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23 The Fluidity at the Finite Speed of Loading
S m = S
∞
m −
T 0
t y
F [τ m (t)]dt.
(23.1)
If the function F is known, the latter formula gives the yield condition at loading
with an arbitrary rate. In a partial case of uniaxial elongation, it coincides with
the Cotrlell–Campbell [2] condition (21.3). Indeed, if we assume in (21.3) τ 0 =
S ∞
m , t = T 0 , and
=
0
a t τ < t y ,
F [τ m (τ )] +
1
2
S m δ(τ − T )
at t y τ T 0 ,
by directly checking, we can make sure that formula (21.3) gives (23.1). (In the
latter formula, the symbol δ designates the Dirac function.)
23.2 Defining the Aging Function
As per the definition (Sect. 22.1), when reaching (T 0 ), the moment of yield stress
(S m ), we have
S m = τ m (T 0 ).
Substituting the yield conditions into the latter equation (23.1) gives
S
∞
m − τ m (T 0 ) =
T 0
t y
F [τ m (t)]dt.
(23.2)
Assume that the stress τ m > τ y is applied instantaneously and before the
moment (T 0 ) of yield occurrence, it remains constant. Then we find as follows from
formula (23.2):
S
∞
m − τ m = F [τ m (t)]t,
(23.3)
where t = T 0 − t y is the time of yield delay. From formula (23.3), we obtain
F [τ m ] =
S ∞
m − τ m
(τ m )
.
(23.4)
In this manner, in case the dependency t ∼ τ m is known, formula (23.4) defines
the aging function.