Journal of Nuclear
North-Holland
Materials
208 (1994) 232-242
Criteria for fracture initiation at hydrides in zirconium alloys
I. Sharp crack tip *
S.-Q. Shi and M.P. Puls
Materials and Mechanics Branch, AECL Research, Whiteshell Laboratories, Pinawa, Manitoba, Canada ROE IL0
Received
29 July 1993; accepted
16 September
1993
A theoretical
framework for the initiation of delayed hydride cracking (DHC) in zirconium is proposed for two different
types of initiating sites, i.e., a sharp crack tip (considered
in this part) and a shallow notch (considered
in part II). In the
present part I, an expression
for K,, is derived which shows that K,, depends on the size and shape of the hydride
precipitated
at the crack tip, the yield stress and elastic moduli of the material and the fracture stress of the hydride. If the
hydride at the crack tip extends in length at constant thickness, then K,,
increases as the square root of the hydride
thickness. Thus a microstructure
favouring the formation of thicker hydrides at the crack tip would result in an increased
K,,. K,, increases slightly with temperature
up to a temperature
at which there is a more rapid increase. The temperature
at which there is a more rapid increase in K,, will increase as the yield stress increases. The model also predicts that an
increase in yield stress due to irradiation will cause an overall slight decrease in K,, compared to unirradiated material.
There is good agreement between the overall predictions of the theory and experimental results. It is suggested that more
careful evaluations of some key parameters are required to improve on the theoretical estimates.
1. Introduction
In hydride-forming
metals, the presence of hydrides
may lead to a process of slow crack propagation
called
delayed hydride cracking (DHC) [l-31. In this process,
hydrogen
atoms diffuse to a region of high tensile
stress (e.g., at a notch or crack tip) and form hydrides.
The hydride formed at the crack tip may fracture
under suitable conditions.
Once the hydride has fractured, the main crack advances and the cycle repeats at
the extended
crack tip. This intermittent
process of
crack growth has been confirmed
most directly by
observations
under a transmission
electron microscope
(TEM) for niobium 141, titanium 151, vanadium 161 and
zirconium [7], although there are also numerous indirect observations
confirming this.
Modelling of DHC is challenging, not only because
it involves three distinct phenomena,
i.e., diffusion,
phase transformation.and
fracture, but also because in
realistic situations, it depends on factors such as material geometry, impurity and hydrogen content, thermal
* Issued
as AECL-10941,
COG-93-362.
Elsevier Science B.V.
SSDI 0022-3 115(93)E0278-H
history and microstructure.
In the present study, our
primary concern is with DHC in the alloy Zr-2SNb
used in the present pressure tubes of CANDU@ nuclear reactors. In modelling DHC initiation, it is useful
to distinguish between three types of crack initiating
sites [8]: (il a sharp crack; (ii) a shallow, smooth notch;
and (iii) a nominally smooth surface. DHC criteria
were experimentally
found to be different for the three
types of sites. Of the three types of sites, the first two
are, by far, the most important.
This study (part I)
focusses on the sharp crack situation while the accompanying paper (part II) deals with the blunt notch case.
Experimentally,
it has been observed [3] that for
sharp cracks, over a certain range of the stress intensity
factor, K,, the DHC velocity is only weakly dependent
on K,. At K,-values
below this range, the velocity
decreases
rapidly with decreasing
K,. A threshold
stress intensity factor, K,,,
is defined as the limiting
stress intensity value below which the DHC velocity
becomes vanishingly small, see fig. 1. Any crack or
sharp notch subjected to a K, less than K,, will not
be able to initiate DHC over the lifetime of the tube
and, therefore,
tubes containing
such defects will be
safe to operate in a nuclear reactor. The total extant
S.-Q. Shi, M.P. Puls / Criteria for fracture initiation. I
233
2. Theory of K,,
OK
Fig. 1. DHC velocity
IH
as a function
of K,: definition
of K,,.
database yields a mean Km-value of 8.2 MPa 6
with
a 95% confidence
limit of a single observation
in the
range of 4.3 to 12.0 MPa 6.
Other experimental
facts
include: (1) the observation
that K,, increases slightly
with temperature;
(2) there is no clear relation between K,, and hydrogen concentration;
and, (3) irradiated materials
(which have a higher yield stress)
seem to have a slightly lower K,,
than unirradiated
materials. It is essential to physically understand
what
factors determine
K,,, both in order to predict and to
raise its value, the latter by providing guidance
on
material property changes that would be required in
the manufacture
of new pressure tubes. Such understanding will also make it possible to provide a quantitative explanation
of the experimental
data and allow
for a less conservative
limit on K,,.
It should be noted that a diffusion model developed
in the past [9], has provided
explanations
for many
significant features of the observed cracking behavior,
such as the dependence
of DHC velocity on temperature and stress intensity factor K, above K,,. However, this diffusion model was not designed to predict
K,,
nor was it structured
to contain a threshold
K,level for DHC propagation.
Nevertheless,
this diffusion
model could possibly provide a way of modeling K,,
by recognizing that at low K,, the hydride precipitation
(causing volume expansion) might effectively cancel the
tensile stress gradient around the crack tip. This could
eventually
inhibit further hydrogen
diffusion
to the
crack tip. An analysis of this type [lo] has shown
promise for predicting
a lower bound value for K,,.
However, due to the lack of linkage to other types of
DHC initiation analyses, in the following we adopt a
more general approach that relies on the concept of a
critical stress to fracture a hydride platelet as the basis
for the model. It is the purpose
of this paper to
establish a quantitative,
physical model of K,,. In the
next section, details of the model are described.
In
section 3, some preliminary
comparisons
with experimental results and discussions
are given. Finally, in
section 4, our conclusions are summarized.
The diffusion process is ignored here. We assume
that it is possible for hydrogen atoms to diffuse to the
crack tip and form at least one hydride in front of the
sharp crack. Experimentally,
it is observed [ll] that
zirconium hydrides formed at crack tips are platelets.
Most often, these platelets lie in, or close to, the crack
plane (see fig. 2). Our objective is to determine at what
applied stress level this hydride will crack. The basic
idea of the theory is that there is a critical threshold
stress for hydride fracture. Hence, when the local stress
at some point inside the hydride, u,_,, is larger than
the stress needed for hydride fracture, cr”, i.e.,
local 2 6 )
(1)
then a crack initiates in the hydride. We further assume that the stresses remain sufficiently
high over
most of the hydride so that the crack traverses the
entire length of the hydride. This approach is assumed
also to be applicable for hydrides located at the other
two types of DHC initiating sites.
A further simplification
in the present analysis is
that the local stress in the hydride is given by a linear
superposition
of an externally applied stress calculated
assuming there is no hydride present (we are interested
in the effective value at the hydride) a& and a stress
inside the hydride, ah, created only by the hydride
formation process in the absence of the external stress.
Therefore,
eq. (1) becomes
(2)
where cfh is assumed to have a definitive value and is a
material property of the hydride. This will be discussed
in section 3. Eq. (2) could give an overestimate
of the
stress on the hydride compared
to the more realistic
case in which the elastic-plastic
behaviour of the two
materials is taken into account. In the following discus-
Fig. 2. Schematic
showing
crack tip hydride
and stresses.
S.-Q. Shi, M.P. Puls / Criteria for fracture initiation. I
234
sion we present arguments that allow us to estimate
u& and uh in the above equation.
2.1. u&
Analytical solutions of the tensile stress in front of a
sharp crack tip without hydrides have been found previously (e.g., see ref. [12]). Here we are only interested
in the stress component normal to the crack plane,
uL . The basic characteristic of this stress component is
the following. In the immediate neighbourhood of the
crack tip there is usually a plastic zone whose extent is
given by the length rpZ along the crack plane. This
plastic zone size, rpZ, is a function of the stress intensity factor K, (which is determined by the externally
applied stress, crack geometry and the crack size) and
the yield stress of the material, oy. Along the crack
plane it is given by
(3)
where b is a proportionality coefficient. Theoretically,
b = 1 for the plane stress case and b = (1 - 2~)~ for an
infinitely sharp crack under plane strain conditions,
with v the Poisson’s ratio [13].
Inside the plastic zone, the tensile stress is given by
the IIutchinson, Rice and Rosengren (HRR) singular
solution [14,15]. At distances closer than 26 to the
crack tip, the HRR solution ceases to be valid and the
stresses decrease somewhat due to crack-tip blunting.
According to Rice and Johnson [16], for an elastic-perfectly plastic material (such as zirconium)
2(12s =
V2)KF
EUy
,
(4)
where E is Young’s modulus of the material. From 26
to the plastic zone boundary, rgzr the stress level decreases slightly. It will be convenient to choose the
stress level at rpZ to be the same as the maximum
stress at 26 because the expression is much simpler
and the error introduced is small compared to all the
other uncertainties in the model. This yields, for plane
strain,
Outside the plastic zone, beyond a certain distance
from rpzr the stress decreases as a function of r-1/2.
When a hydride is formed within the plastic zone,
because of the volume expansion around the zirconium
hydride, the local stresses will be relaxed to some level.
However, ignoring these unloading effects on the
stresses, we assume that, inside the hydride, the local
stress, gloca,, is given by
fflocal=ul
+ah.
(6)
Note that CT, and ah have different signs: cr is
positive (tensile) and gh is negative (compressive).
Ignoring the possible unloading effects implies that
%ff=
=rL.
2.2. cTh
As defined, gh is the stress created only by the
hydride formation process due to its misfit strain with
respect to the surrounding matrix. It is obvious that
this stress is dependent on the shape of the hydride
and the volume expansion parameters. Previously we
showed that crh also depends on the yield stress of the
surrounding matrix and the hydride 1171. For plateshaped hydrides, general analytical solutions are either
not possible or too complicated, and finite element or
other numerical solutions are required. However to
produce a more transparent solution, we simplify the
problem by assuming that
(i) The hydride is a flat, disk-shaped inclusion that
generates only a purely elastic deformation inside the
hydride. This assumption is not strictly correct, but
because the hydride is formed in a three dimensional
tensile stress field in a crack tip region, the amount of
plastic deformation due to the misfit strain during its
nucleation and growth may not be as large as might be
expected in the regions where no external stress is
applied.
(ii) The hydride inclusion only generates a stress
free strain normal to the disk (E, ) while all other
components are zero. This assumption appears not to
be correct, but it considerably simplifies the mathematical treatment. When more accurate values are required or warranted, the full, three-dimensional misfit
strain field can be used, but at the price of greater
complexity.
(iii> The elastic moduli are the same for both matrix
and hydride (see part II of this study for the treatment
of different elastic moduli).
(iv) The effect of the free surface of the crack on
the stress distribution inside the hydride is first ignored
for simplicity, and then it is taken into account as
shown in Appendix A. A finite element simulation has
S.-Q. Shi, M.P. Puls / Criteria for fracture
0.5
shown that the free surface effect of an infinitely sharp
crack (without applied stress) on the normal stress, u h,
inside a hydride of finite length is small at a distance of
2 1 pm away from the crack tip [18]. We will also show
in the Appendix that the free surface effect can be
included for a very long hydride with one end touching
the crack tip and demonstrate that its effect is small
for most of the situations of concern.
2
g
I
235
I
0.2
0.1
a
0
.s -0.1
1
z" 1:::
-0.4
Assuming a flat, ellipsoidally-shaped hydride precipitate and using Eshelby’s method [19], a simple expression for the internal, normal stress is
-0.5
-0.6
Fig.
where CYis a proportionality coefficient, t and L are
the thickness and diameter of the hydride disk, respectively. For t +z L (disc-shaped inclusion), (Y= a/4 and
for t = L (spherical inclusion), a = 8/15. One of Eshelby’s most important results is that for an ellipsoidally-shaped inclusion the stress field becomes uniform for all interior points of the inclusion.
A more realistic option is to assume that the hydride has a rectangular shape (e.g., t XL, x L,, where
t is the thickness and L, and L, are the lengths of the
sides of the rectangle). Such a shape would appear to
be a better idealization of the hydride shapes that are
actually observed by TEM or metallography (in fact,
the hydride appears to be thickest at the crack tip).
General solutions for the stress field of interior points
of a rectangular inclusion have been obtained by several researchers [20-231. An important difference
compared to the ellipsoidal inclusion is that the stress
field is not uniform inside the inclusion. Examples
of numerical
calculations
of normalized
stress,
(1 - V%h/EE,
) are shown in fig. 3 using the methods
given by Chiu [23]. In these analyses, to simplify the
calculations, it is assumed that there is only one component of stress-free strain, ll # 0 and L, = L, = L.
This figure shows the normal stress distribution along
the L direction for half of the inclusion. One can see
from this figure that the stress reaches its lowest compressive values (less negative) at the centre point (in
the length direction) of the inclusion. Analyzing the
numerical values of this centre stress shows that this
stress can be expressed using the same form as that of
eq. (71, with the approximations for the proportionality
constant (Y= 0.45. A general expression of uh as a
function of x (position inside the hydride) is complicated. But, as we will discuss later, simple analytical
expressions are possible if L X- t.
initiation. I
3.
Normalized stress profiles of a rectangularly-shaped
hydride (t X L X I,).
2.3. K,,:
single hydride platelet model
The threshold condition for DHC initiation
given K, is given by the relation
u& + uh = u;.
at a
(8)
Qualitatively, we see that when K, is small, then 26
and rPz are also small (see eqs. (3) and (4)). This means
that only a short part of the hydride is subjected to the
maximum tensile stress and the maximum tensile stress
will be located close to one end of the hydride plate
where the compression stress ah is also a maximum
(more negative) for a rectangularly-shaped
hydride.
Therefore, in order to fracture the hydride, at least
one of the two conditions should be met, i.e., (1) K,
must be high enough (26 and rPz are larger), or (2) the
hydride must be long enough (crh is less negative). To
illustrate these points, it is instructive to calculate the
stress profile in a hydride under external stress (i.e.,
a& + crh). Fig. 4 shows the results of the calculation of
(a& + ah) and assuming a rectangular hydride at a
crack tip with t = 2 p,rn and L = 10 km under different levels of K, (the free surface effect is ignored in
this case). Other parameters used in this figure are
given in table 1 for a Zr-2SNb pressure tube alloy. It
can be seen that, at low K,-values, the stress state
inside the hydride is below the fracture stress or!’ (even
negative at very low K,-values). When K, increases,
the maximum applied stress moves toward the centre
(in the length direction) of the hydride. For sufficiently
high K,, the combined internal stress will increase to
the point where mt is exceeded. This is a necessary
condition for fracture initiation. Fig. 5 is another way
S.-Q. Shi, A4.P. Puls / Criteria for fracture initiation. I
236
than K,, for fracture initiation. However, to calculate
this critical hydride length, we need to know the profile
of oh across the entire length of the hydride and this
profile will change when the value of L/t changes. In
principle, we may still use eq. (7) for gh, but now LYis
also a function of position inside the hydride. By substituting eqs. (5) and (7) into eq. (8) we find that,
L
A=
ffEEl
t
-2500
'
0
12
3
K,=Ifl
4
5
6
7
1
8
910
1
x WM
Fig. 4. Profiles of ~~~~~~
= U& + ah at crack tip for a hydride
(2 x 10 x 10 pm31 under different K,.
Table 1
Parameters
used in calculating
E = 95900 - 57.4 (T(K) - 2731
v=0.436-4.8~10~~{T(K)-300)
~~~~~~
MPa
1251
El = 0.054
my (unirradiated)
= 1088 - 1.02TCK)
MPa
(T (irradiated) = 1388 - 1.027XK) MPa
a) = 7.357x
[251
lo-3E
1251
1251
eq. (16)
of showing the relationship.
Here, K, is fixed, but the
L-value is changed. It is clear that there must be a
critical hydride length, L,, for a given K,-due,
higher
(I-vq&-“r)’
Here, (Y may be found numerically or graphically. As a
result, it is possible to obtain a critical hydride shape
factor (L,/t)
for a given stress intensity factor for
DHC initiation.
To find the ultimate limit to K, (i.e., K,,) for DHC
initiation, we need to assume that the hydride is capable of growing to infinite length, or L z+ t (therefore,
((Th 1 will be a minimum). It should be noted that the
true value of rh at one end of the hydride changes
only by about 5% for t/L ranging from lo-’ to lops.
In other words, for the same hydride thickness (t), ah
at one end region of the hydride changes very little
when L/t > 100. This implies that the lower bound of
the experimentally
determined
K,, of 4.3 MPa 6
[81
(involving the fracture of long, but finite-sized crack-tip
hydrides) should be close to the theoretically estimated
value obtained assuming L -+ 00 (for an infinitely long
hydride). Now, if we ignore the free surface effect at
the crack and assume that only Ed f 0 in this long
hydride, then an analytical solution of ah is 123,241
a”=
EEL
-
27r(l - V’)
1500
1000
r
x 2 tan-‘(tP) -
g
3
2
,/-.\
!
O /
-500
‘\
\
,",
il ',
:,’
-1000
+
4X2,t2j
[
500
;;i
(2x/t)
(1
‘,
\
\
L=IOkm
--- L=3.3pl
----~L=Zpm
i
t
EEL
ah=
01234567891
a
(10)
where x is the distance from the front end (at the
crack tip) of the hydride toward the centre of the
hydride. It is evident that when L + 00, at x = 0, uh is
independent
of the thickness of the hydride, i.e., oh =
- OSEe, /(l - v2>. However, this accurate solution is
not easy to use in our K,, analysis. Instead, we generate the following approximate
expression for gh,
-1500
-2000 i
1
- 457(1 -v2)
x+At
(0 IX),
where
x(w)
Fig. 5. Profiles of ~~~~~~
= a& + ah at crack tip for hydrides
having different lengths under K, = 10 MPa 6.
A = &exp(
-6.518x/t)
(11b)
A
0.5 ,\
D
E
G
----.
-.-.-.-
’ i
S.-Q. Shi, M.P. &Is / Criteria for fracture inifiation. I
lls
the region from 0 s x s rpz is important for the K,,
analysis.
In order to derive a simple expression for the critical threshold stress condition (K, = KIH and a& + u h
=crth), we further assume that the value of &+a”
always reaches a maximum at 26. In general, the net
maximum tensile stress could be between 2S and rpz
and a complete numerical solution would have to be
found. Qualitatively, however, the result would not be
too different from the solution derived below, since
both 26 and rpz are functions of K,. By substituting
2S =x into eq. (12), combining with eqs. (8) and (5)
and rearranging terms, we obtain
equation (10)
exacts&tion:
Equation (11)
Equa1ion(12)
0.4 :\
\
OL.‘.‘~.“.““.““‘~.““‘...’
0
0.5
1
1.5
2
2.5
3
E%, t
x/t W-4
Fig. 6. Comparison of normalized stresses inside a rectangularly shaped hydride CL.> t).
and A is small when x > 0.2%. Therefore, a simple
expression for ah may be used at larger values of x,
i.e.,
uh=
-
EEL
L
47r(l - V’) X
(x > 0.2%).
237
(12)
Fig. 6 compares the absolute values of the three expressions (refer to eqs. (lo), (11) and (12)) of the
normalized stress (1 - ~*)cr~/.Ec, , as a function of
x/t. It is evident that at large values of x, the three
expressions give identical results.
So far, we have ignored the free surface effect (or,
image effect) on gh at the crack tip region. We have
proved in Appendix A that the image effect from the
free surfaces at the crack on uh is small if the crack
opening displacement (COD) at the crack tip is larger
than 0.1 urn. It should be noted that there will always
be a certain external tensile stress applied to the crack
before hydrogen atoms start to diffuse to the tip region
to form hydrides. Therefore, the crack tip is never
infinitely sharp. For a Zr-2SNb alloy at 250°C a load
of K, = 2 MPa fi
would cause a COD of 0.1 pm.
Experimentally, K,, is always found to be larger than
2 MPa 6 for Zr alloys. Therefore, the COD is always
larger than 0.1 km, which would result in an even
smaller image effect. Therefore, the free surface effect
can be ignored in our analysis. In the cases when the
free surface effect cannot be ignored, a simple (and
approximate) way to take the free surface effect into
account is to introduce a modification factor (say, n) to
the values of uh in eqs. (lo), (11) or (12), such that
uLw = qah. The constant n can be determined by the
method given in Appendix A and by realizing that only
Uh)* =
8~(1-~~)~(--&$)’
(13)
The condition for this equation to be valid is that
2s > 0.2%. If 26 < 0.2%, then eq. (11) should be used
to yield
tKrd2=3
X
’
(14)
where A is a small number depending on 2s which in
turn depends on K,,. For example, if 26 = 0.20t, A =
0.043. The expressions in eqs. (13) and (14) show that,
with the stated assumptions, Km increases as t0.5 and
decreases with increase in uY. Thus the thickness of the
hydride platelet and the yield stress of the matrix, are
important variables controlling K,,.
Hence, a microstructure (or texture) that favours the precipitation
of thick hydrides would raise the II,, of the material,
while an increase in cY of the matrix would decrease it.
K,, depends on temperature through the temperature
dependence of the elastic moduli, the strength of the
matrix (alloy) and the fracture stress of the hydride
platelet.
2.4. K,,:
~~lt~-~yd~de platelets model
As has been discussed, the single hydride platelet
model of K,, derived above, when combined with a
lower bound value for uti’, should give a lower limit of
Km-values than the experimentally
observed Kffip
238
S.-Q. Shi, M.P. Puls / Criteria for fracture initiation. I
value. There appear to be at least two factors that
could increase the predicted value of K,, towards that
of KFGp. (1) The formation of multi-layered
hydrides at
the front of the crack tip and (2) incomplete
and
possibly uneven coverage of hydrides across and along
the front of the crack tip. To take account of the first
possibility, we modify the thickness parameter,
t, in
eqs. (13) or (14), to be an effective thickness, T, which,
in the simplest case, could be the sum of the thicknesses of each layer (i.e., r = C,ti). The second possibility may be accounted for by introducing a weighting
factor, f, which could, for instance, be equivalent to
the area fraction of hydride coverage in front of the
crack tip. This would give that
K,H=fKSH+(l-f)Kf?,
(15)
where Kc is the crack initiation threshold for a zirconium alloy containing no, or a very low area fraction of
coverage of small hydrides,
and K& is the single
platelet K,, derived in eqs. (13) or (14). It should be
noted that the possible values of f are determined
by
local microstructural
features (primarily the texture)
and not by the volume fraction of hydrides that may or
may not be present in the bulk of the material. That is,
DHC is possible whenever the level of hydrogen dissolved in the bulk of the material is sufficient to allow
the hydrogen concentration
at the crack tip to build up
to levels such that hydride precipitation
can occur
there, but the actual hydride coverage at the crack tip
is expected to be governed mainly by the microstructural features prevailing at the crack tip.
3. Comparison
sion
with experimental
results and discus-
In order to evaluate K,,,
the value of crt as a
function of temperature
is needed. Unfortunately,
there
is no data that directly gives the fracture stress of
zirconium hydride in the literature.
This is partially
due to the difficulty of preparing suitable solid hydride
specimens. The early work of Barraclough
and Beevers, based on uniaxial compression
tests [26] showed
that a: may be in the range of 100 to 200 MPa in the
temperature
range between 22 and 453°C for ZrH,.,,
bulk specimens.
These values are possibly too low
because their samples likely contained pre-existing microcracks. Recently, Puls and Rabier [27] conducted
a
series of confined and unconfined compression
tests at
temperatures
between 50 to 400°C for solid zirconium
hydride specimens of macroscopic size having stoichiometric compositions
ranging from ZrH,,, to ZrH,,,. It
is found that for most of the specimens, the yield stress
of these solid hydrides ranges from about 600 MPa at
50°C to about 250 MPa at 400°C. At room temperature, yield stress values ranging from about 730 to 1000
MPa are obtained.
In a number of unconfined
compression tests, fracture of the hydride samples occurred
at about 750 MPa at room temperature.
However,
these types of tests cannot be used to give reliable
results of the fracture
stress of a hydride platelet
confined in a zirconium matrix. Choubey and Puls [28]
have used acoustic emission to detect the onset of
cracking from long hydride platelets, located in uniaxially stressed tensile specimens of Zr-2.5Nb
pressure
tube material, to determine
the fracture stress of hydrides over a range of temperatures.
This work forms
an extension of earlier room temperature
tests [29-311.
The results, when combined
with estimates
of the
compressive
transformation
stress inside the hydride
obtained
from elastic-plastic,
finite element
calculations [17] suggest that a lower-bound
value for the
fracture stress of a hydride platelet ranges from 600 to
550 MPa between ambient and lOO”C, respectively.
Physically, it might be reasonable to assume that the
fracture strength of a brittle material such as zirconium
hydride is related to its bond strength, a measure of
which is reflected in the magnitude of Young’s modulus, E [32]. Therefore we have chosen to express err’ in
terms of E as follows,
o:(MPa)
= 7.357 X lo-“E,
(16)
with the constant of proportionality
chosen to give a
fracture strength of 600 MPa at 250°C. E should be the
Young’s modulus
of hydrides.
Here we used the
Young’s modulus for Zr-2.5Nb
alloy because we do
not have a value for the modulus of solid zirconium
hydride. Expressing
the fracture strength of the hydride in terms of E provides a convenient
way of
extending its currently measured temperature
dependence. Here, we have ignored any possible dependence
that qrr’ might have on impurity content, microstructure, hydride size and shape, and irradiation.
In the
following calculations,
the other parameters
used are
those given in table 1 for a Zr-2.5Nb
pressure tube
alloy.
Fig. 7 shows the Km-values (lines) calculated using
eq. (13) together with the total available experimental
data (symbols) collected
over the last 20 years for
Zr-2.5Nb.
Here an average hydride thickness,
t= 2
km is assumed. As expected, the theoretical
Km-values are smaller compared to the experimental
results.
This is reasonable
because in real cases, hydrides can
never grow to infinite lengths and the hydride coverage
S.-Q. Shi, M.P. Pzds / Criteria for fracture initiation. I
initiation at hydrides, is the increase in yield stress. It
is also assumed that the fracture stress of the hydride
remains
unaffected
by the irradiation.
A statistical
analysis by Sagat [35] shows that the experimental
value of K,, for irradiated
Zr-2.5Nb
pressure tube
material is slightly smaller than that for unirradiated
material. This is qualitatively consistent with the theoretical prediction.
12
10
$8
3
x
239
2;6
4
4. Conclusions
F-
?-
400
300
500
600
700
T W
Fig. 7. Comparison
..-.-..
themy radiated Zr-2.5Nh
--c
exptmmmal results (average)
theory unimdiated Zr-2 5Nb
of theoretical
data.
K,,
with experimental
at the front of a crack tip may not be 100%. According
to experimental
observations
by Luo [33] and eq. (15),
we may assume that f is about 80-95% and that Kc is
around 40 MPa&
at 250°C. From eq. (15), K,,
would be around 6.2-11.5 MPaG,
which is about
what has been observed experimentally.
The theory also predicts that K,, increases more
rapidly above 3OO”C, but is still low enough so that
DHC is possible. This prediction
is in conflict with
experimental
observations
of Smith and Eadie [34],
which show that above N 32O”C, DHC is difficult to
initiate despite the fact that the specimens contained
hydrogen well in excess of the terminal solid solubility
at the test temperature
(i.e., the bulk of the material
would have contained copious quantities of hydrides at
the test temperature).
The lack of agreement could be
because,
above that temperature,
eq. (16) does not
describe the temperature
dependence
of u: very well.
It may be that the ductility, as well as the fracture
strength of hydride, approaches
that of the Zr alloy
matrix above that temperature.
One of the reasons for
the increase of K,, with temperature
is due to the
decrease of the value of l/(1 - 2~) - u~:/u,, in eqs.
(13) and (14). When the value of uF/cY approaches (or
even is larger than) the value of l/(1 - 2v), DHC is
impossible. Also shown in this figure is the theoretical
K,,-value
for irradiated Zr-2.5Nb
pressure tube alloy
(the dashed line) which was calculated
using an increased yield stress typical for this alloy (see table 1). It
is assumed, in this case, that the most important effect
of the irradiation
from the point of view of crack
A model for DHC initiation criterion,
K,,, for a
sharp crack tip in zirconium alloys is derived. Preliminary results show good agreement with the experimental data if suitable values for some key material parameters are chosen. This model depends sensitively on
material parameters
such as a:, uY, as well as on
elastic and crystalline
properties
such as E, v and
Ed. More careful evaluations of these parameters
are
required to improve the accuracy of the predictions.
Appendix A. Image effect due to crack surfaces
A.1. Review of the linear-elastic theory for crack tip
stress analysis
Consider a sharp crack with a total crack length of
2a in an infinitely large and elastically isotropic plate.
If the crack is opened by two pairs of splitting point
forces, P, acting against the crack surfaces at y = 0,
x = fb (see fig. 8a), then the K, value at each crack
tip can be calculated by the following equation [36],
(A.1)
Assume next that P(b) = u( 1f b 1) db which acts on
the whole crack surfaces, then eq. (A.11 changes to
K,=$=&$!$.
(A.21
For a constant stress u, eq.
known solution K, = 06.
After K, is determined,
vicinity of each crack tip can
an infinitely sharp crack (see
KI
= -costs{
uyy
fi
(A.2) leads
to the well
the stress field in the
be determined,
e.g., for
fig. 8b for notation),
1 + sin+0 sin@}.
(A.3)
S.-Q. Shi, M.P. Puls / Criteria for fracture initiation. I
240
simply the addition of an extra term when compared to
eq. (A.3). Moreover, K, in eq. (A.4) is exactly the same
as given in eq. (A.3). Therefore,
the formulas for K, for
sharp cracks are fully applicable to the tips of slender
cracks such as the one in fig. 9. Now, the maximum
stress at the crack tip is finite, i.e., (with r’ = p/2 and
ef=o)
ma'= 2K,/fi.
UYY
A.2.
\
(4
Y
-!.L
r
(b)
0
!8
X
Fig. 8. (a) Two pairs of the splitting point force on a crack. (b)
Coordinate
system for the crack tip stress field.
(-4.5)
The image effect
Now consider two identical long hydrides (t K L)
situated at each crack tip as in fig. 10a. We are interested in the case when a z+ t. Then the interaction
between
the stress fields from the two hydrides is
negligible. Therefore
we can focus the discussion on
one half of the whole system (see fig. lob for notation).
The stress field created
by this hydride along the
centre horizontal line, in the absence of the crack, is
compressive on the right-hand side (inside the hydride,
i.e., uh in eqs. (10) or (11)) and tensile on the left-hand
side (outside the hydride). This tensile stress on the
by the same eq.
left-hand side, CT&,,can be represented
(10) or eq. (ll), but with different sign, e.g.,
t
EEL
One can see that, within the linear-elastic
regime,
oYyv+ +m at the crack tip (r = 0). However, if the
crack tip has a finite curvature (or, root radius), p (as is
always true in the DHC initiation experiments),
then it
has been proved [37] that the stress field becomes (see
fig. 9 for notation),
=uyy
&
d,,,(b)= 4ir(l-
v’)
b + At
at
O<b<m.
(A.6)
The image stress (a,, tensile) within the hydride due to
the introduction
of the crack surfaces can be calculated
+-v&7K,
50ste'
2r'
x cos$e’{ i + sin@
sin@‘} .
(A4
Note that, by selecting the centre of coordinates
at the
point p/2 from the tip, the expansion in eq. (A.4) is
ty
Tk
P’2_
r’
,/
oh
““t
8’
,/’
‘j
: o
Fig. 9. Deep
slender
notch and coordinate
stress field.
4
Y
-
oh
X
system
for the
Fig. 10. (a) A crack with two rectangularly-shaped
hydrides
(t X L). (b) Coordinate
system for stress analysis.
S.-Q. Shi, M.P. P&s / Criteria forfracture
initiation. I
241
by using the tensile stress a&, applied to the crack
surfaces. First, by using eq. (A.2), one can find an
effective X, at the crack tip due to a& Then by using
eq. (A.4), one gets ur.
In order to evaluate the integral in eq. (A.2), we
further simplify the stress function in eq. (A.6) to high
accuracy as follows,
%%(~)= a 4P(
Eel
t
1 - v’)
at
0 5 6 5 0.2%
at
0.25t<bsa,
EeL t
= 4a(l-
V*)b
(A.7)
4a(l
.
““.‘I
100
Root Radius, p (pm)
‘....A
10'
Fig. 11. The image effect: j3 in eq. (A.131 as a function of
crack tip root radius p.
-v’)
0.251
a!
i
2a
=-
10-1
EEL t
Kx= &iix
“.“.‘(
IV”
where a >> t and LYis calculated in such a way that the
average value of a$ in the range of 0: b < 0.25t is
obtained, i.e., a = (2/t) x 2.478. Substituting eq. (A.7)
into eq. (A.21, one has
2a
0’
J0
We also know that the maximum compressive stress of
uh at the same location is simply
db
&q7
+ C25t,&&
h
&ii- 4P(l_
v’)
Since a 2> t, therefore,
sin-‘(0.2%/a)
= 0.2%/a
and
EeL t
K1 = 27r(l-
V’)&z
EEL
(A.12)
2(1-l?)’
By comparing the absolute
mums, one has
= a. Eq. (A.8) simplifies to
{_
_
%3x - -
E.cL t
(1.239 + In 8a/t).
Fig. 11 shows the ratio
radius p by assuming t
can see that, generally
small. For example, if
values of the two maxi-
p as a function of the root
= 2 pm and u = 15 mm. One
speaking, the image effect is
the external load on a pre-
(A-9)
Therefore, from eq. (A.41, the image stress is (with
6’= 0 and r’=p/2 +x)
or=
d&
2(P +x)
p + 2x
with
05x,
(A.lO)
where K, is given by eq. (A.9). Therefore, the final
stress profile inside the hydride is the sum of or
(tensile) and c h (compressive).
Now, we examine how significant the image effect is
at the crack tip. Using eqs. (A.9) and (A.lO), the
maximum image stress (tensile) at the crack tip (x = 0)
is
UI
‘-=2=
EEL t
T2(1 _ v2jJap
/
.......---- hydride stress
---image stress
i
-0.5 L
0
Il.239 + In sa/tl.
(A.ll)
-
0.002
combined stress
0.004
0.006
0.008
x (mm)
Fig.
12. The image effect for p = 0.1 pm,
0.01
242
S.-Q. Shi, M.P. Puls / Criteria for fracture initiation. I
cracked specimen of Zr-2.5Nb
pressure tube material
results in a value of K, = 2 MPa 6
at the crack tip,
such K, will cause roughly _ 0.1 urn crack tip opening
at 250°C which corresponds,
approximately,
to a crack
tip root radius p = 0.05 km. The image effect at the
crack tip for this case is only 18%. Any K, larger than
2 MPaG
will cause a larger crack tip opening and,
therefore,
will result in even smaller image effects.
Fig. 12 shows a comparison
of three normalized stress
profiles
inside
the
hydride,
(1 - v2)ah/Ec,,
and the combination
of the two,
(1 - V?(r,/EE,
(1 - v2Xah + U,)/EEI,
by assuming t = 2 ym, a = 15
mm and p = 0.1 km. As expected, the image effect is
small over most of the range concerned.
[lo]
1111
[12]
[13]
[14]
[15]
[16]
[17]
[18]
Acknowledgements
We would like to thank Professor E. Smith of the
University of Manchester,
B.W. Leitch of AECL Research, Whiteshell Laboratories
and Dr. D.R. Metzger
of Ontario Hydro Research Division for many useful
discussions.
This work was funded by the CANDU
Owners Group (COG) under WPIR 2-31-6530.
References
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[l] D.N. Williams, J. Inst. Metals 91 (1962) 147.
[2] R. Dutton,
K. Nuttall, M.P. Puls and L.A. Simpson,
Metall. Trans. A 8 (1977) 1553.
[3] L.A. Simpson and M.P. Puls, Metall. Trans. A 10 (1979)
1979.
[4] H. Matsui, N. Yoshikawa and M. Koiwa, Acta Metall. 35
(1987) 413.
[5] H.Z. Xiao, S.J. Gao and X.J. Wan, Scripta Metall. 21
(1987) 265.
[61 S. Takano and T. Suzuki, Acta Metall. 22 (1974) 265.
[71 CD. Cann and E.E. Sexton, Acta Metall. 28 (1980) 1215.
181 M.P. Puls, S. Sagat, D.K. Rodgers, D.A. Scarth and W.K.
Lee, unpublished
research, AECL Research,
Whiteshell
Laboratories,
Pinawa, Manitoba,
Canada,
and Ontario
Hydro Research
Division, Toronto,
Ontario,
Canada,
1992.
on
191 R. Dutton and M.P. Puls, in Effects of Hydrogen
Behaviour of Materials,
eds. A.W. Thompson
and I.M.
Bernstein (TMS-AIME,
New York, 1976) p. 512.
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
S.Q.
Shi, unpublished
research,
AECL
Research,
Whiteshell
Laboratories,
Pinawa,
Manitoba,
Canada,
1992.
Xin Quan Yuan and K. Tangri, J. Nucl. Mater. 105
(1982) 310.
IN. Sneddon, Proc. R. Sot. Al87 (1946) 229.
T.M. Banks and A. Garlick, Eng. Fracture
Mech. 19
(1984) 571.
J.W. Hutchison,
J. Mech. Phys. Solids 16 (1968) 13.
J.R. Rice and G.F. Rosengren,
J. Mech. Phys. Solids 16
(1968) 1.
J.R. Rice and M.A. Johnson, in Inelastic Behaviour
of
Solids, eds. M.F. Kanninen, W.G. Adler, A.R. Rosenfield
and RI. Jaffee (McGraw-Hill,
New York, 1970) p. 651.
B.W. Leitch and M.P. Puls, Metall. Trans. A 23 (1992)
797.
B.W. Leitch, unpublished
research,
AECL Research,
Whiteshell
Laboratories,
Pinawa,
Manitoba,
Canada,
1992.
J.D. Eshelby, Proc. R. Sot. A241 (1957) 376.
G. Faivre, Phys. Status Solidi 35 (1964) 249.
R. Sankaran
and C. Laird, J. Mech. Phys. Solids 24
(1976) 251.
J.K. Lee and W.C. Johnson,
Scripta Metall. 11 (1977)
477.
Y.P. Chiu, J. Appl. Mech. 44 (1977) 587.
E. Smith, private communication,
University of Manchester, Manchester,
UK, 1992.
M.P. Puls, Metall. Trans. A 21 (1990) 2905.
K.G. Barraclough
and C.J. Beevers, J. Mater. Sci. 4
(1969) 518.
M.P. Puls and J. Rabier, unpublished
research,
AECL
Research,
Whiteshell
Laboratories,
Pinawa, Manitoba,
Canada, 1992.
R. Choubey and M.P. PUB, Metall. Trans., accepted.
L.A. Simpson, Metall. Trans. A 12 (1981) 2113.
M.P. Puls, Metall. Trans. A 19 (1988) 1507.
M.P. Puls, Metall. Trans. A 22 (1991) 2327.
A.S. Tetelman
and A.J. McEvily, Jr., Fracture of Structure Materials (Wiley, New York, 1967) p. 47.
L. Luo, unpublished
research,
Ontario Hydro Research
Division, Toronto, Ontario, Canada, 1992.
R.R. Smith and R.L. Eadie, Scripta Metall. 22 (1988)
833.
S. Sagat, unpublished
research,
AECL Research,
Chalk
River Laboratories,
Chalk River, Ontario, Canada, 1992.
H. Tada, P.C. Paris and G.R. Irwin, The Stress Analysis
of Cracks Handbook (Del Research Corporation,
Hellertown, PA, USA, 1985).
M. Creager
and P.C. Paris, Int. J. Fracture
Mech. 3
(1967) 247.