3. 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.

3.1. 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 \(t_0, t_1, \dots, t_{n+1}=t_n+\Delta t\). Over the interval \([t_n,t_{n+1}]\), the evolution of the internal variables is approximated by an Euler-implicit schema:

\[\]

dot {boldsymbol {epsilon}}} ^papproxepsilon} ^pboldsymbol {epsilon} ^p_ {n}} {epsilon}} {epsilon}} {Delta t}} {Delta t},quadapproxdot {xi}frac {xi-xi_n} {Delta t}

label:: discretisation_euler_implicit

The values of the two internal variables at time \(t_{n+1}\), which are \(\boldsymbol{\epsilon}^p_{n+1}\) and \(\xi_{n+1}\), are sought by solving the following local variational principle:

\[\begin{split}(\ boldsymbol {\ epsilon} _ {n+1} ^p,\ xi_ {n+1}) =\ underset {\ boldsymbol {\ epsilon} ^p,\ xi} {\ mathrm {argmin}}}\ left\ {\ argmin}}}\ left\ {\ argmin}}}\ left\ {\ argmin}}}\ left\ {\\ psi (\ boldsymbol {\ epsilon} _ {n+1},\ boldsymbol {\ epsilon} ^p,\ xi) +\ Delta t\ left\ langle\ langle\ phi\ left (\ frac {\ boldsymbol {\ epsilon} ^p-\ boldsymbol {\ epsilon} ^p_n} {\ Delta t} {\ Delta t},\ frac {\ delta t},\ frac {\ xi-\ xi_n} {\ xi_n} {\ delta t};\ xi (\ tau)\ right)\ right\ rangle\ right\} :label: energie_totale_incrementale_\end{split}\]

The last term on the right-hand side, in square brackets, refers to a consistent approximation of the \(\phi\) dissipation potential averaged over the \([t_n,t_{n+1}]\) interval. According to the developments presented in [Bacq23] _ (see also [BRAV24] _), a legitimate expression for this approximation is written as:

\[\]

Delta tleftlanglephileft (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)rightrangle =phileft},frac {xi-xi_n} {symbol {epsilon} ^p-boldsymbol {epsilon} ^p_n,xi-xi_nright) +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_nright) ^2right)

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 \([t_n,t_{n+1}]\) by freezing the dependence on the state at time \(t_n\), that is to say at \(\xi(\tau) = \xi_{n}\). 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 \(H_n\) is a positive or zero quantity estimated by the solution established at the time \(t_n\). His expression is as follows:

\[H_n =\ max\ left\ {\ frac {R' (\ xi_n)} {R (\ xi_n)}\ left (\ sigma_ {m, n} +p_c (\ xi_n)\ right) ,0\ right\} :label: expression_module_tangent\]

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 \(R'(\xi_n)\neq 0\), this dependence being, let us recall, the source of the isotropic work hardening of the model.

3.2. Optimality equations#

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

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

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

\[\]

begin {align} &boldsymbol {sigma} _ {n+1} =frac {partialpsi_e} {partialboldsymbol {epsilon} _ {n+1}}},quad p_ {n+1}}},quad p_ {c, {n+1}}} =-frac {partialpsi_h} {partialpsi_h} {partialxi_ {n+1}}},quad p_ {n+1}},quad p_ {c, {n+1}}},quad p_ {c, {n+1}}},quad p_ {c, {n+1}}} =-frac {partialpsi_h} {partialxi_ {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_Ndeltalambdadaright) & (text {plasticity criterion})\ &Deltaboldsymbol {epsilon} ^p =Deltalambdafrac {partial f_ {n+1}} {partialboldsymbol {boldsymbol {sigma} {sigma} _ {n+1}}},quaddeltalambdafrac {frac { partial f_ {n+1}} {partial p_ {c, {n+1}}}}} & (text {flow laws})\ &Deltalambdageq 0,quad f_ {n+1}leq 0,leq 0,leq 0,quaddeltadeltalambda f_ {n+1} =0& (text {coherence condition})end {n+1}}leq 0,quadpartial p_ {n+1}leq 0,leq 0,leq 0,quaddeltadeltalambda f_ {n+1} =0& (text {coherence condition})end {align}

label:: equations_optimality

having concisely noted \(\Delta\boldsymbol{\epsilon}^p=\boldsymbol{\epsilon}^p_{n+1}-\boldsymbol{\epsilon}^p_n\), etc., the increments of \(\boldsymbol{\epsilon}^p\), \(\xi\), and \(\lambda\) out of \([t_n,t_{n+1}]\).

Note:

On line equations_optimalite -2 relating to the definition of the plasticity criterion, the radius of the reversibility domain is expressed as \(R(\xi_n)+H_n\Delta\lambda\), being in the general case slightly different from \(R(\xi_{n+1})\) 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 potentiel_dissipation_incrementale_2. On the other hand, let us emphasize the fact that wanting to solve the second line of the equations_optimalite system, by inserting the radius expression \(R(\xi_{n+1})\) instead of \(R(\xi_n)+H_n\Delta\lambda\), is in general not a condition for the optimality of a minimization problem (see for example [BoLe90] _ on this subject).