4. Variational formulation

Before giving the expression for the variational formulations of the equilibrium equations presented above, we will give the definition of the approximation spaces of the fields of displacement, of pressures (of the massif and of the interface), of the flows \({q}_{1}\) and \({q}_{2}\) and of the Lagrange multipliers \(\mathrm{\lambda },\mathrm{\mu }\) and of the Lagrange multipliers and the displacement jump \(w\) useful for the cohesive zone model:

• the space of the kinematically admissible fields of movement on the border of domain \(\Omega\) is such that:

\({U}_{0}=\{{u}^{\text{*}}\in {H}_{1}(\Omega )/{u}^{\text{*}}\text{}\text{discontinu à travers}{\Gamma }_{c},{u}^{\text{*}}=0\text{}\text{sur}{\Gamma }_{u}\}\)

• the space of the admissible pressure fields on the border of domain \(\Omega\) is such that:

\({P}_{0}=\{{p}^{\text{*}}\in {H}_{1}(\Omega )/{p}^{\text{*}}\text{}\text{discontinue à travers}{\Gamma }_{c},{p}^{\text{*}}=0\text{}\text{sur}{\Gamma }_{p}\}\)

• the space of unknowns \({q}_{1}\) and \({q}_{2}\) is such that:

\({Q}_{1}=\{{q}_{1}^{\text{*}}\in {H}^{-1/2}({\Gamma }_{c})/{q}_{1}^{\text{*}}\in {\Gamma }_{1}\}\) and \({Q}_{2}=\{{q}_{2}^{\text{*}}\in {H}^{-1/2}({\Gamma }_{c})/{q}_{2}^{\text{*}}\in {\Gamma }_{2}\}\)

• the pressure unknown space \({p}_{f}\) is such that:

\({F}_{0}=\{{p}_{f}^{\text{*}}\in {H}^{-1/2}({\mathrm{\Gamma }}_{c})/{p}_{f}^{\text{*}}\text{continue sur}{\mathrm{\Gamma }}_{c}\}\)

• the space of the unknowns of the Lagrange multipliers \(\mathrm{\lambda },\mathrm{\mu }\) and the movement jump \(w\) is such that:

\({L}_{0}=\{{\mathrm{\lambda }}^{\text{*}}\in {H}^{-1/2}({\mathrm{\Gamma }}_{c})/{\mathrm{\lambda }}^{\text{*}}\text{continue sur}{\mathrm{\Gamma }}_{c}\}\)

4.1. Weak formulation of the mechanical problem

As explained in [R7.02.19], as part of the « mortar » formulation for the cohesive zone model, the displacement jump \(w\) is introduced as a new unknown to the problem, which will not be discretized as \(⟦u⟧\) but will be a projection onto a reduced space \({M}_{h}\) (see § 5.2.2). The total energy of the problem is then written as:

\(E(u,\mathrm{\lambda },w)=\frac{1}{2}{\int }_{\mathrm{\Omega }}\mathrm{ϵ}(u)\mathrm{:}C\mathrm{:}\mathrm{ϵ}(u)d\mathrm{\Omega }-{\int }_{{\mathrm{\Gamma }}_{t}}t\cdot ud{\mathrm{\Gamma }}_{t}+{\int }_{{\mathrm{\Gamma }}_{c}}\mathrm{\Pi }\left(w,\mathrm{\lambda }\right)d{\mathrm{\Gamma }}_{c}\)

\(\mathrm{\Pi }(w,\mathrm{\lambda })\) is the surface energy density and \(t\) is the surface forces imposed on \({\mathrm{\Gamma }}_{t}\). The Lagrange multiplier \(\mathrm{\lambda }\) will be discretized on the same space as \(w\) (confer [R7.02.19]).

The solution to the ongoing problem consists in minimization under the constraints of equality \(\left(u,w,\mathrm{\lambda }\right)=\underset{{w}^{\text{*}}=⟦{u}^{\text{*}}⟧}{\mathit{argmin}}E\left({u}^{\text{*}},{\mathrm{\lambda }}^{\text{*}},{w}^{\text{*}}\right)\). We can write the associated Lagrangian as:

\(L(u,w,\mathrm{\lambda },\mathrm{\mu })=\frac{1}{2}{\int }_{\mathrm{\Omega }}\mathrm{ϵ}(u)\mathrm{:}C\mathrm{:}\mathrm{ϵ}(u)d\mathrm{\Omega }-{\int }_{{\mathrm{\Gamma }}_{t}}t\cdot ud{\mathrm{\Gamma }}_{t}+{\int }_{{\mathrm{\Gamma }}_{c}}\mathrm{\Pi }(w,\mathrm{\lambda })d{\mathrm{\Gamma }}_{c}+{\int }_{{\mathrm{\Gamma }}_{c}}\mathrm{\mu }\cdot \left(⟦u⟧-w\right)d{\mathrm{\Gamma }}_{c}\)

The Lagrange multiplier \(\mathrm{\mu }\) will also be discretized on the reduced space \({M}_{h}\). Writing the optimality conditions for this Lagrangian leads to the following variational formulation:

Equation of balance	\(\forall {u}^{\text{*}}\in {U}_{\mathrm{0,}}{\int }_{\mathrm{\Omega }}\mathrm{\sigma }(u)\mathrm{:}\mathrm{ϵ}({u}^{\text{*}})d\mathrm{\Omega }-{\int }_{{\mathrm{\Gamma }}_{t}}t\cdot {u}^{\text{*}}d{\mathrm{\Gamma }}_{t}+{\int }_{{\mathrm{\Gamma }}_{c}}\mathrm{\mu }\cdot ⟦{u}^{\text{*}}⟧d{\mathrm{\Gamma }}_{c}=0\)
Move jump projection	\(\forall {\mathrm{\mu }}^{\text{*}}\in {L}_{0},{\int }_{{\mathrm{\Gamma }}_{c}}\left(⟦u⟧-w\right)\cdot {\mathrm{\mu }}^{\text{*}}d{\mathrm{\Gamma }}_{c}=0\)
Expression of cohesive force	\(\forall {w}^{\text{*}}\in {L}_{\mathrm{0,}}-{\int }_{{\mathrm{\Gamma }}_{c}}\left[\mathrm{\mu }-{t}_{c}(\mathrm{\lambda }+rw)\right]\cdot {w}^{\text{*}}d{\mathrm{\Gamma }}_{c}=0\)
Interface law	\(\forall {\mathrm{\lambda }}^{\text{*}}\in {L}_{\mathrm{0,}}-{\int }_{{\mathrm{\Gamma }}_{c}}\frac{\left[\mathrm{\lambda }-{t}_{c}(\mathrm{\lambda }+rw)\right]}{r}\cdot {\mathrm{\lambda }}^{\text{*}}d{\mathrm{\Gamma }}_{c}=0\)

\(r\) is the increase parameter (confer [R7.02.19]). Recall that:

• \(\sigma ={\sigma }^{\text{'}}-\mathit{bp}1\)

• \({t}_{c}={t}_{c}^{\text{'}}-{p}_{f}n\)

4.2. Weak formulations of the hydrodynamic problem

4.2.1. Weak formulation for the massif

The weak formulation of the mass conservation equation in the case of the massif is written as:

\(\begin{array}{c}-{\int }_{\mathrm{\Omega }}\frac{\partial {m}_{w}}{\partial t}{p}^{\text{*}}d\mathrm{\Omega }\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}M\cdot \nabla {p}^{\text{*}}d\mathrm{\Omega }\hfill \\ \phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{F}}{M}_{\text{ext}}{p}^{\text{*}}d{\mathrm{\Gamma }}_{F}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{1}}{q}_{1}{p}^{\text{*}}d{\mathrm{\Gamma }}_{1}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{2}}{q}_{2}{p}^{\text{*}}d{\mathrm{\Gamma }}_{2}\hfill \end{array}\) \(\forall {p}^{\text{*}}\in {P}_{0}\)

with \({M}_{\mathit{ext}}\) the normal flows imposed on part \({\Gamma }_{F}\) of \(\partial \Omega\).

4.2.2. Weak formulation for the interface

The weak formulation of the mass conservation equation in the case of the interface is written as:

\(\begin{array}{c}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{c}}\frac{\partial w}{\partial t}{p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{c}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{c}}W\cdot \nabla {p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{c}\hfill \\ \phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{f}}{W}_{\text{ext}}{p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{f}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{1}}{q}_{1}{p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{1}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{2}}{q}_{2}{p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{2}\hfill \end{array}\) \(\forall {p}_{f}^{\text{*}}\in {F}_{0}\)

with \({W}_{\mathit{ext}}\) the normal flows imposed on part \({\Gamma }_{f}\) of \({\Gamma }_{c}\).

The weak formulation of the pressure continuity condition \({p}_{f}\) at the interface level is written as:

\({\int }_{{\Gamma }_{1}}\left({p}^{\text{sup}}-{p}_{f}\right){q}_{1}^{\text{*}}d{\Gamma }_{1}=0\) \(\forall {q}_{1}^{\text{*}}\in {Q}_{1}\)

\({\int }_{{\Gamma }_{2}}\left({p}^{\text{inf}}-{p}_{f}\right){q}_{2}^{\text{*}}d{\Gamma }_{2}=0\) \(\forall {q}_{2}^{\text{*}}\in {Q}_{2}\)

Note:

The condition of continuity of pressure \({p}_{f}\) at the level of each of the lips of the interface involves two linear relationships of the type \({p}^{\text{sup}}-{p}_{f}=0\) on \({\Gamma }_{1}\) and \({p}^{\text{inf}}-{p}_{f}=0\) on on*:math:{Gamma }_{2}*. In Code_Aster, in order to manage this type of relationship (which is in fact a boundary condition* and not an equilibrium equation, such as the mass conservation equation), we resort to the introduction of Lagrange multiplier fields. In this case, the multipliers (which are referred to below as hydraulic Lagrange multipliers) used in these two variational formulations are in fact the virtual flows \({q}_{1}^{\text{*}}`* and*:math:`{q}_{2}^{\text{*}}\) and .