Institut für Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany.
Adv Mater. 2011 Jun 17;23(22-23):2633-7. doi: 10.1002/adma.201004094.
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)proptophin and G(phi)proptophim 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.
通过基于体素的有限元计算分析无序网络固体的有效线性弹性模量。我们分析了由泊松-沃罗诺伊过程和共聚焦显微镜成像的胶原蛋白纤维网络结构给出的网络固体。通过调整纤维半径来改变固体体积分数 ϕ，同时保持基础网络的结构网格或孔径固定。对于中等的 ϕ，体积模量和剪切模量通过经验幂律 K(phi)proptophin 和 G(phi)proptophim 来近似，其中 n≈1.4，m≈1.7。胶原蛋白和泊松-沃罗诺伊网络固体的指数相似，并且接近于之前基于体素的有限元研究中泊松-沃罗诺伊系统不同边界条件下发现的 n=1.22 和 m=2.11 的值。然而，这些经验幂律的指数与 n=1 和 m=2 的解析值不一致，这对于在薄梁极限下的低密度多孔结构有效。我们提出了一种 K(ϕ)的函数形式，该形式模拟了从低密度的幂律到高密度多孔固体的交叉；对该函数形式的数据拟合得到渐近指数 n≈1.00，这是预期的。此外，泊松-沃罗诺伊过程的强度和样品中的胶原蛋白浓度都会改变典型的孔径或网格尺寸，它们只会通过固体体积分数的变化来影响有效模量。这些发现表明，具有胶原蛋白网络结构的网络固体可以通过泊松-沃罗诺伊过程进行定量建模。
