A new theoretical approach which describes the formation of banded textures in liquid crystalline polymers is introduced. The approach incorporates the concept of extended curvature elastic free energy in describing the response of the director field to strain recovery. Physically, this corresponds to accounting for an additional energetic penalty for deformation of the director field over very short distances, resulting from the natural persistence of orientation in rigid polymer chains. Both analytical and numerical results indicate that periodic patterns are spontaneously generated whenever the magnitude of the second-order gradient terms exceeds a critical value. Numerical investigations of two-dimensional systems reveal that elastic anisotropy has a significant effect on the aspect ratio and orientation of the patterns formed.