In this paper, two new methods are proposed to study wave propagation in materials with constitutive law that have nonlinear terms. In the first method, the gauge transformation is used to derive the dynamic shape function. A perturbation method is then applied in order to derive an equation for the wavenumber. The influence of the nonlinearity takes the form of a dependence of the wavenumber on the magnitude of the corresponding frequency component. Under the small amplitude and weak nonlinearity assumptions of the perturbation method, the wavenumber is incorporated into the spectral finite element method (SFEM). The second approach is a numerical method based on alternating frequency-time (AFT) iterations. The nonlinear term represented as a residual nonlinear force term is reduced through the alternating iterations between the time-domain and the frequency-domain. Finally, response behaviors under impact loading predicted with these methods are studied and compared to equivalent linear response behavior.

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