An easy-to-implement numerical simulation method for adhesive contact problems involving

asymmetric adhesive contact

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J. Phys. D: Appl. Phys. 44 (2011) 405303 (12pp) doi:10.1088/0022-3727/44/40/405303

An easy-to-implement numericalsimulation method for adhesivecontact problems involving asymmetricadhesive contactCongrui Jin1, Anand Jagota2 and Chung-Yuen Hui1,3

1 Field of Theoretical and Applied Mechanics, Sibley School of Mechanical and Aerospace Engineering,Cornell University, Ithaca, NY 14850, USA2 Department of Chemical Engineering, Lehigh University, Bethlehem, PA 18015, USA

E-mail: ch45@cornell.edu

Received 7 July 2011, in final form 17 August 2011Published 15 September 2011Online at stacks.iop.org/JPhysD/44/405303

AbstractA novel numerical method to solve asymmetric adhesive contact problems in rectangularcoordinates has been developed. Surface interaction is modelled using an interface potential,deformation is coupled using Green’s functions for a half space, and the resulting system ofequations is solved by a relaxation technique. The method can handle arbitrary surfacetopography and properties. Compared with previous methods, this numerical scheme is mucheasier to implement and is just as accurate. Here, it is applied to two adhesive contactproblems: one between a sphere and a cylinder; and the other between two identical cylindersin oblique contact. The numerical results reveal inaccuracies in elliptical contact theory whenthe skew angles between the two cylinders are small and the resulting contact is highlyeccentric. The pull-off forces show an indiscernible decrease with decreasing value of theskew angle, which is quite different from the elliptical JKR theory. This technique can be usedto solve adhesive contact problems that involve partial contact or complex geometry, such asrippled or rough surfaces.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

The canonical problem of adhesive contact between twoelastic spheres has been studied extensively over the lastfew decades, in particular, using two continuum mechanicsmodels, namely the Johnson–Kendall–Roberts (JKR) [1] andthe Derjaguin–Muller–Toporov (DMT) models [2, 3]. TheJKR model modifies the Hertz equations for the adhesionlesscontact of elastic spheres by accounting for surface energy. Inthe JKR theory the surface energies cause an infinite tensilestress to act at the contact edge, while in the DMT theory,the tensile stress is finite in a region outside the contact edgebut is zero inside it. The two seemingly contradictory modelswere reconciled first by Tabor [4] who suggested that they

3 Author to whom any correspondence should be addressed.

described two extremes of a certain dimensionless parameter(later dubbed the Tabor parameter µ, which is defined laterin this paper). Tabor showed that the JKR model is suitablefor large, compliant spheres with strong adhesion (whereµ is large), while the DMT model applies to small, stiffspheres with weak adhesion (where µ is small). Maugis[5] then developed a closed form solution for the transitionbetween the JKR and DMT models by applying the Dugdale–Barenblatt model to approximate the surface interaction. Theparameter used by Maugis, quantifying the transition betweenthe JKR and the DMT limit, is commonly referred to asMaugis parameter, which is effectively the same as Taborparameter µ if one identifies ε (defined later in this paper)in the Tabor parameter with Wad/σ0 in the Maugis parameter,where Wad is the work of adhesion and σ0 is the strength of theinterface.

0022-3727/11/405303+12$33.00 1 © 2011 IOP Publishing Ltd Printed in the UK & the USA

J. Phys. D: Appl. Phys. 44 (2011) 405303 C Jin et al

The first numerical simulation for the adhesive contactbetween spheres was presented by Muller et al in 1980[6]. They used the Lennard-Jones potential to model surfaceinteraction and showed a continuous transition from the JKRto the DMT theory as the Tabor parameter decreased. Morenumerical computations were performed by Attard and Parker[7]. They showed a puzzling non-monotonic trend of the pull-off force versus the Tabor parameter, but they did not treatthe singular integrands in the governing equation correctly,resulting in their predicted trend and the quantitative validity oftheir pull-off forces being questioned. A complete numericalsolution was obtained by Greenwood [8]. He pointed outthe existence of singular integrands in the governing equationand found S-shaped load-approach curves for values of µ

greater than one, leading to jumps into and out of contact.Feng [9, 10] proposed a more efficient numerical methodand a more accurate treatment of the singular integrands.He used Newton’s method to solve the nonlinear equationsand applied Keller’s algorithm [11, 12] of the arc-lengthcontinuation to track the solution branches around the turningpoints to determine the jumping-on and jumping-off behaviour.Since spheres are axisymmetric, in all the above numericalcomputations, the number of nonlinear equations is of ordern, where n is the number of elements for tessellation of theproblem domain.

There are many important applications in which thecontact area is not axisymmetric, e.g. a sphere in contactwith a cylinder or two cylinders oriented at a skew angle.Yang [13] analysed the adhesive contact between an ellipticalrigid flat-ended punch and an elastic half space using the energymethod. The separation was found to initiate at the edges of themajor axis, which would lead the initially elliptical contact toevolve to a more circular shape. Johnson and Greenwood [14]proposed an approximate JKR theory for the adhesive contactof smooth elastic bodies whose relative radii of curvature areunequal, in which they assumed that the energy release rate isapproximately constant along an elliptical contact line. In thefollowing, their model is referred to as elliptical JKR theory.

There are even more challenging but important problems,such as the adhesive contact between rough surfaces, where thecontact area is multiply connected and quite irregular. Solvingthese problems with a general numerical technique, such asthe finite element method, is computationally prohibitive. If,as can often be assumed, the surfaces do not deviate stronglyfrom planarity, a significant simplification can be achieved bydiscretizing only the surface and accounting for interactionsbetween them using known contact Green’s functions. Evenwith this simplification, the absence of radial symmetrypresents difficulty in the numerical solution of adhesivecontact, since the number of nonlinear equations increasesfrom n to n2 (see section 3). To bypass these difficulties,Wu [15] proposed an elegant numerical method to solve anasymmetric adhesive contact problem, which combines thefast Fourier transform, the bi-conjugate stabilized method,a preconditioning technique and a path-following method.Elliptical adhesive contact was studied experimentally bySumer et al [16] utilizing two polydimethylsiloxane (PDMS)cylinders placed at different skew angles with respect to each

other. They found that the difference of the maximum adhesionforce between experiments and elliptical JKR theory increasedas the contact line went from mildly elliptical to slim elliptical.Despite the feasibility and sophistication of the techniqueemployed by Wu [15], it is complicated to implement and stillrequires significant computing time and memory resources. Inthis work, a numerical method, much simpler to implementand as accurate as that introduced by Wu [15], is presented.This method is used to investigate two asymmetric adhesivecontact problems: (1) the adhesive contact between a sphereand a long cylinder, as shown in figure 1(a); and (2) theadhesive contact between two identical long cylinders placedat a skew angle θ with respect to each other [16], as shownin figure 1(b). While the problems studied in this papereach have a single asymmetric contact region, this techniquecan handle the problems that have multiple arbitrarily shapedcontact regions. (Those examples will be shown elsewhere.)

2. Governing equations

The equations governing frictionless adhesive interactionbetween two smooth, isotropic, linearly elastic non-conforming bodies are well known and can be found in[8, 15, 17]. The implicit assumption is that the contact is smallcompared with the size of the elastic bodies, and as a result, theinitial air gap ho, interpreted as the separation of the surfacesin the absence of applied and adhesive forces, can be written as

ho = 1

2R′ x2 +

1

2R′′ y2 (1)

where R′ and R′′ are called principal relative radii of curvature.For the two problems to be investigated in this paper (seeappendix A)

ho =

x2

2Rs+

y2

2RsRc/(Rs + Rc)Problem 1

x2

2Rc/(1 − cos θ)+

y2

2Rc/(1 + cos θ)Problem 2

(2)

where Rs and Rc are the radius of the sphere and the cylinderrespectively. Table 1 shows a summary of the dimensional anddimensionless parameters used in this paper.

In the presence of adhesive forces and external load,the surfaces deform and the separation or air gap betweenthem depends on the surface interaction. As in [15], surfaceinteraction is assumed to be governed by the Lennard-Jonespotential (integrated between one surface and the opposinghalf space). In dimensionless form, the local pressure P isrelated to the air gap H by

P = 8

3

[1

(H + 1)9− 1

(H + 1)3

]. (3)

According to (3), the local pressure is compressive (P > 0)

when H < 0 and is tensile (P < 0) when H > 0. At H = 0,P = 0. Derjaguin’s approximation [18] of assuming that (3)can be applied for small areas of surfaces even when they areinclined or curved is used. Following [15], the normalized air

2

J. Phys. D: Appl. Phys. 44 (2011) 405303 C Jin et al

Figure 1. Schematic of the two elliptical adhesive contact problems. The contact area is expected to have an elliptical shape as illustrated inthe top views. (a) Adhesive contact between a sphere and a cylinder. (b) Adhesive contact between two identical cylinders placed at a skewangle θ .

gap after deformation H is related to the pressure distributionP by

H = −D + U +8µ3/2

3π

×∫ ∫

�

[H(X′, Y ′) + 1]−9 − [H(X′, Y ′) + 1]−3√(X − X′)2 + (Y − Y ′)2

dX′ dY ′

(4)

where � is the entire X–Y plane, µ is the Tabor parameter[4, 19], and U is the normalized initial gap given by

U(X, Y )

=

1

2√

(1 + β)[X2 + (1 + β)Y 2] Problem 1

1

2 sin θ[X2(1 − cos θ) + Y 2(1 + cos θ)]. Problem 2

(5)

Note that (4) involves only quantities on the surface of thecontacting bodies, i.e. it assumes that the surfaces are nearlyflat. The dimensionless normal load F acting on the bodies istherefore

F = 1

3π

∫ ∫�

P (X, Y ) dX dY . (6)

3. Numerical method

Since the pressure drops very rapidly as the air gap increases,the region over which the problem is solved, �, can be takento be a finite rectangle � = [−a, a] × [−b, b]. Partition thisrectangle into 2N1 ×2N2 rectangles, each with area ab/N1N2.When a mesh element is small enough, the contact pressure at

each element can be treated as a constant. Since both problemshave symmetry that P(X, Y ) = P(−X, Y ) = P(X, −Y ) =P(−X, −Y ), only the rectangle � = [0, a]×[0, b] needs to beconsidered. Divide � into N1 × N2 rectangles �ij , assumingthat P(x, y) is constant, Pij , in each �ij , where

Pij = 8

3

[1

[H(Xi, Yj ) + 1]9− 1

[H(Xi, Yj ) + 1]3

]. (7)

(4) is evaluated at the mid-point of �kl , denoted by (Xk, Yl),and this results in

H(Xk, Yl) = −D + U(Xk, Yl) +µ3/2

πCij (Xk, Yl)Pij (8)

where

Cij (X, Y ) ≡∫ ∫

�ij

dX′ dY ′√(X − X′)2 + (Y − Y ′)2

. (9)

The summation convention is used for repeated indices in(8) to sum over index i from 1 to N1 and index j from 1to N2. The integrals Cij (X, Y ) can be evaluated exactly [20],which is given in appendix B. By defining Cklij = Cij (Xk, Yl),Hij = H(Xi, Yj ) and Uij = U(Xi, Yj ), (8) can be written as

Hkl = −D + Ukl +µ3/2

πCijklPij . (10)

To write (10) in a more familiar form, define column vectors�H = [Hij ], �U = [Uij ], �P = [Pij ] in R

N1×N2 and a(N1 × N2) × (N1 × N2) matrix C = [C(ij)(kl)] so that

�H = − �D + �U +µ3/2

πC �P (11)

where �D is the constant vector [D, D, . . . , D]T.

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J. Phys. D: Appl. Phys. 44 (2011) 405303 C Jin et al

Table 1. Summary of dimensional and dimensionless parameters.

Dimensional parameters:Principal relative radii of curvature: R′ and R′′ (m)Equivalent radiusa: Re (m)Radius of the sphere: Rs (m)Radius of the cylinder: Rc (m)Initial air gap: ho (m)Air gap after deformation: h (m)Length parameter on the order of interatomic spacing: ε (m)Approach of the surfaces with respect to the zero force position h = ε: δ (m)Position along x-direction: x (m)Position along y-direction: y (m)Pressure: p (N m−2)

Effective modulusb: E∗ (N m−2)

Work of adhesion: Wad (N m−1)Total normal load: f (N)Semi-major and -minor axis of the contact line (assumed to be an ellipse): ac and bc (m)Mean contact radius: cc = √

acbc (m)

Dimensionless parameters:Skew angle: θ (radian)Ratio of the radii of the sphere and the cylinder: β = Rs/Rc

Ratio of the principal relative radii of curvature of the bodies: λ = √R′′/R′

Normalized initial air gap between surfaces: U = hoε

Normalized air gap after deformation: H = h

ε− 1

Normalized approach or displacement: D = δ

ε

Normalized position along x-direction: X = x√εRe

Normalized position along x-direction: Y = y√εRe

Normalized pressure: P = pε

Wad

Tabor parameter: µ =(

ReW2ad

E∗2ε3

)1/3

Normalized normal load: F = f

3πReWador F0 = f

3πRsWad

Reference contact radius: cr =[

9πR2e Wad

4E∗

]1/3or c0 =

[9πR2

s Wad4E∗

]1/3

Normalized mean contact radius: C = cccr

or C0 = ccc0

Normalized approach or displacement: = δRec2

ror 0 = δRs

c20

Axes ratio of the ellipse: g = bc/ac

Eccentricity of the ellipse: e2 = 1 − g2

a Re = √R′R′′ =

{Rs/

√1 + (Rs/Rc) Problem 1

Rc/ sin θ Problem 2.

b 1E∗ = 1−ν2

1E1

+1−ν2

2E2

where Ei and νi are Young’s modulus and Poisson’s ratio of body i,respectively.

Since �P is a highly nonlinear function of �H, (11) is asystem of N1N2 nonlinear equations. Even for moderatelysmall mesh size where N1 = N2 = 102, the number ofelements of C is 108, and thus computation can rapidly becomeintractable. To solve (11) numerically, a virtual state relaxationmethod by interposing a virtual dash-pot in the mechanicalsystem described by (11) is proposed. Then (11) is transformedinto the following evolution equation for the dynamical systemdefined by

d �Hdt

+ �H = −D + �U +µ3/2

πC �P . (12)

The equilibrium solution of this dynamical system isdetermined by d �H/dt = �0, which is the solution of (11).For large separations (D large and negative), the surfaceforces are weak ( �P ≈ �0) and barely deform the surfaces,thus one can start with an initial condition of �H(t = 0) =−D + �U to obtain the solution in the next time step. The

basic idea is that D is gradually increased and the �H vectorobtained from the previous step is used as the initial guess forcomputing �H in the next step. In each step, let time evolveuntil equilibrium is reached within a prescribed tolerance.This method plots only the stable equilibriums for eachD, and thus the load–displacement curves are discontinuousat unstable jumps. All the numerically generated force–displacement curves in this paper are obtained by a two-stageprocess: one as D increases from minimum to the maximumdisplacement, and the other one as it decreases back to theminimum. The two numerical stages essentially generate thetwo experimentally measured force–displacement branches,i.e. approach and detachment, under displacement-controlledloading.

While it has not been mathematically proved that theabove numerical procedure will always converge to the correctequilibrium solution, irrespectively of initial conditions,important insight on how this numerical procedure work canbe gained by considering a one-dimensional version of (12).

4

J. Phys. D: Appl. Phys. 44 (2011) 405303 C Jin et al

–1 0 1 2 3 4 5 6 7 8 9 10–10

0

10

x

x

(a) c1=2.129, c

2=–3

→→ → ← ←←←

–1 0 1 2 3 4 5 6 7 8 9 10–10

0

10

x

x

(b) c1=10, c

2=1

→ ← ← ←←←← ←

–1 0 1 2 3 4 5 6 7 8 9 10–10

0

10

x

x

(c) c1=10, c

2=–3

← →→ ← ←←

–1 0 1 2 3 4 5 6 7 8 9 10–10

0

10

x

x

(d) c1=10, c

2=–5

←→→ →→ ← ←

Figure 2. Illustration of stable (solid dots) and unstable (open dots) equilibrium points associated with (13a) and (13b). Arrows indicate thedirection of the flow.

For this case, it is convenient to rewrite (12) as

dx

dt= c1φ(x) − (x + c2) ≡ f (x) (x > −1) (13a)

where

φ(x) ≡[(

1

x + 1

)9

−(

1

x + 1

)3]

(13b)

and c1(>0) and c2 are dimensionless parameters. Thenonlinear ordinary differential equation (13a) can be viewedas representing a nonautonomous dynamical system in whichx can be interpreted as the position of a particle. The functionf defined by (13a) specifies the particle velocity dx/dt ateach position x. It can be shown that, for c1 < c∗

1 (smallTabor parameter), f is strictly decreasing resulting in exactlyone stable equilibrium or fixed point (see figure 2(a)). Whenc1 > c∗

1 (large Tabor parameter), f first decreases rapidly, thenincreases to a maximum, and then decreases monotonically.For this case, the number of fixed points depends on c2 (seefigures 2(b)–(d)). It is easy to show that, as c2 falls below acritical point c∗

2, whose value depends on c1, a new fixed pointmaterializes and splits into two, one stable and one unstable(see figure 2(c)). This is called a saddle-node bifurcationat c∗

2(c1). As a result, there are three fixed points, two arestable and the middle one is unstable (see figure 2(c)). Furtherdecreasing c2 results in another saddle-node bifurcation wherethe middle unstable fixed point and the stable fixed point on theleft move towards each other, collide, and mutually annihilate.Thus, for c2 sufficiently negative, there is only one fixed point(see figure 2(d)). This behaviour is summarized in figure 2,

the phase portrait of the dynamical system, where x is plottedagainst x and the direction of flow near the fixed point(s) isindicated. This portrait shows that, for a given initial condition,the relaxation method allows the dynamical system to evolve tothese stable equilibrium solutions. Figure 2 shows that the longtime solution converges to the stable fixed points irrespectiveof the initial condition. The N -dimensional system, i.e. (12),is much more complicated. However, it is important to notethat at the beginning of each iteration, the numerical solutionis close to the equilibrium solution, hence one only needs toanalyse the behaviour of the phase portrait near the equilibriumsolution. This analysis is given in appendix C.

To check this method, the problem of adhesive contactbetween a sphere and a half space is solved, and it shows thatthe results are consistent with Wu’s results [15]. The non-dimensional load–displacement curves are shown in figure 3.It can be seen that the load–displacement curves for very smallvalues of Tabor parameter (µ < 0.1) converge to the predictionof rigid-sphere model:

F = 2 − 8(1 − D)6

9(1 − D)8(14)

which is referred to as the Bradley curve [9]. For large valuesof Tabor parameter the load–displacement curves becomeS-shaped, leading to jumps into and out of contact. The load–displacement curves generated by the present method compriseonly the stable solutions that can be observed in experiments.The accuracy and capability of extension to higher values ofTabor’s parameter was further confirmed in [21], in which the

5

J. Phys. D: Appl. Phys. 44 (2011) 405303 C Jin et al

–6 –5 –4 –3 –2 –1 0 1 2

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

D

F

Bradley CurveNumerical Simulation: µ=0.01Numerical Simulation: µ=0.1Numerical Simulation: µ=0.5Numerical Simulation: µ=1.0Numerical Simulation: µ=2.0Numerical Simulation: µ=3.0Numerical Simulation: µ=3.5

Figure 3. Numerical results for normalized load F versusnormalized displacement D for adhesive contact between a sphereand a half space.

adhesive contact problem between a spherical indenter andrippled surface was solved and compared with an exact solutionprovided by Guduru [22].

4. Results and discussion

Since the numerical results will be compared with theapproximate elliptical JKR theory, a few of its basic results [14]are summarized here. In the elliptical JKR theory, the contactline where the energy release rate is equal to the work ofadhesion is assumed to be an ellipse with semi-major and minoraxis ac and bc, respectively. The normal load f , separationδ and mean contact radius cc ≡ √

acbc in normalized form(F, , C) are

F = f

3πReWad= 8

3π

[g(1 − g1/2)2

(�2g2 − �1)2

]

×[�1 − �2g

5/2

1 − g1/2− 1

3(�2g

2 + �1)

](15a)

= δRe

c2r

=(

27/2

9π2

)2/3 [g(1 − g1/2)2

(�2g2 − �1)2

]2/3

×[

2K(e)�1 − �2g

5/2

1 − g1/2− �1B(e) − g2�2D(e)

](15b)

C = cc

cr=

(4√

2

3π

g5/4(1 − g1/2)

�2g2 − �1

)2/3

(15c)

where cr is a reference contact radius defined by cr =[9πR2

e Wad/4E∗]1/3 and g is the axes ratio defined by g =bc/ac. The axes ratio is related to the eccentricity e of theellipse by e2 = 1 − g2. The dimensionless quantities �1 and�2 in (15a)–(15c) are weighting parameters accounting forthe effect of adhesion energy:

�1 = λ + λ−1 − B(e)�2

D(e),

�2 = λ2C(e) + D(e) + C(e)

λ[(D(e) + C(e))(B(e) + g2C(e)) − g2C(e)2](15d)

–0.5 0 0.5 1 1.5 2–0.5

0

0.5

1

1.5

2

∆

F

Elliptical JKR Model: R’/R’’=130Elliptical JKR Model: R’/R’’=70Elliptical JKR Model: R’/R’’=30Elliptical JKR Model: R’/R’’=10Elliptical JKR Model: R’/R’’=7Elliptical JKR Model: R’/R’’=3Elliptical JKR Model: R’/R’’=1

Figure 4. Normalized load F versus normalized displacement fordifferent values of R′/R′′ based on elliptical JKR theory.

where K(e), C(e), B(e) and D(e) are complete elliptic integrals[23] that depend only on the eccentricity e, and λ is the ratioof the principal relative radii of curvature of the bodies:

λ =√

R′′/R′ (R′′ < R′, λ < 1). (16)

Based on (15a) and (15b), the normalized force–displacementcurves for different values of R′/R′′ are plotted in figure 4.In a load-controlled experiment, the two surfaces willspontaneously separate once the pull-off force is reached.Following the standard convention in contact mechanics, theabsolute value of minimum on the F() curve is called asthe pull-off or adhesion force, Fc. Figure 4 shows that thenormalized pull-off force Fc monotonically decreases withincreasing values of R′/R′′.

When R′/R′′ = 1, the results given by (15a)–(15d) reduceto the classical JKR theory for circular contact [1, 17], whichcan be written as

F = C3 −√

2C3, = C2 − 2

3

√2C. (17a)

In the Hertz contact theory, only compressive stresses can existin the contact area, which gives the following equations [17]:

F = C3, = C2. (17b)

For the DMT model, the force is the Hertz force inside thecontact area plus the adhesion force outside the contact area,which gives [2, 17]

F = C3 − 2

3, = C2. (17c)

For the two problems to be investigated in this paper,

R′/R′′ =

1 + β Problem 11 + cos θ

1 − cos θProblem 2.

(18)

The variation in the ratio of principal relative radii of curvatureR′/R′′ versus β or θ is plotted for the two problems in figure 5,which shows that the ratio is not large in problem 1 evenfor very large values of β (large sphere and small cylinder),whereas R′/R′′ can be very large in problem 2 for small skewangles.

6

J. Phys. D: Appl. Phys. 44 (2011) 405303 C Jin et al

0 0.5 1 1.50

50

100

150

θ

0 5 10 15 20 25 300

50

100

150

β

Figure 5. The variation in the ratio of principal relative radii ofcurvature R′/R′′ versus β = Rs/Rc (problem 1) or θ (problem 2).

–1 –0.5 0 0.5 1 1.5

–0.5

0

0.5

1

1.5

2

∆0

F0

Numerical Simulation: R’/R’’=1.0Numerical Simulation: R’/R’’=1.7Numerical Simulation: R’/R’’=8.0JKR Model (Elliptical JKR Model: R’/R’’=1.0)Elliptical JKR Model: R’/R’’=1.7Elliptical JKR Model: R’/R’’=8.0

Figure 6. Adhesive contact between a sphere and a cylinder:normalized load F0 versus normalized displacement 0.

Problem 1 is simulated for β ≡ Rs/Rc = 0.0, 0.7 and7.0, and the corresponding values of R′/R′′ are 1.0, 1.7 and8.0, respectively. As shown in figure 4, to investigate theregion for small values of R′/R′′ (1 < R′/R′′ < 10), adifferent scaling is needed to distinguish the difference, ifany, between the elliptical JKR solution and the numericalresults. Thus Re is replaced by Rs in the dimensionlessvariables, and F0 ≡ f/(3πRsWad) and 0 ≡ αRs/c

20 where

c0 ≡ (9πR2s Wad/4E∗)1/3 are defined. The F0–0 curve is

plotted for µ = 2.2 in figure 6, which indicates that thenumerical results agree very well with the elliptical JKR theory.Numerical simulation was also performed for µ = 3.0, but thecurve is not shown in this figure, since it is quite similar to thatfor µ = 2.2.

Problem 2 is simulated for θ = π/2, π/3, π/6, π/9and π/18, and the corresponding values of R′/R′′ are 1.0,3.0, 13.9, 32.2 and 130.6, respectively. Note that the case ofR′/R′′ = 1 corresponds to classical JKR model for circularcontact, i.e. (17a). Figure 7 plots the normalized forceF versus normalized displacement for µ = 2.2, whichshows that for large values of R′/R′′ (small skew angles), thenumerical results deviate significantly from the elliptical JKRtheory. For example, when R′/R′′ = 32.2 and 130.6, the

–0.5 0 0.5 1 1.5

–0.5

0

0.5

1

1.5

2

∆

Numerical Simulation: R’/R’’=1.0Numerical Simulation: R’/R’’=3.0Numerical Simulation: R’/R’’=13.9Numerical Simulation: R’/R’’=32.2Numerical Simulation: R’/R’’=130.6JKR ModelElliptical JKR Model: R’/R’’=3.0Elliptical JKR Model: R’/R’’=13.9Elliptical JKR Model: R’/R’’=32.2Elliptical JKR Model: R’/R’’=130.6

Figure 7. Adhesive contact between two identical cylinders placedat a skew angle θ : normalized load F versus normalizeddisplacement .

0 0.5 1 1.50

0.5

1

1.5Numerical ResultsElliptical JKR ModelExperimental Results

Figure 8. Adhesive contact between two identical cylinders placedat a skew angle θ : numerical results for the pull-off force fc as afunction of θ in a dimensional scale superimposed with theprediction of the elliptical JKR model and experimental resultsfrom [16].

numerical results predict much larger values of tensile forcefor a given indentation depth. In particular, the pull-off forcesFc are nearly independent of the values of R′/R′′ (showingan indiscernible decrease with increasing value of R′/R′′),which is quite different from the elliptical JKR theory whichshows that the pull-off force Fc decreases with decreasingskew angle. One may argue that this discrepancy is due tothe fact that elliptical JKR theory assumes that µ → ∞, andtherefore it does not explicitly account for surface interaction.However, numerical simulation for the case of µ = 3.0 showsquite similar result to that for µ = 2.2. The percentage ofrelative deviation is less than 2.23%, much smaller than thedeviation from the elliptical JKR solution. This discrepancycan be explained as follows. In the elliptical JKR theory, thecrack front (where the energy release rate equals to the workof adhesion) is assumed to be an ellipse. However, the pull-offinstability is very sensitive to the shape of the crack front, sincea slight change in the shape of crack front can cause the crackto go unstable (see also comment in figures 11 and 12(a)).

Based on the experimental parameters provided by [16],figure 8 plots the pull-off force fc as a function of the

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J. Phys. D: Appl. Phys. 44 (2011) 405303 C Jin et al

–1 –0.5 0 0.5 1 1.50

0.5

1

1.5

2

Numerical Simulation: R’/R’’=1.0Numerical Simulation: R’/R’’=1.7Numerical Simulation: R’/R’’=8.0JKR ModelElliptical JKR Model: R’/R’’=1.7Elliptical JKR Model: R’/R’’=8.0

Figure 9. Adhesive contact between a sphere and a cylinder:normalized mean contact radius C0versus normalized loadF0.

skew angle θ in a dimensional scale superimposed with theprediction of the elliptical JKR model and experimental data.The numerical simulation shows that the elliptical JKR modelis fairly accurate for small values of R′/R′′. The numericalresults start to deviate from the elliptical JKR solution asthe skew angle decreases, i.e. the value of R′/R′′ increases.Neither of them shows exact fit to the experimental dataalthough our numerical results provide a better fit to the datafor 0.2 � θ � 1.

To compare the shape of the contact area with theelliptical JKR theory, contact needs to be defined. UsingGreenwood’s definition [8], the location of the peak tensilestress is considered as the contact edge. Denote the locationof the peak tensile stresses along x- and y-axis as ac and bc,respectively. The mean contact radius cc ≡ √

acbc obtainedusing the numerical results will be compared with the ellipticalJKR theory as given in (15c). For problem 1, figure 9 plots thenormalized force F0 ≡ f/(3πRsWad) versus normalized meancontact radius C0 ≡ cc/c0 for µ = 2.2 and for different valuesof R′/R′′. For problem 2, figure 10 plots the normalized forceF versus normalized mean contact radius C for µ = 2.2. It canbe seen that, near the final pull-off, the discrepancy betweennumerical curves and the elliptical JKR model becomes largerwith increasing value of R′/R′′.

For problem 1, figure 11 plots a series of normalizedpressure distributions P for β = 7.0 (the corresponding valueof R′/R′′ is 8.0) and µ = 1.0 at D = −1.8, −1.0,0.0 and1.0. For problem 2, figure 12(a) plots the normalized pressuredistributions for θ = π/18 (the corresponding value of R′/R′′

is 130.6) and µ = 1.0 at D = −1.8, −1.0 and 0.0. Whenthe two bodies are approaching each other from a noncontactstate, e.g. at D = −1.8, the surfaces barely deform withpressure being nearly zero everywhere. As the two bodiesapproach each other one step further, e.g. at D = −1.0,the surfaces jump to a new equilibrium state suddenly witha nonzero contact area, and the pressure becomes compressive(>0) in the central region and tensile (<0) at the contact edge.

To check how elliptical the contact lines are, the ellipsesbased on the location of the peak tensile stresses along x- andy-axis are plotted in magenta lines, as shown in figures 11 and12(a). It is found that the contact lines are approximately

–0.5 0 0.5 1 1.50

0.5

1

1.5

2

2.5

Numerical Simulation: R’/R’’=1.0Numerical Simulation: R’/R’’=3.0Numerical Simulation: R’/R’’=32.2Numerical Simulation: R’/R’’=130.6JKR ModelElliptical JKR Model: R’/R’’=3.0Elliptical JKR Model: R’/R’’=32.2Elliptical JKR Model: R’/R’’=130.6

Figure 10. Adhesive contact between two identical cylinders placedat a skew angle θ : normalized mean contact radius C versusnormalized load F .

elliptical. However, a closer examination (by zooming inon the y-axis) reveals that at the tip of the major axis, thereal contact shape is always a little blunter as compared withan ellipse, as shown in figure 12(b), in which a comparisonbetween the magenta lines and the real contact lines at D =−1.2 is presented for two different skew angles: θ = π/6 andθ = π/18. This deviation from an ellipse is found to increaseas the skew angle is reduced, which is expected since the initialassumption of Hertzian elliptical boundary in the solution ofelliptical JKR theory does not satisfy the requirement that thestress intensity factor be equal around the periphery, especiallyfor large values of R′/R′′ [14]. This deviation is consistentwith previous experimental observation, shown in [16], whichshows that the nucleation sites for the detachment starts at thetip of the major axis, resulting in a nonelliptical shape of thecontact area.

So far, the simulation has been mostly carried out in theJKR adhesion regime, i.e. for large values of Tabor parameter(µ � 1.0), in which the key assumption is the absence ofsurface interaction outside the contact area. Attractive forcesoutside the contact edge become important for stiff materials,small spheres or weak adhesion. In these cases, the contactmechanics is better captured by the DMT model [2, 17], whichassumes that molecular forces act only in a ring-shaped zoneof noncontact adhesion. To explore the DMT and JKR–DMTtransition regimes, problem 2 is simulated for θ = π/6 (thecorresponding value of R′/R′′ is 13.9) for different valuesof Tabor parameter: µ = 0.1, 0.3 and 1.0. Figure 13 plotsthe normalized force F versus normalized displacement .Figure 14 plots the normalized force F versus normalizedmean contact radius C. The classical JKR model (17a), Hertzcontact theory (17b) and DMT model (17c) for circular contactare superimposed for comparison. It can be seen that withdecreasing value of Tabor parameter, the force–displacementcurve becomes closer to the prediction by the DMT model.

5. Summary

A new numerical technique for analysis of asymmetricadhesive contact problems in rectangular coordinates has

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J. Phys. D: Appl. Phys. 44 (2011) 405303 C Jin et al

Figure 11. Adhesive contact between a sphere and a cylinder: pressure distribution P for D = −1.8, −1,0,1, β = 7.0 (the correspondingvalue of R′/R′′ is 8.0) and µ = 1.0. Magenta lines show the ellipses based on the location of the peak tensile stresses along x- and y-axis.

been developed. Adhesive interactions are representedby an interaction potential and surface deformations arecoupled using half-space Green’s functions discretized on thesurface. The resulting set of nonlinear equations is solvedby a relaxation technique. Because it can handle surfacetopography and spatial variation in adhesive properties, thistechnique allows more efficient modelling of a number ofproblems in adhesive contact mechanics that have been difficultto analyse so far.

In this paper, the new numerical method has been appliedto two adhesive contact problems: the adhesive contactbetween a sphere and a cylinder; and the adhesive contactbetween two identical cylinders placed at a skew angle θ withrespect to each other. The results are compared with Johnsonand Greenwood’s approximate elliptical JKR model [14]. Forsmall values of R′/R′′, i.e. the ratio of the principal relativeradii of curvature of the bodies, the elliptical JKR model workswell, but for large values of R′/R′′, the discrepancy betweennumerical results and the elliptical JKR model becomeslarge. The pull-off forces show an indiscernible decrease withincreasing value of R′/R′′ (nearly independent of the valueof R′/R′′), which is quite different from the elliptical JKRtheory. This deviation is expected since the initial assumptionof Hertzian elliptical boundary in the solution of elliptical JKRtheory does not satisfy the requirement of the stress intensityfactor to be equal around the periphery. The numerical resultsreveal that at the tip of the major axis, the real contact shapeis always a little blunter as compared to an ellipse. This isconsistent with previous experimental observation [16], whichshows that the nucleation sites for the detachment starts atthe tip of the major axis. This numerical technique is muchsimpler to implement and as accurate as the method introducedby Wu [15]. This technique has also been proven to bevery efficient in solving adhesive contact problems betweena spherical indenter and rippled surfaces that involve partialcontact and large value of Tabor’s parameter [21, 22, 24].

Acknowledgments

This work is supported by the US Department of Energy, Officeof Basic Energy Science, Division of Material Sciences andEngineering under Award (DE-FG02-07ER46463).

Appendix A

There is a unique relation between R′, R′′, the principal radii ofcurvature of the two bodies, and the angle θ between the axesof principal curvature of each surface as follows (see appendix2 in [17])

1

R′ +1

R′′ = 1

R′1

+1

R′′1

+1

R′2

+1

R′′2

(A1)

1

R′′ − 1

R′ =[(

1

R′1

− 1

R′′1

)2

+

(1

R′2

− 1

R′′2

)2

+ 2

(1

R′1

− 1

R′′1

) (1

R′2

− 1

R′′2

)cos 2θ

]1/2

. (A2)

Let Rs and Rc be the radius of the sphere and the cylinder,respectively. For problem 1, in the case of a uniform cylinder,first principal curvature takes the value of the radii of thecylinder R′

1 = Rc, and the second principal curvature becomesinfinite R′′

1 = ∞. Thus, R′ = Rs and R′′ = RsRc/(Rs + Rc).Likewise with problem 2, R′ = Rc/(1 − cos θ) and R′′ =Rc/(1 + cos θ).

Appendix B

For a rectangular element of dimension 2a∗ × 2b∗ with centreat (Xk, Yl), Cij (Xk, Yl) is given by

Cij (Xk, Yl) = χ(Xi − Xk, Yj − Yl) + χ(−Xi − Xk, Yj − Yl)

+χ(Xi − Xk, −Yj − Yl) + χ(−Xi − Xk, −Yj − Yl)

(B1)

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J. Phys. D: Appl. Phys. 44 (2011) 405303 C Jin et al

Figure 12. (a) Adhesive contact between two identical cylinders placed at a skew angle θ : pressure distribution P for D = −1.8, −1, 0,θ = π/18 (the corresponding value of R′/R′′ is 130.6) and µ = 1.0. Magenta lines show the ellipses based on the location of the peaktensile stresses along x- and y-axis. (b) Adhesive contact between two identical cylinders placed at a skew angle θ : Pressure distribution Pfor D = −1.2 and µ = 1.0 for two different skew angles θ = π/6 and θ = π/18. Magenta lines show the ellipses based on the location ofthe peak tensile stresses along x- and y-axis. The region coloured by deep blue represents the real contact line.

where

χ(x1, x2)

= (x1 + a∗) ln(x2 + b∗) +

√(x2 + b∗)2 + (x1 + a∗)2

(x2 − b∗) +√

(x2 − b∗)2 + (x1 + a∗)2

+(x2 + b∗) ln(x1 + a∗) +

√(x2 + b∗)2 + (x1 + a∗)2

(x1 − a∗) +√

(x2 + b∗)2 + (x1 − a∗)2

+(x1 − a∗) ln(x2 − b∗) +

√(x2 − b∗)2 + (x1 − a∗)2

(x2 + b∗) +√

(x2 + b∗)2 + (x1 − a∗)2

+(x2 − b∗) ln(x1 − a∗) +

√(x2 − b∗)2 + (x1 − a∗)2

(x1 + a∗) +√

(x2 − b∗)2 + (x1 + a∗)2.

(B2)

Appendix C

The N -dimensional dynamical system described by equa-tion (12) is much more complicated and a full analysis such as

the one for one-dimensional case is very difficult. Fortunately,in the numerical procedure, the initial conditions chosen arealways close to an equilibrium solution. Therefore, only thelocal stability of (12) needs to be studied,

d �Hdt

= − �H − �D + �U +µ3/2

πC �P( �H). (C1)

Letting �φ = �H + �D − �U gives

d �φdt

= −�φ +µ3/2

πC ��( �φ) (C2)

where ��( �φ) ≡ �P( �φ − �D + �U). At equilibrium, d �φ/dt = 0.

There can be many equilibrium solutions of (C2) which aredenoted by �φ∞, i.e.

− �φ∞ +µ3/2

πC ��( �φ∞) = 0. (C3)

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J. Phys. D: Appl. Phys. 44 (2011) 405303 C Jin et al

–0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

–0.5

0

0.5

1

1.5

∆

F

JKR ModelElliptical JKR Model: R’/R’’=13.9Hertz ContactDMT ModelNumerical Simulation: R’/R’’=13.9, µ=0.1Numerical Simulation: R’/R’’=13.9, µ=0.3Numerical Simulation: R’/R’’=13.9, µ=1.0

Figure 13. Adhesive contact between two identical cylinders placedat a skew angle θ : normalized load F versus normalizeddisplacement .

–0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5JKR ModelElliptical JKR Model: R’/R’’=13.9Hertz ContactDMT ModelNumerical Simulation: R’/R’’=13.9, µ=0.1Numerical Simulation: R’/R’’=13.9, µ=0.3Numerical Simulation: R’/R’’=13.9, µ=1.0

Figure 14. Adhesive contact between two identical cylinders placedat a skew angle θ : normalized mean contact radius C versusnormalized load F .

The stability at these equilibrium solutions can be examined by

d( �φ − �φ∞)

dt= −( �φ − �φ∞) − �φ∞ +

µ3/2

πC ��( �φ − �φ∞ + �φ∞).

(C4)

Denote �φ = �φ− �φ∞ a small disturbance from the equilibriumposition and expand ��( �φ + �φ∞) as

��( �φ + �φ∞) = ��( �φ∞) +∂ ��∂ �φ

∣∣∣∣∣ �φ∞

�φ. (C5)

Then (C4) can be rewritten as

dφi

dt= −φi + �

N1×N2j=1

µ3/2

πCij

∂�j

∂φj

∣∣∣∣φ∞j

φj (C6)

where φ∞j is the j th component of �φ∞. A new matrix [J ] isdefined

[J ] = −[I ] +µ3/2

π[C][�] (C7)

where [I ], [C] and [�] are square matrices with N1N2 rows.Specifically, [I ] is the identity matrix, [C] is the matrix [Cij ],

and [�] is the diagonal matrix with �jj = ∂�j

∂φj

∣∣∣φ∞j

(no sum

on j ) on its diagonal. Note that

�jj = 8

3

[−9

(1

φ∞j − D + Uj + 1

)10

+3

(1

φ∞j − D + Uj + 1

)4]

. (C8)

Equation (C6) becomes

d �φdt

= [J ] �φ. (C9)

The stability of an equilibrium solution depends on theeigenvalues of J which are determined by the competitionbetween the two terms on the right-hand side of (C7). Whenthe Tabor parameter is so small that the first term dominates,the solution is always stable since the eigenvalues of [J ] areclose to −1. As the Tabor parameter becomes large, as inthe one-dimensional case, the eigenvalues of J depend on D.Since H > −1, the condition H∞j = φ∞j − D + Uj > −1must be satisfied and this condition constrains D to satisfy

φ∞j + Uj + 1 > D > −∞. (C10)

The following situations are anticipated:

(1) When D is negative and its magnitude is sufficiently large,�jj is positive and small (�jj → 0 as φ∞j − D + Uj →+∞), so the first term on the RHS of (C7) dominates andthe solution is stable.

(2) For small φ∞j − D + Uj , �jj is negative (�jj → −∞as φ∞j − D + Uj → −1) and since the matrix [C] ispositive definite, all eigenvalues of [J ] are negative andthe solution is stable.

(3) For intermediate values of D, �jj can be positive andsome of its values can be so large that the second termdominates, this scenario gives rise to the instability thataccounts for the observed S-shaped load-approach curves.

Note that since [C] is symmetric and [�] is diagonal,[C][�] is symmetric, and hence there exists a time independentorthogonal matrix [Q] such that [C][�] = [Q][�][Q]T, where[�] is a diagonal matrix with the eigenvalues of [C][�] onits diagonal. Substituting [C][�] = [Q][�][Q]T into (C9),multiplying the resulting equation on both sides by [Q]T, andnoting that the elements of [Q] are independent of time, it gives

d �ψdt

= − �ψ +µ3/2

π[�] �ψ (C11)

where �ψ ≡ [Q]T �φ. Using this transformation of basis thedifferential equations in (C9) can be decoupled as

dψj

dt= −

(1 − µ3/2

π�j

)ψj (no sum on j) (C12)

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J. Phys. D: Appl. Phys. 44 (2011) 405303 C Jin et al

where �j is the j th eigenvalue of [C][�]. Therefore, near theequilibrium solution, the local phase portraits are extremelysimple and essentially one dimensional. [�] can have positiveeigenvalue, which may result in 1 − (µ3/2/π)�j < 0, and theψj associated with these positive eigenvalues is responsiblefor the observed S-shaped load-approach curves.

For example, when µ = 1, as shown in figure 3, astwo bodies move closer from a large separation, no nearbystable equilibrium state exists for D = −1.63 and thesolution branch folds back to the states representing twobodies separated further away. Such a turning point indicatesthe jumping-on of contacting surfaces when they moveinfinitesimally closer. On the other hand, when two bodiesare pulled off from a contact state, the turning point at D =−1.74 corresponds to the jumping-off of contacting surfaces.Consistent with mathematical and physical considerations, themiddle part of the solution branch between the two turningpoints represents unstable equilibrium states, indicating thatthe curves generated using the present technique are only thepart that can be observed in experiments.

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