Formulation of the model
=====================

The modified Cam-Clay model is formulated according to the construction usual for Generalized Standard Materials, based on an energy potential and a plasticity criterion (or a dissipation potential). The case of linearized deformations and isothermal conditions is assumed.

State laws
-----------

The state variables are the total strain tensor :math:`\boldsymbol{\epsilon}`, the plastic strain tensor :math:`\boldsymbol{\epsilon}^p`, and a scalar work hardening variable :math:`\xi`. Its interpretation will be specified when the laws of evolution are established.

The free energy potential :math:`\psi` (volume density) is written in the form:

.. math::
\ psi (\ boldsymbol {\ epsilon}},\ boldsymbol {\ epsilon} ^p,\ xi) =\ psi_e (\ boldsymbol {\ epsilon} -\ boldsymbol {\ epsilon} ^p) +\ psi_h (\ xi)
    :label: free_energy

where :math:`\psi_e` refers to the energy returned by elastic discharge and :math:`\psi_h` represents the energy stored by work hardening. Their expressions are indicated later.

The expression for intrinsic dissipation volume density :math:`D` is obtained as:

.. math::
 D=\ mathbf {\ boldsymbol {\ sigma}}:\ dot {\ boldsymbol {\ epsilon}} -\ dot {\ psi} =
    \ mathbf {\ boldsymbol {\ sigma}}:\ dot {\ boldsymbol {\ epsilon}} -\ left (\ frac {\ partial\ psi} {\ partial\ sigma}} {\ partial\ boldsymbol {\ epsilon}}}:
    \ dot {\ boldsymbol {\ epsilon}}} +\ frac {\ partial\ psi} {\ partial\ boldsymbol {\ epsilon} ^p}:\ dot {\ boldsymbol {\ epsilon}} ^p
    +\ frac {\ partial\ psi} {\ partial\ xi} {\ partial\ xi}}\ right) =\ boldsymbol {\ sigma}:\ dot {\ boldsymbol {\ epsilon}}} ^p + p_c\ dot {\ xi}
    :label: intrinsic_dissipation

the latter equality being due to the non-dissipative nature of the total deformation :math:`\boldsymbol{\epsilon}` (the plasticity has no viscous effect).

Above, we will also have defined :math:`\boldsymbol{\sigma}` the stress tensor and :math:`p_c`, which is called the critical pressure. These two thermodynamic forces are naturally derived by the following state laws:

.. math::
\ begin {align}
    \ boldsymbol {\ sigma} &=\ frac {\ partial\ psi} {\ partial\ psi} {\ partial\ boldsymbol {\ epsilon}} =\ frac {\ partial\ psi_e} {\ partial\ boldsymbol {\ boldsymbol {\ epsilon}}} =-\ frac {\ partial\ psi_e} {\ partial boldsymbol {\ epsilon} ^p}\\
    p_c &=-\ frac {\ partial\ psi} {\ partial\ psi} {\ partial\ psi} {\ partial\ psi_h} {\ partial\ psi} {\ partial\ xi}
    \ end {align}
    :label: state_law

Elastic energy
^^^^^^^^^^^^^^^^^^

.. _target to energie_elastique:

The elastic energy potential :math:`\psi_e`, established in [Bour97] _, and taken advantage of here, takes the following expression:

.. math::
\ psi_e (\ boldsymbol {\ epsilon} ^e) =\ frac {\ epsilon} ^e) =\ frac {K} {K} {\ kappa^2}\ left (\ boldsymbol {\ epsilon} ^e) -1\ upsilon (\ boldsymbol {\ epsilon} ^e) -1\ right)\ qquad\ text {with}\ qe) +\ kappa\ mathrm {tr} (r) (\ boldsymbol {\ epsilon} ^e) -1\ right)\ qquad\ text {with}\ qquad\ Upsilon (\ boldsymbol {\ epsilon} ^e) =\ exp\ left (-\ kappa\ mathrm {tr} (\ boldsymbol {\ epsilon} ^e) +\ frac {\ mu} {K} (\ kappa\ boldsymbol {\ epsilon} ^e_d) :(\ kappa\ boldsymbol {\ epsilon} ^e_d) :(\ kappa\ boldsymbol {\ boldsymbol {\ epsilon} ^e) :(\ kappa\ boldsymbol {\ epsilon} ^e) :(\ kappa\ boldsymbol {\ epsilon} ^e) :(\ kappa\ boldsymbol {\ epsilon} ^e) :(\ kappa\ boldsymbol {\ epsilon} ^e_d)\ right)
   :label: elastic_energy


where :math:`\boldsymbol{\epsilon}^e=\boldsymbol{\epsilon}-\boldsymbol{\epsilon}^p` designates the elastic deformation tensor, having noted :math:`\boldsymbol{\epsilon}^e_d=\boldsymbol{\epsilon}^e-\cfrac{\mathrm{tr}(\boldsymbol{\epsilon}^e)}{3}\boldsymbol{I}` as its deviator (:math:`\boldsymbol{I}` tensor second order identity).


The expression for the stress tensor is derived from the state law :eq:`lois_etat` -1:

.. math::
\ boldsymbol {\ sigma} =\ Upsilon (\ boldsymbol {\ epsilon} ^e)\ left (-\ frac {K} {\ kappa}\ boldsymbol {I} +2\ mu\ boldsymbol {\ epsilon} {\ epsilon} ^e_d\ right) +\ frac {K} {\ kappa}\ boldsymbol {I} +2\ mu\ boldsymbol {I} +2\ mu\ boldsymbol {I} +2\ mu\ boldsymbol {I}
   :label: constraint-expression


In :eq:`lois_etat` and :eq:`expression_contrainte`, :math:`K` and :math:`\mu` are the *initial* compressibility and shear modules respectively. The coefficient :math:`\kappa\geq 0` introduces non-linearity through the effect of the mean stress :math:`\sigma_m` on the *tangent* compressibility and shear moduli. In particular, this non-linear elasticity generalizes the more classical elasticity of the modified Cam-Clay model, only on the volume behavior (see [:ref:`r7.01.14 <R7.01.14>`] _), to the deviatoric behavior. This effect is detailed [Bacq23] _ on isotropic and shear loads.

 **Note:**

 The elastic energy potential :eq:`energie_elastique` is well defined for :math:`\kappa=0` since using an expansion limited to order two in :math:`\kappa\boldsymbol{\epsilon}^e`, we have:

 .. math:
    \ psi_e (\ boldsymbol {\ epsilon} ^e)\ underset {\ |\ kappa\ boldsymbol {\ epsilon} ^e\ |\ ll 1} {=}\ frac {K} {2} {2}\ mathrm {tr} (\ boldsymbol {\ epsilon} ^e) ^2 +\ mu\ boldsymbol {\ epsilon} ^e) ^2 +\ mu\ boldsymbol {\ epsilon} ^e) ^2 +\ mu\ boldsymbol {\ epsilon} ^e) ^2 +\ mu\ boldsymbol {\ epsilon} ^e e_ d:\boldsymbol {\ epsilon} ^e_d +\ mathcal {O} (\ |\ kappa\ boldsymbol {\ epsilon} ^e\ |^3)

 So for :math:`\kappa=0`, the elastic energy potential :eq:`energie_elastique` predicts isotropic elastic linear behavior.

Energy stored by work hardening
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The energy potential stored by work hardening :math:`\psi_h` is written as:

.. math::
\ psi_h (\ xi) =\ frac {p_ {c0}} {\ beta}\ left (\ exp (-\ beta\ xi) -1\ right)
   :label: energie_stored

From this we deduce the expression of critical pressure according to state law :eq:`lois_etat` -2:

.. math::
 p_c = p_ {c0}\ exp (-\ beta\ xi)
   :label: critical_pressure_expression

:math:`\beta\geq 0` is called the plastic incompressibility index and :math:`p_{c0}>0` is the initial critical pressure.


Laws of evolution
----------------

.. _target to lois_evolution:

The expression for the plasticity criterion is the equation of an ellipse in the stress meridian plane:

.. math::
 f (\ boldsymbol {\ sigma}, p_c;\ xi) =\ sqrt {\ left (\ frac {\ sigma_ {eq}} {M}\ right) ^2 +\ left (\ sigma_m+p_c\ right) ^2} - R (\ xi)
    :label: plasticity_criteria

with :math:`\sigma_m=\cfrac{\mathrm{tr}(\boldsymbol{\sigma})}{3}` the hydrostatic stress and :math:`\sigma_{eq}=\sqrt{\cfrac{3}{2}\boldsymbol{\sigma}_d:\boldsymbol{\sigma}_d}` the equivalent von Mises stress, having noted :math:`\boldsymbol{\sigma}_d=\boldsymbol{\sigma}-\sigma_m \boldsymbol{I}` the stress tensor deviator.

In :eq:`critere_plasticite`, the parameter :math:`M`, called the critical state slope, modulates the ratio of the axes of the reversibility domain in the stress meridian plane. Geometrically, the critical pressure :math:`p_c` positions the center of the ellipse along the hydrostatic axis, and finally the function :math:`R(\xi)` specifies its size.

 **Note:**

 Following the expression of criterion :eq:`critere_plasticite`, it should be noted that :math:`p_c` acts as a hydrostatic restoring force.
 Its variation therefore predicts kinematic work hardening along the hydrostatic axis. In addition, the criterion is set, as specified in the rating
 :math:`f( ; \xi)`, via the :math:`R(\xi)` function. Its evolution therefore leads to the prediction of work hardening.
 of the isotropic type. Thus, the modified Cam-Clay model has combined kinematic-isotropic work hardening. These two mechanisms are both driven by the
 scalar work hardening variable :math:`\xi`.

The flow rule of the modified Cam-Clay model respects the law of normality, so the plastic deformation tensor :math:`\boldsymbol{\epsilon}^p` and the variable :math:`\xi` evolve as:

.. math::
\ begin {align}
    \ dot {\ boldsymbol {\ epsilon}}} ^p &=\ dot {\ lambda}\ frac {\ partial f} {\ partial\ boldsymbol {\ sigma}} =\ frac {\ cfrac {3} {3} {2M^2} {2M^2}\ boldsymbol {\ sigma} _d +\ left (\ sigma_m+p_c\ right)\ cfrac {\ c\ right)\ cfrac {\ c\ right)\ cfrac {\ c\ right boldsymbol {I}} {3}} {\ sqrt {\ left (\ cfrac {\ sigma_ {eq}} {M}\ right) ^2 +\ left (\ sigma_m+p_c\ right) ^2}}\\
    \ dot {\ xi} &=\ dot {\ lambda}\ frac {\ partial f} {\ partial p_c} =\ frac {\ sigma_m+p_c} {\ sqrt {\ left (\ cfrac {\ sigma_ {eq}}\ frac {\ sigma_ {eq}}} {M}\ right) ^2}}
    \ end {align}
    :label: normal_flow

where :math:`\dot{\lambda}` is the plastic multiplier given by the following consistency condition:

.. math::
\ dot {\ lambda}\ geq 0,\ quad f\ leq 0,\ quad\ dot {\ lambda} f=0
    :label: condition_coherence

From flow law :eq:`ecoulement_normal`, we deduce that :math:`\dot{\xi}=\mathrm{tr}(\dot{\boldsymbol{\epsilon}}^p)`. The scalar work hardening variable of the modified Cam-Clay model is therefore the volume plastic deformation.

 **Note:**

 It is proposed to establish the expression of the dissipation potential :math:`\phi` of the model, using the
 Legendre-Fenchel transformation of the indicator function of the reversibility domain defined by the criterion of
 plasticity in :eq:`critere_plasticite`. We get:

 .. math:
    \ phi (\ dot {\ boldsymbol {\ epsilon}}} ^p,\ dot {\ xi};\ xi) =\ sup_ {f (\ boldsymbol {\ sigma}, p_c;\ xi)\ leq 0}\ {
    \ boldsymbol {\ sigma}:\ dot {\ boldsymbol {\ epsilon} ^p} ^p} +p_c\ dot {\ xi}\} =R (\ xi)\ sqrt {\ left (M\ dot {\ epsilon} (M\ dot {\ epsilon}} _ {\ epsilon}} _ {eq} ^p\ right) ^2+\ mathrm {tr} (\ dot {\ boldsymbol {\ epsilon}} ^p) ^2}
     + I_ {\ {0\}}\ left (\ mathrm {tr} (\ dot {\ boldsymbol {\ epsilon}}} ^p) -\ dot {\ xi}\ right)
    :label: potential_dissipation

 with the indicator function :math:`I_{\{0\}}(x)=0` if :math:`x=0`, :math:`I_{\{0\}}(x)=+\infty` otherwise, and :math:`\dot{\epsilon}_{eq}^p=\sqrt{\cfrac{2}{3}\dot{\boldsymbol{\epsilon}}^p_d: 		\boldsymbol{\epsilon^p_d}}` the equivalent von Mises plastic deformation rate, having noted :math:`\dot{\boldsymbol{\epsilon}}^p_d=\dot{\boldsymbol{\epsilon}}^p-\cfrac{\mathrm{tr}(\dot{\boldsymbol{\epsilon}}^p)}{3}\boldsymbol{I}` its deviator.


Expression of parametrization to the state
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The formulation of the modified Cam-Clay model ends by specifying the definition of function :math:`R(\xi)`, which is responsible for isotropic work hardening with the volume plastic deformation :math:`\xi=\mathrm{tr}(\boldsymbol{\epsilon}^p)`. Recall that this work hardening comes from a parametrization to the state in the plasticity criterion :eq:`critere_plasticite` (and in the dissipation potential :eq:`potentiel_dissipation`). Its expression is deduced from the constant maintenance of a traction limit noted :math:`\sigma_0\geq 0` regardless of the value of :math:`\xi`. This condition is therefore expressed as follows:

.. math::
\ forall\ xi,\ quad f (\ boldsymbol {\ sigma} =\ sigma_0\ boldsymbol {I}, p_c;\ xi) = 0
    :label: condition_sig0

Based on the expression of the plasticity criterion :eq:`critere_plasticite` as well as the expression of the critical pressure :eq:`expression_pression_critique`, the equality :eq:`condition_sig0` is verified if:

.. math::
 R (\ xi) =\ sigma_0+p_c=\ sigma_0+p_ {c0}\ exp (-\ beta\ xi)
    :label: Expression_r

Geometrically, the reversibility domain of the modified Cam-Clay model is therefore delimited along the hydrostatic axis by segment :math:`[-p_c-R(\xi),-p_c+R(\xi)]=[-\sigma_0-2p_c,\sigma_0]` in the meridian plane of the stresses. The following figure shows it for various values of volume plastic deformation :math:`\xi`.

.. figure:: images/domaine_elasticite_MCC.svg
 :align: center
 :width: 480
 :height: 360

 Reversibility domain of the modified Cam-Clay model in the stress meridian plane for several values of volume plastic deformation :math:`\xi`. A decrease in :math:`\xi` (contracency) leads to an increase in the range of reversibility. An increase in :math:`\xi` (dilatance) reduces it. All along, the isotropic tensile limit remains constant equal to :math:`\sigma_0` because :eq:`condition_sig0` is verified.

Critical state equations
----------------------------

In this section, the critical state equations for the modified Cam-Clay model are established. This state corresponds to a monotonous loading in deformation for which the state of the stresses and the volume deformation remain constant. For this, the volume of plastic deformation rate :math:`\dot{\xi}=\mathrm{tr}(\dot{\boldsymbol{\epsilon}}^p)` is cancelled out. Starting with flow law :eq:`ecoulement_normal` -2, this condition occurs when:

.. math::
\ sigma_m+p_c=0
    :label: deformation_volumic_critical_state

The average pressure :math:`-\sigma_m` is thus equal to the critical pressure :math:`p_c`. In addition, using the expression for plasticity criterion :eq:`critere_plasticite`, we deduce that the equivalent stress :math:`\sigma_{eq}` is in this situation:

.. math::
\ sigma_ {eq} = MR (\ xi)
   :label: critical_status_equivalent_constraint

The function expression :math:`R(\xi)` in :eq:`expression_R` allows us to conclude using :eq:`deformation_volumique_etat_critique`:

.. math::
\ sigma_ {eq} = M (\ sigma_0+p_ {c})\ Longrightarrow\ sigma_ {eq} + M (\ sigma_m-\ sigma_0) =0
   :label: right_critical_state

In summary, the critical state predicted by the modified Cam-Clay model is defined by the following set of equations:

.. math::
\ begin {align}
      \ dot {\ boldsymbol {\ sigma}} =\ boldsymbol {0}\\
      \ mathrm {tr} (\ dot {\ boldsymbol {\ epsilon}}) =0\\
      \ sigma_m+p_c = 0\\
      \ sigma_ {eq} +M (\ sigma_m-\ sigma_0) =0
      \ end {align}
      :label: critical_status_system

It is an *isochore* state of plasticity (without volume variation) and *perfect* (without stress variation). Geometrically, the state of the stresses in the critical state is positioned on the half-line with slope :math:`M` intersecting the surface of the elliptical reversibility domain at its vertex of abscissa :math:`-\sigma_m=p_c`.