The hardness of a material can be measured in many ways.  The Knoop hardness test, a method of microindentation hardness, is the most reproducible for dense ceramics.  The Vickers hardness test and superficial Rockwell scales (e.g., 45N) can also be used, but tend to cause more surface damage than Knoop.  The Brinell test is suitable for ductile metals, but not ceramics.  In the Knoop test, a diamond indenter in the shape of an elongated pyramid is forced into a polished (but not etched) surface under a predetermined load, typically 500 or 1000 g.  The load is held for some amount of time, say 10 s, and the indenter is retracted.  The indention long diagonal (d, μm, in Fig. 4) is measured under a microscope, and the Knoop hardness (HK) is calculated from the load (P, g) and the square of the diagonal length in the equations below.  The constants account for the projected area of the indenter and unit conversion factors.  Most oxide ceramics have a Knoop hardness in the range of 1000&ndash;1500 kg/mm (10 &ndash; 15 GPa), and many carbides are over 2000 (20 GPa).  The method is specified in ASTM C849, C1326 & E384.  Microindentation hardness is also called microindentation hardness or simply microhardness.  The hardness of very small particles and thin films of ceramics, on the order of 100 nm, can be measured by nanoindentation methods that use a Berkovich indenter.
: (kg/mm) and  (GPa)
The toughness of ceramics can be determined from a Vickers test under a load of 10 &ndash; 20 kg.  Toughness is the ability of a material to resist  crack propagation.  Several calculations have been formulated from the load (P), elastic modulus (E), microindentation hardness (H), crack length (c in Fig. 5) and flexural strength (σ).  Modulus of rupture (MOR) bars with a rectangular cross-section are indented in three places on a polished surface. The bars are loaded in 4-point bending with the polished, indented surface in tension, until fracture.  The fracture normally originates at one of the indentions.  The crack lengths are measured under a microscope.  The toughness of most ceramics is 2&ndash;4 MPa, but toughened zirconia is as much as 13, and cemented carbides are often over 20. The toughness-by-indention methods have been discredited recently and are being replaced by more rigorous methods that measure crack growth in a notched beam in bending.
: initial crack length
: indention strength in bending