2. Formulation of the law of behavior#
It is proposed to use an additive decomposition of deformations. The mechanical deformation is then expressed as:
\({\underline{\varepsilon }}_{m}={\underline{\varepsilon }}_{e}+{\underline{\varepsilon }}_{p}+{\underline{\varepsilon }}_{i}+{\underline{\varepsilon }}_{g}\) eq 2-1
Mechanical deformation, \({\underline{\varepsilon }}_{m}\) is defined as total deformation minus thermal deformation: \({\underline{\varepsilon }}_{m}=\underline{\varepsilon }-{\varepsilon }_{\text{th}}\). The other components are: elastic deformation \({\underline{\varepsilon }}_{e}\), plastic deformation \({\underline{\varepsilon }}_{p}\), irradiation creep deformation \({\underline{\varepsilon }}_{i}\) and swelling deformation \({\underline{\varepsilon }}_{g}\).
Elastic deformation is linked to stress by Hooke’s law: \(\underline{\sigma }=\underline{\underline{E}}:{\underline{\varepsilon }}_{e}\). The elasticity tensor \(\underline{\underline{E}}\) depends on the temperature \(T\).
Plastic deformation is given by a Von-Misès type law. The flow area is expressed as:
\(f={\sigma }_{\text{eq}}-{\sigma }^{f}\) eq 2-2
where \({\sigma }_{\text{eq}}\) is the Von-Misès constraint and \({\sigma }^{f}\) is the flow limit. This depends on the cumulative plastic deformation, \(p\), on the temperature \(T\), and on the fluence \(\Phi\). The plastic flow is given by the rule of normality so that:
\({\underline{\dot{\varepsilon }}}_{p}=\dot{p}\underline{n}\) with \(\underline{n}=\frac{3}{2}\frac{\underline{s}}{{\sigma }_{\text{eq}}}\) eq 2-3
where \(\underline{s}\) is the constraint deviator. \(\dot{p}\) is given by the consistency condition \(f=0\) and \(\dot{f}=0\). The laws of radiation creep are expressed (for a test with constant stress and flow) [ref]:
\(\varepsilon =\text{max}({A}_{i}\cdot \sigma \cdot \Phi -{A}_{0})\) eq 2-4
The coefficient \({A}_{0}\) reflects a threshold effect. In differential form, the previous law can be rewritten as:
\(\dot{\varepsilon }={A}_{i}\sigma \cdot \varphi\) if \({\eta }_{i}>{A}_{0}/{A}_{i}\) with \({\dot{\eta }}_{i}=\sigma \cdot \varphi\) eq 2-5
where \(\varphi\) is the flow (\(\varphi =\dot{\Phi }\)). An additional state variable, \({\eta }_{i}\), is therefore introduced to describe the threshold effect. This uniaxial law must be extended to the multiaxial case. Since irradiation creep occurs without volume variation, this mechanism will be described by a viscoplastic model based on equipotentials given by the Von-Misès constraint. It will also be assumed that the evolution of the variable \({\eta }_{i}\) is managed by this same constraint. The following laws are then obtained:
\({\dot{\eta }}_{i}={\zeta }_{f}\cdot \sigma \cdot \varphi\) eq 2-6
\({\dot{p}}_{i}={A}_{i}\cdot \sigma \cdot \varphi\) if \({\eta }_{i}>{\eta }_{i}^{s}\) otherwise \({\dot{p}}_{i}=0\) eq 2-7
\({\underline{\dot{\varepsilon }}}_{i}={\dot{p}}_{i}\underline{n}\) eq 2-8
where \({p}_{i}\) is the equivalent irradiation creep deformation. \({A}_{i}\) and \({\eta }_{i}^{s}\) are coefficients of the model that may depend on temperature (we can consider a dependence on fluence but the model will be more complex to identify). The function \({\zeta }_{f}\) makes it possible to introduce a dependence on temperature into the law of evolution of the variable \({\eta }_{i}\) (see equation). It should be noted that the model thus formulated will be able to operate for variable temperatures and flows.
Note 1:
Note that irradiation « creep » is entirely controlled by fluence and not by time. In the event that a thermal creep mechanism is added, time would of course play an explicit role.
Note #2:
To ensure the successful crossing of the threshold \({\eta }_{i}^{s}\) , (from a numerical point of view) it is necessary to define an error criterion. This criterion « TOLER_ET » corresponds to the% of exceeding the threshold that is authorized during digital integration. Irradiation creep deformation begins as soon as the threshold \({\eta }_{i}^{s}\) is crossed, but Code_Aster is forced to respect a reasonable evolution of the variable \({\eta }_{i}\) in the vicinity of the threshold in the vicinity of the threshold \({\eta }_{i}^{s}\) . If during the calculation the criterion is not met, Code_Aster subdivides the time steps, provided that the subdivision of the time steps is allowed.
Swelling can be described by laws like:
\(\frac{\Delta V}{{V}_{0}}={F}_{g}(\Phi )\) eq 2-9
where \(\Delta V\) is the volume change and \({V}_{0}\) is the initial volume. By differentiating this equation and assuming that the volume variation remains low (\(\Delta V\ll {V}_{0}\)) we obtain:
\(\frac{\dot{V}}{V}=\frac{{\text{dF}}_{g}}{d\Phi }\varphi\) eq 2-10
This rate of variation is identified with the trace of the swelling speed tensor: \(\dot{V}/V=\mathrm{trace}({\underline{\dot{\varepsilon }}}_{g})\). It is assumed here that the swelling occurs isotropically and therefore that \({\underline{\dot{\varepsilon }}}_{g}\) can be expressed as:
\({\underline{\dot{\varepsilon }}}_{g}=\dot{g}\underline{1}\) eq 2-11
where \(\underline{1}\) is the unit tensor. So we have \(\dot{V}/V=3\dot{g}\)
That is: \(\dot{g}={A}_{g}\varphi\) eq 2-12
with \({A}_{g}=\frac{1}{3}{\text{dF}}_{g}/d\Phi\). \({A}_{g}\) is a new material setting. This depends on temperature and fluence. In a problem with variable temperature, it is therefore important to use the equation and not the equation (the equation being implicitly written for a constant temperature). A possible coupling between stress state and swelling is overlooked.
The definition of the model is based on five internal variables: \(p\), \({\eta }_{i}\),, \({p}_{i}\), \(g\), and \({\underline{\varepsilon }}_{e}\). Temperature and fluence are considered to be parameters imposed for a given calculation. These values come from neutron and thermal calculations. The equations of the model are shown in the table.
Internal variables |
Evolution equations |
p |
coherence \(\dot{p}\) |
\({\eta }_{i}\) |
|
\({p}_{i}\) |
|
g |
\(\dot{g}={A}_{g}\varphi\) |
\({\underline{\varepsilon }}_{e}\) |
|
Table 2.1: The equations of the model.