*3.3. Tresca Flow Rule*

In the next example, the ML yield function is trained with data from a reference material with a Tresca yield function, which is fundamentally different from the elliptical Hill-like yield functions. The Tresca equivalent stress is defined as

$$
\sigma\_{eq}^{\text{Tresca}} = \sigma\_{\text{I}} - \sigma\_{\text{III}} \,\,\,\,\,\tag{26}
$$

where *σ*<sup>I</sup> is the largest principle stress and *σ*III is the smallest principle stress [23]. Using the Tresca equivalent stress in the yield function (2) leads to an isotropic plastic deformation of the material, however, with very different characteristics than for a J2 equivalent stress. Hence, it is a critical test for the new method developed in this work to apply it to such yield functions. The training data for this yield function has been produced in the same way as before. However, with *n*ang = 600 data points for the polar angle, a much larger number of training data points has been required to follow the subtle features of the Tresca flow rule. Furthermore, the training parameters *C* = 50 and *γ* = 9 have been applied to allow for sufficient flexibility of the ML yield function to approximate the abrupt changes of the yield behavior in the deviatoric stress space, *cf.* Figure 8. The scores with test and training data are above 99%, and the *R* 2 -value on test data is 96%. Even with this training procedure, the sharp corners of the Tresca yield locus are slightly rounded off by the ML yield function, leading to somewhat lower yield strengths of the material in these directions, as seen in Figure 9a, where the resulting stress-strain curves are plotted as J2 equivalent stress vs. equivalent total strain. In Figure 9b the yield loci of the Tresca reference material, the trained ML yield function and an isotropic J2 material are plotted in comparison. Furthermore, the flow stresses resulting from the FEA are plotted in this graph to verify that they lie on the ML yield locus, as expected for ideal plasticity.

**Figure 8.** Field plot of the ML yield function trained with data generated from a reference material with a Tresca yield criterion, where areas in purple color shades represent negative values and brown shades represent positive values. The isoline, where the ML yield function is zero, is plotted as black line. The test data points are plotted as brown circles, for stresses in the plastic regime, and a s light blue circles for stresses in the elastic regime.

**Figure 9.** Results of FEA on the material with an ML flow rule trained with data obtained from a Tresca yield criterion: (**a**) Equivalent J2 stress plotted over the equivalent total strain, for the four different load cases given in Table 2. (**b**) Flow stresses obtained with the ML yield function plotted as yellow circles in the *σ*<sup>1</sup> -*σ*<sup>2</sup> principle stress space, together with the yield loci of the ML flow rule, the Tresca flow rule and an isotropic J2 flow rule.

With these numerical examples, the applicability of the ML yield function developed in this work has been demonstrated for two fundamentally different kinds of flow behavior: anisotropic Hill-like plasticity and isotropic Tresca plasticity. The new formulation has proven to be numerically stable. The numerical effort is somewhat higher than that for mathematical closed-form yield functions, because the calculation of the predictor step requires a higher effort in numerically evaluating the distance of a given point in stress space to the yield locus. However, in conjunction with the implementation of the ML yield function in a compiled computer code, FEA even for large engineering models seems to be feasible with the new ML yield functions.
