*3.2. Application of The Trained ML Yield Function in FE Analysis*

The ML yield function trained and analyzed in the previous step shall now be applied in FEA to demonstrate its usefulness for this purpose. The numerical examples provided here have been conducted with the Python library "pyLab-FE" created by the author, which is provided in the supplementary materials together with a Jupyter notebook following the work-flow defined in this work. The known parameters and support vectors resulting from the training process of the ML yield function together with the mathematical formalism laid out in Section 2 allow a rather straightforward evaluation of the yield function as sum over the support vectors convoluted with the kernel function, such that they can also be used for implementing a user material subroutine (UMAT) for common commercial FEA tools in any compiler language.

As numerical examples, four different load cases are simulated with FEA: (i) uniaxial stress in horizontal direction, (ii) uniaxial stress in vertical direction, (iii) equibiaxial strain under plane-stress conditions, and (iv) pure shear strain under plane-stress conditions. The simple finite element model used to study these load cases consists of four quadrilateral elements with linear shape functions and full integration, as shown in Figure 5. For all load cases, plane stress conditions with *σ*<sup>3</sup> = 0 are enforced and the normal degrees of freedom (dof) for the boundary nodes are prescribed, while all boundary nodes are allowed to relax to their equilibrium positions along the boundary. The bottom and the left-hand-side nodes are always restricted to a normal displacement of zero. For the uniaxial load cases, tensile displacements are prescribed either on the top or on the right-hand-side (rhs) nodes, while the other boundary is force-free, resulting in a uniaxial stress. For equibiaxial strain, the top and the rhs boundary nodes are subjected to identical displacements; whereas for pure shear, the rhs nodes are loaded with the negative displacement applied on the top nodes. By virtue of these boundary conditions, the deformation causes only normal stresses and strains, but no shear components. Hence, the restrictions of the formulation of the ML flow rule are fulfilled, and the material axes remain aligned with the directions of the principal stresses.

**Figure 5.** The finite element model on which four different load cases are studied consists of four quadrilateral elements (green) with linear shape function, and at total of nine nodes (red) situated at the corners of the elements. The bottom and left-hand-side boundary nodes are restricted to zero normal displacement (blue triangles), and the loading is applied on top and right-hand-side-nodes (blue arrows), as described in the text.

These four load cases are applied to the reference material as well as to the material with the ML yield function, and the resulting yield stresses and plastic strains at the end of each load step are compared in Table 2. In Figure 6, the resulting global equivalent stresses and equivalent total strains

$$
\epsilon\_{\text{eq}} = \sqrt{\mathfrak{e} : \mathfrak{e}} \tag{25}
$$

for each load case are plotted for both materials, where the different definitions of the equivalent stress have been applied to the reference material.

**Table 2.** Yield stresses (YS) obtained for Hill-like yield function, with parameters given in Table 1, and machine learning (ML) yield function under the specified load cases. The relative errors in yield stress and equivalent plastic strain (PE) at maximum load are also specified.


**Figure 6.** Stress strain curves obtained for elastic-ideal plastic material behavior under the loading conditions specified in the legend: (**a**) Equivalent total strain vs. equivalent Hill-stress, (**b**) equivalent total strain vs. equivalent J2-stress for Hill-like yield function, and (**c**) equivalent total strain vs. equivalent J2-stress for ML yield function.

To further demonstrate the correctness of the plastic behavior resulting from the ML flow rule, the flow stresses of the material, i.e., the stresses occurring during plastic deformation, are plotted together with the yield locus. For ideal plasticity, treated in this work, it is expected that the flow stress

remains on the yield locus, since the material does not sustain larger stresses. In Figure 7 it is shown that this expectation is fulfilled to a very good degree, by comparing the yield loci and the element solutions of the flow stresses obtained for the ML yield function with those of the reference material with a Hill-like flow rule.

**Figure 7.** Stress states obtained for the four different plane-stress load cases are plotted in the *σ*<sup>1</sup> -*σ*<sup>2</sup> plane together with the yield loci of the trained ML flow rule and the Hill-like reference material. The flow stresses resulting from the ML yield function are plotted as small yellow circles and those from anisotropic Hill plasticity as large blue circles.
