Geometry-defined tensile deformation sensing via electromagnetic phase delay in a liquid-metal conduit

Reliable measurement of tensile deformation in soft and stretchable systems is challenging because elongation in practical settings is rarely isolated and is commonly accompanied by localized bending, compression, and temperature/humidity variations. Conventional strain sensors infer deformation from changes in electrical response amplitude, which are sensitive to such local perturbations and, therefore, provide ambiguous readout under complex deformation states. Here, tensile deformation is measured through the electromagnetic phase delay accumulated by a guided wave propagating along a deformable liquid-metal conduit. The phase delay depends on the total propagation length, establishing a direct correspondence between phase and global geometric elongation. Localized bending, compression, and moderate temperature variations affect the conduit only over confined regions and, therefore, contribute negligibly to the path-integrated phase delay, allowing the readout to reflect global elongation. The phase-based measurement exhibits a highly linear response to tensile deformation (R2 > 0.998), supports flexible definition of the reference state, and enables stable length monitoring on pre-strained, curved, and dynamically deforming surfaces. This approach provides a physical route for length measurement in soft systems under complex deformation states.