We characterize homogeneous patterns, their stability, and phase transitions in nematic liquid crystal polymers (LCPs) with imposed elongational flows. We combine the flow-induced analysis of order parameters by See et al. [J. Chem. Phys. 92, 792 (1990)], Bhave et al. [J. Rheol. 37, 413 (1993)], Rey [Macromol. Theory Simul. 4, 857 (1995)], and Wang [J. Non-Newtonian Fluid Mech. 22, 147 (1997)], with the pure nematic, full tensor analysis of Shimada et al. [J. Chem. Phys. 88, 7181 (1988)]. To make contact with these seminal studies, we select a moment-averaged Doi kinetic model for flows of rod-like nematic LCPs with a quartic short-range intermolecular potential; the connection with alternative kinetic and continuum models for flows of LCPs is noted. New elongation-induced director instabilities are revealed for patterns previously identified as candidates for stable pattern selection. From a full tensor analysis, we determine the complete phase diagram for homogeneous patterns in the parameter space of LCP concentration and elongation rate. With respect to experimental predictions, in axial extension, biaxial patterns exist but they are all unstable and the only stable patterns are uniaxial; in planar extension, above a moderate concentration the only stable nematic patterns are biaxial. © 2000 American Institute of Physics.