Incremental formulation
========================

The formulation of the modified Cam-Clay model is now discretized in time in order to implement a numerical solution approach based on an implicit integration scheme of behavioral equations. These are obtained according to an incremental variational principle ([Mial86] _, [ORSt99] _, etc.) explained below.

Principle of incremental minimization
-------------------------------------

To implement an incremental resolution of the behavioral equations of the modified Cam-Clay model, we consider a discretization of moments :math:`t_0, t_1, \dots, t_{n+1}=t_n+\Delta t`. Over the interval :math:`[t_n,t_{n+1}]`, the evolution of the internal variables is approximated by an Euler-implicit schema:

.. math::
\ dot {\ boldsymbol {\ epsilon}}} ^p\ approx\ epsilon} ^p\ boldsymbol {\ epsilon} ^p_ {n}} {\ epsilon}} {\ epsilon}} {\ Delta t}} {\ Delta t},\ quad\ approx\ dot {\ xi}\ frac {\ xi-\ xi_n} {\ Delta t} 
    :label: discretisation_euler_implicit
    
The values of the two internal variables at time :math:`t_{n+1}`, which are :math:`\boldsymbol{\epsilon}^p_{n+1}` and :math:`\xi_{n+1}`, are sought by solving the following local variational principle:

.. math::
 (\ boldsymbol {\ epsilon} _ {n+1} ^p,\ xi_ {n+1}) =\ underset {\ boldsymbol {\ epsilon} ^p,\ xi} {\ mathrm {argmin}}}\ left\ {\ argmin}}}\ left\ {\ argmin}}}\ left\ {\ argmin}}}\ left\ {\\ psi (\ boldsymbol {\ epsilon} _ {n+1},\ boldsymbol {\ epsilon} ^p,\ xi) +\ Delta t\ left\ langle\ langle\ phi\ left (\ frac {\ boldsymbol {\ epsilon} ^p-\ boldsymbol {\ epsilon} ^p_n} {\ Delta t} {\ Delta t},\ frac {\ delta t},\ frac {\ xi-\ xi_n} {\ xi_n} {\ delta t};\ xi (\ tau)\ right)\ right\ rangle\ right\}
    :label: energie_totale_incrementale_
    
The last term on the right-hand side, in square brackets, refers to a consistent approximation of the :math:`\phi` dissipation potential averaged over the :math:`[t_n,t_{n+1}]` interval. According to the developments presented in [Bacq23] _ (see also [BRAV24] _), a legitimate expression for this approximation is written as:
 
.. math::
\ Delta t\ left\ langle\ phi\ left (\ frac {\ boldsymbol {\ epsilon} ^p-\ boldsymbol {\ epsilon} ^p_n} {\ Delta t} {\ delta t},\ frac {\ xi-\ xi_n} {\ xi-\ xi_n} {\ delta t} {\ delta t};\ xi (\ tau)\ right)\ right\ rangle =\ phi\ left},\ frac {\ xi-\ xi_n} {\ symbol {\ epsilon} ^p-\ boldsymbol {\ epsilon} ^p_n,\ xi-\ xi_n\ right) +\ frac {H_n} {2}\ left (\ left (\ left (M\ |\ left (M\ |\ boldsymbol {\ epsilon} ^p-\ boldsymbol {\ epsilon} ^p_ {n}\\ left (\ left (M\ |\ left (M\ | |\ boldsymbol {\ epsilon} ^p-\ boldsymbol {\ epsilon} ^p_ {n}\ |_ {eq}\ right) ^2 +\ mathrm {tr}\ left (\ boldsymbol {\ epsilon} ^p-\ boldsymbol {\ epsilon} ^p_n\ right) ^2\ right)
    :label: incremental_potential_dissipation
    
In a few words, the first term on the right-hand side above is interpreted as the dissipation potential integrated over the interval :math:`[t_n,t_{n+1}]` by freezing the dependence on the state at time :math:`t_n`, that is to say at :math:`\xi(\tau) = \xi_{n}`. The second term approaches the consequences of the evolution of this dependence on the integration step, in the case of positive work hardening, for which the module :math:`H_n` is a positive or zero quantity estimated by the solution established at the time :math:`t_n`. His expression is as follows:

.. math::
 H_n =\ max\ left\ {\ frac {R' (\ xi_n)} {R (\ xi_n)}\ left (\ sigma_ {m, n} +p_c (\ xi_n)\ right) ,0\ right\}
    :label: expression_module_tangent
    
It should be noted that this possibly non-zero term only exists as a result of the parametrization to the state of the Cam-Clay model modified from :math:`R'(\xi_n)\neq 0`, this dependence being, let us recall, the source of the isotropic work hardening of the model.
 
Optimality equations
---------------------- 

The incremental total energy potential expressed by :eq:`energie_totale_incrementale`, starting with :eq:`potentiel_dissipation_incrementale`, can be rewritten as follows:

.. math::
 (\ boldsymbol {\ epsilon} _ {n+1} ^p,\ xi_ {n+1}) =\ underset {\ boldsymbol {\ epsilon} ^p,\ xi} {\ mathrm {argmin}}}\ left\ {\ argmin}}}\ left\ {\ argmin}}}\ left\ {\ argmin}}}\ left\ {\\ psi (\ boldsymbol {\ epsilon} _ {n+1},\ boldsymbol {\ epsilon} ^p,\ xi) +\ phi\ left (\ boldsymbol {\ epsilon} ^p-\ boldsymbol {\ epsilon} ^p_n,\ xi-\ xi_n\ right) +\ frac {H_n} {right) +\ frac {H_n} {2} {2}\ left (\ left (M\ |\ boldsymbol {\ epsilon} ^p-\ boldsymbol {\ epsilon} ^p-\ boldsymbol {\ epsilon} ^p-\ boldsymbol {\ epsilon}} ^p_ {n}\ |_ {eq}\ right) ^2 +\ mathrm {tr}\ left (\ boldsymbol {\ epsilon} ^p-\ boldsymbol {\ epsilon} ^p_n\ right) ^2\ right)\ right\}
    :label: potential_dissipation_incrementale_2

Solving this problem then involves obtaining first-order optimality equations. They are given as such:

.. math::
\ begin {align} &\ boldsymbol {\ sigma} _ {n+1} =\ frac {\ partial\ psi_e} {\ partial\ boldsymbol {\ epsilon} _ {n+1}}},\ quad p_ {n+1}}},\ quad p_ {c, {n+1}}} =-\ frac {\ partial\ psi_h} {\ partial\ psi_h} {\ partial\ xi_ {n+1}}},\ quad p_ {n+1}},\ quad p_ {c, {n+1}}},\ quad p_ {c, {n+1}}},\ quad p_ {c, {n+1}}} =-\ frac {\ partial\ psi_h} {\ partial\ xi_ {n+1}}} state laws})\\ &f_ {n+1} =\ sqrt {\ left (\ frac {\ sigma_ {eq, {n+1}}} {M}\ right) ^2+\ left (\ sigma_ {m, n+1}} =\ sqrt {m, n+1}} =\ left (\ xi_n) ^2+\ left (\ sigma_ {m, n+1}} = ^2+\ left (\ sigma_ {m, n+1}}) ^2+\ left (\ sigma_ {m, n+1}}) ^2+\ left (\ sigma_ {m, n+1}}) ^2+\ left (\ sigma_ {m, n+1}} +h_N\ delta\ lambda\ da\ right) & (\ text {plasticity criterion})\\ &\ Delta\ boldsymbol {\ epsilon} ^p =\ Delta\ lambda\ frac {\ partial f_ {n+1}} {\ partial\ boldsymbol {\ boldsymbol {\ sigma} {\ sigma} _ {n+1}}},\ quad\ delta\ lambda\ frac {\ frac { partial f_ {n+1}} {\ partial p_ {c, {n+1}}}}} & (\ text {flow laws})\\ &\ Delta\ lambda\ geq 0,\ quad f_ {n+1}\ leq 0,\ leq 0,\ leq 0,\ quad\ delta\ delta\ lambda f_ {n+1} =0& (\ text {coherence condition})\ end {n+1}}\ leq 0,\ quad\ partial p_ {n+1}\ leq 0,\ leq 0,\ leq 0,\ quad\ delta\ delta\ lambda f_ {n+1} =0& (\ text {coherence condition})\ end {align}
   :label: equations_optimality
   
having concisely noted :math:`\Delta\boldsymbol{\epsilon}^p=\boldsymbol{\epsilon}^p_{n+1}-\boldsymbol{\epsilon}^p_n`, etc., the increments of :math:`\boldsymbol{\epsilon}^p`, :math:`\xi`, and :math:`\lambda` out of :math:`[t_n,t_{n+1}]`.
   
 **Note:**
    
 On line :eq:`equations_optimalite` -2 relating to the definition of the plasticity criterion, the radius of the reversibility domain is expressed as :math:`R(\xi_n)+H_n\Delta\lambda`, being in the general case slightly different from :math:`R(\xi_{n+1})` for a non-zero time step. This difference results from the consistency of the behavior equations as conditions of optimality in solving the incremental minimization problem in :eq:`potentiel_dissipation_incrementale_2`. On the other hand, let us emphasize the fact that wanting to solve the second line of the :eq:`equations_optimalite` system, by inserting the radius expression :math:`R(\xi_{n+1})` instead of :math:`R(\xi_n)+H_n\Delta\lambda`, is in general not a condition for the optimality of a minimization problem (see for example [BoLe90] _ on this subject).