# 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 $$\int_{{}^{t+\Delta t}V}{}^{t+\Delta t}\tau_{ij}\delta{}_{t+\Delta t}e_{ij}d{}^{t+\Delta t}V={}^{t+\Delta t}\mathscr{R}$$ 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

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

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 $\boldsymbol{u}\delta{\boldsymbol{f}^B}$?

## 1 Answer

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 $\mathbf{u}$.

• Thanks Amit - that's exactly what I would expect. The book is quite long - do you recall in what section that discussion takes place? – OskarM Dec 9 '17 at 16:32
• In the second edition of the book, it is just mentioned in a signle line immediately after equation 6.84. Kindly, consider accepting my answer if it answered your question. – user26248 Dec 9 '17 at 16:51
• Ah, I think you are right. Here's what he says: In order to obtain an iterative scheme that usually converges in fewer iterations, the effect of the unkown incremental displacements in the load terms needs to be included in the stiffness matrix. Depending on the loading considered, a nonsymmetric stiffness matrix is then otained. The logic is a little strange - he's saying that you could neglect those terms, and the iterations would still converge, although that would take more iterations. On the other hand, including the terms will make each iteration more expensive. – OskarM Dec 9 '17 at 16:55
• Yes. There is another line above equation 6.83 where he says that one could integrate the load step of time t+dt on the volume of the previous time step. So he still accounts for the changing load but since the stiffness matrix is not exactly consistent with the force, the convergence would be slower. – user26248 Dec 9 '17 at 17:04
• I settled this - it's in the original paper - Bathe, Klaus‐Jürgen, Ekkehard Ramm, and Edward L. Wilson. "Finite element formulations for large deformation dynamic analysis." International Journal for Numerical Methods in Engineering 9.2 (1975): 353-386. Indeed, the authors suggest omitting the load-dependent terms from the stiffness matrix. – OskarM Apr 24 '18 at 20:23