ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
Volume 105, Issue 5 e70091
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Refinement to: “Buckling of a nano-rod with taken into account of surface effect”

Anatolii Bochkarev, 

Corresponding Author

Anatolii Bochkarev

Faculty of Applied Mathematics and Control Processes, Saint Petersburg State University, Saint Petersburg, Russia

Correspondence

Anatolii Bochkarev, Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199804, Russia.

Email: [email protected]

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First published: 28 May 2025
https://doi.org/10.1002/zamm.70091

Abstract

A refinement of the previously constructed models of bending of the Timoshenko and Euler–Bernoulli nano-rod with taking into account the Steigmann–Ogden surface elasticity is considered. This refinement increases the coefficient of the lower derivative of the deflection in the bending equation by 50%, which can significantly affect the size effect. Updated results of modeling of buckling of a simply supported nano-rod and a nano-cantilever under its own weight.

1 INTRODUCTION

The article [1] presented a model of bending of the Timoshenko and Euler–Bernoulli nano-rod incorporating the Steigmann–Ogden surface elasticity [2] in comparison with the Gurtin–Murdoch surface elasticity [3]. The results of modeling were illustrated by the examples of buckling of a simply supported nano-rod and a nano-cantilever under its own weight.

In this paper, some refinement of the model formulation is made to take into account surface tension on the lateral surface of a nano-rod. This refinement is specific to the bending of nano-rods, but not to nano-plates, where the effect of surface tension at the ends of the plate is not considered.

2 REFINEMENT OF THE STRAIN ENERGY AND GOVERNING EQUATIONS

In Section 2.2 [1], the potential energy on the lateral surface of the nano-rod is derived from the surface elasticity [2, 3]. In particular on the lateral surface y = ± b / 2 $y =\pm b /2$ , the term u x , z = − ϑ $u_{x,z}=-\vartheta$ was omitted in the group of gradient terms with the coefficient τ 0 $\tau _0$ in Equation (16). After integrating over the thickness of the lateral surface y = ± b / 2 $y =\pm b /2$ , a coefficient appears, expressed by the line integral over the perimeter p $p$ of a cross section  N y = ∮ p n y d s $N_y=\oint _p n_yds$ , where n y $n_y$ is the projection of the normal onto the y $y$ -axis. In particular, for a rectangular cross-section  N y = 2 c $N_y=2c$ . With this term, the total energy of the Timoshenko nano-rod has the form
U = 1 2 ∫ 0 l p τ 0 2 + 2 ε + u ′ 2 + w ′ 2 + N y τ 0 ϑ 2 + S E t ∗ ε 2 + S μ b ∗ γ 2 + I E b ∗ ϑ ′ 2 d x $$\begin{equation} U=\frac{1}{2} \int _0^l {\left(p\tau _0{\left(2+ 2\varepsilon +u^{\prime 2}+w^{\prime 2}\right)}+N_y\tau _0\vartheta ^2+SE^*_{\rm t}\varepsilon ^2+S\mu ^*_{\rm b}\gamma ^2+IE^*_{\rm b}\vartheta ^{\prime 2}\right)} dx \end{equation}$$ (17)
and the total energy of the Euler–Bernoulli nano-rod correspondently
U = 1 2 ∫ 0 l p τ 0 2 + 2 ε + u ′ 2 + ( p + N y ) τ 0 w ′ 2 + S E t ∗ ε 2 + I E b ∗ w ′ ′ 2 d x $$\begin{equation} U=\frac{1}{2} \int _0^l {\left(p\tau _0{\left(2+ 2\varepsilon +u^{\prime 2}\right)}+(p+N_y)\tau _0w^{\prime 2}+SE^*_{\rm t}\varepsilon ^2+IE^*_{\rm b}w^{\prime \prime 2}\right)} dx \end{equation}$$ (18)
In Section 2.3 [1], the governing equations are derived from the Hamilton principle and the additional gradient term will also affect these equations. Omitting the intermediate expressions, I will indicate the final form of the governing equation for the Timoshenko nano-rod (the change concerns only the equation for the bending moment)
0 = ( T ∗ ) ′ + P x 0 = T ∗ w ′ ′ + ( Q ∗ ) ′ + P z = T ∗ w ′ ′ + p τ 0 w ′ ′ + μ b ∗ S ( w ′ ′ − ϑ ′ ) + P z 0 = ( M ∗ ) ′ − Q ∗ + p τ 0 w ′ + N y τ 0 ϑ + m = − E b ∗ I ϑ ′ ′ − μ b ∗ S ( w ′ − ϑ ) + N y τ 0 ϑ + m $$\begin{align} 0&=(T^*)^{\prime }+P_x \nonumber \\ 0&={\left(T^*w^{\prime }\right)}^{\prime }+(Q^*)^{\prime }+P_z={\left(T^*w^{\prime }\right)}^{\prime }+p\tau _0w^{\prime \prime }+\mu ^*_{\rm b}S(w^{\prime \prime }-\vartheta ^{\prime })+P_z \nonumber\\ 0&=(M^*)^{\prime }-Q^*+p\tau _0w^{\prime }+N_y\tau _0\vartheta +m=-E^*_{\rm b}I\vartheta ^{\prime \prime }-\mu ^*_{\rm b}S(w^{\prime }-\vartheta)+N_y\tau _0\vartheta +m\end{align}$$ (28)
and the new form of the bending equation
E b ∗ I 1 + p τ 0 μ b ∗ S w I V + E b ∗ I μ b ∗ S T ∗ w ′ ′ ′ ′ = 1 + N y τ 0 μ b ∗ S ( T ∗ w ′ ) ′ + p + N y + N y τ 0 μ b ∗ S p τ 0 w ′ ′ + 1 − E b ∗ I − N y τ 0 μ b ∗ S P z + m ′ $$\begin{equation} E^*_{\rm b}I{\left(1+\frac{p\tau _0}{\mu ^*_{\rm b}S}\right)}w^{IV}+\frac{E^*_{\rm b}I}{\mu ^*_{\rm b}S}{\left(T^*w^{\prime }\right)}^{\prime \prime \prime }={\left(1+\frac{N_y\tau _0}{\mu ^*_{\rm b}S}\right)}(T^*w^{\prime })^{\prime }+{\left(p+N_y+\frac{N_y\tau _0}{\mu ^*_{\rm b}S}p\right)}\tau _0w^{\prime \prime }+{\left(1-\frac{E^*_{\rm b}I-N_y\tau _0}{\mu ^*_{\rm b}S}\right)}P_z+m^{\prime } \end{equation}$$ (30)
Correspondently in the case of the Euler–Bernoulli nano-rod, the governing equation and the bending equation have the form
0 = ( T ∗ ) ′ + P x 0 = T ∗ w ′ ′ + ( Q ∗ ) ′ + P z = T ∗ w ′ ′ + p τ 0 w ′ ′ + Q ′ + P z 0 = ( M ∗ ) ′ − Q ∗ + ( p + N y ) τ 0 w ′ + m = − E b ∗ I w ′ ′ ′ − Q + N y τ 0 w ′ + m $$\begin{align} 0&=(T^*)^{\prime }+P_x \nonumber\\ 0&={\left(T^*w^{\prime }\right)}^{\prime }+(Q^*)^{\prime }+P_z={\left(T^*w^{\prime }\right)}^{\prime }+p\tau _0w^{\prime \prime }+Q^{\prime }+P_z \nonumber\\ 0&=(M^*)^{\prime }-Q^*+(p+N_y)\tau _0w^{\prime }+m=-E^*_{\rm b}Iw^{\prime \prime \prime }-Q+N_y\tau _0w^{\prime }+m \end{align}$$ (33)
I E b * w IV = ( T * w ′ ) ′ + ( p + N y ) τ 0 w ′ ′ + P z + m ′ $$\begin{equation} I{E}_{b}^{\ast}{w}^{\textit{IV}}={{({T}^{\ast}{w}^{\prime})}^{\prime}}+(p+{N}_{y}){\tau}_{0}{w}^{\prime \prime}+{P}_{z}+{m}^{\prime} \end{equation}$$ (35)
It is clearly seen from the bending Equation (35) that the additional gradient term changes the coefficient at w ′ ′ $w^{\prime \prime }$ : p + N y = 2 b + 4 c $p+N_y=2b+4c$ , that is, one and a half times more than without N y $N_y$ .

3 REFINED NUMERICAL RESULTS AND CONCLUSIONS

In Table 1 (united Tab. 2 and 3 of Section 4.3 [1]), the refined results of calculation of the inverse maximal eigenvalue (proportional to the Euler critical load) are presented. Compared with the previous results, an increase in the size effect of approximately to 18% and to 6%−13% between the Steigmann–Ogden and Gurtin–Murdoch models is seen.

TABLE 1. Inverse maximal eigenvalue 1 / max Λ i $1/\max \Lambda _i$ of buckling of a nano-rod in dependence in the thickness.
Model Ag Simple supported nano-rod Nano-cantilever
2 nm 10 nm 100 nm 2 nm 10 nm 100 nm
Euler–Bernoulli
Steigmann–Ogden < 100 > $<100>$ 111.38 35.26 20.27 118.22 32.33 10.66
Gurtin–Murdoch < 100 > $<100>$ 97.99 35.18 20.27 103.72 32.22 10.66
Steigmann–Ogden < 110 > $<110>$ 67.12 37.17 20.90 69.88 34.80 11.68
Gurtin–Murdoch < 110 > $<110>$ 63.41 36.43 20.89 65.73 33.84 11.66
Timoshenko × l 2 S μ b ∗ I E b ∗ 2 $ \times \left(\frac{l^2S\mu ^*_b}{IE^*_b}\right)^2$
Steigmann–Ogden < 100 > $<100>$ 113.46 34.58 19.66 120.68 32.06 10.56
Gurtin–Murdoch < 100 > $<100>$ 99.41 31.49 19.66 105.52 31.96 10.56
Steigmann–Ogden < 110 > $<110>$ 66.89 36.38 20.27 70.17 34.48 11.56
Gurtin–Murdoch < 110 > $<110>$ 62.96 35.61 20.27 65.83 33.52 11.56

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