**6. Example 3: homotopy perturbation method to dynamic behavior of piezoelectric nanobeam embedded in linear and nonlinear elastic Foundation in a thermal-magnetic environment**

Consider a nanobeam embedded in linear and nonlinear elastic media as shown in **Figure 3**. The nanobeam is subjected to stretching effects and resting on Winkler, Pasternak and nonlinear elastic media in a thermo-magnetic environment as depicted in the figure.

*Perturbation Methods to Analysis of Thermal, Fluid Flow and Dynamics Behaviors of… DOI: http://dx.doi.org/10.5772/intechopen.96059*

**Figure 3.**

*A nanobeam embedded in linear and nonlinear elastic media (note: Only the bottom side of the elastic media is shown).*

Following the nonlocal theory and Euler-Bernoulli theorem, the governing equation of the structure is developed as

*EI <sup>∂</sup>*<sup>4</sup>*<sup>w</sup> ∂x*<sup>4</sup> � � þ *ρAc ∂*2 *∂t* <sup>2</sup> *<sup>w</sup>* � ð Þ *<sup>e</sup>*0*<sup>a</sup>* <sup>2</sup> *<sup>∂</sup>*<sup>2</sup> *w ∂x*<sup>2</sup> � � <sup>þ</sup> *kw <sup>w</sup>* � ð Þ *<sup>e</sup>*0*<sup>a</sup>* <sup>2</sup> *<sup>∂</sup>*<sup>2</sup> *w ∂x*<sup>2</sup> � � � *kp ∂*2 *<sup>∂</sup>x*<sup>2</sup> *<sup>w</sup>* � ð Þ *<sup>e</sup>*0*<sup>a</sup>* <sup>2</sup> *<sup>∂</sup>*<sup>2</sup> *w ∂x*<sup>2</sup> � � <sup>þ</sup>*k*<sup>2</sup> *<sup>w</sup>*<sup>2</sup> � ð Þ *<sup>e</sup>*0*<sup>a</sup>* <sup>2</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>w</sup>*<sup>2</sup> � � *∂x*<sup>2</sup> � � <sup>þ</sup> *<sup>k</sup>*<sup>3</sup> *<sup>w</sup>*<sup>3</sup> � ð Þ *<sup>e</sup>*0*<sup>a</sup>* <sup>2</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>w</sup>*<sup>3</sup> � � *∂x*<sup>2</sup> � � � *<sup>η</sup>AcH*<sup>2</sup> *x ∂*2 *<sup>∂</sup>x*<sup>2</sup> *<sup>w</sup>* � ð Þ *<sup>e</sup>*0*<sup>a</sup>* <sup>2</sup> *<sup>∂</sup>*<sup>2</sup> *w ∂x*<sup>2</sup> � � þ *EAc αx*Δ*T* 1 � 2*ν* � � *∂*<sup>2</sup> *<sup>∂</sup>x*<sup>2</sup> *<sup>w</sup>* � ð Þ *<sup>e</sup>*0*<sup>a</sup>* <sup>2</sup> *<sup>∂</sup>*<sup>2</sup> *w ∂x*<sup>2</sup> � � � *EAc* 2*L* ð*L* 0 *∂w ∂x* � �<sup>2</sup> *dx* ! *∂*<sup>2</sup> *w <sup>∂</sup>x*<sup>2</sup> � ð Þ *<sup>e</sup>*0*<sup>a</sup>* <sup>2</sup> *<sup>∂</sup>*<sup>4</sup>*<sup>w</sup> ∂x*<sup>4</sup> " # � � ¼ 0 (68)

It is assumed that the midpoint of the nanobeam is subjected to the following initial conditions

$$
\overline{w}(\overline{\varkappa},0) = \overline{w}\_o, \quad \frac{\partial \overline{w}(\overline{\varkappa},0)}{\partial \overline{t}} = 0 \tag{69}
$$

The following boundary conditions for the multi-walled nanotubes for simply supported nanotube is given,

$$
\overline{w}(0,\overline{t}) = 0, \quad \frac{\partial^2 \overline{w}(0,\overline{t})}{\partial^2 \overline{x}} = 0, \quad \overline{w}(L,\overline{t}) = 0, \quad \frac{\partial^2 \overline{w}(L,\overline{t})}{\partial^2 \overline{x}} = 0. \tag{70}
$$

Using the following adimensional constants and variables

$$\infty = \frac{\overline{\mathcal{X}}}{L};\ \ w = \frac{\overline{w}}{r};\ \ t = \sqrt{\frac{EI}{\rho A\_c L^4}};\ \ r = \sqrt{\frac{I}{A\_c}};\ \ h = \frac{e\_0 a}{L};\ \ a\_t^d = \frac{N\_{thermal} L^2}{EI};\ \ A = \frac{\overline{w}\_o}{r} $$

$$K\_w = \frac{k\_w L^4}{EI};\ \ K\_p = \frac{k\_p L \Omega}{EI};\ \ Ha\_m = \frac{\eta A\_c H\_x^2 L^2}{EI};\ \ K\_2^d = \frac{k\_2 r L^4}{EI};\ \ K\_3^d = \frac{k\_3 r^2 L^4}{EI}.\tag{71}$$

The adimensional form of the governing equation of motion for the nanobeam is given as

$$\begin{aligned} &\left[\mathbf{1} + K\_p \hbar^2 + H a\_m \hbar^2 - a\_t^d \hbar^2 + \frac{\hbar^2}{2} \right] \left(\frac{\partial w}{\partial \mathbf{x}}\right)^2 dx \left[\frac{\partial^4 w}{\partial \mathbf{x}^4} + \left[a\_t^d - K\_w \hbar^2 - K\_p - H a\_m - \frac{1}{2} \right] \left(\frac{\partial w}{\partial \mathbf{x}}\right)^2 dx\right] \frac{\partial^2 w}{\partial \mathbf{x}^2} \\ &+ K\_w w + \frac{\partial^2 w}{\partial t^2} - h^2 \frac{\partial^4 w}{\partial \mathbf{x}^2 \partial t^2} + K\_2^d \left[w^2 - h^2 \frac{\partial^2 (w^2)}{\partial \mathbf{x}^2}\right] + K\_3^d \left[w^3 - h^2 \frac{\partial^2 (w^3)}{\partial \mathbf{x}^2}\right] = 0 \end{aligned} \tag{72}$$

And the boundary conditions become

$$w(\mathbf{0},t) = \mathbf{0}, \quad \frac{\partial^2 w(\mathbf{0},t)}{\partial^2 \mathbf{x}} = \mathbf{0}, \quad w(\mathbf{1},t) = \mathbf{0}, \quad \frac{\partial^2 w(\mathbf{1},t)}{\partial^2 \mathbf{x}} = \mathbf{0}.\tag{73}$$
