# Nanoindentation of Historic and Artists' Paints
## 3 Instrumented Indentation Testing  
### Quasi-Static Method  
An indentation test consists of pressing an indenter of a well-defined geometry into the sample surface, with the resulting load-displacement curve serving as the'mechanical fingerprint' of the material. The schematic representation of the process is presented in Figure 1.  
While both the loading and unloading responses are recorded, the latter is used to determine the contact stiffness (\(S\)) of the tested material using a curve fitting procedure from the widely accepted Oliver-Pharr method [7; 18]. The depth of the material in contact with the indenter, or the contact depth (\(h_{\text{c}}\)), may be predicted from the displacement (\(h_{\text{max}}\)), load (\(P_{\text{max}}\)) and contact stiffness at the onset of unloading as \(h_{\text{c}}=h_{\text{max}}-0.75P_{\text{max}}/S\)[18]. The projected area of the contact (\(A_{\text{p}}\)) is obtained as a function of contact depth based on knowledge of the indenter geometry [7; 18]. The hardness (H) is defined as the ratio of the maximum load to the projected contact area, or \(H=P_{\text{max}}/A_{\text{p}}\). Using [PERSON]'s solution [22], the reduced elastic modulus (\(E_{\text{r}}\)) is given as the contact stiffness normalized to the lateral dimension of the contact, or \(E_{\text{r}}=S/(2\beta a)\), where \(a\) (\(=\sqrt{A_{\text{p}}/\pi}\)) is the equivalent contact radius of a pyramidal indenter, and \(\beta\) is a geometric constant of the indenter on the order of unity [18; 23]. In the case of an axisymmetric indenter geometry such as a cone, sphere or flat-ended cylindrical punch, \(\beta\) equals unity. The obtained \(E_{\text{r}}\) value is related to the [PERSON]'s modulus (\(E\)) and [PERSON]'s ratio (\(v\)) of the tested material and those of the indenter itself (indicated by the subscript i) through the relation \(1/E_{\text{r}}=\left(1-v^{2}\right)/E+\left(1-v_{1}^{2}\right)/E_{\text{i}}\). Practically, since the [PERSON]'s modulus of the indenter (the probe itself) is several orders of magnitude higher than that of the tested paint, the last term in the above equation can be omitted so that it reduces to \(E_{\text{r}}=E/\left(1-v^{2}\right)\). Determination of the [PERSON]'s modulus is limited by the lack of knowledge about the Poisson ratio of the tested material; therefore, the reduced modulus is often reported instead.  
For time-dependent materials (such as artists' paints), it is possible to visualize their creep or stress-relaxation behavior by adding a pause between the loading and unloading phases. Creep is determined by measuring the deformation under a constant load (in a load-controlled experiment), while stress-relaxation examines change of load under a constant displacement (in a depth-controlled experiment). While virtually any nanoindentation instrument is capable of measuring the time-course creep displacement, its precise measurement requires accounting for thermal drift. One way of ensuring correct measurements of the indenter tip position over an extended period of time is to use a reference probe placed in close proximity to the tested surface. Such a measuring system [24] was proven to be particularly effective for characterizing the creep behavior of an acrylic paint [16]. However, since creep parameters determined by nanoindentation do not directly correspond to those measured in uniaxial testing, its usefulness may be limited for comparative purposes within similar materials or test systems.