3. Calculation of material parameters#

The parameters of the law of behavior can be calculated from creep tests carried out for various stress and temperature levels. To do this, a one-dimensional law of behavior is used because the stress on a cylindrical specimen under tension can be modelled in dimension 1. The stress tensor is reduced to its axial component.

\(\underline{\underline{\sigma }}\mathrm{=}(\begin{array}{ccc}{\sigma }_{0}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array})\) and \({\underline{\underline{\sigma }}}^{\text{'}}\mathrm{=}{\sigma }_{0}(\begin{array}{ccc}\frac{2}{3}& 0& 0\\ 0& \mathrm{-}\frac{1}{3}& 0\\ 0& 0& \mathrm{-}\frac{1}{3}\end{array})\)

So we have: \({J}_{0}(\underline{\underline{\sigma }})={J}_{1}(\underline{\underline{\sigma }})={J}_{2}(\underline{\underline{\sigma }})={\sigma }_{0}\)

\(\chi (\underline{\underline{\sigma }})={\sigma }_{0}\forall (\alpha ,\beta )\)

The system of equations to be solved is then:

\(\begin{array}{}{\underline{\underline{\dot{\varepsilon }}}}^{\text{vp}}=\dot{p}(\begin{array}{ccc}1& 0& 0\\ 0& -\frac{1}{2}& 0\\ 0& 0& -\frac{1}{2}\end{array})\\ \dot{p}=\frac{\dot{r}}{(1-D)}\\ \dot{r}={\langle \frac{{\sigma }_{0}-{\sigma }_{y}(1-D)}{(1-D)K{r}^{\frac{1}{M}}}\rangle }^{N}\\ \dot{D}={\langle \frac{{\sigma }_{0}}{A}\rangle }^{R}(1-D{)}^{-k({\sigma }_{0})}\end{array}\)

This system of equations is integrable, which makes it possible to have a single equation for the cumulative viscoplastic deformation rate (which can be assimilated to the total deformation by neglecting the elastic deformations).

We can then correlate this expression to the experimental data to adjust the coefficients, but the number of parameters and the non-linearities make this difficult (in addition, there is no uniqueness).

It is therefore necessary to use a correlation method using « physical » hypotheses on the phenomenon of creep, the curve of which is represented below.

_images/Object_58.svg

Figure 3 -a: The different creep phases on a creep curve

The deformation curve as a function of time obtained after a creep test is divided into three parts:

  • a part called primary creep where the damage is negligible.

  • a part called secondary creep where the deformation rate is substantially constant.

  • a part called tertiary creep where work hardening is saturated and where damage phenomena are predominant.

A method for calculating parameters using experimental data \((e,t)\) (\((\dot{\varepsilon },t)\) is also used (is also used, which is deduced by a numerical procedure) for different stress levels and different temperatures was developed at CEA. It uses the expressions found above in the case of a homogeneous and one-dimensional constraint by making assumptions according to the part of the curve where the data are taken. For example, in the primary creep phase, we make the assumption \(D=0\) and in the secondary creep phase, we use the fact that \(\dot{\varepsilon }\) is constant.

The complete description and examples of calculations performed on German 22MoNiCr37 tank steel can be found in the references [bib3] and [bib4].