use plain latex syntax without preamble

docs/joss/paper.md

@@ -12,10 +12,6 @@ authors:
   - name: Prashant Kumar Jha
     orcid: 0000-0003-2158-364X
     affiliation: 1
-output:
-  bookdown::pdf_document:
-    includes:
-      in_header: preamble.tex
 affiliations:
  - name: Department of Mechanical Engineering, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA
    index: 1
@@ -41,7 +37,7 @@ PeriDEM model was introduced in [@jha2021peridynamics], where it demonstrated th
 ## Brief Introduction to PeriDEM Model
 
 ![Motion of particle system.\label{fig:schemMultiParticles}](./files/multi-particle.png)
-Suppose a fixed frame of reference and $\{\be_i\}_{i=1}^d$ are orthonormal bases. Consider a collection of $N_P$ particles $\Pscript{\Omega}{p}_0$, $1\leq p \leq N_P$, where $\Pscript{\Omega}{p}_0 \subset \bbR^d$ with $d=2,3$ represents the initial configuration of particle $p$. Suppose $\Omega_0 \supset \cup_{p=1}^{N_P} \Pscript{\Omega}{p}_0$ is the domain containing all particles; see \autoref{fig:schemMultiParticles}. The particles in $\Omega_0$ are dynamically evolving due to external boundary conditions and internal interactions; let $\Pscript{\Omega}{p}_t$ denote the configuration of particle $p$ at time $t\in (0, t_F]$, and $\Omega_t \supset \cup_{p=1}^{N_P} \Pscript{\Omega}{p}_t$ domain containing all particles at that time. The motion $\Pscript{\bx}{p} = \Pscript{\bx}{p}(\Pscript{\bX}{p}, t)$ takes point $\Pscript{\bX}{p}\in \Pscript{\Omega}{p}_0$ to $\Pscript{\bx}{p}\in \Pscript{\Omega}{p}_t$, and collectively, the motion is given by $\bx = \bx(\bX, t) \in \Omega_t$ for $\bX \in \Omega_0$. We assume the media is dry and not influenced by factors other than mechanical loading (e.g., moisture and temperature are not considered). The configuration of particles in $\Omega_t$ at time $t$ depends on various factors, such as material and geometrical properties, contact mechanism, and external loading. 
+Suppose a fixed frame of reference and $\{\boldsymbol{e}_i\}_{i=1}^d$ are orthonormal bases. Consider a collection of $N_P$ particles ${\Omega}^{(p)}_0$, $1\leq p \leq N_P$, where ${\Omega}^{(p)}_0 \subset \mathbb{R}^d$ with $d=2,3$ represents the initial configuration of particle $p$. Suppose $\Omega_0 \supset \cup_{p=1}^{N_P} {\Omega}^{(p)}_0$ is the domain containing all particles; see \autoref{fig:schemMultiParticles}. The particles in $\Omega_0$ are dynamically evolving due to external boundary conditions and internal interactions; let ${\Omega}^{(p)}_t$ denote the configuration of particle $p$ at time $t\in (0, t_F]$, and $\Omega_t \supset \cup_{p=1}^{N_P} {\Omega}^{(p)}_t$ domain containing all particles at that time. The motion ${\boldsymbol{x}}^{(p)} = {\boldsymbol{x}}^{(p)}({\boldsymbol{X}}^{(p)}, t)$ takes point ${\boldsymbol{X}}^{(p)}\in {\Omega}^{(p)}_0$ to ${\boldsymbol{x}}^{(p)}\in {\Omega}^{(p)}_t$, and collectively, the motion is given by $\boldsymbol{x} = \boldsymbol{x}(\boldsymbol{X}, t) \in \Omega_t$ for $\boldsymbol{X} \in \Omega_0$. We assume the media is dry and not influenced by factors other than mechanical loading (e.g., moisture and temperature are not considered). The configuration of particles in $\Omega_t$ at time $t$ depends on various factors, such as material and geometrical properties, contact mechanism, and external loading. 
 Essentially, there are two types of interactions present in the media:
 \begin{itemize}
     \item[(1.)] {\it Intra-particle interaction} that models the deformation and internal forces in the particle and
@@ -51,61 +47,61 @@ In DEM, the first interaction is ignored, assuming particle deformation is insig
 
 The balance of linear momentum for particle $p$, $1\leq p\leq N_P$, takes the form:
 \begin{equation}
-    \Pscript{\rho}{p} \Pscript{\ddot{\bu}}{p}(\bX, t) = \Pscript{\bff}{p}_{int}(\bX, t) + \Pscript{\bff}{p}_{ext}(\bX, t), \qquad \forall (\bX, t) \in \Pscript{\Omega}{p}_0 \times (0, t_F)\,,
+    {\rho}^{(p)} {\ddot{\boldsymbol{u}}}^{(p)}(\boldsymbol{X}, t) = {\boldsymbol{f}}^{(p)}_{int}(\boldsymbol{X}, t) + {\boldsymbol{f}}^{(p)}_{ext}(\boldsymbol{X}, t), \qquad \forall (\boldsymbol{X}, t) \in {\Omega}^{(p)}_0 \times (0, t_F)\,,
 \end{equation}
-where $\Pscript{\rho}{p}$, $\Pscript{\bff}{p}_{int}$, and $\Pscript{\bff}{p}_{ext}$ are density, and internal and external force densities. The above equation is complemented with initial conditions, $\Pscript{\bu}{p}(\bX, 0) = \Pscript{\bu}{p}_0(\bX), \Pscript{\dot{\bu}}{p}(\bX, 0) = \Pscript{\dot{\bu}}{p}_0(\bX), \bX \in \Pscript{\Omega}{p}_0$. 
+where ${\rho}^{(p)}$, ${\boldsymbol{f}}^{(p)}_{int}$, and ${\boldsymbol{f}}^{(p)}_{ext}$ are density, and internal and external force densities. The above equation is complemented with initial conditions, ${\boldsymbol{u}}^{(p)}(\boldsymbol{X}, 0) = {\boldsymbol{u}}^{(p)}_0(\boldsymbol{X}), {\dot{\boldsymbol{u}}}^{(p)}(\boldsymbol{X}, 0) = {\dot{\boldsymbol{u}}}^{(p)}_0(\boldsymbol{X}), \boldsymbol{X} \in {\Omega}^{(p)}_0$. 
 
 ### Internal force - State-based peridynamics
 
 Since all expressions in this paragraph are for a fixed particle $p$, we drop the superscript $p$, noting that material properties and other quantities can depend on the particle $p$.
-Following [@silling2007peridynamic] and simplified expression of state-based peridynamics force in [@jha2021peridynamics], the internal force takes the form, for $\bX \in \Pscript{\Omega}{p}_0$,
+Following [@silling2007peridynamic] and simplified expression of state-based peridynamics force in [@jha2021peridynamics], the internal force takes the form, for $\boldsymbol{X} \in {\Omega}^{(p)}_0$,
 \begin{equation}
-    \Pscript{\bff}{p}_{int}(\bX, t) = \int_{B_{\epsilon}(\bX) \cap \Pscript{\Omega}{p}_0} \left( \bT_{\bX}(\bY) - \bT_{\bY}(\bX) \right) \, \dd \bY\,,
+    {\boldsymbol{f}}^{(p)}_{int}(\boldsymbol{X}, t) = \int_{B_{\epsilon}(\boldsymbol{X}) \cap {\Omega}^{(p)}_0} \left( \boldsymbol{T}_{\boldsymbol{X}}(\boldsymbol{Y}) - \boldsymbol{T}_{\boldsymbol{Y}}(\boldsymbol{X}) \right) \, \dd \boldsymbol{Y}\,,
 \end{equation}
-where $\bT_{\bX}(\bY) - \bT_{\bY}(\bX)$ is the force on $\bX$ due to nonlocal interaction with $\bY$. Let $R = |\bY - \bX|$ be the reference bond length, $r = |\bx(\bY) - \bx(\bX)|$ current bond length, $s(\bY, \bX) = (r - R)/R$ bond strain, then $\bT_{\bX}(\bY)$ is given by \cite{silling2007peridynamic, jha2021peridynamics}
+where $\boldsymbol{T}_{\boldsymbol{X}}(\boldsymbol{Y}) - \boldsymbol{T}_{\boldsymbol{Y}}(\boldsymbol{X})$ is the force on $\boldsymbol{X}$ due to nonlocal interaction with $\boldsymbol{Y}$. Let $R = |\boldsymbol{Y} - \boldsymbol{X}|$ be the reference bond length, $r = |\boldsymbol{x}(\boldsymbol{Y}) - \boldsymbol{x}(\boldsymbol{X})|$ current bond length, $s(\boldsymbol{Y}, \boldsymbol{X}) = (r - R)/R$ bond strain, then $\boldsymbol{T}_{\boldsymbol{X}}(\boldsymbol{Y})$ is given by [@silling2007peridynamic, @jha2021peridynamics]
 \begin{equation}
-    \bT_{\bX}(\bY) = h(s) J(R/\epsilon)\, \left[R \theta_{\bX} \left(\frac{3\kappa}{m_{\bX}} - \frac{15 G}{3 m_{\bx}}\right) + (r - R) \frac{15 G}{m_{\bX}}\right] \frac{\bx(\bY) - \bx(\bX)}{|\bx(\bY) - \bx(\bX)|}\,,
+    \boldsymbol{T}_{\boldsymbol{X}}(\boldsymbol{Y}) = h(s) J(R/\epsilon)\, \left[R \theta_{\boldsymbol{X}} \left(\frac{3\kappa}{m_{\boldsymbol{X}}} - \frac{15 G}{3 m_{\boldsymbol{X}}}\right) + (r - R) \frac{15 G}{m_{\boldsymbol{X}}}\right] \frac{\boldsymbol{x}(\boldsymbol{Y}) - \boldsymbol{x}(\boldsymbol{X})}{|\boldsymbol{x}(\boldsymbol{Y}) - \boldsymbol{x}(\boldsymbol{X})|}\,,
 \end{equation}
 where
 \begin{equation}
     \begin{split}
-        m_{\bX} &= \int_{B_\epsilon(\bX) \cap \Pscript{\Omega}{p}_0} R^2 J(R/\epsilon) \, \dd \bY\,,\\
-        \theta_{\bX} &= h(s) \frac{3}{m_{\bX}} \int_{B_\epsilon(\bX) \cap \Pscript{\Omega}{p}_0} (r - R) \, R \, J(R/\epsilon) \, \dd \bY\,,\\
+        m_{\boldsymbol{X}} &= \int_{B_\epsilon(\boldsymbol{X}) \cap {\Omega}^{(p)}_0} R^2 J(R/\epsilon) \, \dd \boldsymbol{Y}\,,\\
+        \theta_{\boldsymbol{X}} &= h(s) \frac{3}{m_{\boldsymbol{X}}} \int_{B_\epsilon(\boldsymbol{X}) \cap {\Omega}^{(p)}_0} (r - R) \, R \, J(R/\epsilon) \, \dd \boldsymbol{Y}\,,\\
         h(s) &= \begin{cases}
-            1\,, &\qquad \text{if } s < s_0 := \sqrt{\frac{\calG_c}{\left(3 G + (3/4)^4 \left[\kappa - 5G/3\right]\right)\epsilon}}\,, \\
+            1\,, &\qquad \text{if } s < s_0 := \sqrt{\frac{\mathcal{G}_c}{\left(3 G + (3/4)^4 \left[\kappa - 5G/3\right]\right)\epsilon}}\,, \\
             0\,, & \qquad \text{otherwise}\,.
         \end{cases}
     \end{split}
 \end{equation}
-In the above, $J: [0, \infty) \to \bbR$ is the influence function, $\kappa, G, \calG_c$ are bulk and shear moduli and critical energy release rate, respectively. These parameters, including nonlocal length scale $\epsilon$, could depend on the particle $p$.
+In the above, $J: [0, \infty) \to \mathbb{R}$ is the influence function, $\kappa, G, \mathcal{G}_c$ are bulk and shear moduli and critical energy release rate, respectively. These parameters, including nonlocal length scale $\epsilon$, could depend on the particle $p$.
 
 ### DEM-inspired contact forces
-The external force density $\Pscript{\bff}{p}_{ext}$ is generally expressed as
+The external force density ${\boldsymbol{f}}^{(p)}_{ext}$ is generally expressed as
 \begin{equation}
-    \Pscript{\bff}{p}_{ext} = \Pscript{\rho}{p}\bb + \bff^{\Omega_0, (p)} + \sum_{q\neq p} \PQscript{\bff}{q}{p}\,,
+    {\boldsymbol{f}}^{(p)}_{ext} = {\rho}^{(p)}\boldsymbol{b} + \boldsymbol{f}^{\Omega_0, (p)} + \sum_{q\neq p} {\boldsymbol{f}}^{(q),(p)}\,,
 \end{equation}
-where $\bb$ is body force per unit mass, $\bff^{\Omega_0, (p)}$ and $\PQscript{\bff}{q}{p}$ are contact forces due to interaction between particle $p$ and container $\Omega_0$ and neighboring particles $q$, respectively. In \citenb{jha2021peridynamics, jha2024peridynamics}, the contact between two particles is applied locally where the contact takes place; this is exemplified in \autoref{fig:peridemContact} where contact between points $\by$ and $\bx$ of two distinct particles $p$ and $q$ is activated when they get sufficiently close. The contact forces are shown using a spring-dashpot-slider system. To fix the contact forces, consider a point $\bX\in \Pscript{\Omega}{p}_0$ and let $\PQscript{R}{q}{p}_c$ be the critical contact radius (points in particles $p$ and $q$ interact if the distance is below this critical distance). Further, define the relative distance between two points $\bY \in \Pscript{\Omega}{q}_0$ and $\bX \in \Pscript{\Omega}{p}$ and normal and tangential directions as follows:
+where $\boldsymbol{b}$ is body force per unit mass, $\boldsymbol{f}^{\Omega_0, (p)}$ and ${\boldsymbol{f}}^{(q),(p)}$ are contact forces due to interaction between particle $p$ and container $\Omega_0$ and neighboring particles $q$, respectively. In [@jha2021peridynamics, @jha2024peridynamics], the contact between two particles is applied locally where the contact takes place; this is exemplified in \autoref{fig:peridemContact} where contact between points $\boldsymbol{y}$ and $\boldsymbol{x}$ of two distinct particles $p$ and $q$ is activated when they get sufficiently close. The contact forces are shown using a spring-dashpot-slider system. To fix the contact forces, consider a point $\boldsymbol{X}\in {\Omega}^{(p)}_0$ and let ${R}^{(q),(p)}_c$ be the critical contact radius (points in particles $p$ and $q$ interact if the distance is below this critical distance). Further, define the relative distance between two points $\boldsymbol{Y} \in {\Omega}^{(q)}_0$ and $\boldsymbol{X} \in {\Omega}^{(p)}$ and normal and tangential directions as follows:
 \begin{equation}
     \begin{split}
-        \PQscript{\Delta}{q}{p}(\bY, \bX) &= \vert \Pscript{\bx}{q}(\bY) - \Pscript{\bx}{p}(\bX) \vert - \PQscript{R}{q}{p}_c\,, \\
-        \PQscript{\be}{q}{p}_N(\bY, \bX) &= \frac{\Pscript{\bx}{q}(\bY) - \Pscript{\bx}{p}(\bX)}{\vert \Pscript{\bx}{q}(\bY) - \Pscript{\bx}{p}(\bX) \vert}\,, \\
-        \PQscript{\be}{q}{p}_T(\bY, \bX) &= \left[ \bI - \PQscript{\be}{q}{p}_N(\bY, \bX) \otimes \PQscript{\be}{q}{p}_N(\bY, \bX) \right]\frac{\Pscript{\dot{\bx}}{q}(\bY) - \Pscript{\dot{\bx}}{p}(\bX)}{\vert \Pscript{\dot{\bx}}{q}(\bY) - \Pscript{\dot{\bx}}{p}(\bX) \vert} \,.
+        {\Delta}^{(q),(p)}(\boldsymbol{Y}, \boldsymbol{X}) &= \vert {\boldsymbol{x}}^{(q)}(\boldsymbol{Y}) - {\boldsymbol{x}}^{(p)}(\boldsymbol{X}) \vert - {R}^{(q),(p)}_c\,, \\
+        {\boldsymbol{e}}^{(q),(p)}_N(\boldsymbol{Y}, \boldsymbol{X}) &= \frac{{\boldsymbol{x}}^{(q)}(\boldsymbol{Y}) - {\boldsymbol{x}}^{(p)}(\boldsymbol{X})}{\vert {\boldsymbol{x}}^{(q)}(\boldsymbol{Y}) - {\boldsymbol{x}}^{(p)}(\boldsymbol{X}) \vert}\,, \\
+        {\boldsymbol{e}}^{(q),(p)}_T(\boldsymbol{Y}, \boldsymbol{X}) &= \left[ \boldsymbol{I} - {\boldsymbol{e}}^{(q),(p)}_N(\boldsymbol{Y}, \boldsymbol{X}) \otimes {\boldsymbol{e}}^{(q),(p)}_N(\boldsymbol{Y}, \boldsymbol{X}) \right]\frac{{\dot{\boldsymbol{x}}}^{(q)}(\boldsymbol{Y}) - {\dot{\boldsymbol{x}}}^{(p)}(\boldsymbol{X})}{\vert {\dot{\boldsymbol{x}}}^{(q)}(\boldsymbol{Y}) - {\dot{\boldsymbol{x}}}^{(p)}(\boldsymbol{X}) \vert} \,.
     \end{split}
 \end{equation}
-Then the force on particle $p$ due to contact with particle $q$ can be written as \cite{jha2021peridynamics}:
+Then the force on particle $p$ due to contact with particle $q$ can be written as [@jha2021peridynamics}]:
 \begin{equation}
-    \PQscript{\bff}{q}{p} (\bX, t) = \int_{\bY \in \Pscript{\Omega}{q}_0 \cap B_{\PQscript{R}{q}{p}}(\bX)} \left( \PQscript{\bff}{q}{p}_N(\bY, \bX) + \PQscript{\bff}{q}{p}_T(\bY, \bX) \right)\, \dd \bY\,,
+    {\boldsymbol{f}}^{(q),(p)} (\boldsymbol{X}, t) = \int_{\boldsymbol{Y} \in {\Omega}^{(q)}_0 \cap B_{{R}^{(q),(p)}}(\boldsymbol{X})} \left( {\boldsymbol{f}}^{(q),(p)}_N(\boldsymbol{Y}, \boldsymbol{X}) + {\boldsymbol{f}}^{(q),(p)}_T(\boldsymbol{Y}, \boldsymbol{X}) \right)\, \dd \boldsymbol{Y}\,,
 \end{equation}
-with normal and tangential forces following \citenb{jha2021peridynamics, desai2019rheometry} given by
+with normal and tangential forces following [@jha2021peridynamics, @desai2019rheometry] given by
 \begin{equation}
-    \PQscript{\bff}{q}{p}_N(\bY, \bX) = \begin{cases}
-        0\,, & \quad \text{if } \PQscript{\Delta}{q}{p}(\bY, \bX) \geq 0\,, \\
-        \left[ \PQscript{\kappa}{q}{p}_N \PQscript{\Delta}{q}{p}(\bY, \bX) - \PQscript{\beta}{q}{p}_N \PQscript{\dot{\Delta}}{q}{p}(\bY, \bX)  \right] \PQscript{\be}{q}{p}_N\,, & \quad \text{otherwise}\,,
+    {\bff}^{(q),(p)}_N(\boldsymbol{Y}, \boldsymbol{X}) = \begin{cases}
+        0\,, & \quad \text{if } {\Delta}^{(q),(p)}(\boldsymbol{Y}, \boldsymbol{X}) \geq 0\,, \\
+        \left[ {\kappa}^{(q),(p)}_N {\Delta}^{(q),(p)}(\boldsymbol{Y}, \boldsymbol{X}) - {\beta}^{(q),(p)}_N {\dot{\Delta}}^{(q),(p)}(\boldsymbol{Y}, \boldsymbol{X})  \right] {\boldsymbol{e}}^{(q),(p)}_N\,, & \quad \text{otherwise}\,,
     \end{cases}
 \end{equation}
 and
 \begin{equation}
-    \PQscript{\bff}{q}{p}_T(\bY, \bX) = -\PQscript{\mu}{q}{p}_T \, \vert \PQscript{\bff}{q}{p}_N(\bY, \bX) \vert\, \PQscript{\be}{q}{p}_T\,.
+    {\boldsymbol{f}}^{(q),(p)}_T(\boldsymbol{Y}, \boldsymbol{X}) = -{\mu}^{(q),(p)}_T \, \vert {\boldsymbol{f}}^{(q),(p)}_N(\boldsymbol{Y}, \boldsymbol{X}) \vert\, {\boldsymbol{e}}^{(q),(p)}_T\,.
 \end{equation}
 
 # Implementation