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Secondary creep for non-constant loading conditions
Journal of Nuclear Materials 80 (1979) 364-365 o Norm-Holl~d Publishing company
LETTER TO THE EDITORS - LET-IRE AUX ~DACTE~ SECONDARY CREEP FOR NON-CONSTANT LOADING CONDITIONS
it follows that for stable deformation, it should be
The results of a tensile creep experiment depends on many test conditions [l J. In the present letter, the influence of the test procedure on the shape of the creep curve is investigated for the case of a non steady load. Surpose that in a certain stress and temperature range, the strain rate e for high temperature creep at constant stress can be represented by a “power law”. For creep at a time dependent stress, it is t(t) = A (1”)o(t)” )
n/e < 1 ,
which in reality is not obeyed. It is evident from eq. (7) that t: increases with en. Plotting In i vs. fn E, the creep curve is represented by a straight line of slope n. We now consider the case that instead of the stress rate, the loading rate b,, = b(t = 0) is kept constant, Inserting this in eq. (l), we have to compensate for the reduction in cross section in the way that
(1)
u(t) = o. t bo(l f eo)t ,
where 6 and a are considered as true values. Assume o(r)=oe
tbt,
(94
where e. is the engineering strain. For
(2)
t = 7 >> uo,‘bo(l f eO),
with b =
constant Xfit); it follows from eq. (1) by intergrating from t = 0 - t E=A(T)
@b)
eq. (9) can be replaced by
(u. + bt)n+l = P(uo f bt) b(n + 1) b(n + 1) ’
o(f) 2 bo(l + eo)t ;
WI
and inserting this into eq. (I), it is
or
[bo(l + co)t]” .
k(t);A(t)
(101
Recalling that
If b >> a,,/? ,
e=hl(l
where T is the life time at non-constant stress, it is
one obtains from eq. (10)
E A (n + 1) e(t)/t ,
tea),
(5)
(W
from there is follows that or for small strains
di -=_ n de t’
dlnE’ n F&Z’
(6)
(1 lb)
and further dlnE ___ =-n de e’
which is identical with the result for an experiment with a constant stress rate b [see eq. (73. The stability requires that
(7)
Applying Hart’s stability criterion [Z], &.>
dln 6 -61 de
n
I I --
n-1
stable ,
=-_
n
n-l
(12)
Plotting In E’vs. ln E, a curve will result which at low 364
M. BoEek /Secondary creep for non-constant loading conditions Table 1 Results of different test procedures Test procedure at constant:
Steady state slope
._~_ Stress Load Stress rate Load rate
0
365
“acceleration” expressed by the slope d ln P/de in the secondary stage. Contrary to creep at constant stress or load, respectively, creep curves result in a ramp experiment, for which the slope in the secondary stage is always dependent on strain.
n & n(1 + e-l)
References strains has a constant slope n. In table 1, the results of different test procedures are summarized in a sequence of increasing strain rate
Received 19 January 1979
[ 1] B. Ilschner, Hochtemperaturplastizitlt
(Springer, Berlin,
1973). [2] E. H&t, Acta Met. 15 (1967) 351.
M. Bbcek Institut firr Material-und Festk&perforschung II, Kernfor. schungszentrum Karlsruhe GmbH, Fed. Rep. Germany
LETTER TO THE EDITORS - LET-IRE AUX ~DACTE~ SECONDARY CREEP FOR NON-CONSTANT LOADING CONDITIONS
it follows that for stable deformation, it should be
The results of a tensile creep experiment depends on many test conditions [l J. In the present letter, the influence of the test procedure on the shape of the creep curve is investigated for the case of a non steady load. Surpose that in a certain stress and temperature range, the strain rate e for high temperature creep at constant stress can be represented by a “power law”. For creep at a time dependent stress, it is t(t) = A (1”)o(t)” )
n/e < 1 ,
which in reality is not obeyed. It is evident from eq. (7) that t: increases with en. Plotting In i vs. fn E, the creep curve is represented by a straight line of slope n. We now consider the case that instead of the stress rate, the loading rate b,, = b(t = 0) is kept constant, Inserting this in eq. (l), we have to compensate for the reduction in cross section in the way that
(1)
u(t) = o. t bo(l f eo)t ,
where 6 and a are considered as true values. Assume o(r)=oe
tbt,
(94
where e. is the engineering strain. For
(2)
t = 7 >> uo,‘bo(l f eO),
with b =
constant Xfit); it follows from eq. (1) by intergrating from t = 0 - t E=A(T)
@b)
eq. (9) can be replaced by
(u. + bt)n+l = P(uo f bt) b(n + 1) b(n + 1) ’
o(f) 2 bo(l + eo)t ;
WI
and inserting this into eq. (I), it is
or
[bo(l + co)t]” .
k(t);A(t)
(101
Recalling that
If b >> a,,/? ,
e=hl(l
where T is the life time at non-constant stress, it is
one obtains from eq. (10)
E A (n + 1) e(t)/t ,
tea),
(5)
(W
from there is follows that or for small strains
di -=_ n de t’
dlnE’ n F&Z’
(6)
(1 lb)
and further dlnE ___ =-n de e’
which is identical with the result for an experiment with a constant stress rate b [see eq. (73. The stability requires that
(7)
Applying Hart’s stability criterion [Z], &.>
dln 6 -61 de
n
I I --
n-1
stable ,
=-_
n
n-l
(12)
Plotting In E’vs. ln E, a curve will result which at low 364
M. BoEek /Secondary creep for non-constant loading conditions Table 1 Results of different test procedures Test procedure at constant:
Steady state slope
._~_ Stress Load Stress rate Load rate
0
365
“acceleration” expressed by the slope d ln P/de in the secondary stage. Contrary to creep at constant stress or load, respectively, creep curves result in a ramp experiment, for which the slope in the secondary stage is always dependent on strain.
n & n(1 + e-l)
References strains has a constant slope n. In table 1, the results of different test procedures are summarized in a sequence of increasing strain rate
Received 19 January 1979
[ 1] B. Ilschner, Hochtemperaturplastizitlt
(Springer, Berlin,
1973). [2] E. H&t, Acta Met. 15 (1967) 351.
M. Bbcek Institut firr Material-und Festk&perforschung II, Kernfor. schungszentrum Karlsruhe GmbH, Fed. Rep. Germany