A lower bound solution for final plastic deformation of rectangular plates under localised impulsive loading

Abstract

A general approach for constructing a lower bound solution to predict the final plastic deformation of rectangular plates under localized impulsive loading has been developed. This approach employs the modal approximation technique to calculate the initial modal velocity. The average strain rate effect is determined by estimating the maximum strain rate field within the rectangular plate when half of the initial modal kinetic energy has been dissipated. A formula for calculating the final plastic deformation, considering strain rate effects, is derived based on the principle of energy conservation, given a specific deformation profile. To validate the proposed model, the cosine deformation profile, which serves as the exact solution of the field equation, is initially selected. The calculated results of the analytical solution, both with and without strain rate effects, are compared against experimental data to assess the model’s reliability. Subsequently, the effects of two commonly used parabolic and Bessel deformation profiles are analyzed in terms of their influence on the calculated final plastic deformation and average variations. Building on the simplified mode solutions, three new dimensionless numbers are introduced. Using experimental data from square and rectangular plates, four effective linear fitting formulas are established to predict the dimensionless final plastic deformation based on these new dimensionless numbers. Furthermore, the new dimensionless numbers are compared with Jacob’s dimensionless number under various loading scenarios, demonstrating that the dimensionless numbers proposed are more reasonable despite introducing an added level of complexity. The limitations of Jacob’s dimensionless number are identified, and its applicable range is suggested.