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Post-buckling and Snap-through Behavior of Inclined Slender Beams

In MEMS fields, a need arises in engineering practice to predict accurately the nonlinear response of slender post-buckling beams, especially the nonlinear transverse stiffness. The bistability of the post-buckling beams is excellent in reducing power consumption of micro-devices or micro-systems. However, the major difficulty in analyzing the post-buckling and snap-through response is the intractability of the geometric nonlinear control equations of large deflection beams.

The post-buckling analysis of the easily fabricated MEMS beams with both ends fixed is really important in the design of micro structures. To solve the post-buckling problems of small deformation beams, the predominant approach is to use a nonlinear finite element (FE) model, or to resort to a linear analytical model based on the small deflection theory, which agrees only with experiments in a relatively limited range of loadings. The apex displacement of the bent-beam can only be reduced in nearly linear proportion to the vertical force in the design of microactuators  and the error is large compared with the experimental results. This lack of a simple and yet accurate tool for analyzing the post-buckling bent beams results in a poor initial “guess” of the desired geometry and multi-design iterations, thus being unable to provide an insight into the deformation problems. Therefore, the large deflection buckling theory is needed to solve the intractability of the geometric nonlinear control equations of the

Aiming at designing a novel MEMS threshold acceleration switch with post-buckling beams, we study the post-buckling behavior in the snap-through process of the large deflection inclined beam with both ends fixed, and establishes the nonlinear governing equations of the post-buckling beam under combined forces acting on the ending point. These ordinary non-linear differential equations consist of the boundary-value conditions, in which six unknown functions are contained and the length of the deformed beam is considered as one of the unknown functions. By using the implicit compatibility conditions to express the nonlinear statically indeterminate problems of elastic beams, the strongly nonlinear equations formulated in terms of elliptic integrals are directly solved in the numerical sense. Through an incremental displacement method, we can obtain the equilibrium paths of the deformed beam in the snap-through process, and describe explicitly the nonlinear force-and-displacement relations of the central point of the bent-beam structure. For further applications of the nonlinear stiffness of the post-buckling beams, an electronic testing device is designed. The simulation results are in good agreement with those by experiments.

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Zhigang Suo's picture

Interesting problem.  Would you be willing to attach a pre-print of this work?

Thank you very much for your comments!

The work about the post-buckling and snap through behavior has been attached in my blog.

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