Forming limit diagrams of strain-rate-dependent sheet metals

This paper presents a predictive model of localized necking for strain-rate-dependent sheet metals. This is achieved by the development of a modified vertex theory for use as a localized necking criterion. The vertex theory is von Mises based and isotropic. The study is motivated by the apparent discrepancies between the measured forming limit diagrams (FLDs) of strain-rate-dependent sheet metals and those predicted using the conventional vertex theory. The modified theory considers the rate-dependent power-hardening material rule. A novel form of quasi-linear stress–strain relation based on the power-hardening rule is analytically derived to model the constitute behavior of the rate-dependent material. The localized necking criterion of the sheet metals is based on the vertex theory, which assumes that localized necking occurs simultaneously with the initiation of a vertex on the yield surface. The stress–strain relation of rate-dependent material is coupled into the vertex theory to deduce the critical conditions for localized necking on both sides of the FLD. Numerical results indicate that the forming limits of rate-dependent sheet metals follow quite different rule as that for rate-independent sheet metals. The ratio between the strain-rate-hardening index and the strain-hardening index plays an important role in the strain-rate effect on FLDs. A typical strain-rate-dependent metal, AKDQ steel, is chosen for validation of the modified vertex theory. The tensile stress–strain curves of AKDQ are tested on both regular and high-speed MTS, covering a wide range of strain rate from 10−5 to 10 s−1. The hardening law of AKDQ is found to depend greatly on the strain rate, while the strain-rate-hardening index is not a material constant but dependent on the strain rate. Forming limit tests are also performed for AKDQ sheets to measure the FLD. The measured FLD of AKDQ is compared with the prediction based on the developed theory and good agreement is observed.