Impact of nano crack and loading direction on the tensile features of FeCr alloy: a molecular dynamics analysis

Ferrochrome is an important additive element in the making of carbon steel and its byproducts due to its better materials features. But its features are greatly modified, due to the resultant effect of crack and fluctuations in the loading directions which leads to altering its material performance. The prediction of material performance and its features through Molecular dynamics (MD) is effective and also it helps to analyze the material performance up to an atomic level for effective understanding. According to Han and Meyers, the tensile characteristics of nanoscale materials differ more from coarse and micro-grained materials [1–4]. Due to the presence of nanoscale experiments, Li et al detailed that the MD is an effective approach to anticipate the material characteristics up to an atomic scale [5]. On nickel–aluminum nanowire, Alavi et al performed MD simulations and successfully forecast the material characteristics using MD analysis [6]. The dissimilarity between the surface and interior of copper nanowires has been found on the material characteristics, according to Wu et al [7]. Gowthaman and Jagadeesha have confirmed that the occurrence of defects in the polycrystal leads to offer alterations in the material features [8]. The knowledge of underlying atomic-level deformation processes, according to Chokshi, improves the material properties [9, 10]. Due to its un-stability and dislocation starvation, Pastor-Abia et al and others discovered that material flaws have a higher impact on the material and mechanical properties of nanoscale materials [11]. Furthermore, Cao and Wei have identified a link between material defects and the creation of fractional dislocations in polycrystals, implying that material deficiencies play an important role in the elastic performance and the formation of shear strain [12]. However, the occurrence of more severe flaws, including voids, causes a wider divergence between simulation and experimental results [13, 14]. The production of material flaws over any material results in a significant difference in the material's properties and performance during use, as proven by Pastor-Abia et al [11, 12, 15–17]. The production of many dislocations during the tensile examination, according to Weinberger and Cai, results in superior distortion resistance and a higher energy barrier [18]. Furthermore, Gowthaman et al showed the significant impact of randomly distributed vacancies on the tensile performance of nickel–aluminum nanowire and claimed that the formation of vacancies causes changes in the material's properties due to the development of microstructure and lattice transformation through pronounced atomic movements [19, 20]. Talebi et al have stated that modeling is an effective way and also it helps to predict the fracture model effectively [21]. Also, Mortazavi et al have stated that the selection of inter-atromic potential playing a crucial the classical MD simulation [22]. Through the experimentation, Fu et al have confirmed that the existence of crack on concrete and many other materials leads to invoke greater impact on its material characteristics [23–26].

Through the aforementioned extensive literature research, it is clear that little attention has been paid to the effect of temperature without the existence of crack under various loading directions on the tensile performance of FeCr alloy via MD analysis. Also, it is found that the impact of crack under various circumstances has not been studied so far. The main goal of this research is to see how the crack existence and change in loading directions affect the tensile performance and properties of FeCr polycrystals as a function of temperature through MD. The significance of this study is to analyse the creck effect of material behavior and its features as a function temperature and loading condition which leads to avoid the catastrophic failure of a material.

In this examination, the FeCr (lattice parameters: a = 2.923 Å, b = 3.881 Å, c = 4.022 Å, α = β = γ = 90°) (figure 1(a)) has been taken using a CIF file and then the polycrystal (100 Å × 100 Å × 100 Å) under constant grain size (3 nm) has been created using LAMMPS software followed by the energy minimization process using velocity verlet method which helps to optimize the potential energy of an atoms [27] (figure 1(b)) (WOC). The average grain size of a polycrystal has been measured using the following relationship

$$\begin{equation*}\bar d = {\text{ }}\sqrt[3]{{\frac{{3V}}{{4\pi N}}}}\end{equation*}$$

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N and V are the number of grains in the model and volume of the model, respectively

The crack (WC) has been invoked along the Z-direction (crack size of 5 Å × 5 Å × 5 Å on both sides) to determine its outcome on the tensile performance of FeCr alloy using MEAM potential [28]. (Symbolizations are represented in brackets). The energy of an entire system is generally equal to the sum of the energies of all its atoms, with each atom's energy made up of (1) Embedding energy $\left( {{F_i}\left( {\bar \rho {{\text{ }}_i}} \right)} \right)$ (2) A pair interaction energy ($\frac{1}{2}{\text{ }}\mathop {\mathop \sum \nolimits }\nolimits_{j \ne i} {\emptyset _{zz}}({R_{ij}}))$, and (3) a screening function (${S_{ij}})$ as follows:

$$\begin{equation*}{E_{MEAM{\text{ }}}} = {\text{ }}{F_i}\left( {\bar \rho {{\text{ }}_i}} \right) + \frac{1}{2}{\text{ }}\mathop {\mathop \sum \nolimits }\limits_{j \ne i} {\emptyset _{zz}}\left({R_{ij}}\right).{S_{ij}}\end{equation*}$$

• i)  
  The embedding function, which measures the energy needed to position an Z type of element such as Fe or Cr at a site with $\bar \rho {{\text{ }}_i}$ background electron density, has the following expression:

  $$\begin{equation*}{F_i}\left( {\bar \rho {{\text{ }}_i}} \right) = {\text{ }}{A_z}E_Z^0\frac{{\bar \rho {{\text{ }}_i}}}{{\bar \rho _z^O}}{\text{ ln}}\left( {\frac{{\bar \rho {{\text{ }}_i}}}{{\bar \rho _z^O}}} \right)\end{equation*}$$

  ${A_z}{\text{ }}$ is a parameter that changes depending on the type of element ${\text{ }}E_Z^0$ is the cohesive energy of the element type Z reference structure, and $\bar \rho {{\text{ }}_i}$ is the total background electron density at the location of atom I which is the sum of the partial electron densities of a spherically symmetric term and three angular terms. $\bar \rho _z^O$ is the equilibrium background electron density for the reference structure [29].
• ii)  
  The universal equation of state yields the energy of the reference structure, E_u (UEoS)

  $$\begin{equation*}{\emptyset _{zz}}\left( {{R_{ij}}} \right) = {\text{ }}\frac{2}{{Z_{1,ZZ}^0}}.\left( {E_{ZV}^U({R_{ij}}} \right) - {F_z}\bar \rho _z^O\end{equation*}$$

  $E_{ZV}^U$, $Z_{1,ZZ}^0$, ${F_z}$ and $\bar \rho _z^O$ are represents the Pair interaction energy, 1NN coordination number of the reference structure and the background electron density
• iii)  
  The overall screening function is composed of a radial cutoff function and three-body terms impacting each of the other atoms in the system:

  $$\begin{equation*}{S_{ij}} = {\text{ }}\overline {{S_{ij}}} {f_c}\left( {\frac{{{R_c} - {\text{ }}{R_{ij}}}}{{\Delta r}}} \right)\end{equation*}$$

  where S_ij represents the sum of all screening criteria. R_c is the radial cutoff distance, f_c is a smooth cutoff function, and $\Delta r$ is a parameter that determines how far away from R_ij = R_c the radial cutoff is smoothed from 1 to 0.

The tensile characterization has been carried out on cracked and non-cracked polycrystals (figures 1(b)–(e)) using LAMMPS package under various temperatures such as 300 K, 500 K, 700 K and 900 K and loading directions (parallel and normal to the crack cross-section) at a strain rate (constant) of 8 × 10¹⁰ s⁻¹ with periodic boundary conditions to examine the impact of temperature, loading direction and crack over the tensile features of bimetallic FeCr polycrystals. Because of the provision of periodic boundary conditions in 3D periodic space, the results observed are efficient and accurate. The simulations have been repeated three times to predict the uncertainty on the material features. The proportionality of temperature over the diffusion is listed as follows which helps to predict the diffusion as a function of temperature and it is measured through a shear strain contour map, and displacement contour map.

$$\begin{equation*}D = {\text{ }}{D_{0{\text{ }}}}\exp \left(\frac{{ - Q}}{{RT}}\right)\end{equation*}$$

where D₀ is the frequency factor, Q is the activation energy for creep, and R the universal gas constant. Additionally, the shear strain contour map, and displacement contour map have been presented using the Ovito package and analyzed to confirm the material behavior modifications owed to the existence of crack under numerous temperatures [30]. The strain of a polycrystal has been measured using the following equations:

$$\begin{equation*}\varepsilon = {\text{ }}\frac{{l - {\text{ }}{l_0}}}{{{l_0}}}\end{equation*}$$

where, l and l₀ are the initial and final length of a polycrystal along the loading directions, respectively.

The shear strain has been measured using the following equations

$$\begin{equation*}E = {\text{ }}\frac{1}{2}\left( {{F^T}F - I} \right) = \text{Green-Lagrangian strain tensor} \end{equation*}$$

where,

$$\begin{align}F = {\text{ }}\left( {\begin{array}{*{20}{c}} {{F_{xx}}}&{{F_{xy}}}&{{F_{xz}}} \\ {{F_{yx}}}&{{F_{yy}}}&{{F_{yz}}} \\ {{F_{zx}}}&{{F_{zy}}}&{{F_{zz}}} \end{array}} \right) = \text{Atomic deformation gradient tensor} \end{align}$$

$$\begin{equation*}Shear{\text{ }}strain = {\text{ }}{\left( {E_{xx}^2 + E_{yy}^2 + E_{zz}^2 + \frac{1}{6}{\text{ }}\left( {{E_{xx}} - {E_{yy}}} \right) + \left( {{E_{xx}} - {E_{xz}}} \right) + \left( {{E_{yy}} - {E_{zz}}} \right)} \right)^{0.5}}.\end{equation*}$$

Moreover, the material features such as yield strength, yield strain, and youngs modulus of cracked and non-cracked polycrystals

The tensile characterization of FeCr Alloy has been conducted under various loading directions and conditions such as crack and temperature and evaluated as follows:

The temperature has offered greater significance on the tensile behavior, owing to the formation of the superior kinetic energy (KE) of both iron and chromium atoms, which helps to overcome the energy barrier and cause greater modifications in the tensile features, as shown in figure 2. It is also clear that the presence of crack and loading direction has raised greater consequence on the tensile performance of FeCr polycrystal. A comparable kind of propensity has been observed irrespective of loading direction and the existence of crack on the surface owing to the primer of dominant KE over the atoms. Additionally, it is found that the existence of a crack leads to invoking a modest impression on the tensile performance followed by the non-existence of a crack, due to the occurrence of severe stress concentration effect nearby the discontinuity and causes a superior reduction in the distortion resistance as a function of increment in strain. Moreover, it is enumerated that the loading direction has offered modest consequences on the tensile behavior trailed by the existence of crack, owing to the formation of discontinuity and stress concentration nearby the cracked surface which primes to decay in the distortion resistance amid the iron and chromium atoms [8, 31–36]. Amongst various loading directions, the loading normal to the crack cross-section direction has introduced a dominant impression on the tensile behavior compared to the loading parallel to the crack cross-section direction, due to the occurrence of larger stress concentration and decline in the distortion resistance amidst the atoms. Moreover, the non-existence of stress concentration on the non-cracked material leads to shows better tensile performance associated with the cracked material and causes better tensile features [9–11]. Furthermore, the impact of temperature, the existence of crack, and loading direction on the shear strain formation have been presented and discussed as follows.

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According to figures 3–6, the crack, loading direction, and temperature have a greater impact on the formation of greater shear strain and displacement of atoms during tensile characterization. It is also stated that the temperature has a superior impression on the formation of greater shear strain and dislocation when compared to the loading direction and crack, due to the occurrence of more KE among the atoms. The loading direction and existence of the crack also offer the intermediate and lowermost impact on the shear stain formation, owing to the establishment of decrement in the distortion resistance and stress concentration near the discontinuity or crack [36–42]. Further, it is observed that under greater temperatures and the loading along the normal to the cross-section of the crack offers superior shear strain generation which primes to stimulus the tensile characteristics of FeCr polycrystal. The fluctuations in the tensile characteristics such as Young's modulus, yield stress, and yield strain are discussed as follows.

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According to figure 7, the production of KE among the atoms has given the temperature a bigger influence on the material characteristics than on the existence and lodging orientations. The quantified observed maximum yield strain (with the uncertainty of ±0.001), yield strength (with the uncertainty of ±0.01 GPa), and young's modulus (with the uncertainty of ±0.02 GPa), value under the temperature of 100 K and without the exitance of the crack condition under the loading direction (normal to the crack cross section) are 0.1024, 7.2019 GPa and 85.56 GPa respectively and subsequently the augmentation in temperature under similar conditions (up to 500 K) it shows the value of 0.0852, 4.694 GPa and 69.56 GPa, respectively (10%−12% lower), due to the production of more KE and a decrease in the atoms resistance to deformation. Moreover, it originated that the existence of the crack and the loading parallel to the cross-section of the crack does not show a greater impact on material features, owing to the formation of the better deformation resistance and lowermost stress concentration amid the atoms. Amongst the existence of crack and loading directions, the existence of a crack offers a superior impression on the tensile features followed by the loading directions, due to the existence of bigger stress concentration and discontinuity amid the atoms. The quantified observed maximum yield strain, yield strength and young's modulus value under the temperature of 100 K and with the exitance of crack condition are 0.091, 5.30 GPa and 80.22 GPa respectively which clearly confirms the generation of bigger stress concentration and lowermost distortion resistance amid the atoms. But loading parallel to the cross-section of the crack offers a greater impact on the material features, due to the occurrence of greater stress concentration which leads to increases in the shear strain behavior but compared to the loading along perpendicular to the crack cross-section it offers better results, due to the minimal stress concentration and shear strain [12, 43]. The quantified observed maximum yield strain, yield strength, and Young's modulus value under the temperature of 100 K and without the existence of crack condition under the loading direction (parallel to the cross section of crack) are 0.155, 9.30 GPa, and 115.02 GPa respectively which clearly confirms the reductions in the distortion resistance amid the atoms [11–14]. Furthermore, it is stated that the greater temperature along with the existence of crack and loading along normal to the crack cross-section offers greater reductions in the tensile features of FeCr polycrystal, owing to the interactive effect of superior KE and discontinuity among atoms. Over the above learning, it is understood that under bottommost temperature and the loading along the parallel to the cross-section of the crack offer superior material features under the existence of the crack condition, and for the non-cracked material, the lowermost temperature always offers better material features which is irrespective of loading direction.

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The tensile characterization on the cracked and non-cracked FeCr polycrystal under various temperatures and loading directions are conducted and the comments are as follows:

• 1.  
  The impact of temperature and the existence of crack on the FeCr polycrystal under various loading directions has been analyzed and discovered that the temperature has a greater influence on the tensile characteristics due to the introduction of improved atomic KE.
• 2.  
  The Non-cracked FeCr polycrystal offers better material features followed by the cracked polycrystals (10%–12% higher), due to the non-generation of discontinuity and stress concentration on the surface.
• 3.  
  The loading parallel to the crack cross-sectional has offered the lowermost impact on the material behavior trailed by the loading normal to the cross-section, owed to the lowermost stress concentration amid the atoms.
• 4.  
  Further, it is confirmed that under cracked polycrystal, the lowermost temperature and loading parallel to the crack cross-section of crack invoke superior material features, due to its lowermost KE formation and stress concentration.