1. Reference problem

1.1. Geometry

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):

	=	Characteristics relating to the austenitic phase
	=	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

\({\alpha }_{\mathit{fbm}}=15.{10}^{-6}°{C}^{-1}\) \({\alpha }_{\mathit{aust}}=23.5{10}^{-6}°{C}^{-1}\)

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:

\({\sigma }_{y}^{\mathit{fbm}}={\sigma }_{0}^{\mathit{fbm}}+{s}^{\mathit{fbm}}(T-{T}^{0})\) with \({\sigma }_{0}^{\mathit{fbm}}=400.{10}^{6}\mathit{Pa}\) and \({s}^{\mathit{fbm}}=0.5{10}^{6}\mathit{Pa.}°{C}^{-1}\)

\({\sigma }_{y}^{\mathit{aust}}={\sigma }_{0}^{\mathit{aust}}+{s}^{\mathit{aust}}(T-{T}^{0})\) with \({\sigma }_{0}^{\mathit{aust}}=530.{10}^{6}\mathit{Pa}\) and \({s}^{\mathit{aust}}=0.5{10}^{6}\mathit{Pa.}°{C}^{-1}\)

• Thermoplastic parameters (law with linear work hardening)

Tangent modules: \({E}_{T}^{\mathit{fbm}}\) and \({E}_{T}^{\mathit{aust}}\) are chosen such that:

\({H}^{\mathit{fbm}}={H}_{0}^{\mathit{fbm}}+{\lambda }^{\mathit{fbm}}(T-{T}^{0})\) with \({H}_{0}^{\mathit{fbm}}=-50.{10}^{6}\mathit{Pa}\) and \({\lambda }^{\mathit{fbm}}=-5.{10}^{6}\mathit{Pa.}°{C}^{-1}\)

\({H}^{\mathit{aust}}={H}_{0}^{\mathit{aust}}+{\lambda }^{\mathit{aust}}(T-{T}^{0})\) with \({H}_{0}^{\mathit{aust}}=1250.{10}^{6}\mathit{Pa}\) and \({\lambda }^{\mathit{aust}}=-5.{10}^{6}\mathit{Pa.}°{C}^{-1}\)

We remind you that \(H=\frac{{\mathit{EE}}_{T}}{E-{E}_{T}}\)

• Parameters for transformation plasticity:

Reminder:

In the case of a metallurgical evolution of the bainitic type, the model of transformation plasticity is as follows: \({\dot{\epsilon }}^{\mathit{pt}}=\frac{3}{2}\stackrel{̃}{\sigma }{k}^{\mathit{fbm}}{F}^{\text{'}}({Z}_{\mathit{fbm}}){\dot{Z}}_{\mathit{fbm}}\)

Model parameters: \({k}^{\mathit{fbm}}=1.{10}^{-10}{\mathit{Pa}}^{-1}\) and \({F}^{\text{'}}({Z}_{\mathit{fbm}})=2(1-{Z}_{\mathit{fbm}})\)

1.3. Boundary conditions and loads

• \({u}_{Z}=0\) on the \(\mathit{AB}\) side (symmetry condition).

• traction imposed on the side \(\mathit{CD}\)

\(p(t)=\{\begin{array}{cc}{p}_{0}t& \mathit{si}t<{\tau }_{1}\\ {p}_{0}{\tau }_{1}& \mathit{si}t\ge {\tau }_{1}\end{array}\) with \({p}_{0}={6.10}^{6}\mathit{Pa}\) and \({\tau }_{1}=\mathrm{60s}\)

• temperature imposed on the whole structure:

\(T(t)={T}^{0}+\mu t\) with \(\mu =-5°{\mathit{C.s}}^{-1}\)

1.4. Initial conditions