Influence of surface effect on post-buckling behavior of graded porous nanobeam subjected to follower force

Abstract

As the fundamental structural element of the nano electro mechanical system (NEMS), the mechanical properties of nano-beams are often affected by surface effects. In this study, we investigate the buckling and the post-buckling of graded porous nano-beams subjected to follower force, considering surface stiffness and surface residual stress. Considering two kinds of porosity models with symmetrical and asymmetric distribution in gradient porous material, it was assumed that the material properties change continuously in thickness. In order to capture the surface effect, the Gurtin–Murdoch surface elasticity theory, combined with geometrically non-linear beam theory, was utilized to establish the post-buckling control differential equations. And then, by using the shooting method to solve the coupled ordinary differential equations with simply supported nano-beams, some quantitative results of the post-buckling equilibrium path and configuration under different parameters are obtained. The results show that the pore distribution pattern, pore coefficient, surface elastic parameters, and residual surface stress have essential effects on the yield strength of nano-porous materials.

1 Introduction

With the development of micro/nano-technology, nanostructures play an essential role in improving different high-tech equipment such as nano electro mechanical systems (NEMS). Nanostructures have been used in engineering applications, for example, atomic force microscopes, biomedical applications, gyroscopes, and so on (Ayanoglu et al. 2022; Alibakshi et al. 2022; Malikan et al. 2022; Dastjerdi et al. 2021; Dastjerdi et al. 2020). Therefore, researchers studying the mechanical behavior of nanostructures came into being (Fang et al. 2020; Li et al. 2021).

Researchers found that the classical continuum theories are insufficient to describe the mechanical behavior of micro/nano‐sized structures. Therefore, implementation of the size-dependent elasticity theory is inevitable, for example, Eringen’s nonlocal theory (Eringen and Edelen 1972; Eringen 1972, 1983), strain gradient theory (Mindlin 1964, 1965), nonlocal strain gradient theory (Lim et al. 2015), couple stress theory (Toupin 1962), modified couple stress theory (Yang et al. 2022), the higher-order continuum theory (Faghidian et al. 2022a, b, c; Faghidian et al. 2023a, b; Faghidian et al. 2022a, b, c; Faghidian et al. 2023a, b; Faghidian and Elishakoff 2022), the continuum theory of elastic material surfaces (Gurtin and Murdoch 1975) and so on. We only investigate the mechanical behavior associated with surface effects in this article. In the literature survey, we limit our literature review to those most directly related to the current study.

Experiments and atomic simulations have shown that when the structure size becomes smaller and reaches the micro/nano-scale, the influence of the small size becomes significant (Fan et al. 2018). It is found that for submicron devices, surface effects can lead to significant changes in mechanical, physical, and electrical properties, and are a key factor in evaluating the performance of effective structural materials. The surface effect is one of the reasons for the size-related properties of nanostructures because the surface phase of the structure has a different atomic arrangement and larger specific surface area compared with the bulk phase, and the properties of the surface phase are different from the bulk phase. Therefore, the surface effect can well explain the performance of size-related nanomaterials or structures, or the surface plays an important role in affecting the overall performance of nanomaterials or structures (Wang et al. 2011).

Many mechanical models of devices in NEMS can be simplified into nanoscale beams or plates. Especially with the successful manufacture of nanowires and nanorods, it is crucial to study the mechanical properties of one-dimensional nanoscale beam rod structures. Wang and Feng (2010) analyzed the effect of surface stress on the vibration and buckling behavior of piezoelectric nanowires. In addition, as the most promising nanopore, the pole will lose stability when the pressure on its end exceeds the critical load. Zhao et al. (2018) studied the prediction of vibration characteristics of micro/nanobeams. On this basis, the precise frequency equations of Timoshenko beams under different boundary conditions were derived, including surface elasticity and residual surface stress. According to Gibbs (1906), the surface stress tensor is related to the surface energy density. Samaei et al. (2012) gave an analytical expression for the critical buckling load of piezoelectric nanowires based on the Timoshenko beam theory, considering the effect of surface effects. Guo et al. (2021) investigated the transverse wave propagation in viscoelastic single-walled carbon nanotubes (SWCNTs) adhered by surface material. Peng et al. (2020) used a simple Taylor series expansion method to perform a static deformation analysis of a micro cantilever beam with arbitrary axial nonuniformity and varying cross sections. Akgöz and Civalek (2022) analyzed the buckling problem of nonhomogeneous microbeams with a variable cross-section.

In studying the surface effects of elastic materials, the surface elastic model proposed by Gurtin and Murdoch (1975) is groundbreaking and has been widely used to explain the size-dependent behavior of nanostructures (Ermeyev 2016). For example, Yang et al. (2021a, b) used the Gurtin–Murdoch theory to study the effect of surface stress on the bending and vibration response of nano-circular plates. On the basis of the Gurtin–Murdoch theory and Euler beam theory, Miandoab (2021) examined the effect of surface on the stiffness and mass of nanobeams. Recently, Abdelrahman and Eltaher (2022) employed the modified Gurtin–Murdoch surface elasticity to examine the bending and buckling behaviors of perforated nanobeams. The results show that the surface effect is essential in studying nanobeams’ bending and buckling behavior. The nanobeams and nanoplates studied above are all isotropic materials.

In order to meet the particular requirements of materials in aerospace and other fields, a new material, called graded porous material, is obtained by combining functional gradient material (FGM) with a porous material. Graded porous materials are a kind of lightweight materials, whose material properties change with the position of the structure in one or more directions to achieve the desired purpose. A remarkable feature of this material is the gradient distribution of internal pores. Graded porous materials are widely used in aerospace, marine, and mechanical fields because of their unique advantages. Therefore, in the past two decades, the application of gradient porous materials has led to many studies on the mechanical behavior of these materials (Magnucki and Stasiewicz 2004; Jabbari et al. 2015; Chen et al. 2015; Wattanasakulpong et al. 2018; Yusuf and Akbaş 2019; Akbaş et al. 2021; Çelik and Artan 2020).

With the rapid development of science and technology, functional gradient nanomaterials have emerged in the engineering neighborhood to replace ordinary materials. The application of graded porous nanomaterials is particularly noteworthy. However, according to the literature, the research on graded porous nanostructures is limited. Awrejcewicz et al. (2022) derived mathematical modeling and applied it to analyze the mechanical behavior of FG porous nanobeams in temperature fields. The results show that the porous nanobeam not only has a significant size dependence but also has a challenging influence on classical calculation methods such as thee finite difference method, finite element method, etc. Based on nonlocal strain gradient theory and Gurtin Murdoch surface theory, Shahzad et al. (2020a, b) investigated the static deflection, buckling, and vibration responses of graded porous nanobeam using the Navier method. The results showed that the types of porosity distribution patterns, porosity coefficient and tensional residual surface stress have important effects on the mechanics responses. After that, he and his collaborators extended this problem to various shear beam problems, and used differential quadrature method for numerical solution (Shahzada et al. 2020a, b). Faghidian et al. (2022a, b, c) researched the wave dispersion in functionally graded porous Timoshenko-Ehrenfest nanobeams. Jankowski et al. (2020) investigated the effects of porous material on bifurcation buckling and natural vibrations of nanobeams based on the higher‐order nonlocal strain gradient theory.

Through an in-depth investigation of the literature, we found that the research on the mechanical behavior of gradient porous nanostructures with surface effects has just started, and the external loads designed in the research are mainly conservative loads or thermal loads. In engineering practice, many structures or components are subjected to non-conservative forces. For example, the familiar follower force is a non-conservative force, because the direction of action of this force will change with the deformation of the structure. Airflow in aerospace can also be regarded as a tangential random force with uniform distribution. In addition, the pressure of the liquid in the oil tank and the viscous resistance of the medium in the transmission pipeline is follower forces (i.e., non-conservative forces). Li and Zhou (2005) studied the post-buckling behaviors of a hinged-fixed beam under such forces. Glabisz and Jarczewska (2020) published the stability problem of nanobeams subjected to nano-conservative loading. Unfortunately, the surface effect caused by small-scale is not considered.

It can be understood from the public literature that although there are many studies on the bending and buckling behavior of nanobeams, the number of studies is still limited, especially for graded material nanobeams with porosity. In particular, the research on the buckling and post-buckling characteristics of nanobeams under follower loads has not been seen. The main innovation of this paper is to establish the large deformation post-buckling control differential equation of the nanobeam, considering the surface effect. On the other hand, the primary purpose is for the graded porous nanobeams buckling and post-buckling analysis, and it provides a numerical method to solve this kind of multivariable coupling strong nonlinear differential equation. On account of the Gurtin–Murdoch surface elasticity theory and geometrically nonlinear beam theory, governing equations of post-buckling of a simply supported nanobeam subjected to follower force were established. Then, using the shooting method, the nonlinear governing equations were solved numerically, and the shooting method obtained the critical load and post-buckling equilibrium path curves under various factors. The influence of surface effect and non-surface effect on the post-buckling behavior of porous nanobeams was analyzed, which provides a reference for the stability study of nanostructures.

2 Problem formulation

Figure 1 shows a graded porous materials nanobeam with initial length $l$ and rectangular cross-section of width $b$ and height $h$. It is assumed that the nanobeam is subjected to distributed follower force $\tilde{q}$. Here, we use rectangular Cartesian coordinates with the $x$-axis originating from the left edge of the beam and the positive $z$-axis pointing upward along the thickness. Therefore, an arbitrary material point of the central axis in undeformed state denoted by $(x,z) = (x,0)$. When the nanobeam is in post-buckled state, the point of the central axis becomes $C:(x + u,w)$. Where, $u(x)$ and $w(x)$ are the displacement of the point $(x,0)$ in the $x$- and $z$-directions, respectively, $\theta (x)$ is the angle formed by the $x$-axis and the line tangent to the deformed central axis.

Fig. 1

Schematic of porous circular nanobeam with surface effects (left) and Kinematics parameters (right)

Since nanomaterial has a great specific surface area, the surface effects strongly affect the mechanical behaviors of nanoscale structures and should be considered for analyzing buckling and post-buckling nanobeams. The micro/nano structure considering surface effect generally adopts the model of surface delamination, that is, microstructure plus surface layer. The surface effect is simulated by a skinny layer, regardless of thickness which adheres to the bulk material without slipping. For this reason, we appeal to the Gurtin–Murdoch surface elasticity theory to capture the mechanical behavior of the surface phase and the conventional elasticity to describe those of the bulk phase in the beam interior. The parameters $\lambda^{S}$ and $\mu^{S}$ indicate the surface Lame constants, while the classical Lame constants are shown by $\lambda$ and $\mu$. The surface stresses which can be calculated using surface constitutive equations are as follows (Gurtin and Murdoch 1975; Yang et al. 2021a, b)

$$\sigma_{\alpha \beta }^{S} = \sigma_{{0}} \delta_{\alpha \beta } { + }(\lambda^{S} + \sigma^{S} )\varepsilon_{\gamma \gamma }^{S} \delta_{\alpha \beta } + 2(\mu^{S} - \sigma^{S} )\varepsilon_{\gamma \gamma }^{S} + \sigma^{S} u_{\alpha ,\beta }^{S}$$

(1)

where $\sigma^{S}$ is the surface residual stress, the top and bottom surfaces of nanobeams at $z = \pm h/2$ are denoted by $S^{ + }$ and $S^{ - }$, respectively.

The constitutive equations for the bulk material read

$$\sigma_{ij} = \lambda \varepsilon_{kk} \delta_{ij} + 2\mu \varepsilon_{ij}$$

(2)

$$\varepsilon_{ij} = \frac{1}{2}(u_{i,j} + u_{j,i} ),\quad \lambda = \frac{E\nu }{{{1 - }\nu^{{2}} }},\quad \mu = \frac{E}{{2({1 + }\nu )}}$$

(3)

2.1 Materials properties of FGFM

Two types of non-uniform porosity distribution patterns, shown in Fig. 2, are considered in the present study. The first is a symmetric material distribution model defined by Beam-I, and the second is asymmetric material distribution defined by Beam-II.

Fig. 2

Porosity distributions patterns in beam cross-sections

Suppose the mechanical properties of porous materials vary along the whole thickness of nanobeam. The variations of young's modulus, $E$, surface elastic modulus, $E^{S}$, and surface residual stress, $\sigma^{S}$, of graded porous materials with these two functions are given by

$$E(z) = E_{1} \left[ {1 - e_{0} \psi (z)} \right],\quad E^{S} (z) = E_{{0}}^{S} \left[ {1 - e_{0} \psi (z)} \right],\quad \sigma^{S} (z) = \sigma_{{0}}^{S} \left[ {1 - e_{0} \psi (z)} \right]$$

(4)

where $e_{0}$ is known as the porosity coefficient. When $e_{0}$ equals zero, the graded porous nanobeam degenerates into a uniform material nanobeam. For Beam-I, $E_{2}$ is the minimum Young's modulus, which is obtained at the min-plane surface (i.e. $z = 0$), and $U_{r} {(}r,z,t{) = }u{(}r{) - }z\partial w/\partial r$ is the corresponding material property maximum (i.e., material properties of pure nanobeams without pores) in the thickness direction at the upper and lower surfaces (i.e., $z = \pm 0.5h$). For Beam-II, the minimum and maximum values of Young's modulus occur at the bottom (i.e., $z = - 0.5h$) and top surfaces (i.e., $z = + 0.5h$). It is worth mentioning that in porous materials, Poisson’s ratio is generally considered to be a constant. The porosity coefficient $e_{0}$ in Eq. (4) can be calculated as follows:

$$e_{0} = 1 - E_{{2}} /E_{1} ,\quad 0 < e_{0} < 1$$

(5)

For symmetric and asymmetric porosity distributions, $\psi (z)$ can be expressed as

$$\psi (z) = \left\{ {\begin{array}{*{20}l} {\cos \left( {\frac{\pi }{h}z} \right)} \hfill & {\text{Beam-I}} \hfill \\ {\cos \left( {\frac{\pi }{{{2}h}}z{ + }\frac{\pi }{{4}}} \right)} \hfill & {\text{Beam-II}} \hfill \\ \end{array} } \right.$$

(6)

Figure 3 appears the changing curves of Young’s modulus with several values of $e_{0}$ computed by the following formulas in the thickness direction of the nanobeam.

Fig. 3

Young’s modulus Variation along the thickness of the nanobeam associated with different values of $e_{0}$

2.2 Governing equations of the problem

By accurately considering the axial extension, the geometric equations of the deformed central axis of the beam are derived as follows (Sun et al. 2016)

$$\frac{{\text{d}s}}{{\text{d}x}} = R(x),\quad \frac{{\text{d}u}}{{\text{d}x}} = R(x)\cos \theta - 1,\quad \frac{{\text{d}w}}{{\text{d}x}} = R(x)\sin \theta$$

(7)

where $R(x)$ is the extension rate of the axis line, $s$ is the arc length of the deformed axial line, $u$ and $w$ are the displacements of the central axial in longitudinal(i.e.$x$-axis) and transverse directions ($z$-axis), respectively.

According to Kirchhoff’s assumption, the normal strain, $\varepsilon$, at any point $(x,z)$ of the bulk phase can be given in terms of

$$\varepsilon \left( {x,z} \right) = R - 1 - z\frac{{{\text{d}}\theta}}{{{\text{d}}x}}$$

(8)

According to Eqs. (3) and (5), the constitutive relationship of the bulk layer can be written as:

$$\sigma_{xx} = E\left( z \right)\left( {R - 1 - \frac{{{\text{d}}\theta}}{{{\text{d}}x}}z} \right)$$

(9)

Given the surface constitutive relation (1) for the nanobeam with GM surface material, we have the constitutive relationship of the surface layer.

$$\sigma_{xx}^{S} = \sigma_{{0}} { + (}\lambda^{S} { + 2}\mu^{S} {)}\left( {R - 1 - \frac{{\text{d}\theta }}{{\text{d}x}}z} \right)$$

(10)

The resultant axial, $N_{\alpha \alpha }$, and bending moment, $M_{\alpha \alpha }$ can be deduced by proper integration

$$N_{\alpha \alpha } = \iint_{A} {\sigma_{\alpha \alpha } {\text{d}}A}{ + }\oint_{\Gamma } {\sigma_{\alpha \alpha }^{S} {\text{d}}s}$$

(11)

$$M_{\alpha \alpha } = \iint_{A} {(z - z_{0} )\sigma_{\alpha \alpha } {\text{d}}A}{ + }\oint_{\Gamma } {(z - z_{0} )\sigma_{\alpha \alpha }^{S} {\text{d}}s}$$

(12)

in which $z_{0}$ represented the physical neutral surface of the graded porous nanobeams, which can be determined as

$$z_{0} = \int_{{ - h{/2}}}^{{h{/2}}} {zE({\text{z}})} \text{d}z\left( {\int_{{ - h{/2}}}^{{h{/2}}} {E({\text{z}})} \text{d}z} \right)^{{ - 1}}$$

(13)

Obviously, for Beam-I mode, $z_{0} = 0$, and Beam-II mode, $z_{0} = e_{0} h(\pi - 4)/(2e_{0} \pi - \pi^{2} )$. For function $K(z)$ can be written as (Shahzada et al. 2020a, b)

$$\oint_{\Gamma } {K(z){\text{d}}s} = 2\int_{ - h/2}^{h/2} {K(z)dz + b\left[ {K(z)\left( \frac{h}{2} \right) + K(z)\left( { - \frac{h}{2}} \right)} \right]}$$

(14)

Therefore, Eqs. (11) and (12) can be written as

$$N_{\alpha \alpha } = \iint_{A} {\sigma_{\alpha \alpha } {\text{d}}z}{ + }b{(}\sigma_{\alpha \alpha }^{S + } + \sigma_{\alpha \alpha }^{S - } )$$

(15)

$$M_{\alpha \alpha } = \iint_{A} {(z - z_{0} )\sigma_{\alpha \alpha } {\text{d}}z}{ + }\frac{bh}{{2}}(\sigma_{\alpha \alpha }^{S + } - \sigma_{\alpha \alpha }^{S - } )$$

(16)

After inserting Eqs. (9) and (10) into Eqs. (15) and (16), can be obtained:

$$N = A(R - 1) + 2b(\lambda^{S} { + 2}\mu^{S} )(R - 1) + 2\sigma_{{0}}^{S} = (R - 1)(A + \lambda_{1} ) + 2b\sigma_{{0}}^{S}$$

(17)

$$M = D\frac{{{\text{d}}\theta }}{{{\text{d}}x}} + \frac{{bh^{2} }}{2}(\lambda^{S} { + 2}\mu^{S} )\frac{{{\text{d}}\theta }}{{{\text{d}}x}} = (D + \lambda_{2} )\frac{{{\text{d}}\theta }}{{{\text{d}}x}}$$

(18)

where $A$ is the stretching stiffness and $D$ is the bending stiffness coefficients, they are given by

$$(A,D) = \iint_{A} {(1,z^{2} )}E(z)\text{d}A = bhE_{1} (f_{1} ,h^{2} f_{3} )$$

(19)

$$\lambda_{1} = 2b(\lambda^{S} { + 2}\mu^{S} ),\quad \lambda_{2} = bh^{2} (\lambda^{S} { + 2}\mu^{S} )/2$$

(20)

It can be seen that there is no stretching-bending coupling in constitutive equations, which is precisely due to the introduction of the middle plane of physics.

The equilibrium equations of forces, horizontal internal force $H$ and vertical internal force $V$, and moments $M$ can be derived by considering the deformed segment ${\text{d}}s$, as follows (Li et al. 2006):

$$\frac{{\text{d}H}}{{\text{d}x}} = - Rq\cos \theta ,\quad \frac{{\text{d}V}}{{\text{d}x}} = - Rqs{\text{in}}\theta ,\quad \frac{{\text{d}M}}{{\text{d}x}} = R( - H\sin \theta + V\cos \theta )$$

(21)

The axial force $N$ related to $H$ and $V$ can be expressed by:

$$N = - H\cos \theta - V\sin \theta$$

(22)

By substituting Eq. (15) into Eq. (17), and combining Eq. (16), we have

$$\frac{{{\text{d}}\theta }}{{{\text{d}}x}} = \frac{1}{{(D + \lambda_{2} )}}M$$

(23)

$$R = 1 - \frac{1}{{(A + \lambda_{1} )}}(H\cos \theta + V\sin \theta ) - \frac{{2b\sigma_{{0}}^{S} }}{{(A + \lambda_{1} )}}$$

(24)

By introducing the following dimensionless variables

$$\begin{aligned} (\xi ,S,U,W) = (x,s,u,w)/l,\quad \delta = l/h,\quad (P_{H} ,P_{V} ) = \frac{{12l^{2} }}{{bh^{3} E_{1} }}(H,V) \hfill \\ m = \frac{12lM}{{bh^{3} E_{1} }},\quad q = \frac{{12l^{3} \tilde{q}}}{{bh^{3} E_{1} }},\quad \gamma = \frac{{E_{S} }}{{hE_{1} }},\quad \eta = \frac{{\sigma_{0}^{S} }}{{hE_{1} }}. \hfill \\ \end{aligned}$$

where $\gamma$ is surface elastic parameters and $\eta$ is dimensionless residual surface stress parameters.

In terms of the above dimensionless parameters, one can obtain the following governing differential equations and boundary conditions regarding dimensionless variables

$$\frac{{\text{d}S}}{{\text{d}\xi }} = R,\quad \frac{{\text{d}U}}{{\text{d}\xi }} = R\cos \theta - 1,\quad \frac{{\text{d}W}}{{\text{d}\xi }} = R\sin \theta$$

(25)

$$\frac{{\text{d}\theta }}{{\text{d}\xi }} = \frac{1}{{12\left( {f_{3} + \frac{1}{{2(1 - \nu^{2} )}}\gamma } \right)}}m$$

(26)

$$\frac{{\text{d}P_{H} }}{{\text{d}\xi }} = - Rq\cos \theta ,\quad \frac{{\text{d}P_{V} }}{{\text{d}\xi }} = - Rqs{\text{in}}\theta$$

(27)

$$\frac{{\text{d}m}}{{\text{d}\xi }} = R( - P_{H} \sin \theta + P_{V} \cos \theta )$$

(28)

$$R = 1 - \frac{1}{{12\delta^{2} (f_{1} + \frac{2}{{(1 - \nu^{2} )}}\gamma )}}(P_{H} \cos \theta + P_{V} \sin \theta ) - \frac{2}{{(f_{1} + \frac{2}{{(1 - \nu^{2} )}}\gamma )}}\eta$$

(29)

where the dimensionless coefficient for Beam-I can be written as:

$$f_{1} = 1 - \frac{{2e_{0} }}{\pi },\quad f_{3} = \frac{1}{12} - e_{0} \left( {\frac{1}{2\pi }{ - }\frac{4}{{\pi^{3} }}} \right);$$

for Beam-II,

$$f_{1} = 1 - \frac{{2e_{0} }}{\pi },\quad f_{3} = \frac{{e_{{0}} }}{{{(2}e_{{0}} - \pi {)}}}\left( {\frac{{{48}e_{{0}} }}{{\pi^{{3}} }}{ + }\frac{{4}}{\pi }{ + }\frac{{2}}{{3}} - \frac{{{16}e_{{0}} }}{{\pi^{2} }} - \frac{16}{{\pi^{2} }} - \frac{\pi }{{12e_{{0}} }}} \right).$$

According to Fig. 1, its left end is pinned, with horizontal displacement, vertical displacement, and bending moment all zero; The right end is hinged and the corresponding vertical displacement, bending moment, and horizontal internal force are also zero. Therefore, the dimensionless expression for the corresponding boundary condition can be written as

$$S(0) = U(0) = W(0) = m(0) = 0$$

(30)

$$W(1) = P_{H} (1) = m(1) = 0$$

(31)

Let $\theta (0) = \beta$, here $\beta$ is the rotational angle of the left end cross-section.

3 Introduction to shooting method

Due to the strong non-linearity and coupling in Eqs. (25)–(28), finding any analytical solution to the boundary-value problem is complicated. Herein, a shooting method seeks numerical solutions to the problem. It should be pointed out that compared to the finite element method and differential quadrature method, the shooting method has incomparable advantages in solving two-point boundary value problems of nonlinear ordinary equations, because it does not rely on mesh generation. The basic idea of the shooting method is to first convert the boundary value problem of ordinary differential equations into the corresponding initial value problem of ordinary differential equations, in which the initial value conditions contain the same number of arbitrary parameters as the terminal boundary conditions. For the convenience of expression, the nonlinear governing Eqs. (25)–(28) and boundary conditions are recorded as the standard form of the two-point boundary value problem of the following ordinary differential equations:

$$\frac{{{\text{d}}{\mathbf{Y}}}}{{{\text{d}}\xi }} = {\mathbf{H}}(\xi ,{\mathbf{Y}}),\quad (\xi_{1} < \xi < \xi_{2} )$$

(32a)

$${\mathbf{B}}_{1} {\mathbf{Y}}(\xi_{1} ) = {\mathbf{b}}_{1} ,\quad {\mathbf{B}}_{{2}} {\mathbf{Y}}(\xi_{{2}} ) = {\mathbf{b}}_{{2}}$$

(32b, c)

Therefore, the boundary conditions (30) and (31) can be expressed as

$$\begin{aligned} & {\mathbf{B}}_{1} = \left[ {\begin{array}{*{20}c} {1} & {0} & {0} & {0} & {0} & {0} & {0} & {0} \\ {0} & {1} & {0} & {0} & {0} & {0} & {0} & {0} \\ {0} & {0} & {1} & {0} & {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {1} & {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} & {0} & {0} & {1} & {0} \\ \end{array} } \right], \\ & {\mathbf{B}}_{{2}} = \left[ {\begin{array}{*{20}c} {0} & {1} & {0} & {0} & {0} & {0} & {0} & {0} \\ {0} & {0} & {1} & {0} & {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {1} & {0} & {0} & {0} & {0} \\ \end{array} } \right] \\ \end{aligned}$$

(33)

$${\mathbf{b}}_{1} = \left\{ {\begin{array}{*{20}c} {0} & \beta & {0} & {0} & {0} \\ \end{array} } \right\}^{T} ,\quad {\mathbf{b}}_{2} = \left\{ {\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} } \right\}^{T}$$

(34)

In order to solve the two-point boundary value problem with the shooting method, let

$${\mathbf{Y}} = (y_{1} ,y_{2} ,y_{3} ,y_{4} ,y_{5} ,y_{6} ,y_{7} ,y_{8} )^{T} = (S,U,W,\theta ,P_{H} ,P_{V} ,m,q)^{T}$$

(35)

Thus, we get

$$\begin{aligned} &\frac{{{\text{d}}y_{1} }}{{{\text{d}}\xi }} = R,\quad \frac{{{\text{d}}y_{2} }}{{{\text{d}}\xi }} = R\cos y_{4} - 1,\quad \frac{{{\text{d}}y_{{3}} }}{{{\text{d}}\xi }} = R\sin y_{4} \hfill \\ &\frac{{{\text{d}}y_{{4}} }}{{{\text{d}}\xi }} = \frac{1}{{12(f_{3} + {1/}2(1 - \nu^{2} )\gamma )}}y_{7} ,\quad \frac{{{\text{d}}y_{{5}} }}{{{\text{d}}\xi }} = - Ry_{{8}} {\text{cos}}y_{4} , \hfill \\ &\frac{{{\text{d}}y_{6} }}{{{\text{d}}\xi }} = - Ry_{{8}} {\text{sin}}y_{4} ,\quad \frac{{{\text{d}}y_{7} }}{{{\text{d}}\xi }} = R( - y_{{5}} {\text{sin}}y_{4} + y_{6} {\text{cos}}y_{4} ),\quad \frac{{{\text{d}}y_{{8}} }}{{{\text{d}}\xi }} = {0} \hfill \\ \end{aligned}$$

Here, the load parameter $q$ is taken as an unknown function of $\xi$ to form a standard procedure for dealing with such non-linear boundary-value problems.

Then, consider the initial value problem corresponding to the boundary value problem Eqs. (32)

$$\frac{{{\text{d}}{\mathbf{Z}}}}{{{\text{d}}\xi }} = {\mathbf{H}}(\xi ,{\mathbf{Z}})$$

(36a)

$${\mathbf{Z}}\text{(0)} = (0,0,0,\beta ,V_{1} ,V_{2} ,0,V_{3} )^{T} = {\mathbf{I}}\text{(}\beta ,{\mathbf{D}}\text{)}$$

(36b)

where ${\mathbf{D}} = (\begin{array}{*{20}c} {V_{1} } & {V_{2} } & {V_{3} } \\ \end{array} )^{T}$ is any parameter related to the initial value not given in Eq. (32c). For any set of initial parameters ${\mathbf{D}}$, the solution of the initial value problem (36) can be formally expressed as

$${\mathbf{Z}} = {\mathbf{Z}}\text{(}\xi ,{\mathbf{D}}\text{) = }{\mathbf{I}}\text{(}\xi_{1} ,{\mathbf{D}}\text{) + }\int_{{\xi_{1} }}^{\xi } {\mathbf{G}} \text{(}\xi ,{\mathbf{Z}}\text{)d}\xi$$

(37)

The main task of the shooting method is to adjust the initial parameter ${\mathbf{D}}$ so that the solution of the initial value problem (37) can meet the boundary conditions of the endpoint.

4 Numerical results and discussions

4.1 Comparison studies

To ensure the accuracy and effectiveness of the present method, two examples are presented to solve the buckling and post-buckling of uniform isotropic beams and gradied porous beams. The size are set to $l = 43\,{\text{nm}}$, $h = 1\,{\text{nm}}.$

Example 1: Firstly, let $e_{0} = 0$, $\eta = 0$, $\gamma = 0$, the graded porous nanobeam can be transformed into a isotropic beam with simply supported edge and subjected to distributed tangential follower force. Comparisons of the results are provided in Table 1. It can be clearly be seen that the obtained nondimensional load are perfect match with the results of known results (Li et al. 2004).

Table 1 Comparisons of the non-dimensional load $q$ for isotropic simply supported beams corresponding to different values of $\beta$

Example 2: Using the proposed formula, the buckling and post-buckling of graded porous beams are analyzed, and verified by the two porosity distributions mentioned earlier. To this end, let $\eta = 0$, $\gamma = 0$, the study in this part degenerates into the post-buckling problem of the graded porous nanobeam without considering the surface effect. Critical load values of graded porous are tabulated and compared in Table 2 with a set of results (Li et al. 2022). Table 2 shows the good agreement of the results compared well with existing resluts.

Table 2 Comparisons of the non-dimensional load $q$ for graded porous simply supported beams corresponding to different values of $\beta$

Figure 4 shows the relationship between the left rotational angle $\beta$ and follower force $q$ for symmetric and asymmetric cases. It can be seen that bifurcation instability will occur in both symmetric and asymmetric cases. In particular, when $e_{0} = 0$, the beam degenerates into a uniform material beam and the critical load $q_{cr} = 18.971$ obtained in this paper agrees very well with the solutions of Leipholz (1980).

Fig. 4

$\beta$ versus $q$ for different porosity $e_{0}$ without surface effect

In Fig. 5, we have displayed the critical load relationship curves under different porosity coefficients. It is clear that the critical load decreases linearly with the increase of porosity coefficient for the symmetric porosity model beam, and nonlinearly with the increase of porosity for the asymmetric porosity model beam. The critical load in the asymmetric case is lower than that in the symmetric case. With the increase of the porosity coefficient, the critical load difference between the two models becomes larger and larger. It shows that the distribution of porosity and porosity coefficient influences the critical load of the graded porous beam.

Fig. 5

$q_{{\text{cr}}}$ versus $e_{0}$ relation curve

4.2 Surface parametric effects on stability

With the confidence of the established model, we will show the influence of the surface effects on the post-buckling behaviors of graded porous material nanobeams. In the following, we will use another pair of material constants: surface Young’s modulus and surface Poisson’s ratio $E^{S}$,$\nu^{S}$ in place of surface Lame constants. The bulk and surface properties are listed in Table 3 (Yang et al. 2021a, b).

Table 3 Material constants of bulk and surface layer

Now fixing $e_{0} = 0.2$, when the surface effect is considered first, but the surface residual stress is not considered, Figs. 6 and 7 depict the post-buckling path of the left rotation angle $\beta (0)$ and the constraint force $P_{V} (0)$ versus nondimensional follower force $q$. For both symmetric and asymmetric cases, the critical load with surface effect is larger than that without surface effect. It shows that for porous materials made of silver, the surface effect will increase or decrease the stiffness of the overall structure. After buckling, i.e., entering the post-buckling state, one load will correspond to two equilibrium paths.

Fig. 6

Influence of surface effect on $\beta - q$ relation curve $(e_{0} = 0.2)$

Fig. 7

Influence of surface effect on $\beta - P_{V} (0)$ relation curve $(e_{0} = 0.2)$

Figure 8 represents the effect of $\gamma$ on the post-buckling of graded porous nanobeams subjected to follower force. From equation $\gamma = E_{S} /hE_{1}$, we know that the surface elastic parameters are related to the section height $h$. It can be seen that increasing the thickness of the beam section leads to a decrease in the critical load of the graded porous nanobeam. It shows that the surface elastic parameter is also an important factor affecting the buckling and post-buckling equilibrium path of graded porous beams. When $\gamma$ is fixed as $0.016$ (i.e. $h = 1\,{\text{nm}}$), Fig. 9 shows the curves of the post-buckling path for the graded porous beams versus $e_{0}$. It is clearly seen that $\beta$ increases first when $q$ increases from $q_{{{\text{cr}}}}$. When $\beta$ increases to about $120^{{\text{o}}}$, $q$ reaches the maximum. After that, it decreases with the increase of $\beta$.

Fig. 8

The effect of $\gamma$ on the post-buckling path of graded porous nanobeam $(e_{0} = 0.1)$

Fig. 9

The effect of $e_{0}$ on the post-buckling equilibrium path of graded porous nanobeam (Beam I, $\gamma = 0.016$)

The length of the nanobeam is kept at $l = 43.3\,{\text{nm}}$, and the section height is taken as $h = 1\,{\text{nm}}$ and $2\,{\text{nm}}$ respectively. The relevant parameters calculated are $\gamma = 0.016$, $\eta = 0.0117$, $\delta = 43.3$ and $\gamma = 0.008$, $\eta = 0.00585$, $\delta = 21.65$, respectively. Obviously, the residual surface stress parameter $\eta$ is related to the surface elasticity parameter $\gamma$. When considering surface residual stresses $\sigma_{0} = 0.89$, Fig. 10 shows size of the porous nanobeam on post-buckling behavior. It is observed that for a given $e_{0}$, increasing $h$ increases critical load. The figure shows that the size of the porous nanobeam significantly impacts the buckling and post-buckling of the nanobeams. When the slenderness ratio of the beam increases, the critical load of buckling of the porous beam decreases.

Fig. 10

The effect of $h$ on the post-buckling equilibrium path of graded porous nanobeam (Beam I)

Figure 11 shows the effect of surface elasticity parameters on post-buckling behavior. The solid line represents the case when the surface effect is not considered. It can be seen that surface elasticity parameters $\gamma = E^{S} /hE$ can increase or decrease the buckling critical load. A positive $\gamma$ leads a greater critical load and a negative $\gamma$ leads to a lower critical load compared with the one without surface effect. From the post-buckling equilibrium path, we can easily see that for a positive $\gamma$, gradient porous nanobeam appear to become harder, resulting in smaller deformations. For negative surface elasticity parameters, an opposite trend is observed.

Fig. 11

The effect of $\gamma$ on the post-buckling equilibrium path of graded porous nanobeam (Beam I)

Next, the influences of surface residual stress on the buckling and post-buckling of porous nanobeams are investigated. Figures 12, 13 and 14 show the characteristic relationship between relevant physical quantities, such as the left rotational angle, the left-end horizontal internal force $P_{H} (0)$ and the left-end vertical internal force $P_{V} (0)$ and dimensionless loads, respectively. It is revealed that increasing $\sigma_{0}$ leads to raising critical buckling load, which indicates that when considering the surface effect stress, it is equivalent to increasing the rigid of graded porous nanobeam. It should be noted that the above research results are based on the Beam-I model. It can be inferred that under the Beam-II model, the influence of surface stress parameters on the buckling and post-buckling of graded porous nanobeams is similar. Also, from these equilibrium path curves in Figs. 12, 13 and 14, it can be seen that during the increasing post-buckling deformation, the load parameters first monotonically increase from a critical value to a maximum value. Then the load monotonically decreases as the deformation increases.

Fig. 12

Effect of surface residual stress on $\beta - q$ ($e_{0} = 0.3$)

Fig. 13

Effect of surface residual stress on $P_{H} (0) - q$ ($e_{0} = 0.3$)

Fig. 14

Effect of surface residual stress on $P_{V} (0) - q$ ($e_{0} = 0.3$)

To better understand the specific values of the effects of residual surface stress on buckling and post-buckling. Table 4 lists the critical buckling loads $q_{cr}$ under different surface residual stresses and the maximum loads $q_{\max }$ that can be reached as the deformation increases after post-buckling. It can be seen from the table that with the increase of residual surface stress $\sigma_{0}$, the dimensionless critical load decrease, and the maximum load received during post-buckling deformation is also increasing. In particular, it should be noted that the maximum load corresponding to the post-buckling deformation process occurs at the left-end corner of 118.1°. In general, the effect of residual surface stress on the buckling and post-buckling of nanobeams is consistent with the situation reflected in Figs. 12, 13 and 14. This is because the residual surface stress increases the stiffness of the graded porous nanobeam.

Table 4 Effect of surface residual stress on critical load (Beam I, $e_{0} = 0.3$)

It is clear from the above analysis that after buckling, one load corresponds to two post-buckling equilibrium paths. We call them the first post-buckling equilibrium configuration and the second post-buckling equilibrium configuration. Figure 15 depicts the post-buckling equilibrium configurations of the graded porous nanobeams ($e_{0} = 0.2$) for different surface residual stress $\sigma_{0}$ and a given porosity coefficient $e_{0}$. As $\sigma_{0}$ increases, the deformation of the beam decreases, as expected. When $e_{0} = 0.2$ and $\sigma_{0} = 0.1$, Fig. 16 shows the effect of porosity distribution on post-buckling equilibrium configuration. It is shown that under the same conditions, the deformation of the nanobeam with asymmetric porosity distribution is greater than that with symmetrical porosity distribution.

Fig. 15

Effect of surface residual stress on equilibrium configurations of porous nanobeams ($e_{0} = 0.2$)

Fig. 16

Effect of pore distribution pattern on equilibrium configurations of porous nanobeams

5 Conclusions

In this paper, both buckling and post-buckling of graded porous nanobeams subjected to follower force with simply supported edges are presented by solving coupled equations, considering the small-scale effect. We introduce the Gurtin–Murdoch theory to study the surface effects of nanobeams made of graded porous materials. It should be mentioned that it is not common to consider porous nanobeams or plates to study buckling and post-buckling. Among them are two models, namely, the symmetrical and asymmetrical non-uniform distribution of pores along the thickness of porous materials. The coupled nonlinear ordinary differential equations, including surface elastic parameters and residual surface stress parameters, are numerically solved by the shooting method. From the numerical analysis in this study, one can draw several conclusions:

• 1. The nonlinear control differential equations of graded porous nanobeam with surface effects were derived. It provides a very convenient method for us to analyze the buckling and post-buckling behavior of nanobeams.

• 2. As the porosity coefficient $e_{0}$ increases from zero, the critical buckling follow force $q_{cr}$ decreases rapidly. In the symmetrical case, the decrease of a critical load is linear with the increase of the porosity coefficient, while in the asymmetrical case, it is nonlinear.

• 3. Surface effects can increase or decrease the effective stiffness of nanobeams, which well explains the softening or hardening of the structure. Whether surface effect is considered or not, the post-buckling equilibrium path is not a monotone function or a single value function of the load $q_{cr}$.

• 4. The greater the surface effect, the greater the critical load for buckling of porous nanobeams. Especially, the influence of surface residual stress on the buckling and post buckling behavior of porous nanobeams plays an important role. The greater the residual stress σ₀, the greater the critical load q_cr