## Mechanical Engineering

Describe and provide citations for biaxial pure shear stress tests for the characterization of sheet metals. Some citations are described for rubber and plastic but these do not behave the same as sheet metals in plastic deformation.

The paper considers the shear loading condition for constitutive characterization of sheet metals. The authors attempted to highlight several existing challenges associated with shear testing of the materials. The paper does not contain grammatical deficits, and given the short length of the paper, the presentation of the results is reasonably good. However, some of the statements in the paper should be revised and there are a few key points that need to be addressed before possible inclusion of the paper into the proceedings of IDDRG 2018:

• In the paper, the shear states are classified into: “simple shear”, “pure shear (strain)” and “pure shear (stress)”. Did the authors borrow these terms from the literature? If yes, relevant citations need to be included. The common practice in Continuum Mechanics is using either simple or pure shear to describe shear deformation of materials.

• The authors mentioned that pure shear “stressing” can be accomplished by using a biaxial test machine. Are there any published results in the literature describing such a test for sheet metals? To the Reviewers knowledge, the pure shear state is severely hard to achieve experimentally, and even torsion tests of tubes is close to the simple shear condition (Tvegaard, 2015) rather than the pure shear which is claimed by the authors. Buckling of thin metallic sheets would likely occur.

• It appears that the authors are over-concerned with the normal stress components (ση and σξ) in simple shear. The normal stresses mostly develop during elastic deformation and remain negligibly small compared to the shear stress for large deformation in the plastic region. For a rigid plastic material, this assumption is entirely admissible and when coupled with the logarithmic objective rate, the normal stresses are indeed vanishingly small. The authors are referred to Butcher and Abedini (2017) where some of these confusions are addressed.

Anisotropic yield functions that do not satisfy the plastic constraints for shear loading based on the plastic potential may predict larger normal stresses but this is an artefact as discussed in detail in Abedini et al. (2017).

• It is worth mentioning in the paper that anisotropic yield criteria are mostly calibrated at the onset of initial yielding or small plastic strains below 0.10 which is in line with the guidelines of the authors for the range of strains where the non-coaxiality is negligible. The rotations can be quantified and coaxiality to be only several degrees and also not affect the equivalent strain or lead to large normal stresses (Butcher and Abedini, 2017; Abedini et al., IJSS, 2017). In short, for constitutive characterization of sheet metals, the relatively low strains related to necking in tensile tests provide the limit for calibrating an anisotropic yield function. The shear stress would be used in the calibration based on the same level of plastic work as in the tensile tests at the UTS.

• Regarding the remarks on equal and opposite principal stresses and strains, the authors assumed that the Hill48 or Gotoh’s models are capable of accurately predicting the responses of materials in the shear regions and since these models, in their general forms, do not provide the condition of equal and opposite principal stresses and strains, the calibration constraint is problematic.

This is an incorrect assumption since these models are “phenomenological” by definition and they were developed focusing on the tensile regions (the first quadrant of yield loci) by design. There is no justification as to their accuracy in general shear loading.

As a historical example, when the Hill48 model was proposed in 1948 it resulted in good predictions for conventional steel alloys popular in 40s and 50s. Later, with the increasing adoption of aluminum alloys featuring low r-values, it was observed that the Hill48 model is inaccurate in the equal-biaxial region for aluminum alloys. Initially, this behaviour was labelled as “anomalous behaviour of aluminum alloys” because it was not in accordance with the Hill48 predictions. But in fact, aluminum alloys behaved based on their physics and they did not have any anomalous responses. Evidently, the problem was actually the inaccuracy of the Hill48 to capture this response. A similar story can be applied to the shear state as well.

The authors should clearly state that their interpretation of problematic issues of equal and opposite principal stresses and strains in shear state is a consequence of the assumption that the phenomenological Hill48 and Gotoh’s models are universally correct in shear loading.

Although the Hill48 and Gotoh models would necessarily become planar isotropic to satisfy the shear constraint in a general orientation, this is not the case for other anisotropic functions such as yld2000 as demonstrated in Abedini et al. (2017) where planar isotropy does not result.

A distinction should also be made upon the choice of coordinate frame for the anisotropic behavior in the Hill/Gotoh models which is different than Barlat-type and other more advanced models for sheet metals that employ a principal space representation.

References:

Tvegaard V, Study of localization in void-sheet under stress states near pure shear, International Journal of Solids and Structures, 2015, 75-76:134-142.

Butcher C, Abedini A, Shear confusion: Identification of the appropriate equivalent strain in simple shear using logarithmic strain measure, International Journal of Mechanical Sciences, 2017, 134:273-283.