Effective Medium Theory for Rigidity Percolation and Jamming Transitions

Effective Medium theory (EMT) has proven to be a powerful tool for the study of properties of random electronic and elastic media. Though it does not provide (except in some cases) accurate predictions of critical exponents, it does provide detailed and often sophisticated predictions of qualitative behavior, including crossover functions near mechanical critical points. The jamming transition from a floppy state with vanishing bulk and shear moduli to a rigid state is characterized by a discontinuous jump in the bulk modulus but a smooth increase of the shear modulus, whereas the rigidity-percolation transition from the same floppy state is characterized by continuous growth of both the bulk and shear moduli. An often-noted weakness of the EMT treatment is that it generally predicts rigidity-percolation but not jamming transitions. This talk will introduce a series of hybrid periodic lattices composed of two (or more) sub-lattices with shared sites connected by central-force springs and exhibiting distinct elastic properties. The EMT and numerical simulations of spring-diluted versions of these model systems consisting of a honeycomb lattice decorated with next-nearest neighbor bonds yields a Jamming-like critical point terminating a line of rigidity percolation transitions and (to my knowledge) the first analytical expression for the discontinuous growth of the Bulk modulus and crossover at the jamming-rigidity-percolation multi-critical point. If time permits, some results for rigidity transitions in model systems with bending forces at nodes as well as central forces between sites will be reviewed. 

References: 

Liarte, Mao, Stenull, and TCL, PRL 122, 128006 (2019). 

D. B. Liarte, O. Stenull, T. C. Lubensky, PRE 101, 063001 (2020).

Part of the NSCS webinar series

https://softmatterisrael.wixsite.com/nscs