A Finite-Deformation Shell Theory for Carbon Nanotubes Based on the Interatomic Potential—Part I: Basic Theory

2008 ◽  
Vol 75 (6) ◽  
Author(s):  
J. Wu ◽  
K. C. Hwang ◽  
Y. Huang ◽  
J. Song
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Constitutive Relation ◽  
Finite Deformation ◽  
Shell Theory ◽  
Interatomic Potential ◽  
Bending Moments ◽  
Equilibrium Equations ◽  
Potential Part ◽  
Asymmetric Stress

A finite-deformation shell theory for carbon nanotubes (CNTs) is established directly from the interatomic potential for carbon to account for the effect of bending and curvature. Its constitutive relation accounts for the nonlinear multibody atomistic interactions and therefore can model the important effect of CNT chirality and radius. The equilibrium equations and boundary conditions are obtained for the symmetric stresses and bending moments, which are different from many existing shell theories that involve asymmetric stress and bending moments. The theory is used in Part II of this paper to study the instability of carbon nanotubes subjected to different loadings.

A Finite-Deformation Shell Theory for Carbon Nanotubes Based on the Interatomic Potential—Part II: Instability Analysis

2008 ◽  
Vol 75 (6) ◽  
Author(s):  
J. Wu ◽  
K. C. Hwang ◽  
J. Song ◽  
Y. Huang
Keyword(s):  
Carbon Nanotubes ◽  
Internal Pressure ◽  
Critical Load ◽  
Finite Deformation ◽  
Shell Theory ◽  
Interatomic Potential ◽  
External Pressure ◽  
External Pressures ◽  
Tension And Compression ◽  
Potential Part

Based on the finite-deformation shell theory for carbon nanotubes established from the interatomic potential in Part I of this paper, we have studied the instability of carbon nanotubes subjected to different loadings (tension, compression, internal and external pressures, and torsion). Similar to the conventional shells, carbon nanotubes may undergo bifurcation under compression/torsion/external pressure. Our analysis, however, shows that carbon nanotubes may also undergo bifurcation in tension and internal pressure, though the bifurcation modes for tension and compression are very different, and so are the modes for the internal and external pressures. The critical load for instability and bifurcation depends on the interatomic potential used.

Buckling Analyses of Double-Wall Carbon Nanotubes: A Shell Theory Based on the Interatomic Potential

2010 ◽  
Vol 77 (6) ◽  
Author(s):  
W. B. Lu ◽  
J. Wu ◽  
X. Feng ◽  
K. C. Hwang ◽  
Y. Huang
Keyword(s):  
Carbon Nanotubes ◽  
Shell Theory ◽  
Interatomic Potential ◽  
Outer Wall ◽  
Critical Buckling Strain ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes ◽  
Double Wall ◽  
The Continuum ◽  
Vdw Interaction

Based on the finite-deformation shell theory for carbon nanotubes established from the interatomic potential and the continuum model for van der Waals (vdW) interactions, we have studied the buckling of double-walled carbon nanotubes subjected to compression or torsion. Prior to buckling, the vdW interactions have essentially no effect on the deformation of the double-walled carbon nanotube. The critical buckling strain of the double-wall carbon nanotubes is always between those for the inner wall and for the outer wall, which means that the vdW interaction decelerates buckling of one wall at the expenses of accelerating the buckle of the other wall.

Hyperelastic Thin Shells: Equilibrium Equations and Boundary Conditions

2015 ◽  
Vol 82 (9) ◽  
Author(s):  
Chi Zhang ◽  
Jian Wu ◽  
Keh-Chih Hwang
Keyword(s):  
Boundary Conditions ◽  
Finite Deformation ◽  
Deformation Analysis ◽  
Thin Shells ◽  
Membrane Stress ◽  
Equilibrium Equations

The equilibrium equations and boundary conditions in terms of the second Piola–Kirchhoff membrane stress and moment are given in this note, which are necessary for the finite deformation analysis of shells.

A VARIATIONAL PRINCIPLE APPROACH FOR BUCKLING OF CARBON NANOTUBES BASED ON NONLOCAL TIMOSHENKO BEAM MODELS

NANO ◽  
2011 ◽  
Vol 06 (04) ◽  
pp. 363-377 ◽  
Author(s):  
YANG YANG ◽  
C. W. LIM
Keyword(s):  
Carbon Nanotubes ◽  
Variational Principle ◽  
Size Effect ◽  
Boundary Conditions ◽  
Shear Deformation ◽  
Timoshenko Beam ◽  
Numerical Solutions ◽  
Dynamics Simulation ◽  
Beam Model ◽  
Equilibrium Equations

Based on the nonlocal elastic theory and variational principle, new Timoshenko beam models and analytical solutions for buckling of carbon nanotubes considering nanoscale size effect and shear deformation are established. New equilibrium equations and higher-order boundary conditions are derived and the buckling behavior of carbon nanotubes is numerically investigated. The numerical solutions confirm that nanotube stiffness is enhanced by nanoscale size effect and reduced by shear deformation. It is also concluded that nanotubes with different boundary conditions show varying sensitivity to changes in nanoscale and dimension. Comparison with molecular dynamics simulation results verifies the accuracy and reliability of this new analytical nonlocal Timoshenko beam model.

scholarly journals Rayleigh-Ritz Vibrational Analysis of Multiwalled Carbon Nanotubes Based on the Nonlocal Flügge Shell Theory

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
H. Rouhi ◽  
M. Bazdid-Vahdati ◽  
R. Ansari
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Shell Model ◽  
Multiwalled Carbon Nanotubes ◽  
Shell Theory ◽  
Scale Effects ◽  
Ritz Method ◽  
Elastic Shell ◽  
Small Scale ◽  
Multiwalled Carbon

A nonlocal elastic shell model considering the small scale effects is developed to study the free vibrations of multiwalled carbon nanotubes subject to different types of boundary conditions. Based on the nonlocal elasticity and the Flügge shell theory, the governing equations are derived which include the interaction of van der Waals forces between adjacent and nonadjacent layers. To analytically solve the problem, the Rayleigh-Ritz method is employed. In the present analysis, different combinations of layerwise boundary conditions are taken into account. Some new intertube resonant frequencies and the associated noncoaxial vibrational modes are identified owing to incorporating circumferential modes into the shell model.

scholarly journals Vibration of rotating circular cylindrical shells with distributed springs

2021 ◽  
Vol 37 ◽  
pp. 346-358
Author(s):  
Fuchun Yang ◽  
Xiaofeng Jiang ◽  
Fuxin Du
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Shell Theory ◽  
Rotation Speed ◽  
Cylindrical Shells ◽  
Free Vibrations ◽  
Governing Equations ◽  
Natural Response ◽  
Circular Cylindrical Shells ◽  
The Galerkin Method

Abstract Free vibrations of rotating cylindrical shells with distributed springs were studied. Based on the Flügge shell theory, the governing equations of rotating cylindrical shells with distributed springs were derived under typical boundary conditions. Multicomponent modal functions were used to satisfy the distributed springs around the circumference. The natural responses were analyzed using the Galerkin method. The effects of parameters, rotation speed, stiffness, and ratios of thickness/radius and length/radius, on natural response were also examined.

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