## 2 Materials and Methods
### Virtual Dataset under Investigation  
For the comparative study, the LS-DYNA material model MAT_187_SAMP-1 [27; 28] combined with the failure model GISSMO [28; 29] was chosen, which is suitable to describe specific material characteristics, such as the tension-compression asymmetry and strain-rate dependency, of thermoplastics. The direct inverse MPI method applied is not limited to any particular material model, although the use of such a sophisticated material model allows an extensive investigation and comparison of different architectures and settings due to the complexity of the parameter identification task. As a database for this work, a virtual generated dataset was created by numerical simulations. The underlying material behavior was based on experimental investigations of an additively processed ABS [26] and conforms to the description in [7]. In the following, the continuum mechanical formulations of MAT_187_SAMP-1 relevant to this paper are briefly introduced. For in depth information on MAT_187_SAMP-1 and GISSMO, we refer to [27; 29]. For further information regarding a numerical simulation with the commercial solver LS-DYNA, we refer to [30]. Depending on the provided input of the material card, three different yield curve formulations are achievable with MAT_187_SAMP-1. If tension, compression, and shear test data are provided to the material card, the so-called SAMP-1 yields surface definition results:  
\[f=\sigma_{vm}^{2}-A_{0}-A_{1}p-A_{2}p^{2}\leq 0. \tag{1}\]  
\(\sigma_{vm}\) represents the von Mises stress and can be formulated as:  
\[\sigma_{vm}=\sqrt{\frac{3}{2}[(\sigma_{xx}+p)^{2}+(\sigma_{yy}+p)^{2}+(\sigma_{ zz}+p)^{2}+2\sigma_{xy}^{2}+2\sigma_{yz}^{2}+2\sigma_{xz}^{2}]}. \tag{2}\]  
Here, \(p\) is the first stress invariant and can be expressed as:  
\[p=-\frac{\sigma_{zz}+\sigma_{yy}+\sigma_{zz}}{3}. \tag{3}\]Providing the tensile, compression, and shear test data, the unknown constants \(A_{0}\), \(A_{1}\), and \(A_{2}\) can be calculated as functions of the test results:  
\[A_{0}=3\sigma_{s}^{2}, \tag{4}\]  
\[A_{1}=9\sigma_{s}^{2}\bigg{(}\frac{\sigma_{c}-\sigma_{t}}{\sigma_{c}\sigma_{t}} \bigg{)}, \tag{5}\]  
\[A_{2}=9\bigg{(}\frac{\sigma_{c}\sigma_{t}-3\sigma_{s}^{2}}{\sigma_{c}\sigma_{t }}\bigg{)}. \tag{6}\]  
Although shear test data were generated in respective virtual experiments, they were not used as input for the material card, but only as a target curve of the parameter identification task and thus as input for the NN. Consequently, only tensile and compression input data were provided to generate the yield surface, which is why the Drucker-Prager yield surface from MAT_187_SAMP-1 was used and the remaining curve was calculated internally from the other two available curves:  
\[\sigma_{s}=\frac{2\sigma_{c}\sigma_{t}}{\sqrt{3}(\sigma_{t}+\sigma_{c})}. \tag{7}\]  
Depending on the definition of the plastic Poisson's ratio (PPR), MAT_187_SAMP-1 offers the possibility to define an associated or a non-associated flow rule. Since in this work the PPR changing under loading is to be considered in the simulation, the change in PPR is defined as a function of plastic strain, thereby generating the non-associated yield rule. The yield rule is given by:  
\[g=\sqrt{q^{2}+\alpha p^{2}}, \tag{8}\]  
with \(\alpha\) being the angle between the hydrostatic axis and plastic potential and thus a function of plastic strain. By the following formula, \(\alpha\) can be obtained:  
\[\alpha=\frac{9}{2}\cdot\frac{1-2\
u_{p}}{1+\
u_{p}}. \tag{9}\]  
For strain-rate dependency, the load curve defining the yield stress in uniaxial tension is replaced by a table definition, assuming dynamic test data are available. Thus, the table contains multiple load curves corresponding to the respective plastic strain rate. In MAT_187_SAMP-1, these rate effects are assumed to be similar to tension for compression and shear. For a more detailed description, please refer to [27; 28].