I am recently informed about the large deformation theory, and its concepts like curvilinear coordinates. But so far I understand in an updated Lagrangian formulation, the reference configuration is the last converged configuration and in the incremental setting the deformation between the last converged configuration and the current configuration would be small, given the increment can be controlled in such a way. Then why would we need to use the large deformation theory and its concepts in an updated Lagrangian setting ?
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Good question. Not sure I have a definitive answer for you, but, a couple of things come to mind.
Boundary conditions might be specified in the reference configuration, so you'd have to use large def theory to map them into the current config. That's one place you might need to use large deformation theory.
Even though you solved it in the current configuration, you may want to map the results back to the reference configuration for analyis. That's another place you will need to use large deformation theory.
Edit: Also, I remember a statement from a set of notes I used: The stiffness matrices (consistent tangents) from the updated Lagrangian and the total Lagrangian formulation are identical.
It took me a while to believe in this statement.
