2. Benchmark solution#

2.1. Reference solution for modeling A#

It is based on [bib.1] and [R5.03.11]. An analytical solution is found under the hypotheses:

  • the stress tensor \(\sigma\) is known (constraints imposed on a material point)

  • interaction matrix \({a}_{\mathrm{ij}}\) is composed only of 1.

For each sliding system, the resolved split is calculated by: \({\tau }_{s}=\sigma :{m}_{s}\)

with \({m}_{s}\) the orientation tensor defined by: \({({m}_{s})}_{\mathrm{ij}}=\frac{1}{2}({({n}_{s})}_{i}\cdot {({l}_{s})}_{j}+{({l}_{s})}_{i}\cdot {({n}_{s})}_{j})\). \({n}_{s}\) designating the normal to the sliding plane of the \(s\) system and \({l}_{s}\) the sliding direction. The evolution of plastic sliding is given for each system \(s\) by:

\(\dot{{\gamma }_{s}}=\dot{{p}_{s}}\frac{{\tau }_{s}}{∣{\tau }_{s}∣}\) where \(\dot{{p}_{s}}=\dot{{\gamma }_{0}}({(\frac{∣{\tau }_{s}∣}{{\tau }_{f}+{\tau }_{s}^{\mathrm{forest}}})}^{n}-1)\) if \(∣{\tau }_{s}∣\ge {\tau }_{0}+{\tau }_{s}^{f}\), otherwise \(\dot{{p}_{s}}=0\)

with \({\tau }_{s}^{\mathrm{forest}}(\omega )=\mu C(\omega )\sqrt{\sum _{j=\mathrm{1,12}}{a}_{\mathrm{sj}}\langle {\omega }_{j}\rangle }\) where \({\omega }_{s}\) is related to the dislocation density \({\rho }_{s}\) by:: \({\omega }_{s}={b}^{2}\ast {\rho }_{s}\). Since \({\tau }_{s}\) is known, \(\dot{{\gamma }_{s}}\) is therefore only a function of \({\omega }_{s}\).

The evolution of \({\omega }_{s}\) is given by the differential equation: \(\dot{{\omega }_{s}}=\dot{{p}_{s}}{h}_{s}(\langle \omega \rangle )\) with \({h}_{s}(\omega )=(A\frac{\sum _{j\in \mathrm{forest}(s)}\sqrt{{a}_{\mathrm{sj}}}\langle {\omega }_{j}\rangle }{\sum _{j=\mathrm{1,12}}\sqrt{{a}_{\mathrm{sj}}\langle {\omega }_{j}\rangle }}+BC(\omega )\sum _{j\in \mathrm{copla}(s)}\sqrt{{a}_{\mathrm{sj}}\langle {\omega }_{j}\rangle }-\frac{y}{b}\langle {\omega }_{s}\rangle )\)

\(C(\omega )=0.2+0.8\frac{\mathrm{ln}(\alpha \sqrt{\sum _{i=\mathrm{1,12}}\langle {\omega }_{i}\rangle })}{\mathrm{ln}(\alpha b\sqrt{{\rho }_{\mathrm{ref}}})}\).

For the chosen orientation, i.e. \(\text{1-5-9}\), the Schmid factors, relating the stress tensor to the various resolved splits \({\tau }_{s}\) are, for the 12 octahedral systems of \(\text{CFC}\) [R5.03.11]: \(\begin{array}{c}\text{[}0.45784855,0.22892428,0.22892428,0.15261618,0.26707832,0.11446214,\\ \text{}0.19840104,0.29760156,0.4960026,0.04578486,0.11446214,0.16024699\text{]}\end{array}\)

We can therefore see that the first activated sliding system will be number 9 (\(\text{A3}\)), and the second will be number 1 (i.e. \(\text{B4}\)). The splits resolved for these two systems are:

  • system \(\text{A3}\) (number 9): \({\tau }_{s}\mathrm{=}\mathrm{49,6}\mathit{MPa}\)

  • system \(\text{B4}\) (number 1): \({\tau }_{s}=\mathrm{45,785}\mathrm{MPa}\)

For these two systems, \({\tau }_{s}\) being known, it suffices to solve the differential equation \(\dot{{\omega }_{s}}=\dot{{p}_{s}}{h}_{s}(\langle \omega \rangle )\) to know the set of variables. This is done digitally, using scipy’s « odeint » module (see file SSND110A .22).

2.2. Benchmark solution for B modeling#

In the case of \(\text{CC}\), for the chosen orientation, i.e. \(\text{1-5-9}\), the first sliding system (family CUBIQUE) activated will be number 8, and the second will be number 5. The resolved splits for these two systems are (at a temperature of 300K):

  • system number 8: \({\tau }_{s}=\mathrm{49,6}\mathrm{Mpa}\)

  • system number 5: \({\tau }_{s}=\mathrm{45,785}\mathrm{MPa}\)

2.3. Reference solution for C modeling#

_images/10000201000004630000031940CC2B6D3CCDC59C.png

The experimental data are summarized by the smoothed curve below:

For more details, we can refer to [2] and [3].

2.4. Benchmark solution for 3D modeling#

Validation consists in verifying that the sliding systems activated are in fact those that are expected, and in comparing the results between the explicit and the implicit integrations.

2.5. Reference solution for E modeling#

Validation consists in verifying that the stress-strain curve obtained with irradiation does indeed present over-work hardening compared to the non-irradiated case, then softening.

2.6. Bibliographical references#

[1] N.Rupin Note EDF -R&D: HT24 -2010-01128-en « implementation of a new constitutive law based on dislocation dynamics for fcc materials »

[2] J.M. Stephan Note EDF -R&D: HT24 -2010-01329-EN « Project ANR AFGRAP — Average monotonic and cyclic tensile curves of steel AISI 316LN (Tole T252) supplied by AREVA »

[3] J. Schwartz: « Non-local approach in crystalline plasticity: application to the study of the mechanical behavior of AISI 316LN steel in oligocyclic fatigue ». Thesis from the Ecole Centrale de Paris, June 2011.

[4] G.Monnet: « Crystal plasticity constitutive law for irradiated RPV steel » Note EDF R&D H-T27-2011-02738-EN, December 2011.