A method for calculating the stress–strain state in the general boundary value problem of metal forming—part 2. Impact of a bar against a rigid obstacle

✍ Scribed by V.L Kolmogorov; L.F Spevak; A.V Gorshkov

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 406 KB
• Volume: 36
• Category: Article
• ISSN: 0020-7683
• DOI: 10.1016/s0020-7683(98)00012-2

✦ Synopsis

The method for calculating stressÐstrain state and fracture proposed by Kolmogorov "0884# and in Part 0 of this present paper is illustrated by the simple problem of a thin bar impacting a rigid obstacle[ Known exact solutions are used to test the method[ On the basis of the stability theory\ the one!dimensional solution has been shown to be legitimate[ Mathematical simulation of bar fragmentation resulting from impact has been carried out[ Þ 0887 Elsevier Science Ltd[ All rights reserved[ 0[ Calculation of stressÐstrain state on elastic bar impact Suppose a thin bar of length L moves at a rate v and at t t 9 9 begins to interact with a rigid obstacle "Fig[ 0#[ The stress and strain states for t × t 9 need to be de_ned[ The bar is assumed to be incompressible and isotropic[ Besides\ the mass forces "excepting inertial# are assumed to be negligible\ and the constitutive equations "physical coupling equations# for deviators are rep! resented by some functionals[ In the subsequent discussion\ the Lagrangian variables x\ 9 ¾ x ¾ L will be used[ As a Lagrangian coordinate we take the coordinates of the particles at the instant t t 9 \ and in the problem discussed in this section they will coincide with Eulerian ones\ as the deformations are small[ In our problem\ the variational equation for the principle of virtual velocities and stresses "see e[g[ Kolmogorov\ 0884^Kolmogorov\ 0875# at an arbitrary _xed instant of time has the form dI d 6g L 9 "s ij j? ij ¦rw i v? i # dx 7 9[ "0#