*3.2. Constitutive Equations*

The constitutive equations describe the relationship between flow stress and hotworking parameters, such as strain, deformed temperatures, and strain rates. Sellars and McTegart developed a hyperbolic sine model in 1960 [24]. This model has been widely used to describe the workability of different alloys. Workability depends on the initiated microstructures, chemical compositions, and processing histories. For example, the annealed materials (O temper) exhibit better workabilities than the deformed materials (H temper). It should also be noted that the friction between the sample's edges and the dies, such as adiabatic heating during the deformation, may significantly influence the true stress–true strain curves. This could be corrected by introducing lubricant during the

deformation process or empirically corrected by estimation [25,26]. Since we used graphite paste on both ends of the samples, the friction effect can be negligible in this case.

In the 1960s, Sellars and McTegart proposed that the isothermal stress–strain relation is based on Arrhenius equations [24]:

$$
\dot{\varepsilon} = f(\sigma) \exp\left(-\frac{Q}{RT}\right) \tag{4}
$$

where the strain rate unit is s<sup>−</sup>1; *R* is gas constant, *Q* is activation energy for hot deformation, unit in kJ/mol; T is isothermal temperature, unit in Kelvin; and *f*(*σ*) is the strain-related equation, also called Zener–Hollomon parameters in some publications.

It turns out that the alloy may exhibit different deformation mechanisms within other strain regions. Therefore, many publications proposed describing the data using different subfunctions for different intervals:

$$\text{When } \alpha \sigma < 0.8, \dot{\varepsilon} = A\_1 \sigma^{\eta\_1} \exp\left(-\frac{Q}{RT}\right) \tag{5}$$

$$\text{When } \alpha \sigma > 1.2, \ \dot{\varepsilon} = A\_2 \exp(\beta \sigma) \exp\left(-\frac{Q}{RT}\right) \tag{6}$$

$$\text{For all other } \mathfrak{a}\sigma, \text{ } \dot{\varepsilon} = A[\sin \mathfrak{h}(a\sigma)]^n \exp\left(-\frac{Q}{RT}\right) \tag{7}$$

where *n*1, *β*, and α are the material's constants, and α = *β*/*n*1. Taking the natural logarithm on both sides of the equation yields:

$$
\ln \dot{\varepsilon} = \ln A\_1 + n\_1 \ln \sigma - Q/RT \tag{8}
$$

$$
\ln \dot{\varepsilon} = \ln A\_2 + \beta \sigma - Q / RT \tag{9}
$$

$$
\ln \dot{\varepsilon} = \ln A + n \ln [\sin \mathbf{h}(\alpha r)] - Q/RT \tag{10}
$$

According to Equation (5), the slope value of the linear relationship between ln . *ε* and ln(*σ*) is the *n*<sup>1</sup> value. According to Equation (6) at different temperatures, the slope value of the linear relationship between ln . *ε* and *σ* is *β*. The α can be calculated accordingly. The detailed plots are shown in Figure 3a–c. The activation energy *Q* value for hot deformation can be extrapolated by linear fitting for 1/T and ln[sin h(*ασ*)] at a given strain rate condition.

**Figure 3.** *Cont*.

**Figure 3.** (**a**) linear fitting for ln(*σ*) and ln . *ε* at different temperatures; (**b**) linear fitting for σ and ln . *ε* at different temperatures; (**c**) linear fitting for ln[sin h(*ασ*)] and ln . *ε* ; and (**d**) linear fitting for 1/T and ln[sin h(*ασ*)] at given strain rate conditions.

Applying this methodology to other strain conditions, one can calculate the activation energy Q values for studied alloys (as shown in Figure 4). It was demonstrated that an alloy with a Zn/Mg ratio of 10.8 exhibited deficient activation energy compared to the other two studied alloys. This could be an indication that this alloy is easy to be deformed. The activation energy for the other two alloys with close Zn/Mg ratios (Zn/Mg ratio of 6.3 and Zn/Mg ratio of 8.3) demonstrated very similar values. However, their values show a different trend with increasing strain.

**Figure 4.** Calculated Activation energy value under different nominal strain conditions for three studied alloys with different Zn/Mg ratios.
