From f695f333d3c422fe7c48c033a10a822aa5abe809 Mon Sep 17 00:00:00 2001
From: "Prashant K. Jha" <pjha.sci@gmail.com>
Date: Thu, 17 Oct 2024 20:24:28 -0600
Subject: [PATCH] fix pdf compile error

---
 docs/joss/paper.md | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/docs/joss/paper.md b/docs/joss/paper.md
index 6a65f2bf..f8fec6c5 100644
--- a/docs/joss/paper.md
+++ b/docs/joss/paper.md
@@ -56,7 +56,7 @@ where ${\rho}^{(p)}$, ${\boldsymbol{f}}^{(p)}_{int}$, and ${\boldsymbol{f}}^{(p)
 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 $\boldsymbol{X} \in {\Omega}^{(p)}_0$,
 \begin{equation}
-    {\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}\,,
+    {\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) \, \mathrm{d} \boldsymbol{Y}\,,
 \end{equation}
 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}
@@ -65,8 +65,8 @@ where $\boldsymbol{T}_{\boldsymbol{X}}(\boldsymbol{Y}) - \boldsymbol{T}_{\boldsy
 where
 \begin{equation}
     \begin{split}
-        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}\,,\\
+        m_{\boldsymbol{X}} &= \int_{B_\epsilon(\boldsymbol{X}) \cap {\Omega}^{(p)}_0} R^2 J(R/\epsilon) \, \mathrm{d} \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) \, \mathrm{d} \boldsymbol{Y}\,,\\
         h(s) &= \begin{cases}
             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}\,.
@@ -90,7 +90,7 @@ where $\boldsymbol{b}$ is body force per unit mass, $\boldsymbol{f}^{\Omega_0, (
 \end{equation}
 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)\, \dd \boldsymbol{Y}\,,
+    {\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}
 with normal and tangential forces following [@jha2021peridynamics, @desai2019rheometry] given by
 \begin{equation}