1. Reference problem

1.1. Geometry

The stresses and deformations are homogeneous in the volume element. This can be represented by a plane or a solid element, for example:

1.2. Material properties

Material properties are described by the following parameters:

Thermo-metallurgical parameters:

• Zircaloy:

  • \(\rho {C}_{p}={2.0E}^{-3}{\mathit{J.mm}}^{-3.}°{C}^{-1}\)

  • \(\lambda =9.9999{\mathit{W.mm}}^{-1.}°{C}^{-1}\)

• Coefficient for metallurgy:

  • \(\mathit{teqd}\mathrm{=}809°C\), \(K\mathrm{=}1.135{E}^{\mathrm{-}2}\), \(n\mathrm{=}2.187\)

  • \(\mathit{t1c}\mathrm{=}831°C\), \(\mathit{t2c}\mathrm{=}0.\), \(\mathit{qsr}\mathrm{=}14614\), \(\mathit{Ac}\mathrm{=}1.58E\mathrm{-}4\)

  • \(m\mathrm{=}4.7\), \(\mathit{t1r}\mathrm{=}\mathrm{949,1}°C\), \(\mathit{t2r}=0.\), \(\mathit{Ar}\mathrm{=}\mathrm{-}5.725\), \(\mathit{Br}\mathrm{=}0.05\)

Thermo-metallo-mechanical parameters:

• Thermo-metallo-elastic parameters:

  • Young’s module: \(E=195000\mathit{MPa}\)

  • Poisson’s ratio: \(\nu =0.3\)

  • Average thermal expansion coefficient for cold phases: \({\alpha }_{f}=15E-6\)

  • Average thermal expansion coefficient of hot phases: \({\alpha }_{\gamma }=23E-6\)

  • Expansion coefficient definition temperature: \({T}_{\mathit{ref}}=600°C\)

  • Choice of the reference metallurgical phase: \(\mathit{Froide}\)

  • Deformation of the non-reference phase compared to the reference phase, at \({T}_{\mathit{ref}}\): \(\Delta {\epsilon }_{f\gamma }^{{T}_{\mathit{ref}}}={2.52E}^{-3}\)

  • Cold phase elasticity limit 1: \({\sigma }_{y\mathrm{,1}}=181\mathit{MPa}\)

  • Elastic limit of the cold phase 2: \({\sigma }_{y\mathrm{,2}}=181\mathit{MPa}\)

  • Elastic limit of the hot phase: \({\sigma }_{y,\gamma }=0\mathit{MPa}\)

  • Mixture function (calculation of the elastic limit of the multiphase material): \(\mathit{fonction}\mathit{identité}\)

• Thermo-metallo-plastic parameters (law with linear work hardening)

  • Slope of the traction curve for cold phase 1: \({E}_{T\mathrm{,1}}=1930\mathit{MPa}\)

  • Slope of the traction curve for cold phase 2: \({E}_{T\mathrm{,2}}=1930\mathit{MPa}\)

  • Slope of the hot phase traction curve: \({E}_{T,\gamma }=0\mathit{MPa}\)

1.3. Boundary conditions and loads

• Modeling A:

The volume element is locked along \(\mathit{Ox}\) along the \([\mathrm{2,4}]\) side while being subjected to a pull \({\sigma }^{D}\) and a shear force \({\tau }^{D}\).

\({\tau }^{D}\) The loading path is as follows:

\({\sigma }^{D}\)

A

B

	\({\sigma }^{D}\text{[MPa]}\)	\({\tau }^{D}\text{[MPa]}\)
\(A\)	151.2	93.1
O \(B\)	257.2	33.1

OA path from \(t=0\) to \(\mathrm{1s}\).

AB route from \(t=1\) to \(\mathrm{2s}\).

BO route from \(t=2\) to \(\mathrm{3s}\).

The temperature is imposed to be constant and equal to 600° C.

• B modeling:

B modeling is the exact equivalent of A modeling taking into account large deformations via the keyword DEFORMATION =” SIMO_MIEHE “.