**5. Conclusions**

In this work, a new formulation of a data-oriented constitutive model for plasticity has been derived and applied within finite element analysis. The central element of this new constitutive model is a support vector characterization (SVC) algorithm serving as yield function. This SVC algorithm is trained by using deviatoric stresses as input data and the information whether a given stress state leads to purely elastic or rather to elastic-plastic deformation of the material as result data. In this way, a machine learning (ML) yield function is obtained, which can determine whether a given stress state lies inside or outside of the elastic regime of the material. Furthermore, the yield locus, i.e., the hyperplane in stress space on which plastic deformation occurs, can be reconstructed from the SVC, and the gradient on this yield locus can be conveniently calculated. Therefore, the standard formulations of continuum plasticity, as the return mapping algorithm, can be applied in finite element analysis in the usual way. Thus, it has been demonstrated that the new ML yield function can replace conventional yield functions in finite element analysis. The main advantage of such data-oriented constitutive models over the conventional ones is that they can be used with higher-dimensional feature vectors combining mechanical stresses with microstructural parameters of a material. In forthcoming work, it will thus be demonstrated how a single ML yield function can be trained to be used as a constitutive rule for a material in different microstructural states. The production of training data by

micromechanical models, based on crystal plasticity and a discrete representation of the material's microstructure, allows the ML flow rule to serve as efficient homogenization scheme, which offers new possibilities in scale-bridging material modeling.

**Supplementary Materials:** Supporting material in the form of a Python library for finite element analysis and a Jupyter notebook with the codes that have been used to produce the results presented in this work are available online at http://www.mdpi.com/1996-1944/13/7/1600/s1 and as a public repository on https: //github.com/AHartmaier/pyLabFEA.git under the GNU General Public License v3.0.

**Funding:** This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 190389738—TRR 103.

**Acknowledgments:** Helpful discussion with Tobias Glasmachers of Ruhr-Universität Bochum are gratefully acknowledged. Machine learning algorithms have been adopted from the scikit-learn platform (https://scikitlearn.org/stable/).

**Conflicts of Interest:** The author declares no conflict of interest.
