Spatial structure of states of self stress in jammed systems

Abstract

States of self stress, organizations of internal forces in many-body systems that are in equilibrium with an absence of external forces, can be thought of as the constitutive building blocks of the elastic response of a material. In overconstrained disordered packings they have a natural mathematical correspondence with the zero-energy vibrational modes in underconstrained systems.While substantial attention in the literature has been paid to diverging length scales associated with zero-and finite-energy vibrational modes in jammed systems, less is known about the spatial structure of the states of self stress. In this work we define a natural way in which a unique state of self stress can be associated with each bond in a disordered spring network derived from a jammed packing, and then investigate the spatial structure of these bond-localized states of self stress. This allows for an understanding of how the elastic properties of a system would change upon changing the strength or even existence of any bond in the system. * dsussman@sas.upenn.edu 1 arXiv:1601.03416v1 [cond-mat.soft]

Directional mechanical response in the bulk of topological metamaterials

“…It has already been shown that even in regular lattices topological polarization can become wavenumberdependent [47] and can acquire, e.g., half-integer values in the original formulation as the crystalline symmetry of the lattice is reduced [17]. Self stresses in jammed packings have been shown to have a length scale that diverges as the isostatic point is approached from above [53][54][55], consistent with the infinitely long-range self stresses in our crystal lattices. Other isostatic systems without any lattice symmetry should have significant response to bond swelling, and one could examine rigidity percolation models representing fibrous systems [56], disordered versions of triangulated origami [19], etc.…”

Quasilocalized states of self stress in packing-derived networks

“…The identification of c as the relevant lengthscale is further established in Fig. 2b, where we test the scaling suggested in [31,34] by plotting r 4 φ 2 (r) √ p vs. rp 1/3 to find a clear misalignment of the data. Up to this point we have established that the normalization factors c α of QLS (see Eq.…”

Topological boundary modes in jammed matter

“…The argument, which is supported by numerical results, assumes that the characteristic volume of the dipolar response field scales as 1/∆Z, where ∆Z is the distance of the lattice from the jamming transition, and that the magnitude of the response is of similar magnitude everywhere in this volume and does not decay strongly with distance. That the response field has a volume that scales as 1/∆Z is supported both by direct studies of the response to bond elongation [60] and also recent work looking at the SSS associated with the existence of a given bond in a hyperstatic packing [61]. The evidence that the magnitude of the dipolar response is roughly constant in this volume may need further study, as there is evidence that typical states of self stress have a (stretched-) exponentially decaying spatial profile in the volume of interest.…”