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.