23.2 Defining the Aging Function
329
Let us assume that the dependency chart [4] between the yield delay time and
the instantaneous applied stress (Fig. 21.4) is obtained from the experiments. For
analytical representation, we will approximate this dependency by the formula
τ m = D · exp
−η((t)
N
+ τ y ,
(23.5)
where η, D, N, and τ y are constant values defined as coefficients of approximation
of the dependency t ∼ τ m using formula (23.5). Assuming in formula (23.5)
t = 0, τ m = S ∞
m , we obtain D = S ∞
m − τ y , e.g.
τ m =
S
∞
m − τ y
exp
−η((t)
N
+ τ y ,
or
t =
1
η
ln
S ∞
m − τ y
τ m − τ y
1/N
.
(23.6)
Let us find for the aging function:
F [τ m (t)] =
S ∞
m − τ m (t)
1
η
ln
S ∞
m − τ y
τ m (t) − τ y
1/N .
(23.7)
23.2.1 Example
Assume that the proportional loading of the specimen is done under the constant
loading rate v, e.g. τ m = τ y + vt. Then, by substituting the function (23.7) into the
yield condition (23.1), we obtain the following dependency between the yield stress
(S T ) and loading rate:
v =
1
S ∞
m − S T
S T
τ y
S ∞
m − x
dx
1
η
ln
S ∞
m − τ y
τ − τ y
1/N .
(23.8)
Circles in Figs. 23.1 and 23.2 indicate experimental data obtained by Gendrikson
and Wood [4] in experiments for elongation of samples made of annealed steel
containing 0.17% of carbon. Continuous lines in these figures depict diagrams
built using formulas (23.8) and (23.6). The following constant values are adopted:
S ∞
m = S ∞
p = 240 MPa, N = 0.379, η = 3.767 s −1 , and τ y = 150 MPa. As figures
show, the dependencies (23.8) and (23.6) well approximate experimental data.