Length scale dependent elasticity in random three-dimensional fiber networks

Abstract

The mechanics of fibrous materials are complicated by complex deformation mechanisms that result from the discrete fibrous structure. Though it is known that fibrous materials deform primarily by rotating and bending of fibers, the implications of rotating and bending on the mechanics of the network are not fully clear. Previous studies have investigated some effects of the fiber rotation and bending, observing fibers to rotate into directions of maximum principal stress, resulting in strain stiffening, and fibers under compression to buckle, resulting in compression softening. Nonlinear constitutive models have recently been developed to account for these deformation mechanisms, but the classical constitutive models that account for only stress and strain cannot fully account for fiber rotations and bending. Here, we interpret the mechanics of fibrous materials through micropolar elasticity, also called Cosserat elasticity, which differs from classical elasticity in that it accounts for local moments caused by rotation of points within a material. The resulting equations can be written in terms of characteristic lengths that cause stress to depend on both strain and the length scale of the material. We simulated three-dimensional networks of fibers and observed a strong effect of length scale on the stiffness of networks in bending with a more mild effect in torsion. The length-scale dependence of stiffness is consistent with micropolar elasticity and can be described in terms of characteristic lengths of the material. Factors affecting the characteristic length were investigated by altering the fiber density, alignment, and bending stiffness. Although density was found to have no effect on characteristic length, increased fiber alignment led to a decrease in characteristic length, and increased fiber bending stiffness led to an increase in characteristic length. These findings suggest that the characteristic length is increased by factors that increase bending moments supported by the fibers. Although our parameter study manipulated the magnitude of characteristic length, no combination of parameters gave a characteristic length of zero, indicating that the mechanics of fibrous materials depend on length scale.