$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

The time derivatives, $\partial\mathbf{v}/\partial t$ which appears in the expression for $\mathbf{a}$ in eq.(2.11.1-2), and $\partial J/\partial t$ which similarly appears in $\dot{J}$ , may be discretized upon the choice of a time integration scheme, such as the generalized-$\alpha$ method [42] (Section 3.9↓). In this scheme, $\delta W$ is evaluated at an intermediate time step $t_{n+\alpha}=\alpha t_{n+1}+\left(1-\alpha\right)t_{n}$ between the current time step $t_{n+1}$ and previous time step $t_{n}$ , though different values of $\alpha$ are used for the primary variables and their time derivatives. The velocity and volume ratio are evaluated as $\mathbf{v}_{n+\alpha_{f}}$ and $J_{n+\alpha_{f}}$ at the intermediate time step $t_{n+\alpha_{f}}$ , whereas their time derivatives are evaluated as $\left(\partial\mathbf{v}/\partial t\right)_{n+\alpha_{m}}$ and $\left(\partial J/\partial t\right)_{n+\alpha_{m}}$ at the intermediate time step $t_{n+\alpha_{m}}$ . The parameters $\alpha_{f}$ and $\alpha_{m}$ are evaluated from the spectral radius for an infinite time step, $\rho_{\infty}$ , as described in Section 3.9↓. The solution of the nonlinear equation $\delta W=0$ is obtained by linearizing this relation as $$\begin{equation} \delta W+D\delta W\left[\Delta\mathbf{v}\right]+D\delta W\left[\Delta J\right]\approx0\,,\label{eq:linearized-virtual-work} \end{equation}$$ where the operator $D\delta W\left[\cdot\right]$ represents the directional derivative of $\delta W$ at $\left(\mathbf{v},J\right)$ along an increment $\Delta\mathbf{v}$ of $\mathbf{v}$ , or $\Delta J$ of $J$ [23]. The aim of this analysis is to solve for the velocity $\mathbf{v}_{n+1}$ and volume ratio $J_{n+1}$ at the current time step $t_{n+1}$ . Using the split form of $\delta W$ between external and internal work contributions, this relation may be expanded as $$\begin{equation} \begin{aligned} & D\delta W_{int}\left[\Delta\mathbf{v}\right]+D\delta W_{int}\left[\Delta J\right]-D\delta W_{ext}\left[\Delta\mathbf{v}\right]\\ & -D\delta W_{ext}\left[\Delta J\right]\approx\delta W_{ext}-\delta W_{int}\,. \end{aligned} \label{eq:linearized-work-split} \end{equation}$$