Verification of Analytical Solution for a Problem of High-Velocity Transversal Impact on a Prismatic Beam

Abstract

In the present article ray method application for a problem of high-velocity transversal impact on a prismatic beam is considered. The problem is solved for an elastic spherical projectile and elastoplastic destructible targets for velocities up to 500 m/sec. For the targets rate-dependence of the material properties and structural inhomogeneity are taken into consideration. Verification is aimed to prove principal opportunity of receiving accurate analytical solution for the problem, if contact law is known. As a reference solution results of finite element modelling are considered.

Impact response of a Timoshenko-type viscoelastic beam considering the extension of its middle surface

“…In the same page of (p. 345 in Vershinin 2014 ), the equation of motion of the contact zone was presented as where N is the longitudinal force, w and Q are transverse displacement and force, respectively, P ( t ) is the contact force, z is the coordinate directed along the beam axis, a dot denotes the time-derivative, a is the radius of the contact zone, in so doing the author of (Vershinin 2014 ) has considered that plane transient waves (surfaces of discontinuity) propagate from the contact zone during the process of impact. However, if the contact domain is a circular disk with a volume , then the waves travelling from a circular contact zone are diverging circles.…”

“…However, in 2014, Vershinin ( 2014 ) published a conference paper, wherein the problem of impact of an elastic sphere against an elastic Timoshenko beam was considered with due account for extension of the target middle surface, in so doing the solution was based on completely wrong formulas and equations. Thus, in page 345 of (Vershinin 2014 ) a reader could find incorrect formula (6) where [ v ] is the discontinuity in the velocity of longitudinal displacement on the plane transient wave propagating along the beam during the impact process with the velocity , E is Young’s modulus, is the density, is Poisson’s ratio, is the beam thickness, and is the local bearing the target and impactor’s materials. The above formula is valid for an elastic thin plate, but it is invalid for a beam.…”

“…In the same page of (p. 345 in Vershinin 2014 ), the final equation was presented as where , , , K is the shear coefficient, and is the shear modulus. This equation is also incorrect.…”

“…The correct equation neglecting the inertia of the contact zone (this important assumption was not mentioned in Vershinin ( 2014 ) ) has the form with the correct coefficient .…”

“…Based on the aforesaid, the further numerical treatment presented in Vershinin ( 2014 ) occurs to be invalid.…”