Geometry and the onset of rigidity in a disordered network

Abstract

Disordered spring networks that are undercoordinated may abruptly rigidify when sufficient strain is applied. Since the deformation in response to applied strain does not change the generic quantifiers of network architecture - the number of nodes and the number of bonds between them - this rigidity transition must have a geometric origin. Naive, degree-of-freedom based mechanical analyses such as the Maxwell-Calladine count or the pebble game algorithm overlook such geometric rigidity transitions and offer no means of predicting or characterizing them. We apply tools that were developed for the topological analysis of zero modes and states of self-stress on regular lattices to two-dimensional random spring networks, and demonstrate that the onset of rigidity, at a finite simple shear strain ${\gamma }^{\star }$, coincides with the appearance of a single state of self stress, accompanied by a single floppy mode. The process conserves the topologically invariant difference between the number of zero modes and the number of states of self stress, but imparts a finite shear modulus to the spring network. Beyond the critical shear, we confirm previously reported critical scaling of the modulus. In the sub-critical regime, a singular value decomposition of the network's compatibility matrix foreshadows the onset of rigidity by way of a continuously vanishing singular value corresponding to nascent state of self stress.

arXiv comment6 pages, 6 figues

References in corpus (13)

• Topological Boundary Modes in Isostatic Lattices
• Modeling semiflexible polymer networks
• Elasticity of Stiff Polymer Networks
• On the Rigidity of Amorphous Solids
• A coordination-based approach to elasticity of floppy and stiff random networks
• Strain-controlled criticality governs the nonlinear mechanics of fibre networks
• Isostaticity, auxetic response, surface modes, and conformal invariance in twisted kagome lattices
• Non-affine response: jammed packings versus spring networks
• Length-Controlled Elasticity in 3D Fiber Networks
• Jammed frictionless discs: connecting local and global response
• Strain-driven criticality underlies nonlinear mechanics of fibrous networks
• Fiber networks below the isostatic point: fracture without stress concentration
• Curvature-induced cross-hatched order in two-dimensional semiflexible polymer networks

Cited by in corpus (23)

• A geometrically controlled rigidity transition in a model for confluent 3D tissues
• A minimal-length approach unifies rigidity in under-constrained materials
• Anomalous glassy dynamics in simple models of dense biological tissue
• Scaling theory for mechanical critical behavior in fiber networks
• Conformational Control of Mechanical Networks
• Rigidity-Controlled Crossover: From Spinodal to Critical Failure
• Stress-stabilized sub-isostatic fiber networks in a rope-like limit
• Finite size effects in critical fiber networks
• Non-affine deformation of semiflexible polymer and fiber networks
• Athermal Fracture of Elastic Networks: How Rigidity Challenges the Unavoidable Size-Induced Brittleness
• Shear-induced phase transition and critical exponents in 3D fiber networks
• Stiffening of under-constrained spring networks under isotropic strain
• Mechanics of fiber networks under a bulk strain
• Non-affinity and fluid-coupled viscoelastic plateau for immersed fiber networks
• Scaling theory of critical strain-stiffening in disordered elastic networks
• Mechanical criticality of fiber networks at a finite temperature
• Mean field fracture in disordered solids: statistics of fluctuations
• Self-stresses control stiffness and stability in overconstrained disordered networks
• Structural features and nonlinear rheology of self-assembled networks of cross-linked semiflexible polymers
• Effective Medium Theory for Mechanical Phase Transitions of Fiber Networks
• The strain-stiffening critical exponents in polymer networks and their universality
• Rigidity transitions in zero-temperature polygons
• Viscoelastic scaling regimes for marginally-rigid fractal spring networks