THEORETICAL BASIS OF THE SED MODEL

The derivation of the SED model is fashioned after the well-known path independent J-integral approach developed by Rice in the sixties [5] for the analysis of strain concentration by notches and cracks, which revolutionized the field of Fracture Mechanics. A brief review of the J-integral development is first presented, followed by an analysis restricting the derivation to conditions unique to fuel cladding.

2.1. Review of path-independent J-integral

Consider a homogeneous body subjected to a two-dimensional deformation field, containing a defect or a crack that can be represented by a notch of the type shown in Fig. 1. Define the strain energy density U:

ij ij d

U =òI A (1)

where _Iijand _Aijare the stresses and the strains respectively. The J-integral is defined by

ds F

Udy

J x

ò -ò u

=

G G ¶

· ¶ (2)

In eq. 2, the integration is performed over the curve surrounding the notch tip. F is the traction vector such that it is positive in the direction of the outward normal along _/, i.e. f_i =I_ijn_j, u is the displacement vector, and ds is an element of arc length along /. The integral in eq. 2 is path independent; i.e. the value of J does not change if another contour enclosing the notch is chosen; proof of path dependence of the J-integral is given by Rice in his classic paper [5].

FIG. 1. Flat surface notch in two-dimensional deformation field.

2.2. Application to fuel rod analysis

During the course of irradiation, damage accumulation by two sources, fast-neutron fluence and hydrogen absorption, decreases the ability of the cladding to withstand mechanical loading by PCMI or pressure forces. The damage accumulation by fast-neutron fluence occurs rapidly following the start of irradiation and reaches saturation within a short time (<1 year). The effects of fast fluence are observed as an increase in the yield strength of Zircaloy and a decrease in the material ductility. The damage caused by hydrogen absorption is proportional to the Zircaloy corrosion process and requires long residence times and high duty. The damage caused by the presence of hydrogen in high burnup cladding is of two types. Circumferentially oriented platelets of hydrides distributed in the cladding in a radially varying concentration gradient. In some cases, the hydrides are driven towards the cladding outer region, under the effects of temperature and stress gradients, forming a continuous hydride outer rim beneath the corrosion layer [3]. The average hydrogen concentration in this hydride rim can exceed 2000 ppm, with a gradient that varies from almost pure hydride (16000 ppm) at the hydride/oxide interface to one or two hundred ppm in the interior. The second type of damage is the formation of hydride lenses as a result of oxide spallation [2].

The geometric effects of a coherent corrosion layer and the presence of hydrides (as distributed platelets or as denser outer rim) on the cladding performance is equivalent to a thickness reduction. Hydride lenses, on the other hand, are localized discontinuities, which under certain conditions can penetrate to almost mid wall. They have the effect of a notch, causing strain concentration in the surrounding zircaloy material. Such hydride concentration can be characterized as a notch-type discontinuity, which lends itself quite naturally to the application of the J-integral.

To that end we consider the geometric representations of a typical OD defect as shown in Fig.

2. The choice of a discrete notch-type form of damage is mainly to facilitate the mathematical derivations, but the exact form of the damage is not particularly important. We wish to estimate the value of J without analyzing the cracked body.

FIG. 2. Cross-section of fuel rod with contour line surrounding defect tip.

Figure 2 is a cross-section of a fuel rod with a notch-type cladding defect. Owing to the path independence of the J-integral, we are free to take the contour shown by the dashed line in Fig. 2. Now,

F = 0 at r = r0 F » 0 at r = ri

F⁺ = F^- at q = 0

Therefore, the traction term in eq. 2 drops out, and J becomes, using Green’s theorem,

x dxdy

Working in polar coordinates for convenience, we have:

dr

Integrating eq. 4 by parts, and simplifying terms, we obtain:

Near the crack, U(r,q) varies, as shown in the figure below,

0 U

F G

U(ri,G)

U(ro,G) U

2F

where U is the strain energy density in the uncracked body. The presence of the crack will cause U to differ from U by the quantity f _G . f _G would depend on the depth of the crack, as illustrated in the following figure.

0 f(q)

p q

Long Crack U

2p Short Crack

G

G U f

, r

U _i = + , ^U

Substituting eq. 6 into eq. 5, we obtain:

F r r f cos d J = -²ò _i + _o

0

(7)

Now, it remains to estimate a value for f. It should be equal to U at G =F , and zero away from G =F. A simple estimate for f would be a triangular shape with values,

j - +

= U

f for p -j <q <p , and f =0 for q <p -j (8)

This is illustrated in the figure below.

0 f(q)

p q U

2p

p-q p+q

The integral in eq. 7 becomes:

The parameter j depends on the crack length and has a maximum value in the range p

j<

<

0 . A maximum value of _j =0.742_p is obtained using Newton iteration. Substituting this value in eq. 9 and carrying out the integration gives the following upper bound on J:

.

J =145 _i+ _o (10)

Equation 10 states :

Failure of a cladding with a flaw or a defect can be predicted by calculating the strain energy density in the material without modeling the flaw or performing fracture analysis.

From eq. 10 we note that, at the time of failure initiation, J becomes JC (the fracture toughness), and correspondingly U becomes U_c, which we designate as the critical strain energy density (CSED). Using the definition in eq.1, this can be can be written as,

ij

where Af is the material failure strain, which is the total elongation in a uniaxial test; in multi-dimensional tests, A^f becomes a function of the biaxial or triaxial stress ratios. Having thusly established equivalence between CSED and JC, we can henceforth concentrate on the development of the CSED as a failure model for cladding material. Equation 11 states that the CSED is quantified from stress-strain data obtained from material property tests. The functional dependence of CSED on damage mechanisms, such as hydride lenses, hydride rim, spalled oxide, ID or OD cracks, etc. is reflected in the material property data for irradiated cladding with representative corrosion and hydride conditions to those encountered in high burnup fuel rods.

The first application of the CSED model is in the analysis of the RIA tests recently conducted in France and Japan [2,3]. The analyses were carried out using the transient fuel behavior code FALCON [6,7,8]. The code calculates the strain energy density U (SED) at each Gauss (integration) point in the finite element grid as function of time. By comparing the calculated SED to the material failure limit in the CSED model, the failure times can be predicted. As will be discussed later in this paper, the quality of the predictions indicates that the CSED/FALCON methodology can be reliably used to evaluate burnup extensions beyond presently approved limits.

3. CSED DATABASE