THE PLASTICITY OF SOLIDS

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that as aggregates of particles are formed in the process of
collisions, and the size of these aggregates increases as the
concentration of solid increases, there must come a time when
such aggregates or clots will touch each other and form an arch or
bridge across the space through which the flow is taking place.
At that concentration the friction will have a finite value, and the
material may be said to have a structure just as was the case of
the jelly or foam already considered.
The pore space may vary between very wide limits, but
if the suspended particles are assumed to be uniform spheres, it
can easily be calculated that cubical close-packing, would leave
a pore space of 1 — Tr/6 or 47.64 per cent by volume, irrespective
of the size of the particles. It is possible to get the particles
still closer together until with tetrahedral close-packing, which
we have in a pile of cannon-balls, the pore space is 1 — 7r/3\/2 or
25.96 per cent by volume, but in this case the particles are
interlocked and no true flow is possible but rupture, with dis-
integration of the particles. When the pore space is roughly 50
per cent, the mobility is zero, and it is only as the pore space
is in excess of this figure that the mobility has a finite value.
This excess pore space thus plays a role which is analogous to
the free volume of liquids.
As there is a minimum in the allowable pore space in a plastic
solid, so there is a maximum, for as the pore space increases the
substance finally ceases to become a solid. This concentration
of zero friction was found for a certain English china clay to be
19.5 per cent by volume when suspended in water containing
one-tenth of 1 per cent of potassium carbonate. If the particles
of clay were spheres of uniform size, suspensions of this material
would show plasticity in concentrations of solid from 19.5 to
47.64, i.e., over a range of roughly 30 per cent. Colloidal
graphite exhibits zero fluidity when there is only 5.4 per cent
in suspension, hence it has a plasticity range of concentrations
of over 40 per cent. On the other hand, suspensions of many
coarse materials have a plasticity range which is much con-
stricted, which for practical purposes, is sometimes a serious
disadvantage.
There is abundant evidence that as the diameter of the
particles is decreased, the opportunity for the particles touching