## 2 Numerical Models
### Plasticity Models  
Among the several numerical models that can describe flow stress in a material as a function of strain, strain rate and temperature, two different formulations are considered: the Johnson and Cook (JC) [32] and the Zerilli-Armstrong (ZA) [5]. The JC is a phenomenological model based on Mises Plasticity that accounts for strain hardening, strain rate sensitivity and thermal softening. For this reason, it is generally used in adiabatic transient dynamic analyses. The constitutive relation of the model is given by:  
\[\sigma=[A+B\epsilon^{n}]\left[1+Cln\left(\frac{\dot{\epsilon}}{\dot{\epsilon_{0} }}\right)\right]\left[1-\left(\frac{T-T_{0}}{T_{melt}-T_{0}}\right)^{m}\right] \tag{1}\]  
where \(\epsilon\) is the plastic equivalent strain, \(\dot{\epsilon}\) is the plastic strain rate, \(\dot{\epsilon_{0}}\) is a reference strain rate, \(T_{0}\) is the room temperature and \(T_{melt}\) s the melting temperature of the material. The first square bracket of Equation (1) refers to strain hardening, and the terms in the second and third square brackets account for strain rate and temperature, respectively. The model is characterized by five material constants: \(A\) is the yield strength, \(B\) is the hardening coefficient, \(n\) is the hardening exponent, \(C\) is the sensitivity strain rate coefficient and \(m\) is the thermal softening exponent. The limitation of this model is the lack of interaction between the effects of strain rate and temperature, which are regarded as decoupled. It is a model that works best at low temperatures and moderate strain rates (\(10^{2}\) s\({}^{-1}\)).  
On the contrary, the second model considered above, i.e., ZA (see Equation (2)), which is based on dislocation mechanisms, regards the effects of strain rate and temperature as coupled, and it is better suited to higher strain rates.  
\[\sigma=\sigma_{0}+C_{1}e^{\left(-C_{3}T+C_{4}Tln~{}(\dot{\epsilon})~{}\right) }+C_{5}\epsilon^{n} \tag{2}\]  
Furthermore, it is a model that considers the effects of the internal structure of the material on its plastic behavior. In fact, there are two different expressions, one for materials with a Body-Centered Cubic structure (BCC) and one with a Face-Centered Cubic structure (FCC). Equation (2) is for BCC structures and is characterized by six material constants: \(\sigma_{0}\), \(C_{1}\), \(C_{3}\), \(C_{4}\), \(C_{5}\) and \(n\), the strain hardening exponent.  
The material constants of both models can be derived through the following inverse procedure with the aid of the finite element method. A suitable number of dynamic experimental tests are carried out under different conditions, namely, a tensile test on round bar (RB) and round notched bar specimens (RN), and a test under plane strain (PS) conditions, and the experimental load-displacement curves are derived. Afterwards, a finite element model of the experimental tests is carried out in which the material behavior is described by the plasticity model to be calibrated. Through a trial and error minimization procedure, several finite element simulations are performed by varying the material constants until a load-displacement curve as close as possible to that found experimentally is obtained.