1. Reference problem#

1.1. Geometry#

Figure 1.1-a : Geometry of the problem, cube with a side of 1mm.

1.2. Material properties#

The material parameters used in this test case should not be used for studies. They do not correspond to real characteristics.

The various material properties are given below, with:

\(T\): temperature in \(°C\)
\(\mathrm{IRRA}\): irradiation in \(\mathrm{dpa}\) [R5.03.23]

Young’s module: \(E=210000.0–30.0T\) in \(\mathrm{MPa}\)

Poisson’s ratio: \(\nu =0.30+5.0E-05T\)

Coefficient of thermal expansion: \(\alpha =\left(15.0+0.002\right)1.0E-06\)

Plastic part

\(\kappa =0.8\)

Elastic limit at 0.2% in \(\mathrm{MPa}\): \({R}_{02}={R}_{02}^{0}\mathrm{.}{C}_{w}\text{\_}{R}_{e}\mathrm{.}{I}_{r}\text{\_}{R}_{e}\)

with

\({R}_{02}^{0}=270.0-0.65T+0.0010{T}^{2}\)

\({C}_{w}\text{\_}{R}_{e}=1.0\)

\({I}_{r}\text{\_}{R}_{e}=\left(4.0–3.0{e}^{\frac{-\mathit{IRRA}}{3}}\right)\)

Ultimate compulsion in \(\mathrm{MPa}\): \({R}_{m}={R}_{02}(T,\mathrm{IRRA})+({R}_{m}^{0}–{R}_{02}^{0})\mathrm{.}{C}_{w}\text{\_}{R}_{m}\mathrm{.}{I}_{r}\text{\_}{R}_{m}\)

with

\({R}_{m}^{0}=600.0-1.5T+0.010{T}^{2}\)

\({C}_{w}\text{\_}{R}_{m}=0.50\)

\({I}_{r}\text{\_}{R}_{m}=0.005-0.0020\left(1.0-{e}^{\frac{-\mathit{IRRA}}{8.0}}\right)+{e}^{\frac{-\mathit{IRRA}}{3.0}}\)

Distributed elongation: \({ϵ}_{u}=\mathrm{ln}(1.0+{ϵ}_{u}^{0}.{C}_{w}\text{\_}{ϵ}_{u}.{I}_{r}\text{\_}{ϵ}_{u}1.0E-02)\)

with

\({ϵ}_{u}^{0}=50.0-0.15T+0.007{T}^{2}\)

\({C}_{w}\text{\_}{ϵ}_{u}=1.0\)

\({I}_{r}\text{\_}{ϵ}_{u}={e}^{-\mathit{IRRA}}\)

Irradiation part

Swelling part

\({A}_{\mathit{i0}}=3.0E-06{\text{MPa}}^{-1}{\text{dpa}}^{-1}\) \({\eta }_{\mathit{is}}=1000\text{MPa.dpa}\)

\(R=0.0020{\text{dpa}}^{-1}\)

\(\alpha =1.0\) \({\varphi }_{0}=45.0\text{dpa}\) or \({\varphi }_{0}=1.0\text{dpa}\)

1.3. Boundary conditions and loads#

Modeling A

For nodes 1, 2, 3, 4: \(\mathrm{DX}=\mathrm{DY}=\mathrm{DZ}=0\)

For nodes 5, 6, 7, 8: \(\mathrm{DX}=\mathrm{DY}=0\)

In addition, a linear temperature ramp with a maximum of \(400°C\) as well as a linear irradiation ramp with a maximum of \(140\mathrm{dpa}\) (figure) is applied.

The material characteristics for swelling are those of the lower bound.

B Modeling Travel conditions \(\mathrm{DX}=0\), nodes: 1, 4, 5, 5, 8 \(\mathrm{DY}=0\), nodes: 1, 2, 5, 6 \(\mathrm{DZ}=0\), knots: 1, 4, 2, 3 Nodes: 2, 3, 6, 6, 7 Nodes: 2, 3, 6, 6, 7 \(\mathrm{DX}=0.0\) \(t=\mathrm{0.0}\mathrm{sec}\) \(\mathrm{DX}=3.0E-03\) \(t=\mathrm{10.0}\mathrm{sec}\) \(\mathrm{DX}=3.0E-03\) \(t=\mathrm{90.0}\mathrm{sec}\)

Figure 1.3-a : Temperature and irradiation conditions at all nodes as a function of time.

The modeling is done a first time with the swelling characteristics that correspond to the upper bound and then a second time with the swelling characteristics that correspond to the lower bound.