## 1 Introduction
The Johnson-Cook model is the most widely used because it is simple to identify and use and has few parameters to determine [25; 26]. Once the choice has been made concerning the type of flow law to be used for a material, it is then necessary, from a set of experimental tests carried out in the laboratory under conditions close to those of the structure in service, to identify the parameters of these flow laws by machine learning methods based on approaches of minimization of the calculated experiment. Therefore, the use of the Johnson-Cook flow law defined by Equation (2) requires the identification of 5 material parameters. The main problem that researchers are confronted with after the phase of realization of the experimental tests concerns the choice of the flow law to use according to the observations made on these test results. This choice of flow law is also restricted by the FE code used and the availability of such flow laws. Thus, a user of the Abaqus FE code will turn more particularly to a Johnson-Cook [8] flow law, where it is natively implemented in this software. The choice of another form of flow law, Zerilli-Armstrong, or Arrhenius, for example, obliges the user to program himself the computation of the flow stress \(\sigma^{y}\) of the material through a VUMAT subroutine in FORTRAN 77 as proposed by [PERSON] et al. [27], [PERSON] et al. [28] for a Johnson-Cook flow law, or [PERSON] et al. [24] for an Arrhenius type flow law with the following expression:  
\[\sigma^{y}=\frac{1}{\alpha(\varepsilon)}\ln\Bigg{\{}\left(\frac{Z(\varepsilon )}{A(\varepsilon)}\right)^{1/n(\varepsilon)}+\sqrt{1+\left(\frac{Z(\varepsilon )}{A(\varepsilon)}\right)^{2/n(\varepsilon)}}\Bigg{\}} \tag{3}\]  
with  
\[Z(\varepsilon)=\dot{\varepsilon}\exp\Big{(}\frac{Q(\varepsilon)}{RT}\Big{)}, \tag{4}\]where \(Z\) is the Zenner-Hollomon parameter [29], \(Q(\varepsilon)\) is the apparent activation energy (J mol\({}^{-1}\)), \(R\) is the universal gas constant (8.314 J mol\({}^{-1}\)K\({}^{-1}\)). \(Q(\varepsilon)\), \(A(\varepsilon)\), \(\alpha(\varepsilon)\) and \(n(\varepsilon)\) are expressed as a function of the strain \(\varepsilon\) through polynomial functions of degree \(m\) (varying from 1 to 9), which leads to the identification of up to 36 material parameters. Implementing the flow law as a VUMAT FORTRAN subroutine requires the computation of the derivatives \(\partial\sigma^{y}/\partial\varepsilon^{y}\), \(\partial\sigma^{y}/\partial\hat{\varepsilon}\) and \(\partial\sigma^{y}/\partial T\) of the flow stress \(\sigma^{y}\), which can quickly become relatively complex as the complexity of the flow law increases, i.e., the relative complexity of the Arrhenius flow law defined by Equations (3) and (4), regarding the relative simplicity of the Johnson-Cook model defined by Equation (2), one can refer to the work proposed by [PERSON] et al. [24] for details concerning this implementation using the safe version of the Newton-Raphson method proposed by [PERSON] et al. [28]. The choice of the flow law to use for a problem is therefore doubly guided by the behavior of the material on the one hand, but a more important aspect is the list of flow laws implemented natively in the FE code we plan to use for the numerical simulation. At this time, there is not yet a flow law generic enough to cover a wide range of material behavior that is simple to implement and use. As we have seen in the previous paragraph, the choice of the flow law to use is guided mainly by the list of flow laws available in the finite element code used, and very often, this choice is made at the expense of the quality of the model. For example, [PERSON] et al. [14], proposed the identification of the flow law of a GCr15 alloy for a continuous casting bloom with heavy reduction application as introduced by [PERSON] et al. [30], who performed compression tests on this material. In their study, [PERSON] et al.