Fig. 3 Example cross-sectional midsurface deformation profiles at axial station (3L)/4 of the double-layer, two-material, variable-thickness, clamped-ended, example conical shell.Both materials are initially elastic and isotropic, and exhibit rate-independent elastic, perfectly-plastic behavior; accordingly, only one sublayer is needed in the mechanical sublayer model to represent each material. It should be noted that the mechanical sublayer material model used in Refs. 1-7 accounts for strain hardening, strain rate, and the Bauschinger effect.
Forcing FunctionFrontal cosine, meridionally uniform, external pressure loading is used with t p = 10 jusec, and the peak external pressure at 0 = 0° is 25,000 psi.Twenty meridionally-uniform space meshes are used; circumferential space meshes of A® = 9° are employed with symmetry boundary conditions imposed at 0 = 0° and 0 = 180°. Each end of the conical shell is treated as ideally clamped.Shown in Fig. 2 are the Y 3 displacement time histories of the (midsurface) mesh points at axial stations L/4, L/2, and 3/4L for 0 = 0°. It is seen that this intense frontal-cosine external pressure loading has produced a substantial amount of permanent deformation, with peak responses being reached by between 100 and 200 jusec. After about 400 jusec, it appears that little or no further plastic work has occurred, and the subsequent response is essentially an elastic one "centered" about the "permanent deformation state." Shown in Fig. 2 are the crown line (0 = 0°) profiles at t = 0, 40, and 600 #sec. Cross-sectional midsurface profiles at axial station 3/4L are shown in Fig. 3 at t = 0, 40, and 600 /xsec. All of these results appear to be plausible, but, unfortunately, independent predictions and/or appropriate experimental results are not available for comparison.For 300 time steps of this calculation on the IBM 360/65 system at MIT, 48.4 min of computing time were used. The time increment used in the finite-difference integration was equal to 2.0 //sec. These 300 time steps correspond to 600 jusec of real-time response.
