Morphology and Linear‐Elastic Moduli of Random Network Solids

The effective linear‐elastic moduli of disordered network solids are analyzed by voxel‐based finite element calculations. We analyze network solids given by Poisson‐Voronoi processes and by the structure of collagen fiber networks imaged by confocal microscopy. The solid volume fraction ϕ is varied by adjusting the fiber radius, while keeping the structural mesh or pore size of the underlying network fixed. For intermediate ϕ , the bulk and shear modulus are approximated by empirical power‐laws $ K(\phi) \propto \phi^n$ and $ G(\phi) \propto \phi^m$ with n ≈ 1.4 and m ≈ 1.7. The exponents for the collagen and the Poisson‐Voronoi network solids are similar, and are close to the values n = 1.22 and m = 2.11 found in a previous voxel‐based finite element study of Poisson‐Voronoi systems with different boundary conditions. However, the exponents of these empirical power‐laws are at odds with the analytic values of n = 1 and m = 2, valid for low‐density cellular structures in the limit of thin beams. We propose a functional form for K ( ϕ ) that models the cross‐over from a power‐law at low densities to a porous solid at high densities; a fit of the data to this functional form yields the asymptotic exponent n ≈ 1.00, as expected. Further, both the intensity of the Poisson‐Voronoi process and the collagen concentration in the samples, both of which alter the typical pore or mesh size, affect the effective moduli only by the resulting change of the solid volume fraction. These findings suggest that a network solid with the structure of the collagen networks can be modeled in quantitative agreement by a Poisson‐Voronoi process.