Reference solutions
======================


Calculation method used for reference solutions
----------------------------------------------------------

The whole of this demonstration can be read in more detail in the [:ref:`bib1 <bib1>`] document.

In the case of a linear isotropic viscoelastic material, the behavior over time can be described using two functions

.. image:: images/Object_6.svg
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.. _RefImage_Object_6.svg:

and

.. image:: images/Object_7.svg
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.. _RefImage_Object_7.svg:

so that the deformations and the stresses can be written as:


.. image:: images/Object_8.svg
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.. _RefImage_Object_8.svg:

where :math:`{I}_{3}` refers to the identity matrix of rank 3

and :math:`\text{*}` the convolution product:

.. image:: images/Object_10.svg
    :width: 29
    :height: 21


.. _RefImage_Object_10.svg:

The equivalent thermoelastic problem, using the Laplace transform, is:


.. image:: images/Object_11.svg
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.. _RefImage_Object_11.svg:

By eliminating the "+" sign:


.. image:: images/Object_12.svg
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.. _RefImage_Object_12.svg:

either,


.. image:: images/Object_13.svg
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.. _RefImage_Object_13.svg:


.. image:: images/Object_14.svg
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    :height: 21


.. _RefImage_Object_14.svg:

According to the equilibrium equation, we have

.. image:: images/Object_15.svg
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    :height: 21


.. _RefImage_Object_15.svg:

, we get:


.. image:: images/Object_16.svg
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.. _RefImage_Object_16.svg:

,


.. image:: images/Object_17.svg
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.. _RefImage_Object_17.svg:

,


.. image:: images/Object_18.svg
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.. _RefImage_Object_18.svg:

which when integrating with respect to r gives:


.. image:: images/Object_19.svg
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    :height: 21


.. _RefImage_Object_19.svg:

,

Boundary conditions

.. image:: images/Object_20.svg
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.. _RefImage_Object_20.svg:

give:


.. image:: images/Object_21.svg
    :width: 29
    :height: 21


.. _RefImage_Object_21.svg:

Using the initial notations, we therefore have:


.. image:: images/Object_22.svg
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    :height: 21


.. _RefImage_Object_22.svg:

Or, taking the inverse transform,


.. image:: images/Object_23.svg
    :width: 29
    :height: 21


.. _RefImage_Object_23.svg:

We deduce

.. image:: images/Object_24.svg
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    :height: 21


.. _RefImage_Object_24.svg:

and :math:`w`:


.. image:: images/Object_25.svg
    :width: 29
    :height: 21


.. _RefImage_Object_25.svg:


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

Move :math:`\mathrm{DX}` on node :math:`B`


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

:math:`\text{0\%}`: analytical solution


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

Ph. Of BONNIERES, two analytical solutions to axisymmetric problems in linear viscoelasticity and with unilateral contact, Note HI-71/8301