2. Laws of behavior#

2.1. Mechanics#

2.1.1. General writing#

\(\{\begin{array}{c}{\sigma }^{\text{+}}={\sigma }^{\text{+}}({\epsilon }^{\text{+}},{p}_{1}^{\text{+}},{p}_{2}^{\text{+}},{T}^{\text{+}};{\epsilon }^{\text{-}},{p}_{1}^{\text{-}},{p}_{2}^{\text{-}},{T}^{\text{-}},{\sigma }^{\text{-}},{\chi }^{\text{-}})\\ {\chi }^{\text{+}}={\chi }^{\text{+}}({\epsilon }^{\text{+}},{p}_{1}^{\text{+}},{p}_{2}^{\text{+}},{T}^{\text{+}};{\epsilon }^{\text{-}},{p}_{1}^{\text{-}},{p}_{2}^{\text{-}},{T}^{\text{-}},{\sigma }^{\text{-}},{\chi }^{\text{-}})\end{array}\) eq 2.1.1-1

2.1.2. Case of effective constraints#

In the case of the hypothesis of effective constraints, this function will be decomposed in the form:

\(\begin{array}{c}\sigma =\sigma \text{'}+{\sigma }_{p}I\\ \sigma \text{'}\text{est le tenseur des contraintes effectives:}\\ {\sigma }_{p}\text{est un scalaire}\end{array}\)

\(\{\begin{array}{c}\sigma {\text{'}}^{\text{+}}=\sigma {\text{'}}^{\text{+}}({\epsilon }^{\text{+}},{T}^{\text{+}};{\epsilon }^{\text{-}},{T}^{\text{-}},\sigma {\text{'}}^{\text{-}},{\chi }_{\sigma }^{\text{-}})\\ {\chi }_{\sigma }^{\text{+}}={\chi }_{\sigma }^{\text{+}}({\epsilon }^{\text{+}},{T}^{\text{+}};{\epsilon }^{\text{-}},{T}^{\text{-}},\sigma {\text{'}}^{\text{-}},{\chi }_{\sigma }^{\text{-}})\end{array}\) eq 2.1.2-1

\(\{\begin{array}{c}{\sigma }_{p}^{\text{+}}={\sigma }_{p}^{\text{+}}({p}_{1}^{\text{+}},{p}_{2}^{\text{+}};{p}_{1}^{\text{-}},{p}_{2}^{\text{-}},{\chi }_{H}^{\text{-}})\\ {\chi }_{H}^{\text{+}}={\chi }_{H}^{\text{+}}({p}_{1}^{\text{+}},{p}_{2}^{\text{+}};{p}_{1}^{\text{-}},{p}_{2}^{\text{-}},{\chi }_{H}^{\text{-}})\end{array}\) eq 2.1.2-2

We note that in this decomposition:

  1. the thermal dependence has been left in the effective constraints; typically, it is believed that the laws on effective constraints are written as in classical thermomechanics:

\(\sigma {\text{'}}^{\text{+}}=\sigma {\text{'}}^{\text{+}}({\epsilon }^{\text{+}}-{\alpha }^{\text{+}}{T}^{\text{+}};{\epsilon }^{\text{-}}-{\alpha }^{\text{-}}{T}^{\text{-}},\sigma {\text{'}}^{\text{-}},{\chi }_{\sigma }^{\text{-}})\)

  1. we distinguished the internal variables relating to the law of behavior on effective constraints, which we wrote \({\chi }_{\sigma }\), the internal variables of hydraulic origin that we wrote \({\chi }_{H}\) and the internal variables of thermal origin that we wrote \({\chi }_{T}\) (see the following paragraphs).

2.1.3. Choice of constraints#

Because of the fairly frequent use of the hypothesis of effective constraints, it is decided that the stress vector for the mechanical part contains in all cases the effective stress tensor \(\sigma \text{'}\) and the scalar \({\sigma }_{p}\). In the general case where the hypothesis of effective constraints is not retained, we will simply have:. \({\sigma }_{p}=0\)

It is therefore up to the module for integrating balance equations to do the sum: \(\sigma =\sigma \text{'}+{\sigma }_{p}I\).

2.2. Hydraulics#

The law of hydraulic behavior will provide the following relationships:

\(\{\begin{array}{c}{m}_{c}^{p\text{+}}={m}_{c}^{p\text{+}}({\mathrm{\epsilon }}^{\text{+}},{p}_{1}^{\text{+}},{p}_{2}^{\text{+}},{T}^{\text{+}};{\mathrm{\epsilon }}^{\text{-}},{p}_{1}^{\text{-}},{p}_{2}^{\text{-}},{T}^{\text{-}},{m}_{d}^{q\text{-}},{\text{M}}_{d}^{q\text{-}},{\mathrm{\chi }}_{H}^{\text{-}})\\ {\text{M}}_{c}^{p\text{+}}={\text{M}}_{c}^{p\text{+}}\left(\begin{array}{c}{\mathrm{\epsilon }}^{\text{+}},{p}_{1}^{\text{+}},\nabla {p}_{1}^{\text{+}},{p}_{2}^{\text{+}},\nabla {p}_{2}^{\text{+}},{T}^{\text{+}},\nabla {T}^{\text{+}};\\ {\mathrm{\epsilon }}^{\text{-}},{p}_{1}^{\text{-}},\nabla {p}_{1}^{\text{-}},{p}_{2}^{\text{-}},\nabla {p}_{2}^{\text{-}},{T}^{\text{-}},\nabla {T}^{\text{-}},{\text{M}}_{d}^{q\text{-}},{\mathrm{\chi }}_{H}^{\text{-}}\mathrm{:}{\text{F}}^{m\text{+}}\end{array}\right)\\ {\mathrm{\chi }}_{H}^{\text{+}}={\mathrm{\chi }}_{H}^{\text{+}}({\mathrm{\epsilon }}^{\text{+}},{p}_{1}^{\text{+}},{p}_{2}^{\text{+}},{T}^{\text{+}};{\mathrm{\epsilon }}^{\text{-}},{p}_{1}^{\text{-}},{p}_{2}^{\text{-}},{T}^{\text{-}},{m}_{1}^{\text{-}},{m}_{2}^{\text{-}},{\mathrm{\chi }}_{H}^{\text{-}})\end{array}\begin{array}{c}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}\end{array}\}\forall c\mathit{et}\forall p\phantom{\rule{2em}{0ex}}\mathit{de}1à{\mathit{np}}_{c}\) eq 2.2-1

We note that the gravity field is a data of the law of hydraulic behavior because the evolution of the flow vector follows relationships of the type: \(\text{M}={\lambda }_{H}{\rho }^{\mathit{fl}}[-\nabla P+{\rho }^{\mathit{fl}}{\text{F}}^{m}]\).

2.3. Thermal#

The laws of behavior will provide:

\(\begin{array}{c}\{\begin{array}{c}Q{\text{'}}^{\text{+}}=Q{\text{'}}^{\text{+}}({\epsilon }^{\text{+}},{p}_{1}^{\text{+}},{p}_{2}^{\text{+}},{T}^{\text{+}};{\epsilon }^{\text{-}},{p}_{1}^{\text{-}},{p}_{2}^{\text{-}},{T}^{\text{-}},S{\text{'}}^{\text{-}})\\ {h}_{cm}^{p\text{+}}={h}_{cm}^{p\text{+}}({\epsilon }^{\text{+}},{p}_{1}^{\text{+}},{p}_{2}^{\text{+}},{T}^{\text{+}};{\epsilon }^{\text{-}},{p}_{1}^{\text{-}},{p}_{2}^{\text{-}},{T}^{\text{-}},{s}_{\mathit{dm}}^{q\text{-}})\forall c\mathit{et}\forall p\mathit{de}1à{\mathit{np}}_{c}\\ {\text{q}}^{\text{+}}={\text{q}}^{\text{+}}({\epsilon }^{\text{+}},{p}_{1}^{\text{+}},{p}_{2}^{\text{+}},{T}^{\text{+}},\nabla {T}^{\text{+}};{\epsilon }^{\text{-}},{p}_{1}^{\text{-}},{p}_{2}^{\text{-}},{T}^{\text{-}},\nabla {T}^{\text{-}},{\text{q}}^{\text{-}})\\ {\chi }_{T}^{\text{+}}={\chi }_{T}^{\text{+}}({\epsilon }^{\text{+}},{p}_{1}^{\text{+}},{p}_{2}^{\text{+}},{T}^{\text{+}},\nabla {T}^{\text{+}};{\epsilon }^{\text{-}},{p}_{1}^{\text{-}},{p}_{2}^{\text{-}},{T}^{\text{-}},\nabla {T}^{\text{-}},{\chi }_{T}^{\text{-}})\end{array}\\ \text{Avec}{h}_{\mathit{dm}}^{q\text{-}}=({h}_{\mathrm{1m}}^{1\text{-}},{h}_{\mathrm{1m}}^{2\text{-}},{h}_{\mathrm{2m}}^{1\text{-}},{h}_{\mathrm{2m}}^{2\text{-}})\end{array}\) eq 2.3-1

2.4. Homogenized density#

\({r}^{\text{+}}={r}_{0}+{m}_{1}^{1\text{+}}+{m}_{1}^{2\text{+}}+{m}_{2}^{1\text{+}}+{m}_{2}^{2\text{+}}\) eq 2.4-1