Nonlinear Bending of Piezoelectric Cylindrical Shell Reinforced with BNNTs under Electro-Thermo-Mechanical Loadings

Under combined electro-thermo-mechanical loadings, the nonlinear bending of piezoelectric cylindrical shell reinforced with boron nitride nanotubes (BNNTs) is investigated in this paper. By employing nonlinear strains based on Donnell shell theory and utilizing piezoelectric theory including thermal effects, the constitutive relations of the piezoelectric shell reinforced with BNNTs are established. Then the governing equations of the structure are derived through variational principle and resolved by applying the finite difference method. In numerical examples, the effects of geometric nonlinear, voltage, temperature, as well as volume fraction on the deflection and bending moment of axisymmetrical piezoelectric cylindrical shell reinforced with BNNTs are discussed in detail.


Introduction
BNNTs are similar to CNTs in structure and their extraordinary mechanical properties, but are different in that BNNTs possess higher temperature resistance to oxidation and stronger piezoelectric characteristics.Also, unlike CNTs, BNNTs have stable semiconducting behavior with a large band gaps regardless of radius and chirality of the structure.This property of BNNTs makes them promising candidate materials in a large variety of nanosized electronic and photonic devices.Therefore, BNNTs seem to be more suitable as reinforcement in composite structures due to their high resistance to oxidation at elevated temperatures [1], outstanding mechanical

Basic Equations
Consider that a piezoelectric cylindrical shell reinforced with BNNTs has midsurface radius R, thickness h, length L and mass density 0 ρ (see Figure 1).The shell is referred to the coordinate system (x, y, z) in which x and y are the axial and circumferential directions of the shell and z is in the direction of the inward normal to the middle surface.The origin of the coordinate system is located at the end of the shell on the middle plane.The shell is subjected to transverse static load q , applied voltage V and a uniform temperature rise T ∆ .

Strain Displacement Relationships
Supposing that , , u v w denote the axial, circumferential and radial displacement of an arbitrary point on the shell, and the corresponding displacement components of middle surface are , u v and w , then the displace- ment components of piezoelectric cylindrical shell can be written as x y u x y z u x y zw x y v x y z v x y zw x y w x y z w x y where the inferior mark (,) denotes the partial derivative to variables coordinate.
Based on classical shell theory with von Kármán-Donnell type kinematic relations, the nonlinear strain-displacement relations can be expressed as

Constitutive Equations
The constitutive relationship of a piezoelectric structure under combined mechanical, thermal and electrical loadings can be expressed as follows [15] where ( ) =  are elastic constants, piezoelectric constant, dielectric constants, respectively.The material constants of the structure can be calculated using "XY (or YX) rectangle model" [17].
The closed-form formula used in "X model" (or "Y model") expressing the mechanical, the thermal and the electrical properties of the material are as follows [15]: where Superscripts r and m refer to the reinforced and matrix components of the composite, respectively.f V is the vol% of the reinforced BNNTs in matrix.

Governing Equations
For the piezoelectric cylindrical shell reinforced with BNNTs, the total potential energy Π can be written as where U represents the strain energy and W represents the work done by the transverse load.The expression of the strain energy is Considering Equations ( 4) and ( 5), as well as the zigzag structure for BNNTs employed here, and the longitudinal arrangement of strips in matrix, makes 0 . Hence, Equation (9) Letting V is the voltage applied on both ends of shell, then The work done by the transverse load ( ) Applying the variational principle ( ) 0 δΠ = , the nonlinear governing equations of piezoelectric cylindrical shell reinforced with BNNTs can be derived as , , 12 22 66 0 0 0 0 in which ( ) In the above equations, the , ij ij A D are the tensile and bending rigidity and they can be defined as Under the axisymmetrical circumstances, the circumferential displacement 0 v = and , u w is only the function of coordinate x .Hence, the second equation of Equation ( 12) is automatic balance and it can be omitted.Then by Equations ( 3), ( 14) and introducing the following dimensionless parameters, , , , , , , the nonlinear governing equations of axisymmetrical piezoelectric shell reinforced with BNNTs under electrothermo-mechanical loadings can be reduced as ( ) ( ) 1 0 where, ( ) Supposing the both ends of the shell are clamped, then the dimensionless boundary conditions are respectively as follows: where ( )

Solution Methodology
For seeking the solution of differential Equation ( 18) with boundary condition ( 19), the dimensionless displacement functions W and U are dispersed in time-space domain to obtain their approximate solution.
Difference method is adopted in space domain.For the disposal of linear item, taking , W ξξ as example, we have Referring to difference scheme, the difference expressions of the other linear items in governing equation can be easily achieved.
Then the nonlinear items of governing equations are linearized and can be written as follows [18], ( ) ( ) ( ) in which ( ) jp y is the value of the former iterative.For the primary iteration, secondary extrapolation method is introduced to obtain the value of ( ) jp y , that is As for different iterations, the coefficients 1 ∆ , 2 ∆ and 3 ∆ are decided as follows: After the equations and conditions are linearized and disposed by using the finite difference method, the nonlinear partial differential equations are transformed into linear algebraical equations expressed by difference schemes.These algebraic equations are solved by using the iteration method.For every step, the iterative lasts until the difference of the present value and the former is smaller than 0.01%, then continues the calculation of the next step.

Numerical Results and Discussion
The nonlinear bending of piezoelectric shell reinforced with BNNTs under electro-thermo-mechanical loadings is investigated in the following calculations.The geometrical parameter of the shell is 5 3, 30 L R R h = = .The material used for matrix is PVDF and the reinforced material is BNNT.The material constants are listed in Table 1.In the following figures (Figures 2-5), the vertical ordinate W and M ξ are the dimensionless def- lection and bending moment of each point on shell along x.
The effects of geometric nonlinear on the bending of piezoelectric shell reinforced with BNNTs are presented in Figure 2 From the two figures, it can be noticed that the dimensionless deflection and bending moment of the shell in linear case is greater than that in nonlinear case, and this phenomenon becomes more evident when the transverse load Q increases.As we know, the linearity case is based on the limited deformation consumption, and the higher order item in the geometric relations is neglected while it is in consideration for the nonlinear case.So in some sense it can be concluded that the linear lowly predicts the stiffness of the structure.In order to reflect the property of the piezoelectric shell reinforced with BNNTs accurately, the consideration of the nonlinear effect is very necessary.
Figure 3 shows the effect of positive and negative voltage on the nonlinear bending of piezoelectric shell reinforced with BNNTs.The volume fraction 0.6 f V = , the temperature rise 0 T ∆ = and the mechanical load is taken as 150 Q = . From the figure, it is observed that applying negative voltage to BNNT decreases the deflection and bending moment.This is due to the fact that applying negative voltage creates polarization in the BNNT in the longitudinal direction, and leads to its contraction.This makes the structure of BNNT more compact and strong, and correspondingly increases the structure's stiffness.Therefore, the deflection and bending moment of the structure decrease.Figure 3 also depicts the results of deflection and bending moment when applying positive voltage.As expected, the deflection and bending moment increase compared to normal situation, and the results can be explained using the similar concept as mentioned above.
The effects of temperature on the nonlinear bending of the shell are presented in As can be seen, the deflection as well as the bending moment increases when the temperature increases.
The effect of volume fraction on the nonlinear bending of piezoelectric shell reinforced with BNNTs is discussed in Figure 5.The voltage is 0 V = , the temperature rise is 0 T ∆ = and the mechanical load is 150 Q =

Conclusions
In present study, the governing equations of nonlinear bending are presented for piezoelectric cylindrical shell reinforced with BNNTs under combined electro-thermo-mechanical loadings.Results indicate that some parameters, including geometric nonlinear, voltage, temperature, volume fraction and so on, have significant influence on the deflection and bending moment of the shell.The following conclusions may be drawn from the present work: 1) The deflection and bending moment of the shell in linear case is greater than that in nonlinear case, and the nonlinear effect enhances when the transverse load Q increases.2) Applying positive and negative voltage to BNNT leads to increase and decrease of the deflection and bending moment.
3) The deflection as well as the bending moment increases with the increase of temperature, and decreases when the volume fraction of BNNT in matrix increases.
are the change values of curvatures on the middle surface, and

Figure 4 .
The the mechanical load is taken as 150 Q = in the figure.

Figure 2 .
Figure 2. Effect of geometric nonlinear on bending of piezoelectric shell reinforced with BNNTs; (a) Deflection of each point along x; (b) Bending moment of each point along x.

Figure 3 .Figure 4 .
Figure 3.Effect of voltage on nonlinear bending of piezoelectric shell reinforced with BNNTs; (a) Deflection of each point along x; (b) Bending moment of each point along x.

Figure 5 .
Figure 5.Effect of volume fraction on nonlinear bending of piezoelectric shell reinforced with BNNT.(a) Deflection of each point along x; (b) Bending moment of each point along x. in this figure.It can be noticed that the deflection and the bending moment decrease when the volume fraction of BNNT in matrix ( ) f V increases.That is due to the fact that the increase of volume fraction would increase the stiffness of the structure, and thus the deflection decreases. ) becomes: 0

Table 1 .
Mechanical, electrical and thermal properties of PVDF and BNNT.