WIP: draft

docs/joss/paper.md

@@ -34,7 +34,7 @@ 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){width=30%}
+![Motion of particle system.\label{fig:schemMultiParticles}](./files/multi-particle.png){width=60%}
 
 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:
@@ -79,15 +79,15 @@ The external force density ${\boldsymbol{f}}^{(p)}_{ext}$ is generally expressed
 \begin{equation}
     {\boldsymbol{f}}^{(p)}_{ext} = {\rho}^{(p)}\boldsymbol{b} + \boldsymbol{f}^{\Omega_0, (p)} + \sum_{q\neq p} {\boldsymbol{f}}^{(q),(p)}\,,
 \end{equation}
-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:
+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], 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}
         {\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 [@jha2021peridynamics}]:
+Then the force on particle $p$ due to contact with particle $q$ can be written as [@jha2021peridynamics]:
 \begin{equation}
     {\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)\, \mathrm{d} \boldsymbol{Y}\,,
 \end{equation}