Journal of Elasticity 49: 1–30, 1997. 1 c 1997 Kluwer Academic Publishers. Printed in the Netherlands. Remarks on the Behavior of Simple Directionally Reinforced Incompressible Nonlinearly Elastic Solids G.Y. QIU and T.J. PENCE Department of Materials Science and Mechanics, Michigan State University, East Lansing, MI 48824-1226, U.S.A. Received 29 January 1997; in revised form 30 June 1997 Abstract. The effect of directional reinforcing in generating qualitative changes in the mechanical response of a base neo-Hookean material is examined in the context of homogenous deformation. Single axis reinforcing giving transverse isotropy is the major focus, in which case a standard reinforcing model is characterized by a single constitutive reinforcing parameter. Various qualitative changes in the mechanical response ensue as the reinforcing parameter increases from the zero- value associated with neo-Hookean response. These include (in order): the existence of a limiting contractive stretch for transverse-axis tensile load; loss of monotonicity in off-axis simple shear; loss of monotonicity in on-axis compression; loss of positivity in the stress-shear product in off-axis simple shear; and loss of monotonicity for plane strain in on-axis compression. The qualitative changes in the simple shear response are associated with stretch relaxation in the reinforcing direction due to finite rotation. Mathematics Subject Classifications (1991): 73B05, 73B40, 73G05. Key words: incompressible nonlinear elasticity, transverse isotropy. 1. Introduction The effect of directional reinforcement on the mechanical response of highly deformable elastic bodies can be modeled macroscopically in the context of anisotropic nonlinear elasticity, in which case the nature of the anisotropy fol- lows from the reinforcing details. Reinforcement by continuous fibers is a standard example, in which case the multiplicity, directionality and packing of fibers gener- ates the symmetry class for the macroscopic continuum description. For example, as is the case in the linear theory, transverse isotropy follows for the case of a single fiber family under either hexagonalor random packing [1]. These two packing types are sufficient to generate macroscopic rotational symmetry in the cross-section for the fiber family under consideration, in which case a macroscopic description for the elastic strain energy density can be posed in terms of a base strain energy density for the unreinforced material, augmented by energetic penalization for stretching in the directions in which reinforcing is supplied. Such energy functions are expressible in terms of the strain invariants associated with the material symme-