Criteria for fracture initiation at hydrides in zirconium alloys I. Sharp crack tip

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
(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
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.

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[37]

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