Campus D, 6613 Saarbrueken, Germany 3 Departments of Materials and Mehanial Engineering, University of California, Santa Barbara, CA 93106, USA 4 Shool of Engineering, University of Aberdeen, King s College, Aberdeen, AB4 UE, UK 5 Department of Materials Siene and Engineering, Saarland University, Campus D, Saarbrueken, Germany Summary The adhesion of miron-sale surfaes due to intermoleular interations is a subjet of intense interest spanning eletronis, biomehanis and the appliation of soft materials to engineering devies. The degree of adhesion is sensitive to the diameter of miro-pillars in addition to the degree of elasti mismath between pillar and substrate. Adhesion-strengthontrolled detahment of an elasti irular ylinder from a dissimilar substrate is predited using a Dugdale-type of analysis, with a ohesive zone of uniform tensile strength emanating from the interfae orner. Detahment initiates when the opening of the ohesive zone attains a ritial value, giving way to rak formation. When the ohesive zone size at rak initiation is small ompared to the pillar diameter, the initiation of detahment an be expressed in terms of a ritial value H of the orner stress intensity. The estimated pulloff fore is somewhat sensitive to the hoie of stik/slip boundary ondition used on the ohesive zone, espeially when the substrate material is muh stiffer than the pillar material. The analysis an be used to predit the sensitivity of detahment fore to the size of pillar and to the degree of elasti mismath between pillar and substrate. 1. Introdution Adhesion plays an important role in ontat problems at small sale, suh as (i) stition of miro-eletromehanial-systems (van Spengen et al. (00)), (ii) wafer bonding of silion layers in eletronis (Plössl and Kräuter (1999)) and (iii) the adhesion of insets 1

2 and animals (suh as the geko) to smooth walls (Arzt et al. (003)). The mehanis of adhesion falls into two ategories: onforming ontats suh as a sphere on half-spae, and non-onforming ontats suh as a flat-bottomed punh on half-spae. We onsider eah in turn. Johnson et al. (1971) developed the so-alled JKR theory to predit the effet of adhesion upon the Hertzian ontat between onforming elasti spheres, with adhesion haraterized by a surfae energy, whih is equivalent to a toughness measure G in frature mehanis. This approah assumes that the proess zone size, over whih adhesive trations exist, is muh less than the ontat size. To assess this, Maugis (199) developed a ohesive zone model for adhesion and idealised the adhesive tration versus separation law by a onstant normal tration for any separation less than a ritial value. The work of adhesion assoiated with this ohesive zone law is G =. The length of ohesive zone is of order E * / where * E is a ombined measure of Young s modulus (as defined in (1.) below). Maugis thereby demonstrated that JKR theory suffies when fration of the ontat width. E * / is a small The mehanis of non-onforming ontats in the presene of adhesion has reeived muh less attention. For example, the adhesion of a flat-ended, fritionless rigid pillar on a half-spae has been explored for both plane and axisymmetri geometries (Kendall (1971) and Maugis (000)). These geometries give rise to an inverse square-root singularity in stress at the interfae-orner. Consequently, the pull-off fore an be obtained by equating the elasti energy release rate of this rak-like singularity to the interfaial work of adhesion, or toughness, G. This inverse square-root singularity is relaxed somewhat upon replaing the rigid pillar by a ompliant pillar. Adams (014) has reently explored the problem of a flatended, fritionless ompliant pillar on a half-spae made from a dissimilar elasti solid. He assumed that the adhesive tration versus separation law omprises a onstant normal tration for any separation less than a ritial value this ohesive zone law is again. The work of adhesion assoiated with G =. Adams (014) assumed that the ohesive zone is suffiiently small that it is embedded within the zone of dominane of the orner singularity. The present study omplements this work by onsidering the ase of a ompliant pillar

3 bonded to a dissimilar half-spae, and by onsidering the ase where the ohesive zone may oupy a signifiant fration of the interfae between pillar and substrate. Our study builds upon the analysis of Khaderi et al. (015) for the adhesion-energyontrolled detahment of an adhered miropillar from a dissimilar elasti substrate. A flatbottomed planar pillar of width D, or a flat-bottomed irular pillar of diameter D, is bonded to a dissimilar half-spae, see Fig. 1. The pillar is made from material 1 and the half-spae is made from material. Both materials are elasti and isotropi, with shear moduli ( 1, ) and Poisson ratios ( 1, ). For later use, the elasti mismath between these two materials is haraterized by the two Dundurs parameters 1( 1) ( 1 1) ( 1) ( 1), and 1( 1) ( 1 1), (1.1) ( 1) ( 1) where 3 4 for materials m =1,. We shall also make use of the ombined modulus m * E as defined by m 16 E * 11 1 (1.) Corner singularity between a stiking pillar and substrate Appliation of an axial tensile stress to the free end of the bonded pillar results in a singularity in stress at the interfae-orner between pillar and substrate, as desribed by Khaderi et al. (015). An eigenvalue analysis reveals that the stress field near the orner is dominated by two singular eigenfields having eigenvalues 1, with orresponding intensities H1, H, as follows. Introdue the polar o-ordinates of radius r from the orner and angle from the interfae. Then, the asymptoti stress ij and displaement u j fields in the viinity of the orner an be written as 11 1 ij H1r fij 1 Hr fij (, ) (, ) (1.3) and 3

4 1 j 1 j 1 j u H r g (, ) H r g (, ) (1.4) in terms of the eigenfuntions f ij and g j, with higher order terms negleted. The first two eigenvalues 1, assoiated with the leading terms in the infinite series of eigenfuntions both lie within the interval [0.5, 1], and imply unbounded stress as r 0 whereas the higher terms in the asymptoti series give ontributions to σ ij that tend to zero as r 0. The values of 1, depend upon the material mismath parameters, as plotted in Fig. 3 of Khaderi et al. (015). These two eigenvalues are suffiiently lose in magnitude that both terms in (1.3) and (1.4) need to be inluded in the present study. Both eigenvalues are real for the full range of when 0. When /4, the eigenvalues 1, are real for 0.86 and are omplex onjugates of eah other for 0.86, see Fig. 3(b) of Khaderi et al. (015). For benhmarking purposes, when 1 equals 0.5, the level of singularity is idential to that of a rak in a homogeneous solid. Analytial expressions exist for the funtions f ij and g j by asymptoti analysis, see for example Knésl and Náhlík (007), Klusák and Náhlík (007), Khaderi et al. (015) and Akisanya and Flek (1997). The singular zone extends from the orner by approximately 10% of the pillar diameter. Dimensional arguments ditate that the orner stress intensities H1, H are related to the remote stress and geometry aording to 1n H D a,, n = 1, (1.5) n n where the alibration fators a n have been reported already by Khaderi et al. (015) using finite element analysis and a domain integral approah. Note that the values of a n differ for the plane strain (D) and axisymmetri (3D) pillars. Assume that the pillar detahes from the substrate by the nuleation of a rak at the pillar-substrate interfae. Then, following the argument of Akisanya and Flek (1997), detahment ours when the value of the orner stress intensity H 1 attains the ritial value material property H, upon negleting the role of H. The H an be measured by performing experiments for any ombination of elasti mismath. Assume that failure initiates at H1 H. Then, (1.5) gives the sensitivity of pull-off stress to pillar dimension D. 4

5 1. Miromehanial origins of ritial stress intensity H Consider the general problem of a pre-existing orner rak of length with a ohesive zone of length at its tip, embedded within an outer singularity in the form of the H 1 field, suh that D, as skethed in Fig.. Detahment of the pillar by rak advane is deemed to be either energy-ontrolled or strength-ontrolled depending upon the rak length in relation to the proess zone size at failure, as follows. Idealise the proess zone by a ohesive zone with normal tration of onstant strength up to a ritial opening, and zero strength at greater openings than. The interfaial toughness, as defined by the area under the tration versus separation urve, follows immediately as G. Reall that the length of the proess zone is only mildly dependent upon the shape of the normal tration versus separation urve, and is given by EG * / Wang and Suo (1990). Here, Λ is a dimensionless parameter that depends on mode mix, elasti mismath and shape of tration versus separation urve; it is typially in the range 0.1, see to 0.6. Now assume typial values of adhesion energy, G =50 mj/m, and of ohesive strength, = 0.1 MPa, for a PDMS pillar and substrate (Tang et al., 005). Then, the range 1 μm to 5 μm. Two detahment mehanisms an now be envisaged, depending upon the ratio /, as follows. (i) Adhesion-energy-ontrolled detahment for their attention to this limit, and obtained an expression for toughness G and flaw length, suh that is in. Khaderi et al. (015) onfined H in terms of the interfaial H 1 d 1 * EG / (1.6) by making use of (10)-(13) in Khaderi et al. (015). The dependene of 1 and of the omplex oeffiient d 1 upon, is plotted in Figs. 3 and 9, respetively, of Khaderi et al. (015). Note that 1 is in the range 0.5 to 0.7, and so there is only a mild dependene of upon defet length. 5 H

6 (ii) Adhesion-strength-ontrolled detahment for. Detahment ours from the interfae of strength and toughness G. We shall explore strength-ontrolled detahment in the present study for an axisymmetri ylindrial pillar and limit our attention to the ase of a vanishing initial defet, / 0. We shall show below that H is given by H * EG k 11 (1.7) where the parameter k depends upon, and is expressed in terms of various oeffiients to be introdued throughout our study, with the form d k N1 N f 1 1 R 1 R d 1 1 f R fr 1 1 (1.8) The derivations of (1.7) and (1.8) are detailed later in the paper, and all parameters inluding k, as given in (1.8), are listed in Table 1 for seleted values of,. The dependene of k upon, is also plotted in Fig. 3, and we note that it takes values in the range 0.4 to 0.8, depending upon,. We emphasise that the present study onsiders both the limits of a small ohesive zone relative to the pillar diameter (suh that the ohesive zone is embedded within the orner singularity), and the more general ase of a large ohesive zone that extends over a large fration of the pillar diameter (beyond the orner singularity).. Problem statement Within the assumptions of linear elastiity theory, and in the absene of a ohesive zone between pillar and half-spae, the appliation of an axial stress to the remote end of the pillar leads to a singularity in stress at the interfae orner. Consequently, detahment of the pillar begins at the interfae orner; this is ommonly observed, see for example Greiner et al. (007) and Del Campo et al. (007). In the present treatment we shall model adhesion-strength-ontrolled detahment by assuming that a starter defet is absent (=0 in Fig. ) but the normal tration on the interfae 6

7 between a pillar and half-spae is moderated by the presene of a ohesive zone of onstant tensile strength on the interfae. We endow the ohesive zone with zero shear tration, suh that slip an freely our between the faes of the ohesive zone, and a normal tensile tration T versus opening response, suh that T for 0, and T=0 for. The toughness of the interfae is G. When a remote tensile stress is applied to the pillar, the interfaial tensile stress at the interfae orner is limited to, see Fig. 1b. The length of the ohesive zone depends upon the magnitude of the remotely applied stress, and is influened by the degree of elasti mismath between pillar and half-spae. The maximum normal and tangential separations of the ohesive zone exist at the interfae orner, and are denoted by N and S, respetively. We shall assume that detahment of the pillar from the half-spae initiates when the maximum normal displaement N attains the ritial value. The pull-off stress required for detahment is alulated by superposition of two problems A and B, following the approah of Dugdale (1960). The ohesive zone is treated as an interfaial rak of length, with rak fae loading and remote loading as follows: Problem A: a remote tensile stress is applied to the top of the pillar, as shown in Fig. 1; and Problem B: A normal tration of magnitude T is applied to the faes of the interfaial rak, see Fig. 1d, suh that T= for 0, and T=0 for. Consider the ase where the applied remote tensile stress is muh less than the ohesive strength. Then, the length of the ohesive zone is muh less than the pillar diameter D, and the zone is fully embedded within the H-field orner singularity for a fully bonded punh on a half-spae (with the ohesive zone absent); the orresponding analysis is referred to as the short rak solution. Alternatively, when to is omparable in magnitude, the ohesive zone extends beyond the domain of the orner singularity, and the orresponding analysis is referred to as the long rak solution. The pull-off stress for the initiation of pillar detahment (at N ) is alulated in eah regime as a funtion of the Dundurs parameters,. We proeed by onsidering problems A and B in turn. F 7

8 .1 Problem A, orner rak under remote tension An interfaial rak of length and a remote tensile stress of magnitude is present at the orner between pillar and half-spae, is applied to the pillar. In the following we summarize the results relevant to short and long rak solutions, as taken from Khaderi et al. (015). Short rak solution: The asymptoti stress field (1.3) and displaement field (1.4) exist at the interfae orner in the absene of a rak. The values of the alibration fator a n in (1.5), as a funtion of the (, ), have been given by Khaderi et al. (015), and values for a 1 are repeated in Table 1 of the present study. Now embed an interfaial rak of length within the H-dominated zone (see Fig. 1(e)). The interfaial stress intensity fator is represented by the omplex quantity K, (see for example Huthinson and Suo (1991)), and the value of K is ditated by the magnitude of the H-field aording to 1 n i R I n n n n1, K H ( d id ), (.1) where is the usual osillatory index that depends on aording to 1 1 log 1 (.) and the non-dimensional alibration fators ( d, d ) depend on (, ). These fators have been tabulated by Khaderi et al. (015), and seleted values for ( d1, d 1 ) are repeated in R n I n R I Table 1 for subsequent use. The normal rak mouth displaement displaement S an be written in the form N and tangential where H H N * n S * n n1, E n1, E n n,, n n N S,, (.3) * E has already been defined in Eq. (1.). 8

9 The alibration fators ( N, S ) are alulated in the present study by finite element n n simulations using ABAQUS ommerial software 1, by following the method of Khaderi et al. R I (015), and we list seleted values of ( d1, d 1 ) in Table 1. The substrate is represented by a irular ylinder of radius and thikness both equal to 40D. Numerial experimentation onfirmed that these substrate dimensions are suffiiently large to mimi a half-spae. The displaement vanishes at the bottom of the substrate and a normal surfae tration, of magnitude, is applied to the top of the pillar. The pillar and substrate are disretised using elements of type CAX8. Long rak solution: Now onsider an interfaial rak of length that extends beyond the H-dominated region. The stress intensity fator and rak opening are alulated by performing finite element simulations of the entire geometry, see Fig. 1(). The omplex stress intensity fator is represented by K and is related to the remote stress aording to i 1/ R I K b,, ib,,, D D (.4) where b, b are alibration fators. The rak mouth displaement an be written as R I N where the alibration fators N, S N,, * E D, S S,, * E D are also funtions of (,, / D ). The alibration fators are alulated by following the omputational proedure of Khaderi et al. (015), with the same finite element details as those desribed above for the short rak ase. (.5). Problem B, orner rak under rak fae loading Now onsider the seond problem of rak-fae loading. 9 A rak of length emanates from the interfae orner and a normal tration of magnitude ats on the rak faes as shown in Fig. 1(d). Again, a short rak regime an be identified, suh that 1 Dassault Systems, Simulia Corporation, Providene, Rhode Island, USA. Version is used to perform the simulations.

10 / D 1, with geometry as speified in Fig. 1(f). The full geometry, as shown in Fig. 1(d), is needed to analyse the long rak ase, for whih the rak length pillar diameter D. The omplex stress intensity fator is written as is omparable to the K f if i 1/ f R,, I,, D D, (.6) in terms of the alibration fators f, f, and the rak mouth displaement reads R I,, E D, f S S,, * f E D f N N * f in terms of the alibration fators Nf, S f. The alibration fators for long raks are also evaluated by the proedure of Khaderi et al. (015). We list the short-rak limits (.7) f R 0 D, f I D 0 and N f D 0 in Table 1 for seleted values of,. 3. Results We return to the problem of ohesive detahment of the ylindrial pillar. We first obtain the relation between the ohesive zone length the pull-off stress as a funtion of the ohesive strength displaement. and remote stress. We then alulate and ritial opening 3.1 Cohesive zone length The stress intensity fator for the orner rak, due to a remote stress and to rak fae loading of magnitude, is given by the net value NET i i i K K K f. Following the usual Dugdale (1960) argument, the length of the ohesive zone is suh that the rak tip tensile stress is bounded and Re NET i K 0. 10

11 First, fous on the short rak limit. Upon equating the real part of K i to the real part of f i K (as expressed by (.1) and (.6), respetively) and by using the relation (1.5) for H n we obtain 1 n 1 R andn fr n1, D (3.1) thereby providing the relation between ohesive zone length and the remote stress. Seond, onsider the ase of a long rak. Upon equating the real part of K i to f i K (from (.4) and (.6), respetively) the relation between applied stress and ohesive length reads fr ( / D). (3.) b ( / D) R The dependene of ohesive zone length upon remote stress is plotted in Fig. 4 for seleted values of, with 0 and /4. The short rak solution is in agreement with the long rak solution for small values of / D. For a given remote stress, the ohesive zone size inreases with inreasing and is relatively insensitive to the magnitude of. In the short rak limit the ohesive zone size when 0. is almost independent of for 3. Critial stress intensity H for a short ohesive zone The ritial value (1.7) for the stress intensity H for a short ohesive zone an now be established. We onsider the ase where the ohesive zone is fully embedded within the orner singularity as defined by the H1 -field, and neglet the ontribution from the less singular H -field. Then, (3.1) simplifies to d f R 1 H1 R 1 11 (3.3) 11

12 and the net rak mouth opening displaement as f N N follows from (.3a) and (.7a) R 1 R H1 d 1 d 1 1 N * 1 Nf E f R f R (3.4) Debonding initiates at H1 H suh that, and (3.4) an then be re-expressed as (1.7) where k is defined by (1.8) and we have made use of the identity G. 3.3 Pull-off stress F We proeed to alulate the remote pull-off stress F as a funtion of the pillar diameter and the Dundurs parameters (, ). The net rak mouth opening displaement is N f N. For the ase of a short ohesive zone, the expressions (.3a) and (.6a) give Hn n N. * n N * f (3.5) E E n1, and further redution via (1.5) provides D D D n annn N f, n1, (3.6) where the harateristi diameter * D is defined as D D / E. Now use the relation (3.1) between / and / D to obtain 1 n1 n R fr andn annn N f n1, n1,, D D D D (3.7) The expressions (3.1) and (3.7) provide the relation between / D and / as parameterized by / D. 1

13 f For long raks, the rak mouth opening displaement N N is fr N N f N N f, D D D br (3.8) via (.5a), (.7a) and (3.). The expressions (3.) and (3.8) provide the relation between / D and / in terms of the intrinsi variable / D. Now assume that adhesion-strength-ontrolled detahment takes plae when the rak mouth opening displaement attains a ritial value. Write the remote pull-off stress as F. We proeed to plot in Fig. 5 the remote pull-off stress F / as a funtion of D / for both short and long raks by making use of (3.7) and (3.1) for short ohesive zones, and (3.8) and (3.) for long ohesive zones. Results are shown for seleted values of, with 0 and /4. In general, the pull-off stress inreases with dereasing D /. For small values of D /, that is for D / 10, the pull-off stress attains the limiting value F. In ontrast, for larger values of D /, the pull-off stress dereases with an inrease in the normalized diameter. For a given value of D /, the pulloff stress dereases with inreasing. The ase 0.99 : Let us fous on the ase 0.99 (i.e. the substrate material is muh stiffer than the pillar material 1). Consider the regime where the remote axial failure stress attains the limiting value F and the ohesive zone length approahes D / in Fig 4. We an gain some insight into this limiting ase as follows. Note that the stress state within an elasti, fritionless pillar on a rigid substrate with D / is idential to that in a pillar under uniform uniaxial tension. In fat, this same solution exists for all values of rak mouth displaement provided the rak opening displaement is uniform over the rak fae for a rak of length D /. The detahment of an elasti pillar from a rigid substrate ( 1) has been analysed previously by Tang et al. (005) using a ohesive zone analysis, with sliding prevented within the ohesive zone. Contrary to our results of Fig. 5, they find that D / vanishes as 13

14 F. In order to onfirm that the differene in responses is a onsequene of the different boundary onditions, we have performed additional simulations for problem A and B, but now assuming no sliding of the rak faes. The preditions are inluded in Fig. 6: for 0.99, we find that D / vanishes for / 1, in agreement with the findings of Tang et al. (005). It is lear that that the pull-off stress is somewhat sensitive to the hoie of stik/slip boundary ondition at small D /. F The ase 1 and D : An analytial solution relating the interfaial adhesion energy to the pull-off stress has been given by Kendall (1971), when the pillar is rigid and fritionless, and the substrate is ompliant ( 1). He noted that the orner singularity in the substrate, adjaent to the edge of the punh, is the same as that for a mode I rak, regardless of the presene or absene of a small orner defet ( release rate reads D ), and the energy 1 G D (3.9) 16 E Upon equating the energy release rate to the interfaial toughness G G, (3.9) an be rearranged to the form * E D 8. (3.10) Our results in Fig. 5 for the short rak limit of the fritionless pillar at 0.99 are in very lose agreement with this analyti solution for 1: the urves oinide to within the thikness of the line, and onsequently the omparison is not given in Fig Analyti solution for a rigid, fritionless axisymmetri pillar adhering to an elasti half spae with Dugdale zone ( 1 and finite / D ) We proeed to extend the Kendall (1971) solution for the the detahment of a rigid, fritionless ylindrial flat-bottomed pillar of radius R=D/ adhered to an isotropi linear elasti half-spae with Young s modulus E and Poisson s ratio. And, we remove the 14

15 restrition that D. Assume that a Dugdale ohesive zone, of strength and normal separation (i.e. interation distane) exists at the orner edge of the pillar. At low values of tensile load P applied to the pillar, a Dugdale ohesive zone (of strength ) exists over the annulus a r R, all measured from the axis of the pillar, see Fig. 7a. The length of Dugdale zone is R a and the opening profile of the ohesive zone inreases from 0 R at r=a to. With inreasing load P, the Dugdale zone spreads as its inner radius a diminishes; and its separation inreases until the pull-off fore P is attained at R. We shall show below that detahment ensues under dereasing applied fore P < P, suh that the ohesive zone migrates over the annulus a r b where b R, see Fig. 7b. The annulus shrinks inwards suh that both a and b derease in value (along with P) as detahment proeeds. An analytial treatment is now developed to quantify this behaviour. Detahment ours in two phases as follows. Phase (i): initial stable detahment under inreasing load suh that the ohesive zone extends from r=a to r=b=r, with a 0 and R, followed by Phase (ii): unstable detahment under dereasing load, suh that the ohesive zone extends from r=a to r=b<r, with a 0 and b. Fundamental solution In order to analyse eah phase, we need the fundamental solution for a rigid, fritionless pillar subjeted to an axial tensile fore P, adhered to a half-spae over 0 r a, with a Dugdale ohesive zone of strength over the outer annulus a r b. This solution is now given, and then applied to eah phase of detahment. is The stress intensity fator at radius a due to an applied load P (see Tada et al. (000)) K P I P a (4.1) a while the rak opening displaement aused by it (Johnson (1985)) is 15

16 P P ae r aros a / r (4.). The stress intensity fator at radius a due to the tration over the where E E/1 ohesive zone (see Tada et al. (000)) is C K I b aros a / b a b a 3/ a (4.3) Sine the total stress intensity fator at radius a must be zero, we add Eq. (4.1) and (4.3) and set the result to zero, to give P aros a / b a / b 1 a / b b (4.4) The rak opening at r = b due to the applied load P as speified in (4.4) is E P b b b / aaros a / b aros a / b 1 a / b (4.5) It remains to determine the rak opening displaement at r = b due to the tration over the ohesive zone. We follow the Buekner/Rie weight funtion approah and first note that the potential energy for a system with two applied loads, F 1 and F, is given by where 1 C11F 1 1 C1F1 F CF (4.6) C ij is the ompliane matrix, suh that u1 C11F1 C1 F F1 (4.7) is the oaxial displaement where F 1 is applied and u C1 F1 CF F (4.8) 16

17 is the oaxial displaement where F is applied, where we bear in mind the fat that the pillar is rigid. (We shall later assume that the fores F 1 and F are point fores applied to the surfae of the half-spae at loations within the ohesive zone). For an axisymmetri ligament of radius a, the energy release rate is given by 1 1 dc11 1 dc1 1 dc G F1 F1F F a a 4 a da a da 4 a da (4.9) Due the Irwin relationship we find that 1 G k F k F E 1 1 (4.10) where the fator of in the denominator arises from the fat that the pillar is rigid, k 1 is the stress intensity fator due to unit load applied at loation 1, and k is the stress intensity fator due to unit load applied at loation. Upon mathing terms with ommon fators we find that dc1 ak1k da E (4.11) where C 1 is the displaement at loation 1 due to a unit load applied at loation and vie versa. From Tada et al. (000) we find that k 1 aros a a a r r a 1 3/ for a unit load applied at r on the rak surfae and (4.1) k 1 aros a a a r r a 3/ for a unit load applied on the rak surfae at r. Thus (4.13) dc 1 aros a a aros a a da E a r r r a r a (4.14) 17

18 where C 1 is now the rak opening at r due to a unit load on the rak surfae applied at r and vie versa. Now integrate (4.14), subjet to C 1 being zero at the smaller of a = r and a r sine the rak opening at the rak tip is zero and a unit load on the rak surfae applied at the rak tip auses zero rak opening. Thus, C a 1 a a a a aros aros da E r r min r, r a r a (4.15) r a 1 opening at r is C Consider ohesive trations applied in the range a r b. It follows that the rak b min r, r 4 1 a a a a r r aros aros dadr E a r r r a r a (4.16) a a Interhanging the order of integration, we obtain C r 4 1 a a a ra r r drda E a r r a r r a aros aros And, upon integration, this beomes C a b (4.17) aros aros a r a a a a a r a (4.18) 1 r b a b a da E r b Therefore, at r = b we have C b a a a a a b a (4.19) 1 b b a b a da E b b aros aros whih an be restated as 18

19 1 C 1 aros aros 1 b ab / x x E b x x x x x dx 1 a a a b a 1 1 aros aros b b b a b (4.0) The latter result is used by Maugis et al. (1976). We now ombine Eq. (4.5) and (4.0) to obtain the total rak opening at b as E b a a a 1 1 aros b b b b 4 (4.1) Taken together, Eq. (4.4) and (4.1) are a parameterized load versus defletion urve where the rak opening displaement at b is the defletion, and the independent parameter is a/b. Appliation of solution to phase (i) of detahment Consider a rigid pillar of radius R=D/ subjeted to a suffiiently small load P that R the ohesive zone over the annulus r=a to r=b=r satisfies displaement now read. The load and P a a a aros 1 R R R R (4.) from (4.4) and E R a a a 1 1 aros R R R R 4 (4.3) from (4.1). Subjet to R suh that a / R satisfies (4.3) with, these results are valid up to a load as given by Eq. (4.) R. Appliation of solution to phase (ii) of detahment 19

20 We proeed to obtain the solution for R the domain r=a to r=b<r, with a 0 and b b to obtain. The ohesive zone now exists over. We return to Eq. (4.0) and set E 4 a a b b a aros (4.4) b This gives a relationship between a and b, so that, impliitly, one an be eliminated in favour of the other and the result inserted into Eq. (4.4) to determine the load as a funtion of the remaining variable. We now determine whether detahment is unstable under monotonially inreasing load, P. To study this, we first observe that the left hand side of Eq. (4.4) is onstant if the fibril is being detahed and thus b is diminishing. As a onsequene, we dedue that during this proess da b 1 db a a b 1 a / b 1 a / b aros / (4.5) Now evaluate dp / db from (4.4) 1 dp a / b a / b da aros a/ b 4b db 1 / 1 / db a b a b (4.6) We substitute Eq. (4.5) into this and obtain aros a/ b dp aros a / b a / b a / b (4.7) b db 4 The right hand side of (4.7) is always positive, and the oeffiient of the derivative on the left hand side is also positive. Therefore dp / db > 0 and when b diminishes so must P. Therefore detahment ours under monotonially dereasing load with the detahment fore R given by Eq. (4.4) subjet to b=r, and to satisfation of Eq. (4.4). 0

21 We an obtain a plot of the pull-off fore as a funtion of by ross-plotting Eq. (4.4) and R Eq. (4.3), upon taking. This plot has been added as a dotted line to Fig. 5a to ompare with the finite element predition for an almost rigid pillar ( 0.99 ); exellent agreement is noted. The length of ohesive zone R a as a funtion of applied fore up to the point of detahment is likewise obtained by plotting Eq. (4.4), upon taking b Again, the agreement with numerial simulations for 0.99 is exellent, see Fig. 4a. R. Conluding Disussion The present study highlights the signifiane of the orner singularity in promoting detahment at a pillar-substrate interfae. This is onfirmed by experiments with artifiial patterned surfaes, see for example Del Campo et al. (007). Mushroom-shaped aps redue the magnitude of the orner stress intensity H 1 and thereby inhibit detahment. This has reently been analysed in some depth by Balijepalli et al. (016a). The ohesion model of the present study highlights the signifiane of both detahment strength and adhesion energy G in the proess of detahment. detahment stress is ditated by either or G as follows. Consider the two limiting ases: * Case (i): For fibrils of suffiiently small diameter, suh that D / E G 3, the ohesive zone spans the fibril, and the axial strength for fibril detahment F equals, reall the urves shown in Figs. 5 and 6. Case (ii): At F / <<1, the ohesive zone is suffiiently short that it is embedded within the so-alled H-field of the orner singularity. Assume that, at the onset of detahment, a orner flaw of length, and its ohesive zone of length are both embedded within the H- field. Reall that detahment is adhesion-energy-ontrolled when and is adhesionstrength-ontrolled when The. Also note that the level of singularity 1 lies in the range of 0.5 to 0.6, depending upon the values of,, see Fig. 3 of Khaderi et al. (015). We shall now show that the magnitude of H for energy-ontrolled detahment almost equals that for strength-ontrolled detahment when 1 1/. The riterion (1.6) for energy- 1

22 ontrolled detahment implies that H sales with 1/ * EG and is independent of flaw size and of ohesive strength. Likewise, for 1 1/, the riterion (1.7) for strengthontrolled detahment implies that H sales with 1/ * EG, where G. Thus, the differene between the riteria for energy-ontrolled detahment and for strength-ontrolled * detahment is minor, and is ditated by the magnitude of EG 1/ on the interfae, for the ase where the ohesive zone length is small ompared to the pillar diameter at the onset of detahment. The sensitivity of detahment strength to fibril diameter has been explored previously by Gao et al. (005) in the ontext of hierarhial strutures in gekos, and the influene of their sizes on adhesion strength. The exeption, in whih the trend of inreasing strength with diameter redution is not followed, is that of a ompliant fibril on a rigid surfae, i.e. when 0.99, with fritionfree ohesion, as an be seen in Fig. 5. In this ase the strength rises as the fibril diameter redues when the strength is low, but then the trend reverses and the asymptoti limit of full ohesive strength is reahed only by the diameter inreasing again. It is perhaps easier to understand this phenomenon through the fat that the trend is equally driven by the ritial interation distane inreasing. The impliation is that for * D / 10 E we did not find a solution that enables the fibril to remain attahed to the substrate until the separation at its perimeter, D /, reahes the ritial interation distane. That is, before D / inreases to the value, the whole bottom surfae of the fibril aquires a separation 0, and thus the tration applied to the fibril everywhere is equal to. We dedue from this that, in a solution we did not find numerially, detahment ours at F for the ases where * D / 10 E. Furthermore, we note that, for * D E / 10, there are solutions for the detahment strength, one of whih is F and one of whih is lower. The solution for whih F is ertainly the jump-into-ontat ondition that ours when the fibril is brought towards the rigid surfae. Our solutions indiate that when the fibril is ompliant and the substrate is rigid, the lak of frition in the ohesive zone enables the jump-on behaviour also to be a detahment ondition. As noted above, stiking onditions in the ohesive zone preludes this behaviour, as is illustrated in Fig. 6.

23 It is also lear from the present study that the detahment strength for both energyontrolled and strength-ontrolled detahment is inreased by making the pillar from a more ompliant material than that of substrate. This points to the use of a ompliant layer on the end of the pillar, but the signifiane of this modifiation to the pillar awaits a full analysis. Some work on this has been arried out reently by Balijepalli et al. (016b) and Fisher et al. (016), but there the enhanement of adhesion assoiated with a soft tip layer is attributed to the stress distribution indued by the onstraint of the stiff stalk on the ompliant material. Our results in the present paper suggest that the high ompliane of the tip material an have a benefiial effet on adhesion in addition to any stress redistribution ahieved. An interesting feature of our results is that when we onsider a rigid fibril adhered to a ompliant half-spae (i.e 0.99 ) with frition-free onditions everywhere at the tip of the fibril, the strength predited in this ase is idential to that omputed for the ase where the fully adhered region of the fibril tip is subjet to stiking frition. It is known that the strength in frition-free onditions is not always the same as that ahieved when stiking frition prevails. A ase in point is the ompliant fibril on a rigid substrate (i.e ), where stress in the fibril is uniform in frition-free onditions and thus detahment ours always at F, in ontrast to the results in Figs. 4 and 5. However, the fat that the frition-free and stiking frition ases have idential detahment strength when 0.99 suggests that there is a range of situations in whih the frition-free model an be used to gain insights into detahment strength more generally. This is a useful inferene as the fritionfree ase is often easier to analyse, and many of the standard frature-mehanis results for raks an be utilised immediately to obtain relevant results. Aknowledgements NAF is grateful for finanial support in the form of an ERC MULTILAT grant , and to the US ONR (N N3, projet manager, Dr. Dave Shifler). NAF and RMM aknowledge support from the Alexander von Humboldt Foundation in the form of their Forshungspreise, whih enabled them to undertake researh at INM-Leibniz Institute for New Materials, Saarbrüken. EA aknowledges funding from the European Researh Counil under the European Union's Seventh Framework Programme (FP/ ) / ERC Grant Agreement n

24 5. Referenes Adams, G.G. (014). Adhesion and pull-off fore of an elasti indenter from an elasti halfspae. Pro. R. So. Lond. A 470, (doi: /rspa ). Akisanya, A., Flek, N.A. (1997). Interfaial raking from the free edge of a long bimaterial strip. International Journal of Solids and Strutures 34 (13), Arzt, E., Gorb, S., Spolenak, R. (003). From miro to nano ontats in biologial attahment devies. Pro. Nat. Aad. Si. USA, 100, Balijepalli, R.G., Begley, M.R., MMeeking, R.M., Flek, N.A., Arzt, E. (016a). Numerial simulation of stress singularity and adhesion strength for a ompliant mushroom fibril attahed to a rigid substrate. Int. J. Solids and Struts., 85 86, Balijepalli, R.G., Fisher, S.C.L., Hensel, R., MMeeking, R.M., Arzt, E. (016b). Numerial Study of Adhesion Enhanement by Composite Fibrils with Soft Tip Layers. J. Meh. Phys. Solids, DOI: /j.jmps (in press). Del Campo, A., Greiner, C., Arzt, E. (007). Contat shape ontrols adhesion of bioinspired fibrillar surfaes. Langmuir: 3, Dugdale, D.S. (1960). Yielding of steel sheets ontaining slits. J. Meh. Phys. Solids, 8, Fisher, S.C.L., Arzt, E. and Hensel, R. (016). Composite pillars with tuneable interfae for adhesion to rough substrates. ACS Appl. Mater. Interfaes (under revision). Gao, H., Wang, X., Yao, H., Gorb, S., Arzt, E. (005). Mehanis of hierarhial adhesion strutures of gekos. Meh. Mater. 37, Greiner C., del Campo, A., Arzt, E. (007). Adhesion of bioinspired miropatterned surfaes: Effets of pillar radius, aspet ratio, and preload. Langmuir, 3, Huthinson, J.W. and Suo, Z., (1991). Mixed mode raking in layered materials. Advanes in Applied Mehanis, ed. J. W. Huthinson and T. Y. Wu, Aademi Press, New York, 9,

25 Johnson, K. L. (1985). Contat mehanis, Cambridge University Press, ISBN: Johnson, K. L., Kendall, K., Roberts, A. D. (1971). Surfae energy and the ontat of elasti solids. Pro. R. So. Lond. A 34, (doi: /rspa ) Kendall, K. (1971). The adhesion and surfae energy of elasti solids. Journal of Physis D: Applied Physis, 4(8): Khaderi, S.N., Flek, N.A., Arzt, E., MMeeking, R.M. (015). Detahment of an adhered miropillar from a dissimilar substrate. J. Meh. Phys. Solids, 75, Klusák, J., Náhlík,L. (007). Crak initiation riteria for singular stress onentrations, part I: a universal assessment of singular stress onentrations. Eng. Meh. 14(6), Knésl, Z., Náhlík, L. (007). Crak initiation riteria for singular stress onentrations, part II: stability of sharp and bi-material nothes. Eng. Meh. 14(6), Maugis, D. (199). Adhesion of spheres: the JKR-DMT transition using a Dugdale model. J. Colloid Interfae Si. 150, (doi: / (9)9085-t) Maugis, D. (000). Contat, adhesion and rupture of elasti solids. Berlin, Germany: Springer. Maugis, D., Barquins, M., Courtel, R. (1976). Griffith s rak and adhesion of elasti bodies. Métaux, Corrosion Industries, 605, Plössl, A., Kräuter, G. (1999). Wafer diret bonding: tailoring adhesion between brittle materials. Materials Siene and Engineering 5 (1-), Tada, H., Paris, P. C., Irwin, G.R. (000). The Stress Analysis of Craks Handbook, 3 rd edn. ASME Press, New York. Tang, T., Hui, C.-Y., Glassmaker, N. J., 005. Can a fibrillar interfae be stronger and tougher than a non-fibrillar one? Journal of The Royal Soiety Interfae (5), van Spengen, W.M., Puers, R., De Wolf, I.(00). A physial model to predit stition in MEMS. J. Miromeh. Miroeng. 1,

26 Wang, J.S., Suo, Z. (1990). Experimental determination of interfaial toughness urves using Brazil-nut-sandwihes. Ata. Metall. Mater. 38,

27 Table 1: Values of various parameters used for evaluating equation 1.8 for seleted values of (a) 0 a 1 1 d 1 R d 1 I N 1 1 S f R 0 D f I 0 N D f f S k (b) /4 a 1 1 d 1 R d 1 I N 1 1 S f R 0 D f I 0 N D f f S k

28 Figure Captions Figure 1. (a) A irular ylindrial pillar of material 1 is attahed to a half-spae of material, with a remote tensile stress applied to the top of the pillar; (b) the tensile tration on the interfae is limited to the value, and this is treated as a ohesive zone of length from the interfae orner. The problem shown in (b) is solved by superposition of two problems () and (d). When the ohesive zone lies within the zone of dominane of the orner singularity, the problems for () and (d) redue to the asymptoti problems as shown in (e) and (f), respetively. Figure. An interfaial rak of length with a rak tip proess zone of uniform ohesive strength and length due to a orner singularity as stipulated by (1.4) for the displaement field u j on the outer boundary. Figure 3. Dependene of k upon,. Figure 4. Cohesive zone length as a funtion of the remote stress for seleted values of, with (a) 0 and (b) /4. Free sliding is allowed in the ohesive zone. The dotted line in part (a) refers to the analyti model as given by Eq. (4.4), upon taking b R. Figure 5. Debond strength / as a funtion of pillar diameter D / for seleted F values of, with (a) 0 and (b) /4. Free sliding is allowed in the ohesive zone. The dotted line in part (a) refers to the analyti model as given by Eq. (4.4) and Eq. (4.3), R. upon taking Figure 6. Debond strength F / as a funtion of the pillar diameter D / for seleted values of, with (a) 0 and (b) /4. No sliding is allowed in the ohesive zone. Figure 7. (a) Phase (i) of detahment, and (b) phase (ii) of detahment, for a rigid, fritionless pillar. 8

29 Figure 1. (a) A irular ylindrial pillar of material 1 is attahed to a half-spae of material, with a remote tensile stress applied to the top of the pillar; (b) the tensile tration on the interfae is limited to the value, and this is treated as a ohesive zone of length from the interfae orner. The problem shown in (b) is solved by superposition of two problems () and (d). When the ohesive zone lies within the zone of dominane of the orner singularity, the problems for () and (d) redue to the asymptoti problems as shown in (e) and (f), respetively. 9

30 Figure. An interfaial rak of length with a rak tip proess zone of uniform ohesive strength and length due to a orner singularity as stipulated by (1.4) for the displaement field u j on the outer boundary / 4 k Figure 3. Dependene of k upon,. 30

31 Figure 4. Cohesive zone length as a funtion of the remote stress for seleted values of, with (a) 0 and (b) /4. Free sliding is allowed in the ohesive zone. The dotted line in part (a) refers to the analyti model as given by Eq. (4.4), upon taking b R. Figure 5. Debond strength / as a funtion of pillar diameter D / for seleted F values of, with (a) 0 and (b) /4. Free sliding is allowed in the ohesive zone. The dotted line in part (a) refers to the analyti model as given by Eq. (4.4) and Eq. (4.3), R. upon taking 31

32 Figure 6. Debond strength F / as a funtion of the pillar diameter D / for seleted values of, with (a) 0 and (b) /4. No sliding is allowed in the ohesive zone. 3

33 Figure 7. (a) Phase (i) of detahment, and (b) phase (ii) of detahment, for a rigid, fritionless pillar. 33