Benchmark solution
=====================


Calculation of the reference solution (cf. bib [:ref:`1 <1>`] and [:ref:`3 <3>`])
-----------------------------------------------------------------------------

For a tensile test in direction :math:`x`, the Kirchhoff :math:`\tau` and Cauchy :math:`\sigma` tensors are of the form:


.. image:: images/Object_22.svg
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and

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with

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The change in volume

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is given by the resolution of


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where

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is thermal deformation. This applies to an austenitic — bainitic transformation:


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**Note:**

*The K coefficient is the compression modulus (not to be confused with the coefficients*)

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*relating to the law of plasticity of transformation)*

In plastic filler, for isotropic work hardening

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linear, such as:


.. image:: images/Object_31.svg
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the cumulative plastic deformation

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Worth


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with


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The gradient tensors of the transformation

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and

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and the plastic deformation tensor

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are of the form:


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The law of evolution of plastic deformation

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is written:


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* For

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, we have

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. There is no transformation plasticity. We then obtain:


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* For

    .. image:: images/Object_44.svg
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, we have

    .. image:: images/Object_45.svg
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.

To integrate the law of evolution of plastic deformation, it must be assumed that the Kirchhoff stress :math:`\tau` varies very little, that is to say that the change in volume

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is very small. Under this hypothesis, we obtain


.. image:: images/Object_47.svg
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The component

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of the transformation gradient is given by the resolution of:


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Finally, the displacement field :math:`u` (in the initial configuration) is of the form

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. The components are given by:


.. image:: images/Object_51.svg
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note
--------

In test case HSNV101 (modeling B), the coefficients of the material were chosen in such a way as to have no classical plasticity

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during the metallurgical transformation that takes place between moments :math:`60` and :math:`\mathrm{122s}`. In fact, if we write the charge—discharge criterion in this time interval, we obtain


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with

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which is cancelled out for only one value of the cumulative plastic deformation

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.

For the law of behavior written in large deformations, the charge—discharge criterion is written between these two moments


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with

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In this case, as long as the variable

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remains less than the value obtained in time :math:`t\mathrm{=}\mathrm{60s}`, we will have

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. However, the value of

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is a function only of the thermal deformation value (stress :math:`\sigma` is constant) and the coefficient

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is independent of metallurgical phases and temperature).

In this time interval, thermal deformation

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is given by the following equation:


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The thermal deformation as well as the volume variation are plotted below.

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, solution of the 3rd degree equation, as a function of time.


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**Thermal deformation as a function of time**


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**Volume variation**

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**depending on the time**

It can be seen that the variable

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decreases and increases in the same way as thermal deformation. In this case, to find out the time from which the variable

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is greater than the value obtained at time :math:`\mathrm{60s}`, it is sufficient to know the instant for which the thermal deformation is identical to that obtained at time :math:`t\mathrm{=}\mathrm{60s}`. By solving the equation above, we find :math:`t\mathrm{=}\mathrm{84.46s}`.


Uncertainty about the solution
---------------------------

The solution is analytical. Two mistakes are made with this solution. The first relates to the calculation of the proportion of bainitic phase created. The preliminary metallurgical calculation does not exactly restore the equation of [:ref:`§1.2 <§1.2>`] giving

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as a function of time, which is why the reference results presented below are calculated with the proportion of bainitic phase calculated by *Code_Aster*.

The second error is the assumption made on the Kirchhoff constraint :math:`\tau` which is not constant over the time interval between :math:`60` and :math:`\mathrm{176s}`. This will impact the displacement calculation.

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and plastic deformation

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.


Benchmark results
----------------------

As reference results, we will adopt the displacement in the direction of tensile loading, the Cauchy stress :math:`\sigma`, the Boolean plasticity indicator :math:`\chi` and the cumulative plastic deformation.

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. The different calculation times are :math:`t\mathrm{=}\mathrm{47,}\mathrm{48,}\mathrm{60,}\mathrm{83,}\mathrm{84,}85` and :math:`\mathrm{176s}`. For the calculation of the displacement, the initial length of the bar in the loading direction is :math:`\mathrm{0.2m}`.

In any case, we have


* :math:`\mathrm{3K}\mathrm{=}500000\mathit{MPa}` (compression module) :math:`\mu \mathrm{=}76923.077\mathit{MPa}`

**At time** :math:`t\mathrm{=}\mathrm{47s}`, we have

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,

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,

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**At time** :math:`t\mathrm{=}\mathrm{48s}`, we have

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,

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,

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**At time** :math:`t\mathrm{=}\mathrm{60s}`, we have

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,

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,

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**At time** :math:`t\mathrm{=}\mathrm{83s}`, we have

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,

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,

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**At time** :math:`t\mathrm{=}84` **s**, we have

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,

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,

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**At time** :math:`t\mathrm{=}\mathrm{85s}`, we have

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,

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,

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**At time** :math:`t\mathrm{=}\mathrm{176s}`, we have

.. image:: images/Object_98.svg
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,

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,

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.. _RefImage_Object_100.svg:


.. image:: images/Object_101.svg
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.. _RefImage_Object_101.svg:


Bibliographical references
---------------------------

Reference may be made to:


1. V. CANO, E. LORENTZ: Introduction to the *Code_Aster* of a model of behavior in large elastoplastic deformations with isotropic work hardening — Internal note EDF DER HI‑74/98/006/0


2. A.M. DONORE, F. WAECKEL: Influence of structural transformations in elasto—plastic behavior laws Note HI—74/93/024


3. F. WAECKEL, V. CANO: Law of behavior of large elasto (visco) plastic deformations with metallurgical transformations [:ref:`R4.04.03 <R4.04.03>`]