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Several degradation mechanisms, including freeze-thaw, some kinds of sulfate attack, and a hypothesized mechanism for so-called delayed ettringite formation [8], involve a uniform, on average, expansion of the matrix. Let us examine what effect this scenario has on stresses and displacements. Take h=0, so that there is effectively no shell phase, and let ε1o = ε, and ε3o = 0. Combining equations (3)-(7) into three equations for the three variables α3, α1, and β1, we obtain
These equations are exact, although, to be consistent with the dilute limit hypothesis, the
expressions
in eq. (8) should really be expanded to O(c). Using these solutions in
equations (1) and
(2) shows
that 
rr is tensile, and has its greatest value right at the matrix-aggregate interface. If
the
stress generated is large enough, the aggregate should break away from the matrix along the
interface. The tangential stress, 
,
is equal to
rr
in the
aggregate,
but is always compressive in the matrix, so there should be no radial cracking in the matrix. If the
aggregate should break away from the matrix, then there will be a non-zero displacement of the
rim
of the matrix. This can be computed using the same equations, but now eq.
(3) can be ignored,
and
eq. (5) becomes the vanishing of radial
stress at the (now) free boundary at r = a. Solving for the
radial
displacement at r = a, we find that u(r = a) =
a.
Since the outer edge of the detached
aggregate will have zero displacement, there will be a gap around the
aggregate of width
a,
which is proportional to the aggregate diameter. In a real concrete with many aggregates
packed
closely together, as long as each aggregate breaks away from the matrix, each gap will, on the
average, have a width proportional to the radius of the aggregate. If some aggregates remain
attached to the cement paste matrix, the stress fields will be changed somewhat, but the overall
result
should still be approximately true.