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The application of variational principles for analyzing problems in the physical sciences is widespread. Cantilever-like problems, where one end is subjected to a fixed value and the other end is free, have been less studied in terms of their stability despite their abundance. In this article, we develop the stability conditions for these problems by examining the second variation of the energy functional using the generalized Jacobi condition. This involves computing conjugate points that are determined by solving a set of initial value problems from the linearized equilibrium equations. We apply these conditions to investigate the nonlinear stability of intrinsically curved elastic cantilevers subject to a tip load. Kirchhoff rod theory is employed to model the elastic rod deformations. The role of intrinsic curvature in inducing complex nonlinear phenomena, such as snap-back instability, is particularly emphasized. This snap-back instability is demonstrated using various examples, highlighting its dependence on various system parameters. The presented examples illustrate the potential applications in the design of flexible soft robot arms and innovative mechanisms. |
