Background
Metamaterials with stiffness tunability have demonstrated great potential in mechanical systems that operate in variable environments. Real-time stiffness visualization, which translates the stiffness information into digital outputs, allows users to know the exact stiffness after tuning without the need for external sensors, and thus holds great application potential in human-robot interaction and adaptive robotic systems. However, most existing stiffness-tunable metamaterials exhibit high-dimensional and nonlinear structure-property relationships, which hinder precise on-demand tuning and real-time stiffness visualization.
Now, a research team of Professor Yan Chen from the Tianjin University presents a reconfigurable hierarchical metamaterial, establishes a linear relationship between its stiffness and the number of active hinges, and embeds mechanical logic circuits into the metamaterial. This enables a wide range of tunable stiffness and real-time stiffness visualization without the need for external sensors, paving the way for intelligent systems with self-sensing and adaptive capabilities.

Research Progress
To address this challenge, researchers conducted a kinematic analysis of a planar four-bar linkage unit composed of four triangular blocks. Leveraging this unit's kinematic bifurcation behavior, researchers tessellated the units in a plane to create a reconfigurable hierarchical metamaterial. This metamaterial can switch between various single- and multi-level configurations while maintaining a single degree of freedom (DOF) throughout its motion paths, except at bifurcation points. The hierarchical levels of the metamaterial can be defined based on the size of the triangular blocks within the four-bar linkage units. As illustrated in Fig. 1, a 2×2 metamaterial can be reconfigured into single-level configurations comprising four level 0 units, two level 1 units, or one level 2 unit. It can also form a multi-level configuration with two level 0 units and one level 1 unit. During reconfiguration, hinges enclosed within larger blocks remain stationary and are considered inactive. Consequently, the number of active hinges varies dynamically during this process, enabling a wide range of tunable stiffness. Through theoretical modeling, researchers revealed a linear relationship between the stiffness of the metamaterial and the number of active hinges, provided that the rotational stiffness of each hinge is identical. This relationship has been confirmed experimentally. For instance, when the metamaterial size reaches 4×4, there are 86 configurations, ranging from 6 to 94 active hinges, resulting in a tunability ratio of 15.67.
To visualize the stiffness of the hierarchical metamaterial in real time, researchers integrate reconfigurable circuits in the metamaterial to map the transitions among units at different levels to the multichannel electrical outputs. This establishes a linear relationship between structure, properties, and information, enabling the metamaterial to display the number of active hinges and stiffness in various configurations without the need for external sensors. The experimental results show that the number of active hinges as the 4×4 metamaterial reconfigures among 10 configurations can be accurately visualized by the lit LEDs and the 7-segment display (Fig. 1), which demonstrate the reliability of the proposed design for real-time stiffness visualization in hierarchical metamaterials without external sensors.

Future Prospects
This research establishes a linearized structure-property relationship by linking the stiffness of the metamaterial to the number of active hinges. It further integrates mechanical logic circuits into the metamaterial, mapping different hierarchical configurations to LED-visualized stiffness values. This integration of structure, property, and information enables a wide range of tunable stiffness and real-time stiffness visualization without external sensors, paving the way for intelligent systems with self-sensing and adaptive capabilities.

The complete study is accessible via DOI:10.34133/research.0874