Stability and Localization of Deformation Delay in Finitely Strained Plates at Arbitrary Strain-Rates

Of interest here is the stability and associated deformation localization of structures when inertial effects are considered. Concerned primarily with the study of necking failure patterns, the prevailing approach in the relevant literature uses the “modal analysis” method to find the wavelength of the structure’s fastest growing eigenmode, an approach that often uses a rate-dependent material response. However, the experimental studies of (Zhang and Ravi-Chandar in Int. J. Fracture 142: 183–217, 2006; Zhang and Ravi-Chandar in Int. J. Fracture 163: 41–65, 2010) on the high strain-rate expansion of thin rings and tubes, show no evidence of a dominant wavelength in their failure mode and no influence of strain-rate sensitivity on the necking strains. Moreover, modal analysis assumes that at all times the entire structure sees the applied eigenmode perturbation in spite of the physical limitation of a finite wave propagation speed. In addition, the closely related problem of stability in dynamically loaded structures, i.e., the time evolution of perturbations introduced at different times during loading, does not seem to have attracted attention. Based on the above-mentioned experimental and theoretical observations, (Ravi-Chandar and Triantafyllidis in Int. J. Solids Struct. 58: 301–308, 2015) proposed a “localized perturbation” approach to study the dynamic stability of an incompressible, nonlinearly elastic bar at different strain-rates by following the evolution of spatially localized small perturbations introduced at different times. The goal of the present work is to study the dynamic stability – linear and nonlinear – of rate-independent biaxially strained thin plates by following the evolution of spatially localized perturbations introduced at different times, to understand the initiation of the corresponding failure mechanisms. Our 2D linearized analysis of a thin plate under plane stress state, shows that these plates are stable until τ_L, the dimensionless limit time corresponding to the loss of the uniformly strained plate’s stability. This result is supported by fully nonlinear calculations. Our nonlinear numerical calculations also show an imperfection amplitude-dependent and biaxiality-dependent delay in the appearance of localization patterns in dynamically loaded plates for dimensionless times well beyond τ_m, corresponding to the onset of loss of ellipticity in the constitutive law. Moreover, the failure patterns of these plates are studied numerically by following the time evolution of randomly distributed imperfections of different amplitudes.