This page describes the DDIF2 stress potential which describes a damaging law used in CEA’ fuel performance codes. See [1–3].

The DDIF2 behaviour is used to describe the brittle nature of nuclear fuel ceramics and is usually coupled with a description of the viscoplasticity of those ceramics (See for example [6]).

Internally the DDIF2 stress potential is derived from the Hooke stress potential, so the definition of the elastic properties follows the same rules. See this page for details.

Local coordinate

This description is currently limited to initially isotropic behaviours, but the damage is described in three orthogonal directions. Those directions are currently fixed with respect to the global system. For \(2D\) and \(3D\) modelling hypotheses, those directions are determined by a material property, which external name is AngularCoordinate, giving the angular coordinate in a cylindrical system.

Material properties associated with damage

The description of damage is based on the following material properties:

• the fracture stresses in each direction. Two options can be used to described them:
  • if the fracture_stress option is used, the fracture stresses are equal in each directions.
  • otherwise, the fracture_stresses keyword can be used to describe the fracture stresses in each of the three directions.
• the softening slopes stresses in each direction. Two options can be used to described them:
  • if the softening_slope option is used, the softening slopes are equal in each directions.
  • otherwise, the softening_slopes keyword can be used to describe the softening slopes in each of the three directions.

In each case, a material property must be given as a value or as an external MFront file.

Fracture energies

Following Hillerborg approach (see [7]), softening slopes can be related to fracture energies by the mesh size. Thus, rather than the softening slopes, the user can provide the fracture energies through one the fracture_energy or fracture_energies options. In this case, an array of three material properties, which external name is ElementSize, is automatically declared.

External pressure effect

The effect of external pressure on the crack surface can be taken into account using the option handle_pressure_on_crack_surface. If this option is true, an external state variable called pr, which external name is PressureOnCrackSurface, is automatically declared.

Resolution algorithm

By default, the DDIF2 damage behaviour is treated using an algorithm based on statuses. In each damage directions, the damage state is kept constant during the Newton iterations. Once converged, the consistency of the damage state with the solution found is tested. If the state is inconsistent, the iterations are restarted with a new state.

The implicit scheme is divided in two steps by default. In the first step, the time step is set to zero before the prediction stage. This is meant to filter all viscoplastic flows and the convergence is thus performed only on the damage state (unless another rate-independent mechanism is considered). Once converged on the damage state, the time step is reset to its original value and the implicit resolution is restarted.

The DDIF2 stress potential (and thus the DDIF2 brick) has two options to change this behaviour:

• use_status_algorithm (boolean value, true by default). If false, the damage state is updated at each iterations.
• first_converge_on_damage (boolean value, true by default). If false, the time step is not set to zero at the prediction stage.

References