Investigation on the effect of through thickness normal stress on forming limit at elevated temperature by using modified M-K model

The formability can be improved in warm/hot sheet hydroforming due to two important factors of temperature and through thickness normal stress. An extension of the M-K model was presented in this paper with the liquid pressure induced through thickness normal stress taken into account. The through thickness normal stress was introduced by using the general form of the Hill48 criterion. The combined effect of temperature and through thickness normal stress on the increase of FLD 0 is achieved by applying a proposed temperature dependent constitutive equation for 5A06 aluminum alloy (a Chinese designation of Al-Mg alloy, corresponding to the ASTM designation of 5056 aluminum alloy) into the M-K approach. The Newton–Raphson method is used in the numerical procedure and it is proved to be stable and correct. The yield loci show a significant dependence on temperature and through thickness normal stress. Both size shrinking caused by the elevated temperature and location shifting due to the increasing thickness normal stress of the yield loci are observed. Comparison of the experimental FLDs of AISI-1012 low carbon steel (an AISI designation of steel) at plane stress and room temperature, STKM-11A steel (a Japanese designation of Steel Tube Kozo Machine) for tube hydroforming at room temperature and 5A90 aluminum alloy (a Chinese designation of Al-Li alloy) at plane stress and elevated temperature with the theoretical results show good agreements. The effect of key parameters, such as inhomogeneity factor f 0 , n value, m value and initial thickness T 0 , grain size d , initial surface roughness R 0 , show a strong dependence on FLDs and increase of FLD 0 . The increase of FLD 0 is formulized in a full quadratic form, which is a function of temperature and through thickness normal stress. The left quadrant of FLD was determined by the cylindrical warm hydromechanical cup deep drawing, which shows a good accordance with the predicted results by using the modified M-K model.