Incremental Deformations and Bifurcation of Elastic Solids Reinforced by Fibers With Intrinsic Extensional, Flexural, and Torsional Elasticity

Abstract

We present a model for incremental deformations of an elastic solid reinforced by a single family of fibers that offer resistance to extension, flexure, and torsion. The theory is cast in the setting of small-on-large deformations and provides a framework for the multiscale analysis of bifurcation of equilibria in fibrous composites. The model is based on a theory of three-dimensional Cosserat elasticity in which fiber kinematics are controlled by a rotation field that is weakly coupled to the bulk deformation through a pointwise fiber materiality constraint. Fiber-matrix interaction forces are explicitly accounted for via the attendant Lagrange multipliers. We demonstrate the utility of the model by investigating the onset of bifurcation in an incompressible fiber-reinforced elastic half-plane. In particular, we study the influence of axial fiber stiffness, flexural stiffness and fiber-matrix interaction forces on planar buckling modes. We envisage a model for the study of buckling problems of biological and industrial relevance with a view to gaining better insight into the roles of fiber bending, twisting, and fiber-matrix interaction forces in regulating the buckling of fibrous composites.