Applied Mathematical Modelling 35 (2011) 3201–3212

Contents lists available at ScienceDirect

Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Frictionless elliptical contact of thin viscoelastic layers bondedto rigid substrates

I. Argatov ⇑, G. MishurisInstitute of Mathematics and Physics, Aberystwyth University, Ceredigion SY23 3BZ, Wales, UK

a r t i c l e i n f o

Article history:Received 26 November 2010Received in revised form 18 December 2010Accepted 11 January 2011Available online 2 February 2011

Keywords:Contact problemThin layerViscoelastic coatingElliptical contact areaExact solution

0307-904X/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.apm.2011.01.029

⇑ Corresponding author.E-mail addresses: iva1@aber.ac.uk (I. Argatov), g

a b s t r a c t

A three-dimensional unilateral contact problem for thin viscoelastic layers bonded to rigidsubstrates shaped like elliptic paraboloids is considered. Two cases are studied: (a)Poisson’s ratios of the layer materials are not very close to 0.5 and (b) the layer materialsare incompressible with Poisson’s ratio of 0.5. Poisson’s ratios are assumed to be time inde-pendent. In the present paper we derive the general solutions to the problems of ellipticalcontact between thin compressible or incompressible layers of arbitrary viscoelastic mate-rials. The approach is based on the analytical method developed by the authors for theelliptical contact of thin biphasic cartilage layers. The obtained analytical solution is validfor monotonically increasing loading conditions.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

Contact problems involving transmission of loads across joints of coated parts are of considerable practical importance inindustry [1]. There are emerging applications for viscoelastic coatings in bearings and seals. The coatings protect the com-ponents from impact damage, suppress noise and increase the wear resistance. Indentation contact problems for thin visco-elastic layers are recently encountered in applications of the atomic force microscope for determining the mechanicalproperties of thin biological samples [2]. Recently, there has been growing interest in the development of thin-layer modelsfor describing mechanical behavior of biological tissues [3].

Many approximate solutions to the axisymmetric indentation problem of unilateral contact interaction of a rigid spher-ical punch with a thin elastic layer are available. Aleksandrov [4] obtained an asymptotic solution when Poisson’s ratio, m, ofthe layer material is not very close to 0.5. The case of an incompressible layer material was first studied by Matthewson [5]using an approximate approach. A rigorous asymptotic solution to the axisymmetric indentation problem for a thin incom-pressible elastic layer was constructed by Chadwick [6] whose solution corroborates Matthewson’s result [5] to leadingasymptotic order.

For dealing with non-axisymmetric, Barber [7] developed an approximate method of Jaffar [8] and Johnson [9] extendingtheir axisymmetric solutions for the three-dimensional problem of elliptical contact. A three-dimensional unilateral inden-tation problem for a thin elastic layer indented by a rigid punch shaped like elliptic paraboloid was studied by Argatov [10]using the distributional asymptotic technique, and the solution was obtained in an explicit form for the case m < 0.5. Recently,Hlavácek [11] extended the approximate method of [5] to the indentation problem of elliptical contact for an incompressibleelastic layer.

. All rights reserved.

gm@aber.ac.uk (G. Mishuris).

Nomenclature

a1, a2 weighted constants for compressible layersb1, b2 weighted constants for incompressible layersC boundary of the contact regiond0 relative approachDx Laplace operator# = E/(2(1 � m2)) elastic constant, Kolosov’s constantm Poisson’s ratioq = E1/E0 ratio of the relaxed and unrelaxed modulis dummy integration variableU1(t), U2(t) creep functionsUa(t) compound creep function for compressible layersUb(t) compound creep function for incompressible layersu(x) elliptic paraboloid gap functionW1(t), W2(t) relaxation functionsWa(t) compound relaxation function for compressible layersWb(t) compound relaxation function for incompressible layersx contact domainA asymptotic constanta major semi-axes of elliptic contact domainB asymptotic constantb minor semi-axes of elliptic contact domainE1 relaxed elastic modulusF total contact forceF0 initial total contact forceH Heaviside step functionh thickness of the layerK kernel of integral equationm = E1/h3

auxiliary contactPðx; tÞ auxiliary function for the maximum contact pressureP0ðtÞ auxiliary function for the maximum contact pressurep contact pressureR1, R2 radii of curvaturer radial coordinates aspect ratio of the contact domainT relaxation timet time variablet⁄(x) time when the contact contour reaches point xw1, w2 vertical displacement functionsx = (x1,x2)

horizontal coordinatesz vertical coordinate

3202 I. Argatov, G. Mishuris / Applied Mathematical Modelling 35 (2011) 3201–3212

Axisymmetric indentation problems for a viscoelastic layer have been considered in papers [12,13] for the specialcases of the Maxwell model and the standard viscoelastic solid model. In the present paper, those partial problems con-sidered in [12,13] are extended in two directions: (1) gap geometry between the contacting surfaces is assumed to berepresented by an elliptic paraboloid, which results in the so-called elliptical contact between thin layers; (2) layersmaterials are assumed to follow arbitrary viscoelastic constitutive equations. Thus, a solution to the axysymetric prob-lems for an arbitrary viscoelestic model is obtained from our general solution for the three-dimensional geometry as aparticular case.

It is interesting that the analogous contact problem for a thin biphasic layer has attracted much more attention. Ateshianet al. [14] obtained an asymptotic solution for the contact problem of two identical biphasic layers attached to two rigidimpermeable spherical substrates of equal radii. Wu et al. [15] extended this solution to a more general model by combiningthe assumption of the kinetic relationship from classical contact mechanics [9] with the joint contact model for the contact oftwo biphasic cartilage [14]. (See more references on the biphasic layer contact problems and related solutions in [16,17].)These solutions have been widely used as theoretical background in modeling the articular contact mechanics. The analysis

I. Argatov, G. Mishuris / Applied Mathematical Modelling 35 (2011) 3201–3212 3203

[15] was extended by Argatov and Mishuris [16] by formulating the refined contact condition which takes into account thetangential displacements at the contact region. Recently, Argatov and Mishuris [17] obtained an exact solution to the three-dimensional unilateral contact problem of elliptical contact of thin biphasic layers.

It can be shown (see, Section 4.5) that a thin biphasic layer in the contact interaction involving transmission of loadsacross joints behaves like an incompressible viscoelastic material following the Maxwell model. Thus, the developed analyt-ical method [17] can be extended by analogy to the case of elliptical contact between thin layers of general viscoelastic mate-rial properties. It is known [14] that, following the general procedure of contact mechanics [9], the problem of contactinteraction of two thin elastic or biphasic layers can be reduced to the corresponding indentation problem with the com-pound parameters. In the viscoelastic case, this reduction procedure is not so straightforward, since it is complicated bythe necessity to introduce the compound creep and relaxation functions that should be reciprocal. That is why, we considerthe viscoelastic interaction contact problem for two dissimilar thin layers, while the indentation problem is obtained as aparticular case by setting one of the elastic modulus equal to infinity.

2. Asymptotic behavior of a thin elastic layer bonded to a rigid substrate in the thin-layer limit

Let us first consider the frictionless elastic indentation problem for a rigid punch in the form of an elliptic paraboloid

z ¼ uðx1; x2Þ; uðxÞ ¼ ð2R1Þ�1x21 þ ð2R2Þ�1x2

2; ð1Þ

indenting an elastic layer of thickness h to an indentation depth d0. Here, R1 and R2 are the radii of curvature of the principalnormal cross-sections of the surface of the punch at its vertex.

Applying the standard two-dimensional Fourier transform technique [18], the contact problem can be reduced to the fol-lowing integral equation with respect to the contact pressure density p(x) (see, for example [4]):

#�1ZZ

xKðx1 � y1; x2 � y2ÞpðyÞdy ¼ d0 �uðxÞ: ð2Þ

Here, # = E/(2(1 � m2)) is an elastic constant with E and m being Young’s modulus and Poisson’s ratio, respectively.In the case of unilateral contact, the boundary C of the contact region x is not known a priori. Instead, we have the

inequality

pðxÞ > 0; x 2 x; ð3Þ

and the condition

pðxÞ ¼ 0; x 2 C; ð4Þ

which determines boundary of the contact region. Note that under the assumption of the convexity of the punch shape (1),the contact region x should be simply connected.

For an elastic layer firmly attached to a rigid substrate, the kernel of the integral equation (2) is given by

Kðx1; x2Þ ¼1

p2h

ZZ þ1

0

LðsÞs

coss1x1

hcos

s2x2

hds1 ds2;

LðsÞ ¼ 2, sinh 2s� 4s2, cosh 2sþ 1þ ,2 þ 4s2 ;

where s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffis2

1 þ s22

q, and , ¼ 3� 4m is Kolosov’s constant.

The function LðsÞ is continuous and positive for s 2 (0, +1) and satisfies the asymptotic relations

LðsÞ ¼ 1þ Oðs2e�2sÞ; s!1;LðsÞ ¼ Asþ Bs3 þ Oðs5Þ ¼ Asð1þm1s2 þm2s4 þ � � �Þ; s! 0; ð5Þ

where we introduced the notation

A ¼ 4ð,� 1Þð,þ 1Þ2

¼ 1� 2m2ð1� mÞ2

; B ¼ 8ð,2 � 5,þ 6Þ3ð,þ 1Þ3

¼ mð4m� 1Þ6ð1� mÞ3

: ð6Þ

Using the distributional asymptotic analysis approach, Argatov [10] obtained a formal asymptotic expansion of the integraloperator on the left-hand side of Eq. (2), which is valid in the inner contact region, that is far from the boundary C. In thedimensional form the inner asymptotic expansion can be written as follows:

ZZ

xKðx� yÞpðyÞdy ’ A

X1n¼0

ð�1Þnmnh2nþ1Dnx pðxÞ: ð7Þ

Here, Dx ¼ @2=@x21 þ @

2=@x22 is the Laplace operator, mn, n = 0,1, . . ., are the Poisson’s ratio-dependent coefficients from the

expansion (5).

3204 I. Argatov, G. Mishuris / Applied Mathematical Modelling 35 (2011) 3201–3212

Keeping in (7) only the main asymptotic term, we obtain

Fig. 1.w1 andbetwee

ZZxKðx� yÞpðyÞdy ’ AhpðxÞ; ð8Þ

where the coefficient A is given by (6). The asymptotic representation (8) is in complete agreement with the approximatesolutions constructed in [7] following [8] and using Johnson’s approximation [9] in the case of a bonded compressible elasticlayer.

It should be observed that the coefficient A vanishes for an incompressible material. Thus, in the case m = 0.5, the asymp-totic expansion (7) yields the following first order asymptotic approximation:

ZZ

xKðx� yÞpðyÞdy ’ �Bh3DxpðxÞ; ð9Þ

where the coefficient B is given by (6).The asymptotic representations (8) and (9) can be used to construct asymptotic models of elastic contact for compressible

and incompressible coatings.

3. Contact problem for bonded thin compressible viscoelastic coatings

3.1. Formulation of the contact problem for thin compressible viscoelastic layers

Consider a frictionless contact between two thin linear viscoelastic layers firmly attached to rigid substrates shaped likeelliptic paraboloids. Introducing the Cartesian coordinate system (x1,x2,z), we write out the equations of the two coating sur-faces (n = 1,2) in the form

z ¼ ð�1ÞnunðxÞ; unðxÞ ¼ 2RðnÞ1

� ��1x2

1 þ 2RðnÞ2

� ��1x2

2; ð10Þ

assuming that the two coating-substrate systems occupy convex domains z 6 �u1(x) and z P u2(x). Here, RðnÞ1 and RðnÞ2 arethe principal radii of curvature of the nth coating surface at its apex.

In the undeformed state, the coatings are in contact at a single point chosen as the coordinate origin. In the deformedstate, the coatings come into contact with one another within the contact region x(t), which is not known a priori. We de-note the vertical approach of the rigid substrates by �d0(t), where t is a time variable.

Let wn(x, t) be the absolute values of the vertical displacement function of the boundary points of the nth coating layer(see Fig. 1). Then, the linearized contact condition that the coating layers do not penetrate one into another under the im-posed deformation can be written as follows:

d0ðtÞ �w1ðx; tÞ �w2ðx; tÞ 6 uðxÞ: ð11Þ

Here the function u(x) = u1(x) + u2(x) determines the initial gap between the contacting surfaces. In view of (10), we have

uðxÞ ¼ ð2R1Þ�1x21 þ ð2R2Þ�1x2

2; ð12Þ

where R1 and R2 are determined by the formula Ri ¼ Rð1Þi Rð2Þi =ðRð1Þi þ Rð2Þi Þ, i = 1,2. In what follows, we assume that R1 P R2

Note also that for the sake of simplicity, we assume that the elliptic paraboloids z = (�1)nun(x) are coaxial. In the generalcase, the parameters R1 and R2 can be determined using the usual approach of contact mechanics [9].

F(t)

F(t)

a(t)

w1

w2

0(t)

x1

1

2

δ

Schematic diagram of the contact of coating surfaces 1 and 2 under load F(t). The dashed lines imply the surfaces’ profiles in the undeformed state;w2 are local deformations of the contacting surfaces 1 and 2, respectively; d0(t) is the relative approach between the rigid substrates. The differencen the sum w1 + w2 and d0(t) is due to the initial gap between the contacting surfaces.

I. Argatov, G. Mishuris / Applied Mathematical Modelling 35 (2011) 3201–3212 3205

The equality in (11) determines the contact region x(t) with the boundary contour C(t), that is

w1ðx; tÞ þw2ðx; tÞ ¼ d0ðtÞ �uðxÞ; x 2 xðtÞ: ð13Þ

For elastic compressible coatings, according to Eqs. (2) and (8), we will have

wnðx; tÞ ¼ #�1n Anhnpðx; tÞ: ð14Þ

Here we introduced the notation

#n ¼En

2ð1� m2nÞ; An ¼

1� 2mn

2ð1� mnÞ2: ð15Þ

Substituting the expressions (14), n = 1, 2, into Eq. (13), we obtain the contact condition in the following form:

#�11 A1h1 þ #�1

2 A2h2� �

pðx; tÞ ¼ d0ðtÞ �uðxÞ: ð16Þ

Eq. (16) allows to find the contact pressure density p(x, t) in the elastic case. The contour C(t) of the contact area x(t) isdetermined from the condition that the contact pressure is positive in the domain x(t) and vanishes at its boundary (see(3) and (4)), that is

pðx; tÞ > 0; x 2 xðtÞ; pðx; tÞ ¼ 0; x 2 CðtÞ: ð17Þ

Note that (17) implies that CðtÞ ¼ xðtÞ nxðtÞ.The equilibrium equation for the whole system is

ZZ

xðtÞpðx; tÞdx ¼ FðtÞ; ð18Þ

where F(t) denotes the external load.In the case of viscoelastic compressible coatings, for the sake of simplicity, we assume that Poisson’s ratios m1 and m2 are

time independent. Then, applying the viscoelastic correspondence principle to the associated elastic Eq. (16), we arrive at thefollowing governing integral equation:

hE1

Z t

0�Uaðt � sÞ @p

@sðx; sÞds ¼ d0ðtÞ �uðxÞHðtÞ: ð19Þ

Here, h = h1 + h2 is the joint thickness, HðtÞ is the Heaviside step function such that HðxÞ ¼ 1 for x > 0 and HðxÞ ¼ 0 for x 6 0,E1 is a harmonic mean of the relaxed elastic moduli E11 and E12 , Ua(t) is the compound creep function given by

UaðtÞ ¼ a1U1ðtÞ þ a2U2ðtÞ; ð20Þ

a1 ¼2E12 h1

eA1ðm1ÞðE11 þ E12 Þðh1 þ h2Þ

; a2 ¼2E11 h2

eA2ðm2ÞðE11 þ E12 Þðh1 þ h2Þ

ð21Þ

E1 ¼2E11 E12

E11 þ E12; eAnðmnÞ ¼

ð1þ mnÞð1� 2mnÞ1� mn

; n ¼ 1;2:

If the layer’s materials follows a standard linear viscoelastic solid model, we will have

UnðtÞ ¼ 1� ð1� qnÞ expð�t=TnÞ; ð22Þ

where Tn is the characteristic relaxation time of strain under applied step of stress, qn is the ratio of E1n to the unrelaxed elas-tic modulus E0

n (modulus En(t) at t = 0), i.e., qn ¼ E1n =E0n.

Finally note that for non-decreasing loads when dF(t)/dt P 0, the contact zone, generally speaking, increases. Thus, weassume that the following monotonicity condition holds: x(t1) �x(t2) for t1 6 t2.

In the next section, we derive an exact solution of the contact problem for thin compressible viscoelastic layers formu-lated by Eq. (19) in the case (12) under the monotonicity condition.

3.2. General solution for the case of elliptical contact: compressible coatings

According to (17), the contact domain x(t) coincides with the region where the right-hand side of Eq. (19) is positive. Inview of (12), we find that x(t) is an elliptical domain with the semi-axes a(t) and b(t) such that aðtÞ=bðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiR1=R2

pand

aðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2R1d0ðtÞ

p; bðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2R2d0ðtÞ

p: ð23Þ

Integrating the both sides of Eq. (19) over the contact domain and taking into account (17), (18) and (23), we obtain

hE1

Z t

0�Uaðt � sÞ dF

dsðsÞds ¼ p

ffiffiffiffiffiffiffiffiffiffiR1R2

pd2

0ðtÞ: ð24Þ

From here it follows that

3206 I. Argatov, G. Mishuris / Applied Mathematical Modelling 35 (2011) 3201–3212

d0ðtÞ ¼h

pffiffiffiffiffiffiffiffiffiffiR1R2p

E1

Z t

0�Uaðt � sÞ dF

dsðsÞds

� �1=2

: ð25Þ

By inverting Eq. (24), we readily get

FðtÞ ¼ ph

ffiffiffiffiffiffiffiffiffiffiR1R2

pE1

Z t

0�Waðt � sÞ d

dsd2

0ðsÞ� �

ds: ð26Þ

Here, Wa(t) is the compound relaxation function determined by the relaxation functions W1(t) and W2(t) of the viscoelasticcompressible coatings as follows (see Eq. (20)):

WaðtÞ ¼1a1

W1ðtÞ þ1a2

W2ðtÞ: ð27Þ

Note that if the layer’s materials follows a standard linear viscoelastic solid model, we will have

WnðtÞ ¼ 1� ð1� 1=qnÞ expð�t=ðqnTnÞÞ: ð28Þ

Using the Heaviside step function HðtÞ, we can represent the contact pressure distribution as follows:

pðx; tÞ ¼ E1h

Z t

0�Waðt � sÞ @

@sd0ðsÞ 1� x2

1

a2ðsÞ �x2

2

b2ðsÞ

!H 1� x2

1

a2ðsÞ �x2

2

b2ðsÞ

!( )ds: ð29Þ

Note that Eqs. (24), (26) and (29) can be further simplified by integration by parts.

3.3. Case of stepwise loading: compressible coatings

Let us assume that

FðtÞ ¼ F0HðtÞ: ð30Þ

Then, Eq. (29) yields

d0ðtÞ ¼hF0

pE1ffiffiffiffiffiffiffiffiffiffiR1R2p UaðtÞ

� �1=2

HðtÞ: ð31Þ

In view of (22), the function d0(t) determined by Eq. (31) is monotonically increasing. Thus, the elliptical contact domain willbe also monotonically increasing according to Eq. (23).

It is clear that if the point x belongs to the initial contact domain x(0+), i.e., 1� x21=a2ð0þÞ � x2

2=b2ð0þÞ > 0, then formula(29) simplifies to

pðx; tÞ ¼ E1h

Z t

0�Waðt � sÞ @

@sd0ðsÞ �uðxÞHðsÞf gds; ð32Þ

where u(x) is given by (12).Finally, Eq. (32) for t > 0 yields

hE1

pðx; tÞ ¼ WaðtÞ d0ð0þÞ �uðxÞð Þ þZ t

0þWaðt � sÞdd0ðsÞ

dsds; ð33Þ

where in view of (31) we have

d0ð0þÞ ¼hF0Uað0ÞpE1

ffiffiffiffiffiffiffiffiffiffiR1R2p

� �1=2

; ð34Þ

dd0ðsÞds

¼ hF0

2pE1ffiffiffiffiffiffiffiffiffiffiR1R2p 1

d0ðsÞdUaðsÞ

ds: ð35Þ

If the point x lies outside of the initial contact zone x(0+), i.e., 1� x21=a2ð0þÞ � x2

2=b2ð0þÞ < 0, then formula (29) can berewritten as

hE1

pðx; tÞ ¼ Wað0Þd0ðtÞ 1� x21

a2ðtÞ �x2

2

b2ðtÞ

!H 1� x2

1

a2ðtÞ �x2

2

b2ðtÞ

!�Z t

t�ðxÞ

@Wa

@sðt � sÞ d0ðsÞ �uðxÞð Þds; ð36Þ

where t⁄(x) is the time when the contour of the contact zone first reaches the point x. The quantity t⁄(x) is determined by theequation

d0ðt�Þ ¼x2

1

2R1þ x2

2

2R2: ð37Þ

Further, integrating by parts in (36), we obtain

I. Argatov, G. Mishuris / Applied Mathematical Modelling 35 (2011) 3201–3212 3207

hE1

pðx; tÞ ¼ Waðt � t�ðxÞÞ d0ðt�ðxÞÞ �uðxÞð Þ þZ t

t�ðxÞWaðt � sÞdd0ðsÞ

dsds: ð38Þ

Finally, according to Eqs. (31) and (37) can be reduced to the following equation:

Uaðt�Þ ¼pE1

ffiffiffiffiffiffiffiffiffiffiR1R2p

hF0

x21

2R1þ x2

2

2R2

� �2

: ð39Þ

From here it follows that

t�ðxÞ ¼ U�1a

pE1ffiffiffiffiffiffiffiffiffiffiR1R2p

hF0

x21

2R1þ x2

2

2R2

� �2 !

; ð40Þ

where U�1a ðxÞ ¼ t is the inverse function with respect to Ua(t) = x.

In the case of coating layers of the same viscoelastic material following the standard viscoelastic solid model in view of(20) and (22), we will have

UaðtÞ ¼ ða1 þ a2Þ½1� ð1� qÞ expð�t=TÞ�;

and, consequently,

U�1a ðxÞ ¼ T ln½ð1� qÞ=ð1� x=ða1 þ a2ÞÞ�: ð41Þ

Thus, formulas (40) and (41) solve Eq. (39).

3.4. Axisymmetric contact problem for compressible coatings

Let us now set R1 = R2 = R. Then, Eqs. (31) and (23), respectively, take the form

d0ðtÞ ¼ d0ð0þÞUaðtÞUað0Þ

� �1=2

HðtÞ; aðtÞ ¼ að0þÞ UaðtÞUað0Þ

� �1=4

HðtÞ; ð42Þ

where

d0ð0þÞ ¼hF0Uað0ÞpE1R

� �1=2

; að0þÞ ¼ 4RhF0Uað0ÞpE1

� �1=4

: ð43Þ

According to Eq. (29), the axisymmetric contact pressure distribution is given by

pðr; tÞ ¼ E12Rh

Z t

0�Waðt � sÞ @

@sa2ðsÞ � r2� �

HðaðsÞ � rÞ

ds; ð44Þ

where r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

1 þ x22

qis the radial coordinate.

Observe that Eq. (44) differs from the analogous equation derived in [12] (see Section 2.2.1, Eqs. (19) and (21), where thefactor HðsÞ is used instead of HðaðsÞ � rÞ, as in our Eq. (44)).

4. Contact problem for bonded thin incompressible viscoelastic coatings

4.1. Formulation of the contact problem for thin incompressible viscoelastic layers

Now, according Eqs. (2) and (9), we have

wnðx; tÞ ¼ �#�1n Bnh3

nDxpðx; tÞ; ð45Þ

where

Bn ¼mnð4mn � 1Þ6ð1� mnÞ3

: ð46Þ

Substituting the expressions (45), n = 1,2, into Eq. (13), we arrive at the elastic contact condition

� #�11 B1h3

1 þ #�12 B2h3

2

� �Dxpðx; tÞ ¼ d0ðtÞ �uðxÞ: ð47Þ

As before, we require that the contact pressure distribution p(x, t) should satisfy the positivity condition (17). Moreover, fol-lowing [7,6], we additionally assume that there is a smooth transition of the contact pressure from the contact regionx 2x(t) to the outside region x R x(t). In other words, we impose the following boundary condition:

@p@nðx; tÞ ¼ 0; x 2 CðtÞ: ð48Þ

3208 I. Argatov, G. Mishuris / Applied Mathematical Modelling 35 (2011) 3201–3212

Here, @/@n is the normal derivative. Note that for determining the initial contact radius in the case of axisymmetric contact,Hlavácek [19] suggested a new approach, which does not make use of the zero pressure gradient condition (48) and is basedon the analytical solutions of the axisymmetric contact problem for an incompressible single-phasic elastic layer [5].

In the case of viscoelastic incompressible coatings with time independent Poisson’s ratios, Eq. (48) generates the follow-ing governing integral equation:

� h3

E1

Z t

0�Ubðt � sÞDx

@p@sðx; sÞds ¼ d0ðtÞ �uðxÞHðtÞ: ð49Þ

Here, Ub(t) is the compound creep function determined by the formulas

UbðtÞ ¼ b1U1ðtÞ þ b2U2ðtÞ; ð50Þ

b1 ¼2E12 h3

1eB1ðm1Þ

E11 þ E12� �

ðh1 þ h2Þ3; b2 ¼

2E11 h32eB2ðm2Þ

E11 þ E12� �

ðh1 þ h2Þ3; ð51Þ

eBnðmnÞ ¼mnð1þ mnÞð4mn � 1Þ

3ð1� mnÞ2; n ¼ 1;2:

In the next section, we derive an exact solution of the contact problem for thin incompressible viscoelastic layers formulatedby Eq. (49) in the case (12) under the monotonicity condition.

4.2. General solution for the case of elliptical contact: incompressible coatings

Let us rewrite Eq. (49) in the form

DxPðx; tÞ ¼ m uðxÞHðtÞ � d0ðtÞð Þ; x 2 xðtÞ; ð52Þ

where we introduced the notation

Pðx; tÞ ¼Z t

0�Ubðt � sÞ @p

@sðx; sÞds; m ¼ E1

h3 : ð53Þ

In view of (17) and (48), the function Pðx; tÞ must satisfy the following boundary conditions:

Pðx; tÞ ¼ 0; x 2 CðtÞ; @P@nðx; tÞ ¼ 0; x 2 CðtÞ: ð54Þ

In the case (12), the right-hand side of Eq. (52) is a quadratic polynomial in variables x1 and x2, and we put

Pðx; tÞ ¼ P0ðtÞ 1� x21

a2ðtÞ �x2

2

b2ðtÞ

!2

; x 2 xðtÞ: ð55Þ

It is not hard to check that the function (55) satisfies the boundary conditions (54) exactly.The representation (55) assumes that the contour C(t) is an ellipse with the semi-axes a(t) and b(t). Moreover, for x R x(t),

we should have Pðx; tÞ ¼ 0. Thus, the function Pðx; tÞ determined by (55) for x 2x(t) can be extended on the entire plane ofvariable x = (x1,x2) by the multiplication of the right-hand side of (55) by the factor H 1� x2

1=a2ðtÞ � x22=b2ðtÞ

� �.

Substituting (55) into Eq. (52), we obtain after some algebra the following system of algebraic equations:

d0ðtÞ ¼4P0ðtÞ

m1

a2ðtÞ þ1

b2ðtÞ

!; ð56Þ

12R1¼ 4P0ðtÞ

ma2ðtÞ3

a2ðtÞ þ1

b2ðtÞ

!; ð57Þ

12R2¼ 4P0ðtÞ

mb2ðtÞ1

a2ðtÞ þ3

b2ðtÞ

!: ð58Þ

The form of the ellipse C0(t) can be characterized by its aspect ratio s defined as follows:

s ¼ bðtÞaðtÞ : ð59Þ

From Eqs. (57) and (58), it immediately follows that

R2

R1¼ s2ð3s2 þ 1Þ

3þ s2 : ð60Þ

I. Argatov, G. Mishuris / Applied Mathematical Modelling 35 (2011) 3201–3212 3209

Eq. (60) can be reduced to a quadratic equation for s2. In this way one can obtain

s2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR1 � R2

6R1

� �2

þ R2

R1

s� ðR1 � R2Þ

6R1: ð61Þ

Further, Eq. (56) in view of (57)–(59) takes the form

d0ðtÞ ¼a2ðtÞ2R1

ðs2 þ 1Þð3s2 þ 1Þ : ð62Þ

From Eq. (57) it follows that

P0ðtÞ ¼ma4ðtÞ

8R1

s2

ð3s2 þ 1Þ : ð63Þ

Finally, in view of (18) and (53), the contact force F(t) satisfies the equation

Z t

0�Ubðt � sÞdFðsÞ

dsds ¼ pm

24R1

s3

ð3s2 þ 1Þ a6ðtÞ: ð64Þ

Thus, Eq. (64) allows to determine the major semi-axis a(t) of the contact domain x(t) as a function of time t as follows:

aðtÞ ¼ pm24R1

s3

ð3s2 þ 1Þ

� ��1=6 Z t

0�Ubðt � sÞdFðsÞ

dsds

� �1=6

: ð65Þ

Now, formulas (62) and (63) allow to determine the quantities d0(t) and P0ðtÞ, respectively.Let us denote by Wb(t) the compound relaxation function for the viscoelastic incompressible coatings, i.e.,

WbðtÞ ¼1b1

W1ðtÞ þ1b2

W2ðtÞ: ð66Þ

Then, according to Eqs. (53) and (55), the contact pressure distribution is given by

pðx; tÞ ¼Z t

0�Wbðt � sÞ @

@sP0ðsÞ 1� x2

1

a2ðsÞ �x2

2

b2ðsÞ

!2

H 1� x21

a2ðsÞ �x2

2

b2ðsÞ

!8<:9=;ds: ð67Þ

Integration by parts in (67) yields

pðx; tÞ ¼ Wbð0ÞP0ðtÞ 1� x21

a2ðtÞ �x2

2

b2ðtÞ

!2

�Z t

0þ

@Wb

@s ðt � sÞP0ðsÞ 1� x21

a2ðsÞ �x2

2

b2ðsÞ

!2

H 1� x21

a2ðsÞ �x2

2

b2ðsÞ

!ds: ð68Þ

Recall that formula (68) assumes that p(x, t) = 0 for x R x(t).Denoting by t⁄(x) the time when the contour of the contact zone first reaches the point x, we rewrite formula (68) as

follows:

pðx; tÞ ¼ Wbð0ÞP0ðtÞ 1� x21

a2ðtÞ �x2

2

b2ðtÞ

!2

�Z t

t�ðxÞ

@Wb

@sðt � sÞP0ðsÞ 1� x2

1

a2ðsÞ �x2

2

b2ðsÞ

!2

ds: ð69Þ

The quantity t⁄(x) is determined by the equation

a2ðt�Þ ¼ x21 þ

x22

s2 : ð70Þ

Substituting the expression (65) into Eq. (70), we arrive at the following equation:

Z t�

0�Ubðt� � sÞdFðsÞ

dsds ¼ pm

24R1

s3

ð3s2 þ 1Þ x21 þ

x22

s2

� �3

: ð71Þ

Finally note that formula (69) assumes that t⁄(x) = 0 for the points of the initial contact domain x(0+).

4.3. Case of stepwise loading: incompressible coatings

Assuming that

FðtÞ ¼ F0HðtÞ; ð72Þ

and using Eqs. (62) and (65), we find

3210 I. Argatov, G. Mishuris / Applied Mathematical Modelling 35 (2011) 3201–3212

d0ðtÞ ¼ d0ð0þÞUaðtÞUað0Þ

� �1=3

HðtÞ; ð73Þ

aðtÞ ¼ að0þÞ UaðtÞUað0Þ

� �1=6

HðtÞ; ð74Þ

P0ðtÞ ¼ P0ð0þÞUaðtÞUað0Þ

� �2=3

HðtÞ; ð75Þ

where

d0ð0þÞ ¼3

pmR21

!1=3ðs2 þ 1Þ

sð3s2 þ 1Þ2=3 F0Ubð0Þ� �1=3

; ð76Þ

að0þÞ ¼ pm24R1

s3

ð3s2 þ 1Þ

� ��1=6

F0Ubð0Þ� �1=6

; ð77Þ

P0ð0þÞ ¼9mp2R1

� �1=3 s2

s2ð3s2 þ 1Þ1=3 F0Ubð0Þ� �2=3

: ð78Þ

Eq. (77) determines the major semi-axis of the initial contact domain x(0+). For x lying outside x(0+), the quantity t⁄(x) sat-isfies Eq. (70). From Eqs. (77) and (70), it follows that

t�ðxÞ ¼ U�1b

Ubð0Þa6ð0þÞ x1 þ

x22

s2

� �2 !

; ð79Þ

where U�1b ðxÞ ¼ t is the inverse function with respect to Ub(t) = x.

4.4. Axisymmetric contact problem for incompressible coatings

Let us now set R1 = R2 = R. Consequently, we obtain s = 1, and Eqs. (76)–(78), respectively, take the form

d0ð0þÞ ¼3

2pmR21

!1=3

F0Ubð0Þ� �1=3

; ð80Þ

að0þÞ ¼ pm96R1

� ��1=6

F0Ubð0Þ� �1=6

; ð81Þ

P0ð0þÞ ¼18mp2R1

� �1=3

F0Ubð0Þ� �2=3

: ð82Þ

According to Eq. (69), the axisymmetric contact pressure distribution is given by

pðr; tÞ ¼ Wbð0ÞP0ðtÞ 1� r2

a2ðtÞ

� �2

�Z t

t�ðrÞ

@Wb

@sðt � sÞP0ðsÞ 1� r2

a2ðsÞ

� �2

ds; ð83Þ

where r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

1 þ x22

qis the radial coordinate, and in view of (79)

t�ðrÞ ¼ U�1b

Ubð0Þa6ð0þÞ r

6� �

: ð84Þ

Observe that Eq. (83) equation derived in [12] (see Section 2.2.3, Eq. (28), where the factorHðsÞ is used instead ofHðaðsÞ � rÞ,as our Eq. (83)) elaborates.

Formulas (80)–(84) are valid for the stepwise loading (72). In the general case of axisymmetric contact problem, the solu-tion can be easily derived using formulas (62), (65), (69) and (71) with s = 1 and x2

1 þ x22 ¼ r2.

4.5. Contact of incompressible coatings in the framework of the Maxwell model

The series combination of an elastic spring and a dashpot represents the Maxwell model, which is characterized by thefollowing non-normalized creep and relaxation functions:

UðtÞ ¼ 1E0

1þ ts0

� �; WðtÞ ¼ E0e�t=s0 : ð85Þ

Here, E0 is the instantaneous elastic modulus, s0 is a relaxation time. For the sake of brevity in notation only, we do notintroduce new notation to distinguish between the non-normalized functions (85) and the dimensionless normalized ones1 + t/s0 and e�t=s0 . Note that the concept of the relaxed state for the Maxwell model is not applicable.

I. Argatov, G. Mishuris / Applied Mathematical Modelling 35 (2011) 3201–3212 3211

Taking into account that p(x,t) = 0 for t 6 0, we can write Eq. (49) as

Dxpðx; tÞ þ 1s0

Z t

0Dxpðx; sÞds ¼ E0

h3 uðxÞHðtÞ � d0ðtÞð Þ: ð86Þ

Observe that for m = 0.5, i.e., in the case of incompressible viscoelastic material, we have E0 = 3G0, where G0 is the instanta-neous shear modulus. Thus, Eq. (86) coincides with the resulting governing integral equation derived as a result of asymp-totic analysis in [14] for describing the short-time contact of thin biphasic cartilage layers.

In view of (85), formula (69) takes the form

pðx; tÞ ¼ P0ðtÞ 1� x21

a2ðtÞ �x2

2

b2ðtÞ

!2

� 1s0

Z t

t�ðxÞ1� x2

1

a2ðsÞ �x2

2

b2ðsÞ

!2

P0ðsÞe�ðt�sÞ=s0 ds: ð87Þ

Formula (84) coincides with the accuracy up to the notation with the solution obtained in [17].

5. Conclusion

The present study results in the exact closed-form solution to the three-dimensional contact problem of elliptical contactfor bonded thin viscoelastic compressible or incompressible layers. The obtained analytical solution is valid for monotoni-cally increasing loading conditions.

As it was observed earlier [20], the asymptotic solution [14] was constructed for the small time scales, and the modelsdeveloped in papers [17,15], which are also based on the asymptotic results [14], are not valid for long times compared withthe typical diffusion time. As a consequence of this, the asymptotic models [17,15] cannot describe the behavior of articularcartilage layers near the compaction point, where all pore fluid is squeezed out and the load is fully carried by the extracel-lular matrix. In the present work, extending the previously developed analytical method for the elliptical contact of thin bi-phasic cartilage layers, we obtained the general solution to the problem of elliptical contact between thin layers of arbitraryviscoelastic materials obtained. Thus, taking the appropriate viscoelastic law from experiments on articular cartilage layers,one can develop different mathematical models that are capable to reflect the behavior of articular cartilage layers near thecompaction point.

In the case of compressible coatings, the elliptical contact domain has the aspect ratio s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiR2=R1

p. In the case of incom-

pressible coatings, the aspect ratio given by (61) is greater. It should be also noted that the constructed asymptotic modelshold true only when the layer thicknesses are small compared with the characteristic size of the contact area, that ismax{h1,h2} < < a0, where a0 = a(0+) is the initial value of the major semi-axis in the case of a stepwise loading. That iswhy, the contact force F0 (and, generally, F(0)) should not take too small values in order do not violate the thin-layerassumption.

Acknowledgements

I.A. gratefully acknowledges the financial support from the European Union Seventh Framework Programme through theMarie Curie Fellowship PIIF-GA-2009-253055. The authors would like to express their gratitude to the anonymous refereesfor their helpful comments.

References

[1] J. Mackerle, Coatings and surface modification technologies: a finite element bibliography (1995–2005), Model. Simul. Mater. Sci. Eng. 13 (2005) 935–979.

[2] E.K. Dimitriadis, F. Horkay, J. Maresca, B. Kachar, R.S. Chadwick, Determination of elastic moduli of thin layers of soft material using the atomic forcemicroscope, Biophys. J. 82 (2002) 2798–2810.

[3] J.D. Humphrey, Review Paper: Continuum biomechanics of soft biological tissues, Proc. R. Soc. Lond. A 459 (2003) 3–46.[4] V.M. Aleksandrov, Asymptotic solution of the contact problem for a thin elastic layer, J. Appl. Math. Mech. 33 (1969) 49–63.[5] M.J. Matthewson, Axi-symmetric contact on thin compliant coatings, J. Mech. Phys. Solids 29 (1981) 89–113.[6] R.S. Chadwick, Axisymmetric indentation of a thin incompressible elastic layer, SIAM J. Appl. Math. 62 (2002) 1520–1530.[7] J.R. Barber, Contact problems for the thin elastic layer, Int. J. Mech. Sci. 32 (1990) 129–132.[8] M.J. Jaffar, Asymptotic behaviour of thin elastic layers bonded and unbonded to a rigid foundation, Int. J. Mech. Sci. 31 (1989) 229–235.[9] K.L. Johnson, Contact Mechanics, Cambridge Univ. Press, Cambridge, UK, 1985.

[10] I.I. Argatov, The pressure of a punch in the form of an elliptic paraboloid on a thin elastic layer, Acta Mech. 180 (2005) 221–232.[11] M. Hlavácek, Elliptical contact on elastic incompressible coatings, Eng. Mech. 15 (2008) 249–261.[12] G.R. Naghieh, Z.M. Jin, H. Rahnejat, Contact characteristics of viscoelastic bonded layers, Appl. Math. Model. 22 (1998) 569–581.[13] G.R. Naghieh, H. Rahnejat, Z.M. Jin, Characteristics of frictionless contact of bonded elastic and viscoelastic layered solids, Wear 232 (1999) 243–

249.[14] G.A. Ateshian, W.M. Lai, W.B. Zhu, V.C. Mow, An asymptotic solution for the contact of two biphasic cartilage layers, J. Biomech. 27 (1994) 1347–1360.[15] J.Z. Wu, W. Herzog, J. Ronsky, Modeling axi-symmetrical joint contact with biphasic cartilage layers—an asymptotic solution, J. Biomech. 29 (1996)

1263–1281.[16] I. Argatov, G. Mishuris, Axisymmetric contact problem for a biphasic cartilage layer with allowance for tangential displacements on the contact surface,

Eur. J. Mech. A/Solids 29 (2010) 1051–1064.[17] I. Argatov, G. Mishuris, Elliptical contact of thin biphasic cartilage layers: Exact solution for monotonic loading, J. Biomech., doi:10.1016/

j.jbiomech.2010.11.010.

3212 I. Argatov, G. Mishuris / Applied Mathematical Modelling 35 (2011) 3201–3212

[18] I.N. Sneddon, Fourier Transforms, Dover, New York, 1995.[19] M. Hlavácek, A note on an asymptotic solution for the contact of two biphasic cartilage layers in a loaded synovial joint at rest, J. Biomech. 32 (1999)

987–991.[20] S.I. Barry, M. Holmes, Asymptotic behaviour of thin poroelastic layers, IMA J. Appl. Math. 66 (2001) 175–194.