Homogenization of random elastic networks with non‐affine kinematics

Abstract

This contribution concerns the mechanics of materials with random network microstructures. It develops a general homogenization approach that allows to link the microscopic deformation of fibers in the disordered network with the macroscopic response of the continuous solid. This link is established by a novel micro-macro relation based on the kinematics of maximal advance paths that constrains the unknown microscopic stretch of fibers with respect to the macroscopic strain. This relation accounts for the topology of the network, in particular, its connectivity and takes for the tetrafunctional networks a clearly interpretable tensorial form. In line with the principle of the minimum averaged free energy the elastic response of the network is obtained by the relaxation of the variable fiber stretch subjected to the kinematic constraint. Network microstructures are commonly encountered in materials of artificial as well as natural origin. Elastomers, hydrogels and soft biological tissues, non-woven fabrics are all on the microscopic level composed of elongated one-dimensional elements, the fibers. When these soft materials are subjected to a macroscopic strain, the underlying microstructure undergoes a peculiar deformation [1]. One can expect that in the case when the material is stretched in a certain direction the fibers initially oriented closely to it will accordingly elongate. A straightforward assumption of affine kinematics is taken by the classical models of rubber elasticity [2,3] and has again been recently adopted for the modeling of biological gels [4]. Nevertheless, as pointed out in [5], the affine stretch assumption is not generally valid and does not explain the experimentally observable relation of the elastic stress upon the strain like the difference in stiffening of rubber-like materials in uniaxial and equibiaxial tension. The the non-affine microsphere model proposed in [5] suggests certain variations of stretch constrained by a specific relation to the macroscopic deformation. The present work is inspired by the approach to the homogenization of elastic networks developed in [5]. The main contribution is the novel construction of the kinematic constraints [6,7] and an averaged description of the network. The fibers are identified by their initial orientation λ 0 uniformly distributed over the unit sphere S 0 . The deformation of the network is described by a vector-valued stretch function λ(λ 0 ). The variation of this unknown microstretch λ is constrained with respect to the macroscopic strain by a kinematic relation established by considering the maximal advance paths. At each junction on the path attached to f fibers with the orientations λ i 0 as shown in Fig. 1 it propagates by the one of the remaining f − 1 fibers with the maximal advance in a certain direction l 0 . Such fibers in the path have orientation vectors λ m 0 with the distributionThe higher is the functionality of the network the more aligned is the path so that the average advance in the path ξ m = (f − 2)/f with ξ m = λ ...

Analysys of Mechanical Properties of Non-Woven Materials Depending on Their Microstructure

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