Phenomenological Modelling of Stress Corrosion Cracking
M. HÉLIE, C. PEYRAT, O. RAQUET, G. SANTARINI, Ph. SORNAY
CEA/CEREM/Service de la Corrosion, d'Électrochimie et de Chimie des Fluides
BP 6, 92265 Fontenay-aux-Roses Cedex, FRANCE
email: helie@cyborg.cea.fr
INTRODUCTION
In spite of numerous works aiming at a better understanding of the mechanisms involved in Stress Corrosion Cracking (SCC), there is still no quantitative mechanistic model able to predict the initiation and growth of SCC cracks.
The difficulty in building such a model can be attributed to the complexity of this phenomenon in which many various parameters are involved such as chemical, electrochemical, metallurgical or mechanical ones, as well as to the difficulty in quantitatively expressing the results of laboratory experiments.
From this consideration, a model for the characterization of SCC tests has been developed by the Commissariat à l'Énergie Atomique (CEA).
This "morphological" model [1] provides a method for a quantitative characterization of SCC without any mechanistic hypothesis. It consists in analyzing the morphological information (shape of the cracks and size distribution) present in a specimen after an SCC test. It is shown that this experimental datum can be used for the calculation of initiation and propagation rates.
Development of this morphological model has been mainly performed using Constant Elongation Rate Tests (CERTs) on 304L austenitic stainless steel in a boiling 44 wt% MgCl2 aqueous solution ([2] to [4]).
The fundamental basis of the model are
presented in this paper together with the results obtained so
far on the system 304L / boiling MgCl2 solution.
THE MORPHOLOGICAL MODEL
The morphological information existing
on a specimen after a CERT can be expressed in terms of a function
giving the evolution of the crack depth
distribution as a function of the test duration t. In this case,
represents the density (number per unit
of specimen area) of cracks whose depth is greater than 
at time t, Z(0,t) being thus the total density of cracks on a
sample at time t.
It can then be easily seen that the function w(t) given by relation (1) represents the density of cracks initiated per unit of time at time t, or in other words the crack initiation rate :
(1)
We will assume that a given crack will always remain deeper than the other ones which initiate later and that, once initiated, a crack will not disappear by merging with another one. Crack propagation will also be supposed to be two dimensional, in a plane perpendicular to the stress direction.
In this case, if we call 
the instant of initiation of a crack of depth 
at time t, the number of cracks whose depth is greater than 
at time t is the total number of cracks initiated between the
beginning of the test and time ti.
This leads to the following relation
for the densities 
and w(t) :
(2)
It can also be seen from this relation that, being the integral of w(t) between 0 and t, the function W(t) is identical to Z(0,t) and represents the total number of cracks existing at time t per unit of specimen area.
The value of 
obviously depends on the depth 
the crack
has reached at time t, after having propagated during t - ti
at a growth rate 
we will suppose to be
the same, at a given time t, for all the cracks of equal depth
at this time.
At time t + dt, this crack will have
a depth equal to 
, its initiation time
obviously remaining the same.
This can be expressed by the following relation :
(3)
Hence for the crack growth rate 
:
(4)
Since we have seen that 
is linked to 
by the function 
in relation (2), we can write the following relations between
and 
:
(5)
Using relations (4) and (5), we obtain
the crack growth rate 
as a function of
:
(6)
Reciprocally, if w(t) and 
are known, calculation of 
and W(t) leads
with relation (2) to the knowledge of 
.
We have then demonstrated that, provided some simple hypotheses
and without any assumption on the mechanisms, there are unequivocal
relations between 
and the crack initiation
and growth rates w(t) and 
. In other words,
the only knowledge of the crack depth distribution function cangive access to the two characteristic functions of the cracking
process, viz. crack initiation and growth.
Theoretically, the function 
can be experimentally obtained by running several CERTs interrupted
after various durations and, for each test, determining the crack
depth distribution on each specimen.
Unfortunately, microscopic examinations
performed on metallographic cuts can only give access to a function
which represents the crack trace depth
distribution function, as shown figure 1.
As a matter of fact, a longitudinal
cut of the tensile specimen after test will lead to the observation
of "crack traces" whose length 
will depend on the distance x between the crack and the examination
plane. Typically, a crack of real depth 
will be seen as a trace 
such as 
,
depending on 
, where 
represents the crack half width.
To derive the crack depth distribution
from the experimental determination of
, it is necessary to assume that cracking
is statistical enough along the sample for any examination plane
to produce similar values of 
, and to
access to the "crack shape" function which can be written
as 
, relation between x, 
for each time t.
The 
crack traces
which, at time t, have a depth comprised between 
and 
are produced by all the cracks of
depth 
which initiated on both sides of
the examination plane at a distance which, for the cracks of depth
, is comprised between x and x + dx with
and thus for dx :
(7)
To consider all the cracks corresponding
to the traces of length comprised between 
and 
, we must sum up the cracks of depth
comprised between 
and 
(
being the
maximum crack depth) and initiated in bands dx respectively situated
on both sides of the examination plane at a distance x comprised
between 0 for the cracks of depth 
and
for the cracks of depth 
.
Hence, using the value of dx given by
relation (7) and after elimination of 
:
(8)
with 
the number
of cracks of depth comprised between 
at time t, the factor 2 accounting for the cracks initiated on
both sides of the examination plane.
If we assume for the cracks a very simple
triangular shape with an homothetic growth (the ratio between
the half width 
of a crack and its depth
remains constant at a value 
),
we have, as represented figure 2 :
(9)
And, after replacing the partial derivative
of 
in relation (8) :
(10)
Or : 
(11)
And, since 
(by definition, there can be no crack deeper than 
)
:
(12)
If we replace this expression of 
in relations (1) and (6), we finally obtain for the crack initiation
and growth rates :
(13)
And : 
(14)
For this simple homothetic triangular
crack shape, there is then an analytical solution to relation
(8) which allows the determination of 
from the experimental function 
obtained
by microscopic examination on longitudinal cuts of the specimens,
leading then to analytical expressions of the characteristic cracking
rates as functions of the experimental crack trace density.
An other determination of 
from experimental data can be performed using the function 
,
which, similarly to 
or 
,
represents the density of cracks whose half width is greater than
at time t. This function offers the advantage
of being directly accessible from the microscopic examination
of the specimen surface. In this case, provided a crack shape
function be known, the relation giving 
can be easily obtained. With 
(see figure
1), we obtain :
(15)
Since 
deals
with real cracks and not with crack traces, Zw(0,t)
represents the total number of cracks existing at time t by unit
of specimen area and we have Z(0,t) = Zw(0,t) for any
function 
.
The fact that, as for the use of 
it still remains necessary to determine the crack shape function
to obtain 
from 
,
points out that the shape of the cracks will be a fundamental
step to achieve in order to obtain, from experimental data, the
function 
leading to an evaluation of
the crack initiation and growth rates.
From this point of view, it is interesting
to note that the validity of a given crack shape function 
which leads to an analytical solution of relation (8) can be checked
by comparison of the experimental functions 
and 
.
It is the case, for instance, of the
homothetic triangular shape which is described by the function
with 
leading
to the analytical solution given relation (12). With 
and relation (15) written with 
instead
of 
, the combination of relations (12)
and (15) leads to :
(16)
With 
, 
,
and 
, it comes to :
(17)
Thus, validity of the homothetic triangular
shape hypothesis implies that on a logarithmic plot 
should superpose on 
after two translations
of respectively 
on the X and Y axes,
incidentally giving a graphical access to 
.
Since an accurate determination of the
crack shape evolution function is one of the key steps to obtain
reliable expressions of the crack initiation and growth rates,
it has been one of the main objectives of the current tests performed
on 304L stainless steel in boiling MgCl2 solution which
are described hereafter.
SCC TESTS ON AISI 304L IN HOT MAGNESIUM CHLORIDE SOLUTION
The study reported in this section takes into account the direct determination of the crack shape. The results presented differ slightly from the previously published ones ([2] to [4]), which were obtained with the use, for crack shape evaluation, of an indirect theoretical method.
Experimental
The tensile specimens, of 30 mm gauge length, were cut from a commercial sheet in the longitudinal direction, then heat treated in vacuum at 1050°C for 0.5 hour and helium quenched. After this treatment, the material structure was purely austenitic. The specimens were then mechanically polished and electropolished in an acetic-perchloric acid bath. In order to remove the film left by the electropolishing process, the specimens were finally pickled in a mixture of HF (0.9 mol.l-1), HNO3 (1.6 mol.l-1) and distilled water, rinsed, and dried with purified air.
The solution was prepared by adding magnesium chloride hexahydrate to distilled water until the boiling point reached 153°C. The corresponding MgCl2 concentration was 44 wt% and the solution pH was about 3.
The system 304L - MgCl2 was chosen because the number of transgranular cracks obtained in this case after a CERT of relatively short duration is large enough to allow the substitution of observed frequencies for mathematical probabilities. In this type of test, the characteristic parameter is the relative elongation rate, defined as the ratio of crosshead speed to the initial length of the specimen.
The tests were carried out in a PTFE cell equipped with a heater and a condenser, and adapted to a tensile testing machine. To measure the electrode potential of the specimen during the test, a saturated calomel electrode (SCE) was dipped into a tube containing saturated MgCl2 solution at room temperature, connected to the corrosion cell through a salt bridge. After immersion of about one hour in the boiling MgCl2 solution, the electrode potential of the tensile specimen reached a steady value of about -300 mV/SCE. Thus, straining of the specimens was initiated after an immersion of two hours in order to achieve the building of a sound passive layer.
The CERTs were performed at a relative elongation rate of 5x10-5 s-1. In these conditions, rupture was observed after 60 min. Since it has been noted that crack coalescence, which can be significant for long test durations, is not compatible with the hypotheses of the model, the next tests were interrupted before significant apparition of this phenomenon which was found to correspond roughly with the maximum nominal stress.
Results
Cracking was characterized for each
test by determining the crack half width distribution 
and the crack traces depth distribution 
.
As previously stated, 
is directly obtained
from microscopic observation of the sample surface and 
is obtained from the microscopic examination of the sample which
is first longitudinally cut, resin mounted and polished.
Experimental limits interfered with
these determinations since cracks of half width smaller than 12.5
µm could not be distinguished from emerging slip steps and
crack traces smaller than 5 µm could not be accurately estimated.
Therefore, 
and 
were determined for 
12.5 µm and
5 µm. Without access to Zw(0,t),
the crack initiation was then characterized by Zw(12.5 µm,t).
Five interrupted CERTs were carried out to obtain the evolution
of Zw(12.5 µm,t) with time. The results obtained
are reported figure 3. It can be seen from this figure that there
is no significant cracking before 20 min of test duration and
that the density of cracks reaches a steady value 10 min after.
This duration being relatively short compared to the time to rupture
of 60 min, initiation was thus considered in a first approximation
to be instantaneous at ti0.
For the function W(t) as defined in relation (2), this leads to :
W(t) = Z0 H(t - ti0) (18)
Where Z0 is the total density of initiated cracks, ti0 the incubation time, and H the Heaviside function.
Crack shape determination
The crack shape was determined by successive mechanical polishing of the specimen. Two methods can actually be used, which consist in polishing in a direction either parallel to the specimen surface or parallel to a longitudinal cut of the specimen. The first method was chosen because it allowed the study of a greater number of cracks. Measurement of the width of each crack after each polishing step gives access to the crack profiles. The thickness removed after each polishing (typically 5 µm) was determined by measuring the decrease in the diagonal of Vickers microhardness imprints, these imprints being also used as identification tags to follow each crack between two polishing operations. This was first performed with a specimen strained for 21 min and the profiles obtained following cracks of various initial width are reported figure 4.
The function 
derived from these profiles is represented figure 5 in terms of
the evolution of the 
ratio as a function
of 
at t = 21 min, this ratio being thus
equal to 
. Each plotted point results
from an average on between 5 and 10 measurements, and the significant
difference with already published results [4] is that with this
more accurate observation we observed an increase of the 
ratio for the small values of 
.
From these results, we obtained for
both functions 
:
(19)
and : 
(20)
It must be noted that relation (20) implies that cracks "initiate" with a half width wi.
In order to obtain the variation of
crack shape with time, the study described above was also performed
on a specimen strained for 33 min. The results showed that the
following combination of parameters 
correctly
described the crack shape for both test durations. It was thus
considered, in a first approximation, that the crack shape did
not depend on time and that the corresponding functions 
could be assimilated at any time to the expressions of 
given relations (19) and (20).
Crack growth rate
Once determined the crack shape, the
function 
can be derived from the experimental
function 
considering relation (15). The
results obtained on 
using the function
of relation (20) are presented figures
6 and 7.
The use of an analytical expression
for 
would lead, with relation (6), to
the knowledge of 
. However, it was preferred
to express the growth rate v as a function of the parameters Z
and t. As a matter of fact, we assumed in the model that a given
crack could never "overtake" another one, which means
that the density of cracks deeper than a given one is assumed
to remain constant all along the test. Thus, at time t = t2,
the cracks of depth 
having 
cracks deeper than them are the cracks whose depth was 
at time t = t1 (t1 < t2),
with 
and t1 such as 
,
and which grew from 
to 
during t2 - t1. The evolution
of 
with time for constant values of Z,
which can be deduced from figure 6 or 7, will then give access
to graphical determination of the crack growth rate v(Z,t). Figure
8 shows that the evolution of crack depth with time can be assimilated
to a power law for each value of Z, which leads to :
v(Z,t) = A (t - ti)B (21)
In this relation, B is about equal to - 0.5 and A is a sharply decreasing function of Z. This means that the growth rate of all cracks decreases with time and, for each crack, depends strongly on the cracks deeper than this one: the greater this number, the slower the growth rate.
CONCLUSION
It has been shown that, provided some
simple hypotheses be made, the two characteristic functions of
an SCC process, crack initiation and growth rates, can be theoretically
determined from the density 
of cracks
whose depth is greater than 
at time t,
without any assumption on the SCC mechanisms.
It must be noted that crack initiation and growth rates obtained by this method contain parameters depending on the experimental test conditions such as temperature, strain rate, or stress level.
However, this type of approach can offer interesting information in two fields :
can
be a powerful tool for the screening of mechanistic models ;
The tests carried out on AISI 304L in
MgCl2 pointed out the experimental difficulties in
obtaining reliable values of the function 
from the microscopic examination of the specimens surface. The
work in progress, concerning transgranular cracking of stainless
steel in chloride solution as well as intergranular cracking of
nickel alloys in Pressurized Water Reactors primary coolant, is
aimed at improving this determination as well as performing specific
experiments to study the influence of the test conditions on the
crack initiation and growth.
REFERENCES
[2] O. Raquet, "Quantitative characterization of initiation and propagation in stress corrosion cracking. An approach of a phenomenological model", PhD Thesis n°1202, University of Bordeaux, France, 1994.
[3] O. Raquet, M. Hélie, G. Santarini, "Initiation and propagation in stress corrosion cracking", International Symposium on Localized Dissolution and Corrosion, Materials Week 1994, ASM and TMS, Rosemont, USA, October 4-6, 1994, p. 165.
[4] O. Raquet, M. Hélie, G. Santarini, "Stress corrosion cracking of type 304L austenitic stainless steel in a hot aqueous chloride solution : a study of initiation and propagation", Corrosion, NACE, to be published.