WIP: joss paper

docs/joss/paper.md

@@ -16,7 +16,7 @@ affiliations:
  - name: Department of Mechanical Engineering, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA
    index: 1
 header-includes:
-   - \usepackage{amsfonts,amssymb,amsmath}
+   - \usepackage{amsfonts,amssymb,amsmath,bm}
 date: 17 October 2024
 bibliography: paper.bib
 
@@ -38,7 +38,7 @@ The PeriDEM model was introduced in [@jha2021peridynamics], demonstrating its ab
 
 ![Motion of particle system.\label{fig:schemMultiParticles}](./files/multi-particle.png){width=60%}
 
-Consider 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. 
+Consider a fixed frame of reference and $\{\bm{e}_i\}_{i=1}^d$ 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 ${\bm{x}}^{(p)} = {\bm{x}}^{(p)}({\bm{X}}^{(p)}, t)$ takes point ${\bm{X}}^{(p)}\in {\Omega}^{(p)}_0$ to ${\bm{x}}^{(p)}\in {\Omega}^{(p)}_t$, and collectively, the motion is given by $\bm{x} = \bm{x}(\bm{X}, t) \in \Omega_t$ for $\bm{X} \in \Omega_0$. 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:
 - *Intra-particle interaction* that models the deformation and internal forces in the particle and
 - *Inter-particle interaction* that accounts for the contact between particles and the boundary of the domain in which the particles are contained.