A new membrane model for the ballistic impact response and V50 performance of multi-ply fibrous systems

This paper develops an analytical model for the ballistic impact response of fibrous materials of interest in body armor applications. It focuses on an un-tensioned 2D membrane impacted transversely by a blunt-nosed projectile, a problem that has remained unsolved for a half a century. Membrane properties are assumed characteristic of the best current body armor materials (Kevlar^®, Spectra^®, Zylon^®, S2 glass), which have very high stiffness and strength per unit weight, and low strain-to-failure. Successful comparisons will be made with extensive experimental data on such material systems as reported by Cunniff [Decoupled response of textile body armor. Proc. 18th Int. Symp. of Ballistics, San Antonio, Texas, 1999a, pp. 814–821; V_s–V_r relationships in textile system impact. Proc. 18th Int. Symp. of Ballistics, San Antonio, Texas, 1999b; Dimensional parameters for optimization of textile-based body armor systems, Proc. 18th Int. Symp. of Ballistics, San Antonio, Texas, 1999c, pp. 1303–1310]. Our mathematical formulation draws on the seminal work of Rakhmatulin and Dem’yanov [Strength Under High Transient Loads, 1961, pp. 94–152]. Under constant projectile velocity we first develop self-similar solution forms for the tensile ‘implosion’ wave and the curved cone wave that develops in its wake. Through matching boundary conditions at the cone wave front, we obtain an accurate approximate solution for the membrane response including cone wave speed and strain distribution. We then consider projectile deceleration due to membrane reactive forces, and obtain results on cone velocity, displacement and strain concentration versus time. Other results obtained are the membrane ballistic limit, or V₅₀ velocity, and the residual velocity when penetrated above this limit. We then derive an exact functional representation of a V₅₀ ‘master curve’ found empirically by Cunniff [ibid] to reduce data for a wide variety of fabric systems impacted by blunt cylindrical projectiles. This curve is given in terms two dimensionless parameters based only on fiber mechanical properties and the ratio of the fabric areal density to the projectile mass divided by its area of fabric contact. Our functional representation has no fitting parameters beyond one reflecting uncertainty in the effective diameter of the impact zone relative to the projectile diameter, and even then the values are consistent across several experimental systems. The extremely successful comparison of our analytical model to experimental results in the literature raises fundamental questions about many long-held views on fabric system impact behavior and parameters thought to be important.