Geometry and the onset of rigidity in a disordered network

Area of Science:

• * Physics
• * Materials Science
• * Network Mechanics

Background:

• * Disordered spring networks exhibit undercoordination, making them susceptible to abrupt rigidity changes under applied strain.
• * Traditional mechanical analyses (e.g., Maxwell-Calladine count, pebble game) fail to predict or characterize these geometric rigidity transitions.
• * The underlying cause of this rigidity is geometric, not solely dependent on network topology (nodes and bonds).

Purpose of the Study:

• * To investigate the geometric origin of rigidity transitions in undercoordinated disordered spring networks.
• * To apply topological analysis tools to predict and characterize these transitions in 2D random spring networks.
• * To understand the mechanical behavior and elastic properties of networks beyond the critical strain.

Main Methods:

• * Application of topological analysis tools for zero modes and states of self-stress, adapted from regular lattices.
• * Analysis of 2D random spring networks subjected to finite simple shear strain.
• * Use of singular value decomposition on the network's compatibility matrix in the subcritical regime.

Main Results:

• * Rigidity onset at a critical shear strain (γ*) coincides with the appearance of one state of self-stress and one floppy mode.
• * The transition conserves the topological invariant difference between zero modes and self-stress states.
• * Networks develop a finite shear modulus and anisotropic elastic properties post-transition, with enhanced resistance to deformation along the shear direction.
• * Critical scaling of the differential shear modulus is confirmed.
• * A vanishing singular value in the compatibility matrix foreshadows the rigidity onset.

Conclusions:

• * Geometric factors, not just network topology, govern rigidity transitions in disordered spring networks.
• * Topological analysis tools can predict and characterize these transitions, revealing the interplay of self-stress and floppy modes.
• * The study provides a framework for understanding the anisotropic mechanical response of networks after rigidification.