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


Calculation method used for the reference solution
--------------------------------------------------------

The analytical solution corresponding to pre-work hardening can be calculated (monotonic traction, modeling A):

The system of equations for the problem with memory effect is written (20 equations) [R5.03.04]:

**Elasticity**: :math:`\tilde{\sigma }\mathrm{=}2\mu (\tilde{\varepsilon }\mathrm{-}{\varepsilon }^{p})`

Plasticity criterion

.. image:: images/Object_3.svg
    :width: 255
    :height: 49


.. _RefImage_Object_3.svg:



Plastic flow:

.. image:: images/Object_4.svg
    :width: 255
    :height: 49


.. _RefImage_Object_4.svg:

 with

.. image:: images/Object_5.svg
    :width: 255
    :height: 49


.. _RefImage_Object_5.svg:

**Isotropic work hardening**:

.. image:: images/Object_6.svg
    :width: 255
    :height: 49


.. _RefImage_Object_6.svg:

**Maximum work hardening memory:**

.. image:: images/Object_7.svg
    :width: 255
    :height: 49


.. _RefImage_Object_7.svg:

where q is determined by:


* a domain

    .. image:: images/Object_8.svg
        :width: 255
        :height: 49


    .. _RefImage_Object_8.svg:

characterizing the maximum plastic deformations, whose radius q measures and

    .. image:: images/Object_9.svg
        :width: 255
        :height: 49


    .. _RefImage_Object_9.svg:

The center


* :math:`\xi` is calculated according to a law of normality, i.e.:

    .. image:: images/Object_11.svg
        :width: 255
        :height: 49


    .. _RefImage_Object_11.svg:

, with

    .. image:: images/Object_12.svg
        :width: 255
        :height: 49


    .. _RefImage_Object_12.svg:

On the surface of the maximum work hardening range, we have :math:`F\mathrm{=}0`. Applying condition :math:`\mathit{dF}\mathrm{=}0`, we get the speed expression:

.. image:: images/Object_13.svg
    :width: 255
    :height: 49


.. _RefImage_Object_13.svg:

For a material point under uniaxial load, the components of the (uniform) fields are:


.. image:: images/1000068A000069D500002BED0D930C5C37981D8D.svg
    :width: 255
    :height: 49


.. _RefImage_1000068A000069D500002BED0D930C5C37981D8D.svg:

In this case, during the first uniaxial load in the x direction:


.. image:: images/Object_14.svg
    :width: 255
    :height: 49


.. _RefImage_Object_14.svg:

In this case,

.. image:: images/Object_15.svg
    :width: 255
    :height: 49


.. _RefImage_Object_15.svg:

, implies that

.. image:: images/Object_16.svg
    :width: 255
    :height: 49


.. _RefImage_Object_16.svg:

. In this case,

.. image:: images/Object_17.svg
    :width: 255
    :height: 49


.. _RefImage_Object_17.svg:

In addition, in the case of a symmetric tensile compression cycle (in plastic deformation), during the first symmetric discharge (with

.. image:: images/Object_18.svg
    :width: 255
    :height: 49


.. _RefImage_Object_18.svg:

):


.. image:: images/Object_19.svg
    :width: 255
    :height: 49


.. _RefImage_Object_19.svg:


.. image:: images/Object_20.svg
    :width: 255
    :height: 49


.. _RefImage_Object_20.svg:


.. image:: images/Object_21.svg
    :width: 255
    :height: 49


.. _RefImage_Object_21.svg:


.. image:: images/Object_22.svg
    :width: 255
    :height: 49


.. _RefImage_Object_22.svg:

which corresponds well to the expected result (cf. [:ref:`bib2 <bib2>`]): domain :math:`F\mathrm{=}0` centered on the origin, and radius the half-amplitude of plastic deformation.

In the case of increasing traction, and if kinematic work hardening is neglected, the equations to be solved become:


.. image:: images/Object_23.svg
    :width: 255
    :height: 49


.. _RefImage_Object_23.svg:

So we have to calculate the function

.. image:: images/Object_24.svg
    :width: 255
    :height: 49


.. _RefImage_Object_24.svg:

, such as:


.. image:: images/Object_25.svg
    :width: 255
    :height: 49


.. _RefImage_Object_25.svg:

with

.. image:: images/Object_26.svg
    :width: 255
    :height: 49


.. _RefImage_Object_26.svg:

.

In addition, we consider that we are in charge, so

.. image:: images/Object_27.svg
    :width: 255
    :height: 49


.. _RefImage_Object_27.svg:


.. image:: images/Object_28.svg
    :width: 255
    :height: 49


.. _RefImage_Object_28.svg:

It is therefore necessary to integrate the differential equation:


.. image:: images/Object_29.svg
    :width: 255
    :height: 49


.. _RefImage_Object_29.svg:

which is integrated as follows:


.. image:: images/Object_30.svg
    :width: 255
    :height: 49


.. _RefImage_Object_30.svg:

=>

.. image:: images/Object_31.svg
    :width: 255
    :height: 49


.. _RefImage_Object_31.svg:

Method for varying the constant:

.. image:: images/Object_32.svg
    :width: 255
    :height: 49


.. _RefImage_Object_32.svg:


.. image:: images/Object_33.svg
    :width: 255
    :height: 49


.. _RefImage_Object_33.svg:


.. image:: images/Object_34.svg
    :width: 255
    :height: 49


.. _RefImage_Object_34.svg:

by integrating:


.. image:: images/Object_35.svg
    :width: 255
    :height: 49


.. _RefImage_Object_35.svg:

From where


.. image:: images/Object_36.svg
    :width: 255
    :height: 49


.. _RefImage_Object_36.svg:

The constant :math:`K` is defined by the initial conditions: for :math:`p\mathrm{=}0`, :math:`R\mathrm{=}0`


.. image:: images/Object_37.svg
    :width: 255
    :height: 49


.. _RefImage_Object_37.svg:

either

.. image:: images/Object_38.svg
    :width: 255
    :height: 49


.. _RefImage_Object_38.svg:

Finally:


.. image:: images/Object_39.svg
    :width: 255
    :height: 49


.. _RefImage_Object_39.svg:

So we're in charge of: :math:`\sigma \mathrm{=}{R}_{0}+R(p)`


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

**Modeling A:**

Value of :math:`\mathit{SIXX}` at the final moment: :math:`\sigma \mathrm{=}{R}_{0}+R(p)`

with

.. image:: images/Object_42.svg
    :width: 255
    :height: 49


.. _RefImage_Object_42.svg:

:math:`t\mathrm{=}\mathrm{8s}`, we need to find :math:`\mathit{SIXX}\mathrm{=}120\mathit{Mpa}`.

To do this, we calculate :math:`R(p)` from the value of :math:`p` at the time :math:`t\mathrm{=}\mathrm{8s}`.

**Modeling B:**

The results obtained with VISC_CIN2_MEMO will be compared with those obtained with VISCOCHAB, at the end of pre-work hardening and after 10 cycles. The curves below highlight the memory effect (in comparison with VISC_CIN2_CHAB, which does not model it): after pre-work hardening, the imposed deformation cycles stabilize at a stress amplitude greater than that obtained without a memory effect:


.. image:: images/1000000000000250000001B2D75C368820175D8E.png
    :width: 56
    :height: 26


.. _RefImage_1000000000000250000001B2D75C368820175D8E.png:


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


* Modeling A: analytical


* B modeling: inter-comparison between VISCOCHAB and VISC_CIN2_MEMO: precision of numerical integration, estimated at less than :math:`\text{1\%}`.


* C modeling: validation of behaviors in 2D AXIS; the results must be identical to those of modeling B.


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


1. R5.03.04 "Elasto-visco-plastic behaviors of J.L.Chaboche".


2. J.M. PROIX "Viscoplastic behavior taking into account the non-proportionality of the load" EDF R&D-CR- AMA12 -284, 12/12/12