Let the rigid body rotation tensor $\boldsymbol{\Lambda}\left(t\right)$ be expressed as $\boldsymbol{\Lambda}\left(t\right)=\exp\left[\boldsymbol{\xi}\left(t\right)\right]$ , and $\boldsymbol{\xi}\left(t\right)$ is the material rotation of the rigid body from its reference configuration. Thus, $\boldsymbol{\Lambda}_{n}=\exp\left[\boldsymbol{\xi}_{n}\right]$ and $\boldsymbol{\Lambda}_{n+1}=\exp\left[\boldsymbol{\xi}_{n+1}\right]$ respectively represent the rigid body rotation tensors at $t_{n}$ and $t_{n+1}$ . (In practice, $\boldsymbol{\xi}$ is stored as a quaternion to facilitate the multiplication of rotation tensors.) These tensors are related by the incremental spatial rotation $\boldsymbol{\theta}$ or material rotation $\boldsymbol{\Theta}$ from $t_{n}$ to $t_{n+1}$ according to $$\begin{equation} \boldsymbol{\Lambda}_{n+1}=\cay\left[\boldsymbol{\theta}\right]\cdot\boldsymbol{\Lambda}_{n}=\boldsymbol{\Lambda}_{n}\cdot\cay\left[\boldsymbol{\Theta}\right]\,.\label{eq:td-incremental-rotation} \end{equation}$$ Here, it should be understood that the material frame for this incremental rotation is the configuration at time $t_{n}$ , while the spatial frame is the configuration at $t_{n+1}$ . For rotational motion, the Newmark scheme is applied in the material frame as $$\begin{equation} \begin{aligned}\mathbf{W}_{n+1} & =\frac{\gamma}{\beta\Delta t}\boldsymbol{\Theta}-\mathbf{W}_{n}+\left(2-\frac{\gamma}{\beta}\right)\left(\mathbf{W}_{n}+\frac{\Delta t}{2}\mathbf{A}_{n}\right)\\ \mathbf{A}_{n+1} & =\frac{1}{\gamma\Delta t}\left(\mathbf{W}_{n+1}-\mathbf{W}_{n}\right)+\left(1-\frac{1}{\gamma}\right)\mathbf{A}_{n}\\ & =\frac{1}{\beta\Delta t}\left(\frac{1}{\Delta t}\boldsymbol{\Theta}-\mathbf{W}_{n}\right)+\left(1-\frac{1}{2\beta}\right)\mathbf{A}_{n} \end{aligned} \,.\label{eq:td-Newmark-mrot} \end{equation}$$ Then, using the relations $\boldsymbol{\omega}=\boldsymbol{\Lambda}\cdot\mathbf{W}$ and $\boldsymbol{\alpha}=\boldsymbol{\Lambda}\cdot\mathbf{A}$ at $t_{n}$ and $t_{n+1}$ , along with (6.3.4.1-2), we may express these relations in the spatial frame as $$\begin{equation} \begin{aligned}\boldsymbol{\omega}_{n+1} & =\cay\left[\boldsymbol{\theta}\right]\cdot\left(\frac{\gamma}{\beta\Delta t}\boldsymbol{\theta}-\boldsymbol{\omega}_{n}+\left(2-\frac{\gamma}{\beta}\right)\left(\boldsymbol{\omega}_{n}+\frac{\Delta t}{2}\boldsymbol{\alpha}_{n}\right)\right)\\ \boldsymbol{\alpha}_{n+1} & =\cay\left[\boldsymbol{\theta}\right]\cdot\left(\frac{1}{\beta\Delta t}\left(\frac{1}{\Delta t}\boldsymbol{\theta}-\boldsymbol{\omega}_{n}\right)+\left(1-\frac{1}{2\beta}\right)\boldsymbol{\alpha}_{n}\right) \end{aligned} \,.\label{eq:td-Newmark-srot} \end{equation}$$

In a nonlinear solution scheme we solve for $\boldsymbol{\Theta}$ incrementally. According to (6.3.1.3-3), the linearization of $\cay\left[\boldsymbol{\Theta}\right]$ along an increment $\Delta\boldsymbol{\Theta}$ is given by $$\begin{equation} D\left(\cay\left[\boldsymbol{\Theta}\right]\right)\left[\Delta\boldsymbol{\Theta}\right]=\cay\left[\boldsymbol{\Theta}\right]\cdot\widehat{\Delta\boldsymbol{\Theta}}\,,\label{eq:td-linearization-rot} \end{equation}$$ so that $$\begin{equation} D\boldsymbol{\Lambda}_{n+1}\left[\Delta\boldsymbol{\Theta}\right]=\boldsymbol{\Lambda}_{n+1}\cdot\widehat{\Delta\boldsymbol{\Theta}}\,.\label{eq:td-linearization-Lambda} \end{equation}$$ The linearizations of $\mathbf{W}_{n+1}$ and $\mathbf{A}_{n+1}$ , as given in (6.3.4.1-3), along an increment $\Delta\boldsymbol{\Theta}$ requires us to first evaluate $D\boldsymbol{\Theta}\left[\Delta\boldsymbol{\Theta}\right]$ . According to Puso [62], $$\begin{equation} D\boldsymbol{\Theta}\left[\Delta\boldsymbol{\Theta}\right]=\mathbf{T}\left(\boldsymbol{\Theta}\right)\cdot\Delta\boldsymbol{\Theta}\,,\label{td-Theta-linearization} \end{equation}$$ where $$\begin{equation} \mathbf{T}\left(\boldsymbol{\Theta}\right)=\mathbf{I}+\frac{1}{2}\hat{\boldsymbol{\Theta}}+\frac{1}{4}\boldsymbol{\Theta}\otimes\boldsymbol{\Theta}\,.\label{eq:td-T-Theta-relation} \end{equation}$$ Thus, $$\begin{equation} \begin{aligned}D\mathbf{W}_{n+1}\left[\Delta\boldsymbol{\Theta}\right] & =\frac{\gamma}{\beta\Delta t}\mathbf{T}\left(\boldsymbol{\Theta}\right)\cdot\Delta\boldsymbol{\Theta}\\ D\mathbf{A}_{n+1}\left[\Delta\boldsymbol{\Theta}\right] & =\frac{1}{\beta\Delta t^{2}}\mathbf{T}\left(\boldsymbol{\Theta}\right)\cdot\Delta\boldsymbol{\Theta} \end{aligned} \,.\label{eq:tf-W-A-linearizations} \end{equation}$$