# Principle of virtual work - extra term needed for deformation dependent loading?

I'm working on a problem in nonlinear elasticity, for which the external forces (loadings) depend on the displacements. Following Klaus-Jürgen Bathe's book "Finite Element Procedures", the virtual work is \begin{equation} \int_{{}^{t+\Delta t}V}{}^{t+\Delta t}\tau_{ij}\delta{}_{t+\Delta t}e_{ij}d{}^{t+\Delta t}V={}^{t+\Delta t}\mathscr{R} \end{equation} where the prescript notation denotes the time at which the quantities are considered, $V$ is the volume, $\boldsymbol{\tau}$ is the Cauchy stress, $\delta\boldsymbol{e}$ is the strain tensor for the virtual displacement $\delta\boldsymbol{u}$, and the external work term on the right-hand side is

\begin{equation} {}^{t+\Delta t}\mathscr{R}=\int_{{}^{t+\Delta t}V}{}^{t+\Delta t}f_i^B\delta u_id{}^{t+\Delta t}V+\int_{{}^{t+\Delta t}S_f}{}^{t+\Delta t}f_i^S\delta u_i^Sd{}^{t+\Delta t}S \end{equation}

where $\boldsymbol{f}^B$ is the externally applied body force, $\boldsymbol{f}^S$ is the externally applied surface traction, $S_f$ is the surface on which the tractions are applied, and $\delta\boldsymbol{u}^S$ is the virtual displacement evaluated on the surface. My problem has no surface tractions so the second term on the right-hand side can be omitted.

Later in the chapter, Bathe considers deformation-dependent loading, e.g. equations (6.83) and (6.84), where he argues that the external work can be approximated by integrating over the most recently calculated volume and surface area.

My question relates to the derivation of the equation for the external work. Specifically, if the body force $\boldsymbol{f}^B$ depends on the displacement $\boldsymbol{u}$, why is there no term in the virtual work that looks like $\boldsymbol{u}\delta{\boldsymbol{f}^B}$?

If I understand correctly, in the book, the author says that the term $\mathbf{u}\,\delta\mathbf{f}^{B}$ can be moved over to the left hand side and included in the stiffness matrix along with other coefficients of $\mathbf{u}$.