In ceramic fuel such as UO, a significant temperature gradient develops from the fuel center to the radial edge. This gradient appears early and is strong enough to induce cracking in the fuel due to the accompanying stress. The cracks reduce the stress in the fuel and increase the effective fuel volume (decrease the gap size).
A smeared cracking model in Bison may be invoked to account for this cracking. A smeared cracking model adjusts the elastic constants at material points as opposed to introducing topographic changes to the mesh, as would be the case with a discrete cracking model.
When the smeared cracking model is active, principal stresses are compared to a critical stress. If the material stress exceeds the critical stress, the material point is considered cracked in that direction, and the stress is reduced to zero. From that point on, the material point will have no strength unless the strain becomes compressive.
The orientation of the principal coordinate system is determined from the eigenvectors of the elastic strain tensor. However, once a crack direction is determined, that direction remains fixed and further cracks are considered in directions perpendicular to the original crack direction. Note that for axisymmetric problems, one crack direction is known _a priori_. The theta or out-of-plane direction is not coupled to the and directions (i.e., no or shear strain/stress exists) and is therefore a known or principal direction.
If we store a scalar value, , for each of the three possible crack directions at a material point, these in combination with the principal directions (eigenvectors or rotation tensor) provide a convenient way to eliminate stress in cracked directions. A value of 1 for indicates that the material point has not cracked in that direction. A value very close to zero (not zero for numerical reasons) indicates that cracking has occurred.
We define a cracking tensor in the cracked orientation as : (1) The rotation tensor is defined in terms of the eigenvectors : (2) This leads to a transformation operator : (3)
is useful for transforming uncracked tensors in the global frame to cracked tensors in the same frame. For example, the cracked stress in terms of the stress is (subscript indicates cracked, local frame, and global frame): (4) When many material points have multiple cracks, the solution becomes difficult to obtain numerically. For this reason, controls are available to limit the number and direction of cracks that are allowed.
The idea behind the Bison isotropic cracking model is to provide a description of cracked fuel in an isotropic mechanical framework. In particular, it is assumed that
The cracking of the fuel occurs in the elastic regime.
The crack length spans the full fuel dimension in the considered direction.
The description of the cracked material is not dependent on the particular crack-pattern but only on the number of cracks.
The principal strains are conserved.
Under these assumptions, the elastic constants (i.e., Young modulus and Poisson ratio ) are scaled depending on the number of cracks . The calculated stresses are scaled accordingly and isotropically. The model allows for multiple fuel cracking, and an empirical correlation for the number of cracks as function of the rod average linear heat rate is developed.
The basis of the herein described model are the constitutive equations describing the elastic behavior in terms of stresses and strains of a non-cracked material (expressed in terms of principal directions of cylindric coordinates , , ), namely (5) When any of the principal stresses reaches the ultimate stress, the fuel cracks along the direction perpendicular to that stress principal direction. For instance, when a crack occurs perpendicular to the circumferential direction, the corresponding stress component vanishes. Thus, the elastic behaviour of the cracked fuel is described by (6)
The comparison of Eqs. Eq. 6 with Eqs. Eq. 5 points out the anisotropy introduced by the presence of the crack. In order to represent the cracked material as isotropic, it is assumed that the elastic constitutive model for an isotropic material holds for the cracked material (7) Compared to Eqs. Eq. 6, the representation of Eqs. Eq. 7 conserves the principal strains and scales the elastic constants to the values and to allow for an isotropic description of the stresses , , .
In order to determine the scaled elastic constants and the square deviation of the scaled stresses is defined (Eqs. Eq. 7) with respect to the stresses in the cracked material (Eqs. Eq. 6), namely (8) which is a function of the principal strains , , . It can be demonstrated that averaging over any strain range symmetric with respect to 0 and minimizing yields (9) (10) These equations are identical to those derived by Jankus and Weeks (1972). Note that Eqs. Eq. 9,Eq. 10 are independent of the principal direction considered for cracking.
The model allows for multiple cracks in the fuel by applying iteratively Eqs. Eq. 9,Eq. 10. For cracks, the elastic constants become (11) (12) The number of cracks is considered as a function of the rod average linear heat rate . In particular, a correlation is developed based on the data reported by Oguma (1983) and Walton and Husser (1983). No dependencies on burnup or on power operation history (i.e., cycling) are considered, in view of lack of data. The resulting correlation for the number of cracks is (13) in which is the linear heat rate required to trigger the first crack. The parameters and kW/m have been determined by fitting Oguma (1983) and Walton and Husser (1983).
Figure 1 reports the correlation (Eq. 13) together with the data used to derive it. The correlation proposed by Oguma (1983) is reported for comparison. The description of crack healing is currently not considered in the cracking model.
As an example, the application of the isotropic cracking model to a 2D axisymmetric model of a single UO fuel pellet under typical PWR irradiation conditions is presented. The calculated principal stresses (i.e., in the considered coordinate system, radial, circumferential and axial stress), hydrostatic stress and Von Mises stress are shown in Figure 2 and Figure 3. Both isotropic cracking and creep models are activated. Results refer to irradiation times of 3 h (solid line) and after 3 years (dashed line) of irradiation, respectively. These results are in line with the expected stress values in a cracked material.
- V. Z. Jankus and R. W. Weeks.
LIFE-II â€“ A computer analysis of fast-reactor fuel-element behaviour as a function of reactor operating history.
Nuclear Engineering and Design, 18:41â€“49, 1972.[BibTeX]
- M. Oguma.
Cracking and relocation behavior of nuclear-fuel pellets during rise to power.
Nuclear Engineering and Design, 76(1):35â€“45, 1983.[BibTeX]
- L.A. Walton and D.L. Husser.
Fuel pellet fracture and relocation.
IAEA Specialists Meeting on Water Reactor Fuel Element Performance Computer Modelling, Preston, UK.[BibTeX]