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The asymptotic properties of random strength and compliance of single-walled carbon nanotubes using atomistic simulation

Mechanical response of deformable bodies is often concerned with either the sum or the extreme of an underlying random process. This paper investigates the asymptotic statistical properties of ultimate strength (σu) and compliance (C) of single-walled nanotubes (SWNTs) containing random defects using the technique of atomistic simulation (AS). The defects considered are of the Stone–Wales (SW) kind and a Matern hard-core random field applied on a finite cylindrical surface is used to describe the spatial distribution of the SW defects. A nanotube can be viewed as consisting of nominally identical segments of equal length possessing a stationary distribution of ultimate strength, σu. Under a weak dependence condition among the segment strengths (that decay to zero with increasing distance between the segments), consistent with the non-local nature of atomic interactions, formalized here in the form of strong mixing, the asymptotic properties of σu (as the extreme of the strong mixing sequence) and C (as the sum of a related strong mixing sequence) are studied with increasing tube length, l. The extremal index, measuring the stochastic dependence in the strength field, is estimated. We simulate a set of displacement controlled tensile loading up to fracture of (6, 6) SWNTs with length between 49 and 492 Å. With increasing l, the distribution of σu is found to shift to the left and become narrower and appears to fit the Weibull distribution rather well; the compliance of the tube increases with increasing l and becomes asymptotically normal. The compliance and strength of the tube are found to become asymptotically uncorrelated. These results appear to validate the strong mixing property of the strength field.