Motivated by recent strain-limiting models for solids and biological fibers,
we introduce the first intrinsic set of nonlinear constitutive relations,
between the geometrically exact strains and the components of the contact force
and contact couple, describing a uniform, hyperelastic, strain-limiting special
Cosserat rod. After discussing some attractive features of the constitutive
relations (orientation preservation, transverse symmetry, and monotonicity), we
exhibit several explicit equilibrium states under either an isolated end thrust
or an isolated end couple. In particular, certain equilibrium states exhibit
Poynting like effects, and we show that under mild assumptions on the material
parameters, the model predicts an explicit tensile shearing bifurcation: a
straight rod under a large enough tensile end thrust parallel to its center
line can shear.

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