Swift Hardening Research Articles

Instability limits to the forming of sheet metals

In this paper forming limit diagrams are constructed employing predictions from local and diffuse plastic instability. Comparisons are made with test results from: (i) hydrostatic bulging through circular and elliptical dies; and (ii) uniaxial in-plane tension. The sheet materials investigated include steels and brass in both the initially isotropic and anisotropic conditions. It is shown that the test results follow either a diffuse instability prediction or a maximum pressure prediction in the positive strain quadrant of the forming limit diagram (FLD). A local instability prediction applies to the tension-compression strain quadrant. Analyses are based upon plane reductions to Hill's homogenous quadratic potential using: (i) experimental r-values for longitudinal and transverse sheet directions: and (ii) Swift's hardening parameters n and ε 0 from the circular bulge test. In general, instability theories underestimate the in-plane fracture strains when the latter are found either from an etched grid or from the sheet thickness around the fracture. Instability predictions are to be regarded as a useful lower bound to the forming-limit diagrams of ductile materials. Since FLD construction depends upon the parameters (i) and (ii) above, it is through these that the influences of strain history and rate of straining on the diagram may be quantified.

The elastic-plastic spherical shell with nonlinear hardening subject to a radial temperature gradient

Subject of this paper is the quasi-analytical treatment of the elastic-plastic spherical shell whose inner surface is heated slowly. The hardening behavior of the material is presumed isotropic, but it need not be specified beyond that. A first plastic region forms at the inner surface, and, for not too thick-walled shells, a second plastic region appears at the outer surface. The general results are specialized to linear hardening and thereafter to Swift's hardening law with the power one half. Numerical results are represented graphically.