# Nanoindentation of Historic and Artists' Paints
## 3 Instrumented Indentation Testing
### Probe Geometries  
Various probe materials and geometries can be employed for indentation tests. Manufacturing precision and desired high hardness and stiffness favor diamond as the material of choice; however, other materials, such as sapphire, hardened steel and tungsten, are also used. These less stiff materials are adequate for measuring historic and artists' paints since their rigidity is many orders of magnitude greater than those of paints, making their compliance negligible when analyzing the test results. With respect to probe shapes, there are three commonly used geometries offering radically different opportunities: pyramid, flat-ended cylindrical punch, and sphere. Sharp indenters are frequently used in nanoindentation. The simple axial symmetry makes a conical geometry attractive from a modelling standpoint, but the difficulty in manufacturing conical diamonds prevents the use of such indenters in practice. In contrast, pyramidal probes, of which the Berkovich indenter is the most popular, are relatively straightforward to manufacture and used in many practical applications. The well-defined three-sided pyramidal geometry allows one to subject the material to a constant strain during the loading phase [30]. This is because the representative strain exerted in the material scales with the tangent of the complimentary angle (\(\alpha\)) between the sample surface and the indenter flank [30]. As previously stated in Section 3.2, the strain rate during sharp indentation can be held constant for a homogeneous material with the use of a specific loading scheme [29]. Indentation with a diamond Berkovich indenter typically induces plastic deformation of the sample as the imposed strain beneath the indenter is sufficient to cause yielding of many materials. The main disadvantage of using a sharp indenter is the difficulty in defining the contact depth when material 'piles up' around the indenter as a consequence of plastic flow outside the contact perimeter. Incorrect evaluation of the contact depth, and hence of the projected contact area and equivalent contact radius, induces errors on the nanoindentation-derived hardness and modulus [31]. A flat-punch indenter has the form of a vertical cylindrical column with a flat circular base. From the moment of full contact between the circular base of the indenter and the material, the contact area remains well-defined and constant, eliminating the uncertainty of defining the contact area/radius encountered in sharp indentation. The main disadvantage of flat-punch indentation is its relatively large contact radius, which makes the test results more sensitive to the stiffness of the surrounding substrate material [19]. Further, unlike in the case of sharp indentation, the strain field beneath the flat-punch tip changes with indentation depth in a complex manner. Deep indentation of a soft and viscoelastic film beyond the limit of linear viscoelasticity results in an erroneous modulus measurement due to the lack of a universally applicable solution for deducing the reduced modulus in the nonlinear viscoelastic deformation regime. A spherical indenter offers another means of probing a material. Unlike in sharp indentation, the strain imposed by a spherical indenter at a limited contact force can be kept small enough to maintain the sample deformation in the elastic regime. With an increasing indentation load, the average strain (dictated by \(\tan\alpha\)) increases progressively with indentation depth, resulting in a transition from elastic to plastic deformation. Defining the exact point of this transition can enable the determination of the yield stress of the tested material [30; 32; 33], although its accuracy may be hindered by the fact that manufacturing perfectly spherical probes at the micrometer scale is inherently difficult. Spherical indentation, when taken well beyond the elastic limit, can suffer the same problem of incorrectly evaluating the contact area as in sharp indentation.