1. Introduction#

Some materials undergo structural transformations when they are subjected to particular thermal evolutions [bia1] _, [bib2] _, [bib3] _. This is for example the case of low-alloy steels during operations such as welding and heat treatment or even zircaloy alloys in fuel sheaths for certain cases of accidental situations (APRP).

These transformations have a more or less strong influence on thermal and mechanical evolutions.

From a thermal point of view, structural transformations are accompanied by a modification of the thermal characteristics (volume calorific capacity, thermal conductivity) of the material that undergoes them, as well as by the production or absorption of energy (latent heats of transformation) [bib2] _. However, the latent heats of transformation in the solid state are relatively low compared to the latent heats of change of liquid-solid state and it is therefore possible, as a first approximation, to consider thermal and structural evolutions as decoupled. This is currently the case with the thermal and metallurgical calculation options implemented in code_aster. [U4.85.01]

From a mechanical point of view, the consequences of structural transformations (in solid state) are of four types [bib2] _:

  • The mechanical characteristics of the material to which they are subjected are modified. More precisely, the elastic characteristics (Young’s modulus and Poisson’s ratio) are little affected while the plastic characteristics (especially the elastic limit) and the thermal expansion coefficient are strongly affected;

  • The volume expansion or contraction that accompanies structural transformations results in a (spherical) « transformation » deformation that is superimposed on the deformation of purely thermal origin. This effect is demonstrated on a dilatometry test and, in general, it is combined with that due to the modification of the expansion coefficient and we speak globally of the influence of transformations on thermal deformation;

A transformation taking place under stress can give rise to an irreversible deformation, even for stress levels much lower than the elastic limit of the material (at the temperature and in the structural state in question). This phenomenon is called « **plasticity of transformation » *;

During metallurgical transformation, a phenomenon of**work-hardening restoration* may occur. The hardening of the mother phase is not transmitted to the newly created phases. These can then be born with a virgin state of work-hardening or inherit only a portion, possibly all, of the work-hardening of the mother phase.

Moreover, the mechanical state also influences metallurgical behavior. In particular, the state of stress can accelerate or slow down the kinetics of transformations and modify the temperatures at which they occur. However, the experimental characterization of this influence, especially in the case of complex situations (three-dimensional, under temperature and state of variable stresses) remains very delicate and it is very common to consider structural evolution as independent of the mechanical state. This is the case of the structural transformations model implemented in code_aster.

If the various mechanical couplings are neglected, the determination of the mechanical evolution associated with a process involving structural transformations therefore requires two successive and decoupled calculations:

  • A thermo-metallurgical calculation (decoupled) allowing the determination of thermal and then structural evolutions;

  • A mechanical calculation (elasto-viscoplastic) taking into account the effects due to thermal and structural changes;

This document presents the mechanical modeling implemented in*code_aster*. Modeling is available for two materials:

  • Steel which undergoes an austenito-ferritic transformation around \(850°\) (transition from cold ferritic phases with a face-centered cubic structure to an austenitic phase when hot with a centered cubic structure). Steel has four possible ferritic phases: ferrite, pearlite, bainite and martensite;

  • Zircaloy alloys which undergo around \(800°C\) a transformation from a cold phase \(\alpha\) with a compact hexagonal structure to a hot phase \(\beta\) with a centered cubic structure (\(\mathrm{cc}\)).

The models are the same for both materials, only the phase number changes.

The model therefore has five phases for steel and three phases for zircaloy. The modeling of the behavior of zircaloy in fact requires considering two cold phases with different mechanical behavior; a phase \(\alpha\) considered to be pure and a phase \(\alpha\) mixed with \(\beta\) [U4.85.01], [bib4] _.

Note Bene:

The basic metallurgical concepts necessary to understand the general problem are gathered in [bia1] _*.*

The elasto-plastic resolution algorithm, without taking into account the effects due to structural transformations, is explained in [bib5] _*.*

This document is partly taken from [biv6] _*where a more detailed presentation of the model and some validation elements is given.*

The presentation of the models made in this document is illustrated with the case of steel.