Volumetric creep formulation for RTL2020-based models

doc/config.yml

@@ -20,6 +20,7 @@ Extensions:
             Examples:
                 Poroelasticity: examples/poroelasticity/index.md
                 Viscoelasticity: examples/viscoelasticity/index.md
+                Viscoplasticity: examples/viscoplasticity/index.md
             Documentation:
                 Beaver syntax: documentation/beaver.md
     MooseDocs.extensions.appsyntax:

doc/content/bib/beaver.bib

@@ -28,4 +28,13 @@ @article{Gerya2007
   pages = {83--105}, 
   number = {1-4}, 
   volume = {163}
+}
+@article{azabou2021rock,
+  title={Rock salt behavior: From laboratory experiments to pertinent long-term predictions},
+  author={Azabou, M and Rouabhi, Ahmed and Blanco-Mart{\'\i}n, L and Hadj-Hassen, F and Karimi-Jafari, M and H{\'e}vin, G},
+  journal={International Journal of Rock Mechanics and Mining Sciences},
+  volume={142},
+  pages={104588},
+  year={2021},
+  publisher={Elsevier}
 }

doc/content/examples/index.md

@@ -1,4 +1,5 @@
 # Examples
 
 !content outline max_level=3 pages=poroelasticity/index.md
-                                   viscoelasticity/index.md
+                                   viscoelasticity/index.md
+                                   viscoplasticity/index.md

doc/content/examples/viscoplasticity/index.md

@@ -0,0 +1,4 @@
+# Viscoplasticity
+
+!content outline max_level=3 pages=lemaitre.md
+                                   munson_dawson.md

doc/content/examples/viscoplasticity/lemaitre.md

@@ -0,0 +1,39 @@
+# Lemaitre's viscoplastic model
+
+This problem considers a squared medium subject to external stress leading to creep. The material deforms following the Lemaitre's viscoplastic constitutive model.
+
+## Setup
+
+The squared medium is subject to a compressive horizontal uniaxial stress at a constant temperature, resulting in a uniaxial deformation. The setup is sketched in [!ref](fig_lemaitre_creep_model_setup).
+
+!media media/linear_kelvin.png style=display:block;margin:auto;width:80%; caption=Setup for the Lemaitre's viscoplastic medium. id=fig_lemaitre_creep_model_setup
+
+## Solutions
+
+The scalar equivalent creep strain evolution is governed by the following constitutive model:
+
+!equation
+\dot{\gamma}_{vp} = A \left( \frac{q}{A_2} \right)^{\frac{\beta}{\alpha}} \gamma_{vp}^{1-\frac{1}{a}},
+
+where $A$ is a time-scaling model parameter unique to the material under consideration. [!cite](azabou2021rock) expressed this parameter in terms of the Arrhenius law as $\alpha \exp\left(\frac{1}{T_r}-\frac{1}{T}\right)$ to describe the behavior of rock salt. 
+In this case, $A \approx \alpha$ for isothermal conditions. $\alpha$ and $\beta$ are material parameters. $A_2$ is a normalizing parameter always taken as 1.0. $\gamma_{vp}$ is the scalar equivalent creep strain.
+
+The analytical solution for this problem is given as:
+
+!equation
+\gamma_{vp} = \left( \frac{A}{\alpha} \right)^{\alpha} \left[ \left( \frac{q}{A_2} \right)^{\beta} \right] t^{a},
+
+where $t$ is time. 
+
+The following creep curve shows a comparison between the analytical and the numerical solutions of the Lemaitre model.
+
+!media media/lemaitre_strain.png style=display:block;margin:auto;width:60%; caption=Creep strain evolution in a Lemaitre's viscoplastic medium. id=fig_lemaitre
+
+## Complete Source Files
+
+- [lemaitre.i](https://github.com/ajacquey/beaver/blob/main/examples/viscoplasticity/lemaitre/lemaitre.i)
+
+!bibtex bibliography
+
+!content pagination use_title=True
+                    next=viscoplasticity/munson_dawson.md

doc/content/examples/viscoplasticity/munson_dawson.md

@@ -0,0 +1,83 @@
+# Modified Munson-Dawson viscoplastic model
+
+This problem considers a squared medium subject to external stress leading to creep. 
+The material deforms following the modified Munson-Dawson's viscoplastic constitutive model described in [!cite](azabou2021rock).
+
+## Setup
+
+The squared medium is subject to a compressive horizontal uniaxial stress at a constant temperature, resulting in a uniaxial deformation. 
+The setup is sketched in [!ref](fig_munson_dawson_creep_model_setup). 
+The uniaxial stress is held constant at 10 MPa for a duration of 15 days before dropping to 5 MPa for another 15 days to simulate a multi-stage creep test. 
+
+!media media/linear_kelvin.png style=display:block;margin:auto;width:80%; caption=Setup for the munson_dawson's viscoplastic medium. id=fig_munson_dawson_creep_model_setup
+
+## Solutions
+
+The scalar equivalent creep strain evolution is governed by the following constitutive model:
+
+\begin{equation}
+  \dot\gamma_{vp} = 
+  \begin{cases} 
+  A \left( 1 - \frac{\gamma_{vp}}{\bar{\gamma}_{vp}} \right)^n R(\sigma, T) & \text{if } \gamma_{vp} \leq \bar{\gamma}_{vp} \\ 
+  -B \left( \frac{\gamma_{vp}}{\bar{\gamma}_{vp}} - 1 \right)^m R(\sigma, T) & \text{if } \gamma_{vp} \geq \bar{\gamma}_{vp} 
+  \end{cases}
+\end{equation}
+
+where, $A$, $B$, $m$, and $n$ are material parameters, with $A > 0$, $B > 0$, and $m > 1$, $n > 1$ [!cite](azabou2021rock). 
+The parameter $\bar{\gamma}_{vp}$ describes the saturation strain, similar to that expressed in the Munson-Dawson model, 
+and corresponds to the threshold of transient deformation. It is defined as:
+
+\begin{equation} 
+  \bar{\gamma}_{vp}(\sigma, T)= \left\langle \frac{q}{A_1} \right\rangle^{n_1}
+  \label{eq:29}
+\end{equation}
+here $A_1$ and $n_1$ are material parameters and $q = \sqrt{\frac{3}{2}\tau_{ij}\tau_{ij}}$ 
+is the equivalent stress, representing the loading function. $\tau_{ij}$ is the deviatoric stress tensor.
+
+The function $R$ corresponds to the Lemaitre’s scalar equivalent creep strain evolution, 
+ensuring that $\dot\gamma_{vp}$ have similar order of magnitude as the Lemaitre's scalar equivalent creep model [!cite](azabou2021rock):
+
+\begin{equation}
+  R(\sigma, T) = \exp \left( A_R \left( \frac{1}{T_r} - \frac{1}{T} \right) \right) \left\langle \frac{q}{A_2} \right\rangle^{n_2}
+\end{equation}
+
+This model is equivalent to the Munson-Dawson model provided the following conditions are satisfied:
+
+- Parameter $A = \exp{\Delta}$
+- Parameter $n =
+        \begin{cases} 
+        -0.0098\Delta^3 + 0.2040\Delta^2 + 0.5622\Delta + 2.0252, & \text{if } \Delta \leq 6 \\
+        1.9987\Delta - 1.4567, & \text{if } \Delta > 6
+        \end{cases}$
+- Parameter $B = 0$ (i.e., the transient strain threshold is not exceeded). Hence, the strain-rate is always transient.
+
+The analytical solution for this problem is given as [!cite](azabou2021rock):
+
+\begin{equation*}
+    \gamma_{vp} \left( t \right) = {\gamma_{vp}}_0 + (\bar{\gamma}_{vp} + {\gamma_{vp}}_0) \, \xi\left(t\right),
+\end{equation*}
+\begin{equation*}
+    \xi \left( t \right) = \left(1 - \frac{1}{1 + V(t - t_0)} \right)^{\frac{1}{k-1}},
+\end{equation*}
+\begin{equation*}
+    V = (k - 1) \, U \, \frac{A_{1}^{n_{1}}}{A_{2}^{n_{2}}} \, q^{n_{2} - n_{1}},
+\end{equation*}
+\begin{equation*}
+  \begin{cases}
+    \text{when } \gamma_{vp} \leq \bar{\gamma}_{vp}; \quad k = n; \quad U = A \left( 1 - \frac{{\gamma_{vp}}_0}{\bar{\gamma}_{vp}} \right)^{n-1} \\
+    \text{when } \gamma_{vp} \geq \bar{\gamma}_{vp}; \quad k = m; \quad U = B \left( \frac{{\gamma_{vp}}_0}{\bar{\gamma}_{vp}} - 1 \right)^{m-1} 
+  \end{cases}
+\end{equation*}
+
+The following creep curve shows a comparison between the analytical and the numerical solutions of the modified Munson-Dawson model.
+
+!media media/munson_dawson_strain.png style=display:block;margin:auto;width:60%; caption=Creep strain evolution in a Munson-Dawson's viscoplastic medium. id=fig_munson_dawson
+
+## Complete Source Files
+
+- [munson_dawson.i](https://github.com/ajacquey/beaver/blob/main/examples/viscoplasticity/munson_dawson/munson_dawson.i)
+
+!bibtex bibliography
+
+!content pagination use_title=True
+                    previous=viscoplasticity/lemaitre.md

doc/moosedocs.py

@@ -1,6 +1,6 @@
-#!/usr/bin/env python
+#!/usr/bin/env python3
 #* This file is part of the MOOSE framework
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+#* https://mooseframework.inl.gov
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