Contact stiffness depth-sensing indentation: Understanding of material properties of thin films attached to substrates

Mechanics of Materials 114 (2017) 172–179

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Research paper

Contact stiffness depth-sensing indentation: Understanding of material properties of thin films attached to substrates Ivan I. Argatov a,∗, Feodor M. Borodich b, Svetlana A. Epshtein c, Elena L. Kossovich c a

Institut für Mechanik, Technische Universität Berlin, 10623 Berlin, Germany School of Engineering, Cardiff University, Cardiff CF24 3AA, UK c Laboratory of Physics and Chemistry of Coals, Mining Institute, National University of Science and Technology “MISiS”, 119049 Moscow, Russia b

a r t i c l e

i n f o

Article history: Received 26 March 2017 Revised 7 July 2017 Available online 16 August 2017 Keywords: Contact stiffness Substrate effect Thin film Depth-sensing Coal

a b s t r a c t The classic version of the depth-sensing indentation techniques assumes the estimation of the elastic contact modulus of a material sample by measuring the slope (the contact stiffness) of the initial part of the unloading branch of the force-displacement curve. This approach assumes that the curve at loading reflects both elastic and plastic deformations of the material, while the unloading is taking place elastically. Therefore, neglecting the plastic deformations, one can assume that the structure of the material is the same at both branches and the assumptions of the Hertz-type contact theory are valid for the unloading branch. However, the contact problem for an elastic film attached to a substrate depends on the properties of the substrate. Hence, the film contact modulus is usually estimated by measuring the slopes of the initial unloading force-displacement curves obtained for different maximal values of indentation depth, and fitting the experimental points by various empirical analytical dimensionless functions of the ratio between the contact radius, a, and the layer thickness, t. Here, analytical analysis of contact problems for coated materials is performed. Both re-scaling and asymptotic techniques are employed. Asymptotic analysis of the contact at the small-scale indentation range (the ratioa/t is small) shows that the formula of the contact stiffness derived for an elastic half-space, has to be multiplied by the socalled indentation scaling factor that is a function of a/t. Thus, the asymptotic approach allows us to take into account analytically the substrate effect. The analytical fitting function obtained agrees with both some known semi-empirical functional forms and the published experimental results on depth-sensing nanoindentation of thin metallic layers, while the function is in a disagreement with results obtained for inhomogeneous films of brittle materials such as coals. It is argued that the disagreement is caused by structural transformations (crushing) of the coals during loading. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction After the depth-sensing nanoindenter was built by Kalei (1968) and Alekhin et al. (1972), depth-sensing indentation (DSI) has become a popular technique for material properties identification (Antunes et al., 2007; Borodich, 2011; Bull, 2005; Fischer-Cripps, 2004; Mencˇ ík, 2007; Poon et al., 2008). In particular, it was used to determine mechanical properties (hardness, elastic modulus, fracture toughness) of rock materials (Ban et al., 2014; Zhu et al., 2007) and coal (Epshtein et al., 2015; Kossovich et al., 2016). Such geological materials are inhomogeneous at the micro-level, and, therefore, special approaches are needed to assess the mechanical properties of their constitutive components. ∗

Corresponding author. E-mail addresses: [email protected], [email protected] (I.I. Argatov). http://dx.doi.org/10.1016/j.mechmat.2017.08.009 0167-6636/© 2017 Elsevier Ltd. All rights reserved.

If one knows the properties of the components and their distributions within the material then the methods of micromechanics may be employed (Nemat-Nasser and Hori, 1999). Recently, Borodich et al. (2015) proposed to combine the depth-sensing nanoindentation technique and transmitted light microscopy in order to test specific components of very thin films (smooth polished films) of inhomogeneous materials. Because the method deals with a thin film/glue substrate system rather than with a bulk material sample, a non-trivial problem arises to extract intrinsic mechanical properties of the film layer. The DSI method, which was originally developed for hardness testing, has been rooted in a few ideas, which crystallized, step by step, during several last decades. Let us briefly overview them, in particular, focusing only on measuring the elastic modulus. First of all, acknowledging fundamental contributions to analytical solutions of the frictionless contact problems for an elastic halfspace obtained for indenters of different shapes, including cylindri-

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cal (Boussinesq, 1885), spherical (Hertz, 1882), conical (Love, 1939), and monomial (power-law) (Galin, 1946) (the further details may be found elsewhere, see, e.g. Argatov and Dmitriev, 2003; Borodich, 2014), it should be accepted that an initial “push” towards practical application of the indentation technique for measuring elastic properties has been given by the idea (Bulychev et al., 1975; 1976) to measure the contact stiffness, S, at the initial moment of unloading with the subsequent evaluation of Young’s modulus, E, from the so-called BASh (Bulychev–Alekhin–Shorshorov) formula. For a rigid indenter, the BASh formula is

√ πS E = (1 − ν 2 ) √ , 2 A

(1)

where ν is Poisson’s ratio of the material, A is the contact area. The incremental contact stiffness, S, is given by

S=

dP . dh

(2)

The second important idea that has given a great stimulus to development of instrumented indentation techniques belongs to Oliver and Pharr (1992) who developed a semi-empirical approach for estimation of the contact area A, which enters Eq. (1), directly from indentation load and displacement measurements without the need to image the indenter imprint (hardness impression). In fact, the BASh formula (1) is an approximation introduced by Bulychev et al. (1976) for the expression

dP = 2E ∗ a, dh

(3)

which is exact in axisymmetric case for isotropic elastic materials. Here a is the contact radius and E∗ is the reduced elastic modulus (contact modulus). Thus, the BASh formula (1) may be written as

√ dP 2 = √ E ∗ A. dh π

(4)

For further detail, see discussions by Pharr et al. (1992), Borodich and Keer (2004b) and Borodich (2014). For isotropic materials, the following well-known formula holds (Johnson, 1985):

E∗ =

E . 1 − ν2

(5)

We note that, for the sake of simplicity, the indenter is assumed to be absolutely rigid, and, therefore, the right-hand side of Eq. (5) depends only on the material properties of the tested material. The effect of the indenter elasticity can be taken into account via the effective elastic modulus, E∗ , defined by

1 − νi2 1 1 − ν2 = + , E∗ E Ei where Ei and ν i are the indenter elastic modulus and Poisson’s ratio. Other important ideas concern the following aspects of the depth-sensing indentation that may be taken as correction factors to the BASh formula (1): the factor β 1 due to the concept of the effective indenter shape, which accounts for distortion of the originally flat sample surface by the formation of the hardness impression (this is the so-called Galanov effect (Galanov et al., 1983; 1984), see also Pharr and Bolshakov, 2002), the introduction of the contact area shape factor β 2 , which extends the BASh formula to the non-axisymmetric case (King, 1987; Hay et al., 1999; Argatov, 2010), and factor (β 3 ) for the effects of friction between the indenter and the half-space (Borodich and Keer, 20 04a; 20 04b). Hence, the BASh formula (1) can be modified as

√ dP 2 = β √ E ∗ A, dh π

(6)

173

where β = β1 β2 β3 is the correction factor. It has been shown in the case of adhesive (no-slip) contact between a rigid indenter and an elastic sample β3 = CNS , where

CNS =

(1 − ν ) ln(3 − 4ν ) . 1 − 2ν

(7)

It was noted (Borodich and Keer, 20 04a; 20 04b) that taking into account that full adhesion preventing any slip within the contact region is not the case for real physical contact and there is some frictional slip at the edge of the contact region, the values of the correction factor β 3 in (6) cannot exceed the upper bound (7). A practically important aspect of the method arises in the depth-sensing indentation testing of thin samples deposited onto elastic substrates (Antunes et al., 2007; Chen and Vlassak, 2001). In the framework of phenomenological approach (Doerner and Nix, 1986; Jung et al., 2004; Mencˇ ík et al., 1997), a number of approximating functions have been proposed to relate the so-called equiv∗ , evaluated from the DSI tests having alent contact modulus, Eeq the different maximum depth of indentation, to the reduced elastic moduli of the film, E ∗f , and that of the substrate, Es∗ . Also, analytical (Gao et al., 1992; Perriot and Barthel, 2004; Xu and Pharr, 2006) and asymptotic (Argatov and Sabina, 2014) approximations were developed in different limit situations (e.g., when the film elastic modulus does not differ much from the substrate modulus, or when the contact area is relatively small with respect to the film thickness). Nevertheless, the problem of accounting for the substrate effect (Yu et al., 1990) is far from being solved completely. In particular, the present study addresses the question of evaluating the contact ∗ . area, which is used in determining Eeq 2. Scaling of the contact stiffness for a bulk elastic material sample The theoretical analysis of contact problems for DSI is usually based on the Hertz-type problem formulation (Bulychev et al., 1976; Hay et al., 1999; Borodich and Keer, 2004b). However, the three-dimensional Hertz contact problem is non-linear because the contact region varies as a non-linear function of the load. On the other hand, it was shown independently by Galanov (1981a) and Borodich (1983) using two different approaches, that the Hertztype contact problems are ofthen self-similar for linear elastic materials. Hence, the non-linear 3D Hertzian contact problem can be considered as steady-state because using the solution to the problem for only one value of the problem parameter (e.g. the load), one can obtain the solutions for any other value of the parameter by simple scaling of the known solution. Galanov discovered also self-similarity of contact problems for non-linear isotropic elastic, visco-elastic and plastic solids (Galanov, 1981b; 1982). Since Galanov had applied the similarity approach to frictionless contact problems for many isotropic materials, the main task was to extend the similarity approach to anisotropic materials. Later Borodich described the general transformations applicable to the 3D Hertz-type contact problems and showed that selfsimilarity transformations are a particular case of the general case when the shape of the indenter is described by a homogeneous function. The conditions under which Hertz-type contact are selfsimilar may be formulated as follows: “The constitutive relationships are homogeneous with respect to the strains or the stresses and the indenter’s shape is described by a homogeneous function whose degree is greater than or equal to unity. It is also assumed that during the contact process, the loading at any point is progressive” (see for details Borodich and Galanov, 2002; Borodich, 2011). Thus, the Hertz-type contact problems may be self-similar for linear (Galanov, 1981a; Borodich, 1983) and non-linear

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elastic (Borodich, 1989), plastic (Galanov, 1981b), creeping (Galanov, 1982), and prestressed (Borodich, 1990) solids. The materials may be anisotropic (Borodich, 1989) and the bondary conditions may be both frictionless and frictional (Borodich, 1993). It was shown later that self-similar approach is also valid in application to specific non-convex (namely, parametrichomogeneous) indenters (see, Borodich, 1998a; 1998b; Borodich and Galanov, 2002). The scaling approach was applied to indentation tests by Borodich (1989) and to nanoindentation tests by Borodich et al. (2003). Here we develop further this approach with regard to DSI.

By excluding the characteristic size l from (14) by means of Eq. (11), we readily get

2.1. Scaling approach and a generalized BASh formula for the contact stiffness

In the case of a linearly elastic material, κ = 1 and the exponents on the right-hand sides of Eqs. (14)–(16) do not depend on the indenter shape exponent d, that is

We consider a hyperelastic material with an elastic potential U, so that the components of the stress tensor are given by

dP P1 (d + 1 ) l = , dh h1 d l1

σi j =

∂U . ∂εi j

(8)

Following Borodich (1989), we assume that U is a homogeneous function of degree κ + 1 with respect to the strain components ε ij . Moreover, let us assume that the indenter shape function f(x1 , x2 ) is a positive homogeneous function of degree d, i.e., for any constant λ ≥ 0, the following relation takes place:

f (λx1 , λx2 ) = λd f (x1 , x2 ).

(9)

It is clear that the shape function from Eq. (9) describes the monomial (power-law) shape and it can be represented in the polar coordinates r and θ as f (x1 , x2 ) = Bd (θ )r d , where r = (x21 + x22 )1/2 is the polar radius, and Bd (θ ) is the shape function having the physical dimension L1−d , with L denoting the dimension of length. The frictionless contact problem for a semi-infinite hyperelastic solid, which occupies a half-space x3 ≥ 0, and a rigid indenter with the shape function satisfying Eq. (9) exhibits the property of self-similarity, and the contact force P is related to the indenter displacement h via the following equations (Borodich, 1989; 2011):

P = P1 h = h1

 l 2+κ (d−1) l1

 l d l1

,

(10)

.

(11)

 l 2 l1

.

(12)

By excluding the relation l/l1 from (10) and (11), we arrive at the force-displacement relation in the following form (Borodich, 1989):

P = P1

 h [2+κ (d−1)]/d h1

.

(13)

Then, it follows from Eqs. (10) and (11) that the incremental contact stiffness (2) is given by the following formula (Argatov et al., 2016):

  P1 [2 + κ (d − 1 )] l 1+(κ −1)(d−1)

dP = dh h1

d

l1

.

(14)

[1+(κ −1)(d−1)]/d

.

(15)

It is noteworthy that Eq. (15) can be also obtained by direct differentiation of Eq. (13) with respect to h. Finally, taking into account Eq. (12), we reduce formula (14) to the following one:



dP P1 [2 + κ (d − 1 )] A = dh h1 d A1

dP P1 (d + 1 ) = dh h1 d



[1+(κ −1)(d−1)]/2



dP P1 (d + 1 ) h = dh h1 d h1

.

1/d

(16)

,

A . A1

(17)

(18)

In the axisymmetric case, formula (18) represents a self-similar analogue of the so-called BASh (Bulychev–Alekhin–Shorshorov) formula (4), which was first derived by Bulychev et al. (1976) for isotropic materials and some special cases of indenters (flat-ended cylindrical, spherical, and conical), and, afterwards, was established by Pharr et al. (1992) for a general axisymmetric indenter with a circular area of contact. Another very important observation is that in the case of pyramidal indenters, when d = 1, both exponent and coefficient on the right-hand side of each equation from (14)–(16) do not depend on the non-linearity parameter κ , that is

dP 2P1 = l, dh h1 l1

dP 2P1 = 2 h, dh h1

dP 2P1 √ =  A. dh h1 A1

(19)

In particular, formulas (19) hold for the Vickers and Berkovich indenters as well for a conical indenter. 2.2. Relation between the contact force and the contact stiffness It is of interest to evaluate the contact stiffness S1 from Eqs. (14)–(16), each of which leads to the following relation:

S1 =

Here, l is a characteristic size of the current contact area, the two sets {P1 , h1 , l1 } and {P, h, l} represent solution of the contact problem in two different states. Since, when shifting from the first state with the contact area A1 to the second state with the contact area A, the contact area expands homothetically, we will have

A = A1



dP P1 [2 + κ (d − 1 )] h = dh h1 d h1

[2 + κ (d − 1 )] P1 . d h1

(20)

In the case of a linearly elastic material, κ = 1 and formula (20) reduces to

S1 =

(d + 1 ) P1 d

h1

.

(21)

From (21), it immediately follows that

h1 =

(d + 1 ) P1 d

S1

.

(22)

Formula (22) will be extremely useful in the analysis of the Oliver– Pharr method (see Section 3). Finally, the substitution of (22) into Eq. (13) with κ = 1 yields

P=



S1 mP1(m−1)/m

m

hm ,

(23)

where

m=

d+1 . d

(24)

We emphasize that the coefficient in Eq. (23) can be directly measured in the experiment.

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175

2.3. Reduced elastic modulus and the contact area shape factor In the axisymmetric case of a linearly elastic material (κ = 1), by reducing Eq. (18) to the form of Eq. (3), we can write

E∗ =

(d + 1 ) P1 2d

h1 a1

(25)

and thereby introduce the reduced elastic modulus denoted by E∗ . Recall that for isotropic materials, the following well-known formula holds: E ∗ = E/(1 − ν 2 ). We note that, for the sake of simplicity, the indenter is assumed to be absolutely rigid, and, therefore, the right-hand side of Eq. (25) depends only on the material properties of the tested material. In the non-axisymmetric case, as it was recently shown by Argatov et al. (2016), the BASh formula (3) is also valid provided the geometric quantity a is interpreted as the harmonic radius of the contact area (see, e.g., Pólya and Szegö, 1951). On the other hand, the original BASh formula (4) requires the introduction (King, 1987; Pharr et al., 1992) of the contact area shape factor β = β2 into formula (6). A number of studies were conducted to evaluate the value of β 2 for different indenters (see the review given by Oliver and Pharr (2004)). 3. Asymptotic modeling of the contact stiffness depth-sensing indentation Asymptotic methods of various types have been successfully used in solving various problems of contact mechanics. The details of the asymptotic technique employed here can be found in Alexandrov and Pozharskii (2001), Argatov and Dmitriev (2003) and Argatov and Sabina (2013). 3.1. Indentation scaling factor accounting for the substrate effect

(26)

does not depend on the level of indentation, and this fact was experimentally verified for a relatively large elastic samples in a number of studies (Bulychev et al., 1976; Fischer-Cripps, 2004; Saha and Nix, 2002). The same statement will be true for non-axisymmetric selfsimilar indenters, as the contact area expands homothetically during indentation and the dimensionless shape factor β by which Eq. (26) differs from (6), remains constant. However, this theorem will no longer hold for a sample of finite thickness, i.e., in the case when the self-similar scaling is not applicable. In the axisymmetric case, for an elastic film layer deposited on an elastic substrate, by analogy with the indentation scaling factor introduced by Hayes et al. (1972) for an elastic layer deposited on a rigid substrate, the following generalization of the BASh formula (3) was established (Argatov and Sabina, 2013):

dP = 2E ∗f ak. dh

(27)

Here, a is the contact radius, E ∗f is the reduced elastic modulus of the film (which is defined by formula (5), where E and ν are replaced by Ef and ν f , respectively), k is the indentation scaling factor, which depends on the film and substrate Poisson’s ratios ν f and ν s , on the relative size of the circular contact area a/t, where t is the film thickness, as well as on the ratio η of the substrate Young’s modulus Es to the film Young’s modulus Ef , i.e.,

η

Es = . Ef

The concept of indentation scaling factor was related to the phenomenological approach based on the weight functions developed in a number of publications (Gao et al., 1992; Jung et al., 2004; Kossovich et al., 2016; Mencˇ ík et al., 1997; Perriot and Barthel, 2004). In the case of small-scale contact, where the contact radius a is relatively small compared to the film thickness t, the following first-order asymptotic model was developed (Argatov, 2010):



k  1 − α0

a t

−1

.

(29)

Here, α 0 is a dimensionless asymptotic constant, which in the case of perfect bonding between the film layer and the substrate is given by the improper integral

α0 =

2

π



∞ 0

L(u ) du

(30)

with the following kernel function (Burmister, 1945):

As it follows from the BASh formula (4), the right-hand side of the equation

√ π dP = E∗ √ 2 A dh

Fig. 1. Indentation scaling factor as a function of the characteristic size of the contact area normalized by the film thickness.

(28)

L (u ) =

2KLe−4u − (L + K + 4uK + 4u2 K )e−2u , 1 − (L + K + 4u2 K )e−2u + KLe−4u

( 3 − 4 νs ) − n ( 3 − 4 ν f ) 1 + νf 1−n , L= , n=η . 1 + n ( 3 − 4ν f ) ( 3 − 4 νs ) + n 1 + νs

K=

Fig. 1 shows the variation of the indentation scaling factor k according to formula (29). The asymptotic behavior of k as η → 0 was studied by Argatov and Sabina (2016). We finally note that by the appropriate choice of α 0 the simple asymptotic model (29) can be extended for different film-substrate interface conditions (Argatov, 2010) and transverse isotropy (Argatov and Sabina, 2013). 3.2. First-order asymptotic model of the contact stiffness for a non-axisymmetric self-similar indenter For a linearly elastic material (κ = 1), according to the firstorder asymptotic analysis of the small-scale indentation problem (Argatov, 2010), the contact pressure under a rigid indenter, p(x1 , x2 ), which is distributed over an a priori unknown contact area ω, serves as a solution to the integral equation

1 π E ∗f

 ω



p(y1 , y2 ) dy1 dy2

( x1 − y1 )2 + ( x2 − y2 )2

˜ − f ( x1 , x2 ), =h

(31)

where the modified indenter displacement

˜ =h+ h

α0 P 2E ∗f t

(32)

depends on the contact force



P= ω

p(y1 , y2 ) dy1 dy2 .

(33)

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For a self-similar indenter with the shape function f(x1 , x2 ) satisfying Eq. (9), using formulas (10) and (11), with κ = 1, we obtain the following relations:

h=

P = P1

(34)

where S is the contact stiffness. Therefore, from (43) and (44), it follows that

(35)

P hs =  , S

h+

 l d+1 l1

α0 P

2E ∗f t

˜1 =h

,

 l d l1

.

˜ = (d + 1 ) P1 , h d S1

(36)

˜ from and now, utilizing (36), we can exclude the quantity h Eq. (35) as follows:

α0 P

2E ∗f t

=

(d + 1 ) P1  l d d

(d + 1 ) P d

S

,

(44)

(45)

where we have introduced the notation

˜ 1 , l1 } is a solution to the self-similar frictionless contact Here, {P1 , h problem for an elastic half-space with a contact area ω1 . Under the assumption that the characteristic size of the contact area ω is small compared to the film layer thickness t, from Eqs. (34) and (35), the force-displacement relation can be derived by a perturbation technique. We note also that in the axisymmetric case, when the indenter shape function follows the power-law form f (x1 , x2 ) = Br d with r = (x21 + x22 )1/2 , the fourth-order asymptotic expansions were developed by Argatov and Sabina (2013). Further, let S1 be the contact stiffness corresponding to the contact force P1 and the characteristic size l1 . Then, making use of formula (22) we obtain

h+

On the other hand, formula (22) readily yields

S1 l1

.

(37)

Finally, by excluding the ratio l/l1 from Eqs. (34) and (37), we arrive at the displacement-force relation in the form

mP1(m−1)/m 1/m α0 P h= P − , S1 2E ∗f t

(38)

where we also utilized the notation (24). 3.3. Contact depth for an axisymmetric self-similar indenter According to the definition given by Oliver and Pharr (1992), the contact depth, hc , which coincides with the vertical distance along which contact is made, is related to the displacement of the surface at the perimeter of the contact, hs , by the formula

hc = h − hs .

=

(d + 1 )  d



22−d (d ) . d [(d/2 )]2

(46)

In particular, for d = 1 and d = 2, formula (46) yields  = 2(π − 2 )/π ≈ 0.72 for conical indenters and  = 3/4 = 0.75 for spherical indenters that coincide with the results obtained by Oliver and Pharr (1992). Finally, observe that in view of (46) formula (43) can be rewritten as

hs =

d h, (d + 1 )

(47)

and it can be shown that  → 1 as d → +∞. 3.4. Accounting for the substrate effect in the contact depth evaluation by the Oliver–Pharr method Let hmax and hf be respectively the maximum indentation depth, which can be measured at peak load Pmax , and the final indenter displacement after complete unloading, which can be determined by curve fitting. Following Oliver and Pharr (1992), we will make use of the relation

hs =

d ( h − hf ), (d + 1 )

(48)

where the geometric constant  for the monomial indenter is given by (46). Note that in the case d = 1 (conical indenter) formula (48) reduces to hs = (1/2 ) (h − hf ), which, in view of the value  = 2(π − 2 )/π provided by Eq. (46), coincides with the form used by Oliver and Pharr (1992). Observe that the quantity h − hf appears in Eq. (48), because it is applied on the unloading stage. Correspondingly, by formula (38) we obtain

mP1(m−1)/m 1/m α0 P P − . S1 2E ∗f t

h − hf =

(39)

Let us consider the case of axisymmetric indenters whose shape is described by monomial (power-law) functions

1−

(49)

Now, following Oliver and Pharr (1992), we substitute the expression of the right-hand side of Eq. (49) into Eq. (48) and noting that the contact area of interest is that at the peak load, we obtain

f ( x1 , x2 ) = Bd r d ,

d

 mP (m−1)/m

P 1/m −

α0 P 

where Bd is the shape constant having the dimension L1−d , r = (x21 + x22 )1/2 is the polar radius. According to the solution given by Galin (1946), we will have

hs =

d 2 [(d/2 )] d+1 P = E Bd a , (d + 1 ) (d )

(40)

S1 =

(41)

Thus, by substituting (50) into Eq. (39) one can evaluate the contact depth hc , provided the value of E ∗f is known. Further, the contact area A, at which the contact stiffness (51) is experimentally determined, can be evaluated from the relation

∗

h = Bd d 2

2 d−1

d−2 [

2

(d/2 )]2 d a , (d )

where a is the contact radius, E ∗ = E/(1 − ν 2 ) is the reduced elastic modulus. By the definition of hs , we obtain

hs = h − Bd ad .

(42)

In light of (41), formula (42) yields



hs = h 1 −



22−d (d ) . d [(d/2 )]2

(43)

(d + 1 )

1

S1

2E ∗f t

,

(50)

where now we assume that



dP  .  dh h=hmax

A = F ( hc ),

(51)

(52)

where F(h) is the indenter cross-section area function, which relates the cross-sectional area of the indenter to the distance from its tip h (see, e.g., Oliver and Pharr (2004)). Finally, since E ∗f is an a priori unknown quantity, some iterative scheme can be developed to account for the substrate effect in evaluating the contact depth and the contact area.

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3.5. Evaluation of the film elastic modulus from a set of the contact stiffness values For the sake of simplicity, we consider the axisymmetric depthsensing indentation of an elastic layer of thickness t deposited on an elastic substrate. Let us assume that the initial unloading contact stiffness values S(ai ) are given for N different radii ai , i = 1, 2, . . . , N. Based on the BASh formula (3), one can compute from the given data, correspondingly, different values of the so-called equivalent ∗ of the film/substrate system by the formula contact modulus Eeq

S (a ) , 2a

∗ Eeq =

(53)

∗ (a ) Eeq i

that is = S(ai )/(2ai ), i = 1, 2, . . . , N. If the indented elastic sample were homogeneous and relatively large (compared to the maximum contact radius max i {ai }), then ∗ (a ) would not vary systematically with the change the values Eeq i in the contact radius or the indentation depth. However, in the case of an elastic film deposited on a rigid or a compliant substrate, according to the generalized BASh formula (27), we have

2aEs 1 k (1 − ν 2f ) η

S (a ) =



η,



a , t

(54)

where η is the substrate/film ratio of the elastic moduli (28), k is the indentation scaling factor. Provided both Poisson’s ratios ν f and ν s are known, for a particular contact radius ai the corresponding value ηi can be evaluated from the equation

1

η

 k

ai t

η,



=

(1 − ν 2f )S(ai ) 2ai Es

,

(55)

so that the corresponding value for the film Young’s modulus E if will be given by the formula

Es

Ef =

η

.

(56)

After that the obtained values E if , i = 1, 2, . . . , N, can be evaluated arithmetically to get an estimate for Ef . Note that the unique solvability of Eq. (56) was discussed in Argatov and Sabina (2014). In the present paper, based on the explicit asymptotic formula (29) for the indentation scaling factor k, the so-called contact stiffness depth-sensing indentation method for evaluating the film Young’s modulus is reduced to the minimization problem

min η

N   1 i=1

where



k

η,

a t



η

 k



η,

ai t



−

= 1 − α0 (η )

(1 − ν 2f ) Es

a t

∗ Eeq ( ai )

2

,

4.1. Nanoindentation of aluminum and tungsten thin films We consider a set of nanoindentation experiments performed by Saha and Nix (2002) with a Berkovich indenter for aluminum (Al) and tungsten (W) films deposited onto different substrates by sputtering. The following substrates have been considered: aluminum (EAl = 75 GPa), glass (Eglass = 73 GPa), silicon (ESi = 172 GPa), and sapphire (Esapphire = 440 GPa). For all these materials, a Poisson’s ratio of 0.25 was used. For a non-axisymmetric indenter, the modified BASh formula (6) should be used and the equivalent contact modulus is evaluated as

√

∗ Eeq =

πS √ , 2β A

(59)

where A is the contact area, β = β2 is the contact area shape factor (for a Berkovich indenter, β = 1.034, Oliver and Pharr, 1992). Correspondingly, according to (29), the indentation scaling factor is taken to be



a k η, t



√



β A −1 = 1 − α0 (η ) √ . πt

−1 (58)

and the function α 0 (η) is given by (30). Finally, it should be underlined that the asymptotic formula (58) is valid in the case of small-scale indentation, when the ratio a/t is relatively small. 4. Examples of application of the contact stiffness depth-sensing indentation method In this section, we utilize the analytical fitting function obtained by the asymptotic approach for understanding experimental results on depth-sensing nanoindentation of thin elasto-plastic layers of metals and inhomogeneous films of brittle materials such as coals.

(60)

For Berkovich and Vickers pyramidal indenters, thecontact area A or the harmonic radius of the contact area a = β A/π , which represents the characteristic size of the contact area, can be determined as a function of the indentation depth h by the Oliver–Pharr method (see Section 3.4). Here due to the lack of reference data,  the following frequently used simple relation a/t ≈ 24.5/π (h/t ) is utilized. Since the asymptotic formula (60) is not applicable for a > t, only the data below 1 were used in solving the best-fit minimization problem (57), (59), and (60). This circumstance is indicated in Fig. 2a and b by means of shading. It is seen from Fig. 2 that in the ∗ /E can be fitted small-scale indentation range, the behavior of Eeq s quite well by the one-parameter analytical approximation based on the first-order asymptotic model (60). 4.2. Nanoindentation of coal thin films As another example of application of the depth-sensing nanoindentation technique, we consider the experimental data (Borodich et al., 2015; Epshtein et al., 2015; Kossovich et al., 2016) recently obtained for very thin layers of coal (with the thickness about 13 μm) glued to a transparent rigid substrate by means of an epoxy adhesive (with the thickness about 15 μm), which is taken as an elastic substrate (Es = 1.68 and νs = 0.4). Fig. 3 presents the experimental data and a best-fit (dashed line) by the Borodich power-law



(57)

177

∗ Eeq = Es∗ + (E ∗f − Es∗ ) 1 −

 h α t

(61)

for Es∗ = 2 GPa with the following parameters: E ∗f = 27.69 GPa and α =√ 92.76. Note that the mentioned above approximate formula a ≈ 24.π h was utilized to relate the characteristic size of the contact area to the indentation depth. Poisson’s ratio of coal was taken to be ν f = 0.4. It is clearly seen from Fig. 3 that the measured values of equivalent contact modulus cannot be fitted by the asymptotic model (60), notwithstanding the condition that the indentation was performed in the small-scale range. In light of the fact that coal has a range of constituent layers with varying strength properties and relatively low fracture toughness KIc in the range of 50 to 450 kNm−3/2 (Danell et al., 1978), this implies the conclusion that the deformation behavior of coal under indentation is accompanied by fracture, which should be taken into account in interpreting the obtained experimental data. It has been shown in the previous section that the measured values of equivalent contact modulus can be well fitted by the

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4.0

a)

1.5

Sapphire Silicon Glass Aluminum

Relative equivalent modulus

Relative equivalent modulus

2.0

1.0 0.5 Al films

00

1

2 3 Relative contact size

4

b)

Sapphire Silicon Glass Aluminum

3.0 2.0 1.0

W films

00

1

2 3 Relative contact size

4

∗ /Es∗ of 0.5 μm Al films (a) and 640 nm W films (b) on different substrates plotted as a function of the characteristic size of the Fig. 2. Relative equivalent contact modulus Eeq contact area normalized by the film thickness. The experimental data were obtained by Saha and Nix (2002).

(complaint) substrates compared to that predicted by the selfsimilar model. As a plausible generalization of the linear elastic model (34) and (35) for the hyperelastic case, the following equations are suggested:

P = P1

 l 2+κ (d−1) l1

,

 l d ˜ h = . ˜1 l1 h

(63)

˜ is related to h via Eq. (62). Correspondingly, by analogy Here, h with the perturbation analysis given in Section 3.2, an analytical approximation for the force-displacement relation can be derived from Eqs. (62) and (63). ∗ /Es∗ as a function of the relative conFig. 3. Relative equivalent contact modulus Eeq tact size a/t.

asymptotic model (60) in the case of elastic-plastic material, however there is an evident disagreement between the model predictions (60) and the best fit to the data by (61). The disagreement may be explained by the existence of structural transformations of the coals during loading. Indeed, the coal components have often very low fracture toughness and it was crushed during indentation, i.e., the brittle material within the indentation zone is no longer a continuous elastic medium but rather a fine powder of crushed material. The closer the model predictions (60) to (61) the more ductile the coal component is. Hence, the DSI indentation may be used for quantitative evaluation of brittleness of the coal components. 5. Discussion of further generalizations The analysis presented above started with the self-similar solution (10) and (11) of the unilateral frictionless contact problem obtained by Borodich (1989) for a certain type of hyperelastic materials, whereas the first-order asymptotic model (31) was developed for a linearly elastic material. We observe that the accounting for the substrate effect in the range of small-scale indentation has resulted in a correction of the indentation depth (see formula (32)). In other words, for a given contact force P, the indenter displacement h differs from that predicted in the framework of the ˜ , by some quantity, which is proportional to P, self-similar model, h that is

˜− h=h

α0 P . 2E ∗f t

(62)

As it was shown previously (Argatov and Sabina, 2014), the value of the asymptotic constant α 0 is positive (negative) for relatively stiff (complaint) substrates. Therefore, the indentation depth predicted by Eq. (62) will be smaller (larger) for relatively stiff

6. Conclusions The depth-sensing indentation in its classic version suggests to test bulk material samples by a sharp indenter and to record the force-displacement curve. Measuring contact stiffness at the initial part of the unloading force-displacement curve, one can estimate the elastic modulus of a sample. However, the direct application of this approach to thin films attached to elastic substrates gives an estimation not for the film modulus but rather the equivalent modulus of the film-substrate structure. Usually a set of DSI experiments with varying maximum values of the indentation depth is performed to evaluate the influence of the substrate and the true modulus of the film is found by fitting the equivalent elastic moduli obtained experimentally by empirical functional forms. In the present paper, it has been suggested take into account the influence of substrate employing the first-order asymptotic model. The first-order asymptotic model means that the derived indentation scaling factor is a dimensionless function of the ratio between the contact radius, a, and the layer thickness, t, while the second and higher-order terms with respect to the ratio a/t are neglected. This model describes the indentation of an elastic layer bonded to an elastic substrate and provides an analytical fitting function to the set of the equivalent elastic moduli. It is shown that the analytical fitting function obtained by the asymptotic approach is in good agreement with both some known semi-empirical functional forms and experimental results on depth-sensing nanoindentation of thin elastic-plastic layers of metals, while the function is in a disagreement with results obtained for inhomogeneous films of brittle materials such as coals. It is argued that the disagreement is caused by structural transformations of the coals during loading. The brittle material within the indentation zone is no longer a continuous elastic medium but rather a fine powder of crushed material. Thus, the DSI indentation may be used to estimate not only the elastic properties of materials but also for evaluation if the material samples are brittle or ductile.

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