Loss of ellipticity of the governing differential equations and the mechanical response of fiber-reinforced nonlinearly elastic materials under plane deformation are related. Loss of ellipticity may be associated with the onset for failure. Hence, the relation allows us to capture the failure behavior of the materials at hand by examining their response curves in plane strain uniaxial stress in the fiber direction. Previous work established that fiber compression failure is expected under any circumstances since loss of ellipticity cannot be avoided. Here, it is established that the loss of ellipticity of the governing differential equations (and hence failure) under uniaxial tensile loading in the fiber direction for the materials at hand requires the loss of convexity of the Cauchy stress-stretch curve in tension. Furthermore, if the Cauchy stress-stretch curve in tension is non-monotonic, then ellipticity has been lost at a previous deformation. If the fiber reinforcement gives the material only additional stiffness in the fiber direction, then, a convex response curve is sufficient to avoid fiber-reinforced failure in fiber extension. This result is linked to reinforced polymers. If the fiber reinforcement introduces sufficiently large additional stiffness in the composite shear modulus, then the result is limited to fiber-reinforced failure behavior in the fiber direction. This result is linked to matrix failure in fiber composites. Parallel results are established for the nominal stress-stretch curve in tension. Both, incompressible and compressible materials are considered.
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