small fixes to Johnson-Cook documentation

doc/content/source/materials/hardening_models/JohnsonCookHardening.md

@@ -6,25 +6,25 @@
 
 The Johnson-Cook plasticity model has the following form
 \begin{equation}
-    \sigma = \sigma_0\left(A+B\left(\frac{\ep}{\ep0}\right)^n\right)\left(1+C\text{ln}\left(\frac{\epdot}{\epdot0}\right)\right)\left(1-\left(\frac{T-T_0}{T_m-T_0}\right)^m\right)
+    \sigma = \sigma_0\left(A+B\left(\frac{\ep}{\varepsilon_0^p}\right)^n\right)\left(1+C\text{ln}\left(\frac{\epdot}{\dot\varepsilon_0^p}\right)\right)\left(1-\left(\frac{T-T_0}{T_m-T_0}\right)^m\right)
 \end{equation}
 
-Where $\sigma_0$ is the unperturbed yield stress, $\ep0$ is the reference plastic strain, $\epdot0$ is the reference plastic strain rate, $T_m$ is the melting temperature, and $T_0$ is the reference temperature.
+Where $\sigma_0$ is the unperturbed yield stress, $\varepsilon_0^p$ is the reference plastic strain, $\dot\varepsilon_0^p$ is the reference plastic strain rate, $T_m$ is the melting temperature, and $T_0$ is the reference temperature.
 $A$,$B$,and $C$ are parameters of the Johnson-Cook model.
 
 When put into a variational framework, the Johnson-Cook model defines the following plastic energy
 \begin{equation}
-\psi_p = \left[\left(1-Q\right)g_p \sigma_0 \ep \left(A\ep + B\ep0\left(\frac{\ep}{\ep0}\right)^{n+1}\frac{1}{n+1}\right)\right]f(T),\\
-\psi_p^* = \left[Q \sigma_0\left(A+B\ep0\left(\frac{\ep}{\ep0}\right)^n\right)+\left(A+B\frac{\ep}{\ep0}\right)\left(C\text{ln}\left(\frac{\epdot}{\epdot0}\right)-C\right)\right]f(T),\\
-f(T) = \left(\frac{T-T_0}{T_m-T_00}\right)
+\psi_p = \left[\left(1-Q\right)g_p \sigma_0 \ep \left(A\ep + B\varepsilon_0^p\left(\frac{\ep}{\varepsilon_0^p}\right)^{n+1}\frac{1}{n+1}\right)\right]f(T),\\
+\psi_p^* = \left[Q \sigma_0\left(A+B\varepsilon_0^p\left(\frac{\ep}{\varepsilon_0^p}\right)^n\right)+\left(A+B{\left(\frac{\ep}{\varepsilon_0^p}\right)}^n\right)\left(C\text{ln}\left(\frac{\epdot}{\dot\varepsilon_0^p}\right)-C\right)\right]f(T),\\
+f(T) = \left(\frac{T-T_0}{T_m-T_0}\right)
 \end{equation}
 
 Additionally, the energy is split into energetic and dissipative portions using the Taylor-Quinney factor, $Q$.\\
 The flow stresses are defined as the following
 
 \begin{equation}
-    Y^{\text{eq}}=\left[\left(1-Q\right)\sigma_0\left(A+B\frac{\ep}{\ep0}\right)^n\right]f(T),\\
-    Y^{\text{vis}}=\left[Q\sigma_0\left(A+B\left(\frac{\ep}{\ep0}\right)^n\right)+\left(A+B\frac{\ep}{\ep0}\right)\left(C\text{ln}\left(\frac{\epdot}{\epdot0}\right)\right)\right]f(T)
+    Y^{\text{eq}}=\left[\left(1-Q\right)\sigma_0\left(A+B\frac{\ep}{\varepsilon_0^p}\right)^n\right]f(T),\\
+    Y^{\text{vis}}=\left[Q\sigma_0\left(A+B\left(\frac{\ep}{\varepsilon_0^p}\right)^n\right)+\left(A+B\frac{\ep}{\varepsilon_0^p}\right)\left(C\text{ln}\left(\frac{\epdot}{\dot\varepsilon_0^p}\right)\right)\right]f(T)
 \end{equation}
 
 

doc/content/theory/constitutive_models.md

@@ -33,6 +33,7 @@ We support both small deformation and large deformation. Several hyperelastic mo
 
 - [Power-law hardening](PowerLawHardening.md)
 - [Arrhenius-law hardening](ArrheniusLawHardening.md)
+- [Johnson-Cook hardening](JohnsonCookHardening.md)
 
 ## Phase-field fracture models