
Citation: Nataliya A. Sakharova, Jorge M. Antunes, Andre F. G. Pereira, Jose V. Fernandes. Developments in the evaluation of elastic properties of carbon nanotubes and their heterojunctions by numerical simulation[J]. AIMS Materials Science, 2017, 4(3): 706-737. doi: 10.3934/matersci.2017.3.706
[1] | Huilin Ge, Yuewei Dai, Zhiyu Zhu, Biao Wang . Robust face recognition based on multi-task convolutional neural network. Mathematical Biosciences and Engineering, 2021, 18(5): 6638-6651. doi: 10.3934/mbe.2021329 |
[2] | Chii-Dong Ho, Gwo-Geng Lin, Thiam Leng Chew, Li-Pang Lin . Conjugated heat transfer of power-law fluids in double-pass concentric circular heat exchangers with sinusoidal wall fluxes. Mathematical Biosciences and Engineering, 2021, 18(5): 5592-5613. doi: 10.3934/mbe.2021282 |
[3] | José M. Sigarreta . Extremal problems on exponential vertex-degree-based topological indices. Mathematical Biosciences and Engineering, 2022, 19(7): 6985-6995. doi: 10.3934/mbe.2022329 |
[4] | Qingqun Huang, Muhammad Labba, Muhammad Azeem, Muhammad Kamran Jamil, Ricai Luo . Tetrahedral sheets of clay minerals and their edge valency-based entropy measures. Mathematical Biosciences and Engineering, 2023, 20(5): 8068-8084. doi: 10.3934/mbe.2023350 |
[5] | Hao Wang, Guangmin Sun, Kun Zheng, Hui Li, Jie Liu, Yu Bai . Privacy protection generalization with adversarial fusion. Mathematical Biosciences and Engineering, 2022, 19(7): 7314-7336. doi: 10.3934/mbe.2022345 |
[6] | Meili Tang, Qian Pan, Yurong Qian, Yuan Tian, Najla Al-Nabhan, Xin Wang . Parallel label propagation algorithm based on weight and random walk. Mathematical Biosciences and Engineering, 2021, 18(2): 1609-1628. doi: 10.3934/mbe.2021083 |
[7] | Xinmei Liu, Xinfeng Liang, Xianya Geng . Expected Value of Multiplicative Degree-Kirchhoff Index in Random Polygonal Chains. Mathematical Biosciences and Engineering, 2023, 20(1): 707-719. doi: 10.3934/mbe.2023032 |
[8] | Fengwei Li, Qingfang Ye, Juan Rada . Extremal values of VDB topological indices over F-benzenoids with equal number of edges. Mathematical Biosciences and Engineering, 2023, 20(3): 5169-5193. doi: 10.3934/mbe.2023240 |
[9] | Stefano Cosenza, Paolo Crucitti, Luigi Fortuna, Mattia Frasca, Manuela La Rosa, Cecilia Stagni, Lisa Usai . From Net Topology to Synchronization in HR Neuron Grids. Mathematical Biosciences and Engineering, 2005, 2(1): 53-77. doi: 10.3934/mbe.2005.2.53 |
[10] | Qiming Li, Tongyue Tu . Large-pose facial makeup transfer based on generative adversarial network combined face alignment and face parsing. Mathematical Biosciences and Engineering, 2023, 20(1): 737-757. doi: 10.3934/mbe.2023034 |
To exemplify the phenomena of compounds scientifically, researchers utilize the contraption of the diagrammatic hypothesis, it is a well-known branch of geometrical science named graph theory. This division of numerical science provides its services in different fields of sciences. The particular example in networking [1], from electronics [2], and for the polymer industry, we refer to see [3]. Particularly in chemical graph theory, this division has extra ordinary assistance to study giant and microscope-able chemical compounds. For such a study, researchers made some transformation rules to transfer a chemical compound to a discrete pattern of shapes (graph). Like, an atom represents as a vertex and the covalent bonding between atoms symbolized as edges. Such transformation is known as molecular graph theory. A major importance of this alteration is that the hydrogen atoms are omitted. Some chemical structures and compounds conversion are presented in [4,5,6].
In cheminformatics, the topological index gains attraction due to its implementations. Various topological indices help to estimate a bio-activity and physicochemical characteristics of a chemical compound. Some interesting and useful topological indices for various chemical compounds are studied in [3,7,8]. A topological index modeled a molecular graph or a chemical compound into a numerical value. Since 1947, topological index implemented in chemistry [9], biology [10], and information science [11,12]. Sombor index and degree-related properties of simplicial networks [13], Nordhaus–Gaddum-type results for the Steiner Gutman index of graphs [14], Lower bounds for Gaussian Estrada index of graphs [15], On the sum and spread of reciprocal distance Laplacian eigenvalues of graphs in terms of Harary index [16], the expected values for the Gutman index, Schultz index, and some Sombor indices of a random cyclooctane chain [17,18,19], bounds on the partition dimension of convex polytopes [20,21], computing and analyzing the normalized Laplacian spectrum and spanning tree of the strong prism of the dicyclobutadieno derivative of linear phenylenes [22], on the generalized adjacency, Laplacian and signless Laplacian spectra of the weighted edge corona networks [23,24], Zagreb indices and multiplicative Zagreb indices of Eulerian graphs [25], Minimizing Kirchhoff index among graphs with a given vertex bipartiteness, [26], asymptotic Laplacian energy like invariant of lattices [27]. Few interesting studies regarding the chemical graph theory can be found in [28,29,30,31,32].
Recently, the researchers of [33] introduced a topological descriptor and called the face index. Moreover, the idea of computing structure-boiling point and energy of a structure, motivated them to introduced this parameter without heavy computation. They computed these parameters for different models compare the results with previous literature and found approximate solutions with comparatively less computations. This is all the blessings of face index of a graph. The major concepts of this research work are elaborated in the given below definitions.
Definition 1.1. [33] Let a graph G=(V(G),E(G),F(G)) having face, edge and vertex sets notation with F(G),E(G),V(G), respectively. It is mandatory that the graph is connected, simple and planar. If e from the edge set E(G), is one of those edges which surrounds a face, then the face f from the face set F(G), is incident to the edge e. Likewise, if a vertex α from the vertex set V(G) is at the end of those incident edges, then a face f is incident to that vertex. This face-vertex incident relation is symbolized here by the notation α∼f. The face degree of f in G is described as d(f)=∑α∼fd(α), which are elaborated in the Figure 1.
Definition 1.2. [33] The face index FI(G), for a graph G, is formulated as
FI(G)=∑f∈F(G)d(f)=∑α∼f,f∈F(G)d(α). |
In the Figure 1, we can see that there are two faces with degree 4, exactly two with five count and four with count of 6. Moreover, there is an external face with count of face degree 28, which is the count of vertices.
As the information given above that the face index is quite new and introduced in the year 2020, so there is not so much literature is available. A few recent studies on this topic are summarized here. A chemical compound of silicon carbides is elaborated with such novel definition in [34]. Some carbon nanotubes are discussed in [35]. Except for the face index, there are distance and degree-based graphical descriptors available in the literature. For example, distance-based descriptors of phenylene nanotube are studied in [36], and in [37] titania nanotubes are discussed with the same concept. Star networks are studied in [38], with the concept of degree-based descriptors. Bounds on the descriptors of some generalized graphs are discussed in [39]. General Sierpinski graph is discussed in [40], in terms of different topological descriptor aspects. The study of hyaluronic acid-doxorubicin ar found in [41], with the same concept of the index. The curvilinear regression model of the topological index for the COVID-19 treatment is discussed in [42]. For further reading and interesting advancements of topological indices, polynomials of zero-divisor structures are found in [43], zero divisor graph of commutative rings [44], swapped networks modeled by optical transpose interconnection system [45], metal trihalides network [46], some novel drugs used in the cancer treatment [47], para-line graph of Remdesivir used in the prevention of corona virus [48], tightest nonadjacently configured stable pentagonal structure of carbon nanocones [49]. In order to address a novel preventive category (P) in the HIV system known as the HIPV mathematical model, the goal of this study is to offer a design of a Morlet wavelet neural network (MWNN) [50].
In the next section, we discussed the newly developed face index or face-based index for different chemical compounds. Silicate network, triangular honeycomb network, carbon sheet, polyhedron generalized sheet, and generalized chain of silicate network are studied with the concept of the face-based index. Given that the face index is more versatile than vertex degree-based topological descriptors, this study will aid in understanding the structural characteristics of chemical networks. Only the difficulty authors will face to compute the face degree of a generalized network or structure, because it is more generalized version and taking degree based partition of edges into this umbrella of face index.
Silicates are formed when metal carbonates or metal oxides react with sand. The SiO4, which has a tetrahedron structure, is the fundamental chemical unit of silicates. The central vertex of the SiO4 tetrahedron is occupied by silicon ions, while the end vertices are occupied by oxygen ions [51,52,53]. A silicate sheet is made up of rings of tetrahedrons that are joined together in a two-dimensional plane by oxygen ions from one ring to the other to form a sheet-like structure. The silicate network SLn symbol, where n represents the total number of hexagons occurring between the borderline and center of the silicate network SLn. The silicate network of dimension one is depicted in Figure 2. It contain total 3n(5n+1) vertices are 36n2 edges. Moreover, the result required is detailed are available in Table 1.
Dimension | |f12| | |f15| | |f36| |
1 | 24 | 48 | 7 |
2 | 32 | 94 | 14 |
3 | 40 | 152 | 23 |
4 | 48 | 222 | 34 |
5 | 56 | 304 | 47 |
6 | 64 | 398 | 62 |
7 | 72 | 504 | 79 |
8 | 80 | 622 | 98 |
. | . | . | . |
. | . | . | . |
. | . | . | . |
n | 8n+16 | 6n2+28n+14 | n2+4n+2 |
Theorem 2.1. Let SLn be the silicate network of dimension n≥1. Then the face index of SLn is
FI(SLn)=126n2+720n+558. |
Proof. Consider SLn the graph of silicate network with dimension n. Suppose fi denotes the faces of graph SLn having degree i. that is, d(fi)=∑α∼fid(α)=i and |fi| denotes the number of faces with degree i. The graph SLn contains three types of internal faces f12, f15, f36, and single external face which is usually denoted by f∞.
If SLn has one dimension then sum of degree of vertices incident to the external face is 144 and when SLn has two dimension then sum of degree of incident vertices to the external face is 204 whenever SLn has three dimension then sum of degree of incident vertices to the external face is 264. Similarly, SLn has n−dimension then sum of degree of incident vertices to the external face is 60n+84.
The number of internal faces with degree in each dimension is mentioned in Table 1.
By using the definition of face index FI we have
FI(SLn)=∑α∼f∈F(SLn)d(α)=∑α∼f12∈F(SLn)d(α)+∑α∼f15∈F(SLn)d(α)+∑α∼f36∈F(SLn)d(α)+∑α∼f∞∈F(SLn)d(α)=|f12|(12)+|f15|(15)+|f36|(36)+(60n+84)=(8n+16)(12)+(6n2+28n+14)(15)+(n2+4n+2)(36)+60n+84=126n2+72n+558. |
Hence, this is our required result.
A chain silicate network of dimension (m,n) is symbolized as CSL(m,n) which is made by arranging (m,n) tetrahedron molecules linearly. A chain silicate network of dimension (m,n) with m,n≥1 where m denotes the number of rows and each row has n number of tetrahedrons. The following theorem formulates the face index FI for chain silicate network.
Theorem 2.2. Let CSL(m,n) be the chain of silicate network of dimension m,n≥1. Then the face index FI of the graph CSL(m,n) is
FI(CSL(m,n))={48n−12if m=1, n≥1;96m−12if n=1, m≥2;168m−60if n=2,m≥2;45m−9n+36mn−42if both m,n are even45m−9n+36mn−21otherwise. |
Proof. Let CSL(m,n) be the graph of chain silicate network of dimension (m,n) with m,n≥1 where m represents the number of rows and n is the number of tetrahedrons in each row. A graph CSL(m,n) for m=1 contains three type of internal faces f9, f12 and f15 with one external face f∞. While for m≥2, it has four type of internal faces f9, f12, f15 and f36 with one external face f∞. We want to evaluate the algorithm of face index FI for chain silicate network. We will discuss it in two different cases.
Case 1: When CSL(m,n) has one row (m=1) with n number of tetrahedrons as shown in the Figure 3.
The graph has three type of internal faces f9, f12 and f15 with one external face f∞. The sum of degree of incident vertices to the external face is 9n and number of faces are |f9|=2, |f12|=2n and |f15|=n−2. Now the face index FI of the graph CSL(m,n) is given by
FI(CSL(m,n))=∑α∼f∈F(CSL(m,n))d(α)=∑α∼f9∈F(CSL(m,n))d(α)+∑α∼f12∈F(CSL(m,n))d(α)+∑α∼f15∈F(CSL(m,n))d(α)+∑α∼f∞∈F(CSL(m,n))d(α)=|f9|(9)+|f12|(12)+|f15|(15)+(9n)=(2)(9)+(2n)(12)+(n−2)(15)+9n=48n−12. |
Case 2: When CSL(m,n) has more than one rows (m≠1) with n number of tetrahedrons in each row as shown in the Figure 4.
The graph has four type of internal faces f9, f12, f15 and f36 with one external face f∞. The sum of degree of incident vertices to the external face is
∑α∼f∞∈F(CSL(m,n))d(α)={18mif n=1, m≥1;27mif n=2, m≥1;30m+15n−30if both m,n are even30m+15n−33otherwise. |
The number of faces are |f9|, |f12|, f15 and |f36| are given by
|f9|={2if m is odd3+(−1)nif m is even.|f12|={2(2m+n−1)if m is odd4(⌊n+12⌋+2m−1)if m is even|f15|=(3m−2)n−m|f36|={(m−12)(n−1)if m is odd(2n+(−1)n−14)(m−22)nif m is even. |
Now the face index FI of the graph CSL(m,n) is given by
FI(CSL(m,n))=∑α∼f∈F(CSL(m,n))d(α)=∑α∼f9∈F(CSL(m,n))d(α)+∑α∼f12∈F(CSL(m,n))d(α)+∑α∼f15∈F(CSL(m,n))d(α)+∑α∼f36∈F(CSL(m,n))d(α)+∑α∼f∞∈F(CSL(m,n))d(α)=|f9|(9)+|f12|(12)+|f15|(15)+|f36|(36)+∑α∼f∞∈F(CSL(m,n))d(α). |
After some mathematical simplifications, we can get
FI(CSL(m,n))={48n−12if m=196m−12if n=1,∀m168m−60if n=2,∀m45m−9n+36mn−42if both m,n are even45m−9n+36mn−21otherwise. |
There are three regular plane tessellations known to exist, each constituted from the same type of regular polygon: triangular, square, and hexagonal. The triangular tessellation is used to define the hexagonal network, which is extensively studied in [54]. A dimensioned hexagonal network THk has 3k2−3k+1 vertices and 9k2−15k+6 edges, where k is the number of vertices on one side of the hexagon. It has 2k−2 diameter. There are six vertices of degree three that are referred to as corner vertices. Moreover, the result required detailed are available in the Table 2.
Dimension | |f12| | |f14| | |f17| | |f18| |
1 | 6 | 0 | 0 | 0 |
2 | 6 | 12 | 12 | 12 |
3 | 6 | 24 | 24 | 60 |
4 | 6 | 36 | 36 | 144 |
5 | 6 | 48 | 48 | 264 |
6 | 6 | 60 | 60 | 420 |
7 | 6 | 72 | 72 | 612 |
8 | 6 | 84 | 84 | 840 |
. | . | . | . | . |
. | . | . | . | . |
. | . | . | . | . |
k | 6 | 12(k−1) | 12(k−1) | 18k2−42k+24 |
Theorem 2.3. Let THk be the triangular honeycomb network of dimension k≥1. Then the face index of graph THk is
FI(THk)=324k2−336k+102. |
Proof. Consider THk be a graph of triangular honeycomb network. The graph TH1 has one internal and only one external face while graph THk with k≥2, contains four types of internal faces f12, f14, f17, and f18 with one external face f∞.
For TH1 the sum of degree of incident vertices to the external face is 18 and in TH2 the sum of degree of incident vertices to the external face is 66. Whenever the graph TH3, the sum of degree of incident vertices to the external face is 114. Similarly, for THk has n−dimension then sum of degree of incident vertices to the external face is 48k−30.
The number of internal faces with degree in each dimension is given in Table 2.
By using the definition of face index FI we have
FI(THk)=∑α∼f∈F(THk)d(α)=∑α∼f12∈F(THk)d(α)+∑α∼f14∈F(THk)d(α)+∑α∼f17∈F(THk)d(α)+∑α∼f18∈F(THk)d(α)+∑α∼f∞∈F(THk)d(α)=|f12|(12)+|f14|(14)+|f17|(17)+|f18|(18)+(48k−30)=(6)(12)+(12(k−1))(14)+(12(k−1))(17)+(18k2−42k+24)(18)+48k−30=324k2−336k+102. |
Hence, this is our required result.
Given carbon sheet in the Figure 6, is made by grid of hexagons. There are few types of carbon sheets are given in [55,56]. The carbon sheet is symbolize as HCSm,n, where n represents the total number of vertical hexagons and m denotes the horizontal hexagons. It contain total 4mn+2(n+m)−1 vertices and 6nm+2m+n−2 edges. Moreover, the result required detailed are available in Tables 3 and 4.
Dimension m | |f15| | |f16| | |f18| | |f∞| |
2 | 3 | 2(n−1) | n−1 | 20n+7 |
Dimension m | |f15| | |f16| | |f17| | |f18| | |f∞| |
2 | 3 | 2(n−1) | 0 | n−1 | 20n+7 |
3 | 2 | 2n | 1 | 3(n−1) | 20n+17 |
4 | 2 | 2n | 3 | 5(n−1) | 20n+27 |
5 | 2 | 2n | 5 | 7(n−1) | 20n+37 |
6 | 2 | 2n | 7 | 9(n−1) | 20n+47 |
. | . | . | . | . | . |
. | . | . | . | . | . |
. | . | . | . | . | . |
m | 2 | 2n | 2m−5 | 2mn−2m−3n+3 | 20n+10m−13 |
Theorem 2.4. Let HCSm,n be the carbon sheet of dimension (m,n) and m,n≥2. Then the face index of HCSm,n is
FI(HCSm,n)={70n+2ifm=236mn−14−2(n−4m)ifm≥3. |
Proof. Consider HCSm,n be the carbon sheet of dimension (m,n) and m,n≥2. Let fi denotes the faces of graph HCSm,n having degree i, which is d(fi)=∑α∼fid(α)=i, and |fi| denotes the number of faces with degree i. A graph HCSm,n for a particular value of m=2 contains three types of internal faces f15, f16, f17 and f18 with one external face f∞. While for the generalize values of m≥3, it contain four types of internal faces f15, f16 and f17 with one external face f∞ in usual manner. For the face index of generalize nanotube, we will divide into two cases on the values of m.
Case 1: When HCSm,n has one row or HCS2,n.
A graph HCSm,n for a this particular value of m=2 contains three types of internal faces |f15|=3, |f16|=2(n−1) and |f18|=n−1 with one external face f∞. For the face index of carbon sheet, details are given in the Table 3. Now the face index FI of the graph NT2,n is given by
FI(HCS2,n)=∑α∼f∈F(HCS2,n)d(α)=∑α∼f15∈F(HCS2,n)d(α)+∑α∼f16∈F(HCS2,n)d(α)+∑α∼f18∈F(HCS2,n)d(α)+∑α∼f∞∈F(HCS2,n)d(α)=|f15|(15)+|f16|(16)+|f18|(18)+20n+7.=3(15)+2(n−1)(16)+(n−1)(18)+20n+7.=70n+2. |
Case 2: When HCSm,n has m≥3 rows.
A graph HCSm,n for generalize values of m≥3 contains four types of internal faces |f15|=2, |f16|=2n, |f17|=2m−5 and |f18|=2mn−2m−3n+3 with one external face f∞. For the face index of carbon sheet, details are given in the Table 4. Now the face index FI of the graph NTm,n is given by
FI(HCSm,n)=∑α∼f∈F(HCSm,n)d(α)=∑α∼f15∈F(HCSm,n)d(α)+∑α∼f16∈F(HCSm,n)d(α)+∑α∼f17∈F(HCSm,n)d(α)+∑α∼f18∈F(HCSm,n)d(α)+∑α∼f∞∈F(HCSm,n)d(α)=|f15|(15)+|f16|(16)+|f17|(17)+|f18|(18)+20n+10m−13.=36mn−2n+8m−14. |
Given structure of polyhedron generalized sheet of C∗28 in the Figure 7, is made by generalizing a C∗28 polyhedron structure which is shown in the Figure 8. This particular structure of C∗28 polyhedron are given in [57]. The polyhedron generalized sheet of C∗28 is as symbolize PHSm,n, where n represents the total number of vertical C∗28 polyhedrons and m denotes the horizontal C∗28 polyhedrons. It contain total 23nm+3n+2m vertices and 33nm+n+m edges. Moreover, the result required detailed are available in Tables 3 and 5.
m | |f14| | |f15| | |f16| | |f17| | |f18| | |f20| | |f35| |
1 | 2n+1 | 2 | 4n−2 | 0 | 0 | 2n−1 | 0 |
2 | 2n+2 | 2 | 8n−2 | 2 | 2n−2 | 4n−2 | 2n−1 |
3 | 2n+3 | 2 | 12n−2 | 4 | 4n−4 | 6n−3 | 4n−2 |
. | . | . | . | . | . | . | . |
. | . | . | . | . | . | . | . |
. | . | . | . | . | . | . | . |
m | 2n+m | 2 | 4mn−2 | 2m−2 | 2mn−2(m+n)+2 | 2mn−m | 2mn−(m+2n)+1 |
Theorem 2.5. Let PHSm,n be the polyhedron generalized sheet of C∗28 of dimension (m,n) and m,n≥1. Then the face index of PHSm,n is
FI(PHSm,n)=210mn−2(3m+5n). |
Proof. Consider PHSm,n be the polyhedron generalized sheet of C∗28 of dimension (m,n) and m,n≥1. Let fi denotes the faces of graph PHSm,n having degree i, which is d(fi)=∑α∼fid(α)=i, and |fi| denotes the number of faces with degree i. A graph PHSm,n for the generalize values of m,n≥1, it contain seven types of internal faces f14,f15,f16,f17,f18,f20 and f35 with one external face f∞ in usual manner. For the face index of polyhedron generalized sheet, details are given in the Table 5.
A graph PHSm,n for generalize values of m,n≥1 contains seven types of internal faces |f14|=2n+m, |f15|=2, |f16|=4nm−2, |f17|=2(m−1), |f18|=2nm−2(m+n)+2, |f20|=2nm−2mn−m, and |f35|=2mn−m−2n+1 with one external face f∞. Now the face index FI of the graph PHSm,n is given by
FI(PHSm,n)=∑α∼f∈F(PHSm,n)d(α)=∑α∼f14∈F(PHSm,n)d(α)+∑α∼f15∈F(PHSm,n)d(α)+∑α∼f16∈F(PHSm,n)d(α)+∑α∼f17∈F(PHSm,n)d(α)+∑α∼f18∈F(PHSm,n)d(α)+∑α∼f20∈F(PHSm,n)d(α)+∑α∼f35∈F(PHSm,n)d(α)+∑α∼f∞∈F(PHSm,n)d(α)=|f14|(14)+|f15|(15)+|f16|(16)+|f17|(17)+|f18|(18)+|f20|(20)+|f35|(35)+37m+68n−35.=210mn−6m−10n. |
With the advancement of technology, types of equipment and apparatuses of studying different chemical compounds are evolved. But topological descriptors or indices are still preferable and useful tools to develop numerical science of compounds. Therefore, from time to time new topological indices are introduced to study different chemical compounds deeply. In this study, we discussed a newly developed tool of some silicate type networks and generalized sheets, carbon sheet, polyhedron generalized sheet, with the face index concept. It provides numerical values of these networks based on the information of faces. It also helps to study physicochemical characteristics based on the faces of silicate networks.
M. K. Jamil conceived of the presented idea. K. Dawood developed the theory and performed the computations. M. Azeem verified the analytical methods, R. Luo investigated and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript.
This work was supported by the National Science Foundation of China (11961021 and 11561019), Guangxi Natural Science Foundation (2020GXNSFAA159084), and Hechi University Research Fund for Advanced Talents (2019GCC005).
The authors declare that they have no conflicts of interest.
[1] |
Robertson J (2004) Realistic applications of CNTs. Mater Today 7: 46–52. doi: 10.1016/S1369-7021(04)00448-1
![]() |
[2] | Dresselhaus MS, Avouris P (2001) Introduction to carbon materials research, In: Dresselhaus, MS, Dresselhaus G, Avouris P, Carbon Nanotubes: Synthesis, Structure, Properties, and Applications, Springer Book Series: Topics in Applied Physics, Germany: Springer-Verlag Berlin Heidelberg, 80: 1–9. |
[3] |
Neubauer E, Kitzmantel M, Hulman M, et al. (2010) Potential and challenges of metal-matrix-composites reinforced with carbon nanofibers and carbon nanotubes. Compos Sci Technol 70: 2228–2236. doi: 10.1016/j.compscitech.2010.09.003
![]() |
[4] |
Lan Y, Wang Y, Ren ZF (2011) Physics and applications of aligned carbon nanotubes. Adv Phys 60: 553–678. doi: 10.1080/00018732.2011.599963
![]() |
[5] | Zhang Y, Zhuang X, Muthu J, et al. (2014) Load transfer of graphene/carbon nanotube/polyethylene hybrid nanocomposite by molecular dynamics simulation. Compos Part B-Eng 63: 27–33. |
[6] | Schulz MJ, Shanov VN, Yin Z (2014) Nanotube superfiber Materials, Oxford (UK): Elsevier, 848. |
[7] |
Wei DC, Liu YQ (2008) The intramolecular junctions of carbon nanotubes. Adv Mater 20: 2815–2841. doi: 10.1002/adma.200800589
![]() |
[8] |
Salvetat JP, Briggs GAD, Bonard JM, et al. (1999) Elastic and shear moduli of single-walled carbon nanotube ropes. Phys Rev Lett 82: 944–947. doi: 10.1103/PhysRevLett.82.944
![]() |
[9] |
Hall AR, An L, Liu J, et al. (2006) Experimental measurement of single-wall carbon nanotube torsional properties. Phys Rev Lett 96: 256102. doi: 10.1103/PhysRevLett.96.256102
![]() |
[10] |
Kallesøe C, Larsen MB, Bøggild P, et al. (2012) 3D mechanical measurements with an atomic force microscope on 1D structures. Rev Sci Instrum 83: 023704. doi: 10.1063/1.3681784
![]() |
[11] |
Wang L, Zhang Z, Han X (2013) In situ experimental mechanics of nanomaterials at the atomic scale. NPG Asia Mater 5: e40. doi: 10.1038/am.2012.70
![]() |
[12] |
Mielke SL, Troya D, Zhan S, et al. (2004) The role of vacancy defects and holes in the fracture of carbon nanotubes. Chem Phys Lett 390: 413–420. doi: 10.1016/j.cplett.2004.04.054
![]() |
[13] |
Hou W, Xiao S (2007) Mechanical behaviors of carbon nanotubes with randomly located vacancy defects. J Nanosci Nanotechnol 7: 4478–4485. doi: 10.1166/jnn.2007.862
![]() |
[14] |
Tserpes KI, Papanikos P (2005) Finite element modeling of single-walled carbon nanotubes. Compos Part B-Eng 36: 468–477. doi: 10.1016/j.compositesb.2004.10.003
![]() |
[15] |
Rafiee R, Heidarhaei M (2012) Investigation of chirality and diameter effects on the Young's modulus of carbon nanotubes using non-linear potentials. Compos Struct 94: 2460–2464. doi: 10.1016/j.compstruct.2012.03.010
![]() |
[16] |
Sakharova NA, Pereira AFG, Antunes JM, et al. (2015) Mechanical characterization of single-walled carbon nanotubes: Numerical simulation study. Compos Part B-Eng 75: 73–85. doi: 10.1016/j.compositesb.2015.01.014
![]() |
[17] |
Rafiee R, Moghadam RM (2014) On the modelling of carbon nanotubes: A critical review. Compos Part B-Eng 56: 435–449. doi: 10.1016/j.compositesb.2013.08.037
![]() |
[18] |
Yengejeh SI, Kazemi SA, Öchsner A (2016) Advances in mechanical analysis of structurally and atomically modified carbon nanotubes and degenerated nanostructures: A review. Compos Part B-Eng 86: 95–107. doi: 10.1016/j.compositesb.2015.10.006
![]() |
[19] |
Dresselhaus MS, Dresselhaus G, Saito R (1995) Physics of carbon nanotubes. Carbon 33: 883–891. doi: 10.1016/0008-6223(95)00017-8
![]() |
[20] |
Barros EB, Jorio A, Samsonidze GG, et al. (2006) Review on the symmetry-related properties of carbon nanotubes. Phys Rep 431: 261–302. doi: 10.1016/j.physrep.2006.05.007
![]() |
[21] | Melchor S, Dobado JA (2004) CoNTub: An algorithm for connecting two arbitrary carbon nanotubes. J Chem Inf Comp Sci 44: 1639–1646. |
[22] |
Ghavamian A, Andriyana A, Chin AB, et al. (2015) Numerical investigation on the influence of atomic defects on the tensile and torsional behavior of hetero-junction carbon nanotubes. Mater Chem Phys 164: 122–137. doi: 10.1016/j.matchemphys.2015.08.033
![]() |
[23] | Yao YG, Li QW, Zhang J, et al. (2007) Temperature-mediated growth of single-walled carbon-nanotube intramolecular junctions. Nat Mater 6: 283–286. |
[24] |
Li M, Kang Z, Li R, et al. (2013) A molecular dynamics study on tensile strength and failure modes of carbon nanotube junctions. J Phys D Appl Phys 46: 495301. doi: 10.1088/0022-3727/46/49/495301
![]() |
[25] |
Sakharova NA, Pereira AFG, Antunes JM, et al. (2016) Numerical simulation of the mechanical behaviour of single-walled carbon nanotubes heterojunctions. J Nano Res 38: 73–87. doi: 10.4028/www.scientific.net/JNanoR.38.73
![]() |
[26] |
Qin Z, Qin QH, Feng XQ (2008) Mechanical property of carbon nanotubes with intramolecular junctions: Molecular dynamics simulations. Phys Lett A 372: 6661–6666. doi: 10.1016/j.physleta.2008.09.010
![]() |
[27] |
Lu Q, Bhattacharya B (2005) The role of atomistic simulations in probing the small scale aspects of fracture-a case study on a single-walled carbon nanotube. Eng Fract Mech 72: 2037–2071. doi: 10.1016/j.engfracmech.2005.01.009
![]() |
[28] |
Yakobson BI, Brabec CJ, Bernholc J (1996) Nanomechanics of carbon tubes: instabilities beyond linear response. Phys Rev Lett 76: 2511–2514. doi: 10.1103/PhysRevLett.76.2511
![]() |
[29] | Lu JP (1997) Elastic properties of carbon nanotubes and nanoropes. Phys Rev Lett 79: 1298–1300. |
[30] |
Jin Y, Yuan FG (2003) Simulation of elastic properties of single-walled carbon nanotubes. Compos Sci Technol 63: 1507–1515. doi: 10.1016/S0266-3538(03)00074-5
![]() |
[31] |
Liew KM, He XQ, Wong CH (2004) On the study of elastic and plastic properties of multi-walled carbon nanotubes under axial tension using molecular dynamics simulation. Acta Mater 52: 2521–2527. doi: 10.1016/j.actamat.2004.01.043
![]() |
[32] |
Bao WX, Zhu CC, Cui WZ (2004) Simulation of Young's modulus of single-walled carbon nanotubes by molecular dynamics. Physica B 352: 156–163. doi: 10.1016/j.physb.2004.07.005
![]() |
[33] |
Zhang HW, Wang JB, Guo X (2005) Predicting the elastic properties of single-walled carbon nanotubes. J Mech Phys Solids 53: 1929–1950. doi: 10.1016/j.jmps.2005.05.001
![]() |
[34] |
Cheng HC, Liu YL, Hsu YC, et al. (2009) Atomistic continuum modelling for mechanical properties of single-walled carbon nanotubes. Int J Solids Struct 46: 1695–1704. doi: 10.1016/j.ijsolstr.2008.12.013
![]() |
[35] |
Kudin KN, Scuseria GE, Yakobson BI (2001) C2F, BN and C nanoshell elasticity from ab initio computations. Phys Rev B 64: 235406. doi: 10.1103/PhysRevB.64.235406
![]() |
[36] |
Wilmes AA, Pinho ST (2014) A coupled mechanical-charge/dipole molecular dynamics finite element method, with multi-scale applications to the design of graphene nano-devices. Int J Numer Meth Eng 100: 243–276. doi: 10.1002/nme.4706
![]() |
[37] |
Hernandez E, Goze C, Bernier P, et al. (1998) Elastic properties of C and BxCyNz composite nanotubes. Phys Rev Lett 80: 4502–4505. doi: 10.1103/PhysRevLett.80.4502
![]() |
[38] |
Zhou X, Zhou J, Ou-Yang ZC (2000) Strain energy and Young's modulus of single-wall carbon nanotubes calculated from electronic energy-band theory. Phys Rev B 62: 13692–13696. doi: 10.1103/PhysRevB.62.13692
![]() |
[39] | Ru CQ (2000) Effective bending stiffness of carbon nanotubes. Phys Rev B 62: 9973–9976. |
[40] |
Pantano A, Parks DM, Boyce MC (2004) Mechanics of deformation of single- and multi-wall carbon nanotubes. J Mech Phys Solids 52: 789–821. doi: 10.1016/j.jmps.2003.08.004
![]() |
[41] |
Kalamkarov AL, Georgiades AV, Rokkam SK, et al. (2006) Analytical and numerical techniques to predict carbon nanotubes properties. Int J Solids Struct 43: 6832–6854. doi: 10.1016/j.ijsolstr.2006.02.009
![]() |
[42] |
Muc A (2010) Design and identification methods of effective mechanical properties for carbon nanotubes. Mater Design 31: 1671–1675. doi: 10.1016/j.matdes.2009.03.046
![]() |
[43] |
Chang TC (2010) A molecular based anisotropic shell model for single-walled carbon nanotubes. J Mech Phys Solids 58: 1422–1433. doi: 10.1016/j.jmps.2010.05.004
![]() |
[44] |
Sears A, Batra RC (2004) Macroscopic properties of carbon nanotubes from molecular-mechanics simulations. Phys Rev B 69: 235406. doi: 10.1103/PhysRevB.69.235406
![]() |
[45] | Gupta SS, Batra RC (2008) Continuum structures equivalent in normal mode vibrations to single-walled carbon nanotubes. Comp Mater Sci 43: 715–723. |
[46] |
Wang Q (2004) Effective in-plane stiffness and bending rigidity of armchair and zigzag carbon nanotubes. Int J Solids Struct 41: 5451–5461. doi: 10.1016/j.ijsolstr.2004.05.002
![]() |
[47] |
Arash B, Wang Q (2012) A review on the application of nonlocal elastic models in modelling of carbon nanotubes and graphenes. Comp Mater Sci 51: 303–313. doi: 10.1016/j.commatsci.2011.07.040
![]() |
[48] |
Odegard GM, Gates TS, Nicholson LM, et al. (2002) Equivalent continuum modelling of nano-structured materials. Compos Sci Technol 62: 1869–1880. doi: 10.1016/S0266-3538(02)00113-6
![]() |
[49] |
Li C, Chou TW (2003) A structural mechanics approach for the analysis of carbon nanotubes. Int J Solids Struct 40: 2487–2499. doi: 10.1016/S0020-7683(03)00056-8
![]() |
[50] |
Ghavamian A, Rahmandoust M, Öchsner A (2013) On the determination of the shear modulus of carbon nanotubes. Compos Part B-Eng 44: 52–59. doi: 10.1016/j.compositesb.2012.07.040
![]() |
[51] |
Chen WH, Cheng HC, Liu YL (2010) Radial mechanical properties of single-walled carbon nanotubes using modified molecular structure mechanics. Comp Mater Sci 47: 985–993. doi: 10.1016/j.commatsci.2009.11.034
![]() |
[52] |
Eberhardt O, Wallmersperger T (2015) Energy consistent modified molecular structural mechanics model for the determination of the elastic properties of single wall carbon nanotubes. Carbon 95: 166–180. doi: 10.1016/j.carbon.2015.07.092
![]() |
[53] |
Natsuki T, Tantrakarn K, Endo M (2004) Prediction of elastic properties for single-walled carbon nanotubes. Carbon 42: 39–45. doi: 10.1016/j.carbon.2003.09.011
![]() |
[54] |
Meo M, Rossi M (2006) Prediction of Young's modulus of single wall carbon nanotubes by molecular-mechanics based finite element modeling. Compos Sci Technol 66: 1597–1605. doi: 10.1016/j.compscitech.2005.11.015
![]() |
[55] |
Giannopoulos GI, Kakavas PA, Anifantis NK (2008) Evaluation of the effective mechanical properties of single-walled carbon nanotubes using a spring based finite element approach. Comp Mater Sci 41: 561–569. doi: 10.1016/j.commatsci.2007.05.016
![]() |
[56] |
Wernik JM, Meguid SA (2010) Atomistic-based continuum modelling of the nonlinear behavior of carbon nanotubes. Acta Mech 212: 167–179. doi: 10.1007/s00707-009-0246-4
![]() |
[57] |
Ranjbartoreh AZ, Wang G (2010) Consideration of mechanical properties of single-walled carbon nanotubes under various loading conditions. J Nanopart Res 12: 537–543. doi: 10.1007/s11051-009-9808-6
![]() |
[58] |
Parvaneh V, Shariati M (2011) Effect of defects and loading on prediction of Young's modulus of SWCNTs. Acta Mech 216: 281–289. doi: 10.1007/s00707-010-0373-y
![]() |
[59] |
Mahmoudinezhad E, Ansari R, Basti A, et al. (2012) An accurate spring-mass model for predicting mechanical properties of single-walled carbon nanotubes. Comp Mater Sci 62: 6–11. doi: 10.1016/j.commatsci.2012.05.004
![]() |
[60] |
To CWS (2006) Bending and shear moduli of single-walled carbon nanotubes. Finite Elem Anal Design 42: 404–413. doi: 10.1016/j.finel.2005.08.004
![]() |
[61] | Papanikos P, Nikolopoulos DD, Tserpes KI (2008) Equivalent beams for carbon nanotubes. Comp Mater Sci 43: 345–352. |
[62] |
Ávila AF, Lacerda GSR (2008) Molecular mechanics applied to single-walled carbon nanotubes. Mater Res 11: 325–333. doi: 10.1590/S1516-14392008000300016
![]() |
[63] |
Shokrieh MM, Rafiee R (2010) Prediction of Young's modulus of graphene sheets and carbon nanotubes using nanoscale continuum mechanics approach. Mater Design 31: 790–795. doi: 10.1016/j.matdes.2009.07.058
![]() |
[64] |
Her SC, Liu SJ (2012) Theoretical prediction of tensile behavior of single-walled carbon nanotubes. Curr Nanosci 8: 42–46. doi: 10.2174/1573413711208010042
![]() |
[65] |
Lu X, Hu Z (2012) Mechanical property evaluation of single-walled carbon nanotubes by finite element modeling. Compos Part B-Eng 43: 1902–1913. doi: 10.1016/j.compositesb.2012.02.002
![]() |
[66] | Mohammadpour E, Awang M (2011) Predicting the nonlinear tensile behavior of carbon nanotubes using finite element simulation. Appl Phys A-Mater 104: 609–614. |
[67] |
Ghadyani G, Öchsner A (2015) Derivation of a universal estimate for the stiffness of carbon nanotubes. Physica E 73: 116–125. doi: 10.1016/j.physe.2015.05.024
![]() |
[68] |
Giannopoulos GI, Tsiros AP, Georgantzinos SK (2013) Prediction of Elastic Mechanical Behavior and Stability of Single-Walled Carbon Nanotubes Using Bar Elements. Mech Adv Mater Struct 20: 730–741. doi: 10.1080/15376494.2012.676714
![]() |
[69] |
Nasdala L, Ernst G (2005) Development of a 4-node finite element for the computation of nano-structured materials. Comp Mater Sci 33: 443–458. doi: 10.1016/j.commatsci.2004.09.047
![]() |
[70] |
Budyka MF, Zyubina TS, Ryabenko AG, et al. (2005) Bond lengths and diameters of armchair single wall carbon nanotubes. Chem Phys Lett 407: 266–271. doi: 10.1016/j.cplett.2005.03.088
![]() |
[71] | Ghadyani G, Öchsner A (2015) On a thickness free expression for the stiffness of carbon nanotubes. Solid State Commun 209: 38–44. |
[72] |
Pereira AFG, Antunes JM, Fernandes JV, et al. (2016) Shear modulus and Poisson's ratio of single-walled carbon nanotubes: numerical evaluation. Phys Status Solidi B 253: 366–376. doi: 10.1002/pssb.201552320
![]() |
[73] |
Krishnan A, Dujardin E, Ebbesen TW, et al. (1998) Young's modulus of single-walled nanotubes. Phys Rev B 58: 14013–14018. doi: 10.1103/PhysRevB.58.14013
![]() |
[74] |
Yu MF, Files BS, Arepalli S, et al. (2000) Tensile loading of ropes of single wall carbon nanotubes and their mechanical properties. Phys Rev Lett 84: 5552–5554. doi: 10.1103/PhysRevLett.84.5552
![]() |
[75] |
Shen L, Li J (2004) Transversely isotropic elastic properties of single-walled carbon nanotubes. Phys Rev B 69: 045414. doi: 10.1103/PhysRevB.69.045414
![]() |
[76] |
Ghadyani G, Soufeiani L, Ӧchsner A (2017) Angle dependence of the shear behaviour of asymmetric carbon nanotubes. Mater Design 116:136–143. doi: 10.1016/j.matdes.2016.11.097
![]() |
[77] |
Xiao JR, Gama BA, Gillespie Jr JW (2005) An analytical molecular structural mechanics model for the mechanical properties of carbon nanotubes. Int J Solids Struct 42: 3075–3092. doi: 10.1016/j.ijsolstr.2004.10.031
![]() |
[78] |
Wu Y, Zhang X, Leung AYT, et al. (2006) An energy-equivalent model on studying the mechanical properties of single-walled carbon nanotubes. Thin Wall Struct 44: 667–676. doi: 10.1016/j.tws.2006.05.003
![]() |
[79] |
Rahmandoust M, Öchsner A (2012) On finite element modeling of single- and multi-walled carbon nanotubes. J Nanosci Nanotechnol 12: 8129–8136. doi: 10.1166/jnn.2012.4521
![]() |
[80] |
Domínguez-Rodríguez G, Tapia A, Avilés F (2014) An assessment of finite element analysis to predict the elastic modulus and Poisson's ratio of singlewall carbon nanotubes. Comp Mater Sci 82: 257–263. doi: 10.1016/j.commatsci.2013.10.003
![]() |
[81] | Popov VN, Van Doren VE, Balkanski M (2000) Elastic Properties of single-walled carbon nanotubes. Phys Rev B 61: 3078–3084. |
[82] |
Gao RP, Wang ZL, Bai ZG, et al. (2000) Nanomechanics of individual carbon nanotubes from pyrolytically grown arrays. Phys Rev Lett 85: 622–625. doi: 10.1103/PhysRevLett.85.622
![]() |
[83] |
Andrews R, Jacques D, Qian D, et al. (2001) Purification and structural annealing of multiwalled carbon nanotubes at graphitization temperatures. Carbon 39: 1681–1687. doi: 10.1016/S0008-6223(00)00301-8
![]() |
[84] |
Terrones M, Banhart F, Grobert N, et al. (2002) Molecular junctions by joining single-walled carbon nanotubes. Phys Rev Lett 89: 075505. doi: 10.1103/PhysRevLett.89.075505
![]() |
[85] |
Scarpa F, Adhikari S, Wang CY (2009) Mechanical properties of non-reconstructed defective single-wall carbon nanotubes. J Phys D Appl Phys 42: 142002. doi: 10.1088/0022-3727/42/14/142002
![]() |
[86] |
Parvaneh V, Shariati M, Torabi H (2012) Bending buckling behavior of perfect and defective single-walled carbon nanotubes via a structural mechanics model. Acta Mech 223: 2369–2378. doi: 10.1007/s00707-012-0711-3
![]() |
[87] |
Rahmandoust M, Öchsner A (2009) Influence of structural imperfections and doping on the mechanical properties of single-walled carbon nanotubes. J Nano Res 6: 185–196. doi: 10.4028/www.scientific.net/JNanoR.6.185
![]() |
[88] |
Ghavamian A, Rahmandoust M, Öchsner A (2012) A numerical evaluation of the influence of defects on the elastic modulus of single and multi-walled carbon nanotubes. Comp Mater Sci 62: 110–116. doi: 10.1016/j.commatsci.2012.05.003
![]() |
[89] |
Ghavamian A, Öchsner A (2012) Numerical investigation on the influence of defects on the buckling behavior of single-and multi-walled carbon nanotubes. Physica E 46: 241–249. doi: 10.1016/j.physe.2012.08.002
![]() |
[90] |
Ghavamian A, Öchsner A (2013) Numerical modeling of eigenmodes and eigenfrequencies of single- and multi-walled carbon nanotubes under the influence of atomic defects. Comp Mater Sci 72: 42–48. doi: 10.1016/j.commatsci.2013.02.002
![]() |
[91] |
Poelma RH, Sadeghian H, Koh S, et al. (2012) Effects of single vacancy defect position on the stability of carbon nanotubes. Microelectron Reliab 52: 1279–1284. doi: 10.1016/j.microrel.2012.03.015
![]() |
[92] |
Sakharova NA, Pereira AFG, Antunes JM, et al. (2016) Numerical simulation study of the elastic properties of single-walled carbon nanotubes containing vacancy defects. Compos Part B-Eng 89: 155–168. doi: 10.1016/j.compositesb.2015.11.029
![]() |
[93] | Sakharova NA, Antunes JM, Pereira AFG, et al. (2015) The effect of vacancy defects on the evaluation of the mechanical properties of single-wall carbon nanotubes: Numerical simulation study, In: Öchsner A, Altenbach H, Springer Book Series On Advanced Structured Materials: Mechanical and Materials Engineering of Modern Structure and Component Design, Germany: Springer, 70: 323–339. |
[94] |
Wong CH (2010) Elastic properties of imperfect single-walled carbon nanotubes under axial tension. Comp Mater Sci 49: 143–147. doi: 10.1016/j.commatsci.2010.04.037
![]() |
[95] |
Rafiee R, Pourazizi R (2014) Evaluating the Influence of Defects on the Young's Modulus of Carbon Nanotubes Using Stochastic Modeling. Mater Res 17: 758–766. doi: 10.1590/S1516-14392014005000071
![]() |
[96] |
Zhang YP, Ling CC, Li GX, et al. (2015) Radial collapse and physical mechanism of carbon nanotube with divacancy and 5-8-5 defects. Chinese Phys B 24: 046401. doi: 10.1088/1674-1056/24/4/046401
![]() |
[97] | Rafiee R, Mahdavi M (2016) Molecular dynamics simulation of defected carbon nanotubes. P I Mech Eng L-J Mat 230: 654–662. |
[98] |
Sharma S, Chandra R, Kumar P, et al. (2014) Effect of Stone-Wales and vacancy defects on elastic moduli of carbon nanotubes and their composites using molecular dynamics simulation. Comp Mater Sci 86: 1–8. doi: 10.1016/j.commatsci.2014.01.035
![]() |
[99] |
Saxena KK, Lal A (2012) Comparative molecular dynamics simulation study of mechanical properties of carbon nanotubes with number of Stone-Wales and vacancy defects. Procedia Eng 38: 2347–2355. doi: 10.1016/j.proeng.2012.06.280
![]() |
[100] |
Yuan J, Liew KM (2009) Effects of vacancy defect reconstruction on the elastic properties of carbon nanotubes. Carbon 47: 1526–1533. doi: 10.1016/j.carbon.2009.01.048
![]() |
[101] |
Xiao S, Hou W (2006) Fracture of vacancy-defected carbon nanotubes and their embedded nanocomposites. Phys Rev B 73: 115406. doi: 10.1103/PhysRevB.73.115406
![]() |
[102] | Kurita H, Estili M, Kwon H, et al. (2015) Load-bearing contribution of multi-walled carbon nanotubes on tensile response of aluminium. Compos Part A-Appl S 68: 133–139. |
[103] | Kharissova OV, Kharisov BI (2014) Variations of interlayer spacing in carbon nanotubes. RSC Adv 58: 30807–30815. |
[104] |
Kiang CH, Endo M, Ajayan PM, et al. (1998) Size effects in carbon nanotubes. Phys Rev Lett 81: 1869–1872. doi: 10.1103/PhysRevLett.81.1869
![]() |
[105] |
Li C, Chou TW (2003) Elastic moduli of multi-walled carbon nanotubes and the effect of van der Waals forces. Comp Sci Tech 63: 1517–1524. doi: 10.1016/S0266-3538(03)00072-1
![]() |
[106] | Fan CW, Liu YY, Hwu C (2009) Finite element simulation for estimating the mechanical properties of multi-walled carbon nanotubes. Appl Phys A-Mater 5: 819–831. |
[107] | Nahas MN, Abd-Rabou M (2010) Finite element modeling of multi-walled carbon nanotubes. Int J Eng Technol 10: 63–71. |
[108] |
Sakharova NA, Pereira AFG, Antunes JM, et al. (2017) Numerical simulation on the mechanical behaviour of the multi-walled carbon nanotubes. J Nano Res 47: 106–119. doi: 10.4028/www.scientific.net/JNanoR.47.106
![]() |
[109] |
Liu Q, Liu W, Cui ZM, et al. (2007) Synthesis and characterization of 3D double branched K junction carbon nanotubes and nanorods. Carbon 45: 268–273. doi: 10.1016/j.carbon.2006.09.029
![]() |
[110] |
Scarpa F, Narojczyk JW, Wojciechowski KW (2011) Unusual deformation mechanisms in carbon nanotube heterojunctions (5, 5)–(10, 10) under tensile loading. Phys Status Solidi B 248: 82–87. doi: 10.1002/pssb.201083984
![]() |
[111] |
Lee WJ, Su WS (2013) Investigation into the mechanical properties of single-walled carbon nanotube heterojunctions. Phys Chem Chem Phys 15: 11579–11585. doi: 10.1039/c3cp51340h
![]() |
[112] |
Kang Z, Li M, Tang Q (2010) Buckling behavior of carbon nanotube-based intramolecular junctions under compression: Molecular dynamics simulation and finite element analysis. Comp Mater Sci 50: 253–259. doi: 10.1016/j.commatsci.2010.08.011
![]() |
[113] |
Kinoshita Y, Murashima M, Kawachi M, et al. (2013) First-principles study of mechanical properties of one-dimensional carbon nanotube intramolecular junctions. Comp Mater Sci 70: 1–7. doi: 10.1016/j.commatsci.2012.12.033
![]() |
[114] |
Xi H, Song HY, Zou R (2015) Simulation of mechanical properties of carbon nanotubes with superlattice structure. Curr Appl Phys 15: 1216–1221. doi: 10.1016/j.cap.2015.07.008
![]() |
[115] |
Yengejeh SI, Zadeh MA, Öchsner A (2014) On the buckling behavior of connected carbon nanotubes with parallel longitudinal axes. Appl Phys A-Mater 115: 1335–1344. doi: 10.1007/s00339-013-7999-2
![]() |
[116] |
Ghavamian A, Öchsner A (2015) A comprehensive numerical investigation on the mechanical properties of hetero-junction carbon nanotubes. Commun Theor Phys 64: 215–230. doi: 10.1088/0253-6102/64/2/215
![]() |
[117] |
Hemmatian H, Fereidoon A, Rajabpour M (2014) Mechanical properties investigation of defected, twisted, elliptic, bended and hetero-junction carbon nanotubes based on FEM. Fuller Nanotub Car N 22: 528–544. doi: 10.1080/1536383X.2012.684183
![]() |
[118] | Rajabpour M, Hemmatian H, Fereidoon A (2011) Investigation of Length and Chirality Effects on Young's Modulus of Heterojunction Nanotube with FEM. Proceedings of the 2nd International Conference on Nanotechnology: Fundamentals and Applications, Ottawa, Ontario, Canada. |
[119] |
Yengejeh SI, Zadeh MA, Öchsner A (2015) On the tensile behavior of hetero-junction carbon nanotubes. Compos Part B-Eng 75: 274–280. doi: 10.1016/j.compositesb.2015.02.001
![]() |
[120] | Yengejeh SI, Zadeh MA, Öchsner A (2014) Numerical characterization of the shear behavior of hetero-junction carbon nanotubes. J Nano Res 26: 143–151. |
1. | Shabana Anwar, Muhammad Kamran Jamil, Amal S. Alali, Mehwish Zegham, Aisha Javed, Extremal values of the first reformulated Zagreb index for molecular trees with application to octane isomers, 2023, 9, 2473-6988, 289, 10.3934/math.2024017 | |
2. | Ali N. A. Koam, Ali Ahmad, Raed Qahiti, Muhammad Azeem, Waleed Hamali, Shonak Bansal, Enhanced Chemical Insights into Fullerene Structures via Modified Polynomials, 2024, 2024, 1076-2787, 10.1155/2024/9220686 | |
3. | Ali Ahmad, Ali N. A. Koam, Muhammad Azeem, Reverse-degree-based topological indices of fullerene cage networks, 2023, 121, 0026-8976, 10.1080/00268976.2023.2212533 | |
4. | Muhammad Waheed Rasheed, Abid Mahboob, Iqra Hanif, Uses of degree-based topological indices in QSPR analysis of alkaloids with poisonous and healthful nature, 2024, 12, 2296-424X, 10.3389/fphy.2024.1381887 | |
5. | Shriya Negi, Vijay Kumar Bhat, Face Index of Silicon Carbide Structures: An Alternative Approach, 2024, 16, 1876-990X, 5865, 10.1007/s12633-024-03119-0 | |
6. | Haseeb AHMAD, Muhammad AZEEM, Face-degree-based topological descriptors of germanium phosphide, 2024, 52, 18722040, 100429, 10.1016/j.cjac.2024.100429 | |
7. | Belman Gautham Shenoy, Raghavendra Ananthapadmanabha, Badekara Sooryanarayana, Prasanna Poojary, Vishu Kumar Mallappa, 2024, Rational Wiener Index and Rational Schultz Index of Graphs, 180, 10.3390/engproc2023059180 |
Dimension | |f12| | |f15| | |f36| |
1 | 24 | 48 | 7 |
2 | 32 | 94 | 14 |
3 | 40 | 152 | 23 |
4 | 48 | 222 | 34 |
5 | 56 | 304 | 47 |
6 | 64 | 398 | 62 |
7 | 72 | 504 | 79 |
8 | 80 | 622 | 98 |
. | . | . | . |
. | . | . | . |
. | . | . | . |
n | 8n+16 | 6n2+28n+14 | n2+4n+2 |
Dimension | |f12| | |f14| | |f17| | |f18| |
1 | 6 | 0 | 0 | 0 |
2 | 6 | 12 | 12 | 12 |
3 | 6 | 24 | 24 | 60 |
4 | 6 | 36 | 36 | 144 |
5 | 6 | 48 | 48 | 264 |
6 | 6 | 60 | 60 | 420 |
7 | 6 | 72 | 72 | 612 |
8 | 6 | 84 | 84 | 840 |
. | . | . | . | . |
. | . | . | . | . |
. | . | . | . | . |
k | 6 | 12(k−1) | 12(k−1) | 18k2−42k+24 |
Dimension m | |f15| | |f16| | |f18| | |f∞| |
2 | 3 | 2(n−1) | n−1 | 20n+7 |
Dimension m | |f15| | |f16| | |f17| | |f18| | |f∞| |
2 | 3 | 2(n−1) | 0 | n−1 | 20n+7 |
3 | 2 | 2n | 1 | 3(n−1) | 20n+17 |
4 | 2 | 2n | 3 | 5(n−1) | 20n+27 |
5 | 2 | 2n | 5 | 7(n−1) | 20n+37 |
6 | 2 | 2n | 7 | 9(n−1) | 20n+47 |
. | . | . | . | . | . |
. | . | . | . | . | . |
. | . | . | . | . | . |
m | 2 | 2n | 2m−5 | 2mn−2m−3n+3 | 20n+10m−13 |
m | |f14| | |f15| | |f16| | |f17| | |f18| | |f20| | |f35| |
1 | 2n+1 | 2 | 4n−2 | 0 | 0 | 2n−1 | 0 |
2 | 2n+2 | 2 | 8n−2 | 2 | 2n−2 | 4n−2 | 2n−1 |
3 | 2n+3 | 2 | 12n−2 | 4 | 4n−4 | 6n−3 | 4n−2 |
. | . | . | . | . | . | . | . |
. | . | . | . | . | . | . | . |
. | . | . | . | . | . | . | . |
m | 2n+m | 2 | 4mn−2 | 2m−2 | 2mn−2(m+n)+2 | 2mn−m | 2mn−(m+2n)+1 |
Dimension | |f12| | |f15| | |f36| |
1 | 24 | 48 | 7 |
2 | 32 | 94 | 14 |
3 | 40 | 152 | 23 |
4 | 48 | 222 | 34 |
5 | 56 | 304 | 47 |
6 | 64 | 398 | 62 |
7 | 72 | 504 | 79 |
8 | 80 | 622 | 98 |
. | . | . | . |
. | . | . | . |
. | . | . | . |
n | 8n+16 | 6n2+28n+14 | n2+4n+2 |
Dimension | |f12| | |f14| | |f17| | |f18| |
1 | 6 | 0 | 0 | 0 |
2 | 6 | 12 | 12 | 12 |
3 | 6 | 24 | 24 | 60 |
4 | 6 | 36 | 36 | 144 |
5 | 6 | 48 | 48 | 264 |
6 | 6 | 60 | 60 | 420 |
7 | 6 | 72 | 72 | 612 |
8 | 6 | 84 | 84 | 840 |
. | . | . | . | . |
. | . | . | . | . |
. | . | . | . | . |
k | 6 | 12(k−1) | 12(k−1) | 18k2−42k+24 |
Dimension m | |f15| | |f16| | |f18| | |f∞| |
2 | 3 | 2(n−1) | n−1 | 20n+7 |
Dimension m | |f15| | |f16| | |f17| | |f18| | |f∞| |
2 | 3 | 2(n−1) | 0 | n−1 | 20n+7 |
3 | 2 | 2n | 1 | 3(n−1) | 20n+17 |
4 | 2 | 2n | 3 | 5(n−1) | 20n+27 |
5 | 2 | 2n | 5 | 7(n−1) | 20n+37 |
6 | 2 | 2n | 7 | 9(n−1) | 20n+47 |
. | . | . | . | . | . |
. | . | . | . | . | . |
. | . | . | . | . | . |
m | 2 | 2n | 2m−5 | 2mn−2m−3n+3 | 20n+10m−13 |
m | |f14| | |f15| | |f16| | |f17| | |f18| | |f20| | |f35| |
1 | 2n+1 | 2 | 4n−2 | 0 | 0 | 2n−1 | 0 |
2 | 2n+2 | 2 | 8n−2 | 2 | 2n−2 | 4n−2 | 2n−1 |
3 | 2n+3 | 2 | 12n−2 | 4 | 4n−4 | 6n−3 | 4n−2 |
. | . | . | . | . | . | . | . |
. | . | . | . | . | . | . | . |
. | . | . | . | . | . | . | . |
m | 2n+m | 2 | 4mn−2 | 2m−2 | 2mn−2(m+n)+2 | 2mn−m | 2mn−(m+2n)+1 |