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We study the elastic properties of phantom networks of Gaussian and nearly Gaussian springs. We show that the stress tensor of a Gaussian network coincides with the conductivity tensor of an equivalent resistor network, while its elastic constants vanish. We use a perturbation theory to analyze the elastic behavior of networks of slightly non-Gaussian springs. We show that the elastic constants of phantom percolation networks of nearly Gaussian springs have a power-law dependence on the distance of the system from the percolation threshold, and we derive bounds on the exponents.

Elasticity of Gaussian and nearly Gaussian phantom networks