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


Calculation method
------------------

The analytical solution is based on the resolution of the two differential equations that govern the deviatory part of the behavior (cf. [:ref:`R7.01.06 <R7.01.06>`] and [:ref:`R7.01.35 <R7.01.35>`]). Choosing the :math:`\kappa` parameter to a very large value ensures equivalence between the two models for the load applied.

Deviatory stresses are at the origin of a sliding mechanism (or mechanism of virtual dislocation) of sheets of CSH in nano-porosity. Under deviatoric stress, creep takes place at constant volume. Moreover, creep law UMLV assumes the isotropy of deviatoric creep. Phenomenologically, the sliding mechanism includes a reversible viscoelastic contribution of water strongly adsorbed to the sheets of CSH and an irreversible viscous contribution of free water:


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.. _RefImage_Object_11.svg:


.. _RefEquation 2.1-1:

                 eq 2.1-1

The :math:`{j}^{\mathrm{ème}}` main component of total deviatoric deformation is governed by equations [:ref:`éq2.1‑2 <éq2.1‑2>`] and [:ref:`éq 2.1-3 <éq 2.1-3>`]:


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.. _RefImage_Object_12.svg:


.. _RefEquation 2.1-2:

                         Eq 2.1-2

where

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    :height: 77


.. _RefImage_Object_13.svg:

refers to the stiffness associated with the capacity of adsorbed water to transmit charges (*load bearing water*);

and

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    :width: 217
    :height: 77


.. _RefImage_Object_14.svg:

the viscosity associated with the water adsorbed by the hydrate sheets.


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    :height: 77


.. _RefImage_Object_15.svg:


.. _RefEquation 2.1-3:

                                 Eq 2.1-3

where

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    :height: 77


.. _RefImage_Object_16.svg:

refers to the viscosity of free water.

In the case of a constraint step

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.. _RefImage_Object_17.svg:

, the corresponding deviatoric creep deformation is immediately deduced:


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    :width: 217
    :height: 77


.. _RefImage_Object_18.svg:


.. _RefEquation 2.1-4:

                         Eq 2.1-4

When the elastic part is added, it follows that the total shear deformation is equal to:


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    :width: 217
    :height: 77


.. _RefImage_Object_19.svg:


.. _RefEquation 2.1-5:

                     Eq 2.1-5


.. image:: images/10000000000002D00000021CCF95B404C870EAC0.png
    :width: 180
    :height: 26


.. _RefImage_10000000000002D00000021CCF95B404C870EAC0.png:


Reference quantities and results
-----------------------------------

The test is homogeneous. The deformation is tested at any node.


Uncertainties about the solution
----------------------------

Analytical solution.


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


*The PAPE Y.: UMLV behavioral relationship for the clean creep of concrete,*Code_Aster* [:ref:`R7.01.06 <R7.01.06>`] 16 p (2002) Reference Documentation.


* FOUCAULT, A.: Behavioral Relationship BETON_BURGER, Code-Aster Reference Documentation [:ref:`R7.01.35 <R7.01.35>`], 2011.