*4.1. Simplified Johnson–Cook Model*

The simplified Johnson–Cook model is widely accepted to describe the coupling factors among stress, strain, and strain rate [25,36]. The profile of the stress–strain curves of FFRCs was similar to those of traditional metals with a well-defined Johnson–Cook model. Owing to the difficulty in extracting temperature data and the slight effect of temperature on constitutive behaviors under low impact energy, for the sake of simplicity, only isotropic hardening and strain-rate hardening effects were considered in this study. Therefore, the dynamic behavior of FFRCs can be expressed as

$$
\sigma = (A + B\varepsilon^n) \left( 1 + C \ln \dot{\varepsilon}^\* \right),
\tag{2}
$$

where *σ* is the stress; *A* is the yield stress; *B* and *n* represent the effect of strain hardening, respectively; *C* is the material constant determined by the specific material, representing the strain rate dependence of the material; *ε* is the equivalent plastic strain and obtained by subtracting the elastic strain from the total strain; .*ε* is the strain rate; and .*ε*∗ is the dimensionless plastic strain rate expressed as .*<sup>ε</sup>*/.*<sup>ε</sup>*0, where . *ε*0 = 0.006 s<sup>−</sup><sup>1</sup> on the basis of quasi-static experiments.

Naturally, in the quasi-static experiment for .*ε*∗ = 1, the constitutive model of Equation (2) can be further simplified to

$$
\sigma = A + B\varepsilon^n,\tag{3}
$$

Taking the logarithm of both sides of Equation (3) may result in the following:

$$
\ln(\sigma - A) = \ln B + n \ln \varepsilon,\tag{4}
$$

Subsequently, Equation (4) is applied to fit the quasi-static experimental data in logarithmic coordinates by the least square method, such that ln *B* represents the intercept of the straight line, and *n* represents the slope. Thus, *B* and *n* can be determined using simple mathematical conversion. At room temperature, *C* can be obtained through the fitting in accordance with Equation (2)

$$\frac{\sigma\_2(\dot{\varepsilon})}{\sigma\_1} - 1 = C \ln \frac{\dot{\varepsilon}}{\dot{\varepsilon}\_0},\tag{5}$$

where *σ*1 = *A* + *Bεn*, *σ*1 represents the yield stress when the strain rate is 0.006 s<sup>−</sup>1, and *<sup>σ</sup>*2- .*ε* is the yield stress at the strain rate of .*ε*.

The simplified Johnson–Cook model (Equation (2)) was used to describe the dynamic rate-dependent constitutive behavior of the FFRCs. The fitting parameters in the constitutive models in accordance with the experimental data are listed in Table 2 and the constitutive relationship was obtained as follow: *σ* = -102.0 + 70.8*ε*0.416<sup>1</sup> + 0.047 ln .*ε* 0.006. The fitting curves of the model and the experimental data are illustrated and compared in Figure 7.

**Figure 7.** Comparison of the fitting curves of the experimental data.
