Emergence of fibrous fan morphologies in deformation directed reformation of hyperelastic filamentary networks

Abstract

Recently, the authors generalized a theory for modelling the scission and reforming of crosslinks in isotropic polymeric materials to include materials in which elastic fibers are embedded in an elastic matrix. The fibers were assumed to dissolve with increasing deformation and then to immediately reassemble in a direction defined as part of the model. The model was illustrated in detail for uniaxial stretching along the direction of the fibers. Fiber reassembly was along the original fiber direction and did not result in a change in fiber alignment. The present work examines the implications of this model when the direction of reassembly is uncorrelated with the original fiber direction. In particular, the fibers are assumed to reassemble in the direction of maximum principal stretch of the matrix. The specific case is treated when the deformation is simple shear and the initial fiber direction is perpendicular to the direction of shear. The resulting fiber elongation with increasing shear results in fiber dissolution over a constitutively determined interval of the amount of simple shear. Newly formed fibers align in the current principal direction of maximum stretch, which is a direction that changes with the amount of simple shear. The resulting interval of alignment angles generates a fan-like fiber morphology at each material point. The formation and structure of the fan is described. In addition, the relation between the shear and normal stresses and the amount of shear is discussed, both during loading and unloading. It is shown that there can be a state of permanent set that is related to the original shape by triaxial extension and shear.

Fiber Remodeling During Torsion of a Fiber Reinforced Hyperelastic Cylinder—Unloading Behavior

“…Thus, prior to microstructural change, W in (2.3) is taken to be in the form W = W m + W f where W m is the matrix energy storage density and W f is the fiber energy storage density. In the present treatment, as in [2][3][4], there is dissolution and reassembly of the fibrous component while the matrix in which the fibers are embedded does not undergo a change in its mechanical properties. Cauchy stress is taken to be in the form…”

“…Both fibrous and matrix components are present in a representative volume element in the continuum treatment, so that quantities of energy and stress are regarded as appropriate homogenized quantities. As in [2][3][4] these homogenized quantities are assumed to involve separately identifiable matrix and fiber contributions. Thus, prior to microstructural change, W in (2.3) is taken to be in the form W = W m + W f where W m is the matrix energy storage density and W f is the fiber energy storage density.…”

“…In previous work [2][3][4], we introduced and studied a constitutive theory for fiber reinforced hyperelastic materials that allows the fibers to undergo microstructural changes. In this theory, increasing deformation of the matrix leads to increasing stretch of the fibers that causes their gradual dissolution.…”

“…The dissolving fibers reassembled in their initial direction. Homogeneous shear in the direction normal to the fibers was studied in [3]. At each instant during increasing shear, dissolved fibers reassembled in the direction of maximum principal stretch of the matrix, which direction differs from that of the fibers.…”

“…The combined axial stretch and twist of a solid hyperelastic circular cylinder reinforced by a single family of helically wound fibers is formulated in Sect. 3. This is specialized in Sect.…”