As far as I have seen, solving problems of nonlinear elasticity using the finite element method proceeds by linearizing, either around the initial configuration (total Lagrangian approach) or around the most recently solved configuration (updated Lagrangian approach). This produces a system of linear equations, and represents an example of the Newton-Raphson method.
My question is about possible generalizations to other methods of solving systems of nonlinear equations. I realize that for engineering problems, the linearization may not cause any problems, because the goal is often to study deformations starting from a known, unstressed configuration, which can then be gradually loaded. In other words, there is a good small parameter, so linearization is fine. However, I'm interested in the case that external forces (loadings) depend on the deformation, and the dependence can exhibit strong nonlinearities. In this case, the global solution may not be anywhere near the undeformed configuration.
I think that attempts to address this problem are known as "globalization strategies", and I'm interested in the application of these to nonlinear elasticity. I know about methods like arc-length control, but I'm interested in other methods such as full multigrid.
I don't know enough about the lingo or the literature to even search the literature properly - so far my naive attempts have not yielded anything. I also don't know good examples of problems with strong nonlinearities that might be solved with finite element - maybe Navier-Stokes? I don't even see how to properly formulate the problem as a system of nonlinear equations - do I just numerically integrate the principal of virtual work?