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Post-buckling involving large deflection of micro-cantilever using the consistent couple stress theory

Abstract

The paper investigates the post-buckling behavior of micro-cantilevers within the framework of consistent couple stress theory. The theoretical foundation for carrying out the analysis is formulated in terms of moment and curvature which is apt for solving thin beam problems. A differential equation governing the deflection of micro-cantilever subjected to different boundary conditions is obtained from the developed moment-curvature relationship. Axially loaded micro-cantilevers are solved by using an efficient semi-analytical method which involves removal of singularity. On the other hand solutions to eccentrically loaded micro-cantilevers are determined with the help of purely numerical approach based on iterative shooting method. Profiles of post-buckled micro-cantilevers subjected to different loading conditions are furnished. The obtained results for the micro-cantilever display strong size-dependency due to presence of the material length scale parameter in the developed model. Also the non-classical theory results tend to those from classical theory when the length scale parameter is negligible as compared to the characteristic length of the structure.

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Notes

  1. The eccentric load is replaced by an equivalent system comprising of an axial force (with direction parallel to x-axis) and a couple (both the couple forces act in a direction parallel to the x-axis) with moment \(M_0=Pecos\phi _0\) (consideration of large deflections makes the perpendicular distance between the two couple forces to be \(e\; cos\phi _0\))

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Acknowledgements

The authors are grateful to Dr. Debojyoti Pandit for the valuable discussions. Also, we would like to thank the Ministry of Human Resource Development, Government of India for providing scholarship.

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Correspondence to Bhakti N. Patel.

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Patel, B.N., Srinivasan, S.M. Post-buckling involving large deflection of micro-cantilever using the consistent couple stress theory. Ann. Solid Struct. Mech. 12, 155–163 (2020). https://doi.org/10.1007/s12356-020-00058-4

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Keywords

  • Post-buckling
  • Micro-cantilever
  • Couple stress theory
  • Moment-curvature relationship