We combine experiments with simulations to investigate the fluid-structure interaction of a flexible helical rod rotating in a viscous fluid, under low Reynolds number conditions. Our analysis takes into account the coupling between the geometrically nonlinear behavior of the elastic rod with a nonlocal hydrodynamic model for the fluid loading. We quantify the resulting propulsive force, as well as the buckling instability of the originally helical filament that occurs above a critical rotation velocity. A scaling analysis is performed to rationalize the onset of this instability. A universal phase diagram is constructed to map out the region of successful propulsion and the corresponding boundary of stability is established. Comparing our results with data for flagellated bacteria suggests that this instability may be exploited in nature for physiological purposes. DOI: 10.1103/PhysRevLett.115.168101 PACS numbers: 47.63.Gd, 46.32.+x, 46.70.Hg, 47.63.mf Bacteria often rely on the deformation of filamentary helical structures, called flagella, for locomotion [1]. The propulsion arises from a complex fluid-structure interaction between the structural flexibility of the flagellum and the viscous forces generated by the flow. This fluid-structure interaction may lead to geometrically nonlinear deformations [2,3], which in turn can be exploited for turning [4], tumbling [5], bundle formation [6], and polymorphic transformations [7,8].Resistive force theories (RFT) [9,10] are often used to model the role of viscous forces on flexible filaments [11], at low Reynolds number. These simplify the viscous loading by introducing local geometry-dependent drag coefficients. More sophisticated descriptions consider nonlocal hydrodynamic effects, albeit typically assuming that the filament is rigid such that elastic forces are ignored [12,13]. The few studies that have coupled long-range hydrodynamics with elasticity either assume small deflections [13] or approximate the filament as a network of springs [14,15], thereby oversimplifying the mechanics of the problem. One exception is the study of buckling of a straight elastic filament loaded by viscous stresses [16]. More recently, a systematic computational study has been performed on a discretized model based on Kirchhoff's theory for elastic rods (in the form of a chain of connected spheres), coupled with RFT [3]. This study was significant in that it was the first, to the best of our knowledge, to report a series of buckling instabilities of the flagellum that arise during locomotion and suggested its relevance to the biological system. Moreover, it addressed the important rotation-translation coupling. However, recent experiments [12,17] Here, motivated by the locomotion of uniflagellated bacteria, we perform a combined experimental and numerical investigation of the dynamics of a helical elastic filament rotated in a viscous fluid. Our goal is to predictively understand the underlying mechanical instabilities. In our precision model experiments, we reproduce and syste...