Mode localization phenomenon of functionally graded nanobeams due to surface integrity

Abstract

This is the first study on the mode localization and surface stress-based deflection phenomena of functionally graded (FG) nanobeams due to surface integrity. A new model for FG nanobeams with engineering surfaces is developed. The engineering surface is considered a different material phase with a surface texture (i.e. waviness and roughness). The initial curvatures of cantilever, simple supported, and clamped–clamped FG nanobeams due to surface residual stresses are determined. It is revealed that the initial curvature increases with an increase in the slope of the surface texture and/or a decrease in the surface roughness. Moreover, the natural frequencies and mode shapes of FG nanobeams are derived depending on the surface’s texture and mechanical properties. It is observed that natural frequencies of FG beams may decrease or increase due surface roughness. Thus, as a first prospect, the surface roughness allows the vibration energy to propagate over the beam length and hence its natural frequency decreases resulting in a zero-frequency mode. As for the other prospect, surface roughness inhibits the propagation of the vibration energy through the beam length leading to a mode localization and an increase in the natural frequency.

Appendices

Appendix 1

The stiffnesses $D\left( x \right)$, $K\left( x \right)$, $B\left( x \right)$, and $q\left( x \right)$ forming the equation of motion of the FG beam [Eq. (15)] are derived as follows. First, Eq. (18) is substituted into Eq. (14), and the integrations are then performed to obtain:

$$\begin{aligned} D_{B} \left( x \right) & = 2\left( {\frac{b}{2} + P\left( x \right)} \right)\left( {\frac{h}{2} + P\left( x \right)} \right)^{3} \left( {\frac{2}{3}E^{ + } + \left( {E^{ - } - E^{ + } } \right)\xi \left( n \right)} \right) \\ D_{S} \left( x \right) & = 2\left( {\frac{b}{2} + P\left( x \right)} \right)\left( {\frac{h}{2} + P\left( x \right)} \right)^{2} \left( {E_{S}^{ + } + E_{S}^{ - } } \right) + 2\left( {\frac{h}{2} + P\left( x \right)} \right)^{3} \left( {\frac{2}{3}E_{S}^{ + } + \left( {E_{S}^{ - } - E_{S}^{ + } } \right)\xi \left( n \right)} \right) \\ q\left( x \right) & = 2\left( {\frac{b}{2} + P\left( x \right)} \right)\left( {\frac{h}{2} + P\left( x \right)} \right)\left( {\sigma_{S}^{ - } - \sigma_{S}^{ + } } \right) + 2\left( {\frac{h}{2} + P\left( x \right)} \right)^{2} \left( {\sigma_{S}^{ - } - \sigma_{S}^{ + } } \right)\kappa \left( n \right) \\ \end{aligned}$$

(51)

where $\xi \left( n \right)$ and $\kappa \left( n \right)$ are two hypergeometric functions only depend on the grading parameter, $n$. These functions are defined as follows:

$$\begin{aligned} \xi \left( n \right) & = \frac{{2^{ - n} }}{n + 3}\left[ {{\text{hypergeom}}\left( {\left[ { - n, - n - 3} \right],\left[ { - n, - 2} \right], - 1} \right) + \left( { - 1} \right)^{n}\, {\text{hypergeom}}\left( {\left[ { - n, - n - 3} \right],\left[ { - n, - 2} \right],1} \right)} \right] \\ \kappa \left( n \right) & = \frac{{\left( { - 1} \right)^{n} \,{\text{hypergeom}}\left( {\left[ { - n - 2} \right],\left[ {} \right],1} \right)}}{{2^{n} \left( {n + 2} \right)\left( {n + 1 } \right)}}\left[ { - 2n - 3} \right] + \frac{2n}{{\left( {n + 2} \right)\left( {n + 1} \right)}} \\ \end{aligned}$$

(52)

According to Eq. (51) and Eq. (16), the stiffnesses, $K\left( x \right)$ and $B\left( x \right)$ can be derived as follows:

$$\begin{aligned} K\left( x \right) & = 4\left( {\frac{2}{3}E^{ + } + \left( {E^{ - } - E^{ + } } \right)\xi \left( n \right)} \right)\left( {\frac{dP\left( x \right)}{dx}} \right)\left[ {3\left( {\frac{b}{2} + P\left( x \right)} \right)\left( {\frac{h}{2} + P\left( x \right)} \right)^{2} + \left( {\frac{h}{2} + P\left( x \right)} \right)^{3} } \right] \\ & \quad + 4\left( {E_{S}^{ + } + E_{S}^{ - } } \right)\left( {\frac{dP\left( x \right)}{dx}} \right)\left[ {2\left( {\frac{b}{2} + P\left( x \right)} \right)\left( {\frac{h}{2} + P\left( x \right)} \right) + \left( {\frac{h}{2} + P\left( x \right)} \right)^{2} } \right] \\ & \quad + 12\left( {\frac{2}{3}E_{S}^{ + } + \left( {E_{S}^{ - } - E_{S}^{ + } } \right)\xi \left( n \right)} \right)\left( {\frac{dP\left( x \right)}{dx}} \right)\left( {\frac{h}{2} + P\left( x \right)} \right)^{2} \\ B\left( x \right) & = 6\left( {\frac{2}{3}E^{ + } + \left( {E^{ - } - E^{ + } } \right)\xi \left( n \right)} \right)\left( {\frac{dP\left( x \right)}{dx}} \right)^{2} \left[ {\left( {\frac{h}{2} + P\left( x \right)} \right)^{2} + 2\left( {\frac{b}{2} + P\left( x \right)} \right)\left( {\frac{h}{2} + P\left( x \right)} \right) + \left( {\frac{h}{2} + P\left( x \right)} \right)^{2} } \right] \\ & \quad + 4\left( {E_{S}^{ + } + E_{S}^{ - } } \right)\left( {\frac{dP\left( x \right)}{dx}} \right)^{2} \left[ {2\left( {\frac{h}{2} + P\left( x \right)} \right) + \left( {\frac{b}{2} + P\left( x \right)} \right)} \right] \\ & \quad + 12\left( {\frac{2}{3}E_{S}^{ + } + \left( {E_{S}^{ - } - E_{S}^{ + } } \right)\xi \left( n \right)} \right)\left( {\frac{dP\left( x \right)}{dx}} \right)^{2} \left( {\frac{h}{2} + P\left( x \right)} \right) \\ \end{aligned}$$

(53)

Appendix 2

The stiffnesses $D$, $K$, and $B$ and the residual loads $S_{f}$, $q$, and $Q$ appeared in Eqs. (20) and (21) are derived utilizing the average parameters of the surface texture defined in Eq. (20) as follows:

$$\begin{aligned} D & = 2\left( {\frac{b}{2} + W_{a} + R_{a} } \right)\left( {\frac{h}{2} + W_{a} + R_{a} } \right)^{3} \left( {\frac{2}{3}E^{ + } + \left( {E^{ - } - E^{ + } } \right)\xi \left( n \right)} \right) \\ & \quad + 2\left( {\frac{b}{2} + W_{a} + R_{a} } \right)\left( {\frac{h}{2} + W_{a} + R_{a} } \right)^{2} \left( {E_{S}^{ + } + E_{S}^{ - } } \right) \\ & \quad + 2\left( {\frac{h}{2} + W_{a} + R_{a} } \right)^{3} \left( {\frac{2}{3}E_{S}^{ + } + \left( {E_{S}^{ - } - E_{S}^{ + } } \right)\xi \left( n \right)} \right) \\ \end{aligned}$$

(54)

$$\begin{aligned} K & = 4\left( {\frac{2}{3}E^{ + } + \left( {E^{ - } - E^{ + } } \right)\xi \left( n \right)} \right)\left( {WS + RS} \right)\left[ {3\left( {\frac{b}{2} + W_{a} + R_{a} } \right)\left( {\frac{h}{2} + W_{a} + R_{a} } \right)^{2} + \left( {\frac{h}{2} + W_{a} + R_{a} } \right)^{3} } \right] \\ & \quad + 4\left( {E_{S}^{ + } + E_{S}^{ - } } \right)\left( {WS + RS} \right)\left[ {2\left( {\frac{b}{2} + W_{a} + R_{a} } \right)\left( {\frac{h}{2} + W_{a} + R_{a} } \right) + \left( {\frac{h}{2} + W_{a} + R_{a} } \right)^{2} } \right] \\ & \quad + 12\left( {\frac{2}{3}E_{S}^{ + } + \left( {E_{S}^{ - } - E_{S}^{ + } } \right)\xi \left( n \right)} \right)\left( {WS + RS} \right)\left( {\frac{h}{2} + W_{a} + R_{a} } \right)^{2} \\ \end{aligned}$$

(55)

$$\begin{aligned} B & = 6\left( {\frac{2}{3}E^{ + } + \left( {E^{ - } - E^{ + } } \right)\xi \left( n \right)} \right)\left( {WS + RS} \right)^{2} \left[ {\left( {\frac{h}{2} + W_{a} + R_{a} } \right)^{2} + 2\left( {\frac{b}{2} + W_{a} + R_{a} } \right)\left( {\frac{h}{2} + W_{a} + R_{a} } \right) + \left( {\frac{h}{2} + W_{a} + R_{a} } \right)^{2} } \right] \\ & \quad + 4\left( {E_{S}^{ + } + E_{S}^{ - } } \right)\left( {WS + RS} \right)^{2} \left[ {2\left( {\frac{h}{2} + W_{a} + R_{a} } \right) + \left( {\frac{b}{2} + W_{a} + R_{a} } \right)} \right] \\ & \quad + 12\left( {\frac{2}{3}E_{S}^{ + } + \left( {E_{S}^{ - } - E_{S}^{ + } } \right)\xi \left( n \right)} \right)\left( {WS + RS} \right)^{2} \left( {\frac{h}{2} + W_{a} + R_{a} } \right) \\ \end{aligned}$$

(56)

$$q = 2\left( {\frac{b}{2} + R_{a} + W_{a} } \right)\left( {\frac{h}{2} + R_{a} + W_{a} } \right)\left( {\sigma_{S}^{ - } - \sigma_{S}^{ + } } \right) + 2\left( {\frac{h}{2} + R_{a} + W_{a} } \right)^{2} \left( {\sigma_{S}^{ - } - \sigma_{S}^{ + } } \right)\kappa \left( n \right)$$

(57)

$$\begin{aligned} Q & = 2\left( {\sigma_{S}^{ - } - \sigma_{S}^{ + } } \right)\left( {WS + RS} \right)\left[ {\left( {\frac{b}{2} + W_{a} + R_{a} } \right) + \left( {\frac{h}{2} + W_{a} + R_{a} } \right)} \right] \\ & \quad + 4\left( {\sigma_{S}^{ - } - \sigma_{S}^{ + } } \right)\left( {WS + RS} \right)\left( {\frac{h}{2} + W_{a} + R_{a} } \right)\kappa \left( n \right) \\ \end{aligned}$$

(58)

$$S_{f} = 4\left( {\sigma_{S}^{ - } - \sigma_{S}^{ + } } \right)\left( {WS + RS} \right)^{2} + 4\left( {\sigma_{S}^{ - } - \sigma_{S}^{ + } } \right)\left( {WS + RS} \right)^{2} \kappa \left( n \right)$$

(59)

It should be mentioned that the higher-order gradients of the surface profile, $P\left( x \right)$, are neglected as recommended by the author in Shaat (2018a).