Merge pull request #659 from rezgarshakeri/rezgar/update-doc

Poroelastodynamic ref and formulation

Author: baagaard-usgs

docs/references.bib

@@ -105,6 +105,16 @@ @Article{Day:Ely:2002
   doi = 	 {10.1785/0120010273}
 }
 
+@article{ding2013fundamental,
+  title={Fundamental solutions of poroelastodynamics in frequency domain based on wave decomposition},
+  author={Ding, Boyang and Cheng, Alexander H-D and Chen, Zhanglong},
+  journal={Journal of Applied Mechanics},
+  volume={80},
+  number={6},
+  year={2013},
+  publisher={American Society of Mechanical Engineers Digital Collection}
+}
+
 @Article{Drucker:Prager:1952,
   author = 	 {Drucker, D.~C. and Prager, W.},
   title = 	 {Soil mechanics and plastic analysis for limit design},

docs/user/governingeqns/poroelasticity/dynamic.md

@@ -37,7 +37,13 @@ We replace the variation of fluid content variable, $\zeta$, with its definition
 \end{gather}
 %
 We write the volumetric strain in terms of displacement, because this dynamic formulation does not include the volumetric strain as an unknown.
-
+Note that for poroelastodynamics we use the generalized Darcy's law {cite}`ding2013fundamental` as
+%
+\begin{equation}
+  \vec{q}(p) = -\frac{\boldsymbol{k}}{\mu_{f}}(\nabla p - \vec{f}_f + \rho_{f} \frac{\partial \vec{v}}{\partial t}),
+\end{equation}
+%
+where the generalized Darcy's law adds the term $\rho_{f} \frac{\partial \vec{v}}{\partial t}$.
 Using trial functions ${\vec{\psi}_\mathit{trial}^{u}}$, ${\psi_\mathit{trial}^{p}}$, and ${\vec{\psi}_\mathit{trial}^{v}}$, and incorporating the Neumann boundary conditions, the weak form may be written as:
 %
 \begin{align}

docs/user/governingeqns/poroelasticity/index.md

@@ -44,7 +44,7 @@ For this case, we will assume that the material properties are isotropic, result
 %
 \begin{equation}
   \boldsymbol{\sigma}(\vec{u},p) = \boldsymbol{C}:\boldsymbol{\epsilon} - \alpha p \boldsymbol{I}
-  = \lambda \boldsymbol{I} \epsilon_{v} + 2 \mu - \alpha \boldsymbol{I} p
+  = \lambda \boldsymbol{I} \epsilon_{v} + 2 \mu  \boldsymbol{\epsilon} - \alpha \boldsymbol{I} p
 \end{equation}
 %
 where $\lambda$ and $\mu$ are Lamé's parameters, $\lambda = K_{d} - \frac{2 \mu}{3}$, $\mu$ is the shear modulus, and the volumetric strain is defined as $\epsilon_{v} = \nabla \cdot \vec{u}$.