The impression remains, but part of the motion returns
Push a hard probe into a metal surface, then gradually remove the force. An impression can remain even as some surrounding material rises again. The lasting impression records unrecovered deformation. The returning motion releases energy stored elastically. Recording force and probe displacement throughout the cycle separates these parts on one graph.
At the start of unloading, a small reduction in force allows a small recovery of displacement. The force change per recovered distance is the contact stiffness. It describes the resistance of this entire contact at that moment. Turning it into a material elastic modulus also requires the size of the contact.
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Hardness needs an area under load
The same load concentrated over a smaller area gives a higher mean pressure. Indentation hardness divides peak load by the projected contact area at that load: the patch viewed along the loading direction. This patch is defined during loading; the impression can change again after the force is removed.
Depth does not automatically supply that area. A sharp tip and a slightly rounded tip can cover different patches at the same penetration. At shallow depths, a small imperfection near the end of the probe can account for a large fraction of the contact.
The surface around the contact can sink down as well. Total probe travel and the height of the tip actually in contact with material are different lengths. An area function relates the latter length to projected area. It needs calibration before it can turn a depth reading into hardness.
Machine motion enters the reading too
Force travels through the loading head, probe, specimen and support. Even a stiff machine flexes slightly along that path. A displacement gauge spanning the flexible structure reads both contact motion and machine motion. Assigning all its travel to the specimen makes the contact appear too compliant.
One useful reference is an elastic contact with a sphere of independently known radius. Its load and displacement follow the Hertz relation under small-deformation assumptions. A reference specimen with known elastic properties can therefore help identify the extra machine travel and the contact zero.
The working sharp tip is then tested at several depths on a known-modulus reference. With that modulus given, unloading stiffness constrains contact area. The different depths together constrain tip shape. Another independent reference exposure checks whether the calibration transfers beyond the curves used to fit it.
Read the curvature of unloading
In 1992, Oliver and Pharr carefully compared loading and unloading in several materials and emphasised that unloading is generally curved. The contact shrinks as force is removed, so its stiffness changes. Fitting that curvature and taking the slope at peak load gives a physical alternative to choosing an arbitrary straight segment.
Combining stiffness with area gives a reduced modulus, which includes the elastic compliance of both specimen and tip. A specimen Young modulus also needs its Poisson ratio and the tip elastic constants. A purely elastic sphere has a finite mean pressure during contact, but no plastic hardness in this model: after complete recovery, no plastic impression remains.
Area interpretation has conditions too. Material piling up around the impression can enlarge the actual contact beyond the area predicted by a smooth sink-in model. Hardness and modulus can then be overestimated. The contact shape must be measured independently before choosing how to interpret that unloading curve.