Poroelasticity

Linear

For the linear poroelasticity model, the strain energy density is given by

\[ \psi = \frac{\firstlame_{u}}{2} (\trace \bm{\varepsilon})^2 + \secondlame_{d} \bm{\varepsilon} \tcolon \bm{\varepsilon} - B \, M \, \trace \bm{\varepsilon} \, \zeta + \frac{1}{2} M \zeta^2, \]

where \(\firstlame_{u} = \firstlame_{d} + B^2 M\) is undrained first Lamé parameter while \(\firstlame_{d}, \secondlame_{d}\) are Lamé parameters measured in drained condition, \(B, M\) are the Biot effective stress coefficient and Biot modulus defined by

(115)\[ \begin{aligned} B &= 1 - \frac{\bulk_{d}}{\bulk_s}, \\ \frac{1}{M} &= \frac{\phi^f}{\bulk_f} + \frac{B - \phi^f}{\bulk_s}, \end{aligned} \]

with mixture bulk modulus \(\bulk_{d}\) measured in drained condition and solid and fluid bulk moduli \(\bulk_s, \bulk_f\), respectively.

To derive the constitutive law (stress-strain relationship) for the linear poroelasticity model we have

(116)\[ \begin{aligned} \bm\sigma(\bm{u}, p) &= \frac{\partial \psi}{\partial \bm{\varepsilon}} = \bm\sigma'(\bm{u}) - B \, p \, \bm{I} = \firstlame_{d} \nabla\cdot \bm{u} \, \bm{I} + 2 \, \secondlame_{d} \, \bm{\varepsilon} - B \, p \, \bm{I}, \\ p &= \frac{\partial \psi}{\partial \zeta} = M \left(\zeta - B \nabla \cdot \bm{u} \right), \end{aligned} \]

where , \(\bm\sigma' = \firstlame_{d} (\trace \bm\varepsilon) \bm{I} + 2 \, \secondlame_{d} \, \bm{\varepsilon}\) is effective stress, \(p\) is pore pressure, and the variation of fluid content \(\zeta\) is

(117)\[ \zeta = \frac{p}{M} + B \nabla\cdot \bm u. \]

Finite strain

For finite strain poroelasticity, by writing Clausius-Duhem inequality we can derive constitutive equations for solid and fluid phases as [ICR24]

\[ \bm{S}' = 2\frac{\partial \psi^s}{ \partial \bm{C}_s}, \quad p = \frac{\rho^{fR}}{B\phi^f}\frac{\partial \psi^f}{\partial \rho^{fR}}, \]

where \(\bm{S}'\) is effective second Piola-Kirchoff stress tensor, \(\psi^s\) is strain energy for solid phase which is defined in previous section for Neo-Hookean, Mooney-Rivlin, and Ogden models. Helmholtz free energy per unit mass for fluid phase \(\psi^f\) is given by

(118)\[ \psi^f = \frac{1}{2}B \phi^f \bulk_f \left(\log \rho^{fR}\right)^2. \]

The total stress for the mixture for Neo-Hookean model (73) is

(119)\[ \begin{aligned} \bm{S} &= J \bm{F}^{-1} \bm{\sigma} \bm{F}^{-T} = \bm{S}' - J B p \bm{C}^{-1}, \\ \bm{S}' &= \firstlame J V' \bm{C}^{-1} + \secondlame \left( \bm{I} - \bm{C}^{-1} \right) = \frac{\firstlame}{2} \left( J^2 - 1 \right)\bm{C}^{-1} + \secondlame \left( \bm{I} - \bm{C}^{-1} \right), \end{aligned} \]

and using (25) we can write symmetric Kirchhoff stress tensor as

(120)\[ \begin{aligned} \bm{\tau} &= \bm{\tau}' - J B p \bm{I}, \\ \bm{\tau}' &= \firstlame J V' \bm{I} + \secondlame \left( \bm{b} - \bm{I} \right) = \frac{\firstlame}{2} \left( J^2 - 1 \right)\bm{I} + \secondlame \left( \bm{b} - \bm{I} \right). \end{aligned} \]