Topological surface states are unique to topological materials and are immune to disturbances. In isostatic lattices, mechanical topological floppy modes exhibit softness depending on the polarization relative to the terminating surface. However, in three dimensions, the polarization of topological floppy modes is disrupted by the ubiquitous mechanical Weyl lines. Here, we demonstrate, both theoretically and experimentally, the fully polarized topological mechanical phases free of Weyl lines. Floppy modes emerge exclusively on a particular surface of the three-dimensional isostatic structure, leading to the strongly asymmetric stiffness between opposing boundaries. Additionally, uniform soft strains can reversibly shift the lattice configuration to Weyl phases, switching the stiffness contrast to a trivially comparable level. Our work demonstrates the fully polarized topological mechanical phases in three dimensions, and paves the way towards engineering soft and adaptive metamaterials.

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• Classical mechanics
• Elasticity
• Phase transitions
• Surface states
• Topological phases of matter
• 3-dimensional systems