Asymptotic of 'rigid-body' motions for nonlinear dynamics: physical insight and methodologies

The purpose of the present work is to show that an adequate basis for understanding the essentially nonlinear phenomena must also be essentially nonlinear however still simple enough to play the role of a basis. It is shown that such types of 'elementary' nonlinear models can be revealed by tracking the hidden links between analytical tools of analyses and subgroups of the rigid-body motions or, in other terms, rigid Euclidean transformation. While the subgroup of rotations is linked with linear and weakly nonlinear vibrations, the translations with reflections can be viewed as a geometrical core of the strongly nonlinear dynamics associated with the so-called vibro-impact behaviors. It is shown that the corresponding analytical approach develops through non-smooth temporal substitutions generated by the impact models.Comment: Presented at 12th DSTA Conference, December 2-5, 2013 {\L}\'od\'z, Polan

Vibrations of an elastic cylindrical shell near the lowest cut-off frequency

A new asymptotic approximation of the dynamic equations in the two-dimensional classical theory of thin-elastic shells is established for a circular cylindrical shell. It governs long wave vibrations in the vicinity of the lowest cut-off frequency. At a fixed circumferential wavenumber, the latter corresponds to the eigenfrequency of in-plane vibrations of a thin almost inextensible ring. It is stressed that the well-known semi-membrane theory of cylindrical shells is not suitable for tackling a near-cut-off behaviour. The dispersion relation within the framework of the developed formulation coincides with the asymptotic expansion of the dispersion relation originating from full two-dimensional shell equations. Asymptotic analysis also enables refining the geometric hypotheses underlying various ad hoc set-ups, including the assumption on vanishing of shear and circumferential mid-surface deformations used in the semi-membrane theory. The obtained results may be of interest for dynamic modelling of elongated cylindrical thin-walled structures, such as carbon nanotubes

Mechanical Properties of Glassy Polyethylene Nanofibers via Molecular Dynamics Simulations

The extent to which the intrinsic mechanical properties of polymer fibers depend on physical size has been a matter of dispute that is relevant to most nanofiber applications. Here, we report the elastic and plastic properties determined from molecular dynamics simulations of amorphous, glassy polymer nanofibers with diameter ranging from 3.7 to 17.7 nm. We find that, for a given temperature, the Young’s elastic modulus E decreases with fiber radius and can be as much as 52% lower than that of the corresponding bulk material. Poisson’s ratio ν of the polymer comprising these nanofibers was found to decrease from a value of 0.3 to 0.1 with decreasing fiber radius. Our findings also indicate that a small but finite stress exists on the simulated nanofibers prior to elongation, attributable to surface tension. When strained uniaxially up to a tensile strain of ε = 0.2 over the range of strain rates and temperatures considered, the nanofibers exhibit a yield stress σy between 40 and 72 MPa, which is not strongly dependent on fiber radius; this yield stress is approximately half that of the same polyethylene simulated in the amorphous bulk.DuPont MIT AllianceDuPont (Firm) (Young Professor Award

Three-dimensional energy channeling in the unit-cell model coupled to a spherical rotator II: unidirectional energy channeling

This work is the second one in a two-part series devoted to the analysis of complex nonlinear mechanism of energy channeling emerging in a locally resonant three-dimensional, unit-cell model, and the current paper considers unidirectional energy channeling. The considered system comprises an external mass subjected to a symmetric three-dimensional linear local potential with an internal spherical rotator. The present study specifically focuses on the analysis of three-dimensional, dissipative mechanism of irreversible (unidirectional) energy transport across mutually orthogonal directions realized in the limit of low-energy excitations. In particular, this study unveils the special transient regimes of three-dimensional partial and complete transformation of in-plane vibrations of the external element to out-of-plane vibrations. Similar to the results reported in the first part of the series, this three-dimensional energy flow is fully governed by the motion of the internal spherical rotator coupled to the external mass. Analysis of this peculiar response regime is based on regular multi-scale asymptotic analysis resulting in a reduced order dissipative slow-flow model. Results of the analysis are substantiated by the numerical simulations of the full model.by K. R. Jayaprakash and Yuli Starosvetsk