**4. Discussion**

In this section, the requirements on training data for the ML yield function will be more closely examined. For the first application of the ML yield function, training data for *n*ang = 36 load cases have been constructed from a reference material with Hill-like anisotropy, and it has been demonstrated that this number of load cases produces rather accurate results in this case, whereas the approximation

of a Tresca yield function required *n*ang = 600 load cases. Numerical studies with larger data sets reveal that for Hill-like anisotropic yield functions the accuracy of the results increases slightly for data sets of up to *n*ang = 200. The accuracy as well as the numerical stability and efficiency of the method remain stable for even larger data sets, which has been tested for up to *n*ang = 600. It is also interesting to see that the accuracy of the method for Hill-like anisotropic yield functions is only slightly reduced when rather small data sets of *n*ang = 8 load cases are used. Even producing the training data under plane-stress conditions with *σ*<sup>3</sup> = 0 does not change the quality of the results significantly, which is a consequence of mapping all stresses onto the deviatoric plane and extending the results by assuming that the material's flow behavior does not depend on hydrostatic stress components, which is fulfilled to a very good degree for metals. If it is assumed, furthermore, that the material under consideration shows a symmetric flow behavior under tension and compression, only one half-space of the deviatoric plane needs to be characterized, and the results can be mirrored into the other half-plane. Thus, only the four load cases given in Table 2, have been sufficient to produce training data from which a useful ML yield function for Hill-like anisotropic plasticity results. This example is also provided in the supplementary material.

From these considerations, it can be concluded that for material with an anisotropic flow behavior that can be described with a Hill-like formulation, a small number of experiments under plane-stress conditions, as they can be performed on a bi-axial test rig, are fully sufficient to produce enough raw data to train the ML yield function. Of course, this experimental data could also be used to calculate the Hill parameters. However, the ML yield function offers a larger flexibility, and the training process for machine learning methods like support vector classification (SVC) is highly efficient.

Materials with a more irregularly shaped yield function, like the Tresca yield function, pose much higher demands on the available data and, furthermore, the parameters for the training of the SVC algorithm need to be adapted to allow for more flexibility. The obtained yield function still rounds off the sharp corners of the Tresca yield function such that the result resembles yield functions that can also be obtained with the 18-parameter model of Barlat et al. [4]. Again, it would be possible to determine these parameters directly from the available data, however, with an even higher effort than for the Hill parameters. Yet, the training effort for the ML yield function remains the same, independent of the volume of training data.

In general, it has been verified that only knowing the full stress tensors at the onset of plastic yielding is sufficient to train the ML yield function, and further information, e.g., on plastic strains, is not required. The only relevant information that needs to be known is whether for a given stress state the material response is purely elastic or elastic-plastic. Concerning the best strategy to produce training data, the raw data should lie in close proximity to the yield surface to cover the onset of yielding in an accurate way. Since support vectors are only produced from training data closest to the hyperplane separating the two categories "elastic" and "plastic", however, using only this raw data would cause the SVC decision function (18) to drop to zero quickly outside the region covered by the support vectors. Hence, it is important to downscale the "elastic" raw data points and to upscale the "plastic" raw data points to finally cover the entire relevant region of the deviatoric stress plane and to produce sufficiently many support vectors in this domain. Comparing Figure 3 (Hill-like yield function) and Figure 8 (Tresca yield function), the influence of the number of training points is quite clearly seen, as in the former case 145 support vectors are created during the training process, whereas in the latter case 1689 support vectors are necessary to describe the sharp corners of the Tresca yield function. This example also demonstrates how the number of 36 vs. 600 raw data points is reflected in the number of generated support vectors.

Data-oriented constitutive modeling thus requires only a rather limited amount of data, as compared to other approaches in data-driven mechanics [9,11,13]. Large data volumes, of course, help to increase the accuracy of the resulting ML yield function, but the results achieved even with small amounts of data provide already a very good estimate of the material's anisotropic flow behavior even for load cases that have not been tested. These comparatively moderate requirements on the volume

of training data are a consequence of exploiting physical symmetry conditions on the material's flow behavior in the formulation of the data-oriented yield function. For materials exhibiting a significant influence of the hydrostatic stress component on the plastic behavior, the method is still applicable, but the requirements on the training data will be higher.

Another aspect to be discussed here is the use of micromechanical models to produce training data. With such models, the mechanical behavior of realistic microstructures can be simulated with crystal plasticity methods [7], providing an accurate description of the plastic properties of polycrystaline metals with different microstructures and crystallographic textures. One disadvantage of such micromechanical models is their tremendously high numerical effort making them unsuited for FEA applications of engineering structures that are much larger than the grain size of a material. However, it is possible to employ relatively small micromechanical models, validated by experimental data, for creating a sufficiently large data volume describing the mechanical properties of the real material under various loading conditions with a high accuracy. By purposefully varying the microstructure or the texture of the model material, micromechanical simulations also serve the purpose of extending experimental data. This hybrid experimental and numerical data can then be used for the training of the ML yield function presented in this work. In this way, material parameters like grain size, grain morphology, and crystallographic texture can be explicitly included into the feature vector of the ML yield functions, in addition to the purely mechanical data used currently as input for the yield function. This microstructure-sensitive ML yield function can then be used in large-scale FEA for the simulation of engineering structures, which holds the possibility to consider the trained ML flow rule as a "digital twin" of the material, containing all relevant information on the material properties. The data-oriented constitutive model developed in this work will, hence, also pave the way for new approaches to scale-bridging materials modeling.

Critical issues that remain to be solved before the new method can be applied generally in FEA, include (i) the augmentation of the formulation with respect to shear components of stress and strain, (ii) regularization of the ML yield function to ensure its convexity and to the reduce noise in its gradient, and (iii) a data-oriented formulation of work-hardening. Concerning the latter point, it is noted that the current formulation allows the use of the standard methods of isotropic and kinematic work hardening if the hardening parameters are known, because it already contains the gradient of the yield function to calculate the plastic strain rate and also the tangent stiffness. However, a data-oriented formulation of work hardening, e.g., following the ideas of Chinesta et al. [13], would be more consistent with the idea of data-oriented constitutive modeling.
