Published by AMSS Press, Wuhan, ChinaActa Mechanica Solida Sinica, Vol. 22, No. 5, October, 2009 ISSN 0894-9166

MICROMECHANICS OF ROUGH SURFACEADHESION: A HOMOGENIZED PROJECTION

METHOD��

Qunyang Li� Kyung-Suk Kim

(Division of Engineering, Brown University, Providence, RI 02912, USA)

Received 3 August 2009, revision received 8 September 2009

ABSTRACT A quasistatic homogenized projection is made to characterize the effective cohesivezone behavior for rough-surface adhesion. In the context of the homogenized projection, the trac-tion versus separation relation for the homogenized cohesive zone (HCZ) of a rough interface canbe highly oscillatory due to instabilities during microscopic adhesion and decohesion processes.The instabilities are found to occur not only individually but also collectively among the adhesivemicro-asperity contacts, leading to extensive energy dissipation. Based on the behaviors of theHCZ relations, a framework for describing instability-induced energy dissipation in rough-surfaceadhesion is proposed to elucidate the effect of roughness on apparent interface adhesion. Two non-dimensional parameters, α related to roughness morphology and n related to flaw distribution,are identified to be most crucial for controlling the energy dissipation. For an interface with ashallow roughness and a strong intrinsic adhesive strength, the interface adhesion can be strongerif we make it rougher (reducing α) or lower its flaw density (increasing n). The HCZ projectionmethod can be potentially extended and employed to bridge the apparent adhesion from intrinsicadhesion properties for engineering surfaces with multi-scale shallow roughness.

KEY WORDS homogenized cohesive zone, interface adhesion, roughness effect, instability, energydissipation

I. INTRODUCTIONInterface adhesion plays an essential role in a broad range of physical phenomena and governs a

lot of key processes in mechanical[1], electronic[2], and biological[3,4] applications. Owing to its steadilyincreasing technical importance, interface adhesion has been widely studied under various contexts[5–7].Nevertheless, the detailed mechanisms of interface adhesion have not yet been fully understood, mainlydue to its multi-physics and multi-scale nature. The adhesive forces can originate from different inter-molecular interactions. These may include simple electrostatic interactions, strong chemical interactions,such as ionic, covalent, metallic bonds, or relatively weak van der Waals interactions. Other interac-tions, like Derjaguin, Landau, Verwey and Overbeek (DLVO) forces, capillary forces and ligand-receptorbinding, also commonly exist for systems under aqueous, liquid-vapor and biological environments[8,9].Each of these interactions typically operates at its own dominating length scale, spanning several ordersof magnitude from a few angstroms to several millimeters. Besides the complexity of these interactions,

� Corresponding author. E-mail: qli@seas.upenn.edu. Currently at the Department of Mechanical Engineering and Ap-plied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA�� Project supported in part by the Nano and Bio Mechanics Program, under award CMS-0511961, and in part by theMRSEC Program, under award DMR-0520651, of the National Science Foundation.

· 378 · ACTA MECHANICA SOLIDA SINICA 2009

adhered interfaces are usually complex in geometry involving roughness of several length scales[10,11] orheterogeneous in composition involving different materials with dissimilar mechanical properties[12,13].As a confluence of these complicated forces, geometries and elastic/plastic deformations, interface ad-hesion is generally material specific and length-scale dependent.1.1. Overview of Mechanical Characterization of Interface Adhesion

Despite its immensely complicated nature, interface adhesion has historically been characterizedby two mechanical approaches. The first one employs a single term, work of separation (if reversible,also known as work of adhesion), to quantify the strength of inter-surface adhesion. This quantity isdefined as the energy needed to separate an interface of interest per unit area, or equivalently, thework required to break the ‘bonds’ along the interface per unit area. The other approach describesinterface adhesion in the context of cohesive zone models (CZM)[14–17]. In a CZM, it is assumed thatduring adhesion/decohesion processes the upper and lower surfaces of an interface interact througha process zone called a cohesive zone. In a cohesive zone, the cohesive traction and the inter-surfaceseparation obey a certain relation called the cohesive zone law. The introduction of the cohesive zonelaw naturally brings extra length scales to the system, which makes CZM more effective when modelingscale-dependent problems.

As mentioned above, constitutive relations of CZM are generally length-scale dependent. At theatomic scale, adhesion/decohesion processes are usually associated with several nanometer-scale defor-mation mechanisms, e.g. dislocation nucleation and motion, interplanar cleavage, surface atom diffusionand relaxation, and lattice trapping[18–21]. The stress fields for these local and non-local deformations aregenerally highly non-uniform or sometimes oscillatory. To objectively define the tractions and stresseswithin and near a cohesive zone, a continuum framework, termed Field Projection Method (FPM),was developed by Hong and Kim[22] and later extended by Choi and Kim[21]. The FPM was based onthe orthogonal eigenfunction expansion of the elastic field surrounding a crack-tip cohesive zone. Thismethod makes it possible for the CZM constitutive laws extracted from experiments to be comparedin a consistent way with those from molecular level computations. As the length scale becomes larger,adhesion/decohesion processes become even more complex due to the interplay between the intrinsiccohesive zone and its elastic surroundings. Cohesive zones can exhibit distinctively different character-istics depending on the dominant mechanism. If the active mechanism is length-scale dependent, onewould expect so for the CZM law[5,11,23].1.2. Roughness Effects on Interface Adhesion/Decohesion

Among all the extrinsic influences, the surface roughness involved in the adhesive contact has attractedthe most attention because of its intimate relation to friction and the great potential of tailoring thesurfaces to optimize mechanical performance of the nominal contact interface[24,25]. Early work byFuller and Tabor[26] showed that the adhesive force between smooth rubber spheres and roughenedPerspex surfaces decreased monotonically with the roughness. The micrometer-scale roughness of theglass substrates could almost eliminate the adhesion completely. A similar trend was observed forcontacts with nanometer-scale roughness involving various materials, e.g. gold and mica[27], polymericsurfaces[28], glass and silicon grating[29]. The general theoretical explanation for the adhesion reductionwas that surface roughness essentially reduces the true contact area to a small fraction of the nominalvalue and this in turn produces a smaller apparent adhesion[10,11,26,30].

Despite the widely observed reduction in adhesion, observations of increase in adhesion due toroughness were not uncommon. Briggs and Briscoe[31] found that rubber stuck to Perspex glass betterif the glasswasmade slightly rough rather than smooth. Later, by considering a rubber wheel rolling downPerspex tracks, Fuller and Roberts[32] confirmed that adhesion would increase initially with roughnessand reach a maximum value before it dropped down eventually. Again, similar results were observed forcontacts between a smooth crosslinked polydimethylsiloxane (PDMS) surface and aluminum surfaceswith nanometer-scale roughness[33]. The theoretical arguments[10,11,34] for this paradox were mostlybased on the fact that the soft surface can deform conformably to its rough counterpart and the truecontact area can be larger than the nominal one leading to an enhancement in adhesion.

Recently, the advent of micro-fabrication and nano-scale sensing techniques enabled the adhe-sion/decohesion processes to be examined more closely and precisely. In an experiment by Verneuilet al.[35], a smooth PDMS bead was pressed against micro-fabricated polymeric substrates with well-defined hexagonal patterns of spherical caps/holes. The authors observed two distinct contact statuses:

Vol. 22, No. 5 Qunyang Li et al.: Micromechanics of Rough Surface Adhesion · 379 ·

intimate contact, where two surfaces came into direct contact (optically indistinguishable), and sus-

pended contact, where an air film remained intercalated within the contact area. They[35] found thata critical height existed for the microstructures below which the contact could jump from suspendedcontact to intimate contact under a certain compressive load. During the unstable jump, the intimatecontact usually occurred at a nucleation point then unstably spread to the whole contact area. Moreimportantly, they[35] observed a noticeable hysteresis in the contact area as well as in the apparent valueof work of adhesion during the loading and unloading processes. The hysteresis was found to be load-history dependent and its effect was more prominent for rough surface contact than for smooth surfacecontact. It was believed that the hysteresis was due to the geometrical patterning of the rough surface[35];however, none of the existing models could give a quantitative description on this phenomenon.

In another experiment[36], PDMS microspheres with 20-30 μm diameters were attached to atomicforcemicroscope (AFM) cantilevers to form the colloidal probes. These probeswere subsequently broughtinto contact with the rough SrTiO3 ceramic substrates and the load-displacement curves were collected.The detailed responses of the highly sensitive AFM cantilever showed that the surface roughness ledto higher contact compliance and the detachment of the microsphere from the rough substrate wasaccomplished via a series of snap-out events. Similar observation was reported by Li and Kim[37], whofound that the adhesion between a rough metal microsphere and a flat mica substrate in humid air wasgoverned by the unstable shrinking-in of the water meniscus.

The above quantitative measurements[33,35–37] suggest that the pure geometrical arguments abouttrue contact area could not tell the whole story of rough interface adhesion. To study the roughness-controlled and load-history dependent interface adhesion, we consider the problem of a rigid indenterwith one-dimensional waviness indenting and subsequently detaching from a smooth elastic substrate.In §II, a homogenized cohesive zone projection was proposed to mathematically define the effectivecohesive zone laws for a rough adhesive interface. In §III, constitutive relations of a homogenized cohesivezone were calculated for interfaces with different roughness configurations. In §IV, the implications ofhomogenized cohesive zone model calculation to load-history and scale dependent cohesive zone lawswere addressed. Finally, a summary of results and some concluding remarks were given in §V.

II. HOMOGENIZED COHESIVE ZONE PROJECTION OF ROUGH INTERFACESA two-dimensional contact of a rigid indenter RI with an elastic half-space RE is considered as

shown in Fig.1. The surface of the half-space is flat and the lower surface of the indenter is roughwith tiny undulations. This paper will particularly focus on those interfaces that are nominally flatyet microscopically rough. A schematic of the magnified interface is shown in the lower right cornerof Fig.1. As illustrated in the schematic, the two surfaces are interacting with each other through thecohesive force inside the intrinsic cohesive zones represented by the regions shaded with vertical lines.The cohesive zone law for the intrinsic cohesive force follows a certain relation as depicted by thecohesive traction σICZ versus surface separation δICZ curve shown in the upper right corner of Fig.1.In many cases for simplicity of modeling, we are not interested in resolving the detailed deformationaround the individual undulations. Instead, it is desirable to find an effective nominal cohesive-zonerelation so that if we regard the interface to be ideally flat neglecting the roughness, the amount ofwork required to open up the flat interface will be the same as that for opening up the original roughinterface. In other words, for an imaginary system with an ideally flat surface interacting through thiseffective nominal cohesive zone, the work done by an external load will have the same value as theload acting on the original system with the rough interface. Such an effective process zone is calledthe homogenized cohesive zone (HCZ) as denoted pictorially by the region enclosed by the two dashedlines in Fig.1. The corresponding cohesive zone relation is called the HCZ law and will be derived inthe rest of this section.

To formulate the HCZ law, the contact between the slightly undulated rigid indenter RI (denotedas the upper half-space in Fig.1) and the elastic half-space RE (the lower half-space) is considered andre-drawn in Fig.2(a) and (b). In the undeformed configuration illustrated in Fig.2(a), the indenter isfar away from the half-space and no interaction exists between the two bodies. The domain of therepresentative half-sapce is depicted by Re; and the upper surface, bottom surface (infinitely below),left boundary and right boundary of the domain are denoted by ∂R+

e , ∂R−e , ∂RLe , and ∂RR

e . As indicatedin the figure, the coordinate system is chosen such that x runs laterally along the top surface of the

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Fig. 1. Contact of a rigid wavy indenter RI with an elastic half-space RE through a homogenized cohesive zone (HCZ)enclosed by the dashed lines; a zoomed-in view of the rough interface is shown at the lower right corner: intrinsic cohesivezones are presented by regions shaded by vertical lines; constitutive relation of the intrinsic cohesive zone is depicted bythe cohesive traction σICZ versus surface separation δICZ curve at the upper right corner.

half-space while z is perpendicular to the surface and positive upward. As the indenter moves down, itcomes into contact with the half-space and reaches equilibrium under a certain displacement uI witha normal load σ̄ (positive for tension, negative for compression) as shown in Fig.2(b). In general, bothtensile and compressive stresses can exist along the interface due to the inter-surface adhesion in themicroscopic state.

As a two-dimensional problem, the amount of work done to the elastic half-space by the indenterwould be unbounded. However, what we are interested in is not the total quantity of the work but thedifference between the work done to the half-space by this rough indenter and that by a flat indenter. Ascan be seen in the following, this difference would be the work of the HCZ for every depth of indentation(or opening of separation), which in turn determines the effective or homogenized cohesive zone law.Physically, this difference is determined by the competition of two mechanisms: producing undulationof the substrate which requires more work versus forming the adhesive bonds along the interface whichreleases extra energy. To quantify the difference an imaginary system is considered as shown in Fig.3(a)and (b), where an ideally flat indenter R′I is in contact with the same elastic half-space through an HCZas illustrated by the region shaded with vertical lines. The symbols and the coordinate system followthe same manner for the half-space as in Fig.2(a). The HCZ has an initial height of δ̄0 as shown inFig.3(a) and after deformation the height becomes δ̄ as shown in Fig.3(b). For the imaginary system tobe work-equivalent to the original system, the equilibrium displacement of the flat indenter u′I should

be the same as uI for any given normal load σ̄. Let S(1) = [u

(1)e , ε

(1)e , σ

(1)e ] and S

(2) = [u(2)e , ε

(2)e , σ

(2)e ]

be the two elastostatic states of the domain corresponding to the original and imaginary systems.Consider a representative domain of the half-space Re, with its finite boundary enclosing a portion ofthe contacting surface and the non-contacting outer boundary approaches infinity in a same way for

Fig. 2. Contact between a slightly undulated rigid indenter RI and an elastic half-space RE: (a) before deformation; Re

depicts the domain of the representative unit; ∂R+e , ∂R−e , ∂RL

e and ∂RRe denote bottom surface (infinitely below), left

boundary and right boundary; (b) after deformation; σ̄ represents the equilibrium stress under displacement uI; tensilestress can exist along the interface due to adhesion.

Vol. 22, No. 5 Qunyang Li et al.: Micromechanics of Rough Surface Adhesion · 381 ·

Fig. 3. Contact between an ideally flat rigid indenter R′I

and an elastic half-space RE through an effective homogenizedcohesive zone (HCZ): (a) before deformation the initial height of HCZ is δ̄0; Re depicts the domain of the representativeunit; ∂R+

e , ∂R−e , ∂RLe and ∂RR

e denote bottom surface (infinitely below), left boundary and right boundary; (b) afterdeformation the height of HCZ becomes δ̄; σ̄ represents the equilibrium stress under displacement u′

I.

these two cases. By invoking the Betti reciprocal theorem we have∫∂R+

e +∂R−e +∂RLe +∂RR

e

(n · σ(1)

e

)· u(2)

e dA =

∫∂R+

e +∂R−e +∂RLe +∂RR

e

(n · σ(2)

e

)· u(1)

e dA (1)

Because of the boundary conditions and symmetry of the problem, the integrals along ∂R−e , ∂RLe , and

∂RRe vanish on both sides of Eq.(1) and the non-trivial parts only come from the contribution along

∂R+e . Therefore Eq.(1) can be re-written as∫

∂R+e

(n · σ(1)

e

)·(u

(2)eI + u

(2)I

)dA =

∫∂R+

e

(n · σ(2)

e

)·(u

(1)eI + u

(1)I

)dA (2)

where u(1)I , u

(2)I are the displacements of the indenter and u

(1)eI , u

(2)eI are the displacements of the half-

space with respect to the indenter for the two systems. It is noticed that(u

(2)eI + u

(2)I

)=

(δ̄0 − δ̄ − u′I

)ez

and(n · σ

(2)e

)= σez on boundary ∂R+

e , where ez is the unit vector along z-direction. Hence, Eq.(2)

requires

δ̄ =

∫∂R+

e

(δ̄0 − u

(1)eI

)dA

∫∂R+

e

dA(3)

In Eq.(3), the initial height of the HCZ, δ̄0, can be further determined by restricting δ̄ to be zero whenthe half-space is fully conformed to the indenter and establishing complete contact.

In summary, the HCZ law for an adhesive interface with tiny undulation as shown in Fig.2 canbe specified by the relation between the homogenized load σ̄ and the homogenized separation δ̄. Thehomogenized separation δ̄ is given by

δ̄ =

∫∂R+

e

hgapdA

∫∂R+

e

dA(4)

where hgap is the gap between the undulated indenter and the half-space under the homogenized loadσ̄.

III. HOMOGENIZED COHESIVE ZONE FOR PERIODIC WAVY INTERFACESAs has been widely recognized, rough surfaces usually make partial contact when they are compressed

against each other to form interfaces[38]. Reduction of true contact area due to partial contact for roughsurfaces and its implications to adhesion and friction have been extensively studied[10,11,26,30,34,38]. In

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this section, another significant effect arising from interface undulation will be studied in the context ofHCZ. For simplicity, we concentrate our efforts on the interfaces with single-scale sinusoidal corrugationbut with different partial contact configurations. For interfaces with multi-scale roughness, a briefdiscussion will be addressed in a later section. In this paper, the intrinsic cohesive interaction is consideredto be characterized by work of adhesion as previously adopted by Johnson, Kendall and Roberts (JKRmodel)[39].

The contact between a sinusoidal rigid indenter and an elastic half-space is considered as shownin Fig.4. The gap between the indenter and the substrate prior to deformation can be described by asingle-scale sinusoidal wave with wavelength λ and amplitude ρ, following the form

h0(x) = ρ [1 − cos(2πx/λ)] (5)

where ρ � λ. When the indenter is graduallypressed into the half-space, depending on the ad-hesion and surface undulation, the two bodies caninitially form a partial contact then adhere com-pletely or they can snap together immediately fromthe beginning. This problem was previously stud-ied by Johnson[40], who found that two surfaceswould always adhere completely when they arebrought close enough. For such kind of perfectly-bonded joint, an infinite stress or, in practice, a

Fig. 4 Contact between a sinusoidal rigid indenter and anelastic half-space: undeformed gap h0(x) is described by asingle-scale sinusoidal wave with wavelength λ and amplitudeρ, where ρ � λ.

stress with a magnitude of the theoretical strength of the adhesive bonds is required to separate theinterface. This predicted strength is much higher than the general observations in reality and the dis-crepancy could be explained by introducing flaws along the interface analogous to those in fracturemechanics[40]. These flaws may originate from trapped third-materials or gaps due to partial contactat a finer scale. For a real rough interface, distribution of the flaws can be very complicated involvingmultiple length scales[10,11]. For the single-scale model considered in this paper, the configurations willbe simplified such that the flaws are located at every n trough(s). For n = 1, flaws are said to beclosely-distributed; for n > 1, flaws are said to be sparsely-distributed.

The adhesion/decohesion process of an interface with periodically-distributed flaws is schematicallyshown in Fig.5(a). The flaws, located at the troughs of every n periods, have an equilibrated widthof 2c under the combined loading from far-field load σ̄ and inter-surface adhesion. This problem canbe regarded as a superposition of problem I and II as illustrated in Fig.5(b) and (c) respectively. Inproblem I, a rigid indenter makes complete contact with an elastic half-space and they remain adheredby adhesion without any external load. The internal traction along the interface is denoted by t. Inproblem II, an elastic half-space is adhered to a rigid flat indenter with periodic cracks existing alongthe interface. These interfacial cracks have an equilibrated width 2c under far-field load σ2 and localpressure p along the crack surfaces. To maintain the boundary conditions in the original problem, thelocal pressure p in problem II should have the same magnitude as the internal traction t along theinterface in problem I but with a different sign.

Problem I was first considered by Westergarrd[41]. If the Young’s modulus and the Poisson’s ratio ofthe elastic half-space are denoted by E and ν, the traction acting along the interface can be expressedby

t(x) = t∗ cos (2πx/λ) (6)

where t∗ =πEρ

(1 − ν2)λ.

In problem II, the stress intensity factor at the edge of the contact can be expressed by two terms:KIa arising from far-field load σ2 and KIb from the local pressure p(x), where p(x) = t∗ cos (2πx/λ).

Vol. 22, No. 5 Qunyang Li et al.: Micromechanics of Rough Surface Adhesion · 383 ·

Fig. 5. (a) A wavy interface between a rigid indenter and an elastic half-space with periodically-distributed flaws: the flawslocated at the troughs of every n periods are in equilibrium with uniform width of 2c under the combined loading fromfar-field load σ̄ and inter-surface adhesion; (b) Problem I: a rigid indenter adheres completely to an elastic half-space byadhesion: t represents the internal traction along the interface; (c) Problem II: an elastic half-space adhered to a rigid flatindenter with periodic cracks existing along the interface: interfacial cracks have an equilibrated width 2c under far-fieldload σ2 and local pressure p along the crack surfaces.

From Tada’s handbook[42], we have

KIa = σ2

√nλ tan (ψc/n) (7)

KIb =2

nλ

√nλ tan

ψc

n

∫ c

0

p(x) cosψx

n√sin2 ψc

n− sin2 ψx

n

dx (8)

where ψc = πc/λ and ψx = πx/λ. The corresponding crack opening displacements due to the far-fieldloading and the local pressure along the crack surfaces (|x| ≤ c) are given by

da(x) = 21 − ν2

πEnλσ2 cosh−1

⎛⎜⎝cos

ψx

n

cosψc

n

⎞⎟⎠ (9)

and

db(x) = 21 − ν2

πE

∫ c

0

p(ξ) ln

∣∣∣∣∣∣∣∣∣∣

√1 −

(cos

ψc

n/ cos

ψx

n

)2

+

√1 −

(cos

ψc

n/ cos

ψξ

n

)2

√1 −

(cos

ψc

n/ cos

ψx

n

)2

−

√1 −

(cos

ψc

n/ cos

ψξ

n

)2

∣∣∣∣∣∣∣∣∣∣dξ (10)

where p(ξ) = t∗ cos (2πξ/λ) and ψξ = πξ/λ.The equilibrium value of σ2 can be determined by equating the elastic strain energy release rate to

the intrinsic work of adhesion w, which requires

(KIa + KIb)2

=2wE

1 − ν2(11)

Substituting Eqs.(7) and (8) into Eq.(11) gives

σ2 = t∗

⎡⎢⎢⎣√√√√ 2 (1 − ν2)wλ

nπ2Eρ2 tanψc

n

−2

nλ

∫ c

0

cos(2ψξ) cosψξ

n√sin2 ψc

n− sin2 ψξ

n

dξ

⎤⎥⎥⎦ (12)

Defining a non-dimensional parameter α as previously adopted by Johnson[40],

α =

√2 (1 − ν2) wλ

π2Eρ2(13)

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the equilibrium far-field load σ̄ can be expressed as a function of the crack length c non-dimensionallyas

σ̄

t∗= F

( c

λ; α, n

)(14)

where F (c/λ; α, n) is given by

F( c

λ; α, n

)=

α√n tan

(π

n

c

λ

) −2

n

∫ c/λ

0

cos(2πη) cos(π

nη)

√sin2

(π

n

c

λ

)− sin2

(π

nη)dη (15)

According to the definition in Eq.(4), the homogenized cohesive zone separation δ̄ corresponding tothe homogenized traction σ̄ can be calculated by

δ̄ =

∫ c

0[da(x) + db(x)]dx

nλ/2(16)

Utilizing Eqs.(9), (10) and (15), Eq.(16) can be simplified and re-arranged non-dimensionally as

δ̄

ρ= 4Da

( c

λ; α, n

)+ 4Db

( c

λ; n

)(17)

where Da (c/λ; α, n) and Db (c/λ; n) are given by

Da

( c

λ; α, n

)= F

( c

λ; α, n

)∫ c/λ

0

cosh−1

⎡⎣ cos

(πτ

n

)cos

(π

n

c

λ

)⎤⎦ dτ (18)

Db

( c

λ; n

)=

1

n

∫ c/λ

0

∫ c/λ

0

cos (2πζ)Dc

(ζ, τ ;

c

λ, n

)dζdτ (19)

Dc

(ζ, τ ;

c

λ, n

)= ln

∣∣∣∣∣∣∣∣∣∣

√1 − cos2

(π

n

c

λ

)/ cos2

(πτ

n

)+

√1 − cos2

(π

n

c

λ

)/ cos2

(πζ

n

)√

1 − cos2(π

n

c

λ

)/ cos2

(πτ

n

)−

√1 − cos2

(π

n

c

λ

)/ cos2

(πζ

n

)∣∣∣∣∣∣∣∣∣∣

(20)

3.1. Interfaces with Closely Distributed Flaws

For the case when n = 1, flaws are closely distributed at every trough. The configuration reducesto that previously considered by Johnson[40]. The HCZ relations for interfaces with different α valuesare presented in Fig.6 as a series of σ̄/t∗ ∼ δ̄/ρ curves. As previously remarked by Johnson[40], theadhesion/decohesion process of such wavy surfaces could be irreversible due to unstable snap-in andsnap-out events of individual crests. This mechanism can be seen more clearly in the context of HCZ,as discussed in the following.

The σ̄/t∗ ∼ δ̄/ρ curve for α = 0.4 is reproduced in Fig.7. This curve depicts the equilibriumrelationship between σ̄ and δ̄ when we open up or close an interface as shown in Fig.5(a). If the systemfollows exactly this equilibrium path when a completely adhered wavy interface is separated, the wholework done by the external load plus the pre-stored elastic strain energy will have the exact amount asthe intrinsic work of adhesion w. This can be verified by calculating the area enclosed by the σ̄ ∼ δ̄curve, which yields ∫

∞

0

σ̄dδ̄ +1

4t∗ρ = w (21)

or in a non-dimensional form ∫∞

0

σ̄

t∗d

(δ̄

ρ

)+

1

4=

πα2

2(22)

Vol. 22, No. 5 Qunyang Li et al.: Micromechanics of Rough Surface Adhesion · 385 ·

Fig. 6. A series of σ̄/t∗ ∼ δ̄/ρ curves representing equilib-rium HCZ relations for interfaces with different α values.

Fig. 7. σ̄/t∗ ∼ δ̄/ρ curve for an interface with α = 0.4.point A to point B, unstable snap-in during compression;point E to point F, unstable snap-out during separation.

Practically, the above reversible process rarely happens due to the instability when opening up orclosing a rough interface. When two surfaces as shown in Fig.5(a) are brought close from far away, theybegin to touch at point A under zero load. Once they get touched, the two surfaces will snap togetherby themselves as indicated by the snap-in process in Fig.7. If the system follows the equilibrium pathfrom point A to point B, reduction of the adhesion energy will be used to increase the strain energy aswell as to do some work to the external environment. However, the energy associated to this amountof work is dissipated in the form of stress waves during the irreversible snap-in process. Therefore theapparent HCZ relation during compression will follow the dotted curve as shown in Fig.7 instead ofthe equilibrium path. Conversely, if we open up an interface with initially closely-distributed flaws,the apparent HCZ relation will follow the equilibrium curve until at point E, where the system losesstability and snap out of contact abruptly to point F. Similarly to the snap-in process, the elastic strainenergy stored in the system at point E is dissipated in the form of stress waves during the snap-outprocess. The apparent HCZ relation during separation is depicted by the dashed curve in Fig.7. It isnoted that the dotted and dashed curves in Fig.7 are shifted slightly above and below the equilibriumcurve for better representation.

3.2. Interfaces with Sparsely Distributed Flaws

For the case when n > 1, flaws are sparsely-distributed along the interface. When such an interface isseparated, the interfacial cracks will pass several periods of undulation before they join together leadingto final detachment. As cracks grow along the corrugated interface, the equilibrated homogenized stressσ̄ fluctuates. A typical σ̄/t∗ ∼ δ̄/ρ curve is shown in Fig.8(a) for α = 0.8 and n = 10. In this figure, thefluctuation of σ̄ is manifested by five finger-shaped branches on the curve, with five instability pointsP1, P2, ...,P5. Similar to the case for an interface with closely-distributed flaws, the system would followthe dashed line experiencing many snap-out’s at those instability points when it is separated. Since eachof the snap-out’s dissipates certain amount of energy, this collective dissipation can consume much moreenergy than individual dissipation. The apparent σ̄/t∗ ∼ δ̄/ρ curve during separation is regenerated asa solid curve in Fig.8(b). By evaluating the area underneath the σ̄/t∗ ∼ δ̄/ρ curve, we can find that theapparent energy cost to completely separate the interface is more than twice of the amount enclosedby the equilibrium path which is equivalently the intrinsic work of adhesion.

In general, the collective energy dissipationdepends on the value ofn. Figure 9(a) shows an equilibriumσ̄/t∗ ∼ δ̄/ρ curve for a system with α = 0.8 and n = 100. This σ̄ ∼ δ̄ curve is composed of 50 finger-shaped branches, with 50 local instability points P1, P2, ...,P50. The apparent σ̄/t∗ ∼ δ̄/ρ curve duringseparation is given in Fig.9(b). From the figure, we can see that as n goes larger and larger the finger-branches are getting sharper and closer. For an interface with large n values, the apparent HCZ relationis essentially determined by the envelop of the instability points. This observation leads to an interestingdeduction: a larger scale HCZ relation can be bridged from a smaller scale intrinsic cohesive zone lawby considering the finer scale instability induced dissipation. The implications of this mechanism toexperimental observations and theoretical understandings will be discussed in the following section.

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Fig. 8. (a) σ̄/t∗ ∼ δ̄/ρ curve representing the HCZ relation for an interface with α = 0.8 and n = 10: the fluctuation ofσ̄ is manifested by five finger-shaped branches with five instability points P1, P2, ...,P5; (b) Apparent σ̄/t∗ ∼ δ̄/ρ curveduring separation for an interface with α = 0.8 and n = 10.

Fig. 9. (a) σ̄/t∗ ∼ δ̄/ρ curve representing the HCZ relation for an interface with α = 0.8 and n = 100: the fluctuation ofσ̄ is manifested by 50 finger-shaped branches with five instability points P1, P2, ...,P50; (b) Apparent σ̄/t∗ ∼ δ̄/ρ curveduring separation for an interface with α = 0.8 and n = 100: only the first 45 instability points are included, after P45

the interface has been fully separated.

IV. DISCUSSION4.1. Effects of Roughness and Flaw Distribution on Single-scale Instability Induced EnergyDissipation

As mentioned previously, instability events were commonly observed during the adhesion/decohesionprocesses of rough interfaces and sometimes accompanied by stress waves/acoustic radiation[35–37,43].From the analysis in §III, we can see that these unstable snap-in and snap-out events can dissipateextra energy and lead to enhancement of apparent adhesion. This is consistent with the experimentalobservations where rough interfaces sometimes had better adhesion compared to smooth ones[31–33,35].The difference between the HCZ relations during compression and separation also explains the adhesionhysteresis observed in the experiments[35]. In general, a rough interface appears to be more adhesiveduring separation than it does during compression.

Despite the qualitative consistency with experiments, if we only consider interfaces with closely-distributed flaws, the energy dissipation arising from instabilities of individual crests is usually relativelylimited, a few percent depending on value of α. This individual instability could not account for the largeadhesion toughening observed in the experiments[31–33,35]. Our analysis in Section §3.2 demonstratesthat, besides the individual events, the instability can occur collectively if the flaws are sparsely dis-tributed along the interface. This collective instability generally dissipates more energy than individualinstability. Let Γ be the total apparent energy consumed to separate a rough interface, which can be

Vol. 22, No. 5 Qunyang Li et al.: Micromechanics of Rough Surface Adhesion · 387 ·

expressed by

Γ =

∫∞

0

σ̄apdδ̄ap +1

4t∗ρ (23)

The first term in Eq.(23) is the energy absorbed by the apparent HCZ and the second term is the pre-stored elastic strain energy for a completely adhered interface. If we define the energy disposal ratio κdis

as the ratio between Γ and intrinsic work of adhesion w, then κdis can be expressed non-dimensionallyas

κdis =W̄a + 1/4

πα2/2(24)

where W̄a =

∫∞

0

σ̄ap

t∗d

(δ̄ap

ρ

)is the non-dimensional energy consumed by apparent HCZ. From §III, we

know that the energy disposal ratio κdis will depend only on two parameters: n and α. The relationshipbetween κdis and n for different α values is plotted in Fig.10.

As shown in Fig.10, the energy disposal ratio κdis increases very fast at the beginning as n increasesfrom unity and it saturates quickly as n gets larger. Beyond n = 100, there are virtually no changesin the value of κdis. Since both σ̄ap/t∗ and δ̄ap/ρ increase linearly with α, it is expected that the non-

dimensional work W̄a =

∫∞

0

σ̄ap

t∗d

(δ̄ap

ρ

)should be a quadratic function of α for a given n value.

Denoted by κ∞dis, the energy disposal ratio when n approaches infinity should take the form of

κ∞dis =c1α

2 + c2α + c3

α2(25)

Fig. 10. Energy disposal ratio κdis as a function of flawdistribution number n for different α values.

Fig. 11. Energy disposal ratio κdis as a function of α forn = 256: fitting based on Eq.(23) gives c1 = 1.054, c2 =1.174 and c3 = 0.490.

To seek an approximate result, the change of the energy disposal ratio κdis upon the variation of α isplotted for n = 256 in Fig.11. Fitting the data with Eq.(25) gives the following results: c1 = 1.054,c2 = 1.174 and c3 = 0.490. Intuitively, we know that if the undulation is very flat, i.e. ρ/λ → 0 therebyα → ∞, energy disposal ratio should approach 1. The fitting results are indeed consistent with thisrequirement within numerical error. From Eq.(25), we can also see that κ∞dis scales with α−2 for smallα values, which means that an interface will become tougher if it is made rougher. It should be notedthat the previous analyses are only applicable for interfaces with small undulations.

4.2. Load-history Dependency of HCZ Laws

The above discussion examines the maximum adhesion enhancement that a rough surface can gainfrom instability induced dissipation. Nevertheless, the apparent adhesion for an interface formed byprior compression is more complicated. Let us consider a case where a rigid indenter with single-scalewaviness is brought into contact with an elastic flat surface. The surface of an indenter is assumed to besticky everywhere except for some small contaminated spots located at the troughs every n0(n0 > 1)periods. We can recognize that this is the configuration discussed previously with n = n0. Interestingly,

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when the indenter is first brought into contact with the flat surface, the actual configuration of theinterface will start from the one with n = 1 instead of n = n0. The configuration will remain sountil the two surfaces are so compressed that complete contact is established locally at those troughswithout contamination. Only after the uncontaminated surfaces are all closed can the configurationof the interface become that with n = n0. Since real rough surfaces rarely have uniformly-distributedroughness, the effective configuration of the interface will gradually evolve from one with a lower nvalue to another one with a higher n value as a rough interface is compressed. This is one of the reasonsthat the apparent adhesion is load-history dependent. Previously, apparent adhesion was calculatedby

∫∞

0 σ̄apdδ̄ap; however, for an interface formed from prior compression it should be calculated by∫∞

δ̄intσ̄apdδ̄ap, where δ̄int is the initial separation of the HCZ. Even for an interface with a fix n value,

the apparent adhesion will depend on how well or tightly an interface is adhered before the interfaceis separated. This is another reason why the apparent adhesion appears to be load-history dependent.

These two mechanisms predict that apparent adhesion of an interface can be larger or smallerthan the intrinsic value depending on the roughness and the loading history of the interface. Thisis consistent with the experimental observations that both toughening and weakening effects wereobserved[26–29,31–36]. The observation that the adhesion hysteresis was more significant for intimatecontacts than for suspended contacts[35] gives a strong support for these two mechanisms.

4.3. Multi-scale Instability Induced Energy Dissipation and Its Implication to Hierarchical HCZProjection

The above discussion provides some guidelineson how roughness and flaws can influence the ap-parent adhesion of an interface with single-scalewaviness. Nevertheless, the apparent adhesion ex-hibited by an interface is usually much more com-plicated in practice. It has long been recognizedthat a wide range of engineering and biologi-cal surfaces can be approximated by self-affinefractals[10,44]. The apparent adhesion of such hi-erarchical interfaces can also be understood viaa renormalization approach described as follows.Figure 12 shows an interface formed by two suchself-affine fractals. Because of the fractal natureof the surfaces, partial contact occurs at differentlength scales. As illustrated in the figure, a regionwhich appears to be in complete contact at a larger

Fig. 12 An interface formed by two self-affine fractal surfacesviewed under different scales (Scale 1 to Scale k): a regionwhich appears to be in complete contact at a larger lengthscale (lower magnification) is actually in partial contact at asmaller length scale (higher magnification); Scale 1 depictsthe atomic scale.

length scale (lower magnification) is actually in partial contact at a smaller length scale (higher mag-nification). A series of magnification processes from Scale 1 to Scale k are presented in Fig.12. Since aphysical surface cannot be a self affine fractal over all length scales, a cut-off scale has to be invokedand is denoted by Scale 1. At Scale 1, the HCZ relation is given by the intrinsic cohesive interactionbetween the two atomically flat surfaces. At Scale 2, based on the roughness characteristic, the HCZrelation at this scale can be projected from the intrinsic cohesive zone relation at Scale 1 using the HCZprojection method presented previously. Similarly, the HCZ relation at a larger scale can be projectedfrom that at a lower scale via the same approach; therefore the apparent adhesion at Scale k can betraced back scale by scale to Scale 1. This iteration technique is similar to the approach previouslyadopted by Yao and Gao[44] to study the attaching strength of gecko toes and is a special application ofthe well-known renormalization theory in statistical mechanics[45]. The renormalization combining theinstability induced dissipation mechanism can provide a mathematical protocol to bridge the macro-scopic HCZ law from the intrinsic atomic interactions and suggest possible strategies of tailoring theinterfaces to our purposes.

V. CONCLUSIONSApparent cohesive zone constitutive behavior for a rough interface with one-dimensional waviness

has been analyzed using a homogenized projection method. The projected homogenized cohesive zone(HCZ) for a rough interface has been found to be strongly affected by instability events during adhesion

Vol. 22, No. 5 Qunyang Li et al.: Micromechanics of Rough Surface Adhesion · 389 ·

and decohesion processes.Besides the individual occurrence, the instability can also take place at a higherlevel collectively leading to more energy dissipation. Based on the results from HCZ projection, a newinstability-induced energy dissipation mechanism has been proposed to elucidate the roughness effect onapparent interface adhesion. This mechanism is mainly governed by two non-dimensional parameters,α and n, depicting the roughness morphology and flaw distribution respectively. Detailed tougheningeffects arising from these two aspects have been obtained quantitatively and discussed. In general, mak-ing an interface rougher (reducing α) or minimizing the flaw density (increasing n) will make a roughinterface stronger. The mechanism also explains the load-history dependency of the apparent cohesivezone relations, which is well consistent with the experimental observations. The instability induced en-ergy dissipation mechanism together with the HCZ projection method can be potentially extended andused to understand the apparent adhesion behavior of an engineering surface with multi-scale roughness.

Acknowledgments Qunyang Li thanks Dr. B.C. Burke for reading the manuscript and providing theconstructive comments. Qunyang Li would like to dedicate this paper to Professor Shouwen Yu on theoccasion of his 70th birthday.

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