Effective method for fitting complex constitutive equations

The formability of advanced high strength steels (AHSS) depends on the temperature-sensitivity of strength and strain hardening at large strain (∼0.4–0.5), in addition to traditionally-measured attributes such as isothermal strain hardening and strain-rate sensitivity. As such additional dimensions are incorporated into a constitutive model, it becomes progressively more complex and challenging to fit uniquely without invoking unrealistic numbers of fitting attempts starting from myriad sets of trial parameters. For 8-parameter constitutive equations similar to the ones investigated here, the literature reports fitting with 28 (256) starting sets of trial parameters, requiring 256 nonlinear optimizations using special programming, thus making such procedures impractical for many applications. An efficient, effective method requiring only 3 optimizations was proposed, developed, and confirmed for a diverse range of AHSS: DP590, DP780, DP980, TRIP780, CP1180 and TWIP. These alloys represent a wide range of hardening mechanisms, tensile elongations (0.05–0.5) and tensile strengths (590–1180 MPa). All results were derived from standard tensile tests, yet the extrapolated stresses of the constitutive models fit with the new method were confirmed independently within approximately 1% standard deviation by balanced-biaxial bulge experiments at strains up to 6 times larger than the tensile range used to generate the constitutive models. Such accuracy at large strains is crucial for predicting the “shear fracture” of AHSS in practice. The final constitutive equations and parameters obtained using the new method are presented for each alloy, including a new form for TWIP steel.