A parametric description of orthotropic plasticity in metal sheets

Experimental analysis and theoretical predictions of the limit strains of a hot-dip galvanized interstitial-free steel sheet

“…The yield function adopted in this work corresponds to the orthotropic plasticity criterion proposed by Ferron et al 35 defined for plane-stress conditions as:…”

“…Firstly, the parameters (A, B and k) are calculated from the stress-ratios (σ b /σ 45 ) and (σ b /τ0) between the equibiaxial yield stress, σ b , and the uniaxial yield stresses at 45 0 with respect to the rolling direction, σ 45 , and the shear yield stress, τ0, along the rolling direction, respectively, along with the plastic anisotropy coefficient, r 45 . For steel sheets, the recommended values for the exponents are m = 2, n = 1 or 2, p = 1 or 2 and q = 1 or 2 35 . In particular, a positive value for the parameter k allows a flattening of the yield surface near plane-strain tension/compression and pure shear stress states.…”

“…The principal plastic-strain increments are defined with the help of the parametric description, Equation 12, as 35 :…”

“…On the other hand, the plastic-strain increments in the orthotropic symmetry axes frame (x,y,z) are determined from Equation 6, which depends upon the effective plasticstrain increment and the yield function partial derivatives. The later are obtained by applying the consistency condition, df = 0, to the Ferron's yield function defined in the form of Equations 9 and 10, that is: (17) and then, by defining the terms dx 1 , dx 2 and dα as a function of the variables (x 1 ,x 2 ) along with the help of the relations sin 2 α = σ xy /x 2 and cos 2α = (σ x -σ y )/(2x 2 ), which provides: (18) Where: the yield function partial derivatives 35 orthotropic yield criterion plotted in the principal stress space (σ 1 ,σ 2 ) normalized by the equibiaxial yield stress (σ b ) as a function of the orientation angle (α) between the principal stress axes and the in-plane orthotropy directions.…”

“…Henceforth, the corresponding true plane-strain intercept predicted value, In order to validate the M-K model developed in the present work, see Section 3.2, we first present the theoretical predictions for linear and bi-linear strain-paths assuming material models as a function of the plastic anisotropy coefficient. Then, the predictions determined with Hill's quadratic 34 and Ferron's 35 yield criteria are compared to the experimental data obtained from linear and bi-linear strain-paths for an AISI-1012 steel sheet with a nominal thickness of 2.5 mm [42] . Table 4 lists the sheet material plastic properties adopted in all the theoretical predictions discussed in the present section.…”