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(φ) α φn and G(φ) α φm 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. The dependence of linear-elastic properties on effective density is studied for porous network solids, by voxel-based finite element methods. The same dependence is found for solid structures derived from Poisson-Voronoi processes and from confocal microscopy images of collagen scaffolds. We recover the power-law for the bulk modulus for low densities and suggest a functional form for the cross-over to a high-density porous solid

Beyond the percolation universality class: the vertex split model for tetravalent lattices

Wepropose a statistical model defined on tetravalent three-dimensional lattices in general and the three-dimensional diamond network in particular where the splitting of randomly selected nodes leads to a spatially disordered network, with decreasing degree of connectivity. The terminal state, that is reached when all nodes have been split, is a dense configuration of self-avoiding walks on the diamond network. Starting from the crystallographic diamond network, each of the four-coordinated nodes is replaced with probability p by a pair of two edges, each connecting a pair of the adjacent vertices. For all values 0 ⩽ p ⩽ 1the network percolates, yet the fraction fp of the systemthat belongs to a percolating cluster drops sharply at pc = 1 to a finite value fp c . This transition is reminiscent of a percolation transition yet with distinct differences to standard percolation behaviour, including a finitemass f > 0 p c of the percolating clusters at the critical point. Application of finite size scaling approach for standard percolation yields scaling exponents for p → pc that are different from the critical exponents of the second-order phase transition of standard percolation models. This transition significantly affects the mechanical properties of linear-elastic realizations (e.g. as custom-fabricated models for artificial bone scaffolds), obtained by replacing edges with solid circular struts to give an effective density ϕ. Finite element methods demonstrate that, as a low-density cellular structure, the bulkmodulus Kshows a cross-over froma compression-dominated behaviour, K (ϕ) ∝ ϕκ with κ ≈ 1, at p = 0 to a bending-dominated behaviour with κ ≈ 2 at p=1

Tuning elasticity of open-cell solid foams and bone scaffolds via randomized vertex connectivity

Tuning mechanical properties of and fluid flow through open-cell solid structures is a challenge for material science, in particular for the design of porous structures used as artificial bone scaffolds in tissue engineering. We present a method to tune the effective elastic properties of custom-designed open-cell solid foams and bone scaffold geometries by almost an order of magnitude while approximately preserving the pore space geometry and hence fluid transport properties. This strong response is achieved by a change of topology and node coordination of a network-like geometry underlying the scaffold design. Each node of a four-coordinated network is disconnected with probability p into two two-coordinated nodes, yielding network geometries that change continuously from foam- or network-like cellular structures to entangled fiber bundles. We demonstrate that increasing p leads to a strong, approximately exponential decay of mechanical stiffness while leaving the pore space geometry largely unchanged. This result is obtained by both voxel-based finite element methods and compression experiments on laser sintered models. The physical effects of randomizing network topology suggest a new design paradigm for solid foams, with adjustable mechanical properties