## Abstract

Using the connection between depth-sensing indentation by spherical indenters and mechanics of adhesive contact, a new method for non-direct determination of adhesive and elastic properties of contacting materials is proposed. At low loads, the force–displacement curves reflect not only elastic properties but also adhesive properties of the contact, and therefore one can try to extract from experiments both the elastic characteristics of contacting materials (such as the reduced elastic modulus) and characteristics of molecular adhesion (such as the work of adhesion and the pull-off force) using a non-direct approach. The direct methods of estimations of the adhesive characteristics of materials currently used in experiments are rather complicated due to the instability of the experimental force–displacement diagrams for ultra-low tensile forces. The proposed method is based on the use of the stable experimental data for the elastic stage of the force–displacement curve and the mechanics of adhesive contact for spherical indenters. Since the experimental data always have some measurement errors, mathematical techniques for solving ill-posed problems are employed.

## 1. Introduction

The depth-sensing indentation, i.e. the continuous monitoring of *P* and *δ* and plotting the *P*−*δ* diagram, where *P* is the applied load and *δ* is the displacement (the approach of the distant points of the indenter and the sample) for both loading and unloading branches, is widely used to analyse properties of materials and to estimate their mechanical characteristics. These techniques are especially important when mechanical properties of materials are studied using very small volumes of materials. High pressures under indenters may cause plastic deformation of the material, and hence the load–displacement curve at loading usually reflects both the elastic and plastic deformation of the material. Elastic characteristics are estimated using either the unloading branch of the *P*−*δ* diagram, assuming that the material deforms elastically at unloading and by employing the so-called BASh formula (Bulychev *et al*. 1975) and its modification (Borodich & Keer 2004*a*), or from the initial elastic stage of the diagram when the material is loaded by spherical indenters (for details see Chaudhri & Lim 2007). These estimations are usually obtained by neglecting the effects of molecular adhesion and using the formulae of the Hertz contact theory or their modifications (see discussion in Borodich & Keer (2004*b*), Fischer-Cripps (2006), Chaudhri & Lim (2007) and references therein).

In depth-sensing indentation problems, researchers are interested in identifying material properties using data on the displacement of the indenter measured at many force values, i.e. these are typical inverse problems in contrast to direct contact problems. Since the estimation of the parameters of an inverse problem requires both a model and measurements (e.g. Becky & Woodburyz 1998), various contact models are employed for the theoretical treatment of depth-sensing indentation. In particular, for indenters whose shape may be described by a homogeneous function *h*_{d} of degree *d*≥1, it was shown that Hertz-type contact problems are self-similar when the tested material is linear elastic (Galanov 1981*a*; Borodich 1983) or plastic with power-law hardening (physically nonlinear elastic medium; Galanov 1981*b*; Borodich 1989, 1998). This result may be applied not only to ideally shaped indenters, such as cones, pyramids and spheres (conical and pyramidal indenters are described by *h*_{1}, while spherical indenters of radius *R* in the Hertz approximation are described by , where *r* is a polar radius), but also to problems of practical importance when the indenter is blunt and its shape is described by *h*_{d} with 1≤*d*≤2 (Borodich *et al*. 2003). The use of dimensionless parameters characterizing indentation of materials with power-law hardening may contribute to solving a long-standing problem of establishing a correlation between the tensile elastic–plastic stress–strain curve and the hardness measurements (e.g. Davidenkov 1943), and allows researchers to find such unknown material parameters as the elastic modulus, the yield strength and the strain-hardening exponent (Dao *et al*. 2001; Lan & Venkatesh 2007).

The estimates obtained using nominally identical samples may depend on small uncertainties in measurements associated with test devices and environment, surface roughness and other aspects of indentation tests. The problem of sensitivity of the parameter estimates with respect to the data measurements is of great interest for the inverse or parameter estimation problems. The papers devoted to inverse and sensitivity analyses of depth-sensing indentation experiments have used various approaches, including the conventional deterministic batch approach (Bolzon *et al*. 2004), the training neural networks with results of finite-element simulations of indentation of materials (Huber *et al*. 2002), and other approaches (e.g. Dao *et al*. 2001; Lan & Venkatesh 2007). Although stochastic approaches are preferable (Zabaras & Ganapathysubramanian 2008), inverse techniques applied to approximate deterministic systems are often employed. Evidently, it would be important to assess the uncertainties associated with the modelling of the indentation process and with the output of the model itself. However, a sensitivity analysis is out of the scope of the present paper and the analysis will be performed in planned further studies.

A probabilistic formulation of an inverse problem may also be considered, when two covariance matrices—*C*_{I} that represents the *a priori* uncertainties on the model parameters and *C*_{O} that represents the uncertainties on the output parameters—are studied (e.g. Tarantola 2004). If one assumes that all uncertainties of both the input and output data are independent Gaussian with zero mean and standard deviations *σ*_{I} and *σ*_{O}, respectively, and if both *C*_{I} and *C*_{O} are diagonal and isotropic, i.e. and , where *I* is an identity matrix, then the probabilistic formulation of an inverse problem reduces to a problem of the Tikhonov regularization (Tikhonov & Arsenin 1977), with parameter . Owing to its simplicity, the least-squares method, which is a particular case of the regularization techniques, is widely used for the resolution of inverse problems. The only drawback of the least-squares approach is the lack of robustness, i.e. its strong sensitivity to a small number of large outliers in a dataset (Tarantola 2004). Both the least-squares (e.g. Bolzon *et al*. 2004) and the Tikhonov regularization (e.g. Cao & Lu 2004) approaches have been applied to depth-sensing indentation problems. The problem of indentation is ill posed according to Hadamard's definition (e.g. Tikhonov & Arsenin 1977), because it is overdetermined due to a possible high sensitivity of the solution of the inverse problem to noise in the measured data. Since sensitivity has not been analysed, a general Tikhonov regularization approach will be employed in this paper.

To the best of our knowledge, all the above-mentioned papers have dealt with depth-sensing indentation problems neglecting the influence of adhesion. The depth-sensing nanoindentation technique was first introduced by Kalei (1967, 1968). He noted that adhesion between the indenter and the sample may affect the experimental *P*−*δ* diagrams. Indeed, at low loads, the load–displacement curves are greatly affected not only by the elasticity of the contacting materials but also by their adhesive properties, and therefore one can try to extract from the experiments both the elastic characteristic of contacting materials and characteristics of molecular adhesion.

Adhesion to a solid surface (substrate) is a very important physical process. Depending on the properties of the adhering material (the sticking medium), one can distinguish adhesion of fluids and solids. The possibility of adhesion at an isothermal process is defined by the reduction of the free surface energy that is equal to the equilibrium work of adhesion *w* (Zimon 1988). In the mechanics of adhesive contact, it is usually assumed that one knows the work of adhesion *w*, which is represented as . The numbers 1, 2 and 3 represent, respectively, the tested material (substrate), the adhering material and the environmental medium (e.g. the air), and *γ*_{13} and *γ*_{23} are the specific surface energies of the tested material and the material of the adhering indenter (in particular, the material of the spherical indenter), respectively, on the boundary with the environmental medium before adhesion, and *γ*_{12} is the specific surface energy on the boundary separating the contacting pair (in particular, the tested plane and the sphere).

The current indentation methods for the determination of adhesive and elastic characteristics, such as the adhesive (pull-off) force, the reduced elastic modulus (*E*^{*}) and the work of adhesion *w* (or the surface energy *γ*), have been reviewed by Wahl and colleagues (Ebenstein & Wahl 2006; Wahl *et al*. 2006). It follows from these papers that (i) both DC and AC measurements are employed (Pethica & Oliver 1987), (ii) only direct measurements of the adhesive force have been developed, and (iii) although detailed information for *P*−*δ* curves has been obtained, only several key points that belong mainly to unstable stages of the curve have been used. One can see that using the current direct methods, even highly skilled researchers need to exercise heightened caution during measurements and their interpretation. One should realize that the direct measurements of the adhesive force and the minimum value of the displacement by the *P*−*δ* diagram are rather difficult. The difficulty is caused not only by the restricted precision of the measurement devices but also by the instability of the diagrams at ultra-low tensile loads. Although it was shown by Maugis & Barquins (1978) that the theoretical diagram for the Johnson-Kendall-Roberts (JKR) theory is stable under some assumptions about the measurement device, there are always problems caused by the instability of a real experimental device. This instability may be clearly observed in experiments for some materials (e.g. Pethica & Oliver 1987; Wahl *et al*. 2006). In addition, the tensile part of the *P*−*δ* diagram may be greatly influenced by roughness. On the other hand, it has been shown by Borodich & Galanov (2002) for non-adhesive contact problems that the trend of the compressive part of the *P*−*δ* diagram is independent of fine distinctions between functions describing roughness. One can assume that the compressive part of the *P*−*δ* diagram for adhesive contact is much less sensitive to roughness than the tensile part of the diagram.

Currently, only the stable parts of the *P*−*δ* experimental diagrams, i.e. for the compressive forces *P*≥0 and corresponding displacements *δ*≥0, can be obtained in a reliable way with high precision. One would need appropriate and significant modernizations of measurement techniques in order to obtain reliable results for unstable parts of the *P*−*δ* diagram. These modernizations would need to not only improve the measurement precision, but also provide the stability of the measurements. The stability of experimental data means that the dispersion of measurements is small in comparison with the average of the measured value. Evidently, the tensile part of the *P*−*δ* diagram where one can observe the jump-out contact is unstable (figure 1).

It is worth noting that, traditionally, the compressed forces are considered in the mechanics of adhesive contact as positive and the tensile forces as negative. Hence, according to figure 1, the critical force is negative and it is denoted as −*P*_{adh}, and the minimum of *δ* is also negative.

In this paper, a new method is proposed for the non-direct determination of the values *P*_{c} and *δ*_{c} that are the main characteristics of contact, i.e. they reflect the interplay of elastic and adhesive forces at contact. *P*_{c} and *δ*_{c} are characteristic scales for the force *P* and the corresponding displacement *δ* at low loads. For the JKR theory, they have a simple physical meaning, namely *P*_{c} and *δ*_{c} are the exact absolute value of the pull-off force and the minimum value of the displacement, respectively. While in direct methods (Ebenstein & Wahl 2006; Wahl *et al*. 2006) *P*_{c} is extracted from the measurements on the unstable part of the diagram (strictly speaking, they extracted the maximum adhesion force *P*_{adh} that has the same value as *P*_{c} because the JKR theory was applicable in their experiments), the proposed method is non-direct because it is based only on the use of the stable experimental data for the elastic stage of the *P*−*δ* diagram, whose points are relatively far from the unstable parts. After the values of characteristic scales *P*_{c} and *δ*_{c} are extracted, the values of the pull-off force *P*_{adh} and the minimum value of the displacement can easily be found. It is assumed that the experimental points are obtained by the so-called DC experiments, i.e. quasi-static deformation of the materials. The new method is based on the use of mechanics of adhesive contact for spherical indenters along with employing mathematical techniques for solving ill-posed problems. The principal difference between the methods developed earlier and the current method is that the compressible part of the *P*−*δ* diagram is used for extracting adhesive properties of tested materials.

## 2. Mechanics of adhesive contact for spherical indenters

Apparently, Bradley (1932) was the first to consider the attraction of a rigid sphere of radius *R* to a flat surface of a rigid body and obtain the following expression for the pull-off force:(2.1)However, he did not consider the elastic deformation of the contacting solid. Derjaguin (1934) presented the first attempt to consider the problem of adhesion between an elastic sphere and a half-space. However, his arguments were not completely correct (Kendall 2001).

For spherical indenters, mechanics of adhesive contact is currently well established. The theories of adhesive contact of spheres are considered, which can be represented as a functional relation of the following type (Maugis 2000):(2.2)where *F* is given by one of the well-established theories that include JKR, Derjaguin-Müller-Toporov (DMT) and Maugis theories, and *P*_{c} and *δ*_{c} are characteristic values for the scale of the force *P* and the corresponding displacement *δ* at low loads and small displacement. It was stated by Tabor (1977) and shown precisely by Muller *et al*. (1980) that one can introduce non-dimensional parameters such that the JKR and DMT theories apply to the opposite ends of the range of any of the parameters. Here, the Maugis parameter *λ* is used. One can represent the Maugis parameter as(2.3)Here, *z*_{0} is the equilibrium separation between surfaces (for a number of materials, *z*_{0} is within the range 0.3–0.5 nm). Maugis estimated that the DMT theory (Derjaguin *et al*. 1975) corresponds to the small values of *λ*, *λ*<0.1, while the JKR theory (Johnson *et al*. 1971) corresponds to the large values of *λ*, *λ*>5 (Carpick *et al*. 1999; Maugis 2000).

For each value of the parameter *λ*, the graph of the functional relation *P*−*δ* is situated between the corresponding graphs for the JKR and DMT theories (Maugis 2000). In (2.2), the Maugis parameter *λ* represents the most suitable theory for the contacting materials and indenters. Here, the Maugis notations are used for the adhesive characteristics(2.4)where *R* is the radius of the sphere in contact with a flat surface of an elastic sample, . *E*^{*} is the reduced Young moduluswhere *E*_{i} and *ν*_{i} (*i*=1, 2) are the Young modulus and the Poisson ratio of the first and the second solids, respectively. For a rigid indenter, i.e. *E*_{2}=∞, one has , where *E*=*E*_{1} and *ν*=*ν*_{1} are the Young modulus and the Poisson ratio of the half-space, respectively.

In the JKR theory, traditionally, *P*_{c} denotes the absolute value of the pull-off force, which is reflected in (2.4), and *δ*_{c} is the modulus of the minimum value of the displacement that occurs due to adhesion. However, as it has been mentioned for other theories of adhesive contact, the values *P*_{c} and *δ*_{c} are characteristic scales for the force and the corresponding displacement at low loads.

The functional expression (2.2) in the Maugis simplified representation of the DMT theory has the form(2.5)while the functional expression in the JKR theory has the form(2.6)

In accordance with the above-mentioned adhesive theories, the DE branch (*P*≥0 and *δ*≥0) of the experimental *P*−*δ* diagram is stable (figure 1). It is assumed that this branch is described by a known functional expression(2.7)The problem is finding the characteristics *P*_{c} and *δ*_{c} using only the experimental points of the branch DE of the *P*−*δ* diagram.

## 3. Determination of the scale characteristics *P*_{c} and *δ*_{c} as an ill-posed problem

### (a) Formulation of the ill-posed problem

If (*P*_{i}, *δ*_{i}), *i*=1, …, *N*, are the experimental values of the compressing load *P*≥0 and the corresponding values of the displacement *δ*≥0, respectively, then it follows from (2.7) that the problem is reduced to the determination of *P*_{c} and *δ*_{c} (two unknown values) from the following system of a large number of nonlinear equations and two inequalities:(3.1)The system (3.1) is overdetermined for *N*>2, and therefore normally there exists no solution of (3.1) in the classic sense. Since the experimental data always have some measurement errors, this problem and the system (3.1) belong to the class of nonlinear ill-posed problems. Methods for solving such problems are known as regularization methods. A simple form of regularization, generally termed the Tikhonov regularization, is essentially a trade-off between fitting the data and reducing a norm of the solution. It is known that the application of the least-squares fitting to such problems often leads to an approximation that is unstable to measurement errors (e.g. Tikhonov & Arsenin 1977), and it may give a very poor approximation of the actual values of *P*_{c} and *δ*_{c}.

According to the method of regularization of ill-posed problems, solving system (3.1) is reduced to the problem of minimization of the following functional:(3.2)where(3.3)where *α*>0 is the regularization parameter and ‖.‖ denotes the Euclidean norm of the vector. The components *F*_{N+1} and *F*_{N+2} reflect the inequalities in the system (3.1), which can be represented by the following equations:(3.4)

It follows from (3.3) and (3.4) that and . Hence, the functional (3.2) can be represented in the following expanded form:(3.5)

Note that the functional *Φ*_{α} should be written in a dimensionless form because the physical dimensions of the load and the displacement are different. Furthermore, it is assumed that the values of the functional are dimensionless. On the other hand, it will be shown that for the JKR and DMT theories, the minimization of the functional may be split into two independent problems for *P*_{c} and *δ*_{c}, respectively, and in this case one can use dimensional variables.

To avoid undesirable instability of the solution, one has to minimize simultaneously the squares of the norms of the discrepancy vector * F* and the solution vector

*in accordance with (3.2). For any value of*

**x***α*, there is a corresponding vector that gives a minimum to the functional

*Φ*

_{α}(e.g. Tikhonov & Arsenin 1977). The parameter

*α*determines the relative weight of each of the terms ‖

*‖ and ‖*

**F***‖ of (3.2).*

**x**### (b) Determination of the regularization parameter *α*

If it is known *a priori* that the experimental values (*P*_{i},*δ*_{i}), *i*=1, 2, …, *N*, are practically exact, i.e. their measurement errors can be neglected, then the regularization parameter *α* should be taken as *α*=0. This means that one may substitute *α*=0 into (3.2) and (3.5), and use the least-squares approach. If the measurement errors cannot be neglected, then the regularization parameter *α* (which is also known as the Tikhonov parameter) should be determined. The determination of the regularization parameter *α* is one of the main difficulties in the regularization of ill-posed problems. Apparently, the most common methods for the determination of *α* are as follows: the discrepancy principle; the *L*-curve method; and the zero-crossing choice (e.g. Tikhonov & Arsenin 1977; Hansen & Oleary 1993; Johnston & Gulrajani 1997; Wach *et al*. 2001).

The use of the discrepancy principle assumes the *a priori* knowledge of the precision of the measurements of (*P*_{i}, *δ*_{i}), and hence this method would be preferable. However, it will not be used in this paper because the precision of the experiments, which will be discussed further in §6, is unknown.

In the *L*-curve method, one has to plot a graph of the norm of the regularization solution ‖**x**_{α}‖ versus the norm of the corresponding discrepancy for the entire range of valid values of the regularization parameter *α*. The vector **x**_{α} gives the minimum to (3.2) for a fixed *α*. In a number of cases, the shape of the graph is in the form of an ‘L’, and hence it is referred to as the *L*-curve. The seeking value of *α* is taken at the point of maximum curvature of the graph.

In the zero-crossing choice method, the value of the regularization parameter *α* is taken as the root of the equation *B*(*α*)=0, whereAs shown by Johnston & Gulrajani (1997), *B*(*α*) usually has two zeros; the smaller zero is the desired one and the larger zero can be safely ignored. They also noted that sometimes this approach breaks down. However, this occurs in a very rare case when measurement noise is absent. The advantage of the zero-crossing choice and the *L*-curve approaches is that they do not depend on *a priori* knowledge of measurement errors (noise levels).

### (c) The iterative linearized optimization technique for a nonlinear ill-posed problem

Since the system under consideration (3.1) is nonlinear, the iterative linearized optimization technique for nonlinear ill-posed problems should be employed. Using the notations in (3.3), the vector form of (3.1) can be written as(3.6)If is an approximation to the solution , where *k*=0, 1, 2, … is the number of the approximation, then (3.6) can be linearized at the point **x**_{k} in order to obtain a linear problem for the determination of the next approximation *x*_{k+1},(3.7)where the matrix *A*_{k} has two columns and (*N*+2) rows, i.e.For a fixed **x**_{k}, the system (3.7) is a linear ill-posed problem. Any of the above-mentioned regularization techniques applicable to linear ill-posed problems may be used to solve the system. Thus, the problem (3.1) can be solved by an iterative linearized optimization technique: and .

## 4. The application of the method for the determination of *P*_{c} and *δ*_{c}

### (a) The DMT theory of adhesive contact

In this case, the overdetermined system (3.1) for the determination of *P*_{c} and *δ*_{c} is written as(4.1)For convenience, it is assumed further that the sequence {*δ*_{i}} is ordered, i.e.(4.2)

In order to separate the equations for the unknown value *P*_{c}, one can use some straightforward calculations and represent the system (4.1) in the following form:(4.3)where the coefficientsdo not depend on *P*_{c} and *δ*_{c}, and the coefficientdepends only on *P*_{c}. This representation means that the value of *P*_{c} can be found independently of *δ*_{c} and the value of *δ*_{c} should be determined after the evaluation of the *P*_{c} value.

In addition, the representation of the system in the form (4.3) states that, for a given *P*_{c}, the discrepancy of the former group of equations (4.3) is approximately estimated with the mean square error of the load measurement *σ*_{P}, while, for a given *δ*_{c}, the discrepancy of the latter group of equations (4.3) is approximately estimated with the mean square error of the displacement measurement *σ*_{δ}. These estimations are valid for the small values of the relative errors and .

It follows from the above and (3.5) that the functional corresponding only to the variable *P*_{c} has the following form:(4.4)where the parameter of regularization *α*>0 should be determined by one of the above-mentioned methods.

The presence of the inequality *P*_{c}≥0 in the system (4.1) (in (4.4), the terms correspond to this inequality) states that the problem of the determination of *P*_{c} is a nonlinear ill-posed problem. Therefore, it is rather difficult to solve. However, solving becomes considerably simplified if, for any *α*>0, the values of are positive, i.e.(4.5)because in this case the problem becomes linear. Indeed, for a fixed *α*, the value of determined by (4.5) minimizes the following functional:(4.6)for a linear ill-posed problem(4.7)It follows from (4.5) that a necessary and sufficient condition of positiveness of is the condition(4.8)

Thus, if the experimental values (*P*_{i}, *δ*_{i}), *i*=1, …, *N*, satisfy the condition (4.8), then there exists the adhesive characteristic *P*_{c} in accordance with the DMT theory, and the inequalities in (4.1) have been already satisfied. Hence, for given experimental values, (4.8) is the necessary and sufficient condition in which there exists an adhesive characteristic *P*_{c} in the framework of the DMT theory. The value of this characteristic is calculated in accordance with formula (4.5), where the value of the regularization parameter *α* is determined by one of the known methods (the discrepancy method, *L*-curve method, and so on). If condition (4.8) is not satisfied, then it is possible that there is no adhesive characteristic in the framework of the DMT theory, while *P*_{c} may exist in accordance with other theories.

After the value of *P*_{c} has been determined, the adhesive characteristic *δ*_{c} is determined from the latter group of equations in (4.3). Similar to (4.5), one obtains(4.9)where the value of the regularization parameter *α* is estimated by one of the known methods and this value differs from the regularization parameter used in (4.5). It is important to note that if there exists the adhesive characteristic *P*_{c} (i.e. *P*_{c}>0) in the framework of the DMT theory, then it follows from (4.9) that there also exists *δ*_{c} (i.e. *δ*_{c}>0).

As it has been mentioned above, if it is known *a priori* that the experimental values (*P*_{i}, *δ*_{i}), *i*=1, 2, …, *N*, are practically exact, i.e. their measurement errors can be neglected, then the regularization parameter *α* should be taken as *α*=0. This means that one may substitute *α*=0 into (4.4)–(4.6) and (4.9).

### (b) The JKR theory of adhesive contact

In this case, the overdetermined system (3.1) for the determination of *P*_{c} and *δ*_{c} has the form(4.10)After some straightforward calculations, the system (4.10) can be represented in the following form:(4.11)where *b*_{i}=*δ*_{1}, and the functional coefficients *a*_{i} and *c*_{i} are(4.12)

This representation means that the value of *P*_{c} can be found independently of *δ*_{c} by solving the following nonlinear ill-posed problem (the problem is nonlinear because *a*_{i}(*P*_{c}) is a nonlinear function of *P*_{c}):(4.13)Then, the value of *δ*_{c} can be determined by solving a linear ill-posed problem(4.14)where it is assumed that the positive value of *P*_{c} has already been found (by solving the problem (4.13)), and therefore the restriction *δ*_{c}≥0 is satisfied.

In (4.13) and (4.14), the discrepancies are determined using the experimental values (*P*_{i}, *δ*_{i}) with the mean square error that should be approximately equal to the mean square error of the displacement measurement *σ*_{δ}.

As it has been mentioned, the nonlinear ill-posed problem (3.6) may be solved using a sequence of linear ill-posed problems (3.7) whose solutions converge to the solution of the nonlinear problem. If *P*_{ck} (*k*=0, 1, 2, …) are the known approximations to the solution of (4.13), then the linearized version of the equation at the point *P*_{c}=*P*_{ck} has the form(4.15)where is the derivative of the function *a*_{i}(*P*_{c}) with respect to *P*_{c} at *P*_{c}=*P*_{ck}, and , and *P*_{c(k+1)} is the next approximation to *P*_{c}. As the initial approximation is *P*_{c0}, one can take that is obtained by the DMT theory.

The solution of the problem (4.15) can be found in the same way as the above-suggested way of solving the problem of the DMT theory, i.e.(4.16)Evidently, formula (4.16) is an analogue of (4.5). It is worthwhile repeating that if the measurement errors of (*P*_{i}, *δ*_{i}), *i*=1, …, *N*, can be neglected, then the regularization parameter *α*_{k+1} in (4.16) can be taken as zero.

### (c) Other theories of adhesive contact

In the general case, the parameter *λ* may belong to the range 0.1<*λ*<5 when neither the JKR nor the DMT models are applicable to the sample. As it has been mentioned, for each value of the parameter *λ*, there is a corresponding functional relation of the type (2.2). The determination of *P*_{c} and *δ*_{c} can be done by solving the following nonlinear ill-posed problem:(4.17)in accordance with the procedure described in §3. It is clear from (2.3) that the parameter *λ* in (4.17) depends on *δ*_{c}.

## 5. Estimations of elastic and molecular adhesion characteristics

If the scale values *P*_{c} and *δ*_{c} have been determined by solving ill-posed problems, then the work of adhesion *w* and the reduced modulus *E*^{*} can be obtained using (2.4). Indeed, one has(5.1)and(5.2)This approach can be used as an alternative to the currently used BASh formula (Borodich & Keer 2004*b*) for the estimation of the elastic modulus of the material.

After the determination of the values *P*_{c} and *δ*_{c}, one can substitute them into (2.2), with *λ* defined from (2.4), and calculate the pull-off force *P*_{adh} as the modulus of the minimum force of the *P*−*δ* diagram and the minimum displacement min *δ* of the diagram. It is worth remembering that the equalities and *P*_{adh}=−*P*_{c} are valid only for the JKR theory. Thus, instead of direct measurements of *P*_{adh} and the minimal value of displacement, a non-direct way for estimation of these values is proposed.

## 6. Examples of extracting the seeking characteristics from the experimental results

The general techniques of extracting the adhesive characteristics *P*_{c} and *δ*_{c}, the work of adhesion *w*, and the elastic modulus *E*^{*} from the experimental data are illustrated by an example. After the examination of some *P*−*δ* graphs presented by Wahl *et al.* (2006), a series of coordinate pairs were generated, which correspond to the compressive part of the *P*−*δ* diagram for the sample DP130. Ten representative pairs of the coordinates *P*_{i} and *δ*_{i} are given in table 1. These numbers are not the original measurements, but they were extracted from the published experimental graph. This dataset has only illustrative purpose.

Since the mean square errors *σ*_{P} and *σ*_{δ} of the measurements *P*_{i} and *δ*_{i} are unknown, the zero-crossing and *L*-curve methods are used further for the determination of the Tikhonov parameter *α* in the example.

The first theoretical model of adhesion employed is the DMT model. Using (4.5) and (4.9), the adhesion characteristics *P*_{c} and *δ*_{c} are determined. The parameters of regularization corresponding to these formulae are estimated by *L*-curves presented in figures 2 and 3. The obtained values of the parameters are equal to *α*=0.001 (figure 2) and *α*=0.002 (figure 3) for (4.5) and (4.9), respectively. The dimensional values corresponding to these *α* are *P*_{c}=1.55 μN and *δ*_{c}=228.4 nm, respectively.

These values *α*, *P*_{c} and *δ*_{c} correspond to the points of maximum curvature of the *L*-curve indicated in figures 2 and 3. It is clear that substituting the obtained value of *δ*_{c} in (2.3), one obtains *λ*≫5, and hence the DMT theory is not applicable. As already mentioned, this value corresponds to the region of applicability of the JKR model. Therefore, the same parameters *P*_{c} and *δ*_{c} should be determined now using the JKR model. The adhesion characteristic *P*_{c} is determined by the iterative linearized technique for nonlinear ill-posed problems (see formulae (4.13), (4.15) and (4.16) in §4.2), while *δ*_{c} is determined by solving a linear ill-posed problem (4.14). To determine the regularization parameter, the zero-crossing method has been applied. After applying this method, the following values for *α* were obtained: *α*=0.095 for *P*_{c} and *α*=0.003 for *δ*_{c}. Using (4.16), (4.9) and (4.12), the following values of *P*_{c} and *δ*_{c} that correspond to the values of *α* were obtained: *P*_{c}=2.63 μN and *δ*_{c}=328.8 nm.

After the calculation of these parameters, one needs to calculate the Maugis parameter *λ* in order to check whether the JKR model is applicable in this case. Taking *z*_{0}=0.5 nm, one obtains *λ*>5. Hence, the JKR model is applicable. In the general case, the parameter *λ* could belong to the range 0.1<*λ*<5 when neither the JKR nor the DMT models are applicable. In this case, one would need to solve the ill-posed problem (4.17).

Substituting the obtained values of *P*_{c} and *δ*_{c} into (5.1) and (5.2), for *R*=10 μm, one obtains *w*=56 mJ m^{−2} and *E*^{*}=1.91 MPa (*E*=1.43 MPa).

To perform some preliminary studies of sensitivity of the proposed techniques, the above characteristics were extracted from the data with computer-generated Gaussian noise with zero mean. These simulations showed that the solution is stable to the noise within a wide range of standard deviations.

One can compare the results extracted from the measurements on the stable branch DE (figure 1) with the results obtained by Wahl *et al*. (2006) using direct measurements. The direct measurements of the pull-off force on the unstable branch DCF gave the following values: *P*_{adh}=1.9−2.5 μN. Assuming that the JKR theory is valid and hence *P*_{adh}=*P*_{c}, they calculated *w*=47±2 mJ m^{−2}. These values correspond to *P*_{c}=2.12−2.31 μN. If one substitutes *P*_{c}=2.5 μN into (5.1), then *w*=53.05 mJ m^{−2}. This is quite close to the value *w*=56 mJ m^{−2} that has been obtained above using the indirect method.

Using the additional AC experiments on stiffness, they obtained *E*^{*}=1.45±0.05 MPa, while their estimations of the same modulus using the BASh formula gave *E*^{*}=1.3 MPa.

## 7. Conclusion

A general method to the non-direct determination of adhesive and elastic properties of materials, based on the use of stable experimental data for the elastic stage of the *P*−*δ* curve and mechanics of adhesive contact for spherical indenters, has been proposed. Since the experimental data always have some measurement errors, it has been proposed that mathematical techniques be employed for solving ill-posed problems. These techniques may greatly enhance the current possibilities of measurement devices. After the values of characteristic scales *P*_{c} and *δ*_{c} have been extracted from the reliable data, other mechanical and adhesive characteristics are calculated. In particular, the value of the pull-off force *P*_{adh} and the minimum value of the displacement are calculated by the analysis of the curve (2.2) with the extracted values *P*_{c} and *δ*_{c}.

The scale characteristics *P*_{c} and *δ*_{c} have been found using only the experimental points of the compressive branch DE of the *P*−*δ* diagram. Although the condition of positiveness of the external load was directly involved in the above presentation of the method, the condition is not a necessary one. If the measurements are reliable for *P*<0, then the experimental data can be added to the data for *P*>0. The method will be the same.

There is a belief that this method is simpler and more reliable than the existing direct methods when the adhesive characteristics are extracted from the tensile branch of the *P*−*δ* diagram. In the above example, the experimental conditions are within the applicability of the JKR theory. In this case, one could use careful direct measurements of the pull-off force *P*_{adh} and substitute this value into (5.1) in order to estimate the work of adhesion. If the experimental conditions are beyond the applicability of the JKR theory, then *P*_{adh} is no longer equal to *P*_{c}, and hence one will have problems in the application of the direct methods because *P*_{adh} cannot be directly substituted into (5.1).

The example shows that the values of the regularization parameter *α* are rather small. This means that one can take *α*=0 and extract the seeking material parameters just by employing the least-squares fitting to the data. However, in the general case, the latter approach may give a very poor approximation of the actual values of *P*_{c} and *δ*_{c} and, in turn, the seeking material parameters. Evidently, further experimental studies should be performed for the development of the introduced approach. The proposed method can be easily implemented in the measuring devices, and it is expected that the corresponding experimental studies will be performed.

## Acknowledgments

Thanks are due to the Royal Society for funding the visit of B.A.G. to the University of Cardiff, during which the above work was initiated. The authors are grateful to the Leverhulme Trust for funding the International Network ADHESINT that enables them to continue their collaboration. The materials of the paper have been presented at the 3rd International Indentation Workshop (Cavendish Laboratory, University of Cambridge, 15–20 July 2007). The authors are also grateful to Dr M. M. Chaudhri (Cavendish Laboratory) for his invitation to deliver a talk on the above results at the workshop.

## Footnotes

↵† Permanent address: Institute for Problems in Materials Science, National Academy of Sciences of Ukraine, 3 Krzhyzhanovsky Street, Kiev 03142, Ukraine.

- Received January 29, 2008.
- Accepted May 12, 2008.

- © 2008 The Royal Society