Table 5.

Comparison of some surface elastic models for surface Young’s modulus of nanowires resonators (The relative parameter descriptions can be seen in the literature).

Model	Formulation	Materials	Theory and Method	Effects
Surface elasticity model [195]	${\left(EI\right)}^{\text{*}}=EI+2E^{s}w{h}^2$	GaAs	Classical beam theory	Surface elasticity
Surface Cauchy-Born (SCB) model [196]	$c_{IJKL}^s=M_{IJKL}^s-A_{IJp}^s{A}_{KLq}^s{\left(D^{-1}\right)}_{pq}^s$	Si	Based on standard bulk Cauchy-Born model	Surface stress
He’s model [191]	${(EI)}^{*}=\{\begin{array}{ll}EI+E_{s}w{h}^2/2+E_s{h}^3/6\hfill & \text{(rectangle)}\hfill \\ EI+\pi E_s{d_c}^3/8\hfill & \text{(circular)}\hfill \end{array}$	Al and Si	Euler-Bernoulli beam theory and Young-Laplace equation	Surface stress with different boundary conditions
High-order surface stress model [190]	${(EI)}^{*}=\{\begin{array}{ll}EI+E^{s}w{h}^2/2+E^s{h}^3/6+E^{s}wh{h}_1+8D^{s}w\hfill & \text{(rectangle)}\hfill \\ EI+\pi E^s{d}^3\text{/8}\hfill & \text{(circular)}\hfill \end{array}$	Si	Generalized Young–Laplace equation	High-order surface stress and surface moment
Liu’s model [197]	${\left(EI\right)}^{*}=\{\begin{array}{cc}EI+\left(2{\mu }_0\text{+}{\lambda }_0\right)\left(2b{h}^2+4h^3/3\right)-2\upsilon I{\tau }_0/H& \text{(rectangle)}\hfill \\ EI+\pi D^3\left(2{\mu }_0\text{+}{\lambda }_0\right)/8-2\upsilon I{\tau }_0/H\hfill & \text{(circular)}\hfill \end{array}$	Al and Si	Gurtin–Murdoch theory	Surface stress, surface elasticity and surface density
Core-shell model [192,193]	${(EI)}^{*}=\{\begin{array}{ll}E{w}_c{h_c}^3/12+E_{s}t{w}_c{(h_c\text{+2t)}}^2+2E_{s}t{(h_c/2+t)}^3/3-2E_s{t}^3(w_c+2t)/3\hfill & \text{(rectangle)}\hfill \\ \pi (E-E_s){d_c}^4/64+\pi E_s{(d_c+2t)}^4\text{/64}\hfill & \text{(circular)}\hfill \end{array}$	ZnO	resonance experiment and linear surface elastic theory	Surface layer thickness
Rudd’s model [198]	$E_{tot}=E_{core}[1+(C_{tot}t/A_{tot})\Delta ]+O(t^2/R^2)$	hydrogen-passivated Si	First-principles density functional theory	Plane-wave cut-off energy
Feng’s model [189]	$E^{*}=\frac{4\sqrt{A}{\tau }_0(E_{0}t+4E_s)}{\sqrt{A}(E_{0}tl+4E_{s}l+4t{\tau }_0+6l{\tau }_0)-4(E_{0}t+4E_s)\tanh (\sqrt{A}l/4)}\left(\frac{t}{l+t}\right)$	nanoporous materials	Gurtin–Murdoch theory	Surface energy and residual surface stress
Yan’s model [199]	${\left(EI\right)}^{*}=\{\begin{array}{ll}\frac{1}{12}(c_{11}+e_{31}^2/{\kappa }_{33})b{h}^3+(c_{11}^s+e_{31}^s{e}_{31}/{\kappa }_{33})(b{h}^2/2+h^3/6)\hfill & \text{(rectangle)}\hfill \\ \frac{\pi D^4}{64}(c_{11}+e_{31}^2/{\kappa }_{33})+\frac{\pi D^3}{8}(c_{11}^s+e_{31}^s{e}_{31}/{\kappa }_{33})\hfill & \text{(circular)}\hfill \end{array}$	piezoelectric materials	Generalized Young–Laplace equations	residual surface stress, Surface elasticity and piezoelectricity
Non-uniform core-shell model [194]	$\begin{array}{l}{(EI)}^{*}=\pi E\{\frac{D^4}{64}{(1-2r_s/D)}^4+\frac{D^3{r}_s}{8{\alpha }_0}[e^{{\alpha }_0}-{(1-2r_s/D)}^3]\\ -\frac{3D^2{r}_s^2}{4{\alpha }_0^2}[e^{{\alpha }_0}-{(1-2r_s/D)}^2]+\frac{3D{r}_s^3}{{\alpha }_0^3}[e^{{\alpha }_0}-(1-2r_s/D)]-\frac{6r_s^4}{{\alpha }_0^4}(e^{{\alpha }_0}-1)\}\end{array}$	ZnO	Nonlinear surface elastic theory	Non-uniform surface elasticity