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Abstract This work theoretically investigates the strain-mediated magnetic domain wall propagation in a magnetostrictive layer, which is perfectly attached to a thick piezoelectric layer under the combined mechanism of nonlinear viscous and dry friction dissipation. The mathematical model of this study is formulated within the framework of the one-dimensional Extended Landau-Lifshitz-Gilbert equation, considering the simultaneous action of the applied magnetic and electric fields, nonlinear dissipations and piezo-induced strains. We use the traveling wave ansatz to derive analytical expressions of the key features like threshold, breakdown, and domain wall velocity in steady and precessional regimes. More precisely, our prime focus is to examine how nonlinear viscous dissipation affects the dynamics of domain walls in both isotropic and anisotropic magnetostrictive materials. The velocity profile of the magnetic domain wall becomes nonlinear due to the inclusion of the nonlinear viscous dissipation, which also significantly affects the Walker breakdown limit, thus leading to the expansion of the steady-state regime. Finally, we present a numerical illustration of the obtained analytical results, which agree well with recent observations. [ABSTRACT FROM AUTHOR]