Reference problem
=====================


Geometry
---------


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.. _RefImage_10000BCC0000144200000EC8A394F75553C853D0.svg:


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|
+ .. image:: images/Object_224.svg                                                    + +                                                                                  +
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+    :height: 190                                                                     + +                                                                                  +
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 **Metallurgical parameters:**

TRC to model a martensitic metallurgical evolution of the form:

:math:`{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 :math:`\varphi =0.03` and :math:`\lambda =-10°{\mathit{C.s}}^{-1}`

**Thermal parameters:**

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

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

**Thermo-mechanical parameters:**


* Thermo-elastic parameters:

Young's module :math:`E=200000{10}^{6}\mathit{Pa}`

Poisson's Ratio :math:`\nu =0.3`

Thermal expansion coefficients

:math:`{\alpha }_{\mathit{fbm}}=15.{10}^{-6}°{C}^{-1}` :math:`{\alpha }_{\mathit{aust}}=23.5{10}^{-6}°{C}^{-1}`

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

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

Elastic limit:

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

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


* Thermoplastic parameters (law with linear work hardening)

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

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

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

We remind you that :math:`H=\frac{{\mathit{EE}}_{T}}{E-{E}_{T}}`


* Parameters for transformation plasticity:

Reminder:


.. csv-table::

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


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

**Viscose parameters complementary to C modeling:**

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


Boundary conditions and loads
-------------------------------------


* :math:`{u}_{Z}=0` on the :math:`\mathit{AB}` side (symmetry condition).


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


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


Initial conditions
--------------------

:math:`{T}^{0}\mathrm{=}900°C`.