I know infinitely hard materials don't actually exist but sometimes the tip is so much harder than the substance it's scratching that it can be treated like one. Suppose a sharp diamond tip is made to look like the vertex of a tetrahedron by fracturing it along its cleavage planes and then it slides along a surface of glass that's nanosmooth. What is the nature of the scratch that was made and why does it come out that way? Is the scratch a fractal of cracks? If so, what type of fractal is it? What power of the scratching speed and total pressing force does the depth of the scratch vary as? Why does the scratch come out as that type of fractal? When the tip scratches the glass, does the scratch end up so deep that a nanosmooth rod of glass the thickness of the scratch could support a much bigger force of tension than the pressing force of the tip because the material gets weakened as a result of beginning to get scratched enabling it to get scratched even deeper? Is it the case that molecular theory is needed to predict how the scratch would come out and that if atoms were half the size with a quarter the strength of their bonds, glass etched nanosmooth would still have extremely close to the same tensile strength but the diamond tip moving at the same speed with the same total pressing force would make a deeper scratch?
How does an infinitely hard tip scratch an amorphous brittle material when it slides along it?
$\begingroup$ "slides" may be a slightly misleading term here: without some force acting on the diamond 'stylus' there can be no penetration into the substrate and hence no scratching. $\endgroup$– GertJan 15, 2017 at 21:56
A point pressed against a brittle surface will have some likelihood of exerting local pressure below the compressive strength of the material (i.e. it might just be supported by an elastically deformed little footprint). At some higher force, the dent will damage the brittle material, and leave behind it, as it travels, a V-shaped channel, with bits of rubble and perhaps fractures and strains radiating from the scratch.
Those strains and fractures also radiate forward ahead of the moving point. It might be hard to start a scratch, but easier to continue it.
Glass-cutting works by scratching, then applying stretching force across the scratch. It is important to complete the break quickly; some of those strains and microfractures apparently will dissipate or heal. At just below glass-softening temperatures, annealing glass is a common glassblowing procedure, and strains can be seen by polarized light until they dissipate.