Contact with stick zone between an indenter and a thin incompressible layer

European Journal of Mechanics A/Solids 30 (2011) 884e892

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Contact with stick zone between an indenter and a thin incompressible layer P. Malits* PERI, Physics and Engineering Research Institute, Ruppin Academic Center, Emek Hefer 40250, Israel

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 November 2010 Accepted 22 April 2011 Available online 12 May 2011

The dominant asymptotic term for the indentation of a thin elastic incompressible layer by an axisymmetric rigid indenter is considered. Complete adhesion is supposed everywhere in the contact area or else in a given inner region surrounded by an annular frictionless zone. Both the problems are formulated in the form of systems of coupled dual integral equations. Using operators transforming kernels of the Hankel transform into kernels of the WebereOrr transform, the dual integral equations are reduced to systems of Fredholm integral equations of the second kind whose structures permit deriving asymptotic solutions. Simple expressions for the contact stresses, the penetration depth, and the contact radius in the case of an unknown contact area are obtained. Explicit formulae, derived for the flat and power law indenter profiles, allow us to analyze how stick and frictionless zones affect mechanical characteristics. Results manifest that the punch penetration exhibits strong sensitivity to contact conditions inspite of the fact that the radial traction is small. A conical indenter is less sensitive than flatended and spherical indenters.
2011 Elsevier Masson SAS. All rights reserved.

Keywords: Contact mechanics Adhesion Incompressible layer

1. Introduction In this paper, we consider the axisymmetric indentation problem of a rigid circular indenter in complete adhesion contact with an incompressible elastic layer when the layer thickness is much less than the extent of the stick zone. The layer, in the frame of the infinitesimal theory of elasticity, is supposed to be isotropic with shear modulus G. The lower surface of the layer z ¼ h is bonded to a rigid foundation. The indenter is pressed against the surface z ¼ 0 under action of the imbedding axial force P > 0. A contact area between the indenter and the solid is a circle of radius a with the centre at the origin. The complete adhesion zone occupies either the whole contact region as R ¼ a or an inner circle of given radius R < a surrounded by an annular frictionless zone. The model of an incompressible thin layer bonded to a rigid foundation arises in contact mechanics for coating/substrate systems when a coating material is very compliant (elastomer). Such coatings are used in various fields of modern engineering. Another field of applications is biomechanics. The frictionless contact problem for an incompressible thin layer was first attacked by McCormick (1978) and Matthewson (1981). Jaffar (1989) used an heuristic assumption due to Johnson (1985) to study the indentation by a frictionless rigid sphere, but his formula for total load is in error by a factor E ¼ 3G. This approach was generalized by Barber (1990) who has indicated the dominant * Hanaviim 12/3, Ashdod, 77473, Israel. E-mail address: [email protected]. 0997-7538/$ e see front matter
2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2011.04.010

asymptotic term of the solution for a rigid ellipsoid. Barber has also indicated the total indenting force for a flat indenter of arbitrary plan-form but his approach gives the incorrect zero contact pressure at the sharp corner of the indenter. The same solution for a flat-ended rigid cylinder was obtained later by Yang (1998) and La Ragione et al. (2008) in another way. Jaffar’s and Barber’s results were restudied with the asymptotic analysis based on the WienereHopf technique by Chadwick (2002) (for a rigid sphere) and Alexandrov (2003) (for a flat-ended cylindrical indenter). The rigorous method, based on a certain special technique for dual integral equations, was suggested in the paper by Malits (2006), where explicit asymptotic formulae for an axisymmetric punch of general profile were established. These asymptotic formulae, except the contact pressure for a flat-ended rigid cylinder, also validate Jaffar’s and Barber’s solutions. A more detailed investigation of the elliptical frictionless contact was given by Hlavácek (2008). The aforesaid results manifest that the asymptotic behaviour of a frictionless rigid indenter in contact with an incompressible thin layer is dramatically different from that for a thin compressible layer. An axisymmetric indenter in complete adhesion contact with a compressible elastic half-space was considered in a number of papers. This problem was first solved by Mossakovski (1954, 1963). The contact problems with stick and slip zones were studied by Spence (1975), Sundelius (1981), and Zhupanska (2009). Argatov (2010) has analyzed the adhesion contact problem of a spherical indenter pressed against a thick compressible layer on an elastic half-space.

P. Malits / European Journal of Mechanics A/Solids 30 (2011) 884e892

It is well known that for an incompressible half-space, unlike a compressible half-space, there is no coupling of both the normal and radial (tangential) contact stresses with each of the interface displacements. But for an incompressible elastic layer such a coupling is present and may significantly affect a solution, especially as a layer is thin. Keeping in mind peculiarities of the frictionless contact problem, one may anticipate to find new interesting effects. The indenting force/penetration depth relation for a rigid flat-ended cylinder bonded to an incompressible film was first found by Yang (1998), but this relation should be corroborated because Yang’s method for asymptotic treatment of dual integral equations gives the unrealistic finite contact traction. The axisymmetric contact problem with complete adhesion between an indenter of arbitrary profile and a thin incompressible layer is considered, to the best of our knowledge, for the first time.

2. Complete adhesion contact between a punch and a layer

ZN

~ðpÞðpÞJ0 ðprÞdp ¼ 0; ps

ZN

p~sðpÞJ1 ðprÞdp ¼ 0;

ur ðr; 0Þ ¼ uðrÞ;

sz ðr; 0Þ ¼ srz ðr; 0Þ ¼ 0;

(2)

R < r < N;

uz ðr; hÞ ¼ ur ðr; hÞ ¼ 0;

(3)

0 < r < N;

(4)

where the constant c is the penetration depth (punch indentation), c > 0, and the non-negative differentiable function wðrÞ is the indenter profile. For a non-flat indenter the explicit form of the radial displacement uðrÞ depends on a model of adhesion (for instance, the self-similarity approach for a compressive half-space by Spence (1968) and the recent contribution by Borodich and Keer (2004)). In a thin layer solution, all terms containing uðrÞ are, as it will be shown further, asymptotically small and do not contribute to the dominant asymptotic term if uðrÞ ¼ OðwðrÞ  cÞ. Since the latter estimate seems to be conventional for any possible case, we are allowed do not specify uðrÞ explicitly in order to study the thin layer/indenter interaction. Introduce the dimensionless coordinates r ¼ rR, z ¼ zR. Denoting sz ðr; 0Þ ¼ sðrÞ, srz ðr; 0Þ ¼ sðrÞ, and their Hankel trans~ðpÞ and ~sðpÞ; forms as s

ZN

sðrÞ ¼ 0

~ðpÞJ0 ðprÞdp; ps

sðrÞ ¼

ZN

p~sðpÞJ1 ðprÞdp;

1 < r < N;

(9)

where Jk ðpÞ is the Bessel function of the first kind, l ¼ h=R is the dimensionless thickness of the layer, and the pliability functions f ðplÞ, gðplÞ, and sðplÞ are given by

sin h2pl  2pl ; 2 cos h2pl þ 2p2 l þ 1

f0 ðplÞ ¼

sin h2pl þ 2pl

f1 ðplÞ ¼

; 2 cos h2pl þ 2p2 l þ 1

(10)

2

(1)

0  r  R;

(8)

0

sðplÞ ¼

Suppose complete adhesion be throughout the contact area. The boundary conditions of axisymmetric strain in the cylindrical coordinates ðr; q; zÞ are

0  r  R;

1 < r < N;

0

2.1. Reduction to regular equations

uz ðr; 0Þ ¼ wðrÞ  c;

885

2p2 l

2

cos h2pl þ 2p2 l þ 1

;

while the conditions (3) and (4) are satisfied. A peculiarity of the problem is caused by the asymptotic behaviour of the pliability functions as l/0

2 3 2 f0 ðplÞw p3 l ; f1 ðplÞw2pl; sðplÞwp2 l ; 3

(11)

that is quite different from the case of a compressible layer. By 2 ~ðpÞ ¼ Oðl3 Þ, one could see from (6) taking ~sðpÞ ¼ Oðl Þ and s and (7) that for l  1 an appropriate small radial stress sðrÞ may make essential or even dominant contributions to the surface displacements of the incompressible layer. We shall solve the dual equations by means of the operators transforming the kernels of the Hankel transforms Jk ðprÞ into the combinations of Bessel functions of the first and second kind cgnmn ðp; tÞ ¼ Ym ðptÞJn ðpgn Þ  Yn ðpgn ÞJm ðptÞ, that are the kernels of the WebereOrr integral transforms

ZN 6ðtÞ ¼

g

^ ðpÞcnm ðp; tÞ pu dp ; þ Yn2 ðpgÞ n

1 2

0

ZN 6ðpÞ ^ ¼

1 2

n ¼ mþ  ;

J 2 ðpgÞ

(12) g s6ðsÞcnm ðp; tÞds:

g

Namely, g

R0 ½J0 ðprÞ ¼ pc210 ðp; tÞ; 2

d R0 ½$ ¼ 2 2 pg0 t dt

t Z g0

0

(5)

g0 >0;

(13)

h i 2 t 2 þ g20  r2 2g20 t 2 rdr ð$Þrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    ðt þ g0 Þ2 r2 ðt  g0 Þ2 r2

(14)

and

0

g

one is permitted to reformulate the boundary conditions (1) and (2) in the form of the coupled dual integral equations

R1 ½J1 ðprÞ ¼ pc111 ðp; tÞ;

ZNh i 2G ~ðpÞf0 ðplÞ J0 ðprÞdp ¼ ½cwðrRÞ; 0 r 1; ~sðpÞsðplÞ s R

2 d R1 ½$ ¼  pg1 dt

(6)

t Z g1

0

g1 >0

(15)





r2 þ g21  t 2 dr

ð$Þrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   : ðt þ g1 Þ2 r2 ðt  g1 Þ2 r2

(16)

0

ZNh i 2G ~ðpÞsðplÞ J1 ðprÞdp ¼ ~sðpÞf1 ðplÞ  s uðrRÞ; 0  r  1; R 0

(7)

The above operators were first introduced in the papers Malits (2006) and Malits (2005), respectively. Readers can find a rigorous theory of such operators and applications to dual integral equations in Malits (2007).

886

P. Malits / European Journal of Mechanics A/Solids 30 (2011) 884e892

In this paper, we take



g0 ¼

16 3p

1 3

4

l;

g1 ¼ l : p

(17)

On applying the operator R0 to (6) and the operator R 1 to (7) these equations become

ZNh

i

s~ðpÞf0 ðplÞ  ~sðpÞsðplÞ pcg210 ðp; tÞdp ¼

2G R ½wðRrÞ  c; R 0

Substitute (20) and (22) into (18) and (19). Taking into account the inversion formula for WebereOrr transforms (12), we obtain the following system of Fredholm integral equations of the second kind

u0 ðtÞ þ L00 u0 þ L01 u1 ¼ j0 ðtÞ;

g0  t  a0 ;

(26)

u1 ðtÞ þ L11 u1 þ L10 u0 ¼ j1 ðtÞ;

g1  t  a1 ;

(27)

0

g0  t  1 þ g0 ;

(18)

ZNh

i

2G R ½uðRrÞ; R 1

s~ðpÞsðplÞ  ~sðpÞf1 ðplÞ pcg111 ðp; tÞdp ¼ 

with the right parts and the integral operators defined as following

j0 ðtÞ ¼

2ct

p

0

g1  t  1 þ g1 ;

(19)

The solution of the system of the dual integral equations (18), (19), (8) and (9) is sought in the form

g0 Rs~ðpÞ ¼ p pG

Za0

g

xu0 ðxÞc210 ðp; xÞdx

(20)

Za0 g0

  u ðxÞ g ; x2 c220 ðp;xÞd 0 x

Lnm u ¼

Za1

L01 ðt;xÞ ¼

(21)

g

cg101 ðp; xÞdðxu1 ðxÞÞ;

0

g

g

p2 sðplÞc111 ðp;xÞc210 ðx;tÞdx;

(30)

0

(31)

  cg210 ðp; tÞcg210 ðx; xÞ 2 2 l ; ¼ O p J22 ðxg0 Þ þ Y22 ðxg0 Þ

(23)

  pg20 2 p g g 2 p sðplÞc111 ðp; tÞc210 ðx; xÞ ¼ pJ1 ðptÞJ1 ðpxÞ þ O p2 l : 2g1 4

2.2. Penetration depth and contact stresses In order to find relations between the penetration depth and the indenting load P, we use the equilibrium condition ~ð0Þ ¼ P that upon substituting (20) becomes 2pR2 s

g

i h 2 2 x2 þ g2  r2 2g2 x2 Hðx  g  rÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼   ; pg2 x2 r2  ðx þ gÞ2 r2  ðx  gÞ2

P ¼ (24)

  2 r2 þ g2  x2 Hðx  g  rÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    pgr r2  ðx þ gÞ2 r2  ðx  gÞ2

þ

gn n cg2n;1 ðp; tÞc2n;1 ðx; xÞ dx; 2 2 J2n ðxg0 Þ þ Y2n ðxg0 Þ

  cg1 ðp; tÞcg111 ðx; xÞ 2 2 l l F11 ðplÞ 2 11 lnðp Þ ; ¼ O p J1 ðxg1 Þ þ Y12 ðxg1 Þ

pc22 ðp; xÞJ0 ðprÞdp

pc10 ðp; xÞJ1 ðprÞdp ¼

pFnn ðplÞ 0

F00 ðplÞ

where an ¼ 1 þ gn and un ðsÞ are certain auxiliary functions possessing an integrable derivative. It will be shown further that the above representations lead to integrable contact stresses. Insert (21) and (23) into (8) and (9). On interchanging the order of integration one can ascertain by means of the discontinuous integrals (see Malits (2005, 2006 and 2007))

g

ZN

(22)

g1

g1

ZN

g30 pg2 L10 ðt;xÞ ¼ 0 3 2g1 g1

(29)

2 ðxg Þ þ Y 2 ðxg Þ  2. where 2Fnn ðplÞ ¼ ppgn fn ðplÞ½J2n 0 0 2n For l/0 the integrands in the integral representations of Lnm ðt; xÞ have the estimates

xu1 ðxÞc111 ðp; xÞdx

Za1

0

xuðxÞLnm ðt; xÞdx; gn

ZN

g1 R~sðpÞ 2u1 ðg1 Þ g ¼  a1 u1 ða1 Þc101 ðp; a1 Þ pG pp

ZN

1þ Z gn

Lnn ðt; xÞ ¼

and

þ

(28)

go

g0 Rs~ðpÞ g ¼ a0 u0 ða0 Þc220 ðp; a0 Þ þ pG

g1 R~sðpÞ ¼ p pG

 g20 R0 ½wðRrÞ; j1 ðtÞ ¼ g21 R 1 ½uðrRÞ;

2

;

pgr

4pGR

g30

s2 u0 ðsÞds;

(32)

g0

The distributions of the contact stresses are evaluated on inserting the substitutions (21) and (23) into the inversion formulae (5). We readily obtain with use of the integrals (24) and (25)



2

g30 R 2g2 ð1þ g0 Þ2  1 r2 þ2g0 ð1þ g0 Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sðrÞ ¼ 0 iffi u0 ð1þ g0 Þ  h 2G ð1þ g0 Þ 1 r2 ð1þ2g0 Þ2 r2

(25)

where HðxÞ is the Heaviside step function, that these equations are satisfied identically.

Za0

1þ Z g0

þ and

rþg0

 2 2   s þ g20  r2 2g20 s2 u0 ðsÞ ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (33) d    s ðsþ g0 Þ2 r2 ðs g0 Þ2 r2

P. Malits / European Journal of Mechanics A/Solids 30 (2011) 884e892

  Rg21 1þ2g1  r2 ð1þ g1 Þ sðrÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i u1 ð1þ g1 Þ  h 2G r 1 r2 ð1þ2g1 Þ2 r2 1þ Z g1

1



r

rþg1

s2  r2  g21 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    dðsu1 ðsÞÞ: ðsþ g1 Þ2 r2 ðs g1 Þ2 r2

If the contact radius is fixed, then the stress intensity factors at the boundary of the contact area are given by

K2 ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pð1 þ g0 Þ u0 ð1 þ g0 Þ; 2pR lim sðrÞ 1  r ¼ 2G r/1 g30 R sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pð1 þ g1 Þ u1 ð1 þ g1 Þ: 2pR lim sðrÞ 1  r ¼ 2G r/1 g31 R

(35)

s3 qðsÞds;

2l P 4 þ 3pGR R4

ZR   s R2  s2 wðsÞds:

(44)

0

Everywhere, except some vicinity of the origin if the derivative of wðrÞ is nonzero at r ¼ 0, the contact stresses can be written in the form

2

 2 2 2 6 2g0  1  r þ 2g0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pg30 R

sðrÞ ¼ 4 

i h 1  r2 ð1 þ 2g0 Þ2 r2

(36)

(37)

(43)

0

3

c ¼

32G

For adhesive contact with varying contact area, when the contact radius depends on the load P, the stresses are not singular at r ¼ R. This demands the additional conditions

u0 ð1 þ g0 Þ ¼ 0; u1 ð1 þ g1 Þ ¼ 0

g30

Z1

This leads to the formula for the penetration depth

(34)

K1 ¼

64GR

P ¼

887

3 7 þ 1  r2 5qð1Þ

 q0 ðrÞ; Z1 q0 ðrÞ ¼ 2

sqðsÞds ¼ r

(45) 1 r2 cþ2 2

Z1 swðsRÞlnðmaxðr;sÞÞds; 0

and

2

that should be satisfied simultaneously.

3

p2 Rg21

1þ2g1  r2 7 qð1Þ sðrÞ ¼ 6 41 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i5 r  rqðrÞ: h  96G 1 r2 ð1þ2g1 Þ2 r2

2.3. Asymptotic solution To find an asymptotic solution, we note that as l/0,

j1 ðtÞ/0; Lnn ðt; xÞ/0; ZN p L01 ðt; xÞ/

L10 ðt; xÞ/

3

0 ZN

p

p 4x

dðx  tÞ; (38)

3 dðx  tÞ; pJ1 ðptÞJ1 ðpxÞdp ¼ px

  pg30 R sðrÞw r2  1 c  4

(39)

p2 Rg21 48G

sðrÞw  rc þ

2

1 w0 ðtÞ ¼ lim g20 R0 ½wðrÞ ¼ 2 t l/0 2t

qð1Þ ¼

rwðRrÞdr:

(49)

For the problem with fixed contact radius the dominant asymptotic terms of the stress intensity factors are given by

(40)

Zt

swðsRÞds: 0

pffiffiffi 12 RG K2 ¼  3=2 qð1Þ;

(50)

ZR   s R2  2s2 wðsÞds:

(51)

l

p

(48)

Zr

r

pffiffiffiffiffiffi 8 3RG K1 ¼  3=2 qð1Þ;

where l/0

swðsRÞlnðmaxðr; sÞÞds; 0

where dðxÞ is the Dirac delta function. Then, as l/0; the system (26)e(27) turns into the system of two equations for the dominant asymptotic terms of un ðtÞ

~ n ðtÞ ¼ lim un ðtÞ; qðtÞ ¼ c=2  w0 ðtÞ; u

Z1

16G

0

p 4t 3 ~ 1 ðtÞ ¼ 0; 0  t  1; ~ 0 ðtÞ þ u ~ ðtÞ þ u ~ ðtÞ ¼ qðtÞ; u u p p 0 4 1

(47)

The expression (45) manifests that the normal contact stress at every point is four times as strong as the normal stress for the frictionless contact with the same penetration depth (see Malits (2006)). Far away from the edge r ¼ R,

n ¼ 0; 1;

pJ1 ðptÞJ1 ðpxÞdp ¼

4

(46)

h3 P 1 þ 3pGR4 R4

l

0

(41)

The above solution becomes especially simple for a flat-ended rigid cylinder when wðrÞ ¼ 0,

0

Thus

3

~ 0 ðtÞ ¼ 16tqðtÞ=p; u

~ 1 ðtÞ ¼ 48tqðtÞ=p2 : u

(42)

It should be emphasized that the dominant asymptotic terms found above are specified only by the indenter profile and do not depend upon the radial displacement uðrÞ. Now, we obtain from (32)

c ¼

2l P ; 3pGR

sðrÞ ¼

 2 2g20  1  r2 þ 2g0 2P ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r iffi; h pR2  1  r2 ð1 þ 2g0 Þ2 r2

(52)

(53)

888

P. Malits / European Journal of Mechanics A/Solids 30 (2011) 884e892

2

sðrÞ ¼

3

r2

2lP 6 1 þ 2g1  7 41  r2  rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iffi5: h  pR2 r 2 1  r2 ð1 þ 2g1 Þ r2

(54)

Comparing the above results for complete adhesion contact with the results for the frictionless flat-ended cylindrical indenter (Malits (2006)), we observe the same distribution of the normal contact stress for the same load P while the penetration depth is fourfold lesser despite of the fact that the radial traction is small. The formula (52) validates that by Yang (1998) although Yang’s solution for the contact stress, coinciding with (48), does not describe the fringing field effect at the edge r ¼ R. This is attributed to the singular fringing field term making an asymptotically negligible contribution to the total force. In the case of adhesive contact with load-dependent contact zone between a non-flat indenter and the layer, the conditions (37) involve qð1Þ ¼ 0. Then the formula for the penetration depth follows from the definition of qðtÞ

Z1 c ¼ 2

swðsRÞds;

(55)

0

and (44) results in the equation for the unknown contact radius

h3 P ¼ 3pG

ZR

  swðsÞ 2s2  R2 ds:

(56)

0

The distributions of the contact stresses become

l3 R 6G

Z1

sðrÞ ¼

h i swðsRÞ r2  1  2 lnðmaxðr; sÞÞ ds;

l R 6G

sðrÞ ¼

1

Zr

Z1 swðsRÞds  r

r 0

swðsRÞds:

For the indenter of power law profile wðrÞ ¼ er m , m > 0, that includes a rigid blunt cone ðm ¼ 1Þ and a rigid sphere of radius R[R (m ¼ 2, 2e ¼ 1=R), we obtain

Rmþ4 ¼

ðm þ 4Þðm þ 3pmeG

;

c ¼

2eRm

mþ2

;

(59)

 i 2ðm þ 4ÞP h  2 sðrÞ ¼ m r  1 þ 2r2 ð1  rm Þ ; 2 mðm þ 2ÞpR 2ðm þ 4ÞhP m sðrÞ ¼ rðr  1Þ: pmR3

(60)

m

4ðm þ 4Þðm þ 2Þh3 P ; 3pmeG

(61)

m

c0 ¼

2eR0 : mþ2

Consider a cylindrical indenter of radius a whose face is described by a differentiable non-decreasing function wðrÞ, wð0Þ ¼ 0. Let an indenter be in complete contact with a thin incompressible layer and h  a. If on loading the stress intensity factor K2 exceeds a certain critical value, then a closed Griffith crack of annular form, R < r  a, is generated between the punch and the layer. On following unloading a contact area of non-flat indenter is expected to shrink, r  b < a, because the layer recedes from the indenter for a sufficiently small load. In the latter situation, either a frictionless zone exists and the contact stresses are bounded at the boundary of the contact area r ¼ b, b > R, or a frictionless zone disappears. It will be further shown that the assumption of frictionless zone leads to physically correct results as the stick zone radius is much large than the layer thickness. One might expect that the aforesaid problem also gives the lower bounds of mechanical characteristics for the problem with the friction over an external slip zone while the upper bounds are given by the complete adhesion solution obtained above. The boundary conditions of the problem

uz ðr; 0Þ ¼ wðrÞ  c; 0  r  b  a; ur ðr; 0Þ ¼ uðrÞ; 0  r  R; sz ðr; 0Þ ¼ 0; b < r < N;

(63)

srz ðr; 0Þ ¼ 0;

(64)

R < r < N;

uz ðr; hÞ ¼ ur ðr; hÞ ¼ 0;

0 < r < N;

(62)

In accordance with intuitive physical expectations, they are larger than those for the adhesive contact problem. In particular, for a rigid sphere R0 =Rz1:26 and c0 =cz1:59, and for a rigid cone

(65)

give rise to the coupled dual integral equations

ZNh i 2G ~ ðpÞf0 ðplÞ J0 ðprÞdp ¼ ~sðpÞsðplÞ  s ½wðrRÞ  c; R 0

0  r  h;

The above formulae for the contact stresses are valid throughout within the contact area as m > 1, and everywhere except some vicinity of the origin as 0 < m  1. The contact radius and penetration depth of the same punch in frictionless contact with a thin incompressible layer follow from the results of the paper Malits (2006)

R0þ4 ¼

3.1. Problem statement and general solution

(58)

0

2Þh3 P

3. Adhesive contact with an external frictionless zone

(57)

0

2

R0 =R ¼ c0 =cz1:32. When, in addition to stick zones, there are slip zones with a priori unknown friction conditions, the relations (59) and (62) give the lower and upper bounds for the contact radius and the penetration depth. The gap between the bounds is rather wide in spite of the fact that the radial traction caused by adhesion is small.

(66)

ZNh i 2G ~ ðpÞsðplÞ J1 ðprÞdp ¼ uðrRÞ; 0  r  1; ~sðpÞf1 ðplÞ s R

(67)

0

ZN

~ðpÞJ0 ðprÞdp ¼ 0; ps

h < r < N;

(68)

p~sðpÞJ1 ðprÞdp ¼ 0;

1 < r < N;

(69)

0

ZN 0

Here and below the parameter h ¼ b=R > 1 and the notations of the preceding section are employed.

P. Malits / European Journal of Mechanics A/Solids 30 (2011) 884e892

Upon applying the operators (14) and (16) to (66) and (67) these equations become

ZNh i ~ ðpÞf0 ðplÞ pcg210 ðp; tÞdp ~sðpÞsðplÞ  s

g1  t  a1 ;

(70)

where b0 ¼ h þ g0 . ~ðpÞ and ~sðpÞ in the form of the substitutions Now, we shall seek s

g0 Rs~ðpÞ ¼ p pG

Za0

Zb0

g0

xu0 ðxÞc21 ðpxÞdxp g0

g0

x4ðxÞc21 ðp;xÞdx

(71)

a0

Að1; g0 Þ½u0 ða0 Þ  4ða0 ÞHð1  rÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ iffi;  h 1  r2 ð1 þ 2g0 Þ2 r2 ð1 þ g0 Þ  2 Aðx; g0 Þ ¼ 2g20 ðx þ g0 Þ2  x  r2 þ 2g0 ðx þ g0 Þ ; (78) and

2   2 Rg21 6 1þ2g1  r a1 u1 ða1 Þ sðrÞ ¼ 4rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iffi  h 2G 1 r2 ð1þ2g1 Þ2 r2

Za0



g

x2 c220 ðp;xÞd

u0 ðxÞ x

g0

þ

 Zb0   4ðxÞ g þ x2 c220 ðp;xÞd ; (72) x a0

and

g1 R~sðpÞ ¼ p pG

Za1

g1

xu1 ðxÞc11 ðp; xÞdx

(73)

g1 R~sðpÞ 2u1 ðg1 Þ g ¼  a1 u1 ða1 Þc101 ðp; a1 Þ pG pp Za1



3

r2 þ g21 s2 d½su1 ðsÞ

7 Hð1 rÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :   ffi5 r 2 2 2 2 g r g r ðsþ ðs Þ  Þ  rþg1 1 1

(79)

The stress intensity at r ¼ b is characterized by

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðh þ g0 Þ ~ ðb0 Þ: 4 K0 ¼ lim sz ðr; 0Þ 2pðb  rÞ ¼ 2G r/b g30 Rh

(80)

At the transition point between the stick and frictionless zones

g1

þ



Za1

g0 Rs~ðpÞ g g ¼½4ða0 Þu0 ða0 Þa0 c220 ðp;a0 Þb0 4ðb0 Þc220 ðp;b0 Þ pG þ

Aðh; g0 Þ4ðb0 Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iffiþ  h h2  r2 ðh þ 2g0 Þ2 r2 ð h þ g0 Þ

 2 2 s þ g20  r2 2g20 s2 dð4ðsÞ=sÞ; s>a0 ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    dðu ðsÞ=sÞ; s  a 0 0 2 2 ðs þ g0 Þ r2 ðs  g0 Þ r2

rþg0

0

2G R ½uðRrÞ; R 0

g30 R sðrÞ ¼ Zb0

2G R ½c  wðRrÞ; g0  t  b0 ; ¼ R 0 ZNh i ~ðpÞsðplÞ pcg111 ðp; tÞdp ~sðpÞf1 ðplÞ  s ¼

The contact stresses are given by

2G

0

889

K1 ¼ lim sz ðr;0Þ r/R0

cg101 ðp; xÞd½xu1 ðxÞ:

(74)

g1

K2 ¼

sffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa0 2pðR  rÞ ¼ 2G ½4ða0 Þ  u0 ða0 Þ; g30 R

lim srz ðr; 0Þ

r/R0

sffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa1 u ða Þ: 2pðR  rÞ ¼ 2G g31 R 1 1

(81)

(82)

This leads to the system of Fredholm integral equations of the second kind 3.2. Asymptotic solution

~ 4 þ M u þ M u ¼ j ðtÞ; a  t  b ; 4ðtÞ þ M 00 00 0 01 1 0 0 0 ~ 4 þ M u þ M u ¼ j ðtÞ; g  t  a ; u0 ðtÞ þ M 00 00 0 01 1 0 0 0 ~ 4 ¼ j ðtÞ; g  t  a ; u1 ðtÞ þ M11 u1 þ M10 u0 þ M 1 1 1 10

Let h=R  1. In the limit l ¼ h=R/0, the system (75) turns into the equations

(75) ~ ðtÞ ¼ 4

~ nm are the integral operator where Mnm and M

Zan Mnm u ¼

~ nm 4 ¼ xuðxÞLnm ðt;xÞdx; M

gn

Zb0 x4ðxÞLnm ðt;xÞdx;

(76)

a0

g30 R2 ~

sð0Þ ¼ P ¼  4pGR 2GR

Za0

qðtÞ;

1  t  h;

(83)

p 4t 3 ~ 1 ðtÞ ¼ 0; 0  t  1; ~ 0 ðtÞ þ u ~ ðtÞ þ u ~ ðtÞ ¼ qðtÞ; u u p p 0 4 1

t 2 4ðtÞdt: a0

~ ðtÞ ¼ lim 4ðtÞ; 4 l/0

(84)

~ n ðtÞ ¼ lim un ðtÞ: u

(77)

l/0

Hence

~ 0 ðtÞ ¼ 16tqðtÞ=p; u

Zb0 t 2 u0 ðtÞdt þ

g0

p

where

the kernels Lnm ðt;xÞ and the right parts jn ðtÞ are defined in section 2. The equation relating the load and the penetration depth is again found in a simple way

g30

4t

~ 1 ðtÞ ¼ 48tqðtÞ=p2 : u

(85)

Note that for a non-flat indenter face and any given number c the function qðtÞ monotonically decreases since

890

P. Malits / European Journal of Mechanics A/Solids 30 (2011) 884e892

dqðtÞ 2 ¼ 3 dt t

Zt swðRsÞds 

wðRtÞ 2wðRtÞ < t t3

0

Zt sds 

wðRtÞ ¼ 0: t

For the flat-ended rigid cylinder of radius a: h ¼ a=R,

(86)

Now, it follows from (77), (83), and (85) that for l  1 3   8l P ¼ h4 þ 3 c  4 3pGR

Zh

8l 8ðh=aÞ3 h4  P ¼  P; 3pGR h4 þ 3 3paG h4 þ 3

sðrÞ ¼

0

2

2

(94)

and

 h  swðRsÞ 3 1  s2 Hð1  sÞ

i ds: þ h s 

3

c ¼

0

2 2  2P h2 2g20 ðh þ g0 Þ  h2  r2 þ 2g0 ðh þ g0 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r Hðh  rÞ i  h h4 þ 3 pa2 h2  r2 ðh þ 2g0 Þ2 r2

(87) þ

By simple manipulations, we recover the following asymptotic expression for the normal stress distribution in the region e< r  h

2 2  6h2 2g20 ð1 þ g0 Þ  1  r2 þ 2g0 ð1 þ g0 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r i Hð1  rÞ;  h h4 þ 3 2 pa ð1 þ g0 Þ 1  r2 ð1 þ 2g0 Þ2 r2 (95)

2

3   2 ðh þ g Þ  h2  r2 þ 2g ðh þ g Þ 2 g 2 3GqðhÞ 7 0 0 0 sðrÞ ¼ 3 6 þ h2  r2 5 4 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iffi h  2l R 2 2 2 2 h  r ðh þ 2g0 Þ r 2 2 2  9Gqð1Þ62g20 ð1 þ g0 Þ  1  r2 þ 2g0 ð1 þ g0 Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  UðrÞ þ 4 i 3 h  2l R 1  r2 ð1 þ 2g Þ2 r2

2

2

0

3 7 þ 1  r2 5Hð1  rÞ;

ð88Þ

with the bounded function UðrÞ defined by

UðrÞ ¼

3G

½q1 ðrÞ½4Hð1  rÞ þ Hðr  1Þ  q1 ðhÞ; 3 2l R

(89)

Z1 q1 ðrÞ ¼ 2

1  r2 cþ2 2

Zr

Z1 swðsRÞds lnr þ 2

swðsRÞlnsds:

(90)

r

0

The above formula is valid at any point of the contact area if the indenter face is smooth at r ¼ 0. The dominant asymptotic term of the radial contact stress is

20

1

3

6G B 1þ2g1 r2 7 Cqð1Þ sðrÞ¼ 2 6 4@1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffiA r rqðrÞ5Hð1rÞ:   l R 2 2 2 1r ð1þ2g1 Þ r

pffiffiffiffiffiffi 3R qð1Þ; 3=2

K1 ¼ 6Gh

l

pffiffiffi 12G R K2 ¼  3=2 qð1Þ;

l

(92)

and at r ¼ b by

sffiffiffiffiffiffi 3R K0 ¼ 2G qðhÞ: 3

l

(97)

everywhere on ½0; hÞ. Now, we infer that if the edge r ¼ a is in contact with the layer and qða=RÞ  0, then the normal traction is compressive everywhere in the frictionless zone 1 < r  a=R. When qða=RÞ < 0, then either the normal traction is compressive in the frictionless zone and vanishes at r ¼ h if there exists a contact radius h˛ð1; a=RÞ such that qðhÞ ¼ 0 or the frictionless zone disappears because of debonding if such a contact radius h is not exist. The aforesaid condition qðhÞ ¼ 0 involves the formula for the penetration depth

c ¼

2

Zh swðsRÞds;

h2

1 < h < a=R;

(98)

0

(91) It can be readily proved that sðrÞ, 3  r< h, and sðrÞ, 3  r <1, are negative if the function qðrÞ is positive for 3  r < h. The dominant asymptotic terms of the stress intensity factors at r ¼ R are given by

(96)

It is seen that the frictionless zone increases the flat punch indentation by a factor of 4h4 =ðh4 þ3Þ that is equivalent to the relative discrepancy 8 ¼ 300ðh4 1Þ=ðh4 þ3Þ per cent; 8 ¼ 15:34 per cent as h ¼ 1:05, 8 ¼ 31:19 per cent as h ¼ 1:1, and 8 ¼ 202:06 per cent for h ¼ 1:5. The normal traction is everywhere compressive. The radial contact stress is negative monotonically decreasing function throughout the stick zone 0< r <1. As wðrÞs0; the relation (93) manifests that the problem is correctly stated only if qðhÞ  0 that implies a non-tensile normal traction in close vicinity to the contact boundary. This involves

qðtÞ>qðhÞ  0

sqðsÞds r

¼

3

8Phh3 Hð1 rÞ6 1þ2g1  r2 7  41 r2  rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  sðrÞ ¼ iffi5: h  pa3 n4 þ3 r 2 2 2 1 r ð1þ2g1 Þ r

and the equation for the unknown contact radius

P ¼ FðhÞ;

3

4l FðhÞ ¼ 3pGR

1 < h < a=R;



Zh swðsRÞ

3

h2

(99)

  h2 þ 2s2 ds

0

Z1 (93)

6 0

  swðsRÞ 1  s2 ds:

(100)

P. Malits / European Journal of Mechanics A/Solids 30 (2011) 884e892

891

Depending upon the value of a positive load P, each of the three above-mentioned types of contact can occur for a given radius R of the stick zone. Indeed,

2 3   Zh 3 4l dF 3 2 3 4 wðhRÞ  2 wðhRÞ sds5 ¼ 0: > þh h 3pGR dh h

(101)

0

In addition, 3

2l Fð1Þ¼ 3pGR

Z1

Z1     s2 s4 dwðsRÞ>0; wðsRÞ 4s3 2s ds¼

0

0

3

2l Fð0Þ ¼ 6 3pGR

Z1

  swðsRÞ 1  s2 ds < 0:

(102)

Fig. 2. Relative normal traction sðrÞ=P for a rigid sphere as h ¼ 0:02R.

(103)

0

Consequently, FðhÞ is a continuous monotonically increasing ~, 0 < h ~ < 1. Therefore, equation function that is positive for h > h (99) possesses a unique solution for any positive load P ¼ Fðh* Þ; 1 < h* < a=R. Now, as P  Fða=RÞ, the normal contact traction is singular at r ¼ a and compressive at every point of the frictionless zone R < r  a. For P ¼ Fðh* Þ, the normal traction is compressive within the frictionless zone R < r < h* R that shrinks on unloading, while the indenter and the layer are debonded for r > h* R. When P ¼ Fðh0 Þ; h0  1, the indenter is debonded from the layer as r > R, and complete adhesion occurs everywhere within the contact circle r  R. The latter situation comprises both the solutions established in the preceding section because, as it follows from (40), (41), (44), and (101),

adhesive contact with fixed contact zone. The normal contact traction is tensile in the vicinity of the contact boundary r ¼ R and may cause additional debonding between the indenter and the layer because of opening mode interface cracking. To demonstrate how a frictionless zone affects the indenter characteristics, we consider a cylindrical indenter with power law face wðrÞ ¼ er m , m > 0, as 1 < h < a=R. One might derive from (98) (99) and (98) that for this indenter profile

Z1   s 1  2s2 wðsRÞds;

In the case of the rigid blunt cone ðm ¼ 1Þ and a rigid sphere (m ¼ 2), the relative size of the contact area h ¼ b=R is presented in Fig. 1 as a function of P ¼ h3 P=GRmþ4 e. The expressions for the contact stress are given by (88) and (91) with

l3 Fðh0 Þ þ qð1Þ ¼ 3pGR

(104)

0 3

l dFðh0 Þ dqð1Þ ¼ >0: dh0 3pGR dh0



3p mhmþ4 þ 3ðm þ 4Þhm  12 2ehm Rm h3 P ; mþ4 ¼ ; m þ 2 GR 3 4ðm þ 2Þðm þ 4Þ

pffiffiffi 6 3GRmþ2 ðhm  1Þh3 12GRmþ2 ðhm  1Þ3 K1 ¼  ; K2 ¼  ; 3=2 m ð þ 2Þh ðm þ 2Þh3=2

(105)

Then qð1Þ ¼ 0 as h0 ¼ 1, and the contact tractions are finite at the contact boundary r ¼ R. We obtain for P ¼ Fð1Þ the same solution as that for adhesive contact with load-dependent contact ~ < h0 < 1, the stress intensity factors are zone. As P ¼ Fðh0 Þ; h positive for the reason that qð1Þ < 0, and we have the solution for

Fig. 1. Dimensionless contact radius h ¼ b=R versus P ¼ h3 P=GRmþ4 e.

c ¼

qðrÞ ¼

eRm m ðh  rm Þ ; mþ2

(106)

(107)

q1 ðrÞ !

 1  rmþ2 eRm hm  1  r2  : ¼ mþ2 2 mþ2

(108)

In the case of the rigid sphere, for h ¼ 0:02R, h ¼ 1:9 and h ¼ 2:5 the relative tractions sðrÞ=P and sðrÞ=P are shown in

Fig. 3. Relative radial traction sðrÞ=P for a rigid sphere as h ¼ 0:02R.

892

P. Malits / European Journal of Mechanics A/Solids 30 (2011) 884e892

Fig. 4. Relative discrepancy Z for the conical and spherical faces of the indenter.

Fig. 2 and Fig. 3. It is seen that the singular terms of the stress exhibit some effect only very close to the boundary between the stick and frictionless zones. Let c* and b* be the penetration depth and the contact radius, respectively, for the same indenter wðrÞ ¼ er m under the same load P in the case of complete adhesion contact with load-dependent contact radius over the whole contact area. Making use of the solution obtained in the preceding section, we find

c ¼ c*



b b*

m ¼

!m=ðmþ4Þ 4mhmþ4 : 3ðm þ 4Þhm þ mhmþ4  12

(109)

The relative discrepancy Z ¼ ðc=c*  1Þ  100% versus h is plotted in Fig. 4 for the conical and spherical faces of the indenter. We observe that a frictionless zone can rather significantly change the punch indentation, affecting the spherical indenter much stronger than the conical indenter. 4. Conclusions In this paper, the influence of complete adhesion on the indenter mechanical characteristics is investigated for an axisymmetric indenter and a thin incompressible layer. In this case, the main trait of the complete adhesion phenomenon is the coupling of both the normal and radial stresses with the components of the interface displacement. The coupling, together with incompressibility, strongly affects mechanical characteristics of the indenter. Our study of these effects was a success due to very simple asymptotic solutions achieved by solving systems of dual integral equations with the special technique. This technique is suggested since, as it is well known, the traditional Cooke-Lebedev and moment methods are absolutely ineffective for a very thin layer. From a physical point of view, it is important to emphasize that our model for the edge frictionless zone is entirely based on the Griffitth crack concept. This concept demands singularity of the contact stresses at the tip r ¼ R of the frictionless zone. Different

mechanisms of cracking, which assume finite stresses at the tip of the frictionless zone, require the condition qð1Þ ¼ 0 that immediately involves the absence of such a zone and the solution which is the same as that for the complete adhesion contact with loaddependent contact area (see section 2). This can be correct only if for the critical load P cr the equation Fð1Þ ¼ P cr possesses the solution R < a. When the solution is R  a, such mechanisms of cracking are impossible, that is we need some other description of the indenter/layer interaction. Probably, a model taking into account friction in the external zone could be one of options. Another option is a model incorporating the interaction at the atomic/molecular level (such as intermolecular repulsion and attraction). Summarizing the results of this paper, we conclude that the indenter behaviour sensitively depends upon the type of contact with a thin incompressible layer. Therefore, our closed form solutions can be helpful for the correct interpretation of measurement data. Note that a blunt conical indenter is much less sensitive to contact conditions than flat-ended and spherical indenters.

References Alexandrov, V.M., 2003. Asymptotic solution of the axisymmetric contact problem for an elastic layer of incompressible material. J. Appl. Math. Mech. (PMM) 67, 589e593. Argatov, I., 2010. Frictionless and adhesive nanoindentation: asymptotic modeling of size effects. Mech. Mater. 42, 807e815. Barber, J.R., 1990. Contact problem for the thin elastic layer. Int. J. Mech. Sci. 31, 129e132. Borodich, F.M., Keer, L.M., 2004. Contact problems and depth-sensing nanoindentation for frictionless and frictional boundary conditions. Int. J. Solids Struct. 41, 2479e2499. Chadwick, R.S., 2002. Axisymmetric indentation of a thin incompressible elastic layer. SIAM J. Appl. Math. 62, 1520e1530. Hlavá cek, M., 2008. Elliptical contact on elastic incompressible coatings. Eng. Mech. 15, 249e261. Jaffar, M.J., 1989. Asymptotic behaviour of thin elastic layers bonded and unbonded to a rigid foundation. Int. J. Mech. Sci. 32, 229e235. Johnson, K.L., 1985. Contact Mechanics. Cambridge Univ. Press, Cambridge. La Ragione, L., Musceo, F., Sollazzo, A., 2008. Axisymmetric indentation of a rigid cylinder on a layered compressible and incompressible half-space. J. Mech. Mat. Struct. 3, 1499e1520. Malits, P., 2005. Effective approach to the contact problem for a stratum. Int. J. Solids Struct. 42, 1271e1285. Malits, P., 2006. Indentation of an incompressible inhomogeneous layer by a rigid circular indenter. Q. J. Mech. Appl. Math. 59, 343e358. Malits, P., 2007. On a certain class of integral equations associated with Hankel transforms. Acta Appl. Math. 98, 135e152. Matthewson, M.J., 1981. Axi-symmetric contact on thin compliant coatings. J. Mech. Phys. Solids 29, 89e113. McCormick, J.A., 1978. Numerical Solutions for General Elliptical Contact of Layered Elastic Solids. MTI. Report No. 78TR52. Lanthum, Mech. Techn. Inc., NY. Mossakovski, V.I., 1954. The fundamental mixed problem of the theory of elasticity for a half-space with a circular line separating the boundary conditions. J. Appl. Math. Mech. (PMM) 18, 187e196 (in Russian). Mossakovski, V.I., 1963. Compression of elastic bodies under conditions of adhesion (axisymmetric case). J. Appl. Math. Mech. (PMM) 27, 630e643. Spence, D.A., 1968. Self similar solution to adhesive contact problems with incremental loading. Proc. R. Soc. Lond. A 305, 55e80. Spence, D.A., 1975. The Hertz contact problem with finite friction. J. Elasticity 5, 297e319. Sundelius, B., 1981. An axisymmetric contact problem with given region of adhesion. IMA J. Appl. Math. 27, 455e475. Yang, F., 1998. Indentation of an incompressible film. Mech. Mater. 30, 275e286. Zhupanska, O.I., 2009. Axisymmetric contact with friction of a rigid sphere with an elastic half-space. Proc. R. Soc. A 485, 2565e2588.