Given the coolant pressure and the irradiation history, i. e. the evolution of the axial distributions of pin power, bulk coolant temperature and fast neutron flux with time, a fuel performance code calculates the evolution of the thermo­mechanical state of a fuel pin. This involves modelling a large number of phenomena. As described in Section 14.2, these include: (a) standard phenomena associated with thermo-mechanical behaviour of the fuel and cladding materials, i. e. heat transfer by conduction, convection and radiation, thermal expansion, creep, elasticity, plasticity, fatigue, phase changes and melting; (b) phenomena related to the presence of a neutron flux, i. e. cladding hardening, embrittlement, axial growth and void swelling; (c) phenomena related to fissioning, neutron capture and the generation of fission products, i. e. (non-uniform) heat generation, the generation and release of fission gas (Xe and Kr) and helium, and fuel densification and swelling; (d) phenomena related to microstructural changes in the fuel, i. e. formation of high burnup structure, grain growth and restructuring; (e) phenomena related to radial temperature gradients in the fuel pellets, i. e. pellet cracking and fuel fragment relocation, pellet wheatsheafing, axial extrusion, dish filling, oxygen migration and plutonium redistribution; (f) chemical phenomena, i. e. fuel-clad bonding, stress-corrosion cracking and cladding oxidation, erosion and dissolution.

The active length of the fuel pin, i. e. the part containing the fuel pellets or bar, is usually represented by a series of axial zones (or segments). In each axial zone the fuel is divided into radial annuli (or rings), usually of equal volume, but sometimes of equal thickness. The cladding may also be divided into two or more annuli, especially if liner or duplex cladding is being simulated. The free volumes associated with the fuel-clad gap, pellet dishes and chamfers, pellet cracks, the pellet/bar bore (if any) and any upper and lower plena are also generally modelled.

Further details of the thermo-mechanical modelling depend upon the sophistication of the simulation, which is typically summarised as 1/-D, 2-D or 3-D, and whether the code is steady state (where stored heat is ignored) or transient (where stored heat is taken into account).

In the 1/-D representation, only radial, i. e. no axial or circumferential, heat flow is assumed and the fuel annuli are all considered to be subject to the same axial strain (the so-called plane strain assumption). The latter, in conjunction with an assumption of axi-symmetry, allows shear stresses to be ignored, such that only the principal stresses along the radial, circumferential and axial directions are non-zero. The thermal (energy conservation) and mechanical (force balance, stress-strain relationship and strain-displacement relationship) equations are typically solved by a finite difference scheme. Coupling between the axial zones (which explains the ‘half dimension’ in the 1/-D) is restricted to the coolant enthalpy, pin internal pressure and gas transport. A 1/-D code cannot simulate phenomena caused by shear stresses, such as pellet wheatsheafing, clad ridging, axial extrusion and pellet cracking (although the fact that the pellets are cracked is taken into account, as described below). An example of a 1/-D code is TRANSURANUS (Lassmann, 1992).

In the 2-D representation, which is effectively only applicable to pelleted fuel, there is radial and axial modelling of a fuel pellet in each axial zone (axi-symmetry, but not plane axial strain, is still assumed). The thermal and mechanical equations are typically solved by a finite element technique. The advantage is that 2-D phenomena such as pellet wheatsheafing, clad ridging and axial extrusion can be modelled explicitly. The disadvantage is the increased complexity and therefore also the slower running time. An example of a 2-D code is FEMAXI (Suzuki and Uetsuka, 2002).

In the 3-D representation, there is full three-dimensional modelling of the fuel pellets/bar and cladding. As in the 2-D representation, the thermal and mechanical equations are typically solved by a finite element technique. The advantage over 2-D codes is that phenomena such as the azimuthal cladding stress concentration over radial fuel cracks, or pellet-cladding eccentricity, which cannot be modelled when axi-symmetry is assumed, can be simulated. The disadvantage is the increased complexity and therefore also the slower running time. Due to the intricacies of the 3-D representation, advanced numerical techniques are generally required in the solution scheme. An example of a 3-D code is BISON (Hansen et al., 2009).

In reality, codes are often some hybrid of the 1/-D, 2-D or 3-D representations. For example, in the ENIGMA code for modelling LWR and AGR oxide fuel, which nominally has a 1/-D representation: (a) the effects of shear stresses are approximated using models for axial extrusion and for pellet wheatsheafing, which feed back calculated strain increments into the main solution scheme (Gates et al., 1998); (b) the azimuthal cladding stress concentration over radial fuel cracks is calculated using a parasitic model (Jackson et al., 1990). Thus, the key phenomena, which cannot be modelled with the 1-D plane axial strain assumption employed in the code’s main solution scheme, are instead modelled by other means.

The cracked nature of fuel pellets complicates the mechanical analysis of ceramic fuel. There are two main approaches to modelling the effects of pellet cracks on the stresses and strains (Bailly et al., 1999), both of which are only approximate. The first method models directionally dependent (anisotropic) fuel elastic constants (Young’s modulus and Poisson’s ratio). The second method models ‘crack strains’ (the dimensions of the cracks as fractions of the corresponding pellet dimensions) in the stress-strain relations, which relieve the stresses when the rupture stress is exceeded. With 2-D and 3-D codes there is a, potentially more accurate, third approach, which is to model the cracking itself, together with the resultant effects on stresses and strains — this has been demonstrated (albeit with a commercial finite element software package) by Williamson and Knoll (2009).

The thermal and mechanical equations and their solution for a typical F/2-D fuel performance code, together with modelling the effects of pellet cracking, are described in detail by Olander (1976).