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Compressive behavior and buckling response of
Carbon Nanotubes
Dianyun Zhang∗, Aswath Rangarajan†, Anthony M. Waas‡∗
College of Engineering, University of Michigan, Ann Arbor, MI-48109, USA
Carbon nanotubes (CNTs) are allotropes of Carbon, which are supposed to possessvery high strength. There has been relatively less analysis on the buckling response andthe compressive behavior of the Carbon Nanotubes. The current study deals with thecompressive behavior and buckling response of CNTs, while dealing with it as a springsystem as well as by performing lattice calculations using GULP (General Utility LatticeProgram) and Finite Element Analysis (FEA) on the Carbon Nanotube structure, whileapproximating them to be structures made of beams. The spring model shows responseswhich are very similar to the buckling response for long nanotubes, as seen in the literature.
I. Introduction
Carbon Nanotubes (CNTs) are allotropes of Carbon, which are supposed to possess very high strength.The possible applications to which such high strength materials can be used in vary from simple items suchas tennis rackets, going right to next generation armors. The stability and buckling of CNTs has beenwidely discussed, but most of the theories do not conform to the experimental results accurately.
There have been a few experimental studies on the axial stability of multi walled CNTs. The work doneby Waters et al. on the compression of CNTs having very low aspect ratios shows the CNTs to have anincreasing equilibrium load after the initial buckling, which occurs around 2.5 µN .
From the work of Akita et al[1], it can be seen that for CNTs having very large aspect ratios, the forcedisplacement curves have a linear slope, and after a decrease in the slope at the buckling load, the linearrelationship between the load and displacement is maintained. This behavior of the long CNTs can be com-pared with spring systems which have been designed to have the same force-displacement characteristics.The results also show that carbon nanotube bundles seem to exhibit super elastic behavior, making themsuitable candidates for the next generation armors.
Hirooka et al[3] have worked on the loading and unloading of CNTs having intermediate aspect ratios(between 40 and 60). These results show that the CNTs come back to their initial states on unloading evenafter the buckling load has been reached. However, one of the CNTs having a length of 800 mm, showsforce-displacement behavior which is similar to the behavior observed by Akita et al[1].
TEM images of buckled configurations of CNTs show ripple like distortions on the CNT . It can also beseen that one side is in tension while the other side is in compression during the buckling of the CNT , andhence, the load bearing capacity of the CNT even upon buckling.
Non Local Continuum theories can be used to create a 1D model of the CNT , but that would not capturethe exact buckling mechanism of the CNT , since the effects due to the shell type of structure of the nanotubeis ignored in this model.
∗Graduate Student, Department of Aerospace Engineering†Graduate Student, Department of Aerospace Engineering‡Professor, Department of Aerospace Engineering and Department of Mechanical Engineering
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II. Models used for analyzing CNTs
A. Spring Model to characterize the CNT :
1. Two DOF Model
Two Degree of freedom spring systems have been found to have buckling characteristics which are similar tothat of long CNTs. Thus, in the analysis which follows, a non linear spring model is used in order to obtainthe buckling characteristics of the CNT .
2. Description of the model
The model consists of spring systems which are used in the place of the CNT . Considering the CNT as ashell, during buckling, one side of the CNT is in tension while the other side is in compression. Hence, therewill be an additional load bearing capacity due to the tensile side of the CNT . This idea is implementedusing linear and rotational springs as follows. The side which is in compression is characterized by a combi-nation of linear translational springs and non linear rotational springs, while the side which is in tension ischaracterized by non linear translational springs. Also, the linear springs do not bend, i.e. they are rigid inbending. The equilibrium curves are obtained by the minimization of the Potential energy.
π =kt1R
2
4+2kr1θ]
2−4kr2θ]4−P (L−(L−R)Cosθ)+
1
2kt2(L−(L−R)Cosθ)2+
1
4kt3(L−(L−R)Cosθ)4 (1)
Figure 1. Two DOF Spring Model
3. Parametrization
The spring stiffnesses are parameterized in order to obtain a correlation to the experimental data. Theparametrization is done as follows.
NON DIMENSIONAL PARAMETERS
p =P
4kr1(2)
r =R
L(3)
p =P
4kr1(4)
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American Institute of Aeronautics and Astronautics
k1 =kt1L
2
32kr1(5)
k2 =kt2L
2
32kr1(6)
k3 =kt3L
2
32kr1(7)
k22 =kr2kr1
(8)
π0 =1
2θ2 − k22θ
4 + 2k1r2 + 4k2(1 − (1 − r)Cosθ)2 + p(1 − r)Cosθ − p+ 2k3(1 − (1 − r)Cosθ)4 (9)
B. GULP
1. Brief overview of GULP:
General Utility Lattice Program (GULP) is used for performing lattice energy calculations. In the presentcase, the program is used to obtain the minimum energy configuration of the CNT . The process is iterative,ie., the minimum energy configuration for a particular length of the CNT is obtained using the previousleast energy configuration of the CNT as a base. GULP helps in obtaining the configuration of the CNTwhich minimizes the lattice energy of the atoms of the CNT .
Figure 2. CNT used in the Simulation
2. Simulations for various CNT configurations
In this study, armchair configurations of CNTs were analyzed using GULP. A detailed description of thevarious configurations of the CNT is given in referenceThe coordinates of the CNT are obtained usingTubeGen, given in the reference In this study, Single Walled CNTs (SWCNT) are studied.
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3. Potential used for the Simulation
In this study, the Tersoff Potential is used for creating the interactions between the atoms of the CNT . TheTersoff potential has been found to be useful for simulating the behavior of molecules which have covalentbonding between the atoms[]. The Tersoff potential is a bond order potential and takes into account thevarious interactions due to the torsion and stretching behavior between the various carbon atoms. It takesinto account Multi Body potentials and hence the non local behavior of the CNT , which is more realisticthan a local model for describing the behavior of the CNT .
A detailed description of the Tersoff Potential can be found in reference [13].
C. Finite Element
1. Model
The Model for the analysis was created using the scripting facility of CATIA (model shown in Fig). Theanalysis and post processing of the model was performed in ABAQUS. The model was meshed using beamelements, and given stiffness parameters of Youngs Modulus(E) of , and a beam radius (r) of . The links areconsidered to be beams, while spring elements are not considered in the current model.
Figure 3. CNT used in the Simulation
2. Buckling Analysis
The buckling analysis of the CNT model using ABAQUS yields the buckling modes as shown below.However, a detailed analysis of a parametrized shell model has to be implemented in order to obtain theelasticity constants which can be used for the ABAQUS model.
III. Results and Discussion
A. Spring Model
On scaling the experimental curve and varying the values of the non dimensional spring constants, a verygood fit of the analytical model to the experimental result is obtained as shown in the figure below.
The dotted line shows the experimental result scaled by a factor of 3, and the thick lines show the equi-librium curves for the spring system. As seen from the curves, there is a very good correlation betweencharacteristics of the spring model and the experimental data. Hence, the behavior of long CNTs (aspect
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Figure 4. Comparison of Spring System with Experiments
ratios¿ 100), can be approximated using the non linear spring model.
B. GULP
The Force Displacement Curves show that there is a rapid decrease in the equilibrium load of the CNT onbuckling, but the load rapidly increases and stabilizes. In this study, only SWCNTsare studied. However,in most of the experiments, the emphasis has been on the study of compression of Multi Walled CNTs(MWCNT ), due to the difficulty of manufacturing a SWCNT . Hence, simulations on MWCNTs wouldgive a better understanding of the nature of buckling and axial stability of the CNT .
1. Force Displacement Curves
Shown in Fig is the Force Displacement curve for a 6-6 CNT having a length of 15A. The aspect ratio for theCNT is The relations to obtain the radius of the CNT can be obtained from the referenceThe energy (E)displacement and force ( F = dE/dx) displacement curves are obtained for the CNT configuration givenabove. We can see that there is a drastic drop in the load at a displacement of , at which the CNT buckles.We can see that after the steep decrease, the axial load increases and then stabilizes for the CNT .
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Figure 5. Energy and Force Displacement Curves
C. FEA
Analysis using ABAQUS gives the mode shapes for the CNT model. However, these results are dependenton the elastic properties assigned to the beam elements in the model. Further, to get results which capturethe physics of compression of the CNT , the joints would need to be modeled with rotational springs, whichwould give a more realistic picture of the buckling of the nanotube. The buckling mode shapes from theanalysis of the CNT are shown in Fig.
Figure 6. First and Second Buckling Modes
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Figure 7. Third and Fourth Buckling Modes
Appendix
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American Institute of Aeronautics and Astronautics
AE 518 Project2 DOF ModelAdd Spring and perform nondimensionNon-linear springs!!!
Potential Energy
Clear��Θ, �x�;urot � Integrate�kr1 �Θ � kr2 �Θ3, �Θ�;utr1 � Integrate�kt1 �x, �x�;�Θ � 2 Θ;
�x �R
2;
��Load displacement���U � L � �L � R� Cos�Θ�;��Add another spring��
utr2 �1
2kt2 �U2 �
1
4kt3 �U4;
u � urot � 2 utr1 � utr2;
w � P �U;
��Total potential energy��� � u � w;
Print�"��", ��
��kt1 R2
4� 2 kr1 Θ2 � 4 kr2 Θ4 � P �L � �L � R� Cos�Θ�� �
1
2kt2 �L � �L � R� Cos�Θ��2 �
1
4kt3 �L � �L � R� Cos�Θ��4
NondimentionΠ � �
4 kr1
r � RL
p � P4 kr1
k1 ��kt1 L2�
32 kr1
k2 ��kt2 L2�
32 kr1
k3 ��kt3 L2�
32 kr1
k22 �kr2
kr1
Π0 �1
2Θ2 � k22 Θ4 � 2 k1 r2 � 4 k2 �1 � �1 � r� Cos�Θ��2 � p �1 � r� Cos�Θ� � p � 2 k3 �1 � �1 � r� Cos�Θ��4;
�u � 1 � �1 � r� Cos�Θ�;��Equlibrium��f1 � Solve�D�Π0, r� 0, p�;f2 � Solve�D�Π0, Θ� 0, p�;pp1 � p �. f1��1��;pp2 � p �. f2��1��;solr � Solve�pp1 pp2, r�
��r �k1 � Cos�Θ� �k1 Θ Csc�Θ� Sec�Θ� � 4 k1 k22 Θ3 Csc�Θ� Sec�Θ� � k12 Sec�Θ�2
2 k1�,
�r �k1 � Cos�Θ� �k1 Θ Csc�Θ� Sec�Θ� � 4 k1 k22 Θ3 Csc�Θ� Sec�Θ� � k12 Sec�Θ�2
2 k1��
r1 � r �. solr��1, 1��;r2 � r �. solr��2, 1��;p1 � pp1 �. r � r1;
p2 � pp1 �. r � r2;
u1 � �u �. r � r1;
u2 � �u �. r � r2;
2 Math_model_v5.nb
plot1 � ParametricPlot�u1 �. k22 � 0.4, k3 � 1, k2 � 1, k1 � 1, p1 �. k22 � 0.4, k3 � 1, k2 � 1, k1 � 1,Θ, 0.001, 1.5, AspectRatio � 1, PlotStyle � Blue, Thick�;
plot2 � ParametricPlot�u2 �. k22 � 0.4, k3 � 1, k2 � 1, k1 � 1,p2 �. k22 � 0.4, k3 � 1, k2 � 1, k1 � 1,Θ, 0.001, 1.5, AspectRatio � 1, PlotStyle � Green, Thick�;
��For Θ�0����u0��u�.Θ�0;��solu0 � Solve���u �. Θ � 0� u0, r�;r0 � r �. solu0��1��;p0 � pp1 �. Θ � 0, r � r0;plot0 � Plot�p0 �. k22 � 0.4, k3 � 1, k2 � 1, k1 � 1, u0, 0, 1, PlotStyle � Red, Thick�;plot00 � ListLinePlot�0, 0, 1.9 � 3.5, 24.5 � 3.5,
1.9 � 3.5, 15 � 3.5, 2.9 � 3.5, 30 � 3.5, PlotStyle � Dashed�;Show�plot0, plot1, plot2, plot00, AspectRatio � 1,
AxesLabel � Displacement, Force, LabelStyle � 16�
0.2 0.4 0.6 0.8 1.0Displacement
5
10
15
20Force
Math_model_v5.nb 3
plot1 � ParametricPlot�u1 �. k22 � 0.5, k3 � 1, k2 � 1, k1 � 1, p1 �. k22 � 0.5, k3 � 1, k2 � 1, k1 � 1,Θ, 0.001, 1.5, AspectRatio � 1, PlotStyle � Blue, Thick�;
plot2 � ParametricPlot�u2 �. k22 � 0.5, k3 � 1, k2 � 1, k1 � 1,p2 �. k22 � 0.5, k3 � 1, k2 � 1, k1 � 1,Θ, 0.001, 1.5, AspectRatio � 1, PlotStyle � Green, Thick�;
��For Θ�0����u0��u�.Θ�0;��solu0 � Solve���u �. Θ � 0� u0, r�;r0 � r �. solu0��1��;p0 � pp1 �. Θ � 0, r � r0;plot0 � Plot�p0 �. k22 � 0.5, k3 � 1, k2 � 1, k1 � 1, u0, 0, 1, PlotStyle � Red, Thick�;plot00 � ListLinePlot�0, 0, 1.9 � 3.5, 24.5 � 3.5,
1.9 � 3.5, 15 � 3.5, 2.9 � 3.5, 30 � 3.5, PlotStyle � Dashed�;Show�plot0, plot1, plot2, AspectRatio � 1, AxesLabel � Displacement, Force, LabelStyle � 16�
0.2 0.4 0.6 0.8 1.0Displacement
5
10
15
20Force
4 Math_model_v5.nb
� CODE FOR CREATING THE CNT MODEL IN CATIA.clear
clc
as � �� CNT COORDINATES COME HERE ��;
fid � fopen �' catia_script _point _final1.txt', ' a'�;
fprintf �fid, ' � s \n � s \n � s \n � s \n � s \n � s \n', ' Language � "VBSCRIPT"', '
Sub CATMain ��', ' Set partDocument1 � CATIA.ActiveDocument ', ...' Set part1 �
partDocument1.Part', ' Set hybridShapeFactory1 � part1.HybridShapeFactory '�;
for df � 1 : size �as, 1�
str1 � �' Set hybridShapePointCoord ' num2str �df�' � hybridShapeFactory1.AddNewPointCoord�' num2str �as �df, 1��', ' num2str �as �df, 2��', ' num2str �as �df, 3��'�'�;
strr1 � �' Set bodies1 � part1.Bodies'�;
strrr1 � �' Set body1 � bodies1.Item �"PartBody"�'�;
str2 � �' body1''.InsertHybridShape hybridShapePointCoord ' num2str �df��;
str3 � �' part1''.InWorkObject � hybridShapePointCoord ' num2str �df��;
str4 � �' part1.Update'�;
fprintf �fid, ' � s \n � s \n � s \n � s \n � s \n � s \n', str1, strr1, strrr1, str2, str3, str4�;� df
� str5 � �''�;
� for fdf � 1 : 5
� dlmwrite �' catia_script.txt', eval ��' str' num2str �fdf���, ' � append '�;� end
end
fprintf �fid, ' � s \n', ' End Sub'�;fclose �fid�
� CODE FOR CREATING POINTS
� CODE FOR CREATING LINES
clear
clc
as � �� CNT COORDINATES COME HERE ��;
count � 1;
for dfx1 � 1 : size �as, 1� � 1for dfx2 � dfx1 � 1 : size �as, 1�if ��as �dfx1, 1� � as �dfx2, 1��^2 �
�as �dfx1, 2� � as �dfx2, 2��^2 � �as �dfx1, 3� � as �dfx2, 3��^2� � 1.43^2pointt �count, :� � �dfx1, dfx2�;count � count � 1;
end
end
end
fid � fopen �' catia_script _CNT _line _final1.txt', ' a'�;fprintf �fid, ' � s \n � s \n � s \n � s \n � s \n � s \n � s \n � s \n �
s \n', ' Language � "VBSCRIPT"', ' Sub CATMain ��', ' Set partDocument1 �
CATIA.ActiveDocument ', ' Set part1 � partDocument1.Part', ...' Set hybridShapeFactory1 �
part1.HybridShapeFactory ', ' Set bodies1 � part1.Bodies', ' Set body1 �
bodies1.Item �"PartBody"�', ...' Set hybridShapes1 � body1.HybridShapes '�;
for df � 1 : size �pointt, 1�
str1 � �' Set hybridShapePointCoord ' num2str �df � 2 � 1�' �hybridShapes1.Item �"Point.' num2str�pointt�df,1�� '"�'�;
str2 � �' Set reference ' num2str �df � 2 � 1�' �part1.CreateReferenceFromObject �hybridShapePointCoord ' num2str �df � 2 � 1�'�'�;
2 CNT.nb
str3 � �' Set hybridShapePointCoord ' num2str �df � 2�' �hybridShapes1.Item �"Point.' num2str�pointt�df,2�� '"�'�;
str4 � �' Set reference ' num2str �df � 2�' �part1.CreateReferenceFromObject �hybridShapePointCoord ' num2str �df � 2�'�'�;
str5 � �' Set hybridShapeLinePtPt ' num2str �df�' � hybridShapeFactory1.AddNewLinePtPt�reference ' num2str �df � 2 � 1�', reference ' num2str �df � 2�'�'�;
str6 � �' body1.InsertHybridShape hybridShapeLinePtPt ' num2str �df��;
str7 � �' part1.InWorkObject � hybridShapeLinePtPt ' num2str �df��;
str8 � �' part1.Update'�;
fprintf �fid, ' � s \n � s \n � s \n � s \n � s \n � s \n � s \n � s \n',
str1, str2, str3, str4, str5, str6, str7, str8�;� df
� str5 � �''�;
� for fdf � 1 : 5
� dlmwrite �' catia_script.txt', eval ��' str' num2str �fdf���, ' � append '�;� end
end
fprintf �fid, ' � s \n', ' End Sub'�;fclose �fid�
CNT.nb 3 4 CNT.nb
���� CODE FOR GULP AUTOMATION ����
�������������������������������
� function CNT_AUTO
dbstop if error
clear
close all
clc
� � � � � INPUTS TO BE GIVEN BY THE USER � � � � �
� input_atoms � dlmread �' input.txt'�;fid_input � fopen �' input.txt', ' r'�;
count � 1;
while�feof �fid_input�linein � fgetl �fid_input�;linein � strrep �linein, ' C', ''�;linein � textscan �linein, ' � f'�;input_atoms �count, :� � linein �1�;count � count � 1;
end
clear count;
fclose �fid_input�;
num_atoms � size �input_atoms, 1�;length_x � max �input_atoms � : , 3�� � min �input_atoms � : , 3��;
� num_atoms � 360;
� length_x � 36.9303;
maxsize � 100;
del_x � length_x � 40;del_x _init � del_x;
� � � � � � � � � END OF INPUTS � � � � � � � � �
flag_optim � 0;
fid � fopen �' nanotube7.in', ' w'�;fprintf �fid, ' � s \n � s \n', ' opti molq conv', ' cell'�;� fprintf �fid, ' � s \n', ' time 50'�;fprintf
�fid, ' � s \n', num2str �maxsize�'' num2str �maxsize�'' num2str �length_x�' 90 � 90 � 90'�;fprintf �fid, ' � s \n', ' cartesian ' num2str �num_atoms��;
fid_input � fopen �' input.txt', ' r'�;for gf � 1 : num_atoms
fprintf �fid, ' � s \n', fgetl �fid_input��;end
fprintf �fid, ' � s \n � s \n', ' library tersoff', ' output arc nanotube ' num2str �length_x��;dos �' a7.bat'�; �� UNCOMMENTpause �10�;fclose �fid�
for auto_runs � 1 : 20
� � � � � FINDING IF OPTIMISATION HAS BEEN ACHIEVED IN PREVIOUS STEP � � � � �
fid_output � fopen �' output.txt'�;count_optim � 0;
length_x1 � length_x
length_x � length_x � del_x;
aline � fgetl �fid_output�;while�feof �fid_output�aline � fgetl �fid_output�;
if�isempty �findstr �aline, ' Optimisation achieved '��count_optim � count_optim � 1;
end
end
if��count_optim � 0�� astr � ' Optimisation achieved '
flag_optim � 1;
else
flag_optim � 0;
end
fclose �fid_output�;� � � � � � � END OF FINDING OPTIMISATION � � � � � � � �
if flag_optim � 1
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
2 CULP.nb
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � STORING THE FILE OUTPUT.TXT IN OUTPUT1.TXT � � � � � � �
if�isempty �dir �' output.txt'��output_fid � fopen �' output.txt', ' r'�;output_fid1 � fopen �' output1.txt', ' w'�;while�feof �output_fid�output_line � fgetl �output_fid�;fprintf �output_fid1, ' � s \n', output_line�;end
fclose �output_fid�;fclose �output_fid1�;end
� � � � � � � STORING THE FILE OUTPUT.TXT IN OUTPUT1.TXT ENDS � � � � � � �
� � � � � � � STORING THE FILE ENERGY.TXT IN ENERGY1.TXT � � � � � � �
if�isempty �dir �' energy.txt'��energy_fid � fopen �' energy.txt', ' r'�;energy_fid1 � fopen �' energy1.txt', ' w'�;while�feof �energy_fid�energy_line � fgetl �energy_fid�;fprintf �energy_fid1, ' � s \n', energy_line�;end
fclose �energy_fid�;fclose �energy_fid1�;end
� � � � � � � STORING THE FILE ENERGY.TXT IN ENERGY1.TXT ENDS � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
fid � fopen �' nanotube7.in', ' w'�;fprintf �fid, ' � s \n � s \n', ' opti molq conv', ' cell'�;� fprintf �fid, ' � s \n', ' time 50'�;fprintf �fid, ' � s \n', ' 100 � 100' num2str �length_x�' 90 � 90 � 90'�;fprintf �fid, ' � s \n', ' fractional ' num2str �num_atoms��;
� � � � � FINDING THE LINES IN THE OUTPUT FILE
WHICH CONTAIN THE FRACTIONAL COORDINATES OF ATOMS � � � � �
fid_output � fopen �' output.txt'�;count_reqd _line � 1;
aline � fgetl �fid_output�;
CULP.nb 3
count_flag � 1;
while count_flag � 1
aline � fgetl �fid_output�;if�isempty �findstr �aline, ' Final fractional coordinates of atoms : '��count_flag � 0;
end
count_reqd _line � count_reqd _line � 1;
end
fclose �fid_output�;� � � � � FINDING THE LINES IN THE OUTPUT FILE WHICH
CONTAIN THE FRACTIONAL COORDINATES OF ATOMS ENDS HERE � � � � �
� � � � � � � READING THE OUTPUT FILE AND WRITING THE REQUIRED DATA INTO THE INPUT FILE � � � � � � �
fid_output � fopen �' output.txt', ' r'�;for hj1 � 1 : count_reqd _line � 5
bline � fgetl �fid_output�;end
for hj2 � 1 : num_atoms
cline � fgetl �fid_output�;dline � textscan �cline, ' � s'�;eline � �dline �1� �2�'' dline �1� �4�'' dline �1� �5�'' dline �1� �6��;fprintf �fid, ' � s \n', eline�;end
� � � � � � � END OF READING THE OUTPUT FILE AND
WRITING THE REQUIRED DATA INTO THE INPUT FILE � � � � � � �
� � � � � � � UPDATE VMD VIEWER FILE � � � � � � �
readarc �num2str �length_x1�, maxsize�;� � � � � � � END OF UPDATE VMD VIEWER FILE � � � � � � �
� � � � � � � STORE ENERGY OF THE SYSTEM IN A SEPARATE FILE � � � � � � � �
� � � � � � � CONSIDERING THE INTERNAL ENERGY TO BE THE LATTICE ENERGY � � � � �
fid_output � fopen �' output.txt'�;count_reqd _line � 1;
aline � fgetl �fid_output�;
count_flag � 1;
count_flag1 � 1;
4 CULP.nb
���� CODE FOR GULP AUTOMATION ����
�������������������������������
� function CNT_AUTO
dbstop if error
clear
close all
clc
� � � � � INPUTS TO BE GIVEN BY THE USER � � � � �
� input_atoms � dlmread �' input.txt'�;fid_input � fopen �' input.txt', ' r'�;
count � 1;
while�feof �fid_input�linein � fgetl �fid_input�;linein � strrep �linein, ' C', ''�;linein � textscan �linein, ' � f'�;input_atoms �count, :� � linein �1�;count � count � 1;
end
clear count;
fclose �fid_input�;
num_atoms � size �input_atoms, 1�;length_x � max �input_atoms � : , 3�� � min �input_atoms � : , 3��;
� num_atoms � 360;
� length_x � 36.9303;
maxsize � 100;
del_x � length_x � 40;del_x _init � del_x;
� � � � � � � � � END OF INPUTS � � � � � � � � �
flag_optim � 0;
fid � fopen �' nanotube7.in', ' w'�;fprintf �fid, ' � s \n � s \n', ' opti molq conv', ' cell'�;� fprintf �fid, ' � s \n', ' time 50'�;fprintf
�fid, ' � s \n', num2str �maxsize�'' num2str �maxsize�'' num2str �length_x�' 90 � 90 � 90'�;fprintf �fid, ' � s \n', ' cartesian ' num2str �num_atoms��;
fid_input � fopen �' input.txt', ' r'�;for gf � 1 : num_atoms
fprintf �fid, ' � s \n', fgetl �fid_input��;end
fprintf �fid, ' � s \n � s \n', ' library tersoff', ' output arc nanotube ' num2str �length_x��;dos �' a7.bat'�; �� UNCOMMENTpause �10�;fclose �fid�
for auto_runs � 1 : 20
� � � � � FINDING IF OPTIMISATION HAS BEEN ACHIEVED IN PREVIOUS STEP � � � � �
fid_output � fopen �' output.txt'�;count_optim � 0;
length_x1 � length_x
length_x � length_x � del_x;
aline � fgetl �fid_output�;while�feof �fid_output�aline � fgetl �fid_output�;
if�isempty �findstr �aline, ' Optimisation achieved '��count_optim � count_optim � 1;
end
end
if��count_optim � 0�� astr � ' Optimisation achieved '
flag_optim � 1;
else
flag_optim � 0;
end
fclose �fid_output�;� � � � � � � END OF FINDING OPTIMISATION � � � � � � � �
if flag_optim � 1
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
2 CULP.nb
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � STORING THE FILE OUTPUT.TXT IN OUTPUT1.TXT � � � � � � �
if�isempty �dir �' output.txt'��output_fid � fopen �' output.txt', ' r'�;output_fid1 � fopen �' output1.txt', ' w'�;while�feof �output_fid�output_line � fgetl �output_fid�;fprintf �output_fid1, ' � s \n', output_line�;end
fclose �output_fid�;fclose �output_fid1�;end
� � � � � � � STORING THE FILE OUTPUT.TXT IN OUTPUT1.TXT ENDS � � � � � � �
� � � � � � � STORING THE FILE ENERGY.TXT IN ENERGY1.TXT � � � � � � �
if�isempty �dir �' energy.txt'��energy_fid � fopen �' energy.txt', ' r'�;energy_fid1 � fopen �' energy1.txt', ' w'�;while�feof �energy_fid�energy_line � fgetl �energy_fid�;fprintf �energy_fid1, ' � s \n', energy_line�;end
fclose �energy_fid�;fclose �energy_fid1�;end
� � � � � � � STORING THE FILE ENERGY.TXT IN ENERGY1.TXT ENDS � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
fid � fopen �' nanotube7.in', ' w'�;fprintf �fid, ' � s \n � s \n', ' opti molq conv', ' cell'�;� fprintf �fid, ' � s \n', ' time 50'�;fprintf �fid, ' � s \n', ' 100 � 100' num2str �length_x�' 90 � 90 � 90'�;fprintf �fid, ' � s \n', ' fractional ' num2str �num_atoms��;
� � � � � FINDING THE LINES IN THE OUTPUT FILE
WHICH CONTAIN THE FRACTIONAL COORDINATES OF ATOMS � � � � �
fid_output � fopen �' output.txt'�;count_reqd _line � 1;
aline � fgetl �fid_output�;
CULP.nb 3
count_flag � 1;
while count_flag � 1
aline � fgetl �fid_output�;if�isempty �findstr �aline, ' Final fractional coordinates of atoms : '��count_flag � 0;
end
count_reqd _line � count_reqd _line � 1;
end
fclose �fid_output�;� � � � � FINDING THE LINES IN THE OUTPUT FILE WHICH
CONTAIN THE FRACTIONAL COORDINATES OF ATOMS ENDS HERE � � � � �
� � � � � � � READING THE OUTPUT FILE AND WRITING THE REQUIRED DATA INTO THE INPUT FILE � � � � � � �
fid_output � fopen �' output.txt', ' r'�;for hj1 � 1 : count_reqd _line � 5
bline � fgetl �fid_output�;end
for hj2 � 1 : num_atoms
cline � fgetl �fid_output�;dline � textscan �cline, ' � s'�;eline � �dline �1� �2�'' dline �1� �4�'' dline �1� �5�'' dline �1� �6��;fprintf �fid, ' � s \n', eline�;end
� � � � � � � END OF READING THE OUTPUT FILE AND
WRITING THE REQUIRED DATA INTO THE INPUT FILE � � � � � � �
� � � � � � � UPDATE VMD VIEWER FILE � � � � � � �
readarc �num2str �length_x1�, maxsize�;� � � � � � � END OF UPDATE VMD VIEWER FILE � � � � � � �
� � � � � � � STORE ENERGY OF THE SYSTEM IN A SEPARATE FILE � � � � � � � �
� � � � � � � CONSIDERING THE INTERNAL ENERGY TO BE THE LATTICE ENERGY � � � � �
fid_output � fopen �' output.txt'�;count_reqd _line � 1;
aline � fgetl �fid_output�;
count_flag � 1;
count_flag1 � 1;
4 CULP.nb
Acknowledgments
We would like to express our gratitude to Professor Anthony M. Waas and Professor Veera Sundararagha-van for their kindness and support during the course of this project.
References
1Seiji Akita, M. Nishiol and Yoshikazu Nakayama,Buckling of carbon nanotubes under axial compression,JJAP ,45,6B:5586-5589,2006.
2Arthur P. Ramirez,Carbon Nanotubes for Science and Technology,Bell Labs Technical Journal,Vol 10, 3:171-185,2005.3Motoyuki Hirooka, Makoto Okai1, Hiroki Tanaka, and Satoshi Sekino,Evaluation of the Stiffness of Carbon Nanotube
Probe by Force Curve Measurements,Mater. Res. Soc. Symp. Proc. Vol. 1081,P13-05,2008.4Hsao W. Yap Roderic S. Lakes and Robert W. Carpick,Mechanical Instabilities of Individual Multiwalled Carbon Nanotubes
under Cyclic Axial Compression,Nano Letters, 7 (5), 1149-1154 ,2007.5Rodney S. Ruoff, Dong Qian, Wing Kam Liu,Mechanical properties of carbon nanotubes theoretical predictions and ex-
perimental measurements,Comptes Rendus Physique,Vol 4, 9:993-1008,2003.6Ling Liu, Guoxin Cao, and Xi Chen,Mechanisms of Nanoindentation onMultiwalled Carbon Nanotube and Nanotube
Cluster,Journal of Nanomaterials, vol. 2008, Article ID 271763, 12 pages, 2008.7H. W. Yap, R. S. Lakes,and R. W. Carpick,Negative stiffness and enhanced damping of individual multiwalled carbon
nanotubes,PHYSICAL REVIEW B 77, 045423, 2008.8Ifat Sidney Konstantin Viktoria ThomasGotthar,On the mechanical behavior of WS2 nanotubes under axial tension and
compression,Pro. Nat Acad Sc of USA,Vol 103, 3:523-528,2006.9P.R. Guduru Z. Xia,Shell Buckling of Imperfect Multiwalled Carbon Nanotubes-Experiments and Analysis,J Exp Mech,
Vol 47, 1:153-161 , 200710J. F. Waters, P. R. Guduru, Jouzi, and J. M. Xu,Shell buckling of individual multiwalled carbon nanotubes using nanoin-
dentation,App Phy. Lett 87, 103109 ,2005.11Min-Feng Yu Strength and Breaking Mechanism of Multiwalled Carbon Nanotubes Under Tensile Load, SCIENCE VOL
287 28 , 2000.12Siddhartha Goknur, Surya , J. Gregory Yury,Viscoelasticity and high buckling stress of dense carbon nanotube
brushes,Carbon , Carbonol 47,8:1969-1976,2009.13Tersoff, J.,Empirical Interatomic Potential for Carbon, with Applications to Amorphous Carbon,Phys. Rev. Lett. ,
61,25:2879–2882,1988.
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American Institute of Aeronautics and Astronautics