1. Reference problem#

1.1. Geometry#

_images/1000069C000023DD0000170C8DA33F3F4F4E49B0.svg

1.2. Material properties#

The following convention is adopted in order to distinguish the parameters of the hot phase (austenitic) from the parameters of the cold phases (ferrito-pearlitic, bainitic and martensitic):

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Characteristics relating to the austenitic phase
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characteristics relating to the ferrito-pearlitic, bainitic and martensitic phases

Metallurgical parameters:

TRC to model a bainitic metallurgical evolution, over the entire structure, of the form:

\({Z}_{\mathit{fbm}}=\{\begin{array}{ccc}0.& \mathit{si}t\le {\tau }_{1}& {\tau }_{1}=60s\\ \frac{t-{\tau }_{1}}{{\tau }_{2}-{\tau }_{1}}& \mathit{si}{\tau }_{1}\le t<{\tau }_{2}& {\tau }_{2}=112s\\ 1.& \mathit{si}t\ge {\tau }_{2}& \end{array}\)

Thermal parameters:

Calorific capacity: \(\rho {C}_{p}=2.{10}^{6}{\mathit{J.m}}^{-3}\mathrm{.}°{C}^{-1}\)

Conductivity: \(\lambda =9999.9{\mathit{W.m}}^{-1}\mathrm{.}°{C}^{-1}\)

Thermo-mechanical parameters:

  • Thermo-elastic parameters:

Young’s module \(E=200000{10}^{6}\mathit{Pa}\)

Poisson’s ratio \(\nu =0.3\)

Thermal expansion coefficients \({\alpha }_{\mathit{fbm}}={\alpha }_{\mathit{aust}}=20.{10}^{-6}°{C}^{-1}\)

Expansion coefficient definition temperature: \({T}_{\mathit{ref}}=900°C\)

Reference thermal deformation state: \(\Delta {\epsilon }_{f\gamma }^{{T}_{\mathit{ref}}}=2.52{10}^{-3}\)

Elasticity limit:

\({\sigma }_{y}^{\mathit{fbm}}=1200.{10}^{6}\mathit{Pa}\)

\({\sigma }_{y}^{\mathit{aust}}={400.10}^{6}\mathit{Pa}\)

  • Thermoplastic parameters (law with linear work hardening)

Tangent modules: \({E}_{T}^{\mathit{fbm}}={E}_{T}^{\mathit{aust}}={2000.10}^{6}\mathit{Pa}\)

So we have: \({H}^{\mathit{fbm}}={H}^{\mathit{aust}}=\frac{{\mathit{EE}}_{T}}{(E-{E}_{T})}=\mathrm{2,04}{.10}^{9}\mathit{Pa}\)

  • Settings for work hardening restoration (full restore): \({\theta }_{\gamma \mathrm{,3}}=0\)

\({\theta }_{\gamma \mathrm{,3}}\) is the rate of work hardening transmitted from austenite to the ferritic phase 3 (bainite).

1.3. Boundary conditions and loads#

  • \({u}_{Y}=0\) on the \(\mathit{AB}\) side; \({u}_{X}=0\) in \(A\).

  • \(T={T}^{0}+\mu t\), \(\mu =-5°{\mathit{C.s}}^{-1}\) throughout the structure.

  • The load on the structure is due to thermal and metallurgical expansion phenomena constrained in the \(z\) direction by the condition of plane deformations.

1.4. Initial conditions#

\({T}^{0}=900°C={T}^{\mathit{ref}}\)