.. _r7.01.17-introduction:

Introduction to the Barcelona model: behavior under isotropic loading
===========================================================================

Isotropic loading paths are visualized by varying the net pressure :math:`p''=-\sigma_m''=-(\sigma_m+p_g)` and the capillary pressure :math:`p_c=p_g-p_l` in order to characterize the behavior of the soil in its elastic and plastic phases. To do this, we denote the void index :math:`e=d\Omega^p/d\Omega^s` (ratio between the volume of the pores :math:`d\Omega^p` and the volume of the solid skeleton :math:`d\Omega^s` in the *VER*), for which we admit the additive decomposition between the elastic and plastic parts :math:`e^e` and :math:`e^p`, respectively.

.. _r7.01.17-chargement_isotrope_pc_constante:

At constant capillary pressure
-------------------------------

Initially, we consider a loading at constant capillary pressure starting from an initial state for which the soil is very loose (i.e. with high porosity), typically :math:`e>1`. Its elastic limit under compression is almost zero. By increasing the net pressure, the incremental relationship is then observed schematically:

.. math::
 of = de^e + de^p = -\ lambda (p_c)\ cfrac {dp "} {p"}
    :label: plastic_compression

The line represented by the previous relationship in plane :math:`(\ln p'',e)` is similar to the consolidation curve for saturated soils. Here, however, slope :math:`\lambda(p_c)` is a decreasing function of capillary pressure. It is proposed in the Barcelona model in the form:

.. math::
\ lambda (p_c) =\ lambda_0 ((1-r)\ exp (-\ beta p_c) +r)
    :label: variation_plastic_coefficient

where :math:`\lambda_0,r` and :math:`\beta` are constants.

At one point on the consolidation curve, a net pressure discharge makes it possible to observe reversible behavior, which can be written schematically:

.. math::
 of = de^e =-\ kappa\ cfrac {dp "} {p"}
    :label: elastic_compression

Unlike :math:`\lambda(p_c)`, the :math:`\kappa` slope in the :math:`(\ln p'',e)` plane is constant. The :numref:`r7.01.17-compression_isotrope` shows the appearance of the answers obtained by :eq:`compression_plastique`, :eq:`variation_coefficient_plastique`, and :eq:`compression_elastique`.

.. _r7.01.17-compression_isotrope:

.. figure:: images/compression_isotrope.svg
 :align: center
 :width: 500

 Variation of the void index with net pressure for two capillary pressures :math:`p_{c2}>p_{c1}`.

The area of elasticity, at constant capillary pressure, represented by the Barcelona model is defined by:

.. math::
\ left| p" -\ frac {p_ {con} (p_c, e^p) -kp_c} {2}\ right|-\ frac {p_ {con} (p_c, e^p) +kp_p) +kp_p) +kp_c} {2}\ leq 0,\ quad\ text {with}\ quad
     p_ {con} (p_c, e^p) = p_r\ left (\ cfrac {p_ {con} (0, e^p)} {p_r}\ right) ^\ cfrac {\ lambda_0-\ lambda_0-\ kappa} {\ lambda (p_c) -\ kappa}
    :label: limits_compression

:math:`k p_c` represents the isotropic traction limit, which increases proportionally with capillary pressure via the parameter :math:`k`. The isotropic compression limit is :math:`p_{con}(p_c,e^p)`, an increasing and non-linear function of capillary pressure, where :math:`p_r` is a parameter. The zero capillary pressure value :math:`p_{con}(0,e^p)` is obtained by the following reasoning.

For this purpose, consider a loading path with zero capillary pressure in which the net pressure increases. When it reaches :math:`p_{con}(0,e^p)`, loading causes a plastic variation in the void index. Using :eq:`compression_plastique` and :eq:`compression_elastique`, we get:

.. math::
 de^p = de-de^e = - (\ lambda_0-\ kappa)\ frac {dp_ {con} (0, e^p)} {p_ {con} (0, e^p)}\ Longrightarrow p_ {p)}}\ Longrightarrow p_ {con} (\ con} (0, e^p)} {\ lambda_0-\ kappa}\ right)
    :label: definition_ecrouissage_1

:math:`p_{con}(0,0)` is the initial consolidation pressure (isotropic compression limit) at zero capillary pressure.

.. _r7.01.17-chargement_isotrope_p_constante:

At constant net pressure
--------------------------

We now consider a loading at constant net pressure. When capillary pressure reaches
Once the maximum value has previously been reached and continues to increase, a plastic variation in the void index is generated, which can be written schematically:

.. math::
 of = de^e + de^p = -\ lambda_s\ cfrac {dp_c} {p_c+p_ {atm}}
    :label: capillary_plastic_compression

with :math:`\lambda_s` constant. In the case of a capillary pressure discharge from the line represented by the line represented by the relationship :eq:`compression_plastique_capillaire` in the plane :math:`(\ln p_c,e)`, the response is reversible and represented by the relationship:

.. math::
 of = de^e = -\ kappa_s\ cfrac {dp_c} {p_c+p_ {atm}}
    :label: capillary_elastic_compression

with :math:`\kappa_s` and :math:`p_{atm}` two constants. The latter, being usually evaluated [bib2] __ as being atmospheric pressure, has the sole advantage of defining :eq:`compression_elastique_capillaire` in saturated conditions where :math:`p_{cap}=0`.

The area of elasticity, at constant net pressure, represented by the Barcelona model is defined by:

.. math::
 p_c-s_0 (e^p)\ leq 0
     :label: capillary_limit

where the threshold :math:`s_0(e^p)` is written, in accordance with :eq:`compression_plastique_capillaire` and :eq:`compression_elastique_capillaire`:

.. math::
 de^p = de-de^p = - (\ lambda_s-\ kappa_s)\ cfrac {ds_0 (e^p)} {s_0 (e^p) +p_ {atm}}\ Longrightarrow s_0 (e^p) =\ left (e^p) =\ left (e^p) =\ left (-\ frac {e^p) =\ left (-\ frac {e^p) =\ left (^p} {\ lambda_s-\ kappa_s}\ right) - p_ {atm}
   :label: definition_ecrouissage_2

By combining :eq:`limites_compression` and :eq:`limite_capillaire`, the elasticity domain under isotropic loading is delimited on the :numref:`r7.01.17-domaine_isotrope`. It gets bigger when the plastic void index :math:`e_p` decreases (i.e. when the pores close).

.. _r7.01.17-domaine_isotrope:

.. figure:: images/domaine_isotrope.svg
 :align: center
 :width: 500

 Elasticity domains under initial isotropic loading and for :math:`e^p<0`.

..
   How to get the gray box here-under
..

 **Note:**

 The :numref:`r7.01.17-domaine_isotrope` shows that :math:`p_{con}(p_c,e^p)` is an increasing function of capillary pressure :math:`p_c`. To be exact, this is only true when :math:`p_{con}(0,e^p)>p_r` in accordance with :eq:`limites_compression` -b. For :math:`p_{con}(0,e^p)<p_r` on the other hand, :math:`p_{con}(p_c,e^p)` decreases with :math:`p_c`. There is no experimental justification for this decrease. It is an artifact of the Barcelona model.

General behavior under isotropic loading
---------------------------------------------

The :numref:`r7.01.17-chargement_isotrope_pc_constante` and :numref:`r7.01.17-chargement_isotrope_p_constante`
now make it possible to summarize the behavior equations of the Barcelona model under isotropic loading as follows:

* Elastic behavior:

.. math::
 de^e = -\ kappa\ cfrac {dp "} {p"} -\ kappa_s\ cfrac {dp_c} {p_c+p_ {atm}}
    :label: elasticite_incrementale_1

* The elasticity domain:

.. math::
\ left| p" -\ frac {p_ {con} (p_c, e^p) -kp_c} {2}\ right|-\ frac {p_ {con} (p_c, e^p) +kp_p) +kp_p) +kp_c} {2}\ leq 0,\ quad
    p_c-s_0 (e^p)\ leq 0
    :label: isotropic_elasticity_domain

* Work hardening functions:

.. math::
 p_ {con} (p_c, e^p) = p_r\ left (\ cfrac {p_ {con} (0,0)\ exp\ left (-\ cfrac {e^p} {\ lambda_0-\ lambda_0-\ kappa 0-\ kappa-\ kappa}\ right)} {p_r}\ right) ^\ cfrac {\ lambda_0-\ kappa} {\ lambda (p_c) -\ kappa},\ quad
    s_0 (e^p) =\ left (s_0 (0) +p_ {atm}\ right)\ exp\ left (-\ frac {e^p} {\ lambda_s-\ kappa_s-\ kappa_s}\ right) - p_ {atm}
    :label: work_working_functions

* The flow of the plastic void index (here written outside the two "corners" of the elasticity domain, see :numref:`r7.01.17-domaine_isotrope`):

.. math::
\ begin {align}
    &de^p < 0\ quad\ text {si}\ quad p" -p_ {con} (p_c, e^p) = 0,\ quad dp" -\ frac {\ partial p
    _ {con}} {\ partial p_c} dp_c >0\ vphantom {\ big {(}}}\\
    &de^p\ geq 0\ quad\ text {si}\ quad p"+kp_c = 0,\ quad dp"+kdp_c = 0\ vphantom {\ Big {(}}}\\
    &de^p < 0\ quad\ text {si}\ quad p_c = s_0 (e^p),\ quad dp_c > 0\ vphantom {\ Big {(}}}\\
    &de^p = 0\ quad\ text {otherwise}\ vphantom {\ Big {(}}}
    \ end {align}
    :label: isotropic_flow

Having addressed the Barcelona model in the case of isotropic loading, his behavioral equations in a three-dimensional formulation
can now be developed in :numref:`r7.01.17-formulation_tridimensionnelle`.

 **Note:**

 The void index :math:`e` can be deduced, in the case that can be linearized all along, from the volume deformation :math:`\varepsilon_v` by the following relationship:

 .. math:
  e=e_0+ (1+e_0)\ varepsilon_v
  :label: equivalence_deformation_indice_of_voids_1

 with :math:`e_0` the initial void index. Likewise, the relationship :eq:`equivalence_deformation_indice_des_vides_1` is assumed to be true on the parts elastic and plastic of the void index and of
 volume deformation, which leads to:

 .. math:
  e^e=e_0+ (1+e_0)\ varepsilon_v^e,\ quad e^p= (1+e_0)\ varepsilon_v^p
  :label: equivalence_deformation_indice_of_voids_2

 In particular, this last relationship has been used since :numref:`r7.01.17-formulation_tridimensionnelle`.