Abstract

Active deformations in biological tissues, such as those driven by growth or contraction, are commonly modeled through a multiplicative decomposition of the deformation gradient. While this framework is well established in forward problems, the inverse task of reconstructing the original configuration from a deformed state remains largely unexplored.

We study three inverse formulations aimed at recovering the reference configuration: algebraic removal of the active tensor (growth removal formulation), reversed application of the full deformation sequence (reverse path formulation), and a variational approach enforcing equilibrium under a reversed active input (inverse formulation). Analytical results show that the first two methods fail to recover the undeformed state due to incompatibility and noncommutativity. The variational formulation, though mechanically consistent, does not ensure reversibility either. Numerical simulations on twenty cases with random small-magnitude active tensors reveal a wide range of reconstruction errors depending on the structure of the active field. We attribute these discrepancies to geometric incompatibilities and path dependence in elastic relaxation.

Our results highlight fundamental limitations of inverse reconstruction and suggest that recovering a true reference configuration from shape data may require additional constraints, regularization, or data-driven inference.

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