**1. Introduction**

Finite element analysis (FEA) is a widespread numerical tool for studying the mechanical behavior of structures. While in many applications it is sufficient to know under which conditions a part of the structure fails plastically or suffers damage or fracture, in some cases, like in sheet forming or for crash simulations, it is important to be able to simulate the plastic deformation during loading and to obtain the shape of the structure after the external load is released. Such non-linear behavior is typically described by constitutive models that relate stress and strain in a material, as described in any textbook on non-linear finite element modeling, e.g., see [1]. Conventionally, constitutive relations for plasticity are formulated as flow rules based on a plastic potential. In the simplest case, the latter is the yield function of the material, determining at which local stress the material starts yielding plastically and which plastic strain increment will result from such plastic deformation. As described in the next section, such yield functions relate the equivalent stress and the yield strength of a material, which needs to be determined experimentally or with the help of more fundamental models in a scale-bridging approach.

Experimentally, the yield strength is typically determined in uniaxial tensile tests, in which materials frequently exhibit an anisotropy in their plastic behavior, i.e., the yield strength depends on the orientation of the loading axis with respect to the material coordinate system, defined for example by rolling, normal, and transverse direction. In conventional approaches such anisotropic plastic behavior is described by defining a proper equivalent stress that takes into account material anisotropy, such that yielding occurs at a constant scalar yield strength. This approach has been introduced by Hill [2] and applied to orthotropic plasticity in sheet metals [3]. The concept has been generalized to linear transformation-based anisotropic yield functions by Barlat et al. [4] and to methods describing distortions of the yield surface caused by anisotropic work hardening [5]. In forming technology, similar ideas have been successfully applied to predict the resulting shape of sheet metals after deep drawing [6]. All these approaches have in common that the information about the material anisotropy is mapped into the definition of the equivalent stress, while they differ in the amount of material parameters that is required to describe the anisotropy in the material's flow behavior. To determine these parameters, a series of experiments with different mutual orientations of loading axis and material axis is necessary. Alternatively, micromechanical models, in which discrete representations of the material's microstructure are used together with quite fundamental crystal plasticity models, can be used to calculate the anisotropic flow behavior of a polycrystalline metal [7,8].

To model non-linear material behavior, in more recent approaches, the method of data-based mechanics has been introduced by Kirchdoerfer and Ortiz [9], in which stress-strain data from experimental tests are used directly, rather than using constitutive rules. While the first approaches have been limited to elastic structures under static loads, recently, this concept has been extended to dynamics [10] and to inelastic material behavior [11]. These methods are based on a fundamental re-formulation of the basic equations of mechanics and thus require completely new mechanical solvers. Other data-driven methods in plasticity are formulated as process models, e.g., for air-bending [12], or focus on the application of data-oriented methods as constitutive models in computational plasticity [13]. The latter idea allows the use of existing FEA solvers for mechanical problems, and is also followed in this work, in which a new formulation of a data-oriented flow rule is introduced that can replace conventional constitutive models—formulated in a mathematical closed form—by machine learning (ML) algorithms. In this data-oriented formulation, the anisotropy of the material's flow behavior is considered in a directionally dependent yield strength of the material rather than by an anisotropic transformation of the stress. Using ML algorithms as yield functions provides a great flexibility to describe arbitrary mathematical functions, and at the same time, holds the potential to handle large data sets and multi-dimensional feature vectors as input. Hence, using ML algorithms as constitutive rules for plastic material behavior offers the possibility to explicitly take into account the microstructural information of the material in constitutive modeling. Furthermore, data resulting from experiment and micromechanical simulations can be hybridized to generate training data sets. An overview on applications of ML and data-mining methods in continuum mechanical simulations of material behavior has been provided by Bock et al. [14].

ML algorithms can be classified into algorithms for supervised and unsupervised learning. The former group can be further categorized into classification algorithms, which divide a multi-dimensional feature space into regions with similar properties, and into regressors, which provide linear or non-linear regression functions for the given multi-dimensional data set. Support Vector Machines (SVM) are successfully applied both as classifiers (SVC) [15] and regressors (SVR) [16,17]; an overview on both applications is given in a technical report by Gunn [18]. Since yield functions in continuum plasticity are also employed to subdivide stress space into elastic and plastic regions, this work aims at investigating the possibility of using SVC for the purpose of constitutive modeling in plasticity. An overview on data-mining methods and statistical learning, also covering the SVM method, is given by Hastie et al. [19].

The present paper is organized as follows: In the next section, the basic concepts of continuum plasticity are briefly summarized, and a new mathematical formulation is introduced, which enables a data-oriented approach to constitutive modeling. Subsequently, a consistent formulation of the SVC method to serve as yield function in continuum plasticity is introduced, which is then trained with artificial data resulting from a Hill-like reference material. Using such data has the advantage of being able to judge the quality of the approximations in an objective way. In the next step, trained ML yield function is applied as constitutive model in simple finite element simulations (Supplementary Materials) to demonstrate its applicability for this purpose. Finally, the results of ML and conventional flow rules are compared and the conclusions drawn from this comparison are presented.
