2.4.3. Arrhenius-Type Model Constitutive Relations

Last, we consider an Arrhenius-type (AR) constitutive model [19], a strain-compensated equation aiming to reproduce the behavior of metals at high temperature. As in the previous constitutive laws, the AR model is a classical *J*<sup>2</sup> plasticity model with an elaborated expression for the yield stress *σy*. In this case, it is defined as

$$
\sigma\_y = \frac{1}{\alpha(\varepsilon\_p)} \sinh^{-1} \left( \frac{Z(\varepsilon\_{p'} \varepsilon\_{p'} T)}{A(\varepsilon\_p)} \right)^{1/n} \tag{25}
$$

where *<sup>α</sup>* : <sup>R</sup> <sup>→</sup> <sup>R</sup> and *<sup>A</sup>* : <sup>R</sup> <sup>→</sup> <sup>R</sup> are two functions employed to represent the influence of the plastic strain on the response and *<sup>n</sup>* is a material exponent. On the other hand, *<sup>Z</sup>* : <sup>R</sup> <sup>×</sup> <sup>R</sup> <sup>×</sup> <sup>R</sup><sup>+</sup> <sup>→</sup> <sup>R</sup> is the so-called Zener–Holloman function, accounts for the effects of strain rate *ε*˙*<sup>p</sup>* and temperature *T*, and is defined as

$$Z(\varepsilon\_p, \varepsilon\_p, T) := \varepsilon\_p^\cdot \exp\left(\frac{Q(\varepsilon\_p)}{RT}\right),\tag{26}$$

where *<sup>R</sup>* is the universal gas constant and *<sup>Q</sup>* : <sup>R</sup> <sup>→</sup> <sup>R</sup> is the activation energy, assumed to be a third-order polynomial.

The scalar functions that enter the definition of the yield function are thus *α*, *A* and *Q*. The three are defined parametrically as

$$\begin{aligned} \alpha(\varepsilon\_p) &= \alpha\_0 + \mathfrak{a}\_1 \varepsilon\_p + \mathfrak{a}\_2 \varepsilon\_p^2 + \mathfrak{a}\_3 \varepsilon\_p^3, \\ Q(\varepsilon\_p) &= Q\_0 + Q\_1 \varepsilon\_p + Q\_2 \varepsilon\_p^2 + Q\_3 \varepsilon\_p^3, \\ A(\varepsilon\_p) &= \exp\left[A\_0 + A\_1 \varepsilon\_p + A\_2 \varepsilon\_p^2 + A\_3 \varepsilon\_p^3\right], \end{aligned} \tag{27}$$

where *α*0, . . . , *α*3, *Q*0, . . . , *Q*3, and *A*0, . . . , *A*<sup>3</sup> are material constants determined experimentally. Depending on the author, these three functions might adopt slightly different forms leading to potentially higher accuracy at the expense of more difficulties for their calibration.
