A directional framework of similarity laws for geometrically distorted structures subjected to impact loads

A directional framework of similarity laws, termed oriented-density-length-velocity (ODLV) system, is suggested for the geometrically distorted structures subjected to impact loads. The distinct feature of the framework is that the newly proposed oriented dimensions, dimensionless numbers and scaling factors for all basic physical quantities are explicitly expressed by three characteristic lengths of spatial directions, which overcomes the inherent defects only with one scalar length in the traditional dimensional analysis. Meanwhile, similarity laws of the directional stresses, strains and displacements are expressed by different power law relationships of the ratios of undistorted characteristic lengths to distorted characteristic lengths. Therefore, the ability of similarity theory to express structural geometric characteristics are effectively developed. Based on the newly proposed framework, the non-scalabilities of geometric and material distortion (including strain hardening and strain rate effects) and the gravity effects could be compensated by correction methods of velocity, density and geometry. The analytical models of beams subjected to impact mass and impulsive velocity are verified. The results show that the proposed framework has excellent performance for expressing various dimensionless response equations and geometrically distorted scaling. A numerical model of circular plate subjected to dynamic pressure pulse is further carried out to verify the geometrically distorted scaling of the directional components of displacement, strain and stress. The refined analysis results show that, structural dimensionless responses in different directions can behave good consistency between the scaled model and the prototype in both the spatial and the temporal fields, with the correction of the directional physical quantities using different powers of the ratios of characteristic lengths.