Analytical and parametric analysis of thermoelastic damping in circular cylindrical nanoshells by capturing small-scale effect on both structure and heat conduction

Abstract

This article intends to examine thermoelastic damping (TED) in circular cylindrical nanoshells by considering small-scale effect on both structural and thermal areas. To fulfill this aim, governing equations are extracted with the aid of nonlocal elasticity theory and dual-phase-lag (DPL) heat conduction model. Circular cylindrical shell is also modeled on the basis of Donnell–Mushtari–Vlasov (DMV) equations for thin shells. By inserting asymmetric simple harmonic oscillations of nanoshell into motion, compatibility and heat conduction equations, the size-dependent thermoelastic frequency equation is obtained. By solving this equation and deriving the frequency of nanoshell affected by thermoelastic coupling, the value of TED can be calculated through complex frequency approach. Results of this investigation are given in two sections. First, to appraise the validity of presented formulation, a comparison study is conducted between the results of this work in special cases and those reported in the literature. Next, by providing several numerical data, a detailed parametric study is performed to highlight the profound impact of nonlocality and dual-phase-lagging on TED value in simply supported cylindrical nanoshells. The influence of some determining factors such as mode number and type of material on TED is also evaluated.

Appendix

The free vibration of a nonlocal Euler–Bernoulli beam is governed by the following equation:

$$EI\frac{{\partial^{4} w}}{{\partial x^{4} }} - \mu \rho A\frac{{\partial^{4} w}}{{\partial x^{2} \partial t^{2} }} + \rho A\frac{{\partial^{2} w}}{{\partial t^{2} }} = 0,$$

(59)

in which $I$ and $A$ represent the area moment of inertia of cross sections and cross section area of the beam. By adopting simple harmonic form $w\left(x,t\right)={\sum }_{m=1}^{\infty }{W}_{m}(x){\mathrm{e}}^{i{\omega }_{m}t}$, substituting it into equation above, and simplifying the result, one can get:

$$\frac{{\partial^{4} W_{m} }}{{\partial x^{4} }} + \mu \lambda_{m}^{4} \frac{{\partial^{4} W_{m} }}{{\partial x^{2} }} - \lambda_{m}^{4} W_{m} = 0,$$

(60)

where

$$\lambda_{m}^{4} = \frac{\rho A}{{EI}}\omega_{m}^{2} .$$

(61)

The general solution of Eq. (60) has the following form:

$${W}_{m}={C}_{1}\mathrm{sin}\left({\gamma }_{1}x\right)+{C}_{2}\mathrm{cos}\left({\gamma }_{1}x\right)+{C}_{3}\mathrm{sinh}\left({\gamma }_{2}x\right)+{C}_{4}\mathrm{cosh}\left({\gamma }_{2}x\right),$$

(62)

in which ${C}_{1}$, ${C}_{2}$, ${C}_{3}$ and ${C}_{4}$ are integration constants. Substitution of relation above into Eq. (60) and solving the obtained equation gives:

$$\gamma_{1}^{2} = \frac{1}{2}\left( {\mu \lambda_{m}^{4} + \sqrt {\mu^{2} \lambda_{m}^{8} + 4\lambda_{m}^{4} } } \right) \quad {\text{and}}\quad \gamma_{2}^{2} = \frac{1}{2}\left( { - \mu \lambda_{m}^{4} + \sqrt {\mu^{2} \lambda_{m}^{8} + 4\lambda_{m}^{4} } } \right).$$

(63)

Boundary conditions of three common types of beams, namely doubly simply supported (SS), doubly clamped (CC) and cantilever (CF) are expressed by [38]:

$$SS: \quad W_{m} \left( 0 \right) = \frac{{{\text{d}}^{2} W_{m} }}{{{\text{d}}x^{2} }}\left( 0 \right) = W_{m} \left( L \right) = \frac{{{\text{d}}^{2} W_{m} }}{{{\text{d}}x^{2} }}\left( L \right) = 0,$$

(64)

$$CC: \quad W_{m} \left( 0 \right) = \frac{{{\text{d}}W_{m} }}{{{\text{d}}x}}\left( 0 \right) = W_{m} \left( L \right) = \frac{{{\text{d}}W_{m} }}{{{\text{d}}x}}\left( L \right) = 0,$$

(65)

$$CF: \quad W_{m} \left( 0 \right) = \frac{{{\text{d}}W_{m} }}{{{\text{d}}x}}\left( 0 \right) = EI\frac{{{\text{d}}^{2} W_{m} }}{{{\text{d}}x^{2} }}\left( L \right) + \mu \rho A\omega_{m}^{2} W_{m} \left( L \right) = EI\frac{{{\text{d}}^{3} W_{m} }}{{{\text{d}}x^{3} }}\left( L \right) + \mu \rho A\omega_{m}^{2} \frac{{{\text{d}}W_{m} }}{{{\text{d}}x}}\left( L \right) = 0.$$

(66)

By inserting Eq. (62) into Eqs. (64)–(66), using Eq. (61) and setting the determinant of the coefficient matrix of the obtained algebraic equations for ${C}_{1}$, ${C}_{2}$, ${C}_{3}$ and ${C}_{4}$ to zero, one can attain the following characteristic equations:

$$SS:\quad \sin (\gamma_{1} L) = 0,$$

(67)

$$CC:\quad 2\cos \left( {\gamma_{1} L} \right)\cosh \left( {\gamma_{2} L} \right) + \left( {\frac{{\gamma_{1} }}{{\gamma_{2} }} - \frac{{\gamma_{2} }}{{\gamma_{1} }}} \right)\sin (\gamma_{1} L)\sinh (\gamma_{2} L) - 2 = 0,$$

(68)

$$CF:\quad 2\cos \left( {\gamma_{1} L} \right)\cosh \left( {\gamma_{2} L} \right) + \left( {\frac{{\gamma_{1} }}{{\gamma_{2} }} - \frac{{\gamma_{2} }}{{\gamma_{1} }}} \right)\sin (\gamma_{1} L)\sinh (\gamma_{2} L) + \left( {\frac{{\gamma_{1}^{2} }}{{\gamma_{2}^{2} }} + \frac{{\gamma_{2}^{2} }}{{\gamma_{1}^{2} }}} \right) = 0.$$

(69)

By considering the relation of ${\gamma }_{1}$ and ${\gamma }_{2}$ with ${\lambda }_{m}$ through relation (63) and solving the equations above, ${\gamma }_{1}$ and ${\gamma }_{2}$ are extracted, and by inserting them in Eq. (62), the mode shape of nonlocal beams with mentioned boundary conditions is obtained. Since the model of Lu et al. [14] has been provided in the context of classical continuum theory (i.e. $\mu =0$) for CC boundary conditions, according to Eq. (63), the comparison study must be conducted on the basis of ${\gamma }_{1}={\gamma }_{2}={\lambda }_{m}$. Hence, by considering Eq. (68), the characteristic equation of a classical CC beam becomes:

$$\cos \left( {\lambda_{m} L} \right)\cosh \left( {\lambda_{m} L} \right) - 1 = 0$$

(70)

On the other hand, by imposing boundary conditions (65) on Eq. (62) and letting $\gamma_{1} = \gamma_{2} = \lambda_{m}$, the mode shape of a classical CC beam is obtained as follows:

$$W_{m} = C_{1} \left[ {(\sin \left( {\lambda_{m} x} \right) - \sinh \left( {\lambda_{m} x} \right)) - \frac{{\sin \left( {\lambda_{m} L} \right) - \sinh \left( {\lambda_{m} L} \right)}}{{\cos \left( {\lambda_{m} L} \right) - \cosh \left( {\lambda_{m} L} \right))}}\left( {\cos \left( {\lambda_{m} x} \right) - \cosh \left( {\lambda_{m} x} \right)} \right)} \right].$$

(71)