Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 4, 1332-1342 2025 Publisher: Learning Gate DOI: 10.55214/25768484.v9i4.6261 © 2025 by the authors; licensee Learning Gate © 2025 by the authors; licensee Learning Gate History: Received: 6 February 2025; Revised: 4 April 2025; Accepted: 8 April 2025; Published: 15 April 2025 * Correspondence: nvtrang@tnut.edu.vn Atomistic simulation of two-dimensional 1T-MnX2 sheets under uniaxial tension Nguyen Huu Tu1, Nguyen Van Thanh1, Dang Van Manh1, Nguyen Van Trang2* 1Faculty of Fundamental Science, Military Academy of Logistics, Hanoi, VietNam. 2Faculty of mechanical engineering, Thai Nguyen University of Technology, Vietnam; nvtrang@tnut.edu.vn (N.V.T.) Abstract: We investigate the mechanical properties of two-dimensional 1T-MnX2 materials through the molecular dynamics finite element method with the Stillinger-Weber potential. The two- dimensional Young’s modulus, Poisson’s coefficient, maximum stress, and strain of these materials are examined for four 1T-MnX2 sheets. The effects of the armchair and zigzag directions on the mechanical properties under uniaxial tension are considered and discussed. Under uniaxial tension, we have determined the main mechanical properties. For 1T-MnO2, the maximum stress (σt) was 16.794 N/m in the armchair direction, and the maximum elastic modulus (Et) was 154.96 N/m in the zigzag direction. For 1T-MnTe2, the maximum Poisson's ratio (ν) observed was 0.181 in the armchair direction. These materials are considered approximately isotropic and are characterized by brittle fracture. Simulation results will help to design and use two-dimensional 1T-MnX2 sheet-based nanocomposites and nanodevices. Keywords: ITstructures, 2d-materials, Atomistic simulation, Stillinger-weber. Uniaxial tension. 1. Introduction This study focuses on determining the mechanical properties of two-dimensional materials that are compounds of transition metal Mn with non-metallic elements existing in the form of hexagonal structure 1T-MnX2 Jiang and Zhou [1] where X is a non-metallic element. This polymorph can exist in nature or be created in the laboratory [1]. The structure of the 1T-MnX2 material is shown in Figure. 1, through the orthogonal projection and the axial projection of a basic cell with the corresponding numbered atoms. Each basic cell (the rectangle drawn with dashed lines in Figure. 1) with side size a (Å) contains 9 atoms including 3 Mn atoms (red) on a plane, each Mn atom is bonded to 6 non-metallic atoms X (blue) distributed on two planes symmetrical to the plane containing Mn, the distance between the plane containing Mn and X is h; meanwhile, an X atom is bonded to 3 surrounding Mn atoms; X is one of the following elements: Oxygen (O), sulfur (S), selenium (Se), tellurium (Te). The material structure is more clearly shown through the material parameters including: the bond distance between two Mn and X atoms is dMn-X; the bond angle between X-Mn-X atoms is the angle MnXX; Mn-X-Mn is the angle XMnMn (see Table 1). 1333 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 4: 1332-1342, 2025 DOI: 10.55214/25768484.v9i4.6261 © 2025 by the authors; licensee Learning Gate Figure 1. Schematic illustration of: 1T-MnX2 structures (X= O, S, Se and Te are non-metallic elements). Table 1. Material network parameters of 1T-MnX2 materials. No. Materials Lattice constant, a (Å) dMn-X (Å) MnXX (°) XMnMn (°) 1 1T-MnO2 2.82 1.88 97.181 97.181 2 1T-MnS2 3.12 2.27 86.822 86.822 3 1T-MnSe2 3.27 2.39 86.330 86.330 4 1T-MnTe2 3.54 2.59 86.219 86.219 Source: Jiang and Zhou [1]. Various the two-dimensional (2D) materials are existed in many different structural forms such as Pucked structure Lê Minh, et al. [2] planar hexagonal graphene Geim [3], low-bucked silicene, BN and AlN [4], 1H and 1T structures [1]. These materials exhibit exceptional properties, including high electrical and thermal conductivity, and remarkable mechanical strength. They exhibit good mechanical properties with high elastic modulus of ~0.5-1 TPa and tensile strength of ~61 GPa [5]. They possess distinguishable chemical and thermal stability with high oxidation resistance up to 9000 C in air [6], wide band-gaps independent of tube structures [7, 8]. Excellent thermal conductivity [9]. They are also an effective violet and ultra-violet light emission material [10, 11]. Interestingly, their mechanical behavior at the nanoscale can differ significantly from macroscopic observations, showcasing phenomena like a negative Poisson's ratio [2, 12]. And extreme hardness (e.g., graphene being harder than diamond) [13]. Furthermore, their bandgap can be tailored through the application of mechanical strain [14-16] or external electric fields, without requiring any chemical alterations [16-18]. Their versatility has led to their incorporation into electronic devices for manufacturing nanometer-sized transistors [1, 3, 16]. electronic storage devices [19, 20]. And applications within the energy [21, 22]. And medical fields [23, 24]. Such applications require a deep comprehension of the mechanical properties and performance of 1T-MnX2 materials under various loading conditions. Characterizing the mechanical properties of nanoscale materials presents significant experimental challenges due to difficulties in manipulation, high costs, and often infeasibility. Consequently, researchers often rely on computational methods like experimental simulation (e.g., atomic finite element method (AFEM) [25-27]. Molecular dynamics (MD) [28]. First-principles [29, 30]. And ab initio calculations [31] to predict these properties. The Mechanical properties of 1T-MnX2 materials seem still unexplored by MDFEM method. It should be noted that the Mechanical properties of these materials has been investigated by molecular dynamics (MD) method see e. g. Jiang and Zhou [1] and references therein. The present work investigates through molecular dynamics finite element method (MDFEM) the mechanical properties of two-dimensional 1T-MnX2 materials under uniaxial tension. The effects of the armchair and zigzag directions to the mechanical properties of these materials under uniaxial tension are studied and discussed. 1334 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 4: 1332-1342, 2025 DOI: 10.55214/25768484.v9i4.6261 © 2025 by the authors; licensee Learning Gate 2. Framework for Analysis While density functional theory (DFT) calculations and molecular dynamics (MD) simulations are time-consuming, molecular dynamic finite element methods (MDFEM), sometime known as atomic- scale finite element methods or atomistic finite element methods, have been developed to analyze nanostructured materials in a computationally efficient way, see e. g. [25, 28]. To achieve the atomic positions of the BN-NT under specific boundary conditions, molecular dynamic finite element method (MDFEM) is here adopted. In MDFEM, atoms and atomic displacements are considered as nodes and translational degrees of freedom (nodal displacements), respectively. Both first and second derivatives of system energy are used in the energy minimization computation, hence it is faster than the standard conjugate gradient method which uses only the first order derivative of system energy as discussed in [32]. The stiffness matrices of these elements are established based upon interatomic potentials. Similar to conventional finite element method, global stiffness matrix is assembled from element stiffness matrices. Hence, relations between atomic displacement and force can be derived by solving a system of equations. For each 1T-MnX2 sheet, the atomic interaction potential is determined through the parameters of the Stillinger-Weber potential function including the direct binding energy of two adjacent atoms (Er, eV) and the binding energy of three adjacent atoms (E, eV) through the following equations: rE E E= + (1) 2 1 m r i E V = = (2) 3 1 n i E V = = (3) ( ) ( )4 2 1 ij/ r r ijV Ae B / r   −  = − max (4) ( ) ( ) ( ) ( )( ) 2 3 0 ij ij ij ik ik/ r r / r r ijkV Ke   − + −  =  −  max ik max cos cos (5) In which, E is total atomic bond energy; E(eV) is total angular bond energy of 3 atoms on the entire membrane; Er(eV) is total linear bond energy between two atoms of the membrane; V2(eV) is linear bond energy of two adjacent atoms; V3(eV) is angular bond energy of 3 adjacent atoms; m and n are number of linear bonds and angular bonds in a calculation model; A(eV) and K(eV) are material coefficients; ρ(Å), B(Å4), ρij(Å), ρik(Å), θo (degrees) are geometric parameters of the material; rij(Å), rik(Å) are bond length between two atoms i, j and k respectively; ijk (degrees) is bond angle between three atoms i, j, k (where i is the middle atom) (Figure. 2). These parameters are summarized in Tables 2 and 3. Figure 2. Element model when using Stillinger-Weber potential function: a) Straight bond between two atoms; b,c) Angular bond between 3 atoms. 1335 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 4: 1332-1342, 2025 DOI: 10.55214/25768484.v9i4.6261 © 2025 by the authors; licensee Learning Gate Table 2. Stillinger-Weber potential parameters for the straight bond of material 1T-MnX2 [1]. No. Materials Bond A, eV , Å B, Å4 rmin, Å rmax, Å 1 1T-MnO2 Mn-O 9.675 1.212 6.246 0.0 2.635 2 1T-MnS2 Mn-S 3.127 1.111 13.276 0.0 3.064 3 1T-MnSe2 Mn-Se 3.422 1.153 16.314 0.0 3.220 4 1T-MnTe2 Mn-Te 4.007 1.246 22.499 0.0 3.488 Table 3. Stillinger-weber potential parameters for angle bond of the 1T-MnX2 [1]. No. Materials Angular bond K, eV 0, (°) 1, Å 2, Å rmin12, Å rmax12, Å rmin13 Å rmax13, Å rmin23, Å rmax23, Å 1 1T-MnO2 MnOO 60.755 97.181 1.212 1.212 0.0 2.635 0.0 2.635 0.0 3.852 OMnMn 60.755 97.181 1.212 1.212 0.0 2.635 0.0 2.635 0.0 3.852 2 1T-MnS2 MnSS 19.765 86.822 1.111 1.111 0.0 3.064 0.0 3.064 0.0 4.262 SMnMn 19.765 86.822 1.111 1.111 0.0 3.064 0.0 3.064 0.0 4.262 3 1T-MnSe2 MnSeSe 19.390 86.330 1.153 1.153 0.0 3.220 0.0 3.220 0.0 4.467 SeMnMn 19.390 86.330 1.153 1.153 0.0 3.220 0.0 3.220 0.0 4.467 4 1T-MnTe2 MnTeTe 19.307 86.219 1.246 1.246 0.0 3.488 0.0 3.488 0.0 4.836 TeMnMn 19.307 86.219 1.246 1.246 0.0 3.488 0.0 3.488 0.0 4.836 For each 1T-MnX2 membrane, the total number of atoms is N; Xi and ui are the initial coordinates and displacement of the i-th atom; then, the coordinates of the atom after deformation are xi=Xi+ui. The atomic interaction potential of the entire membrane calculated by formula (1) is an equation that depends on the coordinates of each atom on the membrane as follows: ( )1 2 NE E , ,...,   = x x x (6) On the other hand, when the membrane is subjected to an external force fi on the i-th atom (considered as nodes), the atoms in the membrane have displacements ui correspondingly, the potential energy of the applied external force is calculated as follows 1 N ext i i i E = = f u (7) The total potential energy of the membrane is extE E = − (8) The membrane is in equilibrium when the total potential energy reaches its minimum value according to the principle of minimum potential energy. Therefore, the first derivative of the total potential energy will be zero as shown in Eq. (9) 0; 1 i i N  = =   u (9) Solving the system of equations (9), the displacement of all atoms in the membrane is determined. In the experimental simulation using the atomic finite element method, solving the system of equations (9) using the Newton–Raphson iteration method, the use of this method has been clearly shown in [25, 27, 28], then the equation in the form of finite elements is K(k).u(k)=F(k) (10) 1336 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 4: 1332-1342, 2025 DOI: 10.55214/25768484.v9i4.6261 © 2025 by the authors; licensee Learning Gate With ( ) 2 ( ) ( ) ( ) ; ; k k k kk ij i i i j i i E F f     = = − = −     K u u u u (11) Equations (10) and (11) are the basic equations of the finite element. With K(k) being the global stiffness matrix, u(k) being the nodal displacement vector and F(k) being the nodal force vector. If each atom is considered as a node, the displacement of the atom is the displacement of the node. Due to the structure of the orthogonal hexagonal membrane, two types of elements are formed on the membrane, i.e., 3-node elements (Valence) are molecules on the boundary of the membrane, each element has 3 bonded atoms, 4-node elements (Improper) are elements inside the membrane, and each element is bonded by 4 atoms (Figure. 3). Since each atom can move in 3 directions, formula (11) is used to calculate the stiffness matrix of each element with dimensions ( ) ( ) 9 9 12 12 ; e e Val Imp       x x K K , corresponding to 3-node and 4-node elements. From the element stiffness matrix, calculate the global stiffness matrix   ( ) 3 3 k N Nx K which is the composite matrix of the element stiffness matrices over the entire membrane. The displacement vectors and force vectors have dimensions corresponding to the global stiffness matrix. a) b) Figure 3. Element model using the atomic finite element method: a) the Valence element is formed by 3 adjacent atoms; b) the Improver element is formed by 4 adjacent atoms. The system of equations (10) is solved by using the Newton–Raphson iteration method with displacement boundary conditions i.e., atoms on the tensile boundary have displacement equal to u(0), and atoms on the retaining boundary (Figure. 4) have no displacement in the tensile direction. This is the experimental simulation method using molecular dynamic finite element method (MDFEM). The The experimental model of the sheet tensile test is shown in Figure. 4 through the displacement method. 1337 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 4: 1332-1342, 2025 DOI: 10.55214/25768484.v9i4.6261 © 2025 by the authors; licensee Learning Gate Figure 4. The sheet with displacement boundary conditions under uniaxial tension: a) the armchair direction; b) the zigzag direction. In the simulation, pull the rectangular sheet with approximately equal side dimensions (the rectangle is considered to be a square); each sheet has 4200 atoms; each stretching step causes the atoms on the stretching edge to move with an increment of = 0.001, repeating until the sheet has a broken bond (the sheet is destroyed as shown in Figure. 6). At each step, equation (10) is solved to determine the displacement and nodal force of all atoms on the sheet at this step. The position of atoms in the next steps is determined as follows: ( ) ( ) ( )1k k kx x u+ = + (12) The iterative process continues until , with a given error, at which point the sheet is considered to have failed. The image of the failed sheet is shown in Figure. 6. The data of the entire experimental process, of all steps, is synthesized to determine the mechanical parameters that need to be calculated. The tensile results of the 1T-MnX2 materials are shown through the stress-strain relationship graph; the two-dimensional elastic modulus is determined by linearizing their relationship with strain in the range from 0 to 0.1; the Poisson's ratio is determined based on the ratio of transverse strain and axial strain; =-y/x; the two-dimensional tensile stress and tensile strain occur at the end of the tensile program. The stress-strain relationship when stretching the intact sheet of the four materials is shown in Figure. 5. 3. Results and Discussion Figure 5 shows the stress-strain curves of the 1T-MnX2 sheets in armchair and zigzag directions calculated by molecular dynamic finite element method (MDFEM). It can be seen from Figure 5 that the stress increases monotonously with an increase of the strain up to a maximum value and then drops suddenly. Therefore, the sheets exhibit a brittle fracture mechanism. Then, the maximum stress and strain at the maximum stress location replace the critical stress and critical strain, respectively. 1338 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 4: 1332-1342, 2025 DOI: 10.55214/25768484.v9i4.6261 © 2025 by the authors; licensee Learning Gate Figure 5. The stress-strain curves of 1T-MnX2 material sheets under uniaxial tension according to armchair and zigzag directions. Table 4 shows the elastic modulus, maximum stress and maximum strain of 1T-MnX2 material sheets according to armchair and zigzag directions calculated by MDFEM method. The results showed that maximum stress: t=16.794N/m under uniaxial tension 1T-MnO2 in the armchair direction with strain =0.208; minimum stress: t=4.752 N/m under uniaxial tension 1T-MnTe2 in the armchair direction with strain =0.256; maximum poisson's ratio: =0.181 under uniaxial tension 1T-MnTe2 in the armchair direction; minimum poisson's ratio: =0.121 under uniaxial tension 1T-MnO2 in the zigzag direction; maximum elastic modulus: Et=154.96 N/m under uniaxial tension 1T-MnO2 in the zigzag direction; minimum elastic modulus: Et=37.385 N/m under uniaxial tension 1T-MnTe2 in the armchair direction. These results showed that two-dimensional elastic modulus and poisson's ratio are approximately equal value when stretched in two directions armchair and zigzag. They are considered as approximately isotropic materials. In Table 4, these results were compared with the research results by Jiang under the same experimental simulation conditions but other method [1]. The comparison results showed that the errors were good agree. 0 2 4 6 8 10 12 14 16 18 0 0.05 0.1 0.15 0.2 0.25  t, N /m Axial strain 1T-MnO2 Armchair Zigzag 0 1 2 3 4 5 6 0 0.05 0.1 0.15 0.2 0.25  t, N /m Axial strain 1T-MnS2 Armchair Zigzag 0 1 2 3 4 5 6 0 0.05 0.1 0.15 0.2 0.25  t, N /m Axial strain 1T-MnSe2 Armchair Zigzag 0 1 2 3 4 5 6 0 0.07 0.14 0.21 0.28  t, N /m Axial strain 1T-MnTe2 Armchair Zigzag 1339 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 4: 1332-1342, 2025 DOI: 10.55214/25768484.v9i4.6261 © 2025 by the authors; licensee Learning Gate Table 4. The mechanical parameters of four 1T-MnX2 materials by MDFEM method. No. Materials Directions Elastic modulus Et, N/m Poisson's ratio Maximum stress t, N/m Tensile strain Evaluations 1 1T-MnO2 AC 153.900 0.123 16.794 0.208 Our results 156.300 0.120 16.800 0.210 by MD at 10 K [1] -1.536 2.250 -0.035 -1.190 error (%) with [1] ZZ 154.960 0.121 15.883 0.224 Our results 155.400 0.120 16.200 0.240 by MD at 10 K [1] -0.283 1.083 -1.957 -6.667 error (%) with [1] 2 1T-MnS2 AC 46.159 0.147 5.468 0.218 Our results 47.100 0.150 5.500 0.210 by MD at 10 K [1] -1.998 -1.800 -0.590 3.571 error (%) with [1] ZZ 46.500 0.145 5.143 0.235 Our results 46.800 0.150 5.300 0.250 by MD at 10 K [1] -0.641 -3.400 -2.959 -6.000 error (%) with [1] 3 1T-MnSe2 AC 42.726 0.162 5.338 0.227 Our results 43.200 0.170 5.400 0.220 by MD at 10 K [1] -1.097 -4.765 -1.147 3.182 error (%) with [1] ZZ 43.083 0.160 4.996 0.246 Our results 42.900 0.170 5.200 0.260 by MD at 10 K [1] 0.427 -6.059 -3.916 -5.385 error (%) with [1] 4 1T-MnTe2 AC 37.835 0.181 5.107 0.239 Our results 38.500 0.190 5.200 0.240 by MD at 10 K [1] -1.727 -4.947 -1.792 -0.417 error (%) with [1] ZZ 38.206 0.178 4.752 0.256 Our results 38.400 0.190 5.000 0.280 by MD at 10 K [1] -0.505 -6.105 -4.963 -8.571 error (%) with [1] Figure 6 shows the destruction of the 1T-MnSe2 sheet when stretching in the armchair direction with a strain of  = 0.227; when stretching in the zigzag direction, at the step  = 0.246, many bonds were broken and the sheet was destroyed. . 1340 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 4: 1332-1342, 2025 DOI: 10.55214/25768484.v9i4.6261 © 2025 by the authors; licensee Learning Gate Figure 6. Destroyed shapes of 1T-MnSe2 sheets: a) uniaxial tension according to armchair; b) uniaxial tension according to zigzag. 4. Conclusions We present the simulation results of the mechanical properties of 1T-MnX2 nanosheets under uniaxial tension with the use of MDFEM with Stillinger-Weber potential. We have found maximum stress: t=16.794N/m under uniaxial tension 1T-MnO2 in the armchair direction with strain =0.208; minimum stress: t=4.752 N/m under uniaxial tension 1T-MnTe2 in the armchair direction with strain =0.256; maximum poisson's ratio: =0.181 under uniaxial tension 1T-MnTe2 in the armchair direction; minimum poisson's ratio: =0.121 under uniaxial tension 1T-MnO2 in the zigzag direction; maximum elastic modulus: Et=154.96 N/m under uniaxial tension 1T-MnO2 in the zigzag direction; minimum elastic modulus: Et=37.385 N/m under uniaxial tension 1T-MnTe2 in the armchair direction. These materials generally behave as approximately isotropic substances and exhibit brittle fracture characteristics. These results will help to design and use two-dimensional 1T-MnX2 sheets based nanocomposites and nanodevices. 1341 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 4: 1332-1342, 2025 DOI: 10.55214/25768484.v9i4.6261 © 2025 by the authors; licensee Learning Gate Transparency: The authors confirm that the manuscript is an honest, accurate, and transparent account of the study; that no vital features of the study have been omitted; and that any discrepancies from the study as planned have been explained. This study followed all ethical practices during writing. Authors’ Contributions: The idea of the paper was proposed by Nguyen Van Trang and Nguyen Huu Tu. The simulation and the calculation were conducted by Nguyen Van Thanh and Dang Van Manh. The manuscript was written by Nguyen Van Trang with support from Nguyen Huu Tu. All authors have read and agreed to the manuscript. Acknowledgements: This word has been supported by the Thai Nguyen University of Technology, Thai Nguyen University. Copyright: © 2025 by the authors. This open-access article is distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). References [1] J.-W. Jiang and Y.-P. Zhou, "Parameterization of Stillinger-Weber potential for two-dimensional atomic crystals," arXiv preprint arXiv:1704.03147, 2017. https://doi.org/10.48550/arXiv.1704.03147 [2] Q. Lê Minh, T. N. Hữu, L. Đ. T. Kim, and T. N. Văn, "Mô phỏng kéo màng vật liệu hai chiều đa nguyên tử cấu trúc nếp gấp," Tạp chí Khoa học Giao thông vận tải, vol. 73, no. 5, pp. 514-526, 2022. [3] A. K. Geim, "Graphene: status and prospects," science, vol. 324, no. 5934, pp. 1530-1534, 2009. [4] H.-T. Nguyen, M.-Q. Le, and V.-T. Nguyen, "Mode-I stress intensity factors of silicene, AlN, and SiC hexagonal sheets," Materials Research Express, vol. 5, no. 6, p. 065025, 2018. [5] R. Arenal, M.-S. Wang, Z. Xu, A. Loiseau, and D. Golberg, "Young modulus, mechanical and electrical properties of isolated individual and bundled single-walled boron nitride nanotubes," Nanotechnology, vol. 22, no. 26, p. 265704, 2011. https://doi.org/10.1088/0957-4484/22/26/265704 [6] Y. Chen, J. Zou, S. J. Campbell, and G. Le Caer, "Boron nitride nanotubes: pronounced resistance to oxidation," Applied physics letters, vol. 84, no. 13, pp. 2430-2432, 2004. https://doi.org/10.1063/1.1667278 [7] B. Baumeier, P. Krüger, and J. Pollmann, "Structural, elastic, and electronic properties of SiC, BN, and BeO nanotubes," Physical Review B, vol. 76, no. 8, p. 085407, 2007. https://doi.org/10.1103/PhysRevB.76.085407 [8] G. Guo, S. Ishibashi, T. Tamura, and K. Terakura, "Static dielectric response and Born effective charge of BN nanotubes from ab initio finite electric field calculations," Physical Review B, vol. 75, no. 24, p. 245403, 2007. [9] C. Chang et al., "Isotope effect on the thermal conductivity of boron nitride nanotubes," Physical review letters, vol. 97, no. 8, p. 085901, 2006. [10] C. Attaccalite, L. Wirtz, A. Marini, and A. Rubio, "Efficient Gate-tunable light-emitting device made of defective boron nitride nanotubes: from ultraviolet to the visible," Scientific reports, vol. 3, 2013. [11] L. H. Li, Y. Chen, M.-Y. Lin, A. M. Glushenkov, B.-M. Cheng, and J. Yu, "Single deep ultraviolet light emission from boron nitride nanotube film," Applied physics letters, vol. 97, no. 14, p. 141104, 2010. [12] L. Yu, Q. Yan, and A. Ruzsinszky, "Negative Poisson’s ratio in 1T-type crystalline two-dimensional transition metal dichalcogenides," Nature communications, vol. 8, no. 1, pp. 1-8, 2017. [13] M.-Q. Le and R. C. Batra, "Mode-I stress intensity factor in single layer graphene sheets," Computational Materials Science, vol. 118, pp. 251-258, 2016. [14] D. Singh, S. K. Gupta, I. Lukačević, and Y. Sonvane, "Indiene 2D monolayer: a new nanoelectronic material," RSC advances, vol. 6, no. 10, pp. 8006-8014, 2016. [15] Z. Zhu and D. Tománek, "Semiconducting layered blue phosphorus: a computational study," Physical review letters, vol. 112, no. 17, p. 176802, 2014. [16] J.-A. Yan, S.-P. Gao, R. Stein, and G. Coard, "Tuning the electronic structure of silicene and germanene by biaxial strain and electric field," Physical Review B, vol. 91, no. 24, p. 245403, 2015. https://doi.org/10.1103/PhysRevB.91.245403 [17] A. Molle, J. Goldberger, M. Houssa, Y. Xu, S.-C. Zhang, and D. Akinwande, "Buckled two-dimensional Xene sheets," Nature materials, vol. 16, no. 2, pp. 163-169, 2017. [18] Z. Ni et al., "Tunable bandgap in silicene and germanene," Nano letters, vol. 12, no. 1, pp. 113-118, 2012. https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.48550/arXiv.1704.03147 https://doi.org/10.1088/0957-4484/22/26/265704 https://doi.org/10.1063/1.1667278 https://doi.org/10.1103/PhysRevB.76.085407 https://doi.org/10.1103/PhysRevB.91.245403 1342 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 4: 1332-1342, 2025 DOI: 10.55214/25768484.v9i4.6261 © 2025 by the authors; licensee Learning Gate [19] F. H. Stillinger and T. A. Weber, "Computer simulation of local order in condensed phases of silicon," Physical review B, vol. 31, no. 8, p. 5262, 1985. [20] J. Tersoff, "Modeling solid-state chemistry: Interatomic potentials for multicomponent systems," Physical review B, vol. 39, no. 8, p. 5566, 1989. https://doi.org/10.1103/PhysRevB.39.5566 [21] P. Vishnoi, K. Pramoda, and C. Rao, "2D elemental nanomaterials beyond graphene," ChemNanoMat, vol. 5, no. 9, pp. 1062-1091, 2019. [22] D. Geng and H. Y. Yang, "Recent advances in growth of novel 2D materials: beyond graphene and transition metal dichalcogenides," Advanced Materials, vol. 30, no. 45, p. 1800865, 2018. https://doi.org/10.1002/adma.201800865 [23] F. Wang et al., "2D library beyond graphene and transition metal dichalcogenides: a focus on photodetection," Chemical Society Reviews, vol. 47, no. 16, pp. 6296-6341, 2018. [24] B. Liu and K. Zhou, "Recent progress on graphene-analogous 2D nanomaterials: Properties, modeling and applications," Progress in Materials Science, vol. 100, pp. 99-169, 2019. [25] B. Liu, Y. Huang, H. Jiang, S. Qu, and K. Hwang, "The atomic-scale finite element method," Computer methods in applied mechanics and engineering, vol. 193, no. 17-20, pp. 1849-1864, 2004. [26] D.-T. Nguyen, M.-Q. Le, T.-L. Bui, and H.-L. Bui, "Atomistic simulation of free transverse vibration of graphene, hexagonal SiC, and BN nanosheets," Acta Mechanica Sinica, vol. 33, pp. 132-147, 2017. [27] Y. Wang et al., "Atomistic finite elements applicable to solid polymers," Computational materials science, vol. 36, no. 3, pp. 292-302, 2006. [28] J. Wackerfuß, "Molecular mechanics in the context of the finite element method," International Journal for Numerical Methods in Engineering, vol. 77, no. 7, pp. 969-997, 2009. https://doi.org/10.1002/nme.2442 [29] B. Mortazavi, O. Rahaman, M. Makaremi, A. Dianat, G. Cuniberti, and T. Rabczuk, "First-principles investigation of mechanical properties of silicene, germanene and stanene," Physica E: Low-dimensional Systems and Nanostructures, vol. 87, pp. 228-232, 2017. https://doi.org/10.1016/j.physe.2016.10.047 [30] F. Hao and X. Chen, "First-principles study of the defected phosphorene under tensile strain," Journal of Applied Physics, vol. 120, no. 16, 2016. [31] J. Pokluda, M. Černý, M. Šob, and Y. Umeno, "Ab initio calculations of mechanical properties: Methods and applications," Progress in Materials Science, vol. 73, pp. 127-158, 2015. https://doi.org/10.1016/j.pmatsci.2015.04.001 [32] B. Liu, Y. Huang, H. Jiang, S. Qu, and K. C. Hwang, "The atomic-scale finite element method," Computer Methods in Applied Mechanics and Engineering, vol. 193, pp. 1849-1864, 2004. https://doi.org/10.1103/PhysRevB.39.5566 https://doi.org/10.1002/adma.201800865 https://doi.org/10.1002/nme.2442 https://doi.org/10.1016/j.physe.2016.10.047 https://doi.org/10.1016/j.pmatsci.2015.04.001