1. Evaluation of the Mechanical Parameters of Nanotubes by Means of Non-classical Theories of Shells
Let me introduce the work on the Evaluation of the Mechanical Parameters of Nanotubes by Means of Non-classical Theories of Shells.
Svetlana M. Bauer, Andrei M. Ermakov, Stanislava V. Kashtanova, Nikita F. Morozov

2. The Asbestos Fiber
The stiffness of bridges and cantilevers made of natural chrysotile asbestos nanotubes has been studied by means of scanning probe microscopy.
1 sm

3. The cross-section of the Asbestos Fiber.
Ankudinov, A.V., Bauer, S.M., Ermakov, A.M., Kashtanova, S.V., Morozov, N.F.: On mechanical properties of asbestos nanotubes (in Russian).
The modulus of elasticity  GPa; The inner radius  nM,
The outside radius  nM

4. The Porous thin lavsan membrane (11.5 microns).
The Lavsan membrane
The Porous thin lavsan membrane (11.5 microns).
(Approximate pore diameter is 0.6 microns).

5. The model of the experiment
The rigidity was understood as the ratio between applied strain and value of bridge deflection formed by the
nanotube, which blocked an opening in porous bottom layer.
The rigidity was understood as the ratio between applied strain and value of bridge deflection formed by then anotube, which blocked an opening in porous bottom layer. .

6. The model of the experiment
The stiffness is defined as a ratio the value of local load (applied to the tube) to the value of the displacement.
The stiffness is defined as
a ratio the value of local load (applied to the tube) to the value of the displacement.

7. Boundary condition
Izotova O.V., Nazarov S.A. Asymptotic solution of the Signorinri problem of a beam laying on a rigid bases . J. Math. Sci. 2006, v. 138, N 2, p

8. The classical models of isotropic beams.
- The Coordinate of applied load,
- The moment of inertia,
E - Young's modulus.
The simplest classical models of isotropic beams and non-classical transversal isotropic models
were considered.

9. The Timoshenko–Reissner generalized model.
-- The Coordinate of applied load,
-- The moment of inertia of the
Cylindrical cros-section,
E -- Young's modulus, G’- Shear modulus,
n = 5/6,
-- Cross-sectional area
Как можно видеть классическая теория прогиба балки не может объяснить экспериментальные результаты, тк не учитывает влияние поперечного сдвига.
Экспериментальные результаты хорошо согласуются при использовании теории Тимошенко-Рейсснера, когда модуль поперечного сдвига меняется от наполнителя.

10. The Rigidity of a nanotube with filler and without.
The experiments show that the stiffness of the tube depends on the materials for filling. The tubes with water are softer than the tubes
without filling materials and the tubes filled with mercury are more rigid than tubes without filling materials.

11. The Effect of Internal Pressure.
The inner cavity of a tube was filled with water, mercury or tellurium under pressure. The
rigidity of nanotubes was measured with the use of scanning probing microscopy. The rigidity
was understood as the ratio between applied strain and value of bridge deflection formed by the
nanotube, which blocked an opening in porous bottom layer. The experiments showed that tube filled with water is substantially
softer than a ”dry” tube ( tube without any filler) and tubes filled with mercury are slightly more
rigid than ”dry” tubes.

12. The geometric model for a tube
Let α and β be cylindrical coordinates on a shell surface, α — the polar angle, β — coordinate
along the tube’s generatrix, h(i) —thicknesses, R(i) —radiuses of medium surfaces of the
shell layers, and L—tube’s length. For definition of coefficients we will use symbol A(i)
j . Low
index j denotes curvilinear coordinate corresponding to the value A, upper index i — denotes
to which layer it corresponds, i.e. if i = 1 then A corresponds to the first inner layer, if i = N
then A corresponds to the last outer layer. E(i)
1 ,E(i)
2 ,E(i)
3 are modules of elasticity in tangential
and normal directions, ν(i)
jk are the Poisson’s ratios.

13. The iterative theory of anisotropic shells of Rodionova-Titaev -Chernykh
Transverse tangential and normal stresses are distributed along the shell thickness according to quadratic and cubic laws respectively,
Tangential and normal components of the displacement vector are distributed along the shell thickness according to quadratic and cubic laws
This theory takes into account rotation of a fiber, its deviation and change of the length.

14. The iterative theory of anisotropic shells of Rodionova-Titaev -Chernykh
Functions describing the displacements of a shell layer are presented as the series in Legendre polynomials.

15. The Paliy-Spiro shells theory
The theory of shells of moderate thickness
Straight fibers of the shell, which are orthogonal to its middle surface before deformation, remain also straight after the deformation;
Cosine of the slope angle of the fiber slope to the middle surface of the deformed shell is equal to the average angle of the transverse shear.
This theory takes into account rotation of a fiber and change of the length

16. The Paliy-Spiro shells theory
Mathematical formulation of these hypotheses gives following equalities:

17. The dimensionless variables:
Let us introduce the following dimensionless variables:

18. The components of deformations
Theory R-T-C
Theory P-S
Shell deformations of the theories under consideration are expressed through the components
of displacements with the use of the following equations.
The differing components of the deformation are underlined.

19. Constitutive relations R-T-C.
Let us substitute the following relations (8) for six components of the displacement in formulas.
Thus we reduce them to dependence on the five main components of the displacement.

20. Constitutive relations P-S.
The same transformation for Paliy-Spiro theory gives:

21. Equations of equilibrium of a cylindrical shell.
By substituting these relations into equilibrium equation of the cylindrical shell one can get a system of five partial differential equations with five unknown functions for both
theories. For the Rodionova-Titaev-Chernykh theory this system of equations is 14 order, and
for the Paliy-Spiro theory is of 10 order. Substituting corresponding components of the deformation
in formulas for displacements one can get all components of the stress-deformed state of the shells
under consideration.

22. The numerical method Displacements
For solving the system of equations in the displacements we will use the following series.

23. The numerical method Components of the load.
External and internal forces, which act on the shell surface can be presented as a product of sectional forces presented in
series.

24. The Expansion of loading functions in Fourier series in cross-sections.
The load localized in a small rectangular area can be represented in the form of a product of the
Fourier series of two loading functions in cross-section and longitudinal section:

25. Conditions on the shells surfaces. Conditions for rigid pinched layers
Pressure on
External
Surfaces
Pressure on
Internal
Surfaces
Conditions for rigid pinched layers

26. The System of resolving equations.
Equations of equilibrium of the cylindrical shells.
k=1…5
we obtain a system of 8N-3 linear algebraic equations for 5N deformation components and 3N-3 forces
of interaction between layers of shells. For realization of the aforementioned numerical method we developed a program based on
code Mathematica 7.0.
Conditions for rigid pinched layers
j=1…3

27. The general picture of deformations

28. Deflection of the multilayered nanotube.Lv
250
220
200
170
150
120
100 70 40
TR1
(Beam)
60.61
59.7
58.07
54.14
50.52
43.65
38.12
28.47
17.24
TR2
59.2
58.31
56.72
52.87
49.34
42.62
37.22
27.79
16.82
RTCH
57.79
56.86
55.21
51.22
47.56
40.62
35.07
25.45
14.4 PS
57.54
56.62
54.97 51
47.35
40.45
34.92
25.34
14.34
Ansys1
54.11
53.26
51.7
48.3
45.1
39.02
34.07
25.37
15.37
Ansys2
52.39
52.38
50.12
46.77
43.71
37.81
33.02
24.6
14.91
“TR1”, “Ansys1” Models of a tube with a hole;
“TR2 ”, “Ansys2 ” Models of a solid cylinder,
S – Area of the applied load

29. Deflection of a single-layer shell with a constant outer radius
h/R
1/15
1/10
1/5
1/4
1/3
1/2
TR1 (Beam)
46.33
31.91
17.55
14.71
11.88
9.13
RTCH
78.09
52.64
27.14
22.13
17.31
12.98 PS
75.28
49.82
24.29
19.26
14.39
9.9
Ansys1
76.36
46.44
20.37
15.9
11.95
8.92
The shell with a constant outer radius, the thickness of the shell gradually increases.
The load (Fv = 1nH) is applied to the center of the tube.

30. Conclusion
Let us compare the results obtained by means of three-dimensional theory, which is used in the Ansys 11 package, with those obtained by means of the non-classical shell theories described above. We consider a one-layer cylindrical shell with constant outer radius. Gradually increasing the thickness of the shell (and, as a consequence, decreasing the radius of the middle surface of the shell), we obtain the values of deflection at the centre of the shells under consideration given in the table.
The results presented in Table 2 show that both theories give close deflections of shells. Moreover, other quantities characterizing stress-strain state of shells are close as well. As the relative thickness of the shell increases, the deflection values obtained by means of the Paliy–Spiro theory approach those obtained by the finite element method.

31. Thank you for your attention!