Mechanics of Anisotropic Spring Networks

Collaborators

T. Zhang$^{1}$, J. M. Schwarz$^{1}$, and Moumita Das$^{2}$

$^{1}$Department of Physics, Syracuse University, Syracuse, New York 13244, USA
$^{2}$School of Physics and Astronomy, Rochester Institute of Technology, Rochester, New York 14623, USA

Motivation

The onset of rigidity in disordered spring networks has been studied as a model for elasticity in disordered materials. Such a model undergoes a phase transition from not-rigid to rigid at some critical fraction of springs (bonds). Numerical simulations on a bond-diluted triangular lattice suggest that the transition is a continuous one with the percolating rigid cluster having a fractal dimension at the transition. To date, one of the few theoretical tools to analyze rigidity percolation is effective medium theory (EMT). We now extend this theoretical framework – the EMT and the numerical simulations – to a disordered spring network with anisotropy. Anisotropic networks abound in nature and man-made materials, and are an important class of soft matter. Consider, for example, an orientationally ordered, but spatially disordered, network made of liquid crystal elastomers. The interplay between the gel-sol transition and the isotropic-nematic transition in both synthetic and biological systems has been studied. Consider also a layered, but disordered, system of granular particles. What are the macroscopic elastic properties of such a system? Some properties have been analyzed in the ordered case, but the disordered case is more complex and has received less attention. Finally, the cytoskeleton, the filamentous scaffolding that provides most animal cells their shape and rigidity, can consist of rather oriented, cross-linked filaments such as in lamellipodia, the broad, thin protrusion at the leading edge of a crawling cell.

While we focus on extending the analytical framework of effective medium theory to disordered linear spring networks with anisotropy, the rigidity transition in “super-elastic” anisotropic central-force networks has been studied in prior work by Roux and Hansen and Wang and Harris. In these “super-elastic” networks, all bonds in a preferred direction are occupied with springs, and bonds in other directions have infinite rigidity with a probability p and a finite rigidity with a probability 1 − p. In addition to determining the rigidity percolation threshold, the notion of splay rigidity, in which only rotational degrees of freedom are frozen out, and a mapping to a random resistor network for the bulk modulus is discussed. Some of their results can be extended to the usual bond-diluted system that we study here. However, there remain open questions about the interplay between anisotropy and rigidity in these systems. In particular, one can ask how does the difference in the directional occupation probability of springs influence the mechanical response of the network, and how does this couple to the direction of the applied deformation?

Approach

To answer these questions, we investigate a triangular lattice based anisotropic bonddiluted network and study how the anisotropy in the occupation of the springs influences the ability of the network to bear stresses using an effective medium theory and numerical simulations.

Our model consists of a triangular lattice with two different bond occupation probabilities, px and py, for the linear springs. We develop an effective medium theory (EMT) to describe the network elasticity as a function of px and py. We find that the onset of rigidity in the EMT agrees with Maxwell constraint counting.

Plot of the phase diagram according to mean field constraint counting argument, with the inset showing the shear modulus G as a function of px and py obtained from the EMT
Plot of the phase diagram according to mean field constraint counting argument, with the inset showing the shear modulus G as a function of px and py obtained from the EMT

We also find beyond linear behavior in the shear and bulk modulus as a function of occupation probability in the rigid phase for small strains, which differs from the isotropic case. We compare our EMT with numerical simulations to find rather good agreement.

The shear modulus $G$ as a function of $p_y$ for different $p_x$ as shown in the legend. The open symbols show data from the simulations and thelled symbols (joined by solid lines in the inset) represent the result from the effective medium theory. The inset shows the same data on a log-linear scale. The system size in the simulation is $N_x=N_y=128$.
The shear modulus $G$ as a function of $p_y$ for different $p_x$ as shown in the legend. The open symbols show data from the simulations and the lled symbols (joined by solid lines in the inset) represent the result from the effective medium theory. The inset shows the same data on a log-linear scale. The system size in the simulation is $N_x=N_y=128$.
The area bulk modulus $K$ as a function of $p_y$ for different $p_x$ as shown in the legend in units of the bare spring constant $\gamma$ (set to unity). The open symbols show data from the simulations and the filled symbols (joined by solid lines in the inset) represent the results from the effective medium theory. The system size in the simulation is set to $N_x=N_y=128$ and the modulus is calculated from part of the network within a box in the center of the system with box size $N_x'=N_y'=108$.
The area bulk modulus $K$ as a function of $p_y$ for different $p_x$ as shown in the legend in units of the bare spring constant $\gamma$ (set to unity). The open symbols show data from the simulations and the filled symbols (joined by solid lines in the inset) represent the results from the effective medium theory. The system size in the simulation is set to $N_x=N_y=128$ and the modulus is calculated from part of the network within a box in the center of the system with box size $N_x'=N_y'=108$.

Acknowledgments

T.Z. and J.M.S. acknowledge support from the Soft Matter Program at Syracuse University, and M.D. acknowledges support from the RIT College of Science via a D-RIG grant. J.M.S. and M.D. also acknowledge the hospitality of the Aspen Center for Physics (supported by NSF PHY-1066293), where some preliminary discussions on the project took place.

Tao Zhang
Tao Zhang
Special Research Fellow

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