Reachability along finite load paths

Determine whether the set of finite displacements reachable along finite load paths in a hyperelastic mechanical network is larger than the set characterized by the tangent-response reciprocity bound.

Background

The paper establishes the reciprocity constraint for the tangent response about any pre-loaded state because the tangent stiffness is a Hessian and therefore symmetric. This governs small-signal responses around deformed configurations, including large-strain states tested numerically.

The finite-load input–output map is different from the tangent map. It remains unresolved whether nonlinear motion along a finite loading path can access responses outside the tangent-response reachable set.

References

The finite-path question stays open. The map from a finite load to a finite displacement is not the tangent map, and whether the set reachable along a load path is larger is open.

— Reciprocity can halve what a mechanical network can learn  (2609.04169 - Vu et al., 3 Sep 2026) in Section 3, subsection “Numerical verification,” paragraph “And it survives in the tangent response about a deformed state”; Limitations