3. Continuous equations

3.1. Mechanics: conservation of momentum

We note \(\sigma\) the Cauchy stress tensor and \(s\) the second (symmetric) Piola-Kirchhoff tensor.

We denote \(P\) as the \({x}_{0}={x}_{S}(0)\to {x}_{S}({x}_{0},t)\) transformation gradient.

\(P=\frac{\partial {x}_{S}({x}_{0},t)}{\partial {x}_{0}}\)

We have: \(s=\text{det}P\text{.}{P}^{-1}\text{.}\sigma \text{.}{P}^{-T}\).

The mechanical balance equations are written in configuration \({\Omega }_{0}\):

\({\text{Div}}_{0}(P\text{.}s)+r{F}^{m}=0\)

We noted \({\text{Div}}_{0}\) as the divergence operator with respect to the \({x}_{0}\) space variables in the \({\Omega }_{0}\) configuration.

Insofar as we hypothesize small displacements and small deformations of the skeleton, this equation can be approximated by:

\(\text{Div}\sigma +r{F}^{m}=0\) eq 3.1-1

We will see later that we are still adopting the \(\sigma =\sigma \text{'}+{\sigma }_{p}I\) decomposition, where \(\sigma \text{'}\) refers to the effective constraint. It is therefore up to the module for integrating balance equations to do the sum: \(\sigma =\sigma \text{'}+{\sigma }_{p}I\).

3.2. Hydraulics: mass conservation

The Eulerian writing for the conservation of fluid mass for the constituent \(c\) is written:

\(\frac{{d}^{\mathrm{fl}}}{\mathrm{dt}}\int {}_{\Omega }\text{}\sum _{p}{\rho }_{c}^{p}\phi {S}^{p}d\Omega =0\)

We can then apply [éq 2.3-1] by taking: \({a}_{s}=0\) and \({{a}^{m}}_{c}^{p}=1\) and [éq 2.3-3] will give:

\(\sum _{p}\frac{{d}^{s}{\rho }_{c}^{p}\phi {S}^{p}}{\mathrm{dt}}+\sum _{p}\text{Div}({w}_{c}^{p})=0\)

Using the definition of mass inputs [éq 2.2.2.3-3], the definition of Lagrangian flows [éq2.2.2.3-2] we find the Lagrangian form of fluid mass conservation:

\(\{\begin{array}{}\dot{{m}_{1}}+{\text{Div}}_{0}({M}_{1})=0\\ \dot{{m}_{2}}+{\text{Div}}_{0}({M}_{2})=0\end{array}\) eq 3.2-1

3.3. Energy equation

For thermodynamic functions, we systematically adopt a decomposition of the [éq2.3-1] type. This corresponds to the fact that the different energies all have a part carried by the solid and a part carried by the fluids. The part carried by the solid is characterized by a volume density while the parts carried by the fluid are characterized by mass densities, as we showed in paragraph [§2.3].

Total internet energy: \(E=\int {}_{\Omega }\text{}({e}_{s}+\sum _{p,c}{\rho }_{c}^{p}\phi {S}^{p}{{e}^{m}}_{c}^{p})d\Omega\) eq 3.3.1

Total entropy: \(S=\int {}_{\Omega }\text{}({s}_{s}+\sum _{p,c}{\rho }_{c}^{p}\phi {S}^{p}{{s}^{m}}_{c}^{p})d\Omega\) eq 3.3.2

Total enthalpy: \(H=\int {}_{\Omega }\text{}({h}_{s}+\sum _{p,c}{\rho }_{c}^{p}\phi {S}^{p}{{h}^{m}}_{c}^{p})d\Omega\) eq 3.3.3

Free energy: \(\{\begin{array}{}\Psi =E-TS\\ {\Psi }_{s}={e}_{s}-{\text{Ts}}_{s}\\ {{\Psi }^{m}}_{c}^{p}={{e}^{m}}_{c}^{p}-T{{s}^{m}}_{c}^{p}\end{array}\) eq 3.3.4

Free enthalpy: \(\{\begin{array}{}G=H-TS\\ {g}_{s}={h}_{s}-T{s}_{s}\\ {{g}^{m}}_{c}^{p}={{h}^{m}}_{c}^{p}-T{{s}^{m}}_{c}^{p}\end{array}\) eq 3.3.5

Finally, by noting \(\dot{Q}(\Omega )\) the heat rate received by a volume \(\Omega\), we have by definition:

\(\dot{Q}(\Omega )=\underset{\partial \Omega }{\int }q\text{.}nd\Gamma +\underset{\Omega }{\int }\Thetad \Omega\) eq 3.3.6

Finally, we recall that the enthalpy of fluids is calculated by the formula:

\(h=e+\frac{p}{\rho }\) eq 3.3.7

3.3.1. The first principle

With the definitions given above, it is written:

\(-\sum _{p,c}\text{Div}\left({{h}^{m}}_{c}^{p}{\mathrm{M}}_{c}^{p}\right)+\mathrm{\sigma }\mathrm{:}\dot{\mathrm{\epsilon }}+\sum _{p,c}{\mathrm{M}}_{c}^{p}\text{.}{\mathrm{F}}^{m}+\mathrm{\Theta }-\text{Div}q=0\) eq 3.3.1-1

This writing corresponds to equation (22) in chapter III -2-3 of [bib1], in which we neglected inertia terms. For homogeneous media, it corresponds to equation (31) in paragraph IV-3-2 of [bib3].

3.3.2. The second principle

Its fairly well known form is:

\(\dot{s}+\sum _{p,c}\text{Div}({{s}^{m}}_{c}^{p}{M}_{c}^{p})+\text{Div}(\frac{q}{T})-\frac{\Theta }{T}\ge 0\) eq 3.3.2-1

Using the classical thermodynamic considerations [bib1] linked to the introduction of free enthalpy [éq 3.3.5], we show that we must necessarily have:

\(\sigma -\frac{\partial \Psi }{\partial \varepsilon }=0\) eq 3.3.2-2

\({{g}^{m}}_{c}^{p}-\frac{\partial \Psi }{\partial {m}_{c}^{p}}=0\) eq 3.3.2-3

\(s+\frac{\partial \Psi }{\partial T}=0\) eq 3.3.2-4

3.3.3. Energy equation

Quite often, it is considered that, since the transformations are reversible, the second principle ultimately provides equality. In addition, in [éq 3.3.2-1] the unknown temperature \(T\) is replaced by a constant value called the reference temperature. It is finally a linearization of [éq 3.3.2-1] justified if the temperature variations are « small ». Note that the transport term \(\sum _{p,c}\text{Div}({{s}^{m}}_{c}^{p}{M}_{c}^{p})\) complicates the treatment of nonlinearity due to the presence of temperature in the denominator of the other terms in [éq 3.3.2-1].

We work in enthalpy in order to overcome this difficulty. We start from the equation of the first principle [éq 3.3.1-1] into which we inject the equations [éq 3.3.2-2], [], [éq 3.3.2-3], [éq 3.3.2-4], and the definition of free enthalpy [éq 3.3-5] into which we obtain:

\(T\dot{s}+\sum _{p,c}({{h}^{m}}_{c}^{p}\dot{{m}_{c}^{p}}-T{{s}^{m}}_{c}^{p}\dot{{m}_{c}^{p}})=-\sum _{p,c}\text{Div}({{h}^{m}}_{c}^{p}{M}_{c}^{p})+\sum _{p,c}{M}_{c}^{p}\text{.}{F}^{m}+\Theta -\text{Div}q\) eq 3.3.3-1

We then ask:

\(\deltaq \text{'}=T\deltas -T\sum _{p,c}{s}^{{m}_{c}^{p}}\delta {m}_{c}^{p}\) eq 3.3.3-2

The quantity \(Q\text{'}\) has the dimension of one energy per unit volume. It represents the heat received by the system in a transformation for which there is no heat input by entering a fluid having an enthalpy. Although \(\deltaq \text{'}\) is not an exact differential, we take this quantity as a state variable.

Finally, the energy equation adopted has the following form:

\(\sum _{p,c}{{h}^{m}}_{c}^{p}\dot{{m}_{c}^{p}}+\dot{Q\text{'}}+\sum _{p,c}\text{Div}({{h}^{m}}_{c}^{p}{M}_{c}^{p})+\text{Div}q-\sum _{p,c}{M}_{c}^{p}\text{.}{F}^{m}=\Theta\) eq 3.3.3-3