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We extend the geometrical theory presented in [5] for collisional andfrictional particle abrasion to include an independent physical equation forthe evolution of mass and volume. We introduce volume weight functions asmultipliers of the geometric equations and use these mutipliers to enforcephysical volume evolution in the unified equations. The latter predict, inaccordance with Sternberg's Law, exponential decay for volume evolution. Wedescribe both the PDE versions, which are generalisations of Bloore's equationsand their heuristic ODE approximations, called the box equations. The latterare suitable for tracking the collective abrasion of large particlepopulations. The mutual abrasion of identical particles, called the self-dualows, play a key role in explaining geological scenarios. We give stabilitycriteria for the self-dual ows in terms of the parameters of the physicalvolume evolution models and show that under reasonable assumptions thesecriteria can be met by physical systems. We also study a naturalgeneralisation, the unidirectional Bloore equation, covering the case ofunidirectional abrasion. We have previously shown that his equation admitstravelling front solutions with circular profiles. More generally, in threedimensions, they are so-called linear or special Weingarten surfaces.

Geometrical and physical models of abrasion