Elasticity of Floppy and Stiff Random Networks

We study 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 (average number of springs per node) relative to that of a marginally rigid network $\delta z$: a floppy network has $\delta z<0$, while a stiff network has $\delta z>0$. Under the influence of an externally applied load, we observe that the response of both floppy and stiff networks is controlled by the critical point corresponding to the onset of rigidity. We use numerical simulations to compute the exponents which characterize the shear modulus, the heterogeneity of the response, and the network stiffening as a function of $\delta z$ and derive these theoretically, thus allowing us to predict aspects of the mechanical response of glasses and fibrous networks.

Figure 1

(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 $z=z_c=4.0$, and the shear strain $\gamma =0.005$. (c) Zooming in shows the presence of large correlated rotational deformations.

Figure 2

(a) Dimensionless stress-strain curves for $z=3.8$, 4.0, and 4.2. (b) Critical strain ${\gamma }^{*}$ vs $\delta z\equiv z-4$. Each point averages over 2 configurations. (c) Log-log plot of dimensionless stress-strain curve for $z=4$. (d) Rescaled stress-strain curves for $z=3.0$, 3.2, 3.4, 3.6, 3.8, 3.9, 3.96, 4.2, 4.4, 4.6, 4.8, and 4.99 show that both floppy and stiff networks can be described in terms of the relative coordination $\delta z$.

Figure 3

(a) The relative nonaffine displacement per unit strain as defined in the text $\delta V_{\mathrm{n}.\mathrm{a}.}^{\perp }$ for $\gamma ={\gamma }^{*}/10$ and (b) the dimensionless shear modulus $G$ vs $-\delta z$, in the floppy regime, for $k_w={10}^{-5}$. The variation in (c) the dimensionless shear modulus $G$ and (d) $\delta V_{\mathrm{n}.\mathrm{a}.}^{\perp }$ vs $\delta z\in [-1,1]$, with $k_w=1/300$, ${\rho }_w=4.1$. Note the large particle displacement rate near $\delta z=0$.