A variable strain hardening model for anisotropic sheet metals

Abstract: A theory is suggested which describes, on a macroscopic scale, the yielding and plastic flow of an anisotropic metal. The type of anisotropy considered is that resulting from preferred orientation. A yield criterion is postulated on general grounds which is similar in form to the Huber-Mises criterion for isotropic metals, but which contains six parameters specifying the state of anisotropy. By using von Mises' concept (1928) of a plastic potential, associated relations are then found between the stress and strain-increment tensors. The theory is applied to experiments of Korber & Hoff (1928) on the necking under uniaxial tension of thin strips cut from rolled sheet. It is shown, in full agreement with experimental data, that there are generally two, equally possible, necking directions whose orientation depends on the angle between the strip axis and the rolling direction. As a second example, pure torsion of a thin-walled cylinder is analyzed. With increasing twist anisotropy is developed. In accordance with recent observations by Swift (1947), the theory predicts changes in length of the cylinder. The theory is also applied to determine the earing positions in cups deep-drawn from rolled sheet.

Abstract: The process of the loss of stability is analysed for sheet metal subjected to biaxial tension when the ratio of the principal stresses 0.5 ⩽ σ 2 /σ 1 ⩽ 1 . The loss of stability manifests itself by a groove running in a direction perpendicular to the larger principal stress. In this groove local strains begin to concentrate gradually. In the initial stage of the process the deepening of the groove is associated with a gradually fading strain in the regions adjacent to the groove. This fading strain attains a certain limiting value e∗. This paper contains both experimental results and a theoretical analysis of the process of the generation of the groove based on anisotropic plasticity theory. The system of equations derived was solved numerically with the aid of a computer, which enabled the limiting strain of the sheet metal to be determined as a function of the following properties of the material: (i) Initial inhomogeneity of the sheet metal, (ii) exponent of the strain-hardening function, (iii) coefficient of normal anisotropy, (iv) initial plastic strain, (v) strain at which the fracture occurs. The results are discussed and the properties are described that influence the drawability of sheet metal used in the stretch-forming process.

Abstract: A yield function that describes the behavior of orthortropic sheets ,metals exhibiting planar anisotropy and subjected to plane stress conditions is proposed. It is shown to give a reasonable approximation to plastic potentials calculated with the Taylor/Bishop and Hill theory of polycrystalline plasticity for plane stress states. Therefore, this formulation should be particulary useful to study, at a low degree of complexity, the influence of polycrystalline textures on the forming performances of metal sheets. In part II, this yield function will be used to study the influence of the yield surface shape on failure behavior of sheet metals.

Abstract: A large number of polycrystalline materials, both manmade and natural, display preferred orientation of crystallites. Such alignment has a profound effect on anisotropy of physical properties. Preferred orientation or texture forms during growth or deformation and is modified during recrystallization or phase transformations and theories exist to predict its origin. Different methods are applied to characterize orientation patterns and determine the orientation distribution, most of them relying on diffraction. Conventionally x-ray polefigure goniometers are used. More recently single orientation measurements are performed with electron microscopes, both SEM and TEM. For special applications, particularly texture analysis at non-ambient conditions, neutron diffraction and synchrotron x-rays have distinct advantages. The review emphasizes such new possibilities. A second section surveys important texture types in a variety of materials with emphasis on technologically important systems and in rocks that contribute to anisotropy in the earth. In the former group are metals, structural ceramics and thin films. Seismic anisotropy is present in the crust (mainly due to phyllosilicate alignment), the upper mantle (olivine), the lower mantle (perovskite and magnesiowuestite) and the inner core (e-iron) and due to alignment by plastic deformation. There is new interest in the texturing of biological materials such as bones and shells. Preferred orientation is not restricted to inorganic substances but is also present in polymers that are not discussed in this review.

Abstract: Deformation by 〈111〉-pencil glide has been analyzed by an upper-bound model which combines a least-shear analysis and Piehler's maximum virtual work analysis. The least-shear analysis gives exact solutions if three 〈111〉 slip systems are active, while the maximum work analysis provides solutions for the case of four active slip systems. Independent checks are used to determine which solution method is appropriate. Computer calculations using this model have been made to determine; (1) the orientation dependence of the Taylor factor for axisymmetric deformation; (2) the yield loci for textured materials having [100], [110] and [111] sheet metals and rotational symmetry; (3) the isotropic yield locus for randomly oriented materials; and (4) flow stresses along critical loading paths for various assumed textures with rotational symmetry. The latter calculations indicate that anisotropic yield loci of textured bcc metals with rotational symmetry are much better approximated by σ x a + σ y a + R¦σ x − σ y ¦ a = (R + 1)Y a where R is the strain ratio and Y is the tensile yield strength with an exponent a = 6 rather than with a = 2 as postulated by Hill. It is not known how well upper-bound calculations like these represent actual yielding behavior.