Elasticity of floppy and stiff random networks

Citation

M. Wyart, H. Liang, A. Kabla and L. Mahadevan

Physical Review Letters 101:215501 (2008)

Abstract

We study the role of connectivity on the linear and nonlinear elastic behavior of amorphous systems using a two-dimensional random network of harmonic springs as a model system. A natural characterization of these systems arises in terms of the network coordination relative to that of an isostatic network δz; a floppy network has δz < 0, while a stiff network has δz > 0. Under the influence of an externally applied load we observe that the response of both floppy and rigid network are controlled by the same critical point, corresponding to the onset of rigidity. We use numerical simulations to compute the exponents which characterize the shear modulus, the amplitude of nonaffine displacements, and the network stiffening as a function of δz, derive these theoretically and make predictions for the mechanical response of glasses and fibrous networks.

Figure sample

(a) Stretching a zigzag chain of springs that are freely hinged costs no energy until they are aligned. (b) A random network of 10 000 particles, when subjected to shear, exhibits a heterogeneous response. The coordination is z=zc=4.0, and the shear strain γ=0.005. (c) Zooming in shows the presence of large correlated rotational deformations.