A general stress decomposition (GSD) method is suggested to analyze the nonlinear behavior in arbitrary nonlinear oscillatory experiments. This method is based on the symmetric properties of different harmonics. For shear stress, it is found that odd harmonics can be separated into a viscous and elastic part, respectively, while even harmonics has a viscoelastic character. The GSD method is validated by a model wave, and applied in nonlinear oscillatory shear of viscous shear thinning fluid, Bingham material, viscoelastic polymers, polymer blends via theoretical analysis and polymer composites via an experiment. The nonlinear behaviors of these materials are studied by the GSD method to show characteristic scaling relations. The GSD method has an advantage of determination the contribution of even harmonics readily by the decomposed stresses under either transient or steady flow conditions.