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7 Remodeling
To compute the change in volume of fiber family n as a function of time, we need
to add all the incremental changes that occur from the initial to the current time. The
volume of constituent n produced between times τ and τ + dτ is
dJ
n + = ˙
J
n + (τ ) dτ
per unit undeformed mixture volume. At some later time, the amount that remains is
dJ n = [ ˙
J n + (τ ) dτ ] q n (t, τ ),
(7.11)
where q n (t, τ ) is the survival function, representing the fraction of n produced at
time τ that remains at time t ≥ τ . For exponential degradation,
q n (t, τ ) = e −k n − (t−τ ) ,
(7.12)
and comparison with Eq. (7.10) reveals that
Q
n (t) = q
n (t, 0).
With this relation, integrating Eq. (7.11) gives the partial volume ratio
J n (t) = J n (0) q n (t, 0) +
t
0
˙
J
n + (τ ) q
n (t, τ ) dτ,
(7.13)
which reduces to Eq. (7.9) for ˙
J n + = 0. It can be shown that this relation is
equivalent to Eq. (7.7) (see Problem 7.1).
Example 7.1 Suppose fiber family n is produced at the constant rate
˙
J
n + = k
n + J
n
0 ,
where J n
0 = J n (0) and k n + is a positive rate constant. If the survival function is
given by Eq. (7.12), compute J n (t).
Solution
Substituting the given relations into Eq. (7.13) yields
J
n (t) = J
n
0 e
−k n − t
+
t
0
k
n + J
n
0 e
−k n − (t−τ ) dτ.
(7.14)