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 martensitic metallurgical evolution of the form:

\({Z}_{\mathit{fbm}}=\{\begin{array}{ccc}0.& \mathit{si}t\le {\tau }_{1}& {\tau }_{1}=25s\\ 1-{e}^{\varphi \lambda (t-{\tau }_{1})}& \mathit{si}{\tau }_{1}\le t<{\tau }_{2}& {\tau }_{2}=40s\\ 1.& \mathit{si}t\ge {\tau }_{2}& \end{array}\)

with \(\varphi =0.03\) and \(\lambda =-10°{\mathit{C.s}}^{-1}\)

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

Elastic limit:

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

\({\sigma }_{y}^{\mathit{aust}}={\sigma }_{0}^{\mathit{aust}}+{s}^{\mathit{aust}}(T-{T}^{0})\) with \({\sigma }_{0}^{\mathit{aust}}=50.{10}^{6}\mathit{Pa}\) and \({s}^{\mathit{aust}}=1.3{10}^{6}\mathit{Pa.}°{C}^{-1}\)

• 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 \({H}_{0}^{\mathit{fbm}}=0.\mathit{Pa}\) and \({\lambda }^{\mathit{fbm}}=-6.{10}^{6}\mathit{Pa.}°{C}^{-1}\)

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

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

• Parameters for transformation plasticity:

Reminder:

In the case of a martensitic metallurgical evolution, 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 \({F}^{\text{'}}({Z}_{\mathit{fbm}})=2(1-{Z}_{\mathit{fbm}})\)

Viscose parameters complementary to C modeling:

The two parameters \(\eta\) and C are taken to be zero so that the viscoplastic model tends towards a plastic model that is independent of time.

1.3. Boundary conditions and loads

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

• traction imposed on the side \(\mathit{CD}\), \(p(t)={p}_{0}t\), \({p}_{0}\mathrm{=}{15.10}^{\mathrm{6 }}\mathit{Pa}\).

• \(T(t)={T}^{0}+\mu t\), \(\mu =–10°\mathit{C.}{s}^{-1}\) throughout the structure.

1.4. Initial conditions

\({T}^{0}\mathrm{=}900°C\).