Determination of Young Modulus and Stress-Strain Curve for Metal Fe and Interstitial Alloy FeC

Abstract

In this research, the numerical calculation for elastic and nonlinear strains of Fe metal and FeC alloy under different pressures has been performed by the statistical moment method SMM with Mie–-Lennard–Jones potential (MLJ) and Embedded-Jones potential Atom Method (EAM). The analysis reveals that an enhancement in the concentration (cC) from 0 to 5% causes a decrement in the Young’s modulus (E) at room temperature (T = 300 K) for FeC. These calculated results are consistent with the experimental results. In addition, the obtained stress-strain curves for Fe are in perfect agreement with the experimental curves. Besides, increasing the cC for a continuous strain decreases the stress, showing that adding C to Fe to form FeC steel will increase strength and hardness, but decrease elasticity and hardness. The results obtained will be very useful not only for experimental studies but also for theoretical studies of metals and their interstitial alloys.

1. Introduction

Today with the rapid development of science and technology, the role of computer science, combined with research methods, is crucial to the success of new materials research. Among the research methods, the Molecular Dynamics Simulation (MDS) method plays the leading role because of its simplicity. This research method has been used since the 1950s, but it was only developed in 1980 based on the first IBM computer system, and the theoretical models were perfected up to now thanks to the development of supercomputing systems combined with new innovative algorithms, to globally optimize the physical-chemical properties of materials, based on the theoretical models of solids [1]. Various studies based on MDS have been reported in the literature [2,3], and scientists are always interested in issues such as the stability of materials, or whether the selection of the proposed parameters in the model is consistent with the experimental data. Furthermore, the sensitive dependence of the model on the initial conditions, which can significantly affect the results, needs to be considered [4]. Scientists have successfully studied the phase transformation, electronic structure, and crystallization of Ni [5], Fe [6,7,8,9,10,11], Ni [12,13,14], Al [15,16], Cu [17], and alloy [18,19], NiFe nanoparticles [20,21], NiCu [22,23,24], AlNi nanoparticles [25], AgAu alloy [26], AuCu bulk [27,28], NiAu alloy [29], polymer [30], Fe₂O₃ [31,32], and FeCoNi [33] using experimental and MDS methods. It is well known that metals and their interstitial alloys have found various applications in a wide variety of technological and industrial areas. Special interest is directed to the assessment of the elastic properties of metals and their interstitial alloys determined experimentally, analytically, and numerically in various conditions [34,35,36,37,38,39,40]. The experimental stress-strain curves obtained for Fe were reported across different studies [19,38,39,40]. From this point of view, there are many studies concerning the dependence of the elastic properties of materials on factors such as pressure (P) and temperature (T). There are several theoretical methods, based on various mathematical models, used to investigate the thermodynamic and mechanical characteristics of materials in the literature [41,42,43].

For example, the Full-Potential linear response Linear-Muffin-Tin Orbital (FP-LMTO) method has been used to investigate the elastic deformation of a single metal Fe, where the elastic moduli of body-centered cubic (BCC) Fe under temperature and pressure were investigated by the First-Principles Quasiharmonic Lattice Dynamics (FP-QLD) method [8].

The elastic deformation of Fe and its alloys were evaluated in Resonance Ultrasound Spectroscopy (RUS) experiments over a range form 3 K to 500 K [9], using pulse-echo technique experiments in the range of components from zero to 10 at. pct. Elastic moduli E and G nonlinearly depend on temperature for alloys as well as Fe in the range from 77 K to 473 K [10].

In recent studies, scientists have successfully investigated the elastic deformation under pressure in BCC and face-centered cubic (FCC) interstitial alloys using the Statistical Moment Method (SMM) [36,37,44,45]. In this study, the SMM was applied to perform numerical calculations for nonlinear and elastic deformations of BCC-Fe and BCC-FeC under pressure.

2. Theoretical Background

In the approximation of three coordination spheres, the cohesive energy ${\mathrm{u}}_0$ and the $\mathrm{k},{\mathsf{\gamma }}_1,{\mathsf{\gamma }}_2,\mathsf{\gamma }$ alloy parameters for the interstitial atom B (in the face centers of the cubic unit cell), the main metal atom A₁ (in the body center of the cubic unit cell), and the main metal atom A₂ (in the vertices of the cubic unit cell) in the BCC interstitial alloy AB can be expressed as [46,47]:

$${\mathrm{u}}_{0\mathrm{B}}=\frac{1}{2}{\displaystyle \sum _{\mathrm{i}=1}^{{\mathrm{n}}_{\mathrm{i}}}{\mathsf{\phi }}_{\mathrm{AB}}{(\mathrm{r}}_{\mathrm{i}})}={\mathsf{\phi }}_{\mathrm{AB}}\left({\mathrm{r}}_1\right)+2{\mathsf{\phi }}_{\mathrm{AB}}\left({\mathrm{r}}_1\sqrt{2}\right)+4{\mathsf{\phi }}_{\mathrm{AB}}\left({\mathrm{r}}_1\sqrt{5}\right),$$

(1)

$${\mathrm{k}}_{\mathrm{B}}=\frac{1}{2}{\displaystyle \sum _{\mathrm{i}}{\left(\frac{{\partial }^2{\mathsf{\phi }}_{\mathrm{AB}}}{\partial {\mathrm{u}}_{\mathrm{i}\mathsf{\beta }}^2}\right)}_{\mathrm{eq}}}={\mathsf{\phi }}_{\mathrm{AB}}^{\left(2\right)}\left({\mathrm{r}}_1\right)+\frac{\sqrt{2}}{{\mathrm{r}}_1}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(1\right)}\left({\mathrm{r}}_1\sqrt{2}\right)+\frac{16}{5\sqrt{5}{\mathrm{r}}_1}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(1\right)}\left({\mathrm{r}}_1\sqrt{5}\right),$$

(2)

$${\gamma }_{\mathrm{B}}=4\left({\gamma }_{1\mathrm{B}}+{\gamma }_{2\mathrm{B}}\right),$$

(3)

$$\begin{array}{ll}{\mathsf{\gamma }}_{1\mathrm{B}}=\frac{1}{48}{\displaystyle \sum _{\mathrm{i}}{\left(\frac{{\partial }^4{\mathsf{\phi }}_{\mathrm{AB}}}{\partial {\mathrm{u}}_{\mathrm{i}\mathsf{\beta }}^4}\right)}_{\mathrm{eq}}}=\frac{1}{24}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(4\right)}& \left({\mathrm{r}}_1\right)+\frac{1}{{8\mathrm{r}}_1^2}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(2\right)}\left({\mathrm{r}}_1\sqrt{2}\right)-\frac{\sqrt{2}}{{16\mathrm{r}}_1^3}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(1\right)}\left({\mathrm{r}}_1\sqrt{2}\right)+\\ & +\frac{1}{150}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(4\right)}\left({\mathrm{r}}_1\sqrt{2}\right)+\frac{4\sqrt{5}}{{125\mathrm{r}}_1}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(3\right)}\left({\mathrm{r}}_1\sqrt{5}\right),\end{array}$$

(4)

$$\begin{array}{l}{\mathsf{\gamma }}_{2\mathrm{B}}=\frac{6}{48}{\displaystyle \sum _{\mathrm{i}}{\left(\frac{{\partial }^4{\mathsf{\phi }}_{\mathrm{AB}}}{\partial {\mathrm{u}}_{\mathrm{i}\mathsf{\alpha }}^2\partial {\mathrm{u}}_{\mathrm{i}\mathsf{\beta }}^2}\right)}_{\mathrm{eq}}=}\frac{1}{{4\mathrm{r}}_1}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(3\right)}\left({\mathrm{r}}_1\right)-\frac{1}{{4\mathrm{r}}_1^2}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(2\right)}\left({\mathrm{r}}_1\right)+\frac{5}{{8\mathrm{r}}_1^3}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(1\right)}\left({\mathrm{r}}_1\right)+\frac{\sqrt{2}}{{8\mathrm{r}}_1}{\mathsf{\varphi }}_{\mathrm{AB}}^{\left(3\right)}\left({\mathrm{r}}_1\sqrt{2}\right)-\frac{1}{{8\mathrm{r}}_1^2}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(2\right)}\left({\mathrm{r}}_1\sqrt{2}\right)+\\ +\frac{1}{{8\mathrm{r}}_1^3}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(1\right)}\left({\mathrm{r}}_1\sqrt{2}\right)+\frac{2}{25}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(4\right)}\left({\mathrm{r}}_1\sqrt{5}\right)+\frac{3}{25\sqrt{5}{\mathrm{r}}_1}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(3\right)}\left({\mathrm{r}}_1\sqrt{5}\right)+\frac{2}{{25\mathrm{r}}_1^2}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(2\right)}\left({\mathrm{r}}_1\sqrt{5}\right)-\frac{3}{25\sqrt{5}{r}_1^3}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(1\right)}\left({\mathrm{r}}_1\sqrt{5}\right),\end{array}$$

(5)

$${\mathrm{u}}_{{0\mathrm{A}}_1}={\mathrm{u}}_{0\mathrm{A}}+{\mathsf{\phi }}_{\mathrm{AB}}\left({\mathrm{r}}_{{1\mathrm{A}}_1}\right),$$

(6)

$${\gamma }_{{\mathrm{A}}_1}=4\left({\gamma }_{{1\mathrm{A}}_1}+{\gamma }_{{2\mathrm{A}}_1}\right),$$

(7)

$${\mathrm{k}}_{{\mathrm{A}}_1}={\mathrm{k}}_{\mathrm{A}}+\frac{1}{2}{\displaystyle \sum _{\mathrm{i}}{\left[{\left(\frac{{\partial }^2{\mathsf{\phi }}_{\mathrm{AB}}}{\partial {\mathrm{u}}_{\mathrm{i}\mathsf{\beta }}^2}\right)}_{\mathrm{eq}}\right]}_{\mathrm{r}={\mathrm{r}}_{{1\mathrm{A}}_1}}}{=\mathrm{k}}_{\mathrm{A}}+{\mathsf{\phi }}_{\mathrm{AB}}^{\left(2\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_1}\right)+\frac{5}{{2\mathrm{r}}_{{1\mathrm{A}}_1}}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(1\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_1}\right),$$

(8)

$${\mathsf{\gamma }}_{{1\mathrm{A}}_1}={\mathsf{\gamma }}_{1\mathrm{A}}+\frac{1}{48}{\displaystyle \sum _{\mathrm{i}}{\left[{\left(\frac{{\partial }^4{\mathsf{\phi }}_{\mathrm{AB}}}{\partial {\mathrm{u}}_{\mathrm{i}\mathsf{\beta }}^4}\right)}_{\mathrm{eq}}\right]}_{\mathrm{r}={\mathrm{r}}_{{1\mathrm{A}}_1}}=}{\mathsf{\gamma }}_{1\mathrm{A}}+\frac{1}{24}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(4\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_1}\right)+\frac{1}{{8\mathrm{r}}_{{1\mathrm{A}}_1}^2}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(2\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_1}\right)-\frac{1}{{8\mathrm{r}}_{{1\mathrm{A}}_1}^3}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(1\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_1}\right),$$

(9)

$${\mathsf{\gamma }}_{{2\mathrm{A}}_1}={\mathsf{\gamma }}_{2\mathrm{A}}+\frac{6}{48}{\displaystyle \sum _{\mathrm{i}}{\left[{\left(\frac{{\partial }^4{\mathsf{\phi }}_{\mathrm{AB}}}{\partial {\mathrm{u}}_{\mathrm{i}\mathsf{\alpha }}^2\partial {\mathrm{u}}_{\mathrm{i}\mathsf{\beta }}^2}\right)}_{\mathrm{eq}}\right]}_{\mathrm{r}={\mathrm{r}}_{{1\mathrm{A}}_1}}}{=\mathsf{\gamma }}_{2\mathrm{A}}+\frac{1}{{2\mathrm{r}}_{{1\mathrm{A}}_1}}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(3\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_1}\right)-\frac{3}{{4\mathrm{r}}_{{1\mathrm{A}}_1}^2}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(2\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_1}\right)+\frac{3}{{4\mathrm{r}}_{{1\mathrm{A}}_1}^3}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(1\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_1}\right),$$

(10)

$${\mathrm{u}}_{{0\mathrm{A}}_2}={\mathrm{u}}_{0\mathrm{A}}+{\mathsf{\phi }}_{\mathrm{AB}}\left({\mathrm{r}}_{{1\mathrm{A}}_2}\right),$$

(11)

$${\gamma }_{{\mathrm{A}}_2}=4\left({\gamma }_{{1\mathrm{A}}_2}+{\gamma }_{{2\mathrm{A}}_2}\right),$$

(12)

$${\mathrm{k}}_{{\mathrm{A}}_2}={\mathrm{k}}_{\mathrm{A}}+\frac{1}{2}{\displaystyle \sum _{\mathrm{i}}{\left[{\left(\frac{{\partial }^2{\mathsf{\phi }}_{\mathrm{AB}}}{\partial {\mathrm{u}}_{\mathrm{i}\mathsf{\beta }}^2}\right)}_{\mathrm{eq}}\right]}_{\mathrm{r}={\mathrm{r}}_{{1\mathrm{A}}_2}}}={\mathrm{k}}_{\mathrm{A}}+2{\mathsf{\phi }}_{\mathrm{AB}}^{\left(2\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_2}\right)+\frac{4}{{\mathrm{r}}_{{1\mathrm{A}}_2}}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(1\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_2}\right),$$

(13)

$$\begin{array}{ll}{\mathsf{\gamma }}_{{1\mathrm{A}}_2}={\mathsf{\gamma }}_{1\mathrm{A}}+\frac{1}{48}{\displaystyle \sum _{\mathrm{i}}{\left[{\left(\frac{{\partial }^4{\mathsf{\phi }}_{\mathrm{AB}}}{\partial {\mathrm{u}}_{\mathrm{i}\mathsf{\beta }}^4}\right)}_{\mathrm{eq}}\right]}_{\mathrm{r}={\mathrm{r}}_{{1\mathrm{A}}_2}}}={\mathsf{\gamma }}_{1\mathrm{A}}+\frac{1}{24}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(4\right)}& \left({\mathrm{r}}_{{1\mathrm{A}}_2}\right)+\frac{1}{{4\mathrm{r}}_{{1\mathrm{A}}_2}}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(3\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_2}\right)-\\ & -\frac{1}{{8\mathrm{r}}_{{1\mathrm{A}}_2}^2}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(2\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_2}\right)+\frac{1}{{8\mathrm{r}}_{{1\mathrm{A}}_2}^3}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(1\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_2}\right),\end{array}$$

(14)

$$\begin{array}{ll}{\mathsf{\gamma }}_{{2\mathrm{A}}_2}={\mathsf{\gamma }}_{2\mathrm{A}}+\frac{6}{48}{\displaystyle \sum _{\mathrm{i}}{\left[{\left(\frac{{\partial }^4{\mathsf{\phi }}_{\mathrm{AB}}}{\partial {\mathrm{u}}_{\mathrm{i}\mathsf{\alpha }}^2\partial {\mathrm{u}}_{\mathrm{i}\mathsf{\beta }}^2}\right)}_{\mathrm{eq}}\right]}_{\mathrm{r}={\mathrm{r}}_{{1\mathrm{A}}_2}}}={\mathsf{\gamma }}_{2\mathrm{A}}+\frac{1}{8}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(4\right)}& \left({\mathrm{r}}_{{1\mathrm{A}}_2}\right)+\frac{1}{{4\mathrm{r}}_{{1\mathrm{A}}_2}}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(3\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_2}\right)+\\ & +\frac{3}{{8\mathrm{r}}_{{1\mathrm{A}}_2}^2}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(2\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_2}\right)-\frac{3}{{8\mathrm{r}}_{{1\mathrm{A}}_2}^3}{\mathsf{\phi }}_{\mathrm{AB}}^{\left(1\right)}\left({\mathrm{r}}_{{1\mathrm{A}}_2}\right),\end{array}$$

(15)

where ${\mathsf{\phi }}_{\mathrm{AB}}$ corresponds to the interaction potential between the A and the B atoms. ${\mathrm{r}}_{1\mathrm{X}}={\mathrm{r}}_{01\mathrm{X}}+{\mathrm{y}}_{0\mathrm{X}}(\mathrm{T})$, and ${\mathrm{r}}_{01\mathrm{X}}$ represents the Nearest Neighbor Distance (NND) between the atom X (X = A, A₁, A₂, B) and other atoms at T and T = 0 K, respectively. ${\mathrm{r}}_{01\mathrm{X}}$ is calculated from the minimum condition of the cohesive energy ${\mathrm{u}}_{0\mathrm{X}},{\mathrm{y}}_{0\mathrm{X}}(\mathrm{T})$ and shows the displacement of atom X from equilibrium position at T, ${\mathsf{\phi }}_{\mathrm{AB}}^{\left(\mathrm{m}\right)}\equiv \frac{{\partial }^{\mathrm{m}}{\mathsf{\phi }}_{\mathrm{A}}{(\mathrm{r}}_{\mathrm{i}})}{\partial {\mathrm{r}}_{\mathrm{i}}^{\mathrm{m}}}$, m = 1, 2, 3, 4, α, β = $\mathrm{x},\mathrm{y},\mathrm{z},$ $\mathsf{\alpha }\ne \mathsf{\beta },{\mathrm{r}}_{\mathrm{i}}$ represents the radii of the ith coordination sphere (i = 1, 2, 3), n_i shows the number of particles on this sphere, ${\mathrm{u}}_{0\mathrm{A}},{\mathrm{k}}_{\mathrm{A}},{\mathsf{\gamma }}_{1\mathrm{A}},{\mathsf{\gamma }}_{2\mathrm{A}}$ are the corresponding quantities in the BCC pure metal A in the approximation of two coordination spheres [46].

The equations of state for the conditions (at T and P, and at 0 K and P, respectively) can be expressed as [46]:

$$\mathrm{Pv}=-{\mathrm{r}}_1\left(\frac{1}{6}\frac{\partial {\mathrm{u}}_0}{\partial {\mathrm{r}}_1}+\mathsf{\theta }\mathrm{xcthx}\frac{1}{2\mathrm{k}}\frac{\partial \mathrm{k}}{\partial {\mathrm{r}}_1}\right),\mathrm{v}=\frac{{4\mathrm{r}}_1^3}{3\sqrt{3}}.$$

(16)

$$\mathrm{Pv}=-{\mathrm{r}}_1\left(\frac{1}{6}\frac{\partial {\mathrm{u}}_0}{\partial {\mathrm{r}}_1}+\frac{\hslash {\mathsf{\omega }}_0}{4\mathrm{k}}\frac{\partial \mathrm{k}}{\partial {\mathrm{r}}_1}\right).$$

(17)

Equation (17) allows us to determine the nearest neighbor distance (NND) ${\mathrm{r}}_{1\mathrm{X}}(\mathrm{P},0\left)\right(\mathrm{X}=\mathrm{A},{\mathrm{A}}_1,{\mathrm{A}}_2,\mathrm{B})$, the parameters ${\mathrm{k}}_{\mathrm{X}}(\mathrm{P},0),{\mathsf{\gamma }}_{1\mathrm{X}}(\mathrm{P},0),{\mathsf{\gamma }}_{2\mathrm{X}}(\mathrm{P},0),{\mathsf{\gamma }}_{\mathrm{X}}(\mathrm{P},0)$ at P and 0 K, and the displacement ${\mathrm{y}}_{0\mathrm{X}}(\mathrm{P},\mathrm{T})$ of atom X from the equilibrium position at T and P [46]. This is the reason why we can calculate the NND ${\mathrm{r}}_{1\mathrm{X}}(\mathrm{P},\mathrm{T})$ and the mean NND $\overline{{\mathrm{r}}_{1\mathrm{A}}(\mathrm{P},\mathrm{T})}$ between two atoms A in AB alloy at T and P, based on the following formulas [47]:

$$\begin{array}{ll}{\mathrm{r}}_{1\mathrm{B}}(\mathrm{P},\mathrm{T})={\mathrm{r}}_{1\mathrm{B}}(\mathrm{P},0)+{\mathrm{y}}_{{\mathrm{A}}_1}(\mathrm{P},\mathrm{T}),{\mathrm{r}}_{1\mathrm{A}}(\mathrm{P},\mathrm{T})& ={\mathrm{r}}_{1\mathrm{A}}(\mathrm{P},0)+{\mathrm{y}}_{\mathrm{A}}(\mathrm{P},\mathrm{T}),\\ & {\mathrm{r}}_{{1\mathrm{A}}_1}(\mathrm{P},\mathrm{T})\approx {\mathrm{r}}_{1\mathrm{B}}(\mathrm{P},\mathrm{T}),{\mathrm{r}}_{{1\mathrm{A}}_2}(\mathrm{P},\mathrm{T})={\mathrm{r}}_{{1\mathrm{A}}_2}(\mathrm{P},0)+{\mathrm{y}}_{\mathrm{B}}(\mathrm{P},\mathrm{T}).\end{array}$$

(18)

$$\begin{array}{ll}\overline{{\mathrm{r}}_{1\mathrm{A}}(\mathrm{P},\mathrm{T})}=\overline{{\mathrm{r}}_{1\mathrm{A}}(\mathrm{P},0)}& +\overline{\mathrm{y}(\mathrm{P},\mathrm{T})},\overline{{\mathrm{r}}_{1\mathrm{A}}(\mathrm{P},0)}=\left(1-{\mathrm{c}}_{\mathrm{B}}\right){\mathrm{r}}_{1\mathrm{A}}(\mathrm{P},0)+{\mathrm{c}}_{\mathrm{B}}{\mathrm{r}\text{′}}_{1\mathrm{A}}(\mathrm{P},0),{\mathrm{r}\text{′}}_{1\mathrm{A}}(\mathrm{P},0)=\sqrt{3}{\mathrm{r}}_{1\mathrm{B}}(\mathrm{P},0),\\ & \overline{\mathrm{y}(\mathrm{P},\mathrm{T})}=\left(1-{7\mathrm{c}}_{\mathrm{B}}\right){\mathrm{y}}_{\mathrm{A}}(\mathrm{P},\mathrm{T})+{\mathrm{c}}_{\mathrm{B}}{\mathrm{y}}_{\mathrm{B}}{(\mathrm{P},\mathrm{T})+2\mathrm{c}}_{\mathrm{B}}{\mathrm{y}}_{{\mathrm{A}}_1}(\mathrm{P},\mathrm{T})+{4\mathrm{c}}_{\mathrm{B}}{\mathrm{y}}_{{\mathrm{A}}_2}(\mathrm{P},\mathrm{T}),\end{array}$$

(19)

For the condition c_A << c_B, the Helmholtz Free Energy (HFE) of BCC interstitial alloy AB is defined by the following expressions [47]:

$$\begin{array}{l}{\mathsf{\Psi }}_{\mathrm{AB}}=\mathrm{N}\left[\left(1-{7\mathrm{c}}_{\mathrm{B}}\right){\mathsf{\psi }}_{\mathrm{A}}+{\mathrm{c}}_{\mathrm{B}}{\mathsf{\psi }}_{\mathrm{B}}{+2\mathrm{c}}_{\mathrm{B}}{\mathsf{\psi }}_{{\mathrm{A}}_1}+{4\mathrm{c}}_{\mathrm{B}}{\mathsf{\psi }}_{{\mathrm{A}}_2}\right]-{\mathrm{TS}}_{\mathrm{c}},\\ {\mathsf{\Psi }}_X{=\mathrm{N}\mathsf{\psi }}_{\mathrm{X}}\approx {\mathrm{U}}_{0\mathrm{X}}+{\mathsf{\Psi }}_{0\mathrm{X}}+3\mathrm{N}\{\frac{{\mathsf{\theta }}^2}{{\mathrm{k}}_{\mathrm{X}}^2}\left[{\mathsf{\gamma }}_{2\mathrm{X}}{\mathrm{Y}}_{\mathrm{X}}^2-\frac{{2\mathsf{\gamma }}_{1\mathrm{X}}}{3}\left(1+\frac{{\mathrm{Y}}_{\mathrm{X}}}{2}\right)\right]+\frac{{2\mathsf{\theta }}^3}{{\mathrm{k}}_{\mathrm{X}}^4}[\frac{4}{3}{\mathsf{\gamma }}_{2\mathrm{X}}{\mathrm{Y}}_{\mathrm{X}}\left(1+\frac{{\mathrm{Y}}_{\mathrm{X}}}{2}\right)-\\ -2\left[{\mathsf{\gamma }}_{1\mathrm{X}}^2+{2\mathsf{\gamma }}_{1\mathrm{X}}{\mathsf{\gamma }}_{2\mathrm{X}}\right]\left(1+\frac{{\mathrm{Y}}_{\mathrm{X}}}{2}\right)\left(1+{\mathrm{Y}}_{\mathrm{X}}\right)]\},\\ {\mathsf{\psi }}_{0\mathrm{X}}=3\mathrm{N}\mathsf{\theta }\left[{\mathrm{x}}_{\mathrm{X}}+\ln \left(1-{\mathrm{e}}^{-{2\mathrm{x}}_{\mathrm{X}}}\right)\right],{\mathrm{Y}}_{\mathrm{X}}\equiv {\mathrm{x}}_{\mathrm{X}}{\cot \mathrm{hx}}_{\mathrm{X}},{\mathrm{x}}_{\mathrm{X}}=\frac{\hslash {\mathsf{\omega }}_{\mathrm{X}}}{2\mathsf{\theta }},{\mathsf{\omega }}_{\mathrm{X}}=\sqrt{\frac{{\mathrm{k}}_{\mathrm{X}}}{\mathrm{m}}},\end{array}$$

(20)

where ${\mathsf{\Psi }}_X{=\mathrm{N}\mathsf{\psi }}_{\mathrm{X}}$ gives the HFE of BCC materials consisting only atoms X and ${\mathrm{S}}_{\mathrm{c}}$ is the configurational entropy of the alloy AB.

The Young modulus, the bulk modulus, the shearing modulus and elastic constants for the alloy AB have the form [47]:

$$\begin{array}{l}{\mathrm{E}}_{\mathrm{AB}}=\frac{1}{{\mathsf{\pi }\mathrm{r}}_{1\mathrm{A}}{\mathrm{A}}_{1\mathrm{A}}}\left(1-{7\mathrm{c}}_{\mathrm{B}}+{\mathrm{c}}_{\mathrm{B}}\frac{\frac{{\partial }^2{\mathsf{\psi }}_{\mathrm{B}}}{\partial {\mathsf{\epsilon }}^2}+2\frac{{\partial }^2{\mathsf{\psi }}_{{\mathrm{A}}_1}}{\partial {\mathsf{\epsilon }}^2}+4\frac{{\partial }^2{\mathsf{\psi }}_{{\mathrm{A}}_2}}{\partial {\mathsf{\epsilon }}^2}}{\frac{{\partial }^2{\mathsf{\psi }}_{\mathrm{A}}}{\partial {\mathsf{\epsilon }}^2}}\right),\\ {\mathrm{A}}_{1\mathrm{A}}=\frac{1}{{\mathrm{k}}_{\mathrm{A}}}\left[1+\frac{{2\mathsf{\gamma }}_{\mathrm{A}}^2{\mathsf{\theta }}^2}{{\mathrm{k}}_{\mathrm{A}}^4}\left(1+\frac{{\mathrm{Y}}_{\mathrm{A}}}{2}\right)\left(1+{\mathrm{Y}}_{\mathrm{A}}\right)\right],\\ \frac{{\partial }^2{\mathsf{\psi }}_{\mathrm{X}}}{\partial {\mathsf{\epsilon }}^2}=\{\frac{1}{2}\frac{{\partial }^2{\mathrm{u}}_{0\mathrm{X}}}{\partial {\mathrm{r}}_{1\mathrm{X}}^2}+\frac{3}{4}\frac{\hslash {\mathsf{\omega }}_{\mathrm{X}}}{{\mathrm{k}}_{\mathrm{X}}}{\left[\frac{{\partial }^2{\mathrm{k}}_{\mathrm{X}}}{\partial {\mathrm{r}}_{1\mathrm{X}}^2}-\frac{1}{{2\mathrm{k}}_{\mathrm{X}}}\left(\frac{\partial {\mathrm{k}}_{\mathrm{X}}}{\partial {\mathrm{r}}_{1\mathrm{X}}}\right)\right]}^2\}{4\mathrm{r}}_{01\mathrm{X}}^2+\left(\frac{1}{2}\frac{\partial {\mathrm{u}}_{0\mathrm{X}}}{\partial {\mathrm{r}}_{1\mathrm{X}}}+\frac{3}{2}\hslash {\mathsf{\omega }}_{\mathrm{X}}{\cot \mathrm{hx}}_{\mathrm{X}}\frac{\partial {\mathrm{k}}_{\mathrm{X}}}{\partial {\mathrm{r}}_{1\mathrm{X}}}\right){2\mathrm{r}}_{01\mathrm{X}},\end{array}$$

(21)

$${\mathrm{K}}_{\mathrm{AB}}=\frac{{\mathrm{E}}_{\mathrm{AB}}}{3\left(1-{2\mathsf{\nu }}_{\mathrm{A}}\right)},$$

(22)

$${\mathrm{G}}_{\mathrm{AB}}=\frac{{\mathrm{E}}_{\mathrm{AB}}}{2\left(1+{\mathsf{\nu }}_{\mathrm{A}}\right)},$$

(23)

$${\mathrm{C}}_{11\mathrm{AB}}=\frac{{\mathrm{E}}_{\mathrm{AB}}\left(1-{\nu }_{\mathrm{A}}\right)}{\left(1+{\nu }_{\mathrm{A}}\right)\left(1-2{\nu }_{\mathrm{A}}\right)},$$

(24)

$${\mathrm{C}}_{12\mathrm{AB}}=\frac{{\mathrm{E}}_{\mathrm{AB}}{\nu }_{\mathrm{A}}}{\left(1+{\nu }_{\mathrm{A}}\right)\left(1-2{\nu }_{\mathrm{A}}\right)},$$

(25)

$${\mathrm{C}}_{44\mathrm{AB}}=\frac{{\mathrm{E}}_{\mathrm{AB}}}{2\left(1+{\mathsf{\nu }}_{\mathrm{A}}\right)}.$$

(26)

where $\mathsf{\epsilon }$ is the strain of the alloy AB. The Poisson ratio of the alloy AB can be expressed as [47]:

$${\nu }_{\mathrm{AB}}={\mathrm{c}}_{\mathrm{A}}{\nu }_{\mathrm{A}}+{\mathrm{c}}_{\mathrm{B}}{\nu }_{\mathrm{B}}\approx {\nu }_{\mathrm{A}},$$

(27)

where ${\nu }_{\mathrm{A}},{\nu }_{\mathrm{B}}$ correspond to the Poisson ratios of materials A and B, respectively.

The NND ${\mathrm{r}}_{1\mathrm{X}}\left(\mathrm{P},0\right)\left(\mathrm{X}={\mathrm{B},\mathrm{A},\mathrm{A}}_1{,\mathrm{A}}_2\right)$ in alloy at P and 0 K, at P and T after deformation, can be expressed as [44]:

$${\mathrm{r}}_{1\mathrm{X}}^{\mathrm{F}}\left(\mathrm{P},0\right)=r_{1\mathrm{X}}\left(\mathrm{P},0\right)\left(1+\epsilon \right).$$

(28)

$${\mathrm{r}}_{1\mathrm{X}}^{\mathrm{F}}\left(\mathrm{P},\mathrm{T}\right)={\mathrm{r}}_{1\mathrm{X}}\left(\mathrm{P},\mathrm{T}\right)+\mathsf{\epsilon }.{\mathrm{r}}_{1\mathrm{X}}\left(\mathrm{P},0\right)\left(2+\mathsf{\epsilon }\right).$$

(29)

The HFE of AB alloy after deformation can be expressed as [44]:

$$\begin{array}{l}{\mathsf{\Psi }}_{\mathrm{A}\mathrm{B}}^{\mathrm{F}}=\frac{1}{\mathrm{N}}\left[\left(1-{7\mathrm{c}}_{\mathrm{B}}\right){\mathsf{\psi }}_{\mathrm{A}}^{\mathrm{F}}+{\mathrm{c}}_{\mathrm{B}}{\mathsf{\psi }}_{\mathrm{B}}^{\mathrm{F}}{+2\mathrm{c}}_{\mathrm{B}}{\mathsf{\psi }}_{{\mathrm{A}}_1}^{\mathrm{F}}+{4\mathrm{c}}_{\mathrm{B}}{\mathsf{\psi }}_{{\mathrm{A}}_2}^{\mathrm{F}}\right]-{\text{TS}}_{\mathrm{c}},\\ {\mathsf{\Psi }}_{\mathrm{X}}^{\mathrm{F}}{=\mathrm{N}\mathsf{\psi }}_{\mathrm{X}}^{\mathrm{F}}\approx {\mathrm{U}}_{0\mathrm{X}}^{\mathrm{F}}+{\mathsf{\Psi }}_{0\mathrm{X}}^{\mathrm{F}}+3\mathrm{N}\{\frac{{\mathsf{\theta }}^2}{{\mathrm{k}}_{\mathrm{X}}^{\mathrm{F}2}}\left[{\mathsf{\gamma }}_{2\mathrm{X}}^{\mathrm{F}}{\mathrm{Y}}_{\mathrm{X}}^{\mathrm{F}2}-\frac{2{\mathsf{\gamma }}_{1\mathrm{X}}^{\mathrm{F}}}{3}\left(1+\frac{{\mathrm{Y}}_{\mathrm{X}}^{\mathrm{F}}}{2}\right)\right]+\frac{{2\mathsf{\theta }}^3}{{\mathrm{k}}_{\mathrm{X}}^{\mathrm{F}4}}[\frac{4}{3}{\mathsf{\gamma }}_{2\mathrm{X}}^{\mathrm{F}2}{\mathrm{Y}}_{\mathrm{X}}^{\mathrm{F}}\left(1+\frac{{\mathrm{Y}}_{\mathrm{X}}^{\mathrm{F}}}{2}\right)\\ -2\left({\mathsf{\gamma }}_{1\mathrm{X}}^{\mathrm{F}2}+2{\mathsf{\gamma }}_{1\mathrm{X}}^{\mathrm{F}}{\mathsf{\gamma }}_{2\mathrm{X}}^{\mathrm{F}}\right)\left(1+\frac{{\mathrm{Y}}_{\mathrm{X}}^{\mathrm{F}}}{2}\right)\left(1+{\mathrm{Y}}_{\mathrm{X}}^{\mathrm{F}}\right)]\},& \\ {\mathsf{\Psi }}_{0\mathrm{X}}^{\mathrm{F}}=3\mathrm{N}\mathsf{\theta }\left[{\mathrm{x}}_{\mathrm{X}}^{\mathrm{F}}+\text{ln}(1-{\mathrm{e}}^{-2{\mathrm{x}}_{\mathrm{X}}^{\mathrm{F}}})\right],{\mathrm{Y}}_{\mathrm{X}}^{\mathrm{F}}\equiv {\mathrm{x}}_{\mathrm{X}}^{\mathrm{F}}{\cot \mathrm{hx}}_{\mathrm{X}}^{\mathrm{F}},& \end{array}$$

(30)

where ${\mathsf{\Psi }}_{\mathrm{X}}^{\mathrm{F}}$ is the HFE of materials consisting of only A atoms with the structure bcc after deformation.

The stress dependence on strain in non-linear deformation can be expressed as [44]:

$${\mathsf{\sigma }}_{\mathrm{lAB}}={\mathsf{\sigma }}_{\mathrm{oAB}}\frac{{\mathsf{\epsilon }}_{\mathrm{F}}^{{\mathsf{\alpha }}_{\mathrm{AB}}}}{{1+\mathsf{\epsilon }}_{\mathrm{F}}},$$

(31)

where σ_oAB and α_AB are constants depending on the nature of the alloy.

The density of deformation energy of alloy AB is defined by the following relation [44]:

$$\begin{array}{ll}{\mathrm{f}}_{\mathrm{AB}}\left(\mathsf{\epsilon }\right)=\frac{{\mathsf{\Psi }}_{\mathrm{AB}}^{\mathrm{F}}}{{\mathrm{V}}_{\mathrm{AB}}^{\mathrm{F}}}-\frac{{\mathsf{\Psi }}_{\mathrm{AB}}}{{\mathrm{V}}_{\mathrm{AB}}}=\frac{1}{\mathrm{N}}& (\frac{{\mathsf{\Psi }}_{\mathrm{AB}}^{\mathrm{F}}}{{\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}-\frac{{\mathsf{\Psi }}_{\mathrm{AB}}}{{\mathrm{v}}_{\mathrm{AB}}})=[\left(1-{7\mathrm{c}}_{\mathrm{B}}\right)\left(\frac{{\mathsf{\psi }}_{\mathrm{A}}^{\mathrm{F}}}{{\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}-\frac{{\mathsf{\psi }}_{\mathrm{A}}}{{\mathrm{v}}_{\mathrm{AB}}}\right)+{\mathrm{c}}_{\mathrm{B}}\left(\frac{{\mathsf{\psi }}_{\mathrm{B}}^{\mathrm{F}}}{{\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}-\frac{{\mathsf{\psi }}_{\mathrm{B}}}{{\mathrm{v}}_{\mathrm{AB}}}\right)+\\ & +2{\mathrm{c}}_{\mathrm{B}}\left(\frac{{\mathsf{\psi }}_{{\mathrm{A}}_1}^{\mathrm{F}}}{{\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}-\frac{{\mathsf{\psi }}_{{\mathrm{A}}_1}}{{\mathrm{v}}_{\mathrm{AB}}}\right)+4{\mathrm{c}}_{\mathrm{B}}\left(\frac{{\mathsf{\psi }}_{{\mathrm{A}}_2}^{\mathrm{F}}}{{\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}-\frac{{\mathsf{\psi }}_{{\mathrm{A}}_2}}{{\mathrm{v}}_{\mathrm{AB}}}\right)]\end{array}$$

(32)

or

$$\begin{array}{l}{\mathrm{f}}_{\mathrm{AB}}(\mathsf{\epsilon })=\left(1-{7\mathrm{c}}_{\mathrm{B}}\right)\{{\mathsf{\psi }}_{\mathrm{A}}\left(\frac{1}{{\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}-\frac{1}{{\mathrm{v}}_{\mathrm{AB}}}\right)+\frac{2\epsilon r_{01\mathrm{A}}^{\mathrm{F}}}{{\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}{\left(\frac{\partial {\mathsf{\psi }}_{\mathrm{A}}^{\mathrm{F}}}{\partial {\mathrm{r}}_{1\mathrm{A}}^{\mathrm{F}}}\right)}_{\mathrm{T}}+\frac{{\mathsf{\epsilon }}^2}{{2\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}[{\left(\frac{{\partial }^2{\mathsf{\psi }}_{\mathrm{A}}^{\mathrm{F}}}{\partial {\mathrm{r}}_{1\mathrm{A}}^{\mathrm{F}2}}\right)}_{\mathrm{T}}{\left({2\mathrm{r}}_{01\mathrm{A}}^{\mathrm{F}}\right)}^2+{\left(\frac{\partial {\mathsf{\psi }}_{\mathrm{A}}^{\mathrm{F}}}{\partial {\mathrm{r}}_{1\mathrm{A}}^{\mathrm{F}}}\right)}_{\mathrm{T}}{2\mathrm{r}}_{01\mathrm{A}}]\}+\\ +{\mathrm{c}}_{\mathrm{B}}\{{\mathsf{\psi }}_{\mathrm{B}}\left(\frac{1}{{\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}-\frac{1}{{\mathrm{v}}_{\mathrm{AB}}}\right)+\frac{2\mathsf{\epsilon }{\mathrm{r}}_{01\mathrm{B}}^{\mathrm{F}}}{{\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}{\left(\frac{\partial {\mathsf{\psi }}_{\mathrm{B}}^{\mathrm{F}}}{\partial {\mathrm{r}}_{1\mathrm{B}}^{\mathrm{F}}}\right)}_{\mathrm{T}}+\frac{{\mathsf{\epsilon }}^2}{{2\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}[{\left(\frac{{\partial }^2{\mathsf{\psi }}_{\mathrm{B}}^{\mathrm{F}}}{\partial {\mathrm{r}}_{1\mathrm{B}}^{\mathrm{F}2}}\right)}_{\mathrm{T}}{\left({2\mathrm{r}}_{01\mathrm{B}}^{\mathrm{F}}\right)}^2+{\left(\frac{\partial {\mathsf{\psi }}_{\mathrm{B}}^{\mathrm{F}}}{\partial {\mathrm{r}}_{1\mathrm{B}}^{\mathrm{F}}}\right)}_{\mathrm{T}}{2\mathrm{r}}_{01\mathrm{B}}]\}+\\ +2{\mathrm{c}}_{\mathrm{B}}\{{\mathsf{\psi }}_{{\mathrm{A}}_1}\left(\frac{1}{{\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}-\frac{1}{{\mathrm{v}}_{\mathrm{AB}}}\right)+\frac{2\mathsf{\epsilon }{r}_{{01\mathrm{A}}_1}^{\mathrm{F}}}{{\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}{\left(\frac{\partial {\mathsf{\psi }}_{{\mathrm{A}}_1}^{\mathrm{F}}}{\partial {\mathrm{r}}_{{1\mathrm{A}}_1}^{\mathrm{F}}}\right)}_{\mathrm{T}}+\frac{{\mathsf{\epsilon }}^2}{{2\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}[{\left(\frac{{\partial }^2{\mathsf{\psi }}_{{\mathrm{A}}_1}^{\mathrm{F}}}{\partial {\mathrm{r}}_{{1\mathrm{A}}_1}^{\mathrm{F}2}}\right)}_{\mathrm{T}}{\left({2\mathrm{r}}_{{01\mathrm{A}}_1}^{\mathrm{F}}\right)}^2+{\left(\frac{\partial {\mathsf{\psi }}_{{\mathrm{A}}_1}^{\mathrm{F}}}{\partial {\mathrm{r}}_{{1\mathrm{A}}_1}^{\mathrm{F}}}\right)}_{\mathrm{T}}{2\mathrm{r}}_{{01\mathrm{A}}_1}]\}+\\ +4{\mathrm{c}}_{\mathrm{B}}\{{\mathsf{\psi }}_{{\mathrm{A}}_2}\left(\frac{1}{{\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}-\frac{1}{{\mathrm{v}}_{\mathrm{AB}}}\right)+\frac{2\mathsf{\epsilon }{\mathrm{r}}_{{01\mathrm{A}}_2}^{\mathrm{F}}}{{\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}{\left(\frac{\partial {\mathsf{\psi }}_{{\mathrm{A}}_2}^{\mathrm{F}}}{\partial {\mathrm{r}}_{{1\mathrm{A}}_2}^{\mathrm{F}}}\right)}_{\mathrm{T}}+\frac{{\mathsf{\epsilon }}^2}{{2\mathrm{v}}_{\mathrm{AB}}^{\mathrm{F}}}[{\left(\frac{{\partial }^2{\mathsf{\psi }}_{{\mathrm{A}}_2}^{\mathrm{F}}}{\partial {\mathrm{r}}_{{1\mathrm{A}}_2}^{\mathrm{F}2}}\right)}_{\mathrm{T}}{\left({2\mathrm{r}}_{{01\mathrm{A}}_2}^{\mathrm{F}}\right)}^2+{\left(\frac{\partial {\mathsf{\psi }}_{{\mathrm{A}}_2}^{\mathrm{F}}}{\partial {\mathrm{r}}_{{1\mathrm{A}}_2}^{\mathrm{F}}}\right)}_{\mathrm{T}}{2\mathrm{r}}_{{01\mathrm{A}}_2}]\}.\end{array}$$

(33)

Let us consider the relation between the density of deformation energy and the strain as follows [44]:

f_AB(ε) = C_AB·σ_AB·ε

(34)

where C_AB represents the proportional factor. The maximum density of deformation energy corresponds to the strain ${\mathsf{\epsilon }}_{\mathrm{AB}}^{\mathrm{F}}$ and therefore we obtain the relation:

$${\mathrm{f}}_{\mathrm{AB}}{(\mathsf{\epsilon }}_{\mathrm{AB}}^{\mathrm{F}})={\mathrm{f}}_{\mathrm{ABmax}}={\mathrm{C}}_{\mathrm{AB}}{\mathsf{\sigma }}_{\mathrm{ABmax}}{\mathsf{\epsilon }}_{\mathrm{AB}}^{\mathrm{F}}.$$

(35)

The maximum stress σ_ABmax and the maximum real stress σ_1Abmax are defined by [44]:

$${\mathsf{\sigma }}_{\mathrm{ABmax}}=\frac{{\mathrm{f}}_{\mathrm{ABmax}}}{{\mathrm{C}}_{\mathrm{AB}}{\mathsf{\epsilon }}_{\mathrm{AB}}^{\mathrm{F}}},{\mathsf{\sigma }}_{1\mathrm{ABmax}}=\frac{{\mathsf{\sigma }}_{\mathrm{ABmax}}}{1+{\mathsf{\epsilon }}_{\mathrm{AB}}^{\mathrm{F}}}=\frac{{\mathrm{f}}_{\mathrm{ABmax}}}{{\mathrm{C}}_{\mathrm{AB}}{\mathsf{\epsilon }}_{\mathrm{AB}}^{\mathrm{F}}{(1+\mathsf{\epsilon }}_{\mathrm{AB}}^{\mathrm{F}})}.$$

(36)

C_AB is calculated from the stress σ_0.2AB in the alloy, which is determined experimentally [48] by the following expression [44]:

$${\mathrm{C}}_{\mathrm{AB}}=\frac{{\mathrm{f}}_{\mathrm{AB}}\left({\mathsf{\epsilon }}_{0.2\mathrm{AB}}\right)}{{\mathsf{\sigma }}_{0.2\mathrm{AB}}{\mathsf{\epsilon }}_{0.2\mathrm{AB}}}\cdot$$

(37)

The limit of elastic deformation of the alloy is defined by the following expression [44]:

$${\mathrm{E}}_{\mathrm{AB}}{\mathsf{\epsilon }}_{\mathrm{dh}}{=\mathsf{\sigma }}_{0\mathrm{AB}}\frac{{\mathsf{\epsilon }}_{\mathrm{ABdh}}^{{\mathsf{\alpha }}_{\mathrm{AB}}}}{{1+\mathsf{\epsilon }}_{\mathrm{Adh}}}\cdot$$

(38)

3. Numerical Calculations and Discussions for Fe and FeC

For FeC, we apply the Mie–Lennard–Jones (MLJ) potential [49]:

$$\mathsf{\phi }\left(\mathrm{r}\right)=\frac{\mathrm{D}}{\mathrm{n}-\mathrm{m}}\left[\mathrm{m}{\left(\frac{{\mathrm{r}}_0}{\mathrm{r}}\right)}^{\mathrm{n}}-\mathrm{n}{\left(\frac{{\mathrm{r}}_0}{\mathrm{r}}\right)}^{\mathrm{m}}\right],$$

(39)

where D represents the depth of potential well corresponding to the equilibrium distance r₀, m and n, is obtained empirically. Thus, the potential parameters for the Fe-C interaction are described by the following expression [50]:

$${\mathrm{D}}_{\mathrm{Fe}-\mathrm{C}}=\sqrt{{\mathrm{D}}_{\mathrm{Fe}-\mathrm{Fe}}{\mathrm{D}}_{\mathrm{C}-\mathrm{C}}},{\mathrm{r}}_{0\mathrm{Fe}-\mathrm{C}}=\sqrt{{\mathrm{r}}_{0\mathrm{Fe}-\mathrm{Fe}}{\mathrm{r}}_{0\mathrm{C}-\mathrm{C}}}.$$

(40)

m_Fe-C and n_Fe-C are found by fitting the theoretical result with the experimental data for the E of FeC_0.2% interstitial alloy at T = 300 K. The obtained MLJ potential parameters for different interactions are listed in

Table 1. The MLJ potential parameters for interactions of Fe-Fe, C-C and Fe-C.

Interaction	D [eV]	r0 [10−10 m]	m	N
Fe-Fe [49]	0.4005	2.4775	7	11.5
C-C [51]	8.43	1.545	3.73	2.21
Fe-C (proposal)	1.84	1.96	2.5	5.5

.

First, we assume the limit case when cC$\to 0$. Figure 1 illustrates the impact of the T on the E parameter at P = 0 for various methods and experimental studies. Note that the experimental samples in studies published in ref. [36,37,47] are monocrystalline ones, and the SMM calculations are applied to polycrystalline samples. Therefore, we use the Voigt–Reuss–Hill (VRH) conversion rule [49] as follows, based on the following expressions:

$$\mathrm{E}=\frac{9\mathrm{KG}}{3\mathrm{K}+\mathrm{G}},\mathrm{K}=\frac{{\mathrm{C}}_{11}^{*}+{2\mathrm{C}}_{12}^{*}}{3},\mathrm{G}=\frac{3{\left({\mathrm{C}}_{11}^{*}-{\mathrm{C}}_{12}^{*}\right)}^2+38\left({\mathrm{C}}_{11}^{*}-{\mathrm{C}}_{12}^{*}\right){\mathrm{C}}_{44}^{*}+{12\mathrm{C}}_{44}^{*2}}{30\left({\mathrm{C}}_{11}^{*}-{\mathrm{C}}_{12}^{*}\right)+{40\mathrm{C}}_{44}^{*}}.$$

(41)

In this equation, the symbol * represents the elastic quantities of monocrystalline material. As clearly noticed in Figure 1, a significant decrease in the E value occurred as the T increased due to an increase in the NND and a decrease in the cohesive energy. The curve E(T) obtained from the LMTO-GGA [8] describes very well the relationship E-T, qualitatively. However, the estimated values of E are very high compared to the measured values [9,10]. In contrast, all errors between the SMM calculations and the experimental data [9,10] in the 0 K to 800 K range are below 10%. However, it can be concluded that the rate of decrease of the E value relative to the T value calculated by the SMM is faster than the normal one. At T = 900 K, the value of E determined using the SMM is up to 14.4% smaller than the experimental data [10]. Since the E values obtained using the SMM are consistent with the experiments (EXPT) and other calculations, the obtained results of the G, K, C₁₁, C₁₂, and C₄₄ quantities are also in good agreement.

K, G, C₁₁, C₁₂ calculations for single Fe metal at different T and p values are illustrated in

Table 2. C₁₁(T,P), C₁₂(T,P), K(T,P) and G(P) for Fe.

	P [GPa]	T [K]	C11 [GPa]	C12 [GPa]	K [GPa]	G [GPa]
LMTO-GGA [8]	0	250	297	148	198	100
EXPT [52]		300	266	140	169	78
EXPT [53]		300	223	127	159	84
SMM (this paper)		300	255	89	146	83
LMTO-GGA [8]	4.6	250	326	167	220	110
SMM (this paper)		250	288	101	163	93
CAL [54]		300	260	154	189	100
SMM (this paper)		300	283	99	160	92
LMTO-GGA [8]	9.8	250	360	188	245	120
SMM (this paper)		250	318	111	180	103
EXPT [53]		300	283	167	206	101
SMM (this paper)		300	313	110	178	102

in comparison with other theoretical and experimental studies.

The results displayed in Figure 1 show that Fe deforms nonlinearly over the temperature (T) range of T = 3 K to T = 900 K and the K, G, C₁₁, and C₁₂ values are determined at T = 250 K, 300 K (Table 2). The results obtained at T = 250 K, 300 K show that there are differences in the values obtained for K, G, C₁₁, and C₁₂ compared with other research methods. Finally, these deviation values are very small compared with the results obtained by the experimental method [52,53,54], and the numerical calculation method [8], so they are acceptable.

In Figure 2 and

Table 3. E(cC,P) [10¹⁰ Pa] for FeC at T = 300 K determined using the SMM.

P [GPa]	cC = 0	cC = 0.2%	cC = 0.4%	cC = 0.6%
1	21.32	21.23	21.13	21.04
3	22.32	22.22	22.12	22.03
5	23.30	23.20	23.10	23.00
7	24.26	24.16	24.06	23.96

, the dependence of E on the concentration (cC) at different T and p values is shown for FeC. An increment in the cC leads to a considerable decrease in the E parameter due to the strong deformation of the crystal lattice. For example, at T = 300 K, E decreases from 20.79 × 10¹⁰ to 18.67 × 10¹⁰ Pa with increases of cC from 0 to 5%. This result is in very good agreement with experimental studies [11] when all errors are below 2%. When T increases, the curve E(cC) for FeC has a smaller slope. The size correlation between interstices in crystalline lattice and the radii of interstitial atoms determines the lattice deformation. It can be concluded that a higher T value determines a stronger atomic vibration, wider interstice, and less lattice deformation. In fact, vacancies are formulated at high T values and interstitial carbon atoms can occupy positions in these vacancies.

Figure 3 and Figure 4, respectively show the relationship between E and T for FeC_0.2% and FeC_0.4% calculated by SMM with the MLJ potential proposed in this paper and with the Embedded-Atom Method (EAM) potential [47] and from experimental data [18]. Clearly, using the parameters of MLJ potential in Table 1 requires extremely low calculation cost but gives equally reliable results as the EAM potential [47]. Moreover, in the range of 73 to 700 K, all deviations between the SMM calculations and experiments [18] are below 5%. Especially in the range of 294 to 533 K, all errors are below 2%. At T = 866 K, the error only is 13%.

To determine the non-linear deformation of FeC, it is first necessary to consider the intensity of the deformation energy $\mathrm{f}(\mathsf{\epsilon }).$ Figure 5 shows the dependence of $\mathrm{f}(\mathsf{\epsilon })$ on the deformation $\mathsf{\epsilon }$ in the case. There is one maximum point (ε_p, f_max) in the graph f(ε). An increase in the T causes a decrease in both ${\mathsf{\epsilon }}_{\mathrm{P}}$ and f_max.

Concretely, at T = 325 K we have ${\mathsf{\epsilon }}_{\mathrm{P}}$ = 6.2% corresponding to f_max = 16.9 GPa; at T = 500 K we have ${\mathsf{\epsilon }}_{\mathrm{P}}$ = 5.2% corresponding to f_max = 14.33 GPa, and at T = 700 K we have ${\mathsf{\epsilon }}_{\mathrm{P}}$ = 4.4% corresponding to f_max = 12.16 GPa. Knowing ${\mathsf{\epsilon }}_{\mathrm{P}}$ and f_max, the maximum value of real stress ${\mathsf{\sigma }}_{1\max }$ can be determined when the non-linear deformation occurs. Note that in our calculation steps, it is proposed that ${\mathsf{\sigma }}_{0.2}$ varies slowly with T and cC of interstitial atoms, and ${\mathsf{\sigma }}_{0.2}$(cC, T) ≈ ${\mathsf{\sigma }}_{0.2}$(cC, 300 K), this is one of the reasons for deviations between SMM calculations and experiments. Numerical calculations will certainly have higher accuracy if experimental data of ${\mathsf{\sigma }}_{0.2}$ in different conditions are available.

Table 4. E (cC, P) [10¹⁰ Pa] for FeC at T = 300 K determined using the SMM.

P [GPa]	cC = 0	cC = 0.2%	cC = 0.4%	cC = 0.6%
1	21.32	21.23	21.13	21.04
3	22.32	22.22	22.12	22.03
5	23.30	23.20	23.10	23.00
7	24.26	24.16	24.06	23.96

indicates the elastic deformation limit for single Fe metal and FeC interstitial alloy at T = 300 K, P = 0, and the maximum value of real stress.

Figure 6 shows the stress-strain relationship used by the SMM calculations. The deformation process of FeC can be considered as two stages.

Initially, the $\mathsf{\sigma }-\mathsf{\epsilon }$ dependence is linear when the strain is within the range ($\mathsf{\epsilon }$ < 0.5%). In this case, FeC is in the elastic deformation stage, which is still a reversible process. However, when $\mathsf{\epsilon }>0.5\%$ FeC is in the non-linear deformation stage, the $\mathsf{\sigma }-\mathsf{\epsilon }$ dependence is defined as a complex curve. It is known that the non-linear deformation stage is an irreversible process. In the limit case, the stress-strain curve obtained for single metal Fe in the present study is quite consistent with the experimental curve reported in [19]. For a constant strain $\mathsf{\epsilon },$ an increase of cC of interstitial atoms gives a rise to a decrease in stress $\mathsf{\sigma }$. Thus, adding C to Fe to form the steel will cause an enhancement in the strength and hardness, but a decrement in the elasticity and toughness.

These results are in full agreement with the results in the joints of composites [55,56]. The results demonstrated in Figure 5 and Figure 6 give good results about applying SMM to study processes of non-linear deformation of metals and alloys. In order to enhance the reliability of calculated results, it is also necessary to consider some important factors such as the rate of deformation, the annealing time, etc. [39] in representing the stress—strain curve.

Based on the obtained results, there is a good agreement between our findings and previous findings reported in theoretical and experimental studies.

4. Conclusions

In this research, the elastic and non-linear deformations quantities of alloy FeC were calculated numerically from the obtained theoretical results by applying the MLJ potential parameters for interactions Fe-Fe, C-C and Fe-C. The obtained results of elastic moduli and constants for main metal Fe were in good accordance with the results derived in other theoretical and experimental studies. At T = 300 K for the FeC interstitial alloy, the results of the analysis revealed that an increase in the cC from 0 to 5% gives rise to a decrease in the E from 20.79 × 10¹⁰ to 18.67 × 10¹⁰ Pa. The dependence of E on T obtained for FeC alloy with cC = 0.2 and 0.4% (obtained using the SMM with MLJ potential and EAM potential) and the stress-strain curve obtained for single Fe were quite consistent with the experimental studies. For a constant strain, increasing cC led to a decrease in stress. From the results obtained, it can be also concluded that adding C to Fe to form the steel will cause an enhancement in the strength and hardness, but a decrement in the elasticity and toughness. The results obtained by using the SMM in this research are expected to be useful for not only experimental but also theoretical studies on metals and their interstitial alloys.