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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
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In this paper we study systems of linear hyperbolic equations on a bounded interval, say,
∂t(υϖ)=(−C+00C−)∂x(υϖ),0<x<1,t>0, | (1a) |
Ξ(υ(0,t),ϖ(1,t),υ(1,t),ϖ(0,t))T=0,t>0, | (1b) |
υ(x,0)=υ˚ | (1c) |
where
\begin{equation} \boldsymbol{\Xi}_{out} (\boldsymbol{{ \upsilon}}(0, t), \boldsymbol{{ \varpi}}(1, t))^T + \boldsymbol{\Xi}_{in}(\boldsymbol{{ \upsilon}}(1, t), \boldsymbol{{ \varpi}}(0, t))^T = 0, \quad t > 0. \end{equation} | (2) |
An important class of such problems arises from dynamical systems on metric graphs. Let
\begin{equation} {\partial}_t \boldsymbol{p}^j+ \mathcal{M}^j{\partial}_x\boldsymbol{p}^j = 0, \quad 0 < x < 1, \; t > 0, \; 1\leq j\leq m, \end{equation} | (3) |
where
Such problems have been a subject of intensive research, both from the dynamics on graphs, [1,10,5,4,11,17,19], and the 1-D hyperbolic systems, [7,21,15,14], points of view. However, there is hardly any overlap, as there seems to be little interest in the network interpretation of the results in the latter, while in the former the conditions on the Riemann invariants seem to be "difficult to adapt to the case of a network", [11,Section 3].
The main aim of this paper, as well as of the preceding one [2] is to bring together these two approaches. In [2] we have provided explicit formulae allowing for a systematic conversion of Kirchhoff's type network transmission conditions to (1b) in such a way that the resulting system(1) is well-posed. We also gave a proof of the well-posedness on any
To briefly describe the content of the paper, we observe that if the matrix
\widehat {\left(\widehat{\boldsymbol{\Xi}_{out}}\right)^T\widehat{\boldsymbol{\Xi}_{in}}} |
is the adjacency matrix of a line graph (where for a matrix
The main idea of this paper is similar to that of [3]. However, [3] dealt with first order problems with (2) solved with respect to the outgoing data. Here, we do not make this assumption and, while(1) technically is one-dimensional, having reconstructed
The paper is organized as follows. In Section 2 we briefly recall the notation and relevant results from [2]. Section 3 contains the main result of the paper. In Appendix we recall basic results on line graphs in the interpretation suitable for the considerations of the paper.
We consider a network represented by a finite, connected and simple (without loops and multiple edges) metric graph
\mathcal{F}^j = \left(\begin{array}{cc} f^j_{+, 1}&f^j_{-, 1}\\ f^j_{+, 2}&f^j_{-, 2} \end{array}\right), |
the diagonalizing matrix on each edge. The Riemann invariants
\begin{equation} \boldsymbol{u}^j = (\mathcal{F}^{j})^{-1} \boldsymbol{p}^j\quad\text{and}\quad \boldsymbol{p}^j = \binom{f^j_{+, 1} u^j_1 + f^j_{-, 1}u^j_2}{f^j_{+, 2}u^j_1 + f^j_{-, 2}u^j_2}. \end{equation} | (4) |
Then, we diagonalize (3) and, discarding lower order terms, we consider
\begin{equation} {\partial}_t\boldsymbol{u}^j = \mathcal{L}^j{\partial}_x\boldsymbol{{u}}^j = \left(\begin{array}{cc}- \lambda^j_+&0\\0&- \lambda^j_- \end{array}\right){\partial}_x\boldsymbol{{u}}^j, \end{equation} | (5) |
for each
Remark 1. We refer an interested reader to [7,Section 1.1] for a detailed construction of the Riemann invariants for a general
The most general linear local boundary conditions at
\begin{equation} \boldsymbol{\Phi}_{ \mathbf{v}} \boldsymbol{p}( \mathbf{v}) = 0, \end{equation} | (6) |
where
\begin{equation} \boldsymbol{\Phi}_{ \mathbf{v}} : = \left(\begin{array}{ccccc}\phi^{j_1}_{ \mathbf{v}, 1}& \varphi^{j_1}_{ \mathbf{v}, 1}&\ldots&\phi^{j_{|J_{ \mathbf{v}}|}}_{ \mathbf{v}, 1}& \varphi^{j_{|J_{ \mathbf{v}}|}}_{ \mathbf{v}, 1}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \phi^{j_1}_{ \mathbf{v}, k_{ \mathbf{v}}}& \varphi^{j_1}_{ \mathbf{v}, k_{ \mathbf{v}}}&\ldots&\phi^{j_{|J_{ \mathbf{v}}|}}_{ \mathbf{v}, k_{ \mathbf{v}}}& \varphi^{j_{|J_{ \mathbf{v}}|}}_{ \mathbf{v}, k_{ \mathbf{v}}}\end{array}\right), \ \end{equation} | (7) |
where
\begin{equation} \boldsymbol{\Psi}_{ \mathbf{v}} \boldsymbol{u}( \mathbf{v}) : = \boldsymbol{\Phi}_{ \mathbf{v}} \mathcal{F}( \mathbf{v})\boldsymbol{u}( \mathbf{v}) = 0. \end{equation} | (8) |
For Riemann invariants, we can define their outgoing values at
Definition 2.1. Let
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Denote by
\begin{equation} k_{ \mathbf{v}} : = \sum\limits_{j\in J_{ \mathbf{v}}} (2(1-\alpha_j)l_j( \mathbf{v}) +\alpha_j ). \end{equation} | (9) |
Definition 2.2. We say that
● a sink if either
● a source if either
● a transient (or internal) vertex if it is neither a source nor a sink.
We denote the sets of sources, sinks and transient vertices by
We observe that if
A typical example of (8) is Kirchhoff's law that requires that the total inflow rate into a vertex must equal the total outflow rate from it. Its precise formulation depends on the context, we refer to [8,Chapter 18] for a detailed description in the context of flows in networks. Since it provides only one equation, in general it is not sufficient to ensure the well-posedness of the problem. So, we introduce the following definition.
Definition 2.3. We say that
To realize the requirement that the outgoing values should be determined by the incoming ones, we have to analyze the structure of
\begin{equation} \{1, \ldots, m\} = : J_1\cup J_2\cup J_0, \end{equation} | (10) |
where
J_{ \mathbf{v}} : = J_{ \mathbf{v}, 1}\cup J_{ \mathbf{v}, 2}\cup J_{ \mathbf{v}, 0}. |
We also consider another partition
\begin{equation} k_{ \mathbf{v}} = \sum\limits_{j\in J^0_{ \mathbf{v}}} \alpha_j + \sum\limits_{j\in J^1_{ \mathbf{v}}} (2-\alpha_j) = |J_{ \mathbf{v}, 1}|+ 2(|J^0_{ \mathbf{v}}\cap J_{ \mathbf{v}, 2}| + |J^1_{ \mathbf{v}}\cap J_{ \mathbf{v}, 0}|). \end{equation} | (11) |
Then, by [2,Lemma 3.6],
We introduce the block diagonal matrix
\begin{equation} \mathcal{{\tilde{{F}}}}_{out}( \mathbf{v}) = {\rm diag}\{\mathcal{{\tilde{{F}}}}_{out}^j( \mathbf{v})\}_{j \in J_{ \mathbf{v}}}, \end{equation} | (12) |
where
\mathcal{{\tilde{F}}}_{out}^j( \mathbf{v}) = \left\{\begin{array} {ccc} \left(\begin{array}{cc}0&0\\0&0\end{array}\right)&\text{if} &j\in (J_{ \mathbf{v}, 0} \cap J_{ \mathbf{v}}^0)\cup (J_{ \mathbf{v}, 2}\cap J_{ \mathbf{v}}^1), \\ \left(\begin{array}{cc}f^j_{+, 1}(l_j( \mathbf{v}))&f^j_{-, 1}(l_j( \mathbf{v}))\\f^j_{+, 2}(l_j( \mathbf{v}))&f^j_{-, 2}(l_j( \mathbf{v}))\end{array}\right)&\text{if}& j\in (J_{ \mathbf{v}, 0} \cap J_{ \mathbf{v}}^1)\cup (J_{ \mathbf{v}, 2}\cap J_{ \mathbf{v}}^0), \\ \left(\begin{array}{cc}f^j_{+, 1}(0)&0\\f^j_{+, 2}(0)&0\end{array}\right)&\text{if} &j\in J_{ \mathbf{v}, 1} \cap J_{ \mathbf{v}}^0, \\ \left(\begin{array}{cc}0&f^j_{-, 1}(1)\\0&f^j_{-, 2}(1)\end{array}\right)&\text{if}& j\in J_{ \mathbf{v}, 1} \cap J_{ \mathbf{v}}^1. \end{array} \right. |
Further, by
In a similar way, we extract from
\widetilde{\boldsymbol{u}}^j_{out}( \mathbf{v}) = \left\{\begin{array} {ccc} (0, 0)^T&\text{if} &j\in (J_{ \mathbf{v}, 0} \cap J_{ \mathbf{v}}^0)\cup (J_{ \mathbf{v}, 2}\cap J_{ \mathbf{v}}^1), \\ (u^j_1(l_j( \mathbf{v})), u^j_2(l_j( \mathbf{v})))^T&\text{if}& j\in (J_{ \mathbf{v}, 0} \cap J_{ \mathbf{v}}^1)\cup (J_{ \mathbf{v}, 2}\cap J_{ \mathbf{v}}^0), \\ (u^j_1(0), 0)^T&\text{if} &j\in J_{ \mathbf{v}, 1} \cap J_{ \mathbf{v}}^0, \\ (0, u^j_2(1))^T&\text{if}& j\in J_{ \mathbf{v}, 1} \cap J_{ \mathbf{v}}^1, \end{array} \right. |
and
Proposition 1. [2,Proposition 3.8] The boundary system (8) at
\begin{equation} \boldsymbol{\Phi}_{ \mathbf{v}} \mathcal{F}_{out}( \mathbf{v}) \boldsymbol{u}_{out}( \mathbf{v}) + \boldsymbol{\Phi}_{ \mathbf{v}} \mathcal{F}_{in}( \mathbf{v})\boldsymbol{u}_{in}( \mathbf{v}) = 0 \end{equation} | (13) |
and hence it uniquely determines the outgoing values of
\begin{equation} \boldsymbol{\Phi}_{ \mathbf{v}} \mathcal{F}_{out}( \mathbf{v})\quad \mathit{\text{is nonsingular}}. \end{equation} | (14) |
In this case,
\begin{equation} \boldsymbol{u}_{out}( \mathbf{v}) = - (\boldsymbol{\Phi}_{ \mathbf{v}} \mathcal{F}_{out}( \mathbf{v}))^{-1} \boldsymbol{\Phi}_{ \mathbf{v}} \mathcal{F}_{in}( \mathbf{v})\boldsymbol{u}_{in}( \mathbf{v}). \end{equation} | (15) |
To pass from (3) with Kirchhoff's boundary conditions at each vertex
\begin{equation} \boldsymbol{\Psi}' \gamma \boldsymbol{u} = 0. \end{equation} | (16) |
We note that the function values that are incoming at
\begin{equation} \boldsymbol{\Psi}^{out} \gamma \boldsymbol{u}_{out} + \boldsymbol{\Psi}^{in}\gamma\boldsymbol{u}_{in} = 0, \end{equation} | (17) |
where
Using the adopted parametrization and the formalism of Definition 2.1, we only need to distinguish between functions describing the flow from
\begin{equation} \begin{split} \boldsymbol{{\upsilon}} &: = \left((u^j_1)_{j\in J_1\cup J_2}, (u^j_2)_{j\in J_2}\right) = ( \upsilon_j)_{j\in J^+}, \\ \boldsymbol{{ \varpi}}& : = \left((u^j_1)_{j\in J_0}, (u^j_2)_{j\in J_1\cup J_0}\right) = ( \varpi_j)_{j\in J^-}, \end{split} \end{equation} | (18) |
where
In this way, we converted
Using this construction, the second order hyperbolic problem (3), (17) was transformed into first order system (1) with (17) written in the form (2). However, it is clear that (1) can be formulated with an arbitrary matrix
how to characterize matrices \boldsymbol{\Xi} that arise from \boldsymbol{\Psi} so that (1) describes a network dynamics?
For a graph
\boldsymbol{{\Psi}}_{ \mathbf{v}}^{out} = (\psi_{ \mathbf{v}, i}^j)_{1\leq i\leq k_{ \mathbf{v}}, j\in \boldsymbol{J}_{ \mathbf{v}}^-}, \qquad \boldsymbol{{\Psi}}_{ \mathbf{v}}^{in} = (\psi_{ \mathbf{v}, i}^j)_{1\leq i\leq k_{ \mathbf{v}}, j\in \boldsymbol{J}_{ \mathbf{v}}^+}. |
Since no outgoing value should be missing, we adopt the following
Assumption 1. No column or row of
These matrices provide some insight into how the arcs are connected by the flow which is an additional feature, superimposed on the geometric structure of the incoming and outgoing arcs at the vertex. In principle, these two structures do not have to be the same, that is, it may happen that the substance flowing from
Definition 3.1. Let
Using this idea, we define a connectivity matrix
\mathsf{c}_{ \mathbf{v}, lj} = \left\{\begin{array}{lcl} 1&\text{if}& \boldsymbol{{\varepsilon}}^j\;\text{flow connects to}\; \boldsymbol{{\varepsilon}}^l, \\ 0&&\text{otherwise}. \end{array} \right. |
Remark 2. We observe that
● the above definition implies that for
●
For an arbitrary matrix
Lemma 3.2. If
\begin{equation} \mathsf{C}_{ \mathbf{v}} = \widehat{\left(\widehat{\boldsymbol{{\Psi}}^{out}_{ \mathbf{v}}}\right)^T \widehat{\boldsymbol{{\Psi}}^{in}_{ \mathbf{v}}}}. \end{equation} | (19) |
Proof. Denote
\sum\limits_{r = 1}^{k_{ \mathbf{v}}} \hat{\psi}_{ \mathbf{v}, r}^i\hat{\psi}^j_{ \mathbf{v}, r} \neq 0. |
This occurs if and only if there is
Let
Definition 3.3. Let
As before, we construct a connectivity matrix
\begin{equation} \mathsf{c}_{ \mathbf{v}, ij} = \left\{\begin{array}{lcl} 1&\text{if}& \boldsymbol{{\varepsilon}}^j\;\text{and}\; \boldsymbol{{\varepsilon}}^i\;\text{are flow connected}, \\ 0&&\text{otherwise}. \end{array} \right. \end{equation} | (20) |
Note that, contrary to an internal vertex, here the connectivity matrix is symmetric. We also do not stipulate that
Lemma 3.4. If
\begin{equation} \mathsf{C}_{ \mathbf{v}} = \widehat{\left(\widehat{\boldsymbol{{\Psi}}^{out}_{ \mathbf{v}}}\right)^T \widehat{\boldsymbol{{\Psi}}^{out}_{ \mathbf{v}}}}. \end{equation} | (21) |
Proof. As before, let
\sum\limits_{r = 1}^{k_{ \mathbf{v}}} \hat{\psi}_{ \mathbf{v}, r}^i\hat{\psi}^j_{ \mathbf{v}, r} \neq 0. |
Certainly, by Assumption 1,
We adopt an assumption that the structure of flow connectivity is the same as of the geometry at the vertex. Thus, if
Assumption 2. For all
\mathsf{C}_{ \mathbf{v}} = \boldsymbol{1}_{ \mathbf{v}} = \left(\begin{array}{cccc}1&1&\ldots&1\\ \vdots&\vdots&\vdots&\vdots\\ 1&1&\ldots&1 \end{array}\right). |
We observe that the dimension of
If
Assumption 3. For all
Remark 3. Assumption 3 is weaker than requiring each two arcs from
Proposition 2. Let
\begin{equation} \boldsymbol{\Psi}_{ \mathbf{v}} \boldsymbol{u}( \mathbf{v}) = 0, \end{equation} | (22) |
contains a Kirchhoff's condition
\begin{equation} \sum\limits_{j \in J_{ \mathbf{v}}} (\psi^j_{ \mathbf{v}, r} u^j_1( \mathbf{v}) + \psi^j_{ \mathbf{v}, r} u^j_2( \mathbf{v})) = 0, \end{equation} | (23) |
with
Proof. Condition (23) ensures that each entry of the
Example 1. Consider the model of [20], analysed in the framework of our approach in [2,Example 5.12], i.e.,
\begin{equation} {\partial}_tp^{j}_1 + K^j {\partial}_x p^j_2 = 0, \quad {\partial}_tp^{j}_2 + L^j {\partial}_x p^j_1 = 0, \end{equation} | (24) |
for
\boldsymbol{p_1}( \mathbf{v}) \in X_{ \mathbf{v}}, \quad T_{ \mathbf{v}}\boldsymbol{p_2}( \mathbf{v}) \in X^\perp_{ \mathbf{v}}, |
that is, denoting
\begin{equation} \sum\limits_{j\in J_{ \mathbf{v}}}\phi^j_r p^j_1( \mathbf{v}) = 0, \quad r \in I_2, \quad \sum\limits_{j\in J_{ \mathbf{v}}} \varphi^j_r \nu^j( \mathbf{v})p^j_2( \mathbf{v}) = 0, \qquad r \in I_1, \end{equation} | (25) |
where
\begin{align*} p^r_1(0) & = u^r_1(0)+ u^r_2(0) = 0, \quad r = n_{ \mathbf{v}}+1, \ldots, |J_{ \mathbf{v}}|, \\ p^r_2(0) & = u^r_1(0)- u^r_2(0) = 0, \quad r = 1, \ldots, n_{ \mathbf{v}}. \end{align*} |
Thus
On the other hand, the Kirchhoff condition,
\begin{equation} \sum\limits_{j\in J_{ \mathbf{v}}} \nu^{j}( \mathbf{v}) p^{j}_2( \mathbf{v}) = 0, \end{equation} | (26) |
see [20,Eqn (4)], satisfies the assumption of Proposition 2, as we have
\begin{align*} 0& = \sum\limits_{j\in J_{ \mathbf{v}}}\nu^j( \mathbf{v})p^j_2( \mathbf{v}) = \sum\limits_{j\in J_{ \mathbf{v}}}\nu^j( \mathbf{v}) (f^j_{+, 2}( \mathbf{v})u^j_1( \mathbf{v}) +f^j_{-, 2}( \mathbf{v})u^j_2( \mathbf{v})) \\& = \sum\limits_{j\in J_{ \mathbf{v}}}\nu^j( \mathbf{v})\sqrt{K^jL^j} (u^j_1( \mathbf{v}) -u^j_2( \mathbf{v})) \\ & = -\sum\limits_{j \in J^0_{ \mathbf{v}}} \sqrt{K^jL^j}u^j_1(0) -\sum\limits_{j\in J^1_{ \mathbf{v}}} \sqrt{K^jL^j}u^j_2(1) \\ &\phantom{x}+\sum\limits_{j \in J^1_{ \mathbf{v}}} \sqrt{K^jL^j}u^j_1(1) +\sum\limits_{j\in J^0_{ \mathbf{v}}} \sqrt{K^jL^j}u^j_2(0), \end{align*} |
where we used [2,Eqn 5.2]. Hence, by Proposition 2, Assumption 2 is satisfied.
Example 2. Let us consider the linearized Saint-Venant system,
\begin{equation} {\partial}_tp^j_1 = -V^j {\partial}_x p^j_1 - H^j{\partial}_x p^j_2, \quad {\partial}_t p^j_2 = -g {\partial}_x p^j_1 -V^j {\partial}_x p^j_2, \end{equation} | (27) |
see [2,Example 1.2], assuming that on each edge we have
\begin{equation} \binom{p^j_1}{p^j_2} = \binom{f^j_{+, 1} u^j_1 + f^j_{-, 1}u^j_2}{f^j_{+, 2}u^j_1 + f^j_{-, 2}u^j_2} = \binom{H^j u^j_1+ H^j u^j_2}{\sqrt{gH^j}u^j_1 -\sqrt{gH^j}u^j_2}. \end{equation} | (28) |
We use the flow structure of [11,Example 5.1], shown in Fig. 1, and focus on
p^j_1(0) = p^1_1(1), \quad p^j_2(0) = p_2^1(1), \qquad j = 2, \ldots, N. |
In terms of the Riemann invariants, they can be written as
\begin{align*} &\left(\begin{array}{ccccccc}H^2&H^2&0&0&\ldots&0&0\\\sqrt{gH^2}&-\sqrt{gH^2}&0&0&\ldots&0&0\\\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&0&\ldots&H^N&H^N\\0&0&0&0&\ldots&\sqrt{gH^N}&-\sqrt{gH^N}\end{array}\right)\left(\begin{array}{c}u_1^2(0)\\u_2^2(0)\\\vdots \\ u_1^N(0)\\u_2^N(0)\end{array}\right)\\ &\phantom{xxxx} = \left(\begin{array}{cc}H^1&H^1\\\sqrt{gH^1}&-\sqrt{gH^1}\\\vdots\\ H^1&H^1\\\sqrt{gH^1}&-\sqrt{gH^1}\end{array}\right)\left(\begin{array}{c}u_1^1(1)\\u_2^1(1)\end{array}\right) \end{align*} |
and it is clear that Assumption 2 is satisfied.
For a matrix
\begin{equation} A = (\boldsymbol{a}^c_{j})_{1\leq j\leq m} = (\boldsymbol{a}^r_{i})_{1\leq i\leq n}, \end{equation} | (29) |
that is, we represent the matrix as a row vector of its columns or a column vector of its rows. In particular, we write
\begin{align*} \boldsymbol{\Xi}_{out} & = (\xi^{out}_{ij})_{1\leq i\leq 2m, 1\leq j\leq 2m} = (\boldsymbol{\xi}^{out, c}_{j})_{1\leq j\leq 2m} = (\boldsymbol{\xi}^{out, r}_{i})_{1\leq i\leq 2m}, \\\boldsymbol{\Xi}_{in} & = (\xi^{in}_{ij})_{1\leq i\leq 2m, 1\leq j\leq 2m} = (\boldsymbol{\xi}^{in, c}_{j})_{1\leq j\leq 2m} = (\boldsymbol{\xi}^{in, r}_{i})_{1\leq i\leq 2m}. \end{align*} |
For any vector
Definition 3.5. We say that the problem (1) is graph realizable if there is a graph
Before we formulate the main theorem, we need to introduce some notation. Let us recall that we consider the boundary system (2), i.e.,
\boldsymbol{\Xi}_{out}(( \upsilon_j(0, t))_{j\in J^+}, ( \varpi_j(1, t))_{j\in J^-}) = - \boldsymbol{\Xi}_{in}(( \upsilon_j(1, t))_{j\in J^+}, ( \varpi_j(0, t))_{j\in J^-}). |
Let us emphasize that in this notation, the column indices on the left and right hand side correspond to the values of the same function. To shorten notation, let us renumber them as
In the second step we will determine additional assumptions that allow
Since we do not want (2) to be under- or over-determined, we adopt
Assumption 4. For all
\boldsymbol{\xi}^{out, c}_j\neq 0\quad\mathit{\text{and}}\quad \boldsymbol{\xi}^{out, r}_j \neq 0. |
Our strategy is to treat
Assumption 5. The matrix
\mathsf{A}: = \widehat {\left(\widehat{\boldsymbol{\Xi}_{out}}\right)^T\widehat{\boldsymbol{\Xi}_{in}}} |
is the adjacency matrix of the line graph of a multi digraph.
For
I : = \{i\in \{1, \ldots, 2m\};\; \boldsymbol{\xi}^{in, r}_i = 0\} |
and adopt
Assumption 6. For all
\mathit{\text{supp}}\; \boldsymbol{\xi}^{out, r}_i \subset V^{out}_j. |
In the next proposition we shall show that
Proposition 3. If Assumptions 4, 5 and 6 are satisfied, then the sets
\begin{equation} \{\{\mathcal{V_i}\}_{1\leq i\leq n}, \mathcal{V_S}\}, \end{equation} | (30) |
where
\begin{align} \mathcal{V_S} & = \{i \in \{1, \ldots, 2m\};\; \mathrm{supp\;} \boldsymbol{\xi}^{out, r}_i \subset V^{out}_{M'}\}, \end{align} | (31) |
\begin{align} \mathcal{V_i} & = \bigcup\limits_{s \in V^{out}_i}\mathrm{supp\;} \boldsymbol{{\xi}}^{out, c}_s, \;1\leq i\leq n, \end{align} | (32) |
form a partition of the row indices of both
Since the proof is quite long, we first present its outline.
Step 1. Reconstruct a multi digraph
Step 2. Identify the rows of
Step 3. Associate other rows of
Step 4. Associate remaining rows with vertices and construct a possible partition of the row indices.
Step 5. Check that the constructed partition has the required properties.
Proof. Step 1. By Assumption 5,
Let us recall that the entry
\mathsf{a}_{ij} = \widehat{\widehat{\boldsymbol{{\xi}}^{out, c}_i}\cdot \widehat{\boldsymbol{{\xi}}^{in, c}_j}} |
and if a row
Step 2. To determine the rows in
\mathcal{V_S} = \{i \in \{1, \ldots, 2m\};\; \text{supp}\; \boldsymbol{\xi}^{out, r}_i \subset V^{out}_{M'}\}. |
For any
Step 3. Now, consider the indices
Step 4. Next, we associate the remaining rows in
\mathcal{V}^{out}_i = \bigcup\limits_{s \in V^{out}_i} \text{supp}\; \boldsymbol{{\xi}}^{out, c}_s, \qquad \mathcal{V}^{in}_j = \bigcup\limits_{q \in V^{in}_j} \text{supp}\; \boldsymbol{{\xi}}^{in, c}_q. |
We first observe that if
\begin{equation} \mathcal{V}^{out}_i\setminus \{s \in \mathcal{V}^{out}_i;\; \boldsymbol{\xi}^{in, r}_s = 0\} = \mathcal{V}^{in}_j. \end{equation} | (33) |
Indeed, let
Step 5. We easily check that this partition satisfies the conditions of the proposition. We have already checked this for
We note that (30) does not contain rows corresponding to sinks and they must be added following the rules described in Appendix A. With such an extension, we consider the multi digraph
\begin{equation} \{\{\mathcal{V_i}\}_{1\leq i\leq n}, \mathcal{V_S}, \mathcal{V_Z}\}, \quad \{\{V^{out}_i\}_{1\leq i\leq n}, V^{out}_{M'}, \emptyset\}, \quad \{\{V^{in}_{j_i}\}_{1\leq i\leq n}, \emptyset, V^{in}_{N'}\}, \end{equation} | (34) |
where the association
\begin{equation} \begin{split} &\boldsymbol{\Xi}^i_{out}(( \upsilon_j(0, t))_{j\in J^+\cap V^{out}_i}, ( \varpi_j(1, t))_{j\in J^-\cap V^{out}_i})\\& \phantom{xxx} = - \boldsymbol{\Xi}^i_{in}(( \upsilon_j(1, t))_{j\in J^+\cap V^{in}_{j_i}}, ( \varpi_j(0, t))_{j\in J^-\cap V^{in}_{j_i}}), \quad 1\leq i\leq n, \\ &\boldsymbol{\Xi}^S_{out}(( \upsilon_j(0, t))_{j\in J^+\cap V^{out}_{M'}}, ( \varpi_j(1, t))_{j\in J^-\cap V^{out}_{M'}}) = 0. \end{split} \end{equation} | (35) |
This system can be seen as a Kirchhoff system on the multi digraph
Let
If we grouped all sources into one node, as before Proposition 5, then, by Lemma 3.4, the flow connectivity in this source would be given by
\mathsf{C}_{ \mathbf{v}}: = \widehat{\left(\widehat{\boldsymbol{\Xi}_{out}^{S}}\right)^T\widehat{\boldsymbol{\Xi}_{out}^{S}}}. |
However, such a matrix would not necessarily satisfy Assumption 3. Thus, we separate the arcs into non-communicating groups, each determining a source satisfying Assumption 3. For this, by simultaneous permutations of rows and columns,
\begin{equation} \boldsymbol{\Xi}^S = {\rm diag}\{ \boldsymbol{\Xi}^S_{i}\}_{1\leq i\leq k}, \end{equation} | (36) |
where
\xi^{out, r}_{S_i j} = \left\{\begin{array}{lcl} 1 &\text{if}& j\in V^{out}_{S_i}, \\0&\text{otherwise}.&\end{array}\right. |
For the sinks, it is simpler as there is no constraining information from (2). We have columns with indices in
\begin{equation} V^{in}_{\mathcal{V_i}} = \{j\in V^{in}_{N'};\; \text{supp}\; \boldsymbol{\xi}^{out, c}_j\cap \mathcal{V_i}\neq \emptyset\}, \quad i = 1, \ldots, n, S_1, \ldots, S_k. \end{equation} | (37) |
For each
\begin{equation} V^{in}_{\mathcal{V_i}} = V^{in}_{i, {1}}\cup\ldots\cup V^{in}_{i, {l_i}}, \quad i = 1, \ldots, n, S_1, \ldots, S_k, \end{equation} | (38) |
where
\xi^{in, r}_{\{i, {l}\}, q} = \left\{\begin{array}{lcl} 1 &\text{if}& q\in V^{in}_{i, {l}}, \\0&\text{otherwise}.&\end{array}\right. |
Remark 4. We expect
Then, as in Remark 5, the incoming and outgoing incidence matrices are
A^+ = \left(\begin{array}{c} \mathsf{A}^+\\\boldsymbol{0}\\\boldsymbol{\xi}^{in, r}\end{array}\right), \qquad A^- = \left(\begin{array}{c} \mathsf{A}^-\\\boldsymbol{\xi}^{out, r}\\\boldsymbol{0}\end{array}\right) |
which, by a suitable permutation of columns moving the sources and the sinks to the last positions, can be written as
(39) |
respectively. Both matrices have
(40) |
where the dimensions of the blocks in the first row are, respectively,
Consider a nonzero pair
\begin{equation} \begin{split} &(a_{ij}\mapsto \{k^{ij}_1, \ldots, k^{ij}_h\}, a_{ji}\mapsto \{k^{ji}_1, \ldots, k^{ji}_e\}) \\ & = (a_{ij}\mapsto \text{supp}\; \boldsymbol{a}^{+, r}_i\cap \text{supp}\; \boldsymbol{a}^{-, r}_j, a_{ji}\mapsto \text{supp}\; \boldsymbol{a}^{+, r}_j\cap \text{supp}\; \boldsymbol{a}^{-, r}_i)\end{split} \end{equation} | (41) |
of columns of
Theorem 3.6. System (2) is graph realizable with generalized Kirchhoff's conditions satisfying Assumptions 1 and 2 for
1. for any
\begin{equation} (2, 0), \; (1, 1), \; (0, 2)\; \mathit{\text{or}}\; (0, 0); \end{equation} | (42) |
2. if
\begin{array}{l} if\;({a_{ij}}, {a_{ji}}) = (2, 0)\;or\;(0, 2), \;\;\;then\;k, l \in {J^ + }\;or\;k, l \in {J^ - }\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;and\;{c_k} \ne {c_l}, \\ if\;({a_{ij}}, {a_{ji}}) = (1, 1), \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;then\;k \in {J^ + }\;and\;l \in {J^ - }\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;or\;k \in {J^ - }\;and\;l \in {J^ + }. \end{array} | (43) |
Proof. Necessity. Let us consider the Kirchhoff system (17). By construction, both matrices
\tilde A = (\tilde a_{ij})_{1\leq i, j\leq 2m} = \widehat{(\widehat{\boldsymbol{\Psi}^{out}})^T\widehat{\boldsymbol{\Psi}^{in}}} |
is block diagonal with blocks of the form
\boldsymbol{\Xi}_{out} = \boldsymbol{\Psi}^{out}P, \qquad \boldsymbol{\Xi}_{in} = \boldsymbol{\Psi}^{in}Q, |
where
A = (a_{ij})_{1\leq i, j\leq 2m} : = \widehat{(\widehat{\boldsymbol{\Xi}_{out}})^T\widehat{\boldsymbol{\Xi}_{in}}} = \widehat{(\widehat{\boldsymbol{\Psi}^{out}}P)^T\widehat{\boldsymbol{\Psi}^{in}}Q} = P^T\tilde AQ |
is a matrix where the indices
Sufficiency. Given (2), we have flows
Now, (42) ensures that there are no loops at vertices and that between any two vertices there are either two arcs or none. If
Finally, the assumption
Example 3. Let us consider the system
\begin{equation} \begin{split} {\partial}_t \upsilon_{j}+c_j{\partial}_x \upsilon_j & = 0, \quad 1\leq j\leq 4, \\ {\partial}_t \varpi_{j}-c_j{\partial}_x \varpi_j & = 0, \quad 5\leq j\leq 6, \end{split} \end{equation} | (44) |
where
\begin{equation} \left(\!\!\begin{array}{cccccc}0&1&1&0&0&0\\ 1&0&0&0&0&0\\ 1&1&0&0&0&0\\ 0&0&1&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{array}\!\! \right)\left(\!\!\!\begin{array}{c} \upsilon_1(0)\\ \upsilon_2(0)\\ \upsilon_3(0)\\ \upsilon_4(0)\\ \varpi_5(1)\\ \varpi_6(1)\end{array}\!\!\!\right) = \left(\!\!\begin{array}{cccccc}0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 1&1&0&0&0&1\\ 0&0&1&1&1&0\end{array}\!\! \right)\left(\!\!\!\begin{array}{c} \upsilon_1(1)\\ \upsilon_2(1)\\ \upsilon_3(1)\\ \upsilon_4(1)\\ \varpi_5(0)\\ \varpi_6(0)\end{array}\!\!\!\right). \end{equation} | (45) |
Thus,
A = \widehat{(\widehat{\boldsymbol{\Xi}_{out}})^T\widehat{\boldsymbol{\Xi}_{in}}} = \left(\begin{array}{cccccc}0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 1&1&0&0&0&1\\ 0&0&1&1&1&0\end{array} \right). |
Thus, there is a multi digraph
\boldsymbol{\Xi}^S_{out} = \left(\begin{array}{cccc}0&1&1&0\\ 1&0&0&0\\ 1&1&0&0\\ 0&0&1&1\end{array}\right) \quad\text{and so}\quad \boldsymbol{\Xi}^S = \widehat{\left(\widehat{\boldsymbol{\Xi}_{out}^{S}}\right)^T\widehat{\boldsymbol{\Xi}_{out}^{S}}} = \left(\begin{array}{cccc}1&1&0&0\\ 1&1&1&0\\ 0&1&1&1\\ 0&0&1&1\end{array}\right). |
This matrix is irreducible and thus we have one source. Therefore
A^+ = \left(\begin{array}{cccccc}1&1&0&0&0&1\\ 0&0&1&1&1&0\\ 0&0&0&0&0&0\end{array}\right), \qquad A^- = \left(\begin{array}{cccccc}0&0&0&0&1&0\\ 0&0&0&0&0&1\\ 1&1&1&1&0&0\end{array}\right) |
and consequently
Further,
\begin{align*} \text{supp}\; \boldsymbol{a}^{+, r}_1& = \{1, 2, 6\}, \; \text{supp}\; \boldsymbol{a}^{+, r}_2 = \{3, 4, 5\}, \\ \text{supp}\; \boldsymbol{a}^{-, r}_1& = \{5\}, \; \text{supp}\; \boldsymbol{a}^{-, r}_2 = \{6\}, \; \text{supp}\; \boldsymbol{a}^{-, r}_3 = \{1, 2, 3, 4\}, \end{align*} |
hence, by (41),
(a_{12}\mapsto \{6\}, a_{21}\mapsto \{5\}), \quad (a_{13}\mapsto \{1, 2\}, a_{31}\mapsto \emptyset), \quad (a_{23}\mapsto \{3, 4\}, a_{32}\mapsto \emptyset). |
To reconstruct
Consider a small modification of (44), (45),
\begin{equation} \begin{split} {\partial}_t \upsilon_{j}+c_j{\partial}_x \upsilon_j & = 0, \quad 1\leq j\leq 5, \\ {\partial}_t \varpi_{6}-c_6{\partial}_x \varpi_6 & = 0, \end{split} \end{equation} | (46) |
\begin{equation} \begin{split} \upsilon_5(0)- \upsilon_1(1)- \upsilon_2(1)- \varpi_6(0)& = 0, \\ \varpi_6(1)- \upsilon_3(1)- \upsilon_4(1)- \upsilon_5(1)& = 0. \end{split} \end{equation} | (47) |
The matrices
\begin{equation} \begin{array}{c} {\partial}_t u^1_1+c_1{\partial}_x u^1_1 = 0, \\ {\partial}_t u^1_2+c_2{\partial}_x u^1_2 = 0, \end{array}\;\begin{array}{c} {\partial}_t u^2_1+c_5{\partial}_x u^2_1 = 0, \\ {\partial}_t u^2_2-c_6{\partial}_x u^2_2 = 0, \end{array}\; \begin{array}{c}{\partial}_t u^3_1+c_3{\partial}_x u^3_1 = 0, \\{\partial}_t u^3_2+c_4{\partial}_x u^3_2 = 0, \end{array} \end{equation} | (48) |
with boundary conditions at
(49) |
Consider a digraph
Proposition 4. [6,Thm. 2.4.1] A binary matrix
For our analysis, it is important to understand the reconstruction of
\mathsf{A} = (\mathsf{{a}}_{ij})_{1\leq i, j\leq m} = (\boldsymbol{{a}}^c_j)_{1\leq j\leq m} = (\boldsymbol{{a}}^r_i)_{1\leq i\leq m}. |
If for some
Using the adjacency matrix of a line digraph, we cannot determine how many sources or sinks the original graph could have without additional information. We can lump all potential sources and sinks into one source and one sink, we can have as many sinks and sources as there are zero columns and rows, respectively, or we can subdivide the arcs into some intermediate arrangement. We describe a construction with one source and one sink and indicate its possible variants.
We introduce
\begin{equation} \begin{split} M: = \left\{\begin{array}{lcl} M' &\text{if}& V^{out}_{M'} = \{j;\; \boldsymbol{a}^r_j \neq 0\}, \\ M'-1 &\text{if} & V^{out}_{M'} = \{j;\; \boldsymbol{a}^r_j = 0\}, \end{array}\right.\\ N: = \left\{\begin{array}{lcl} N' &\text{if}& V^{in}_{N'} = \{j;\; \boldsymbol{a}^c_j \neq 0\}, \\ N'-1 &\text{if} & V^{in}_{N'} = \{j;\; \boldsymbol{a}^c_j = 0\}.\end{array}\right.\end{split} \end{equation} | (A.1) |
Thus, we see that the number of internal (or transient) vertices, that is, which are neither sources nor sinks is
\begin{equation} \mathbf{v}_j = \{V^{in}_j, V^{out}_i\}, \qquad a_{i_pj_r} = 1\;\text{for some/any }i_p \in V^{out}_i, j_r \in V^{in}_j. \end{equation} | (A.2) |
With this notation, we present a more algorithmic way of reconstructing
\begin{equation} \mathsf{A}^+ = (\boldsymbol{a}^r_i)_{i\in \mathbb{I}^+}, \qquad \mathsf{A}^- = \left((\boldsymbol{a}^c_j)_{j\in \mathbb{I}^-}\right)^T. \end{equation} | (A.3) |
We see now that each row of
Proposition 5.
Proof. Since each column of
The adjacency matrix
Remark 5. Assume that
where
\bar A^+(\bar A^-)^T = (A^+P)(A^-P)^T = A^+PP^T(A^-)^T = A^+A^-, |
as
(A.4) |
Example 4 Consider the networks
\begin{equation} \mathsf{A} = \left(\begin{array}{ccccccc}0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 1&1&0&1&0&0&0\\ 0&0&1&0&1&0&0\\ 0&0&0&0&0&0&0\\ 0&0&1&0&1&0&0\\ 0&0&0&0&0&0&0\end{array}\right). \end{equation} | (A.5) |
Then,
\begin{equation} \mathsf{A}^+ = \left(\begin{array}{ccccccc} 1&1&0&1&0&0&0\\ 0&0&1&0&1&0&0\\ 0&0&0&0&0&0&0\end{array}\right) \end{equation} | (A.6) |
and we see that there are two transient (internal) vertices
\begin{equation} \mathsf{A}^- = \left(\begin{array}{ccccccc} 0&0&1&0&0&0&0\\ 0&0&0&1&0&1&0\\ 0&0&0&0&0&0&0\end{array}\right). \end{equation} | (A.7) |
The last row corresponds to sinks and the zero columns inform us that arcs
If we want to reconstruct the original graph with one source and one sink, then
A^+ = \left(\begin{array}{ccccccc} 1&1&0&1&0&0&0\\ 0&0&1&0&1&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&1&1\end{array}\right), \qquad A^- = \left(\begin{array}{ccccccc} 0&0&1&0&0&0&0\\ 0&0&0&1&0&1&0\\ 1&1&0&0&1&0&1\\ 0&0&0&0&0&0&0\end{array}\right) |
and
A^+(A^-)^T = \left(\begin{array}{cccc} 0&1&2&0\\ 1&0&1&0\\ 0&0&0&0\\ 0&1&1&0\end{array}\right), |
which describes the right multi digraph in Fig. 5. On the other hand, we can consider two sinks (maximum number, as there are two zero columns in
A^+ = \left(\begin{array}{ccccccc} 1&1&0&1&0&0&0\\ 0&0&1&0&1&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&1\\ 0&0&0&0&0&1&0\end{array}\right), \qquad A^- = \left(\begin{array}{ccccccc} 0&0&1&0&0&0&0\\ 0&0&0&1&0&1&0\\ 1&0&0&0&0&0&1\\ 0&1&0&0&0&0&0\\ 0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\end{array}\right) |
and
A^+(A^-)^T = \left(\begin{array}{ccccccc} 0&1&1&1&0&0&0\\ 1&0&0&0&1&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0\\ 0&1&0&0&0&0&0\end{array}\right) |
which describes the left multi digraph in Fig. 5.
It is easily seen that both digraphs have the same line digraph, shown on Fig. 6, whose adjacency matrix is
[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. |
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