Hill's Quadratic Criterion Research Articles

An equivalent work-hardening description of an interstitial-free steel sheet based on uniaxial tensile and hydraulic bulge tests

In this work, the plastic behavior of an interstitial-free steel sheet was first evaluated by standard uniaxial tensile and hydraulic bulge tests to reproduce the intermediate and high straining levels commonly observed in sheet metal forming processes. Next, a simple procedure was proposed to obtain an equivalent work-hardening curve, representing the high strain domain instead of the conventional uniaxial tensile data. This procedure involves an anisotropy parameter calculated based on Hill's quadratic criterion using the measured r-values in the rolling and transverse sheet directions. Under the assumption of an isotropic work-hardening, the equivalent curve is defined from the true stress and true thickness plastic-strain results of the hydraulic bulge test. The transformed data were fitted to usual power-law type equations to obtain an adequate work-hardening description for finite element numerical simulation purposes. Based on the goodness-of-fit measures, namely, R-squared, root-mean-square error, and root-mean-square of residual percentage, the combined Swift–Hockett–Sherby equation has proved to be the most suitable choice to describe the plastic behavior of the investigated interstitial-free steel under both intermediate and high strains conditions.

Approximate yield criteria for anisotropic metals with prolate or oblate voids

Following the study of Gologanu et al. (1997) which has extended the well-known approach of Gurson (1975), we propose approximate yield criteria for anisotropic plastic voided metals containing non spherical cavities. The plastic anisotropy of the matrix is described by means of Hill's quadratic criterion. The procedure to establish the closed form expression of approximate macroscopic criteria, in which void shape and plastic anisotropic effects are included, is detailed. The new criteria allow us to recover existing results in the cases of spherical and cylindrical voids in an Hill type plastic matrix. Moreover, they agree with previous criteria for non spherical voids in an isotropic plastic matrix. Finally, for validation purposes, we provide, in the general case of non spherical cavities in the anisotropic matrix, a comparison with the numerical exact two field criteria. To cite this article: V. Monchiet et al., C. R. Mecanique 334 (2006).

Sheet metal forming simulations using a new model for orthotropic plasticity

A description of orthotropic plasticity using a polar-coordinate representation of the yield surface in principal stress space has recently been developed by the authors. Comparisons with micro-macro computations or experimental results show that the new criterion is able to provide an improved description of yield surfaces. The model has been implemented both in specific programs for hydrostatic bulging of a cylindrical diaphragm and for deep-drawing of a cylindrical cup, and in the ABAQUS FEM code by means of the UMAX subroutine. As compared with the predictions of Hill's quadratic criterion, the simulations generally show improved agreement with experimental results concerning load (or pressure), strain distributions and earing in deep drawing.

An analytical micro-macro model for textured polycrystals at large plastic strains

An analytical micro-macro model of evolving plastic anisotropy is presented that is suitable for numerical simulation of forming processes. The model is based on the combination of a polycrystal model and different analytical procedures for writing anisotropic plastic potentials, expressing their coefficients in terms of texture coefficients, and updating the texture coefficients as function of the (tensorial) strain increment. The use of a fourth-order dual plastic potential (“C4”) in the analytical micro-macro model is studied, and this use is compared with that of Hill's [1948] yield criterion and also with the usual run of the Taylor model. The coefficients of the C4 potential depend linearly on the texture coefficients, which are updated using a variational polycrystal model. The analytical operation of this updating lies on the method first proposed by Esling et al. [1984] and is described and checked in some detail. The predictions of the analytical micro-model compare well with measurements of the Lankford coefficient, provided the C4 potential is used. The predicted texture evolution is also in a good experimental agreement: a better one than with the Taylor model, which in some cases, gives a poor updating. The theoretical stress evolution during biaxial or plane-strain tension is experimentally consistent too, although in that case the C4 potential, closer to Taylor's model, makes no improvement as compared with Hill's quadratic criterion.

An Experimental Investigation of Yield Criteria with Planar Anisotropy. 2nd Report, Yield Loci and Equivalent Stress-Strain Curves.

In addition to uniaxial tension and through-thickness compression reported in the previous paper, commercial-purity aluminium and deep-drawing-quality steel sheets are tested in plane-strain compression and simple shear. The yield loci and the equivalent stress-strain flow curves based on Hill's 1948 quadratic, Hil1's 1990 nonquadratic and Gotoh's biquadratic criteria are obtained on the basis of equal amounts of plastic work. The yield loci calculated from Hill's quadratic criterion for case (2) in which the anisotropic parameters are mainly determined with yield stresses and Gotoh's biquadratic criterion are close to each other and provide fairly good fits to the experiments except for the annealed aluminium. Hill's theories for case (2) are also found to be rather reasonable for all the materials used here when the flow curves are compared with each other.