.. _R5.04.03:

**r5.04.03** Second gradient modeling
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**Summary**

Here are presented the second gradient models from the work of Chambon et al ([:ref:`1 <1>`], [], [:ref:`2 <2>`], [:ref:`3 <3>`]) and the second dilation gradient models from the work of Fernandes et al ([:ref:`4 <4>`], [:ref:`5 <5>`]), and Gantier ([:ref:`14 <14>`]).

These models are part of microstructured environments and have the numerical objective of giving convergent results in relation to the fineness of spatial discretization in order to avoid obtaining localized solutions. They must be used when the mechanical rheological laws in question present a softening of the behavior, which reflects the damage or degradation of a material before cracking.

Second expansion gradient modeling is, in fact, a simplified approach to the second gradient restricted to materials subject to the expansion phenomenon. The numerical objective is to significantly reduce calculation times. It lends itself particularly well to geomaterials.

Note: Second gradient modeling has been removed from the code. Only dilation gradient modeling is available.

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.. toctree::
    :hidden:

    self

.. toctree::
    :maxdepth: 2
    :numbered:

    Le_mod_le_second_gradient
    Le_mod_le_second_gradient_de_dilatation
    Int_gration_num_rique_des_mod_les_second_gradient
    Conseils_Proc_dure_pour_l_utilisation_des_mod_les_second_gradient
    V_rification
    Bibliographie