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Dynamic stability of structures under high loading rates by localized perturbation approach

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Of interest here is the stability of structures subjected to high loading rates where inertia is taken into account. The approach currently used in the literature to analyze these stability problems is the method of modal analysis which considers all eigenmodes of the structure and determines the fastest growing one, thus selecting the corresponding wavelength as the critical one that pertains to the structure’s failure mode. This method of analysis is meaningful only when the velocity of material points in the perfect structure is significantly lower than the associated characteristic wave propagation speeds. The novel idea proposed here is to analyze the time-dependent response to perturbations of the transient (high strain rates) states of these structures, in order to understand the initiation of the corresponding failure mechanisms. Three examples will be presented: the first pertains to axially strained bars, motivated by the experimental studies of [1] on the high strain rate extension of thin rings, that show no evidence of a dominant wavelength in their failure mode and no influence of strain-rate sensitivity on the necking strains. In the interest of analytical tractability, we study the extension of a 1D incompressible, nonlinearly elastic bar at different strain rates by following the evolution of localized small perturbations introduced at different times. The second example deals with an externally pressurized ring, a structure that is already unstable under static loading. Experiments on electromagnetically compressed rings by [2] show an irregular failure pattern with no dominant wavenumber, in contrast to the static case where an ellipsoidal mode is the critical one at the onset of bifurcation. The ring’s stability is studied by following the evolution of a localized small perturbation. For small values of the applied loading rate, the structure fails through a global mode, while for large values of the applied loading rate the structure fails by a localized mode of deformation. The third example consists of a thin elasto-plastic membrane under rapid biaxial stretching. A perturbation is placed at the center of the sheet and its stability is studied by following the time-evolution of this perturbation. For as long as the sheet has not reached conditions of loss of ellipticity, the amplitude of perturbation decays with time, indicating that the sheet is dynamically stable. By investigating the times associated with the onset of loss of ellipticity along a particular propagation direction and with the loss of ellipticity along all possible propagation directions, one can establish influence zones for a localized perturbation. Similar techniques can also estimate the ductility increase in a sheet by studying the time necessary for the signal of a perturbation to reach the sheet’s boundary. These results have been calculated for a number of different elasto-plastic constitutive laws.

Work in collaboration with: K. Ravi-Chandar, T. Putelat and G. Wen

[1] Zhang, H. and Ravi-Chandar, K. (2006). On the dynamics of necking and fragmentation – I. real-time and post-mortem observations in al 6061-O. International Journal of Fracture, 142:183–217.
[2] Mainy, A. (2012). Dynamic buckling of thin metallic rings under external pressure. Master’s thesis, University of Texas.

This talk is part of the Engineering - Mechanics and Materials Seminar Series series.

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