1 Introduction

Different reasons such as diseases, trauma, congenital defects and etc. may lead to degeneration of tissues in the human body. Nowadays, via development in tissue engineering, novel approaches have been emerged to regenerate a damaged tissue, in spite of replacing it. In this way, the pore architecture and porosity of scaffolds play an essential role in cell migration and in growth, and recently, several studies have been performed in this research area. Shariful Islam and Todo (2016) discovered the sintering effects on the compressive mechanical properties of the scaffold. Hedayati et al. (2017a) analyzed the fatigue crack propagation in additively manufactured porous biomaterial via an analytical model. Zhang et al. (2017) investigated the influence of three kinds of sterilization method on a porous zein scaffold as a new biomaterial. Bobbert et al. (2017) designed porous metallic biomaterials on the basis of four different types of triply periodic minimal surfaces which cause to mimic the properties of bone to an unprecedented level. Kadkhodapour et al. (2017) utilized triply periodic minimal surfaces to obtain structure-property relations for Ti6Al4 V scaffolds designed.

Nanoscale porous biomaterials have been recently evolved as o new class of porous materials having exciting applications. For instance, the materials utilized to manufacture nanoscaffolds in heart valves are typically packed together with pores of a very small size to direct the colonization and growth of cells in a more efficient way. Due to the high surface to volume ratio as well as size-dependent characteristics, nanoporous materials feature unique behaviour in comparison with conventional porous materials (Beg et al. 2017).

In order to make the continuum mechanical applicable to the analysis of micro/nano-structures, it needs to take small length scales such as lattice spacing and grain size into account. Up to now, various unconventional continuum theories have been established to consider size dependency in mechanical characteristics of micro/nano-structures (Shen 2010; Xu and Deng 2013; Farrokhabadi et al. 2014; Sahmani et al. 2014a, b; Li and Hu 2015a; Sahmani and Bahrami 2015a, b; Ghorbanpour Arani 2015; Sahmani et al. 2015a, b, c, d, e; Sahmani et al. 2016a, b, c, d; Jun Yu et al. 2016; Simsek 2016; Wang et al. 2016; Yang and He 2017; Sahmani and Aghdam 2017a, b, c, d; Korayem and Korayem 2017; Awrejcewicz et al. 2017; Sahmani et al. 2017a, b; Nazemnezhad and Fahimi 2017; He et al. 2017; Mirsalehi et al. 2017a, b; Kheibari and Tadi Beni 2017; Sahmani and Fattahi 2017a, b, c, d, e; Malikan 2017; Kiani 2017; Bornassi and Haddadpour 2017). According to the previous studies, it has been observed that the nonlocal elasticity theory and strain gradient continuum mechanics represent two entirely different side effects including softening-stiffness and hardening-stiffness influences. To overcome this paradox, Lim et al. (2015) developed the nonlocal strain gradient elasticity theory which incorporates simultaneously both features of size dependency. After that, several investigations have been carried out to analyze the size-dependent mechanical behaviour of micro/nano-structures. Li and Hu (2015b) used the nonlocal strain gradient theory of elasticity to develop a size-dependent Euler–Bernoulli beam model for buckling analysis of nanobeams. They also formulated the equations related to the wave motion fluid-conveying viscoelastic carbon nanotubes based upon the nonlocal strain gradient continuum mechanics (Li and Hu 2016). Simsek (2016) examined the size-dependent nonlinear vibrations of functionally graded Euler–Bernoulli nanobeams via nonlocal strain gradient theory of elasticity. Yang et al. (2016) studied the nonlocal strain gradient dynamic pull-in instability of functionally graded carbon nanotube-reinforced nano-actuators. Li et al. (2016) analyzed the longitudinal vibrations of nano-scaled rods on the basis of the nonlocal strain gradient elasticity theory. Tang et al. (2016) predicted the viscoelastic wave propagation in an embedded viscoelastic carbon nanotube based on the theory of nonlocal strain gradient elasticity. Sahmani and Aghdam (2017e) developed a nonlocal strain gradient third-order shear deformable beam model for nonlinear vibration analysis of multilayer functionally graded nanocomposite nanobeams within the prebuckling and postbuckling domains. They also constructed a refined nonlocal strain gradient shell model for axial and radial postbuckling behaviours of multilayer functionally graded nanoshells reinforced with graphene platelets (Sahmani and Aghdam 2017f, g). Xu et al. (2017) explored the nonlocal strain gradient bending and buckling of Euler–Bernoulli nanobeams Based on the weighted residual approaches. Sahmani and Aghdam (2017h, 2018a) captured size effects on the nonlinear instability of axially and radially loaded microtubules surrounded by cytoplasm based upon the nonlocal strain gradient shell model. Lu et al. (2017) proposed a nonlocal strain gradient sinusoidal shear deformable beam model for vibration analysis of nanobeams. Wang et al. (2018) studied the transverse free vibration response of axially moving nanobeams via a nonlocal strain gradient Euler–Bernoulli beam model. Sahmani and Aghdam (2018b) employed the nonlocal strain gradient elasticity theory for the nonlinear axial instability analysis of magneto-electro-elastic micro/nano-shells. El-Borgi et al. (2018) investigated the torsional vibrations of viscoelastic nanorods embedded in an elastic medium based on the nonlocal strain gradient theory of elasticity. Sahmani and Aghdam (2018c) analyzed the size-dependent nonlinear primary resonance of a biological nanoporous micro/nano-beam via the nonlocal strain gradient beam model. Nematollahi et al. (2018) presented a new formulation based upon the higher-order nonlocal strain gradient theory for free vibration analysis of rectangular nanoplates. Sahmani et al. (2018a) explored the nonlinear bending of the nonlocal strain gradient beam made of a functionally graded porous material reinforced with graphene platelets. They also proposed a nonlocal strain gradient plate model for the nonlinear large-amplitude vibration and postbuckling analysis of micro/nano-plates made of the reinforced functionally graded porous material (Sahmani et al. 2018b, c). Xu et al. (2017) employed the weighted residual approaches to present a closed-form solution for the nonlinear boundary value problem of a nonlocal strain gradient Euler–Bernoulli beam. Radwan and Sobhy (2018) proposed a nonlocal strain gradient beam model for dynamic deformation response of graphene nanosheets surrounded by the viscoelastic medium. Ma et al. (2018) reported the wave propagation in magneto-electro-elastic nanoshells on the basis of the nonlocal strain gradient theory of elasticity. Sahmani and Fattahi (2018) examined the axial buckling and postbuckling characteristics of functionally graded composite nanoshells with the aid of the nonlocal strain gradient elasticity theory. Karami et al. (2018) developed a nonlocal strain gradient higher-order shear deformable shell model for size-dependent wave propagation in anisotropic doubly-curved nanoshells.

In the current study, at first, a refined form of the analytical approach developed by Hedayati et al. (2017b) is put to use to construct an explicit expression for mechanical properties of nanoporous biomaterial made from refined truncated cube lattice structure in terms of pore size. Subsequently, based upon the extracted mechanical properties, the nonlocal strain gradient theory of elasticity is utilized to take size dependencies into consideration for the nonlinear instability and nonlinear large-amplitude vibrations within the prebuckling and postbuckling regimes of axially compressed microbeams made of the nanoporous biomaterial. Via the Galerkin method together with an improved perturbation technique, explicit analytical expressions for the nonlocal strain gradient nonlinear mechanical responses of the biological nanoporous microbeam are obtained.

2 Analytical Extracted Properties of Nanoporous Biomaterials

In this work, it is assumed that the biological nanoporous microbeam includes refined truncated cube lattice structure as illustrated in Fig. 1. Accordingly, by repeating the cells, a unit cell surrounding by the truncated cubes is resulted in, each membrane of which is dedicated to a unique refined truncated cube. It is demonstrated in Fig. 2 that because of the geometrical symmetry, the links \(c_{1} a_{1} b_{1} d_{1} a_{2} c_{2}\) and \(c_{1} a_{1} b_{2} d_{2} a_{2} c_{2}\) and \(c_{1} a_{1} b_{3} d_{3} a_{2} c_{2}\) and \(c_{1} a_{1} b_{4} d_{4} a_{2} c_{2}\) of the unit cell have the same mechanical in-plane deformations. Consequently, analysing one of them is enough to obtain the mechanical response of the unit cell. Here, the link \(c_{1} a_{1} b_{1} d_{1} a_{2} c_{2}\) is chosen to be analyzed.

Fig. 1
figure 1

Schematic representation of a biological nanoporous microbeam: a coordinate system and geometric parameters; b a refined truncated cube lattice framework

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Fig. 2
figure 2

A refined truncated cube unit cell

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Based on the refined hyperbolic shear deformable beam model for the links of the unit cell, it yields

$$\bar{E}\bar{I}w_{xxxx} = \bar{E}\bar{I}\left[ {\cosh \left( {1/2} \right) - 12\left( {\cosh \left( {1/2} \right) - 2\sinh \left( {1/2} \right)} \right)} \right]\psi_{xxx} + q\left( x \right)$$
(1a)
$$\begin{aligned} & \bar{E}\bar{I}\left[ {\cosh \left( {1/2} \right) - 12\left( {\cosh \left( {1/2} \right) - 2\sinh \left( {1/2} \right)} \right)} \right]w_{xxx} \\ & \quad \quad = \bar{E}\bar{I}\left[ {\left( {\cosh \left( {1/2} \right)} \right)^{2} + 6\left( {\sinh \left( 1 \right) - 1} \right) - 24\cosh \left( {1/2} \right)\left( {\cosh \left( {1/2} \right) - 2\sinh \left( {1/2} \right)} \right)} \right]\psi_{xx} \\ & \quad \quad\quad - \bar{G}\bar{A}\left[ {\left( {\cosh \left( {1/2} \right)} \right)^{2} + \left( {1/2} \right)\left( {\sinh \left( 1 \right) + 1} \right) - 4\cosh \left( {1/2} \right)\sinh \left( {1/2} \right)} \right]\psi \\ \end{aligned}$$
(1b)

in which \(\bar{E} \cdot \bar{G} \cdot \bar{I} \cdot \bar{A} \cdot w \cdot \psi\) stand for, respectively, the Young’s modulus, shear modulus, moment inertia, cross-section area, deflection and angel of rotation for the links of the unit cell.

Consequently, for a cantilever beam with constructed load \(P\) at the free end, one will have

$$\delta_{p} = w\left( \ell \right) = P\ell^{3} /\left( {3\bar{E}\bar{I}} \right) + \left( {6P\ell /\left( {5\bar{G}\bar{A}} \right)} \right)\left[ {1 + \left( {\cosh \left( {\varGamma \ell } \right) - \sinh \left( {\varGamma \ell } \right) - 1} \right)/\varGamma \ell } \right]$$
(2a)
$$\theta = \varphi \left( \ell \right) = P\ell^{2} /\left( {2\bar{E}\bar{I}} \right) + \left( {6P/\left( {5\bar{G}\bar{A}} \right)} \right)\left[ {1 + \sinh \left( {\varGamma \ell } \right) - \cosh \left( {\varGamma \ell } \right)} \right]$$
(2b)

in which

$$\varGamma = \sqrt {\frac{{\bar{G}\bar{A}\alpha_{3} /\left( {\bar{E}\bar{I}\alpha_{1} } \right)}}{{\alpha_{2} /\alpha_{1} - \alpha_{1} }}}$$
(3)
$$\alpha_{1} = \cosh \left( {1/2} \right) - 12\left[ {\cosh \left( {1/2} \right) - 2\sinh \left( {1/2} \right)} \right]$$
$$\alpha_{2} = \left( {\cosh \left( {1/2} \right)} \right)^{2} + 6\left[ {\sinh \left( 1 \right) - 1} \right] - 24\cosh \left( {1/2} \right)\left[ {\cosh \left( {1/2} \right) - 2\sinh \left( {1/2} \right)} \right]$$
$$\alpha_{3} = \left( {\cosh \left( {1/2} \right)} \right)^{2} + \left( {1/2} \right)\left[ {\sinh \left( 1 \right) + 1} \right] - 4\cosh \left( {1/2} \right)\sinh \left( {1/2} \right)$$

In order to capture the equivalent bending moment at the free end of the strut causing the same rotation, one will have

$$\begin{aligned} M\ell /\left( {\bar{E}\bar{I}} \right) & = P\ell^{2} /\left( {2\bar{E}\bar{I}} \right) + \left( {6P/\left( {5\bar{G}\bar{A}} \right)} \right)\left( {1 + \sinh \left( {\varGamma \ell } \right) - \cosh \left( {\varGamma \ell } \right)} \right) \\ & \to M = P\ell /2 + \left( {6P\bar{E}\bar{I}/\left( {5\ell \bar{G}\bar{A}} \right)} \right)\left( {1 + \sinh \left( {\varGamma \ell } \right) - \cosh \left( {\varGamma \ell } \right)} \right) \\ \end{aligned}$$
(4)

As a result, the lateral deflection due to the applied concentrated load \(P\) and bending moment \(M\) at the free end can be written as

$$\begin{aligned} \delta & = \delta_{P} + \delta_{M} = \frac{{P\ell^{3} }}{{\left( {3\bar{E}\bar{I}} \right)}} + \left( {\frac{6P\ell }{{\left( {5\bar{G}\bar{A}} \right)}}} \right)\left[ {1 + \frac{{\left( {\cosh \left( {\varGamma \ell } \right) - \sinh \left( {\varGamma \ell } \right) - 1} \right)}}{\varGamma \ell }} \right] \\ & \quad \quad \quad \quad \quad \quad - \left[ {\frac{\ell }{2} + \left( {\frac{{6P\bar{E}\bar{I}}}{{\left( {5\ell \bar{G}\bar{A}} \right)}}} \right)\left( {1 + \sinh \left( {\varGamma \ell } \right) - \cosh \left( {\varGamma \ell } \right)} \right)} \right]\left( {\frac{{\ell^{2} }}{{\left( {2\bar{E}\bar{I}} \right)}}} \right) \\ & = P\ell^{3} /\left( {12\bar{E}\bar{I}} \right) + 3P\ell /\left( {5\bar{G}\bar{A}} \right) + \left( {6P\ell /\left( {5\bar{G}\bar{A}} \right)} \right) \\ & \quad \quad \left[ {\left( {\left( {1 + \varGamma \ell /2} \right)\cosh \left( {\varGamma \ell } \right) - \left( {1 + \varGamma \ell /2} \right)\sinh \left( {\varGamma \ell } \right) - 1} \right)/\varGamma \ell } \right] \\ \end{aligned}$$
(5)

As a result, it yields

$$\begin{aligned} P & = \delta /\bigg[\frac{{\ell^{3} }}{{(12\bar{E}\bar{I})}} + \frac{3\ell }{{(5\bar{G}\bar{A})}} + (6\ell /(5\bar{G}\bar{A})) \hfill \\ &\quad \quad \quad [((1 + \varGamma \ell /2)\cosh (\varGamma \ell ) - (1 + \varGamma \ell /2)\sinh (\varGamma \ell ) - 1)/\varGamma ]\bigg] \hfill \\ \end{aligned}$$
(6)

It should be noticed that due to the in-plane deformation, the link \(c_{1} a_{1} b_{1} d_{1} a_{2} c_{2}\) has 18° of freedom. However, by considering the following reasonable assumptions considered by Hedayati et al. (2017b), the number of degrees of freedom can be reduced to 6 as depicted in Fig. 3:

Fig. 3
figure 3

Degrees of freedom for the link \(c_{1} a_{1} b_{1} d_{1} a_{2} c_{2}\) of the unit cell

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  • The vertices of link do not enable to rotate.

  • The points \(a_{1} \cdot a_{2} \cdot c_{1}\) enable only to displace vertically.

  • The points \(b_{1}\) and \(d_{1}\) displace the same vertically, but different horizontally.

  • The point \(c_{2}\) is fixed.

Now, by introducing the degrees of freedom \(\eta_{i} (i = 1,2,\, \ldots \,,6)\) as well as the associated external force \(\lambda_{i} (i = 1,2,\, \ldots \,,6)\), one will obtain

$$\left\{ {\begin{array}{*{20}c} {\lambda_{1} } \\ {\lambda_{2} } \\ {\lambda_{3} } \\ {\lambda_{4} } \\ {\lambda_{5} } \\ {\lambda_{6} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {S_{11} } & {S_{12} } & {S_{13} } & {S_{14} } & {S_{15} } & {S_{16} } \\ {S_{21} } & {S_{22} } & {S_{23} } & {S_{24} } & {S_{25} } & {S_{26} } \\ {S_{31} } & {S_{32} } & {S_{33} } & {S_{34} } & {S_{35} } & {S_{36} } \\ {S_{41} } & {S_{42} } & {S_{43} } & {S_{44} } & {S_{45} } & {S_{46} } \\ {S_{51} } & {S_{52} } & {S_{53} } & {S_{54} } & {S_{55} } & {S_{56} } \\ {S_{61} } & {S_{62} } & {S_{63} } & {S_{64} } & {S_{65} } & {S_{66} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\eta_{1} } \\ {\eta_{2} } \\ {\eta_{3} } \\ {\eta_{4} } \\ {\eta_{5} } \\ {\eta_{6} } \\ \end{array} } \right\}$$
(7)

To extract the elements of the stiffness matrix column by column, the displacements corresponding to each degree of freedom are achieved separately in such a way that the related degree of freedom is supposed to be unit and the other ones are zero.

For \(\varvec{\eta}_{1} = 1\) and \(\varvec{\eta}_{2} =\varvec{\eta}_{3} =\varvec{\eta}_{4} =\varvec{\eta}_{5} =\varvec{\eta}_{6} = 0\):

It means that the point \(c_{1}\) displaces downwards by unity. Consequently, it causes the following associated forces in the struts

$$\lambda_{1} = 2\bar{A}\bar{E}/\ell \; \cdot \; \lambda_{2} = - 2\bar{A}\bar{E}/\ell \; \cdot \;\lambda_{3} = \lambda_{4} = \lambda_{5} = \lambda_{6} = 0$$
(8)

For \(\varvec{\eta}_{2} = 1\) and \(\varvec{\eta}_{1} =\varvec{\eta}_{3} =\varvec{\eta}_{4} =\varvec{\eta}_{5} =\varvec{\eta}_{6} = 0\):

It means that the point \(a_{1}\) (the vertices of links \(a_{1} b_{1} \cdot a_{1} b_{2} \cdot a_{1} b_{3} \cdot a_{1} b_{4}\)) displaces downwards by unity. As a result, it leads to the following associated forces in the struts

$$\lambda_{1} = - 2\bar{A}\bar{E}/\ell \;\; \cdot \;\;\lambda_{4} = \lambda_{6} = 0$$
$$\begin{aligned} \lambda_{2} & = 4(1/[\ell^{3} /(6\bar{E}\bar{I}) + 6\ell /(5\bar{G}\bar{A}) + (12/(5\bar{G}\bar{A}))((1 + \varGamma \ell /2)\cosh (\varGamma \ell ) \\ & \quad \quad \quad - (1 + \varGamma \ell /2)\sinh (\varGamma \ell ) - 1)/\varGamma ] + 2\bar{A}\bar{E}/\ell ) + 2\bar{A}\bar{E}/\ell \\ \end{aligned}$$
$$\begin{aligned} \lambda_{3} & = - 4(1/[\ell^{3} /(6\bar{E}\bar{I}) + 6\ell /(5\bar{G}\bar{A}) + (12/(5\bar{G}\bar{A})) \\ & \quad \quad \quad [((1 + \varGamma \ell /2)\cosh (\varGamma \ell ) - (1 + \varGamma \ell /2)\sinh (\varGamma \ell ) - 1)/\varGamma ]] + 2\bar{A}\bar{E}/\ell ) \\ \end{aligned}$$
(9)
$$\begin{aligned} \lambda_{5} & = 4(1/[\ell^{3} /(6\bar{E}\bar{I}) + 6\ell /(5\bar{G}\bar{A}) + (12/(5\bar{G}\bar{A})) \\ & \quad \quad \quad [((1 + \varGamma \ell /2)\cosh (\varGamma \ell ) - (1 + \varGamma \ell /2)\sinh (\varGamma \ell ) - 1)/\varGamma ]] - \bar{A}\bar{E}/(2\ell )) \\ \end{aligned}$$

For \(\varvec{\eta}_{3} = 1\) and \(\varvec{\eta}_{1} =\varvec{\eta}_{2} =\varvec{\eta}_{4} =\varvec{\eta}_{5} =\varvec{\eta}_{6} = 0\):

It means that the point \(b_{1}\) (similarly, the points \(b_{2} , b_{3} , b_{4}\)) displaces downwards by unity. Therefore, the correspondence forces in the struts become

$$\lambda_{1} = \lambda_{5} = \lambda_{6} = 0$$
$$\begin{aligned} \lambda_{2} & = - 4(1/[\ell^{3} /(6\bar{E}\bar{I}) + 6\ell /(5\bar{G}\bar{A}) + (12/(5\bar{G}\bar{A})) \\ & \quad \quad \quad [((1 + \varGamma \ell /2)\cosh (\varGamma \ell ) - (1 + \varGamma \ell /2)\sinh (\varGamma \ell ) - 1)/\varGamma ]] - \bar{A}\bar{E}/(2\ell )) \\ \end{aligned}$$
(10)
$$\begin{aligned} \lambda_{3} & = 4(1/[\ell^{3} /(12\bar{E}\bar{I}) + 3\ell /(5\bar{G}\bar{A}) + (6/(5\bar{G}\bar{A})) \\ & \quad \quad \quad [((1 + \varGamma \ell /2)\cosh (\varGamma \ell ) - (1 + \varGamma \ell /2)\sinh (\varGamma \ell ) - 1)/\varGamma ]] + \bar{A}\bar{E}/\ell ) \\ \end{aligned}$$
$$\begin{aligned} \lambda_{4} & = - 4(1/[\ell^{3} /(6\bar{E}\bar{I}) + 6\ell /(5\bar{G}\bar{A}) + (12/(5\bar{G}\bar{A})) \\ & \quad \quad \quad [((1 + \varGamma \ell /2)\cosh (\varGamma \ell ) - (1 + \varGamma \ell /2)\sinh (\varGamma \ell ) - 1)/\varGamma ]] - \bar{A}\bar{E}/(2\ell )) \\ \end{aligned}$$

For \(\varvec{\eta}_{4} = 1\) and \(\varvec{\eta}_{1} =\varvec{\eta}_{2} =\varvec{\eta}_{3} =\varvec{\eta}_{5} =\varvec{\eta}_{6} = 0\):

It means that the point \(a_{2}\) (the vertices of links \(a_{2} b_{1} , a_{2} b_{2} , a_{2} b_{3} , a_{2} b_{4}\)) displaces downwards by unity. Consequently, the associated forces in the struts can be written as

$$\lambda_{1} = \lambda_{2} = \lambda_{6} = 0$$
$$\begin{aligned} \lambda_{3} & = - 4(1/[\ell^{3} /(6\bar{E}\bar{I}) + 6\ell /(5\bar{G}\bar{A}) + (12/(5\bar{G}\bar{A})) \\ & \quad \quad \quad [((1 + \varGamma \ell /2)\cosh (\varGamma \ell ) - (1 + \varGamma \ell /2)\sinh (\varGamma \ell ) - 1)/\varGamma ]] + \bar{A}\bar{E}/(2\ell )) \\ \end{aligned}$$
(11)
$$\begin{aligned} \lambda_{4} & = 4(1/[\ell^{3} /(6\bar{E}\bar{I}) + 6\ell /(5\bar{G}\bar{A}) + (12/(5\bar{G}\bar{A})) \\ & \quad \quad \quad [((1 + \varGamma \ell /2)\cosh (\varGamma \ell ) - (1 + \varGamma \ell /2)\sinh (\varGamma \ell ) - 1)/\varGamma ]] + \bar{A}\bar{E}/(2\ell )) + 2\bar{A}\bar{E}/\ell \\ \end{aligned}$$
$$\begin{aligned} \lambda_{5} & = - 4(1/[\ell^{3} /(6\bar{E}\bar{I}) + 6\ell /(5\bar{G}\bar{A}) + \left( {12/(5\bar{G}\bar{A})} \right) \\ & \quad \quad \quad [((1 + \varGamma \ell /2)\cosh (\varGamma \ell ) - (1 + \varGamma \ell /2)\sinh (\varGamma \ell ) - 1)/\varGamma ]] - \bar{A}\bar{E}/(2\ell )) \\ \end{aligned}$$

For \(\varvec{\eta}_{5} = 1\) and \(\varvec{\eta}_{1} =\varvec{\eta}_{2} =\varvec{\eta}_{3} =\varvec{\eta}_{4} =\varvec{\eta}_{6} = 0\):

It means that the point \(b_{1}\) (similarly, the points \(b_{2} , b_{3} , b_{4}\)) displaces horizontally by unity. As a result, the associated forces in the struts are derived as

$$\lambda_{1} = \lambda_{3} = 0$$
$$\begin{aligned} \lambda_{2} & = 4(1/[\ell^{3} /(6\bar{E}\bar{I}) + 6\ell /(5\bar{G}\bar{A}) + (12/(5\bar{G}\bar{A})) \\ & \quad \quad \quad [((1 + \varGamma \ell /2)\cosh (\varGamma \ell ) - (1 + \varGamma \ell /2)\sinh (\varGamma \ell ) - 1)/\varGamma ]] - \bar{A}\bar{E}/(2\ell )) \\ \end{aligned}$$
$$\begin{aligned} \lambda_{4} & = - 4(1/[\ell^{3} /(6\bar{E}\bar{I}) + 6\ell /(5\bar{G}\bar{A}) + (12/(5\bar{G}\bar{A})) \\ & \quad \quad \quad [((1 + \varGamma \ell /2)\cosh (\varGamma \ell ) - (1 + \varGamma \ell /2)\sinh (\varGamma \ell ) - 1)/\varGamma ]] - \bar{A}\bar{E}/(2\ell )) \\ \end{aligned}$$
(12)
$$\begin{aligned} \lambda_{5} & = 4(1/[\ell^{3} /(12\bar{E}\bar{I}) + 3\ell /(5\bar{G}\bar{A}) + (6/(5\bar{G}\bar{A})) \\ & \quad \quad \quad [((1 + \varGamma \ell /2)\cosh (\varGamma \ell ) - (1 + \varGamma \ell /2)\sinh (\varGamma \ell ) - 1)/\varGamma ]] + 5\bar{A}\bar{E}/\ell ) \\ \end{aligned}$$
$$\lambda_{6} = - 8\bar{A}\bar{E}/\ell$$

For \(\varvec{\eta}_{6} = 1\) and \(\varvec{\eta}_{1} =\varvec{\eta}_{2} =\varvec{\eta}_{3} =\varvec{\eta}_{4} =\varvec{\eta}_{5} = 0\):

It means that the point \(d_{1}\) (similarly, the points \(d_{2} \cdot d_{3} \cdot d_{4}\)) displaces horizontally by unity. Thereby, the correspondence forces in the struts can be expressed as:

$$\begin{aligned} \lambda_{1} & = \lambda_{2} = \lambda_{3} = \lambda_{4} = 0 \\ \lambda_{5} & = - 8\bar{A}\bar{E}/\ell \\ \lambda_{6} & = 8\bar{A}\bar{E}/\ell \\ \end{aligned}$$
(13)

Thereafter, the elements of the stiffness matrix can be achieved as presented in Appendix A.

Similar to the assumption considered by Hedayati et al. (2017a, b), It is supposed that the external force acts vertically on point \(c_{1}\) of the refined truncated cube lattice structure, which results in an additional horizontal force equal to \(8\bar{A}\bar{E}\left( {\eta_{6} - \eta_{5} } \right)/\ell\) at point \(d_{1}\).

As a result, one will have

$$\left\{ {\begin{array}{*{20}c} P \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {S_{11} } & {S_{12} } & 0 & 0 & 0 & 0 \\ {S_{21} } & {S_{22} } & {S_{23} } & 0 & {S_{25} } & 0 \\ 0 & {S_{32} } & {S_{33} } & {S_{34} } & 0 & 0 \\ 0 & 0 & {S_{43} } & {S_{44} } & {S_{45} } & 0 \\ 0 & {S_{52} } & 0 & {S_{54} } & {S_{55} } & {S_{56} } \\ 0 & 0 & 0 & 0 & {2S_{65} } & {2S_{66} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\eta_{1} } \\ {\eta_{2} } \\ {\eta_{3} } \\ {\eta_{4} } \\ {\eta_{5} } \\ {\eta_{6} } \\ \end{array} } \right\}$$
(14)

The elastic modulus of the refined truncated cube unit cell can be calculated as:

$$E = F_{u} L_{u} /\left( {A_{u} \delta_{u} } \right) = P/\left[ {\left( {1 + \sqrt 2 } \right)\ell \eta_{1} } \right]$$
(15)

where \(F_{u} \cdot L_{u} \cdot A_{u} \cdot \delta_{u}\) represent, respectively, the applied load, length, cross-sectional area, and shortening of the unit cell.

Through inversion of the above equation, \(\eta_{1}\) can be extracted as a function of \(P\). Therefore, it yields

$$\begin{aligned} E & = (S_{11} S_{22} S_{33} S_{66} S_{45} S_{54} - S_{11} S_{22} S_{33} S_{44} S_{55} S_{66} + S_{11} S_{22} S_{55} S_{66} S_{34} S_{43} + S_{11} S_{33} S_{44} S_{66} S_{25} S_{52} \\ {\kern 1pt} & \quad \quad - S_{11} S_{66} S_{25} S_{52} S_{34} S_{43} + 2S_{11} S_{66} S_{34} S_{45} S_{23} S_{25} + S_{11} S_{44} S_{55} S_{66} S_{23} S_{32} - S_{11} S_{66} S_{45} S_{54} S_{23} S_{32} \\ & \quad \quad + S_{33} S_{44} S_{55} S_{66} S_{12} S_{21} - S_{33} S_{66} S_{45} S_{54} S_{12} S_{21} - S_{55} S_{66} S_{12} S_{21} S_{34} S_{43} + S_{11} S_{22} S_{33} S_{44} S_{56} S_{65} \\ & \quad \quad - S_{33} S_{44} S_{12} S_{21} S_{56} S_{65} - S_{11} S_{44} S_{23} S_{32} S_{56} S_{65} - S_{11} S_{22} S_{34} S_{43} S_{56} S_{65} + S_{12} S_{21} S_{34} S_{43} S_{56} S_{65} ) \\ & \quad \quad /[(1 + \sqrt 2 l(S_{66} S_{22} S_{33} S_{45} S_{54} - S_{66} S_{22} S_{33} S_{55} S_{44} ) + S_{22} S_{55} S_{66} S_{34} S_{43} + S_{33} S_{44} S_{66} S_{25} S_{52} \\ & \quad \quad - S_{66} S_{25} S_{52} S_{34} S_{43} + 2S_{66} S_{45} S_{34} S_{23} S_{25} + S_{44} S_{55} S_{66} S_{23} S_{32} - S_{66} S_{45} S_{54} S_{23} S_{32} \\ {\kern 1pt} & \quad \quad + S_{22} S_{33} S_{44} S_{56} S_{65} - S_{44} S_{23} S_{32} S_{56} S_{65} - S_{22} S_{34} S_{43} S_{56} S_{65} )] \\ \end{aligned}$$
(16)

Moreover, to obtain the Poisson’s ratio, it can be introduced as the ratio of horizontal to vertical displacements in the following form

$$\nu = \frac{{2\eta_{6} }}{{\eta_{1} }}$$
(17)

Consequently, one will have

$$\begin{aligned} \nu & = 2S_{12} S_{56} \left( {S_{33} S_{44} S_{25} - S_{25} S_{34} S_{43} + S_{23} S_{34} S_{45} } \right) \\ & \quad /(S_{22} S_{33} S_{66} S_{45} S_{54} - S_{22} S_{33} S_{44} S_{55} S_{66} + S_{22} S_{55} S_{66} S_{34} S_{43} \\ & \quad \quad + S_{33} S_{44} S_{66} S_{25} S_{52} - S_{66} S_{25} S_{52} S_{34} S_{43} + 2S_{66} S_{23} S_{25} S_{34} S_{45} \\ & \quad \quad + S_{44} S_{55} S_{66} S_{23} S_{32} - S_{66} S_{23} S_{32} S_{45} S_{54} + S_{22} S_{33} S_{44} S_{56} S_{65} \\ & \quad \quad - S_{44} S_{23} S_{32} S_{56} S_{65} + S_{22} S_{34} S_{43} S_{56} S_{65} ) \\ \end{aligned}$$
(18)

Furthermore, the mass density of the nanoporous biomaterial can be achieved as a function of pore size (\(\ell /r\)) as below

$$\rho = V^{ * } \bar{\rho } = \frac{15\pi }{{\left( {1 + \sqrt 2 } \right)^{3} }}\left( {\frac{r}{\ell }} \right)^{2} \bar{\rho }$$
(19)

in which \(\bar{\rho }\) is the mass density of the material without porosity, and \(V^{*}\) denotes the occupied volume of the complete cube to the occupied volume of the truncated cube ratio.

3 Nonlocal Strain Gradient Hyperbolic Shear Deformable Beam Model

Within the framework of the refined hyperbolic shear deformation beam theory, the components of the displacement field along different coordinate directions can be given as

$$\begin{aligned} u_{x} \left( {x \cdot z \cdot t} \right) & = u - zw_{,x} + \left[ {z\cosh \left( {1/2} \right) - h\sinh \left( {z/h} \right)} \right]\psi \\ u_{y} \left( {x \cdot z \cdot t} \right) & = 0 \\ u_{z} \left( {x \cdot z \cdot t} \right) & = w \\ \end{aligned}$$
(20)

in which \(u\) and \(w\) represent the displacement components of the biological microbeam along x- and z-axis, respectively. Moreover, \(\psi\) denotes the rotation relevant to the cross section of the microbeam at neutral plane normal about y-axis.

Thereafter, the non-zero strain components can be governed as

$$\begin{aligned} \varepsilon_{xx} & = \varepsilon_{xx}^{0} + z\kappa_{xx}^{(1)} + \left[ {z\cosh \left( {1/2} \right) - h\sinh \left( {z/h} \right)} \right]\kappa_{xx}^{(2)} \\ & \, = u_{x} + \left( {1/2} \right)\left( {w_{x} } \right)^{2} - zw_{xx} + \left[ {z\cosh \left( {1/2} \right) - h\sinh \left( {z/h} \right)} \right]\psi_{x} \\ \end{aligned}$$
(21a)
$$\gamma_{xz} = \left[ {\cosh \left( {1/2} \right) - \cosh \left( {z/h} \right)} \right]\psi$$
(21b)

in which \(\varepsilon_{xx}^{0}\) denote the mid-plane strain components, \(\kappa_{xx}^{(1)}\) is the first-order curvature component, and \(\kappa_{xx}^{(2)}\) is the higher-order curvature component.

As it has been reported in the specialized literature on the subject of size dependency, it has been indicated that small-scale effects may cause two entirely different influences incorporating hardening-stiffness or stiffening-stiffness features. Motivated by this fact, Lim et al. (2015) proposed a new unconventional continuum theory namely as nonlocal strain gradient elasticity theory which contains the both nonlocal and strain gradient size effects simultaneously. As a result, the total nonlocal strain gradient stress tensor \(\varLambda\) for a beam-type structure can be expressed as below (Lim et al. 2015).

$$\varLambda_{xx} = \sigma_{xx} - \sigma_{xx\; \cdot \;x}^{*}$$
(22a)
$$\varLambda_{xz} = \sigma_{xz} - \sigma_{xz\; \cdot \;x}^{*}$$
(22b)

where \(\sigma\) and \(\sigma^{*}\) are the stress and higher-order stress tensors, respectively, which can be defined as

$$\sigma_{ij} = \int\nolimits_{\varOmega } {\left\{ {\varrho_{1} \left( {\left| {{\mathcal{X}}^{'} - {\mathcal{X}}} \right|} \right)C_{ijkl} \varepsilon_{kl} \left( {{\mathcal{X}}^{'} } \right)} \right\}d\varOmega }$$
(23a)
$$\sigma_{ij}^{*} = l^{2} \int_{\varOmega } {\left\{ {\varrho_{2} \left( {\left| {{\mathcal{X}}^{'} - {\mathcal{X}}} \right|} \right)C_{ijkl} \varepsilon_{kl.x} \left( {{\mathcal{X}}^{'} } \right)} \right\}d\varOmega }$$
(23b)

in which \(C\) is the stiffness matrix, \(\varrho_{1}\) and \(\varrho_{2}\) are, respectively, the principal attenuation kernel function including the nonlocality and the additional kernel function associated with the nonlocality effect of the first-order strain gradient field, \({\mathcal{X}}\) and \({\mathcal{X}}^{'}\) in order represent a point and any point else in the body, and \(l\) stands for the internal strain gradient length scale parameter. Following the method of Eringen, the constitutive relationship corresponding to the total nonlocal strain gradient stress tensor of a beam-type structure can be obtained as

$$\varLambda_{ij} - \mu^{2} \varLambda_{ij,xx} = C_{ijkl} \varepsilon_{kl} - l^{2} C_{ijkl} \varepsilon_{kl\; \cdot \;xx}$$
(24)

where \(\mu\) is the nonlocal parameter. As a result, the nonlocal strain gradient constitutive relations for a hyperbolic shear deformable biological microbeam can be written as

$$\left\{ {\begin{array}{*{20}c} {\sigma_{xx} - \mu^{2} \sigma_{xx\; \cdot \;xx} } \\ {\sigma_{xz} - \mu^{2} \sigma_{xz\; \cdot \;xx} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {Q_{11} } & 0 \\ 0 & {Q_{44} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{xx} - l^{2} \varepsilon_{xx\; \cdot \;xx} } \\ {\gamma_{xz} - l^{2} \gamma_{xz\; \cdot \;xx} } \\ \end{array} } \right\}$$
(25)

in which

$$Q_{11} = \frac{E}{{1 - \nu^{2} }}\,\,.\,\,{\kern 1pt} Q_{44} = \frac{E}{{2\left( {1 + \nu } \right)}}$$
(26)

Therefore, based upon the nonlocal strain gradient hyperbolic shear deformable beam model, the total strain energy of the biological nanoporous microbeam can be expressed as

$$\begin{aligned} \varPi_{s} & = \frac{1}{2}\int_{0}^{L} {\int_{S} {\left( {\sigma_{ij} \varepsilon_{ij} + \sigma_{ij}^{*} \nabla \varepsilon_{ij} } \right)dSdx} } \\ & = \frac{1}{2}\int_{0}^{L} {\left\{ {N_{xx} \varepsilon_{xx}^{0} + M_{xx} \kappa_{xx}^{(1)} + R_{xx} \kappa_{xx}^{(2)} + Q_{x} \gamma_{xz} } \right\}dx} \\ \end{aligned}$$
(27)

where \(S\) is the cross sectional area of the micro/nano-beam, and the stress resultants are in the following forms

$$\begin{aligned} & N_{xx} - \mu^{2} N_{xx\; \cdot \;xx} = A_{11}^{*} \left( {\varepsilon_{xx}^{0} - l^{2} \varepsilon_{xx\; \cdot \;xx}^{0} } \right) \\ & M_{xx} - \mu^{2} M_{xx\; \cdot \;xx} = D_{11}^{*} \left( {\kappa_{xx}^{(1)} - l^{2} \kappa_{xx\; \cdot \;xx}^{(1)} } \right) + F_{11}^{*} \left( {\kappa_{xx}^{(2)} - l^{2} \kappa_{xx\; \cdot \;xx}^{(2)} } \right) \\ & R_{xx} - \mu^{2} R_{xx\; \cdot \;xx} = F_{11}^{*} \left( {\kappa_{xx}^{(1)} - l^{2} \kappa_{xx\; \cdot \;xx}^{(1)} } \right) + H_{11}^{*} \left( {\kappa_{xx}^{(2)} - l^{2} \kappa_{xx\; \cdot \;xx}^{(2)} } \right) \\ & Q_{x} - \mu^{2} Q_{x\; \cdot \;xx} = A_{44}^{*} \left( {\gamma_{xz} - l^{2} \gamma_{xz\; \cdot \;xx} } \right) \\ \end{aligned}$$
(28)

in which

$$\begin{aligned} & \left\{ {N_{xx} \cdot M_{xx} \cdot R_{xx} } \right\} = b\int_{ - h/2}^{h/2} {\varLambda_{xx}^{\left( k \right)} \left\{ {1 \cdot z \cdot z\cosh \left( {1/2} \right) - h\sinh \left( {z/h} \right)} \right\}dz} \\ & Q_{x} = b\int_{ - h/2}^{h/2} {\varLambda_{xz}^{(k)} \left\{ {\cosh \left( {1/2} \right) - \cosh \left( {z/h} \right)} \right\}dz} \\ \end{aligned}$$
(29)

and

$$\begin{aligned} & \left\{ {A_{11}^{*} \cdot F_{11}^{*} } \right\} = bQ_{11} \int_{ - h/2}^{h/2} {\left\{ {1 \cdot z^{2} \cosh \left( {1/2} \right) - zh\sinh \left( {z/h} \right)} \right\}dz} \\ & \left\{ {D_{11}^{*} \cdot H_{11}^{*} } \right\} = bQ_{11} \int_{ - h/2}^{h/2} {\left\{ {z^{2} \cdot \left( {z\cosh \left( {1/2} \right) - h\sinh \left( {z/h} \right)} \right)^{2} } \right\}dz} \\ & \left\{ {A_{44}^{*} } \right\} = bQ_{44} \,\int_{ - h/2}^{h/2} {\left\{ {\cosh \left( {1/2} \right) - \cosh \left( {z/h} \right)} \right\}dz} \\ \end{aligned}$$
(30)

Furthermore, the kinetic energy of a micro/nano-beam modelled via the nonlocal strain gradient hyperbolic shear deformable beam model can be presented as

$$\begin{aligned} \varPi_{T} & = \frac{1}{2}\int_{0}^{L} {\int_{S} {\rho \left\{ {\left( {u_{x \cdot t} } \right)^{2} + \left( {u_{z \cdot t} } \right)^{2} } \right\}dSdx} } \\ & = \frac{1}{2}\int_{0}^{L} {\left\{ {I_{0} \left( {u_{t} } \right)^{2} + I_{2} \left( {w_{xt} } \right)^{2} + I_{3} w_{xt} \psi_{t} + I_{4} \left( {\psi_{t} } \right)^{2} + I_{0} \left( {w_{x} } \right)^{2} } \right\}dx} \\ \end{aligned}$$
(31)

where

$$\left\{ {I_{0} \cdot I_{3} } \right\} = b\rho \int_{ - h/2}^{h/2} {\left\{ {1 \cdot z^{2} \cosh \left( {1/2} \right) - zh\sinh \left( {z/h} \right)} \right\}dz}$$
(32)
$$\left\{ {I_{2} \cdot I_{4} } \right\} = b\rho \int_{ - h/2}^{h/2} {\left\{ {z^{2} \cdot \left( {z\cosh \left( {1/2} \right) - h\sinh \left( {z/h} \right)} \right)^{2} } \right\}dz}$$

Also, the work done by the transverse force can be introduced as below

Thereby, by using the Hamilton’s principle, the governing differential equations in terms of stress resultants can be constructed as

Afterwards, by substituting Eq. (34a), (b) and (c), and using Eq. (30), the nonlocal strain gradient governing differential equations for a hyperbolic shear deformable biological microbeam with immovable boundary conditions can be derived as

Analytical Solving Process for Asymptotic Solutions

First of all, for extracting the asymptotic solutions associated with the present size-dependent problem, the following dimensionless parameters are introduced

where \(A_{00} = Ebh\) and \(I_{00} = \rho bh\). Thus the nonlocal strain gradient governing differential equations of motion for the refined hyperbolic shear deformable biological microbeam can be rewritten in the following dimensionless form

Now, an improved perturbation method namely as two-stepped perturbation technique (Shen and Yang 2015; Shen and Wang 2015; Shen et al. 2016; Sahmani and Aghdam 2017i, j, k, 2018d; Shen and Xiang 2018; Sahmani et al. 2018d) put to use. To continue the solving process, the independent variables are defined as the summations of the solutions corresponding to different orders of the first perturbation parameter, \(\epsilon\), as follows

$$\bar{W}\left( {X \cdot \hat{\tau } \cdot \epsilon } \right) = \mathop \sum \limits_{i = 1} \epsilon^{i} \bar{W}_{i} (X \cdot \hat{\tau }) \;\; \cdot \;\;\bar{\varPsi }\left( {X \cdot \hat{\tau } \cdot \epsilon } \right) = \mathop \sum \limits_{i = 1} \epsilon^{i} \bar{\varPsi }_{i} (X \cdot \hat{\tau })$$
(38)

in which \(\hat{\tau } = \epsilon \tau\) is considered to improve the efficiency of the perturbation approach for capturing the solution of vibration problem. In such a case, the nonlocal strain gradient governing differential equations of motion take the following form

It is assumed that the immovable ends of the biological microbeam are simply supported and the initial conditions are as follow

$$\left. {\bar{W}} \right|_{{\left. {\hat{\tau } = 0} \right|}} = 0\;\; \cdot \;\;\left. {\bar{W}_{{\hat{\tau }}} } \right|_{{\hat{\tau } = 0}} = 0\;\; \cdot \;\;\left. {\bar{\varPsi }} \right|_{{\hat{\tau } = 0}} = 0\;\; \cdot \;\;\left. {\bar{\varPsi }_{{\hat{\tau }}} } \right|_{{\hat{\tau } = 0}} = 0$$
(40)

Substitution of Eq. (38) into Eq. (39a) and (b) and then collecting the expressions with the same order of \(\epsilon\) result in a set of perturbation equations. Subsequently, the asymptotic solutions corresponding to each individual variable can be obtained a

$$\bar{W}\left( {X \cdot \tau \cdot \epsilon } \right) = \epsilon A_{10}^{(1)} \left( \tau \right)\sin \left( {mX} \right) + O\left( { \epsilon^{4} } \right)$$
(41a)
$$\bar{\varPsi }\left( {X \cdot \tau \cdot \epsilon } \right)\,\, = \,\, \epsilon B_{10}^{(1)} \left( \tau \right)\sin \left( {mX} \right) + \epsilon^{3} B_{10}^{(3)} \cos \left( {mX} \right) + O\left( { \epsilon^{4} } \right)$$
(41b)

where

$$\xi_{1} = 1 + \pi^{2} m^{2} {\mathcal{G}}_{1}^{2} \;\; \cdot \;\;\xi_{2} = 1 + \pi^{2} m^{2} {\mathcal{G}}_{2}^{2}$$
(42)

For a free vibration analysis, one will have . As a consequence, after applying the Galerkin method, it yields

$$\begin{aligned} & [((m^{4} \xi_{2} /\xi_{1} )*(d_{11}^{*} + f_{11}^{*} (f_{11}^{*} m^{2} + a_{44}^{*} )/(a_{44}^{*} - h_{11}^{*} m^{2} )))( \epsilon A_{10}^{(1)} (\tau )) \\ & \quad \quad \quad + (\bar{I}_{0} + m^{2} (\bar{I}_{2} - \bar{I}_{3} + ((\bar{I}_{3} - \bar{I}_{4} )\xi_{2} )(f_{11}^{*} m^{2} + a_{44}^{*} )/(a_{44}^{*} \xi_{1} - h_{11}^{*} m^{2} \xi_{1} )) \\ & \quad \quad \quad - (f_{11}^{*} m^{4} \xi_{2} /(a_{44}^{*} \xi_{1} - h_{11}^{*} m^{2} \xi_{1} ))(\bar{I}_{4} - \bar{I}_{3} (f_{11}^{*} m^{2} + a_{44}^{*} )/(a_{44}^{*} - h_{11}^{*} m^{2} ))) \epsilon A_{10\; \cdot \;\tau \tau }^{(1)} (\tau )] \\ & \quad \quad \quad + (\pi^{2} m^{4} a_{11}^{*} /4)( \epsilon A_{10}^{\left( 1 \right)} (\tau ))^{3} = 0 \\ \end{aligned}$$
(43)

Thereby, the nonlinear nonlocal strain gradient frequency of the axially compressed biological nanoporous microbeam can be extracted explicitly as below

$$\omega_{NL} = \omega_{L} \sqrt {1 + \frac{{\left( {3\pi^{2} m^{4} a_{11}^{*} /4} \right)}}{{4\left[ {\left( {m^{4} \xi_{2} /\xi_{1} } \right)\left( {d_{11}^{*} + f_{11}^{*} \left( {f_{11}^{*} m^{2} + a_{44}^{*} } \right)/\left( {a_{44}^{*} - h_{11}^{*} m^{2} } \right)} \right) - m^{2} P} \right]}}W_{max}^{2} }$$
(44)

where the linear nonlocal strain gradient natural frequency can be defined as

$$\omega_{L} = \sqrt {\frac{{\left( {m^{4} \xi_{2} /\xi_{1} } \right)\left( {d_{11}^{*} + f_{11}^{*} \left( {f_{11}^{*} m^{2} + a_{44}^{*} } \right)/\left( {a_{44}^{*} - h_{11}^{*} m^{2} } \right)} \right) - m^{2} P}}{{\bar{I}_{0} + m^{2} \left( {\bar{I}_{2} - \bar{I}_{3} + \frac{{\left( {\bar{I}_{3} - \bar{I}_{4} } \right)\xi_{2} }}{{\xi_{1} }}\frac{{f_{11}^{*} m^{2} + a_{44}^{*} }}{{a_{44}^{*} - h_{11}^{*} m^{2} }}} \right) - \frac{{f_{11}^{*} m^{4} \xi_{2} }}{{\left( {a_{44}^{*} - h_{11}^{*} m^{2} } \right)\xi_{1} }}\left( {\bar{I}_{4} - \bar{I}_{3} \frac{{f_{11}^{*} m^{2} + a_{44}^{*} }}{{a_{44}^{*} - h_{11}^{*} m^{2} }}} \right)}}}.$$
(45)

4 Numerical Results and Discussion

Herein, selected numerical results are presented for the size-dependent nonlinear large-amplitude vibrations of micro/nano-beams made of nanoporous biomaterial including different pore sizes. It is assumed that the biomaterial is made from Ti6Al4 V-ELI Titanium alloy having an elastic modulus of \(\bar{E} = 122.3\;{\text{GPa}}\), and Poisson’s ratio of \(\bar{\nu } = 0.342\) (Ahmadi et al. 2014). Also, the geometrical parameter of the micro/nano-beam are selected as: \(h = b\). Additionally, the links of the refined truncated cubic cells has a circular cross section with a radius of \(r\).

In Fig. 4, the size-dependent nonlinear instability characteristics of an axially compressed biological nanoporous microbeam corresponding to different values of the nonlocal and strain gradient parameters. It can be seen that the nonlocality size effect causes to decrease the critical buckling load, while the strain gradient size dependency leads to enhance it. However, by moving to a deeper part of the postbuckling regime, it is revealed that the influence of both types of size effect tends to decrease smoothly.

Fig. 4
figure 4

Dimensionless postbuckling equilibrium paths of an axially compressed biological nanoporous microbeam corresponding to various small scale parameters (\(\ell /r = 10\)): a \(l = 0\;{\text{nm}}\), b \(\mu = 0\;{\text{nm}}\)

Full size image

Figure 5 shows the influence of the pore size (\(\ell /r\)) on the size-dependent nonlinear instability characteristics of an axially compressed biological nanoporous microbeam. It is found that by increasing the pore size, the critical buckling load of the biological microbeam reduces, but this reduction is more significant for the lower value of aspect ratio cube’s link (\(\ell /r\)).

Fig. 5
figure 5

Influence of the pore size on the nonlocal strain gradient postbuckling equilibrium paths of an axially compressed biological nanoporous microbeam (\(\mu = l = 80\;{\text{nm}}\))

Full size image

Figure 6 illustrates the size-dependent load-frequency response of an axially compressed biological nanoporous microbeam corresponding to different values of the nonlocal and strain gradient parameters. It is observed that within the prebuckling domain, the nonlocality size effect causes to decrease the natural frequency of the axially loaded biological microbeam, while the strain gradient size dependency leads to increase it. However, within the postbuckling regime, this pattern is vice versa, so the nonlocal small scale effect makes an enhancement in the natural frequency of the buckled biological nanoporous microbeam, but the strain gradient size effect makes a reduction.

Fig. 6
figure 6

Dimensionless load-frequency response of an axially compressed biological nanoporous microbeam corresponding to various small scale parameters (\(\ell /r = 10\)): a \(l = 0\;{\text{nm}}\), b \(\mu = 0\;{\text{nm}}\)

Full size image

In Fig. 7, the influence of the pore size (\(\ell /r\)) on the size-dependent load-frequency response of an axially compressed biological nanoporous microbeam. It is demonstrated that within the prebuckling regime, an increment in the value of pore size leads to decrease the natural frequency of the axially loaded biological nanoporous microbeam, especially for the lower value of aspect ratio cube’s link of the porosity unit cell (\(\ell /r\)). However, within the postbuckling domain, the pore size plays an opposite role, so it causes to enhance the natural frequency of the axially loaded biological nanoporous microbeam.

Fig. 7
figure 7

Influence of the pore size on the nonlocal strain gradient load-frequency response of an axially compressed biological nanoporous microbeam (\(\mu = l = 80\;{\text{nm}}\))

Full size image

Figure 8 depicts the size-dependent nonlinear frequency-deflection response of an axially compressed biological nanoporous microbeam corresponding to different values of the nonlocal and strain gradient parameters. It can be found that for a specific value of the maximum deflection, the nonlocality size effect makes a reduction in the value of the nonlinear frequency within the both prebuckling (\(P = P_{cr} /2\)) and postbuckling (\(P = 2P_{cr}\)) domains. However, by taking the strain gradient size dependency into consideration, it is seen that the nonlinear frequency of biological nanoporous microbeam associated with a specific value of the maximum deflection increases within the both prebuckling (\(P = P_{cr} /2\)) and postbuckling (\(P = 2P_{cr}\)) regimes. Also, it is revealed that these patterns are more significant for the higher value of the maximum deflection.

Fig. 8
figure 8

Dimensionless nonlinear frequency-deflection response of an axially compressed biological nanoporous microbeam corresponding to various small scale parameters (\(\ell /r = 10\)): a \(l = 0\;{\text{nm}}\), b \(\mu = 0\;{\text{nm}}\)

Full size image

Figure 9 represents the influence of the pore size (\(\ell /r\)) on the size-dependent nonlinear frequency-deflection response of an axially compressed biological nanoporous microbeam. It is demonstrated that within the both prebuckling (\(P = P_{cr} /2\)) and postbuckling (\(P = 2P_{cr}\)) regimes, an increment in the value of pore size leads to reduce the nonlinear frequency of the axially loaded biological nanoporous microbeam, especially for the lower value of aspect ratio cube’s link of the porosity unit cell (\(\ell /r\)). In addition, it is demonstrated that this pattern of the pore size influence on the nonlinear frequency of the prebuckled and postbuckled biological nanoporous microbeam is more considerable at higher maximum deflection.

Fig. 9
figure 9

Influence of the pore size on the nonlocal strain gradient nonlinear frequency-deflection response of an axially compressed biological nanoporous microbeam \((\mu = l = 80\;{\text{nm)}}\)

Full size image

5 Conclusion

The prime objective of the present investigation was to predict the size-dependent nonlinear instability as well as the nonlinear vibrations within the prebuckling and postbuckling domains of an axially compressed biological nanoporous microbeam. To accomplish this purpose, based upon a refined truncated cube lattice structure, the elastic mechanical properties of the nanoporous biomaterial were achieved explicitly in terms of the pore size. After that, via a developed nonlocal strain gradient beam model and using an improved perturbation technique, analytical solutions were proposed for the nonlinear mechanical characteristics.

It was displayed that the nonlocality size effect causes to decrease the critical buckling load, while the strain gradient size dependency leads to enhance it. However, by moving to the deeper part of the postbuckling regime, it is revealed that the influence of both types of size effect tends to decrease smoothly. Moreover, within the prebuckling domain, the nonlocality size effect causes to decrease the natural frequency of the axially loaded biological microbeam, while the strain gradient size dependency leads to increase it. However, within the postbuckling regime, this pattern is vice versa. It was seen that for a specific value of the maximum deflection, the nonlocality size effect makes a reduction in the value of the nonlinear frequency within both prebuckling and postbuckling domains. However, the strain gradient size dependency plays an opposite role.

Additionally, it was observed that by increasing the pore size, the critical buckling load of the biological microbeam reduces, but this reduction is more significant for the lower value of aspect ratio cube’s link (\(\ell /r\)). It was revealed that within the prebuckling regime, an increment in the value of pore size leads to decrease the natural frequency of the axially loaded biological nanoporous microbeam, especially for lothe wer value of aspect ratio cube’s link of the porosity unit cell. However, within the postbuckling domain, the pore size has an opposite influence. Also, it was found that within the both prebuckling and postbuckling regimes, an increment in the value of pore size leads to reduce the nonlinear frequency of the axially loaded biological nanoporous microbeam.