The development of the crack growth model in zirconium claddings in iodine environment
Introduction
Iodine assisted Stress Corrosion Cracking (SCC) is a major mode of pellet-cladding interaction failure under transient conditions. Now it is well-established that the process of corrosion cracking consists of three steps: crack initiation, quasi-steady-state crack growth and ductile rupture of remained cladding. The main part of the lifetime of cladding during iodine assisted SCC is determined by the first two steps. In the present paper modeling of the quasi-steady-state crack growth is presented.
As a rule the published models of quasi-steady-state crack growth are restricted by a construction of some correlations between the crack growth rate and stress intensity factor (SIF) for certain set of experimental data. Existing experimental data have enough large dispersion and the applicability of these correlations to prediction of cladding behavior is rather limited.
During last decade new experimental results (Bibilashvily et al., 1995, Schuster et al., 1995) have been obtained which led to more deep insight into the mechanism of SCC. In particular it was shown that the behavior of the material at the crack front (along the line perpendicular to the direction of crack propagation) is not homogeneous. The fracture surface consists of alternating regions of brittle quasi-cleavage and ductile rupture (regions accompanied with strong plastic deformations). Below, the microscopic model of corrosion crack growth is presented which is based on these experimental results.
Section snippets
Phenomenological model
In the temperature range up to 860 °C zirconium and its alloys has hexagonal lattice wherein the ratio of c/a is less than the ratio in the closed packed lattice of this type (c and a are edges of the crystal structure). Hence, it follows that plastic strains here are determined by dislocation slip in prismatic planes (Douglass, 1975). If tensile stresses act in the direction normal to the basal plane, the plastic deformations do not take place. In that case pseudo-cleavage occurs for
Brittle and plastic regions, stress distribution
The crack along its front is divided into brittle and plastic regions. Each region is characterized by effective stresses σbr and σpl. Fraction of brittle regions α is taken to be equal to the tangential component of the texture parameter. The fraction of plastic regions is equal to (1−α). The stresses in the plastic regions are approximately equal to the critical ones for which the growth of the ductile crack of length l begins. For calculation of its values δK-model (Panasyuk et al., 1988) is
Iodine concentration near the brittle region of the crack tip
The brittle crack of length l is shown schematically in Fig. 1 for semi-infinite medium which is subjected to the tensile stresses σbr. The length a determines the size of the region in which the action of interatomic forces between the faces of the crack is essential. Outside this region we shall consider that opening of the crack increases in proportion to the distance r from crack tip. It reaches the value h≈lϕ at the point of outlet of the crack on the surface. Here ϕ=σ/πμ, μ is the shift
Determination of iodine flux to the crack tip
If iodine concentration Cg outside the crack is greater than Ct, iodine flux to the crack tip occurs and the crack begins to grow. Transfer of iodine inside the crack takes place through gaseous phase and along the surface and is described by diffusion equations. In the course of diffusion the exchange of iodine atoms between the bulk and the surface of the crack occurs. For given bulk concentration C the equilibrium concentration of atoms adsorbed on the surface neq is defined by the equation:
Results of calculation and comparison with experimental data
For computation of the rate of crack growth the numerical model has been developed which self-consistently takes into account all considered processes. The calculations of the crack growth rate were carried out in the following order. First, from Eq. (1) effective stresses in plastic regions σd are calculated which are necessary for the crack of the length l to grow. For given average stresses σ the stresses in brittle region σbr and corresponding SIF Kbr are calculated. According to , bulk Ct
Conclusion
On the basis of the analysis of experimental data the model of quasi-steady-state growth of corrosion cracks in zirconium claddings in iodine atmosphere under stress is proposed. The regions of two types (brittle and plastic) are formed along the crack front as a result of the anisotropic properties of zirconium and due to texture of the cladding. High stresses in the brittle regions of the crack tip promote iodine atoms to penetrate into the metal that leads to the crack growth. The level of
Acknowledgements
The authors would like to thank Dr Yu.V. Troschiyov for assistance in carrying out of computations. The work was partially supported by Russian Foundation for Basic Researches (Project No. 00-02-16276a).
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