# Nanoindentation of Historic and Artists' Paints
## 3 Instrumented Indentation Testing
### Dynamic Method  
The continuous stiffness measurement (CSM) technique [18; 25; 26; 27; 28] is an effective method for dynamically characterizing the indentation response of a time-dependent material. This technique measures the stiffness dynamically, typically during the loading phase of the sample, by imposing small-amplitude sinusoidal oscillations on the monotonically increasing load. In the case of a time-independent material, for which the oscillations in  
Figure 1: (**a**) Load versus displacement curve and (**b**) deformation of a sample upon indentation by a conical indenter. displacement and load are in phase with each other, the dynamic stiffness (\(S^{*}\)) reduces to the ratio of the load amplitude (\(\Delta P\)) to the displacement amplitude (\(\Delta l\)), or \(S^{*}=\Delta P/\Delta l\). In the case of a viscoelastic material, for which the displacement lags behind the load by a phase angle \(\delta\), the dynamic stiffness consists of in-phase and out-of-phase components expressed as \(S^{\prime}=(\Delta P/\Delta h)\cos\delta\) and \(S^{\prime\prime}=(\Delta P/\Delta h)\sin\delta\), respectively. \(S^{\prime}\) is termed the storage stiffness that describes the stiffness of the contact, while \(S^{\prime\prime}\) is the loss stiffness that equals the damping of the contact multiplied by the angular frequency of the dynamic oscillation. Based on [PERSON]'s solution together with the elastic-viscoelastic correspondence principle, normalizing the storage and loss stiffnesses to the contact dimension (\(2\beta a\)) yields the reduced storage and loss moduli as \(E_{\text{r}}^{\prime}=S^{\prime}/(2\beta a)\) and \(E_{\text{r}}^{\prime\prime}=S^{\prime\prime}/(2\beta a)\), respectively. The loss tangent (or \(\tan\delta\)), defined as the ratio of loss modulus to storage modulus (tan \(\delta=E_{\text{r}}^{\prime\prime}/E_{\text{r}}^{\prime}\)), describes the viscous state of the material; tan \(\delta\) ranges from zero for an ideal elastic solid to infinity for an ideal liquid. The dynamic indentation technique may thus be regarded as comparable to dynamic mechanical analysis (DMA). Considering the definitions of the storage and loss moduli, it is apparent that these measurements are dependent on the accuracy of determining the (equivalent) contact radius \(a\), whereas tan \(\delta\) remains unaffected. An example of indentation testing in the CSM mode using a pyramidal (Berkovich) indenter on a 0.2 mm-thick TiO\({}_{2}\) acrylic paint film glued onto a glass substrate is presented in Figure 2. Dynamic force oscillations are applied during the loading phase of the sample with an exponentially increasing force. Using such an exponential loading scheme with a pyramidal (Berkovich) indenter, a constant strain rate of the sample can be maintained for a homogeneous material [29]. Such a test is equivalent to a uniaxial tensile or compression experiment with a linearly varying displacement, thereby keeping the strain rate constant during the test. It is practically impossible to maintain a constant strain rate during the loading phase with any indenter geometry other than pyramidal, or during the load-hold or unloading phase with any geometry. Figure 2b demonstrates an immediate advantage of the dynamic test and its ability to characterize material properties (reduced storage modulus in this case) at different depths from the surface. If the material is homogeneous, one can define the depth range at which \(E_{\text{r}}^{\prime}\) is affected by the roughness of the sample surface (at small indentation depths) or the substantially greater stiffness of the underlying substrate (at large indentation depths). As a result, dynamic indentation offers the user a means of identifying the valid depth range of indentation at which systematic errors due to the finite size of the sample and its preparation method can be minimized.