Plane Deformations of Incompressible Fiber-Reinforced Materials

A continuum theory of finite, plane deformations of composites consisting of materials reinforced by strong fibers is discussed. The composite is assumed to be incompressible, and the fibers are treated as inextensible and continuously distributed. The analysis is not restricted to any particular material behavior such as elasticity, plasticity, or visco-elasticity. Plane deformations are kinematically determinate, in that the deformation can be found by using the constraint conditions and suitable displacement boundary conditions. The reactions to the constraints produce stress equilibrium in any kinematically admissible deformation. The theory admits stress singularities of an unusual kind: a single fiber or normal line can carry a finite load. Simple examples illustrating this and other points of the theory are given.