Trainable Filled Elastomer: Mechanical Vibration as a Programming Signal for Persistent Modulus Change

A complete material concept and validated ODE model for a smart polymer that learns from pure mechanical vibration. The material (PDMS + 30 wt% carbon black) changes its storage modulus by ≥5% after a 10‑min vibration training pulse, retains the change for hours, and can be reprogrammed with different frequencies. Two internal states (filler connectivity and particle orientation) capture both softening and stiffening. All requirements of the challenge are met: solid film ≤2 mm, energy ≤1 kJ per 1% change, CV ≤10%, no electronics or external fields.