Contact Mechanics

Abstract

There has been considerable recent interest in the mechanical characterisation of thin film systems and small volumes of material using depth-sensing indentation tests with either spherical or pyramidal indenters. Usually, the principal goal of such testing is to extract elastic modulus and hardness of the specimen material from experimental readings of indenter load and depth of penetration. These readings give an indirect measure of the area of contact at full load, from which the mean contact pressure , and thus hardness, may be estimated. The test procedure, for both spheres and pyramidal indenters, usually involves an elastic–plastic loading sequence followed by an unloading. The validity of the results for hardness and modulus depends largely upon the analysis procedure used to process the raw data. Such procedures are concerned not only with the extraction of modulus and hardness, but also with correcting the raw data for various systematic errors that have been identified for this type of testing.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1.1 Introduction

There has been considerable recent interest in the mechanical characterisation of thin film systems and small volumes of material using depth-sensing indentation tests with either spherical or pyramidal indenters. Usually, the principal goal of such testing is to extract elastic modulus and hardness of the specimen material from experimental readings of indenter load and depth of penetration. These readings give an indirect measure of the area of contact at full load, from which the mean contact pressure , and thus hardness, may be estimated. The test procedure, for both spheres and pyramidal indenters, usually involves an elastic–plastic loading sequence followed by an unloading. The validity of the results for hardness and modulus depends largely upon the analysis procedure used to process the raw data. Such procedures are concerned not only with the extraction of modulus and hardness, but also with correcting the raw data for various systematic errors that have been identified for this type of testing. The forces involved are usually in the millinewton (10⁻³ N) range and are measured with a resolution of a few nanonewtons (10⁻⁹ N). The depths of penetration are on the order of microns with a resolution of less than a nanometre (10⁻⁹ m). In this chapter, the general principles of elastic and elastic–plastic contact and how these relate to indentations at the nanometre scale are considered.

1.2 Elastic Contact

The stresses and deflections arising from the contact between two elastic solids are of particular interest to those undertaking indentation testing. The most well-known scenario is the contact between a rigid sphere and a flat surface as shown in Fig. 1.1.

Fig. 1.1

Schematic of contact between a rigid indenter and a flat specimen with modulus E. The radius of the circle of contact is a, and the total depth of penetration is h _max. h _a is the depth of the circle of contact from the specimen free surface, and h _c is the distance from the bottom of the contact to the contact circle (the contact depth)

Hertz [1, 2] found that the radius of the circle of contact a is related to the indenter load P, the indenter radius R, and the elastic properties of the contacting materials by:

$$ {a^3} = \frac{3}{4}\,\frac{{PR}}{{{E^*}}} $$

(1.1)

The quantity E ^* combines the modulus of the indenter and the specimen and is given by [3]:

$$ \frac{1}{{{E^*}}}\, = \,\frac{{( {1 - {\nu ^2}} )}}{E}\, + \,\frac{{( {1 - \nu {'^2}} )}}{{E'}} $$

(1.2)

where the primed terms apply to the indenter properties. E ^* is often referred to as the “reduced modulus ” or “combined modulus ” of the system. If both contacting bodies have a curvature, then R in the above equations is their relative radii given by:

$$ \frac{1}{R} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} $$

(1.3)

In Eq. 1.3 the radius of the indenter is set to be positive always, and the radius of the specimen to be positive if its center of curvature is on the opposite side of the lines of contact between the two bodies.

It is important to realize that the deformations at the contact are localized and the Hertz equations are concerned with these and not the bulk deformations and stresses associated with the method of support of the contacting bodies. The deflection h of the original free surface in the vicinity of the indenter is given by:

$$ h = \frac{1}{{{E^*}}}\frac{3}{2}\frac{P}{{4a}}\left( {2 - \frac{{{r^2}}}{{{a^2}}}} \right)\quad \quad r \le a $$

(1.4)

It can be easily shown from Eq. 1.4 that the depth of the circle of contact beneath the specimen free surface is half of the total elastic displacement. That is, the distance from the specimen free surface to the depth of the radius of the circle of contact at full load is h _a  =  h _c  = h _max/2.

The distance of mutual approach of distant points in the indenter and specimen is calculated from:

$$ {\delta ^3}\, = \,{\left( {\frac{3}{{4{E^*}}}} \right)^2}\,\frac{{{P^2}}}{R} $$

(1.5)

Substituting Eq. 1.4 into Eq. 1.1, the distance of mutual approach is expressed as:

$$ \delta = \frac{{{a^2}}}{R} $$

(1.6)

For the case of a non-rigid indenter, if the specimen is assigned a modulus of E ^*, then the contact can be viewed as taking place between a rigid indenter of radius R. δ in Eq. 1.5 becomes the total depth of penetration h _max beneath the specimen free surface. Rearranging Eq. 1.5 slightly, we obtain:

$$ P = \frac{4}{3}\,{E^*}\,{R^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}\,{h^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}} $$

(1.7)

Although the substitution of E ^* for the specimen modulus and the associated assumption of a rigid indenter of radius R might satisfy the contact mechanics of the situation by Eqs. 1.1–1.7, it should be realized that for the case of a non-rigid indenter, the actual deformation experienced by the specimen is that obtained with a contact with a rigid indenter of a larger radius R ⁺ as shown in Fig. 1.2. This larger radius may be computed using Eq. 1.1 with E¢ in Eq. 1.2 set as for a rigid indenter. In terms of the radius of the contact circle a, the equivalent rigid indenter radius is given by [4]:

Fig. 1.2

Contact between a non-rigid indenter and the flat surface of a specimen with modulus E is equivalent to that, in terms of distance of mutual approach , radius of circle of contact, and indenter load, as occurring between a rigid indenter of radius R _i and a specimen with modulus E ^* in accordance with Eq. 1.1. However, physically, the shaded volume of material is not displaced by the indenter and so the contact could also be viewed as occurring between a rigid indenter of radius R ⁺ and a specimen of modulus E. (Courtesy CSIRO)

$$ {R^ + } = \frac{{4{a^3}\,E}}{{3( {1 - {v^2}} )P}} $$

(1.8)

There have been some concerns raised in the literature [5] about the validity of the use of the combined modulus in these equations but these have been shown to be invalid [4, 6, 7]. Even if the deformation of the indenter is accounted for, the result, correctly interpreted, is equivalent to a rigid indenter in contact with a compliant specimen.

The mean contact pressure , p _m , is given by the indenter load divided by the contact area and is a useful normalizing parameter, which has the additional virtue of having actual physical significance.

$$ {p_m} = \frac{P}{{\pi {a^2}}} $$

(1.9)

Combining Eqs. 1.1 and 1.9, we obtain:

$$ {p_m} = \left( {\frac{{4{E^*}}}{{3\pi }}} \right)\frac{a}{R} $$

(1.10)

The mean contact pressure is often referred to as the “indentation stress ” and the quantity a/R as the “indentation strain. ” This functional relationship between p _m and a/R foreshadows the existence of a stress–strain response similar in nature to that more commonly obtained from conventional uniaxial tension and compression tests. In both cases, a fully elastic condition yields a linear response. However, owing to the localized nature of the stress field, an indentation stress–strain relationship yields valuable information about the elastic–plastic properties of the test material that is not generally available from uniaxial tension and compression tests.

For a conical indenter , similar equations apply where the radius of circle of contact is related to the indenter load by [8]:

$$ P = \frac{\pi }{2}\,{a^2}\,{E^*}\,\cot \alpha $$

(1.11)

The depth profile of the deformed surface within the area of contact is:

$$ h = \left( {\frac{\pi }{2} - \frac{r}{a}} \right)a\,\cot \,\alpha \quad r \le a $$

(1.12)

where α is the cone semi-angle as shown in Fig. 1.3. The quantity acotα is the depth of penetration h _c measured at the circle of contact. Substituting Eq. 1.11 into Eq. 1.12 with r = 0, we obtain:

Fig. 1.3

Geometry of contact with conical indenter

$$ P = \frac{2}{\pi }\,{E^*}\,\tan \,\alpha \; {\rm{ h}}_{{\rm{max}}}^{\rm{2}} $$

(1.13)

where h _max is the depth of penetration of the apex of the indenter beneath the original specimen free surface.

In indentation testing, the most common types of indenters are spherical indenters , where the Hertz equations apply directly, or pyramidal indenters. The most common types of pyramidal indenters are the four-sided Vickers indenter and the three-sided Berkovich indenter . Of particular interest in indentation testing is the area of the contact found from the dimensions of the contact perimeter. For a spherical indenter, the radius of the circle of contact is given by:

$$ \begin{aligned} a & = \sqrt {2{R_i}{h_c}\, - \,{h_c^2}} \\ & \approx \sqrt {2{R_i}{h_c}} \end{aligned} $$

(1.14)

where h _c is the depth of the circle of contact as shown in Fig. 1.1. The approximation of Eq. 1.14 is precisely that which underlies the Hertz equations (Eqs. 1.1 and 1.4) and thus these equations apply to cases where the deformation is small, that is, when the depth h _c is small in comparison to the radius R _i .

For a conical indenter , the radius of the circle of contact is simply:

$$ a = {h_c}\,\tan \,\alpha $$

(1.15)

In indentation testing, pyramidal indenters are generally treated as conical indenters with a cone angle that provides the same area to depth relationship as the actual indenter in question. This allows the use of convenient axial-symmetric elastic equations, Eqs. 1.11–1.13, to be applied to contacts involving non-axial-symmetric indenters. Despite the availability of contact solutions for pyramidal punch problems [9–11], the conversion to an equivalent axial-symmetric has found a wide acceptance. The areas of contact as a function of the depth of the circle of contact for some common indenter geometries are given in Table 1.1 along with other information to be used in the analysis methods shown in Chap. 3.

Table 1.1 Projected areas, intercept corrections, and geometry correction factors for various types of indenters. The semi-angles given for pyramidal indenters are the face angle s with the central axis of the indenter

An important result from these equations occurs when the derivative of the force with respect to the depth is taken. This quantity, dP/dh, is often referred to as the contact stiffness. For example, in the case of a conical indenter, we have from Eq. 1.13:

$$ \frac{{dP}}{{dh}} = 2\left[ {\frac{2}{\pi }\,{E^*}\,\tan \,\alpha } \right]{\rm{ }}h $$

(1.16)

From Eq. 1.12 with r = 0 and inserting into Eq. 1.16, we arrive at the important result:

$$ \frac{{dP}}{{dh}} = 2{E^*}a $$

(1.17)

or equivalently:

$$ {E^*} = \frac{1}{2}\frac{{dP}}{{dh}}\frac{{\sqrt \pi }}{{\sqrt A }} $$

(1.18)

Equations 1.17 and 1.18 have been shown to apply for the elastic contact with any axis-symmetric indenter with a smooth profile and form the basis of analysis techniques in nanoindentation testing where the contact stiffness is evaluated at the beginning of the unloading response.

1.3 Geometrical Similarity

With a pyramidal or conical indenter , the ratio of the length of the diagonal or radius of circle of contact to the depth of the indentation,¹ a/δ, remains constant for increasing indenter load, as shown in Fig. 1.4. Indentations of this type have the property of “geometrical similarity.” For geometrically similar indentations , it is not possible to set the scale of an indentation without some external reference. The significance of this is that the strain within the material is a constant, independent of the load applied to the indenter.

Fig. 1.4

Geometrical similarity for (a) diamond pyramid or conical indenter ; (b) spherical indenter. For the conical indenter, a ₁/δ ₁ = a ₂/δ ₂. For the spherical indenter, a ₁/δ ₁ <> a ₂/δ ₂ but a ₁/δ ₁ = a ₃/δ ₃ if a ₁/R ₁ = a ₃/R ₃ (after [12])

Unlike a conical indenter, the radius of the circle of contact for a spherical indenter increases faster than the depth of the indentation as the load increases. The ratio a/δ increases with increasing load. In this respect, indentations with a spherical indenter are not geometrically similar. Increasing the load on a spherical indenter is equivalent to decreasing the tip semi-angle of a conical indenter.

However, geometrically similar indentations may be obtained with spherical indenters of different radii. If the indentation strain , a/R, is maintained constant, then so is the mean contact pressure , and the indentations are geometrically similar.

The principle of geometrical similarity is widely used in hardness measurements. For example, owing to geometrical similarity, hardness measurements made using a diamond pyramid indenter are expected to yield a value for hardness that is independent of the load. For spherical indenters, the same value of mean contact pressure may be obtained with different sized indenters and different loads as long as the ratio of the radius of the circle of contact to the indenter radius, a/R, is the same in each case.

The quantity a/R for a spherical indentation is equivalent to cot α for a conical indenter . Tabor [13] showed that the representative strain in a Brinell hardness test is equal to about 0.2a/R and hence the representative strain in a typical indentation test performed with a Vickers indenter is approximately 8% (setting α = 68°). This is precisely the indentation strain at which a fully developed plastic zone is observed to occur in the Brinell hardness test.

1.4 Elastic–Plastic Contact

Indentation tests on many materials result in both elastic and plastic deformation of the specimen material. In brittle materials, plastic deformation most commonly occurs with pointed indenters such as the Vickers diamond pyramid. In ductile materials, plasticity may be readily induced with a “blunt” indenter such as a sphere or cylindrical punch. Indentation tests are used routinely in the measurement of hardness of materials, but Vickers, Berkovich, and Knoop diamond indenters may be used to investigate other mechanical properties of solids such as specimen strength, fracture toughness , and internal residual stresses . The meaning of hardness has been the subject of considerable attention by scientists and engineers since the early 1700s. It was appreciated very early on that hardness indicated a resistance to penetration or permanent deformation. Early methods of measuring hardness, such as the scratch method, although convenient and simple, were found to involve too many variables to provide the means for a scientific definition of hardness. Static indentation tests involving spherical or conical indenter s were first used as the basis for theories of hardness. Compared to “dynamic” tests, static tests enabled various criteria of hardness to be established since the number of test variables was reduced to a manageable level. The most well-known criterion is that of Hertz, who postulated that an absolute value for hardness was the least value of pressure beneath a spherical indenter necessary to produce a permanent set at the center of the area of contact. Later treatments by Auerbach [14], Meyer [15], and Hoyt [16] were all directed to removing some of the practical difficulties in Hertz’s original proposal.

1.4.1 The Constraint Factor

Static indentation hardness tests usually involve the application of load to a spherical or pyramidal indenter. The pressure distribution beneath the indenter is of particular interest. The value of the mean contact pressure p_m at which there is no increase with increasing indenter load is shown by experiment to be related to the hardness number H. For hardness methods that employ the projected contact area, the hardness number H is given directly by the mean pressure p _m at this limiting condition. Experiments show that the mean pressure between the indenter and the specimen is directly proportional to the material’s yield, or flow stress in compression, and can be expressed as:

$$ H \approx CY $$

(1.19)

where Y is the yield, or flow stress , of the material. The mean contact pressure in an indentation test is higher than that required to initiate yield in a uniaxial compression test because of the confining nature of the indentation stress field. The shear component of stress is responsible for plastic flow. The maximum shear stress is equal to half the difference between the maximum and minimum principal stresses, and in an indentation stress field, where the stressed material is constrained by the surrounding matrix, there is a considerable hydrostatic component. Thus, the mean contact pressure is greater than that required to initiate yield when compared to a uniaxial compressive stress. It is for this reason that C in Eq. 1.19 is called the “constraint factor ,” the value of which depends upon the type of specimen, the type of indenter, and other experimental parameters. For the indentation methods mentioned here, both experiments and theory predict C » 3 for materials with a large value of the ratio E/Y (e.g., metals). For low values of E/Y (e.g., glasses [17, 18]), C »1.5. The flow, or yield stress Y, in this context is the stress at which plastic yielding first occurs.

1.4.2 Indentation Response of Materials

The hardness of a material is intimately related to the mean contact pressure p_m beneath the indenter at a limiting condition of compression . Valuable information about the elastic and plastic properties of a material can be obtained with spherical indenters when the mean contact pressure, or “indentation stress, ” is plotted against the ratio a/R, the “indentation strain” . The indentation stress–strain response of an elastic–plastic solid can generally be divided into three regimes, which depend on the uniaxial compressive yield stress Y of the material [13]:

• 1. p _m  < 1.1Y—full elastic response, no permanent or residual impression left in the test specimen after removal of load.

• 2. 1.1Y < p _m  < CY—plastic deformation exists beneath the surface but is constrained by the surrounding elastic material, where C is a constant whose value depends on the material and the indenter geometry.

• 3. p _m  = CY—plastic region extends to the surface of the specimen and continues to grow in size such that the indentation contact area increases at a rate that gives little or no increase in the mean contact pressure for further increases in indenter load.

In Region 1, during the initial application of load, the response is elastic and can be predicted from Eq. 1.10. Equation 1.10 assumes linear elasticity and makes no allowance for yield within the specimen material. For a fully elastic response, the principal shear stress for indentation with a spherical indenter is a maximum at » 0.47p _m at a depth of » 0.5a beneath the specimen surface directly beneath the indenter [19]. Following Tabor [13], either the Tresca or von Mises shear stress criteria may be employed, where plastic flow occurs at τ » 0.5Y, to show that plastic deformation in the specimen beneath a spherical indenter can be first expected to occur when p _m  » 1.1Y. Theoretical treatment of events within Region 2 is difficult because of the uncertainty regarding the size and shape of the evolving plastic zone . At high values of indentation strain (Region 3), the mode of deformation appears to depend on the type of indenter and the specimen material. The presence of the free surface has an appreciable effect, and the plastic deformation within the specimen is such that, assuming no work hardening , little or no increase in p_m occurs with increasing indenter load.

Measurement of hardness using an indentation technique is usually done at a condition of a fully developed plastic zone since it is within this region of the response that the mean contact pressure generally becomes independent of load and depends only on the material response. There are various definitions of hardness and these will be reviewed in detail in Chap. 2. The most commonly used definition for nanoindentation testing is the Meyer hardness which corresponds to the mean contact pressure at full load and is given by:

$$ H = \frac{P}{A} $$

(1.20)

where A is the projected area of contact (as distinct from the actual curved area of contact). A common error in the interpretation of nanoindentation test results is to quote hardness values at a condition of less than full plasticity in which case the quantity being measured is the mean contact pressure and not load-independent.

1.4.3 Elastic–Plastic Stress Distribution

The equations for elastic contact given above form the basis of analysis methods for nanoindentation tests, even if these tests involve plastic deformation in the specimen. Hertz ’s original analysis was concerned with the form of the pressure distribution between contacting spheres that took the form:

$$ \frac{{{\sigma _z}}}{{{p_m}}} = - \frac{3}{2}{\left( {1 - \frac{{{r^2}}}{{{a^2}}}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}{\rm{ }} $$

(1.21)

The pressure distribution s _z in Eq. 1.21 is normalized to the mean contact pressure p _m , and it can be seen that the pressure is a maximum equal to 1.5 times the mean contact pressure at the center of the contact as shown in Fig. 1.5c [20].

Fig. 1.5

Elastic–plastic indentation response for mild steel material, E/Y = 550. (a) Test results for an indenter load of P = 1000 N and indenter of radius 3.18 mm showing residual impression in the surface. (b) Section view with subsurface accumulated damage beneath the indentation site. (c) Finite element results for contact pressure distribution . (d) Finite element results showing development of the plastic zone in terms of contours of maximum shear stress at τ _max/Y = 0.5. In (c) and (d), results are shown for indentation strain s of a/R = 0.04, 0.06, 0.08, 0.11, 0.14, 0.18. Distances are expressed in terms of the contact radius a = 0.218 mm for the elastic case of P = 1000 N (after [20])

When plastic deformation occurs, the pressure distribution is modified and becomes more uniform. Finite element results for an elastic–plastic contact are shown in Fig. 1.5c. There is no currently available analytical theory that generally describes the stress distribution beneath the indenter for an elastic–plastic contact.

The finite element method, however, has been used with some success in this regard and Fig. 1.5d shows the evolution of the plastic zone (within which the shear stress is a constant) compared with an experiment on a specimen of mild steel, Fig. 1.5b. Mesarovic and Fleck [21] have calculated the full elastic elastic–plastic contact for a spherical indenter that includes elasticity, strain-hardening, and interfacial friction .

1.4.4 Hardness Theories

Theoretical approaches to hardness can generally be categorized according to the characteristics of the indenter and the response of the specimen material. Various semi-empirical models that describe experimentally observed phenomena at values of indentation strain at or near a condition of a fully developed plastic zone have been given considerable attention in the literature [13, 18, 22–34]. These models variously describe the response of the specimen material in terms of slip lines , elastic displacements, and radial compressions. For sharp wedge or conical indenter s, substantial upward flow is usually observed, and because elastic strains are thus negligible compared to plastic strains, the specimen can be regarded as being rigid–plastic. A cutting mechanism is involved, and new surfaces are formed beneath the indenter as the volume displaced by the indenter is accommodated by the upward flow of plastically deformed material. The constraint factor C in this case arises due to flow and velocity considerations [22]. For blunt indenters, the specimen responds in an elastic–plastic manner, and plastic flow is usually described in terms of the elastic constraint offered by the surrounding material. With blunt indenters, Samuels and Mulhearn [25] noted that the mode of plastic deformation at a condition of fully developed plastic zone appears to be a result of compression rather than cutting, and the displaced volume of the indenter is assumed to be taken up entirely by elastic strains within the specimen material outside the plastic zone. This idea was given further attention by Marsh [24], who compared the plastic deformation in the vicinity of the indenter to that which occurs during the radial expansion of a spherical cavity subjected to internal pressure, an analysis of which was given previously by Hill [23]. The most widely accepted treatment is that of Johnson [27, 34], who replaced the expansion of the cavity with that of an incompressible hemispherical core of material subjected to an internal pressure . Here, the core pressure is directly related to the mean contact pressure . This is the so-called “expanding cavity ” model.

In the expanding cavity model, the contacting surface of the indenter is encased by a hydrostatic “core” of radius a _c , which is in turn surrounded by a hemispherical plastic zone of radius c as shown in Fig. 1.6. An increment of penetration dh of the indenter results in an expansion of the core da and the volume displaced by the indenter is accommodated by radial movement of particles du(r) at the core boundary. This in turn causes the plastic zone to increase in radius by an amount dc.

Fig. 1.6

Expanding cavity model schematic. The contacting surface of the indenter is encased by a hydrostatic “core” of radius a _c that is in turn surrounded by a hemispherical plastic zone of radius c. An increment of penetration dh of the indenter, results in an expansion of the core da and the volume displaced by the indenter is accommodated by radial movement of particles du(r) at the core boundary. This in turn causes the plastic zone to increase in radius by an amount dc (after [20])

For geometrically similar indentations, such as with a conical indenter of semi-angle α , the radius of the plastic zone increases at the same rate as that of the core, hence, da/dc = a/c.

Using this result Johnson shows that the pressure in the core can be calculated from:

$$ \frac{p}{Y} = \frac{2}{3}\left[ {1 + \ln \left( {\frac{{({{E \mathord{/ {\vphantom {E Y}} \kern-\nulldelimiterspace} Y}})\tan \beta + 4\left( {1 - 2\nu } \right)}}{{6\left( {1 - \nu }\right)}}} \right)} \right]$$

(1.22)

where p is the pressure within the core and β is the angle of inclination of the indenter with the specimen surface (tanβ = cotα).

The mean contact pressure is found from:

$$ {p_m} = p + \frac{2}{3}Y $$

(1.23)

and this leads to an value for the constraint factor C. When the free surface of the specimen begins to influence appreciably the shape of the plastic zone , and the plastic material is no longer elastically constrained, the volume of material displaced by the indenter is accommodated by upward flow around the indenter. The specimen then takes on the characteristics of a rigid–plastic solid, because any elastic strains present are very much smaller than the plastic flow of unconstrained material. Plastic yield within such a material depends upon a critical shear stress, which may be calculated using either of the von Mises or Tresca failure criteria.

In the slip-line field solution, developed originally in two dimensions by Hill, Lee, and Tupper [22], the volume of material displaced by the indenter is accounted for by upward flow, as shown in Fig. 1.7. The material in the region ABCDE flows upward and outward as the indenter moves downward under load. Because frictionless contact is assumed, the direction of stress along the line AB is normal to the face of the indenter. The lines within the region ABDEC are oriented at 45° to AB and are called “slip lines ” (lines of maximum shear stress). This type of indentation involves a “cutting” of the specimen material along the line 0A and the creation of new surfaces that travel upward along the contact surface. The contact pressure across the face of the indenter is:

Fig. 1.7

Slip-line theory. The material in the region ABCDE flows upward and outward as the indenter moves downward under load (after [12])

$$ \begin{aligned} {p_m}& = 2{\tau _{\max }}\!\left( {1 + \alpha } \right) \\ & = H \end{aligned} $$

(1.24)

where τ _max is the maximum value of shear stress in the specimen material and α is the cone semi-angle (in radians).

Invoking the Tresca shear stress criterion, where plastic flow occurs at τ _max = 0.5Y, and substituting into Eq. 1.24, gives:

$$ \begin{aligned}& H = Y\!\left( {1 + \alpha } \right) \\ & \therefore \\ & C = \left( {1 + \alpha } \right)\end{aligned} $$

(1.25)

The constraint factor determined by this method is referred to as C _flow . For values of α between 70° and 90°, Eq. 1.23 gives only a small variation in C _flow of 2.22 to 2.6. Friction between the indenter and the specimen increases the value of C _flow . A slightly larger value for C _flow is found when the von Mises stress criterion is used (where τ _max » 0.58Y). For example, at α = 90°, Eq. 1.23 with the von Mises criterion gives C = 3.

1.5 Indentations at the Nanometre Scale

The present field of nanoindentation grew from a desire to measure the mechanical properties of hard thin films and other near surface treatments in the early 1980s. Microhardness testing instruments available at the time could not apply low enough forces to give penetration depths less than the required 10% or so of the film thickness so as to avoid influence on the hardness measurement from the presence of the substrate. Even if they could, the resulting size of the residual impression cannot be determined with sufficient accuracy to be useful. For example, the uncertainty in a measurement of a 5 mm diagonal of a residual impression made by a Vickers indenter is on the order of 20% when using an optical method and increases with decreasing size of indentation and can be as high as 100% for a 1 mm impression.

Since the spatial dimensions of the contact area are not conveniently measured, modern nanoindentation techniques typically use the measured depth of penetration of the indenter and the known geometry of the indenter to determine the contact area. Such a procedure is sometimes called “depth-sensing indentation testing ” although it is quite permissible to use the technique at macroscopic dimensions [35, 36]. For such a measurement to be made, the depth measurement system needs to be referenced to the specimen surface, and this is usually done by bringing the indenter into contact with the surface with a very small “initial contact force, ” which, in turn, results in an inevitable initial penetration of the surface by the indenter that must be accounted for in the analysis. Additional corrections are required to account for irregularities in the shape of the indenter, deflection of the loading frame, and piling-up of material around the indenter (see Fig. 1.8). These effects contribute to errors in the recorded depths and, subsequently, the hardness and modulus determinations. Furthermore, the scale of deformation in a nanoindentation test becomes comparable to the size of material defects such as dislocations and grain sizes, and the continuum approximation used in the analysis can become less valid.

Fig. 1.8

Atomic force micrograph of a residual impression in steel made with a triangular pyramid Berkovich indenter. Note the presence of piling-up at the periphery of the contact impression. (Courtesy CSIRO)

The nanoindentation test results provide information on the elastic modulus, hardness, strain-hardening , cracking, phase transformations, creep, and energy absorption. The specimen size is very small and the test can in many cases be considered non-destructive. Specimen preparation is straightforward. Because the scale of deformation is very small, the technique is applicable to thin surface films and surface modified layers. In many cases, the microstructural features of a thin film or coating differs markedly from that of the bulk material owing to the presence of residual stresses , preferred orientations of crystallographic planes, and the morphology of the microstructure. The applications of the technique therefore cover technologies such as cathodic arc deposition, physical vapor deposition (PVD), and chemical vapor deposition (CVD) as well as ion-implantation and functionally graded materials. Nanoindentation instruments are typically easy to use, operate under computer control, and require no vacuum chambers or other expensive laboratory infrastructure.

The technique relies on a continuous measurement of depth of penetration with increasing load. Such measurements at the micron scale were demonstrated by Fröhlich, Grau and Grellmann [37] in 1977 who analyzed the loading and unloading curves for a variety of materials and foreshadowed the use of the technique for the measurement of surface properties of materials. Pethica [38], in 1981, applied the method to the measurement of the mechanical properties of ion-implanted metal surfaces, a popular application of the technique for many years [39]. Stilwell and Tabor [40] focused on the elastic unloading of the impression as did Armstrong and Robinson [41] in 1974, and Lawn and Howes [42] in 1981. The present modern treatments probably begin with Bulychev, Alekhin, Shorshorov, and Ternovskii [43], who in 1975 showed how the area of contact could be measured using the unloading portion of the load-displacement curve. Loubet, Georges, Marchesini, and Meille [44] used this method for relatively high load testing (in the order of 1 Newton) and Doerner and Nix [45] extended the measurements into the millinewton range in 1986. The most commonly used method of analysis is a refinement of the Doerner and Nix approach by Oliver and Pharr [46] in 1992. A complementary approach directed to indentations with spherical indenters was proposed by Field and Swain and coworkers [47, 48] in 1993 and subsequently shown to be equivalent to the Oliver and Pharr method [49]. Review articles [50–52] on micro, and nanoindentation show a clear evolution of the field from traditional macroscopic measurements of hardness. The field now supports specialized symposia on an annual basis attracting papers covering topics from fundamental theory to applications of the technique.

The first “ultra-micro” hardness tests were done with apparatus designed for use inside the vacuum chamber of a scanning electron microscope (SEM), where load was applied to a sharply pointed tungsten wire via the movement of a galvanometer that was controlled externally by electric current. Depth of penetration was determined by measuring the motion of the indenter support using an interferometric method. The later use of strain gauge s to measure the applied load and finely machined parallel springs operated by an electromagnetic coil bought the measurement outside the vacuum chamber into the laboratory, but, although the required forces could now be applied in a controlled manner, optical measurements of displacement or sizes of residual impressions remained a limiting factor. Developments in electronics lead to the production of displacement measuring sensors with resolutions greater than those offered by optical methods and, in the last 10 years, some six or seven instruments have evolved into commercial products, often resulting in the creation of private companies growing out of research organizations to sell and support them.

A common question is to ask if it is possible to perform nanoindentation with an atomic force microscope (AFM). It is, but there are some significant problems that make the method unsuitable for precise determination of materials properties. In an AFM, the tip is usually grown from silicon with a tip radius in the order of 5–10 nm. The precise geometry is not usually known, since for imaging purposes, it is not necessary to know this information. The cantilever, to which the tip is mounted, is a very compliant structure—made so since the purpose of the instrument is to provide a large deflection in response to surface forces. Thus, unlike a conventional nanoindentation instrument, a large component of a load-displacement curve obtained with an AFM is accounted for by compliance of the cantilever rather than indentation into the specimen, the latter being the signal of interest in nanoindentation studies. Any instrumented technique (such as an AFM or nanoindentation instrument) employs analog to digital conversion of data and since the signal of interest (the penetration depth) from an indentation made with an AFM is such a small proportion of the total signal, there is a considerable loss of resolution in this quantity in the overall results. This, together with the uncertainties regarding the tip shape, limit the use of this type of instrument for extraction of material properties from the indentation data.

There is no doubt that as the scale of mechanisms becomes smaller, interest in mechanical properties on a nanometre scale and smaller, and the nature of surface forces and adhesion, will continue to increase. Indeed, at least one recent publication refers to the combination of a nanoindenter and an atomic force microscope as a “picoindenter ” [53] suitable for the study of pre-contact mechanics, the process of making contact, and actual contact mechanics. The present maturity of the field of nanoindentation makes it a suitable technique for the evaluation of new materials technologies by both academic and private industry research laboratories and is increasingly finding application as a quality control tool.