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Spherical Indentation |
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Theory
The elastic behaviour of contacting spheres was first described by Heinrich Hertz in 1882. Hertzian analysis predicts a contact circle of radius a given by the equation:
a = (3PR/4E*) 1/3
where P is the applied load, R is the radius of the sphere, and E* is the reduced modulus given by:
1/E* = (1-n12)/E1 + (1-n22)/E 2
where the subscripts 1 and 2 refer to the two contacting materials.
In static Hertzian contact, the maximum shear stress occurs at a distance of about half the contact radius immediately below the point of contact. Yielding therefore first takes place below the surface, and generally this occurs when
Pm = 1.1Y
where Pm is the mean pressure and Y is the yield stress. If the mean pressure is less than this, the contact remains elastic.
An indentation process may produce: (i) a reversible, elastic deformation; (ii) a permanent, plastic deformation; or (iii) both plastic and elastic deformation. The detailed behaviour depends on the ratio of the actual strain to the yield strain of the material. Low ratios (< 2) produce elastic behaviour; high ratios (> 50) produce plastic behaviour. The actual strain is given by tanb , where b is the angle between the indenter and sample surfaces, so it is clear that a spherical indenter will behave in a fundamentally different way from a pyramidal indenter. In the case of a Vickers pyramid, for example, b is constant and the strain is therefore constant (about 8%), regardless of depth.
For a spherical indenter, b , and therefore the strain, increases as the indentation depth increases. A series of spherical indentations with progressively higher maximum load can therefore produce results ranging from purely elastic to elastic + plastic, and, furthermore, can yield curves of Stress vs. Strain.
In the NanoTest, multiple load - unload cycles with increasing maximum loads are performed at one point. From this data and the known diamond radius, the contact area is determined and hardness, Young's modulus, stress and strain are calculated. The contact areas are corrected for possible "piling up" or "sinking in" around the contact perimeter.
Experimental
Spherical indentations were produced at two points on a highly polished E52100 steel surface. Five sets of load-unload data were obtained at each point, with the maximum load being increased from P1=2 mN to P5=10 mN. For each indentation, unloading was continued to 10% of the maximum load. The diamond indenter was conical with a spherical endform of 5 µm radius.
A transition from elastic to fully plastic deformation was observed in the load range investigated. A first order approximate value of a', the radius of the contact circle at each maximum load, was calculated, with the plastic depth determined by differentiating the first part of the unloading curve and taking epsilon as 0.75. A plot of log(Pi) vs. log(a') was produced for all experiments (i.e., for i = 1 to 5). This gave Meyer's index 2+1/n as the slope. The "pile up" correction parameter c (actual contact radius = ca') was found to be 0.923, thus indicating "sinking-in" type deformation. |
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