###### Abstract  
Finite element (FE) simulations have been effective in simulating thermomechanical forming processes, yet challenges arise when applying them to new materials due to nonlinear behaviors. To address this, machine learning techniques and artificial neural networks play an increasingly vital role in developing complex models. This paper presents an innovative approach to parameter identification in flow laws, utilizing an artificial neural network that learns directly from test data and automatically generates a Fortran subroutine for the Abaqus standard or explicit FE codes. We investigate the impact of activation functions on prediction and computational efficiency by comparing Sigmoid, Tanh, ReLU, Swish, Softplus, and the less common Exponential function. Despite its infrequent use, the Exponential function demonstrates noteworthy performance and reduced computation times. Model validation involves comparing predictive capabilities with experimental data from compression tests, and numerical simulations confirm the numerical implementation in the Abaqus explicit FE code. constitutive behavior; ANN flow law; numerical implementation; user hardening; activation functions; abaqus +
Footnote †: journal:  
0000-0001-537X/0000-0001-537X/0000-00001-537X/0000-00001-537X/00000-00001-537X/00000-00001-537X/00000-0001-537X/00  
The historical development of behavior laws for simulating hot forming processes, beginning in the 1950s, involved the use of power laws to describe strain/stress relationships, later adapted to incorporate temperature effects. In the 1970s, thermomechanical models evolved to include time dependence, linking flow laws to strain rate and time. Notable models like the Johnson-Cook ([PERSON] and [PERSON], 1971), Zerilli-Armstrong ([PERSON] and [PERSON], 1972), and Arrhenius ([PERSON], 1972) flow laws emerged and are commonly used in forming process simulations. The selection of a flow law for simulating material behavior in finite element analysis is crucial, and it should be based on experimental tests conducted under conditions resembling real-world applications. Researchers face a challenge in choosing the appropriate flow law after characterizing experimental behavior. This decision is influenced by the availability of flow laws within the finite element analysis software being used. For instance, users of Abaqus FE code ([PERSON] and [PERSON], 1971) may prefer the native implementation of the Johnson-Cook flow law. Opting for alternative flow laws like Zerilli-Armstrong or Arrhenius requires users to personally compute material yield stress \(\sigma\) through a VUMAT subroutine in Fortran, which is time-consuming, demands expertise in flow law formulations, numerical integration, Fortran programming, and model development and testing ([PERSON] et al., 2016; [PERSON] et al., 2017; [PERSON] et al., 2017). Developing and implementing a user behavior law on the Abaqus code involves calculating \(\sigma\), defined by Equation (1), and its three derivatives with respect to \(\dot{\epsilon}\), \(\dot{\epsilon}\), and \(T\). This process becomes complex and time-consuming with increasing flow law complexity. The choice of a flow law for simulating thermomechanical forming is influenced by both material behavior and process physics, but primarily by the flow laws available in the FE code. The decision often prioritizes the availability of a flow law over the quality of the model, making the choice of flow law a critical factor in the simulation process. In [PERSON] et al. (2016), several flow laws were investigated through experimental compression tests on a Gleeble-3800 thermomechanical simulator for P20 medium-carbon steel used in foundry mold production. The examined flow laws included Johnson-Cook, Modified Zerilli-Armstrong, Arrhenius, Hensel-Sputtle, and PTM. The findings revealed that, among the tested strain rates and temperatures, only the Arrhenius flow law accurately replicated the compression behavior of the material with a fidelity suitable for industrial applications.