Rigidity Loss in Disordered Systems: Three Scenarios

Abstract

We reveal significant qualitative differences in the rigidity transition of three types of disordered network materials: randomly diluted spring networks, jammed sphere packings, and stress-relieved networks that are diluted using a protocol that avoids the appearance of floppy regions. The marginal state of jammed and stress-relieved networks are globally isostatic, while marginal randomly diluted networks show both overconstrained and underconstrained regions. When a single bond is added to or removed from these isostatic systems, jammed networks become globally overconstrained or floppy, whereas the effect on stress-relieved networks is more local and limited. These differences are also reflected in the linear elastic properties and point to the highly effective and unusual role of global self-organization in jammed sphere packings. DOI: 10.1103/PhysRevLett.114.135501 PACS numbers: 62.20.D-, 63.50.Lm, 64.60.ah Disordered elastic networks and sphere packings represent a large class of amorphous athermal materials, ranging from (bio)polymer networks to granular media and foams [1][2][3]. Random networks of springs lose their rigidity when enough springs are cut; this random bond dilution process is known as rigidity percolation (RP) [4][5][6][7][8]. Packings of soft spheres do the same when their confining pressure is lowered towards zero: This is called (un)jamming [9][10][11][12][13]. These rigidity loss scenarios have been studied extensively, in particular, for the simplest cases of networks of harmonic springs [7,8] or soft frictionless harmonic spheres [10][11][12][13]. In that case, the linear elastic properties of packings can be mapped to those of a spring network, where each contact is replaced by the appropriate spring [14][15][16]. Lowering the pressure, the number of bonds in the equivalent network decreases.Given this close correspondence, it is surprising that the nature of the RP and unjamming transitions, and of their respective marginally rigid states, are significantly different. For packings of a large number (N) of soft spheres, extensive studies have shown that (i) the connectivity, i.e., the average number of contacts z per particle, goes to z c ¼ 2D þ Oð1=NÞ at the marginal point, where D is the space dimension [3,[9][10][11][12][13][17][18][19][20], (ii) the system remains homogeneously jammed up to the point of unjamming (with the exception of individual loose particles called rattlers or very rare small particle clusters) [10], and (iii) the shear modulus G vanishes as Δz ≔ z − z c whereas the bulk modulus K remains finite when Δz → 0 [9-14]. In contrast, in the rigidity percolation of generic networks, extensive studies have revealed that for large systems (i) the connectivity z, which gives the average number of springs per node, approaches z c ¼ 3.9612… < 2D for the bond diluted triangular network [7,8], (ii) the largest rigid cluster takes on a heterogeneous, fractal shape, and (iii) both the shear modulus G and bulk modulus K smoothly vanish at the critical point in a way typ...

Rigidity percolation by next-nearest-neighbor bonds on generic and regular isostatic lattices

“…This is strikingly similar to the phenomena in jamming as reported in Ref. [50]. In addition, our density argument for the first-order-like transition in Sec.…”

Rigidity and auxeticity transitions in networks with strong bond-bending interactions

“…These objects are akin to the so-called states of self stress studied intensively in the context of the jamming transition [2,11,25,26] and the physics of topological metamaterials [27,28]. Simple counting arguments [11,24,26] suggest that in a system of size N with coordination z there are N (z − z c ) orthonormal modes |φ .…”

The Principle of Independent Bond-Level Response: Tuning by Pruning to Exploit Disorder for Global Behavior

“…This is the condition for local stability of a particle in the original jammed packing [23]. As the excess coordination number decreases, the bulk and shear moduli vanish together, so that G ∼ B ∼ ΔZ [4,21,22,24] [see Fig. 2(b)].…”

“…First, we consider the known case of rigidity percolation [4,21,22], where a bond is picked at random and removed. This pruning is repeated until the system becomes unstable at ΔZ ¼ 0.…”