Molecular simulations are carried out to determine the spinodal of crystalline poly(p-phenylene terephthalamide) (PPTA) with respect to temperature and compressive axial stress. The simulations show that the temperature of the spinodal decreases with increasing compressive axial stress, and vice verse. This coupling of the effects of temperature and stress occurs because both increased temperature and compressive axial stress lead to axial contraction in the crystal, and the spinodal corresponds to an Euler buckling instability that occurs when the axial lattice parameter falls below a critical value. The simulation results for the spinodal are very similar to the experimental results for the temperature-dependent compressive strength of Kevlar fibers (which are composed of semicrystalline PPTA), suggesting that the compressive failure of Kevlar fibers arises from the spinodal of the PPTA crystal phase, rather than a defect-mediated mechanism.