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The linearization of the body force term is $$\begin{equation} D\left(\int_{B}\delta\mathbf{v}\cdot\rho_{r}\mathbf{b}\,dV\right)\left[\Delta\mathbf{u}\right]=\int_{b}\delta\mathbf{v}\cdot\alpha_{f}\rho\grad\mathbf{b}\cdot\Delta\mathbf{u}\,dv\,.\label{eq:edy-Dext} \end{equation}$$ The linearization of the traction force term is $$\begin{equation} \begin{aligned} & D\left(\int_{\Gamma_{\eta}}\delta\mathbf{v}\cdot\mathbf{t}\left|\mathbf{g}_{1}\times\mathbf{g}_{2}\right|\,d\eta^{1}d\eta^{2}\right)\left[\Delta\mathbf{u}\right]\\ & =\int_{\Gamma_{\eta}}\alpha_{f}\delta\mathbf{v}\cdot\left(\mathbf{t}\otimes\mathbf{n}\right)\cdot\left(\mathbf{g}_{1}\times\frac{\partial\Delta\mathbf{u}}{\partial\eta^{2}}-\mathbf{g}_{2}\times\frac{\partial\Delta\mathbf{u}}{\partial\eta^{1}}\right)\,d\eta^{1}d\eta^{2}\\ & +\int_{\Gamma_{\eta}}\delta\mathbf{v}\cdot D\mathbf{t}\left[\Delta\mathbf{u}\right]\left|\mathbf{g}_{1}\times\mathbf{g}_{2}\right|\,d\eta^{1}d\eta^{2} \end{aligned} \,.\label{eq:edy-Dext-final} \end{equation}$$ Note that $D\mathbf{t}\left[\Delta\mathbf{u}\right]$ depends on the nature of the surface traction. For a prescribed traction we have $D\mathbf{t}\left[\Delta\mathbf{u}\right]=\mathbf{0}$ . A contact analysis needs more elaborate derivations (not yet implemented as of FEBio 2.7). In the above expression we used $$\begin{equation} \begin{aligned}D\left|\mathbf{g}_{1}\times\mathbf{g}_{2}\right|\left[\Delta\mathbf{u}\right] & =\mathbf{n}\cdot D\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\left[\Delta\mathbf{u}\right]\\ & =\alpha_{f}\mathbf{n}\cdot\left(\mathbf{g}_{1}\times\frac{\partial\Delta\mathbf{u}}{\partial\eta^{2}}-\mathbf{g}_{2}\times\frac{\partial\Delta\mathbf{u}}{\partial\eta^{1}}\right) \end{aligned} \label{eq:edy-Dext-DA} \end{equation}$$ where the unit outward normal is evaluated as $$\begin{equation} \mathbf{n}=\frac{\mathbf{g}_{1}\times\mathbf{g}_{2}}{\left|\mathbf{g}_{1}\times\mathbf{g}_{2}\right|}\,.\label{eq:edy-unit-normal} \end{equation}$$