Dynamic stability of biaxially strained thin sheets under high strain-rates: response to local perturbations

Abstract

This work pertains to the stability of structures under rapid loading rates when inertia is taken into account. In contrast to the widely used approach in the relevant literature, which is based on the method of modal analysis to determine the structure’s fastest growing eigenmode—meaningful only for cases where the velocity of the perfect structure is significantly lower than the associated characteristic wave propagation speeds—the present study analyzes the time-dependent response to spatially localized perturbations of the transient (time-dependent) states of these structures, in order to understand the initiation of the corresponding failure mechanisms. We are motivated by the experimental studies of Zhang and Ravi-Chandar (Int J Fract 142:183-217, 2006), Int J Fract 163:41-65, (2010) on the high strain-rate expansion of thin rings and tubes, which show no evidence of a dominant wavelength in their failure mode and no influence of strain-rate sensitivity on the necking strains. Recently, Ravi-Chandar and Triantafyllidis (Int J Solids Struct 58:301-308, 2014) studied the dynamic stability of an incompressible, nonlinearly elastic bar at different strain-rates by following the evolution of localized small perturbations introduced at different times. The same approach is followed here for the biaxial stretching of thin plates, where we follow the time evolution of spatially localized perturbations and their interactions. The nonlinear time evolution of a such a perturbation is studied numerically and it is shown that these structures are stable until the time when the condition for the loss of ellipticity is reached. An analytical method, based on linearization, is used to define the size of the influence zone of a point-wise perturbation and we study its dependence on constitutive laws and loading conditions.