Two Efficient Approaches for Elastica Problem of Nonlinear Elastic Bars

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Abstract
An elastica post‐buckling response of a uniform, axially compressed, pinned bar of nonlinearly elastic “soft” material is presented. A stability analysis is performed using two approximate analytical approaches based on a strongly nonlinear bending moment‐curvature relationship. The results are compared with those of an “exact” solution, achieved using the Runge Kutta's technique, showing the reliability, range of applicability, and efficiency of both approaches. Conditions of bifurcational instability, depending on the material nonlinearity, are established. It is found that a critical material nonlinearity exists, depending on the slenderness ratio, for which the mechanism of bifurcational buckling changes from stable to unstable and vice versa. Thus, a bar with nonlinear elastic “soft” material might be imperfection‐sensitive. An extensive variety of numerical results, also allowing the study of the effects of various parameters on the post‐buckling response of the bar, proves the validity and efficiency of the proposed approaches.

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