Homogenized elastic response of random fiber networks based on strain gradient continuum models

Abstract

The purpose of this work is to develop anisotropic strain gradient linear elastic continuum models for two-dimensional random fiber networks. The constitutive moduli of the strain gradient equivalent continuum are assessed based on the response of the explicit network representation in so-called windows of analysis, in which each fiber is modeled as a beam and the fibers are connected at crossing points with welded joints. The principle of strain energy equivalence based on the extension to the strain gradient of the Hill–Mandel macro homogeneity condition is employed to identify the classical and strain gradient moduli, based on the application of a sequential set of polynomial displacements on windows of analysis of different sizes. The scaling of the first- and second-order moduli with network parameters, such as network density and the ratio of fiber bending to axial stiffness, is determined. We observe a similar dependency of classical and strain gradient moduli on the same network parameters. The internal length scales associated with the gradient coefficients of the constitutive equation are also defined in terms of the network parameters. The strain gradient moduli prove to be size-independent in the affine regime, and they converge toward a size-independent value in the non-affine deformation regime after a rescaling of physical dimensions by the window size. The obtained results show that the strain gradient moduli scale uniformly with the square of the magnitude of the strain gradients applied to the window of analysis.