Tunable signal penetration and response plateaus in bistable mechanical media

Dynamically processing mechanical signals is crucial for soft robotics and mechanosensing, where classical viscoelastic materials lack intrinsic tunability. We show that internal bistability actively controls the response and signal attenuation in mechanical (meta)materials. In our model, bistable elements switch discretely with a predefined timescale between states distinguished by potential energy $ε$, equilibrium length $Δl$, and spring constant $Δk$. The system is simulated via microscopic Brownian dynamics coupled to Poisson switching with rate $ν$, and described macroscopically by a nonlinear continuum field theory. Crucially, the model yields closed-form analytical solutions for the linear response and spatial penetration depth, revealing two phenomena: a universal screening mechanism (akin to the electrostatic 'skin effect') reducing spatial signal penetration when the driving frequency exceeds the internal relaxation rate, and a frequency-insensitive response plateau from timescale separation. The screening length is controlled primarily by the conformational length change $Δl$, while the attenuation regime and plateau are tuneable via the switching rate $ν$. A systematic parameter study exposes a fundamental design trade-off: larger $Δl$ strengthens dissipation but raises the energy barrier for state transitions, eventually causing state-locking where damping vanishes. Optimal attenuation thus requires a compromise between pronounced bistability and a surmountable barrier. Due to its analytical tractability, our framework provides explicit design rules for fine-tuning the adaptive response of bistable media. It applies to diverse experimental systems-from biopolymers to synthetic catch bonds and metamaterials-enabling the predictive engineering of intelligent soft matter for frequency-selective signal processing.