Nanoindentation: Theoretical Considerations

Whilst studying optical interference rings between two glass lenses in contact Hertz proposed the first satisfactory analysis of the stresses between two spheres in contact.

Hertz hypothesised that the contact area between two elastic solids having profiles of solid bodies of revolution, is, in general, elliptical. To facilitate the calculation of the

local deformations he treated each body as an elastic half space loaded over a small elliptical region of its plane surface. For this to be justified two conditions must be met. The first condition is that the dimensions of the contact area must be small compared with the dimensions of each body. This ensures that any boundaries of the body do not affect the calculated stress field. The second condition is that the dimensions of the contact area should be small compared to the relative radii of curvature of the surfaces, thus ensuring that the surfaces outside the contact region can be approximated as the plane surface of a half space and that the strains are small enough to be within the scope of the linear theory of elasticity. Additionally, the surfaces are assumed to be frictionless so that the only pressure between them is normal (Johnson 1985).

For two elastic spherical bodies in contact, the contact area is circular, with contact radius, a, and this defines the surface boundary wherein pressure gives rise to normal displacements. Hertz relates a to the load, P, as

3

where R^* is the relative radius of curvature of the spheres given by

2

and E^* is the reduced modulus derived from the moduli of both materials and given by

The mutual approach of distant points in the two solids, δ , is given by

The spherical indentation of a flat elastic half space, Figure 2.4-2, is simply a special case of Hertzian contact theory for two spheres in contact. The flat elastic half space has radius infinity so that R^* becomes the radius of the indenter, R and the distance of mutual approach becomes the penetration into the half space, h .

Providing the same conditions are met, i.e. the surfaces are continuous and non-conforming, a<< ; the strains are small, R a/R<0.3(Johnson 1985); and frictional effects can be ignored it can be seen from the above, that for indentation with a spherical indenter, load is related to depth as

3

Using Hertzian theory for elastic contact together with the relationships developed by Sneddon (1965) for indenting a flat elastic half space with tips of different Landingham et al.2000) Sneddon derived relationships between load, displacement

and contact area for any punch that can be described as a solid of revolution of a smooth function (Oliver and Pharr 1992).

Figure 2.4-2 Schematic diagram of the deformation and definition of the contact dimensions of an elastic surface by a sphere

As the indenter is made from a material with high hardness to minimise its potential plastic deformation during a test, it is difficult to produce a perfectly spherical tip.

Consequently, the effective radius, R is generally a function of the depth of penetration in contact with the spherical indenter, hc , see Figure 2.4-2. By indenting into a series of reference materials the tip shape function can be determined (Bushby and Jennett 2001, Zhu, Bushby and Dunstan 2008).

Oliver, Hutchings and Pethica suggested that contact areas be measured from load displacement curves and knowledge of the indenter area function rather than the difficult and time consuming direct imaging of impressions (Oliver and Pharr 1992).

Doerner and Nix put together these ideas and developed a comprehensive method for determining hardness and modulus from indentation load displacement data. This was based on the observation that, for some materials, during the initial stages of unloading, the unloading curves were linear, that is, the elastic behaviour of the

P

R

a a

h

h/2 h/2=hc

indentation contact is similar to that of a flat cylindrical punch (Doerner and Nix 1986). The contact area was determined by extrapolating to zero load the initial linear portion of the unloading curve and using the extrapolated depth with the indenter shape function to determine the contact area.

Oliver and Pharr subsequently extended the method proposed by Doerner and Nix to account for the fact that for a large number of materials their experiments showed non linear unloading even during the initial stages.

Using the method of Oliver and Pharr, from one cycle of loading and unloading, mechanical properties can be determined from measurements of the maximum applied load and maximum penetration depth. From the resulting load/displacement curve of a nanoindentation test, the contact stiffness, S = dP/dh, is obtained from the initial part of the unloading curve. Contact depth, h , is then related, according to the _c tip shape, to the intercept of this tangent, hi, with the x – axis, Figure 2.4-3.

Figure 2.4-3 Glass nanoindentation example showing contact stiffness and contact depth from 0

50 100 150 200 250 300 350

0 200 400 600 800 1000 1200 1400

Penetration / nm

Force / mN

dh dP

hc

hmax

hi

Based on displacement equations from Sneddon’ s analyses for spherical tipped

indenters

S h P

h_c = _max −0.75 ^max ( Sneddon 1965,Van Landingham 2003)

Hardness, H, is determined from the equation H = P/A, where A is the contact area calculated as a function of contact depth. The elastic modulus is determined from the reduced modulus E^*, where ^{E* =}

(

^π⋅S

)

^{/2β A and β} is a constant that depends on the geometry of the indenter, for spherical tipsβ =1.

Generally, the relationship between depth of penetration, h, and load, P, is given as

m

hf

h

P=α( − ) Equation 2-39

where contains geometric constants, the sample elastic modulus, the sample Poisson’ s ratio, the indenter elastic modulus, and the indenter Poisson’ s ratio, hf is the final depth after unloading and m is a power law exponent that is related to the indenter geometry, for a spherical tip m = 3/2. Final depth after unloading is relevant for elastic-plastic indentation and analysis methods have been described fully (Field and Swain 1993), (Oliver and Pharr 2004).

2.4.2 Visco-elastic correspondence with a spherical indentation

In document Nanoindentation as a method to interrogate the mechanical properties of polymer coatings (Page 51-56)