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Using stability analysis of discrete elastic systems to study the buckling of nanostructures

Korobeynikov S. N., Alyokhin V. V., Annin B. D., Babichev A. V.

Archives of Mechanics

2012

Vol. 64, nr 4

367-404

Stability/instability criteria of discrete elastic systems are used to study the buckling of nanostructures. The deformation of nanostructures is simulated by solving the nonlinear equations of molecular mechanics. The external forces applied to the nanostructure are assumed to be dead (that is the directions of their action remain constant throughout nanostructure deformation). We note that the positive-definiteness property of the tangential stiffness matrix of a nanostructure is a universal sufficient stability criterion for both equilibrium states and quasi-static/dynamic motions of the nanostructure. The equilibrium configurations are stable in Lyapunov's sense, and quasi-static/dynamic motions are stable in a finite time interval t "isin" (0, Tcr) in which the positive-definiteness property of this matrix is preserved. For dynamic motions of nanostructures, the stability property in this time interval follows from Lee’s criterion of quasi-bifurcation of solutions of second order ODEs. The non-positive definiteness of the tangential stiffness matrix of nanostructures at a time t > Tcr corresponds to both unstable equilibrium configurations and unstable dynamic motions. Computer procedures for determining the critical time and buckling mode(s) are developed using this criterion and are implemented in the PIONER FE code. This code is used to obtain new solutions for the deformation and buckling of twisted (10, 10) armchair and (10, 0) zigzag single-walled carbon nanotubes