## 1 Introduction  
Numerical methods for simulating the behavior of structures subjected to high thermomechanical loads, as in the case of the high-temperature forming of metallic materials, are generally based on the use of commercial finite element (FE) codes, such as Abaqus, or laboratory codes, such as DynELA [1]. These FE codes are based on two types of equations: conservation equations and constitutive equations. If the first equations are well established on the basis of physics and mechanics, it is not the same for the second type of equations: the constitutive equations. Thus, in a general way, the conservation equations concern the fundamental principles of physics, such as the mass conservation law, the momentum law (fundamental equation) and the energy law (declined as the first and second principles of thermodynamics). By themselves, these laws are not sufficient to describe the behavior of a material or a structure subjected to thermomechanical solicitations because the nature of the material's behavior translated through the behavior laws is not included in the system previously proposed. Therefore, for each type of material, it is necessary to define behavior laws whose formulation is based on observation in order to describe the behavior of this material under external forces. The quality and the accuracy of the results of any numerical simulation depend on the choice of these behavior laws and on the ability of the user to identify the coefficients of these behavior laws for a given material by performing experiments under conditions close to those encountered during the real stress of the structure in service that one wishes to design [2]. Depending on the nature of the solicitations, these tests are based on quasi-static or dynamic tensile or compression tests, tests on thermomechanical simulators such as Gleeble [3] or impact tests using gas launchers or Hopkinson bars [4]. In the thermomechanical simulation of forming processes, these behavior laws define the dependence [5] of the flow stress of the material \(\sigma^{y}\) as a function of the three input variables, which are the plastic strain \(\varepsilon^{p}\), the strain rate \(\dot{\varepsilon}\) and the temperature \(T\) of the material, so that the general form of the flow law can be written with the following expression:  
\[\sigma^{y}=f(\varepsilon^{p},\dot{\varepsilon},T) \tag{1}\]  
These laws, due to the nature of materials and the phenomena involved [6; 7] (work hardening, movement of dislocations, structural hardening, phase transformations, etc.) are highly non-linear, and their validity is restricted to a certain range of strains \(\varepsilon\), strain rates \(\dot{\varepsilon}\) and temperatures \(T\). From the observations made, we can define two main classes of behavior laws: the flow laws based on physics and the empirical flow laws. From the mechanics of continuous media and experimental tests and depending on the materials used, several flow models have been developed in the past, including the Johnson-Cook flow law [8; 9], the Zerilli-Armstrong flow law [10] and their respective derived forms [11; 12; 13; 14; 15; 16; 17; 18; 19], the Hansel-Spittle [20; 21] or the Arrhenius [22; 23; 24] flow laws, to name only a few of the most widely used in the metal-forming processes at high temperature. As an example, and because it is widely used in numerical simulation of metal forming processes, the equation that describes the Johnson-Cook flow law [8] is given as follows:  
\[\sigma^{y}=\Big{(}A+Be^{p^{n}}\Big{)}\Big{[}1+C\ln\Big{(}\frac{\dot{\varepsilon }}{\dot{\varepsilon}_{0}}\Big{)}\Big{]}\Big{[}1-\Big{(}\frac{T-T_{0}}{T_{m}- T_{0}}\Big{)}^{m}\Big{]}, \tag{2}\]  
where \(A\) is the initial elastic limit of the material, \(B\) is the strain hardening coefficient, \(n\) is the strain hardening exponent, and \(C\) and \(m\) are the material constants that describe the strain rate hardening coefficient and the thermal softening coefficient, respectively.