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In this paper, by using classical Faedo-Galerkin approximation andcompactness method, the existence of martingale solutions for the stochastic 3DNavier-Stokes equations with nonlinear damping is obtained. The existence anduniqueness of strong solution are proved for $\beta > 3$ with any $\alpha>0$and $\alpha \geq \frac12$ as $\beta = 3$. Meanwhile, a small time largedeviation principle for the stochastic 3D Navier-Stokes equation with dampingis proved for $\beta > 3$ with any $\alpha>0$ and $\alpha \geq \frac12$ as$\beta = 3$.

Stochastic 3D Navier-Stokes equations with nonlinear damping: martingale solution, strong solution and small time large deviation principles