**2. Elastic, Plastic, and Cracking Aspects of Crystal Nano-Indentations**

Figure 1 provides a background log/log compilation of indentation load, *P*, vs. surface-projected contact diameter, *d*, spanning a range from pioneering nano-indentation measurements to micro-scale indentation fracture mechanics test results obtained on silicon crystals, as identified in the top-left corner of the figure [14]. The figure was constructed to illustrate a *d*<sup>3</sup> dependence of load for elastic contact in pioneering nano-indentation measurements made by Pethica, Hutchings, and Oliver [15], as compared with an expected *d*<sup>2</sup> dependence for a constant material hardness; *D* is the actual or effective spherical tip diameter for each indentation. The *dc* <sup>3</sup>/<sup>2</sup> dependence applies for crack tip-to-crack tip measurements across the enclosed indentations.

**Figure 1.** Load, *P*, vs. either elastic contact diameter, *de*, plastic diamond pyramid diagonal, *dd*, or tip-to-tip crack length, *dc*, for silicon crystal indentations [14], and including additional cracking [16] and inset elastic-plastic spherical Berkovich-type nano-indentation measurements [17].

In Figure 1, more recently added open- and closed-ellipsoidal points [16] have covered many of the earlier data points on the *d*3/<sup>2</sup> fracture mechanics dependence and, likewise, the inset open-square points [17] for both elastic, *de* 3, and plastic, *dp* 2, Berkovich spherically-tipped nano-indentations would have significantly overlapped the previously established hardness dependence. Thus, these points that illustrate a significant "pop-in" behavior at initial plastic yield have been shifted on the abscissa scale for clarity. Close examination of the deviation from the labeled *d*<sup>2</sup> dependence in Figure 1 shows that an increasing hardness applies for a smaller indentation size. Calculations of the stress–strain behaviors both at the onset of plastic yielding and follow-on nano-indentation strain hardening behaviors have been reported very recently for NaCl, MgO, and copper crystals [18], tungsten crystals [19], ammonium perchlorate, and α-iron crystals [20]. Beyond the well-established determination of very high flow stress levels for initial plastic yielding, whether gradual or of pop-in type, the plastic strain hardening behavior has also been shown to be exceptionally high. The dimensionally smaller plastic deformation zones account for both higher values of initial yielding and subsequent flow on the basis of the smaller dislocation line lengths and their interactions.
