1 Introduction

Possessing surprising mechanical and chemical properties, hyper-elastic materials have gained potential applications in bio-engineering and even mechanical systems and structures. Hyper-elastic materials undergo large strains under various loading conditions. Hence, their mechanical characteristics cannot be described using a linear stress–strain relationship [1]. Therefore, hyper-elastic materials exhibit material nonlinearity and there is a need for nonlinear stress–strain relations to express their constitutive laws. Up to now, several models are proposed for modeling of hyper-elastic materials such as neo-Hookean, Mooney–Rivlin, Ishihara, Ogden and Yeoh models [2,3,4]. These models require material parameters to define the stress–strain relation. The simplest approach to obtain these material parameters is performing uniaxial or biaxial tension tests [5, 6]. Then, the constitutive hyper-elastic material models can be fitted to the test data based on specified values of material parameters.

The main difference between elastic and hyper-elastic materials is in the mechanism of deformation. Actually, structural elements constructed from hyper-elastic material show large strains when subjected to different loadings. In investigating the vibration behavior of hyper-elastic structures, it is important to consider large strain effects. Based on von-Kármán’s strain model, many investigations consider large deflections in the analysis of nonlinear structures [7,8,9,10,11,12]. In this model, only large deflection of the structures has been considered, and rotations are considered to be small [13,14,15,16,17]. This assumption is also correct for a wide range of materials when their behavior is near the elastic region.

Many investigations are published on nonlinear vibrations of hyper-elastic structures. Breslavsky et al. [18] studied the nonlinear vibration behavior of thin hyper-elastic plates made of rubber and soft biological tissues. They used neo-Hookean, Mooney–Rivlin, and Ogden hyper-elastic laws to examine the nonlinear vibration of the plate. Soares and Gonçalves [19] examined nonlinear vibrations of hyper-elastic plates resting on an elastic foundation with considering pre-stretch effect. Wang et al. [20] studied the vibration behavior of axially moving beams made of hyper-elastic material accounting for large deformations. Also, Barforooshi and Mohammadi [21, 22] used hyper-elastic material laws to examine the nonlinear vibrations of beam-type resonators. They used von-Kármán strain type in order to capture large deflections of the beam under oscillation.

Based on a higher-order shear deformation beam theory, the present research treats the nonlinear vibration analysis of hyper-elastic beams resting on an elastic foundation. Hyper-elastic beams have many applications, especially at bio-engineering and mechanical engineering fields. Various types of hyper-elastic material models have been introduced including neo-Hookean, Mooney–Rivlin, Ishihara, and Yeoh models. Large deflections of the beam have been considered based on von-Kármán strain model. The nonlinear vibration frequency of the beam is obtained by solving the tube governing equation based on the Hamiltonian approach. The effects of hyper-elastic material parameters, hyper-elastic material law, nonlinear elastic medium, length-to-thickness ratio, and boundary conditions on the vibration frequency of the hyper-elastic beam have been explored.

2 Different models for hyper-elastic materials

Based on the approach of developing strain energy function (W), there are three types of models for hyper-elastic materials [5, 23]:

  1. 1.

    Phenomenological models: They are based on the mathematical development of the strain energy function, for example, Rivlin series.

  2. 2.

    Models based on direct determination of material parameters using experiment.

  3. 3.

    Physically based models: They consider the physics of polymer chains’ network and statistical methods to develop the strain energy function.

In the following, some well-known models of hyper-elastic materials will be introduced and discussed. The strain energy function based on these models is based on strain invariants (\({I}_{1}\), \({I}_{2}\), \({I}_{3}\)) which are [23, 24]:

$$\begin{aligned} I_1= & {} \lambda _1^2 +\lambda _2^2 +\lambda _3^2 =tr(C), \nonumber \\ I_2= & {} \lambda _1^2 \lambda _2^2 +\lambda _1^2 \lambda _3^2 +\lambda _2^2 \lambda _3^2 =\frac{1}{2}\left( {I_1^2 -tr(C^{2})} \right) , \nonumber \\ I_3= & {} \lambda _1^2 \lambda _2^2 \lambda _3^2 =\det (C) \end{aligned}$$
(1)

where \({\lambda }_{{i}}\) (\({i}=1, 2, 3\)) denote the square roots of the right Cauchy–Green strain tensor (C), and physically they are stretch ratios. Also, based on incompressibility assumption of the materials, \({I}_{3}=1\). Also, the right Cauchy–Green strain tensor (C) is related to the strain tensor of the beam (E) as:

$$\begin{aligned} {[}C]=2[E]+[I]. \end{aligned}$$
(2)

2.1 Neo-Hookean model

The neo-Hookean model has only one material constant (\({c}_{10}\)) and is known as simplest model for hyper-elastic materials. It is a physically based model and introduces the strain energy function as follows [23]:

$$\begin{aligned} W=c_{10} (I_1 -3) \end{aligned}$$
(3)

where \({I}_{1}\) is the first strain invariant. For a rubber material, it is found that \({c}_{10}=0.2\) MPa. In the literature, it is reported that this model does not have adequate terms for mathematical modeling of beams or plates to obtain their governing equations. So, this model has been discarded in the present paper due to its limitations.

2.2 Mooney–Rivlin model

As reported by Mooney, the response of rubber subjected to simple shear is linear. Based on two material parameters (\({c}_{10}\) and \({c}_{01}\)), the strain energy function based on this model can be expressed as:

$$\begin{aligned} W=c_{10} (I_1 -3)+c_{01} (I_2 -3) \end{aligned}$$
(4)

where \({I}_{1}\) and \({I}_{2}\), respectively, denote the first and second strain invariants. This model is suitable for moderate deformations, lower than 200%, and is erroneous for larger deformations. Similar to the neo-Hookean model, the Mooney–Rivlin model does not have adequate terms to include nonlinear strains of the beam. So, the governing equations will not be completely derived. So, there is a need for another complete model of hyper-elastic materials.

2.3 Yeoh model

This model only depends on the first strain invariant (\({I}_{1}\)) and is phenomenological type, but introduces three material parameters (\({c}_{10}\), \({c}_{20}\), and \({c}_{30}\)). The strain energy function based on this model can be expressed as [23]:

$$\begin{aligned} W=c_{10} (I_1 -3)+c_{20} (I_1 -3)^{2}+c_{30} (I_1 -3)^{3}. \end{aligned}$$
(5)

This model is successfully used by many researchers. The efficacy of this model in the modeling of hyper-elastic material structures will be figured out in the following sections.

2.4 Ishihara model

This model is an extension of Mooney–Rivlin model with two parameters. Hence, this model contains three material parameters and the strain energy function has the following form:

$$\begin{aligned} W=c_{10} (I_1 -3)+c_{01} (I_2 -3)+c_{20} (I_1 -3)^{2}. \end{aligned}$$
(6)

For unfilled natural rubber [23]: \({c}_{10}= 0.1161\) MPa, \({c}_{01}= 0.0114\) MPa, \({c}_{20}= 0.0136\) MPa.

For Silicon rubber: \({c}_{10}= 315571.10\) Pa, \({c}_{01}= -5931.41\) Pa, \({c}_{20}= 66929.04\) Pa.

3 Mathematical formulation

Based on higher-order shear deformation theory, the displacement field (\(u_{1}\), \(u_{2}=0\), \(u_{3}\)) consists of axial (u), lateral (w) displacements, and also the rotation (\(\psi \)) as:

$$\begin{aligned}&u_1 \left( {x,z,t} \right) =u\left( {x,t} \right) -z\frac{\partial w}{\partial x}+f(z)\psi (x,t), \end{aligned}$$
(7)
$$\begin{aligned}&u_3 (x,z,t)=w(x,t) \end{aligned}$$
(8)

where f(z) is a shear function and may be considered to be: \(f(z)=\frac{h}{\pi }\sin \left( {\frac{\pi z}{h}} \right) \). Based on von-Kármán’s nonlinearity, the axial and shear strains may be defined as [17]:

$$\begin{aligned} \varepsilon _{xx}= & {} \frac{\partial u_1 }{\partial x}=\frac{\partial u}{\partial x}-z\frac{\partial ^{2}w}{\partial x^{2}} +\frac{1}{2}\left( \frac{\partial w}{\partial x}\right) ^{2}+f(z)\frac{\partial \psi }{\partial x}, \nonumber \\ \varepsilon _{xz}= & {} {f}'\psi . \end{aligned}$$
(9)

Based on the above strains and Eq. (1), one can derive the first and second strain invariants as:

$$\begin{aligned} I_1= & {} 2\varepsilon _{xx} +3, \nonumber \\ I_2= & {} 4\varepsilon _{xx} -\varepsilon _{xz}^2 +3. \end{aligned}$$
(10)

The most important issue is that the shear strain (\(\varepsilon _{xz}\)) of the presented beam model only exists in the second strain invariant (\({I}_{2}\)). From Eq. (5), it is clear that the strain energy function of the Yeoh model consists of only \({I}_{1}\). So, it can be concluded that the Yeoh model is not able to consider the shear strain effect in the modeling of a higher-order hyper-elastic beam. However, the Ishihara model can capture the shear strain effect due to incorporation of \({I}_{2}\) in its energy function.

To derive the governing equations of a hyper-elastic tube, the extended Hamilton’s principle may be defined as:

$$\begin{aligned} \int _0^t {\delta (W+V-K) \mathrm{d}t=0}. \end{aligned}$$
(11)

Here, W is the strain energy function, V is the work done by external forces, and K is the kinetic energy.

Based on Yeoh or three-parameter Mooney–Rivlin models, the first variation of the strain energy can be generally defined as:

$$\begin{aligned} \delta W=\delta W_{10} +\delta W_{01} +\delta W_{20} \end{aligned}$$
(12)

where using Eqs. (5) and (6) one can get:

$$\begin{aligned} \delta W_{10}= & {} 2C_{10} \int {\frac{\partial w}{\partial x}} \delta \left( \frac{\partial w}{\partial x}\right) \mathrm{d}V=2A_{xx} C_{10} \frac{\partial w}{\partial x}\delta w\left| {_{x=0}^{x=L} } \right. -2A_{xx} C_{10} \int _0^L {\frac{\partial ^{2}w}{\partial x^{2}}} \delta (w)\mathrm{d}x, \end{aligned}$$
(13)
$$\begin{aligned} \delta W_{01}= & {} 4C_{01} \int \left( \frac{\partial w}{\partial x} \delta \left( \frac{\partial w}{\partial x}\right) \right) \mathrm{d}V-2C_{01} \int {(( {{f}'} )^{2}\psi } \delta (\psi ))\mathrm{d}V \nonumber \\= & {} 4A_{xx} C_{01} \frac{\partial w}{\partial x}\delta w\left| {_{x=0}^{x=L} } \right. -4A_{xx} C_{01} \int _0^L {\frac{\partial ^{2}w}{\partial x^{2}}} \delta (w)\mathrm{d}x-2\tilde{F}C_{01} \int _0^L {\psi \delta (\psi )} \mathrm{d}x, \end{aligned}$$
(14)
$$\begin{aligned} \delta W_{20}= & {} 4C_{20} \int \left( 2\frac{\partial u}{\partial x} \delta \left( \frac{\partial u}{\partial x}\right) +2f\frac{\partial \psi }{\partial x}\delta \left( \frac{\partial u}{\partial x}\right) +2f\frac{\partial u}{\partial x}\delta \left( \frac{\partial \psi }{\partial x}\right) +\left( \frac{\partial w}{\partial x}\right) ^{2}\delta \left( \frac{\partial u}{\partial x}\right) \right. \nonumber \\&+\,2\frac{\partial u}{\partial x}\frac{\partial w}{\partial x}\delta \left( \frac{\partial w}{\partial x}\right) {+}\,2z^{2}\frac{\partial ^{2}w}{\partial x^{2}}\delta \left( \frac{\partial ^{2}w}{\partial x^{2}}\right) {-}2zf\frac{\partial \psi }{\partial x}\delta \left( \frac{\partial ^{2}w}{\partial x^{2}}\right) {-}2zf\frac{\partial ^{2}w}{\partial x^{2}}\delta \left( \frac{\partial \psi }{\partial x}\right) {+}2f^{2}\frac{\partial \psi }{\partial x}\delta \left( \frac{\partial \psi }{\partial x}\right) \nonumber \\&\left. +\,f\left( \frac{\partial w}{\partial x}\right) ^{2}\delta \left( \frac{\partial \psi }{\partial x}\right) +2f\frac{\partial w}{\partial x}\frac{\partial \psi }{\partial x}\delta \left( \frac{\partial w}{\partial x}\right) +\left( \frac{\partial w}{\partial x}\right) ^{3}\delta \left( \frac{\partial w}{\partial x}\right) \right) \mathrm{d}V \nonumber \\= & {} 8C_{20} A_{xx} \frac{\partial u}{\partial x}\delta u\left| {_{x=0}^{x=L} } \right. -8C_{20} A_{xx} \int _0^L {\frac{\partial ^{2}u}{\partial x^{2}}} \delta (u)\mathrm{d}x+8C_{20} E_{xx} \frac{\partial \psi }{\partial x}\delta u\left| {_{x=0}^{x=L} } \right. \nonumber \\&-\,8C_{20} E_{xx} \int _0^L {\frac{\partial ^{2}\psi }{\partial x^{2}}} \delta (u)\mathrm{d}x +\,8C_{20} E_{xx} \frac{\partial u}{\partial x}\delta \psi \left| {_{x=0}^{x=L} } \right. \nonumber \\&-8C_{20} E_{xx} \int _0^L {\frac{\partial ^{2}u}{\partial x^{2}}} \delta (\psi )\mathrm{d}x+4C_{20} A_{xx} \left( \frac{\partial w}{\partial x}\right) ^{2}\delta u\left| {_{x=0}^{x=L} } \right. \nonumber \\&-\,8A_{xx} C_{20} \int _0^L {\frac{\partial ^{2}w}{\partial x^{2}}\frac{\partial w}{\partial x}} \delta (u)\mathrm{d}x +\,8C_{20} A_{xx} \frac{\partial u}{\partial x}\frac{\partial w}{\partial x}\delta w\left| {_{x=0}^{x=L} } \right. \nonumber \\&-8A_{xx} C_{20} \int _0^L {\left( \frac{\partial ^{2}w}{\partial x^{2}}\frac{\partial u}{\partial x}+\frac{\partial ^{2}u}{\partial x^{2}}\frac{\partial w}{\partial x}\right) } \delta (w)\mathrm{d}x\nonumber \\&+\,8C_{20} D_{xx} \frac{\partial ^{2}w}{\partial x^{2}}\delta \left( \frac{\partial w}{\partial x}\right) \left| {_{x=0}^{x=L} } \right. -\,8C_{20} D_{xx} \frac{\partial ^{3}w}{\partial x^{3}}\delta (w)\left| {_{x=0}^{x=L} } \right. +8C_{20} D_{xx} \int _0^L {\frac{\partial ^{4}w}{\partial x^{4}}} \delta (w)\mathrm{d}x\nonumber \\&-8c_{20} F_{xx} \frac{\partial \psi }{\partial x}\delta \left( \frac{\partial w}{\partial x}\right) \left| {_{x=0}^{x=L} } \right. +\,8c_{20} F_{xx} \frac{\partial ^{2}\psi }{\partial x^{2}}\delta w\left| {_{x=0}^{x=L} } \right. \nonumber \\&-\,8c_{20} F_{xx} \int _0^L {\frac{\partial ^{3}\psi }{\partial x^{3}}} \delta (w)\mathrm{d}x-8C_{20} F_{xx} \frac{\partial ^{2}w}{\partial x^{2}}\delta (\psi )\left| {_{x=0}^{x=L} } \right. +8c_{20} F_{xx} \int _0^L {\frac{\partial ^{3}w}{\partial x^{3}}} \delta (\psi )\mathrm{d}x \nonumber \\&+\,8C_{20} H_{xx} \frac{\partial \psi }{\partial x}\delta (\psi )\left| {_{x=0}^{x=L} } \right. -8c_{20} H_{xx} \int _0^L {\frac{\partial ^{2}\psi }{\partial x^{2}}} \delta (\psi )\mathrm{d}x+4C_{20} E_{xx} \left( \frac{\partial w}{\partial x}\right) ^{2}\delta \psi \left| {_{x=0}^{x=L} } \right. \nonumber \\&-\,8E_{xx} C_{20} \int _0^L {\frac{\partial ^{2}w}{\partial x^{2}}\frac{\partial w}{\partial x}} \delta (\psi )\mathrm{d}x+8C_{20} E_{xx} \frac{\partial \psi }{\partial x}\frac{\partial w}{\partial x}\delta w\left| {_{x=0}^{x=L} } \right. \nonumber \\&-\,8E_{xx} C_{20} \int _0^L {\left( \frac{\partial ^{2}w}{\partial x^{2}}\frac{\partial \psi }{\partial x}+\frac{\partial ^{2}\psi }{\partial x^{2}}\frac{\partial w}{\partial x}\right) } \delta (w)\mathrm{d}x+4A_{xx} C_{20} \left( \frac{\partial w}{\partial x}\right) ^{3}\delta (w)\left| {_{x=0}^{x=L} } \right. \nonumber \\&-\,12C_{20} A_{xx} \int _0^L {\frac{\partial ^{2}w}{\partial x^{2}}\left( \frac{\partial w}{\partial x}\right) ^{2}} \delta (w)\mathrm{d}x \end{aligned}$$
(15)

where

$$\begin{aligned}&\{A_{xx} ,D_{xx} ,E_{xx} ,F_{xx} ,H_{xx} \}=\int _{-0.5h}^{+0.5h} {\{1,z^{2}} ,f,zf,f^{2}\}\mathrm{d}z, \nonumber \\&\tilde{F}=\int _{-0.5h}^{+0.5h} {({f}'} )^{2}\mathrm{d}x. \end{aligned}$$
(16)

The first variation of the work done by the applied forces can be written in the form:

$$\begin{aligned} \delta V=\int _0^L {q\delta w} \mathrm{d}x. \end{aligned}$$
(17)

The external transverse load q is expressed by [13]:

$$\begin{aligned} q=k_w w-k_p \frac{\partial ^{2}w}{\partial x^{2}}+k_{nl} w^{3} \end{aligned}$$
(18)

where \(k_w \) and \(k_p \) are Winkler and Pasternak coefficients of the surrounding medium, respectively. Also, \(k_{nl}\) denotes the nonlinear stiffness of the surrounding medium. The variation of the kinetic energy is represented by:

$$\begin{aligned} \delta K= & {} \int _0^L \left( I_0 \left[ \frac{\partial u}{\partial t}\frac{\partial \delta u}{\partial t} +\left( \frac{\partial w}{\partial t}\right) \left( \frac{\partial \delta w }{\partial t}\right) \right] -I_1 \left( \frac{\partial u}{\partial t}\frac{\partial ^{2}\delta w}{\partial x\partial t} +\frac{\partial ^{2}w }{\partial x\partial t} \frac{\partial \delta u}{\partial t}\right) \right. \nonumber \\&\left. +\,I_2 \left( \frac{\partial ^{2}w }{\partial x\partial t}\frac{\partial ^{2}\delta w }{\partial x\partial t}\right) +I_3 \left( \frac{\partial u}{\partial t}\frac{\partial \delta \psi }{\partial t} +\frac{\partial \psi }{\partial t}\frac{\partial \delta u}{\partial t}\right) \right) \mathrm{d}x \end{aligned}$$
(19)

where

$$\begin{aligned} (I_0 ,I_1 ,I_2 ,I_3 )=\int _{-h/2}^{h/2} {(1,z,z^{2},f(z))\rho } \mathrm{d}z. \end{aligned}$$
(20)

The nonlinear governing equations may be derived for a hyper-elastic beam by using Eqs. (13)–(19) as follows:

$$\begin{aligned}&-\,8C_{20} A_{xx} \frac{\partial ^{2}u}{\partial x^{2}}-8C_{20} E_{xx} \frac{\partial ^{2}\psi }{\partial x^{2}}-8A_{xx} C_{20} \frac{\partial ^{2}w}{\partial x^{2}}\frac{\partial w}{\partial x}=I_0 \frac{\partial ^{2}u}{\partial t^{2}}-I_1 \frac{\partial ^{3}w}{\partial x\partial t^{2}}+I_3 \frac{\partial ^{2}\psi }{\partial t^{2}} \end{aligned}$$
(21)
$$\begin{aligned}&-\,2\tilde{F}C_{01} \psi -8C_{20} E_{xx} \frac{\partial ^{2}u}{\partial x^{2}}+8c_{20} F_{xx} \frac{\partial ^{3}w}{\partial x^{3}}-8c_{20} H_{xx} \frac{\partial ^{2}\psi }{\partial x^{2}}-8E_{xx} C_{20} \frac{\partial ^{2}w}{\partial x^{2}}\frac{\partial w}{\partial x} \nonumber \\&\quad =+I_3 \frac{\partial ^{2}u}{\partial t^{2}}, \end{aligned}$$
(22)
$$\begin{aligned}&-\,2A_{xx} C_{10} \frac{\partial ^{2}w}{\partial x^{2}}-4A_{xx} C_{01} \frac{\partial ^{2}w}{\partial x^{2}}-8A_{xx} C_{20} \left( \frac{\partial ^{2}w}{\partial x^{2}}\frac{\partial u}{\partial x}+\frac{\partial ^{2}u}{\partial x^{2}}\frac{\partial w}{\partial x}\right) +8C_{20} D_{xx} \frac{\partial ^{4}w}{\partial x^{4}} \nonumber \\&\qquad -\,8c_{20} F_{xx} \frac{\partial ^{3}\psi }{\partial x^{3}}-8E_{xx} C_{20} \left( \frac{\partial ^{2}w}{\partial x^{2}}\frac{\partial \psi }{\partial x}+\frac{\partial ^{2}\psi }{\partial x^{2}}\frac{\partial w}{\partial x}\right) -12C_{20} A_{xx} \frac{\partial ^{2}w}{\partial x^{2}}\left( \frac{\partial w}{\partial x}\right) ^{2} \nonumber \\&\quad =+\,I_0 \frac{\partial ^{2}w}{\partial t^{2}}+I_1 \left( \frac{\partial ^{3}u}{\partial x\partial t^{2}}\right) -I_2 \nabla ^{2}\left( \frac{\partial ^{2}w}{\partial t^{2}}\right) -k_w w+k_p \nabla ^{2}w-k_{nl} w^{3}. \end{aligned}$$
(23)

Note that \({I}_{1}={I}_{3}=0\) for an isotropic hyper-elastic material.

4 Solution procedure

In this Section, the nonlinear vibration problem of hyper-elastic beams is solved using a Hamiltonian approach and with the help of Galerkin’s method. However, the three displacement components are selected as [25]:

$$\begin{aligned} u= & {} \sum _{i=1}^\infty {U_i \frac{\partial \varphi _i }{\partial x}(x)}, \end{aligned}$$
(24)
$$\begin{aligned} \psi= & {} \sum _{i=1}^\infty {\Psi _i \frac{\partial \varphi _i }{\partial x}(x)}, \end{aligned}$$
(25)
$$\begin{aligned} w= & {} \sum _{i=1}^\infty {W_i \varphi _i (x)} \end{aligned}$$
(26)

where \(W_{i}\) is the vibration amplitude. The boundary conditions based on transverse displacement from Eqs. (13)–(15) can be expressed as:

$$\begin{aligned}&w\left| {_{x=0} } \right. =w\left| {_{x=L} } \right. =0,\quad \frac{\partial ^{2}w }{\partial x^{2}}\left| {_{x=0} } \right. =\frac{\partial ^{2}w }{\partial x^{2}}\left| {_{x=L} } \right. =0 \quad \hbox {Simply Supported}~ \hbox {(S-S)}, \end{aligned}$$
(27)
$$\begin{aligned}&w\left| {_{x=0} } \right. =w\left| {_{x=L} } \right. =0,\quad \frac{\partial w }{\partial x}\left| {_{x=0} } \right. =\frac{\partial w }{\partial x}\left| {_{x=L} } \right. =0\quad \hbox {Clamped-Clamped}~ \hbox {(C-C)}. \end{aligned}$$
(28)

To satisfy the above-mentioned boundary condition, the function \(\varphi _i (x)\) may be selected as [26,27,28]:

$$\begin{aligned} \varphi _i (x)= & {} \sin \left( {\frac{i\pi }{L}x} \right) \quad \hbox {for}~ \hbox {S-S}, \end{aligned}$$
(29)
$$\begin{aligned} \varphi _i (x)= & {} 0.5\left( 1-\cos \left( {\frac{2i\pi }{L}x} \right) \right) \quad \hbox {for}~ \hbox {C-C}. \end{aligned}$$
(30)

Now, placing Eqs. (24)–(26) into the governing Eq. (21) and using Galerkin’s method leads to the following simplified nonlinear equation:

$$\begin{aligned}&K_{11} U+G_1 W^2 +K_{21} \Psi =0, \end{aligned}$$
(31)
$$\begin{aligned}&K_{12} U+K_{22} \Psi +K_{32} W+G_2 W^2 =0, \end{aligned}$$
(32)
$$\begin{aligned}&K_{23} \Psi +K_{33} W+G_3 W^3 +G_4 UW+G_5 \Psi W+M\ddot{W} =0 \end{aligned}$$
(33)

in which M is the mass matrix and \({K}_{{i}}\) are structural stiffnesses:

$$\begin{aligned} K_{11}= & {} -8C_{20} A_{xx} \Phi _{31}, \nonumber \\ K_{12}= & {} K_{21} =-8C_{20} E_{xx} \Phi _{31}, \nonumber \\ K_{22}= & {} -2\tilde{F}C_{01} \Phi _{11} -8c_{20} H_{xx} \Phi _{31}, \nonumber \\ K_{32}= & {} +8c_{20} F_{xx} \Phi _{31}, \nonumber \\ K_{23}= & {} -8c_{20} F_{xx} \Phi _{40}, \nonumber \\ K_{33}= & {} -2A_{xx} (C_{10} +2C_{01} )\Phi _{20} +8C_{20} D_{xx} \Phi _{40} +k_w \Phi _{00} -k_p \Phi _{20}, \nonumber \\ G_1= & {} -8A_{xx} C_{20} \Phi _{211}, \nonumber \\ G_2= & {} -8E_{xx} C_{20} \Phi _{211}, \nonumber \\ G_3= & {} -12C_{20} A_{xx} \Phi _{2110} +k_{nl} \Phi _{0000}, \nonumber \\ G_4= & {} -8A_{xx} C_{20} (\Phi _{220} +\Phi _{310} ), \nonumber \\ G_5= & {} -8E_{xx} C_{20} (\Phi _{220} +\Phi _{310} ), \nonumber \\ M= & {} I_0 \Phi _{00} -I_2 \Phi _{20}. \end{aligned}$$
(34)

In the above equation, the constructed integrals are:

$$\begin{aligned} \Phi _{00}= & {} \int _0^L { \varphi _i \varphi _i \mathrm{d}x}, \nonumber \\ \Phi _{20}= & {} \int _0^L { \varphi _i^{{\prime }{\prime }} \varphi _i \mathrm{d}x}, \nonumber \\ \Phi _{40}= & {} \int _0^L { \varphi _i^{{\prime }{\prime }{\prime }{\prime }} \varphi _i \mathrm{d}x}, \nonumber \\ \Phi _{11}= & {} \int _0^L { \varphi _i^{\prime } \varphi _i^{\prime } \mathrm{d}x}, \nonumber \\ \Phi _{31}= & {} \int _0^L { \varphi _i^{{\prime }{\prime }{\prime }} \varphi _i^{\prime } \mathrm{d}x}, \nonumber \\ \Phi _{0000}= & {} \int _0^L { \left( {\varphi _i } \right) ^{4}\mathrm{d}x}, \nonumber \\ \Phi _{2110}= & {} \int _0^L { \varphi _i^{{\prime }{\prime }} \varphi _i^{\prime } \varphi _i^{\prime } \varphi _i \mathrm{d}x}. \end{aligned}$$
(35)

With the help of Eqs. (31) and (32), one can find that:

$$\begin{aligned} U= & {} T_1 W+T_2 W^{2}=\frac{K_{21} K_{32} }{K_{11} K_{22} -K_{12} K_{21} }W+\frac{+G_2 K_{21} -G_1 K_{22} }{K_{11} K_{22} -K_{12} K_{21} }W^{2}, \end{aligned}$$
(36)
$$\begin{aligned} \Psi= & {} T_3 W+T_4 W^{2}=\frac{-K_{11} K_{32} }{K_{11} K_{22} -K_{12} K_{21} }W+\frac{G_1 K_{12} -G_2 K_{11} }{K_{11} K_{22} -K_{12} K_{21} }W^{2}. \end{aligned}$$
(37)

Inserting Eqs. (36) and (37) into Eq. (33) yields:

$$\begin{aligned} (K_{33} +T_3 )W+\left( {G_3 +G_4 T_2 +G_5 T_4 } \right) W^3 +(G_4 T_1 +G_5 T_3 +T_4 )W^{2}+M\ddot{W} =0. \end{aligned}$$
(38)

4.1 Extended Hamiltonian method

The nonlinear governing equation of a hyper-elastic tube, Eq. (38), contains cubic and quadratic terms. The Hamiltonian approach [29] is mostly used for Duffing-type oscillators having only cubic nonlinearity. In order to solve Eq. (38), an extended version of the Hamiltonian method is used in the presented research [30].

The Hamiltonian of a conservative system will be constant during vibrations. Hence, the Hamiltonian of the governing equation of the beam can be expressed as:

$$\begin{aligned} H= & {} +\frac{1}{2}M\dot{W}^2 _ +\frac{1}{2}(K_{33} +T_3 )W^2 +\left( {G_3 +G_4 T_2 +G_5 T_4 } \right) \left( \frac{1}{4}\tilde{W}^4 \right) \nonumber \\&+\,(G_4 T_1 +G_5 T_3 +T_4 )\left( \frac{1}{3}\tilde{W}^3 \right) =\hbox {Constant}=H_0. \end{aligned}$$
(39)

The above relation gives that \(H-H_0 =0\). Therefore, from the mathematical point of view, it can be concluded that [31,32,33]:

$$\begin{aligned} \frac{\partial H}{\partial W }=0. \end{aligned}$$
(40)

By defining the following new function, it is possible to express that:

$$\begin{aligned} \tilde{H}(t)=\int _0^{T/4} H \mathrm{d}t=\frac{1}{4}TH \end{aligned}$$
(41)

where T is the period of the nonlinear oscillator. It is clear from Eq. (41) that

$$\begin{aligned} \frac{\partial \tilde{H}(t)}{\partial T}=\frac{1}{4}H. \end{aligned}$$
(42)

According to Eqs. (40) and (42), it may be deduced that

$$\begin{aligned} \frac{\partial }{\partial \tilde{W} }\left( {\frac{\partial \tilde{H}(t)}{\partial T}} \right) =0. \end{aligned}$$
(43)

The presented Eq. (43) can be also represented as:

$$\begin{aligned} \frac{\partial }{\partial \tilde{W} }\left( {\frac{\partial \tilde{H}(t)}{\partial \left( {1/\omega _{NL} } \right) }} \right) =0. \end{aligned}$$
(44)

For nonlinear harmonic oscillation of the beam, the vibration amplitude can be defined as [31,32,33]:

$$\begin{aligned} W_ (t)=\tilde{W} \cos (\omega t). \end{aligned}$$
(45)

Now, with the use of Eq. (45), one can express Eq. (41) in detailed form as:

$$\begin{aligned} \tilde{H}(\omega _{NL} )= & {} \frac{1}{\omega _{NL} }\int _0^{\pi /2} \left[ +\left( \frac{1}{2}M\tilde{W}^{2}\right) \omega _{NL}^2 \sin ^{2}(\theta )+\left( \frac{1}{2} (K_{33} +T_3 )\tilde{W}^{2}\right) \cos ^{2}(\theta ) \right. \nonumber \\&+\left( {G_3 +G_4 T_2 +G_5 T_4 } \right) \left( \frac{1}{4}\tilde{W}^{4}\right) \cos ^{4}(\theta )\nonumber \\&\left. +\,(G_4 T_1 +G_5 T_3 +T_4 )\left( \frac{1}{3}\tilde{W}^{6}\right) \cos ^{3}(\theta )\right] d\theta . \end{aligned}$$
(46)

The stationary conditions \(\partial \tilde{H}/\partial \tilde{W}=0\) result in

$$\begin{aligned} \frac{(K_{33} +T_3 )}{M}\tilde{W}+\frac{3}{4}\frac{\left( {G_3 +G_4 T_2 +G_5 T_4 } \right) }{M}(\tilde{W})^{3}+\frac{8}{3\pi }\frac{(G_4 T_1 +G_5 T_3 +T_4 )}{M}(\tilde{W})^{2}-\omega ^{2}\tilde{W}=0. \end{aligned}$$
(47)

Finally, the nonlinear frequency of hyper-elastic beam is derived as:

$$\begin{aligned} \omega _{NL} =\sqrt{\frac{(K_{33} +T_3 )}{M}+\frac{3}{4}\frac{\left( {G_3 +G_4 T_2 +G_5 T_4 } \right) }{M}(\tilde{W})^{2}+\frac{8}{3\pi }\frac{(G_4 T_1 +G_5 T_3 +T_4 )}{M}(\tilde{W})}. \end{aligned}$$
(48)

In the present article, results are presented based on the following normalized quantities:

$$\begin{aligned} \Omega ={\upomega }_{NL} L\sqrt{\frac{{\uprho }}{c_{10} }} ,K_w =k_w \frac{L^{4}}{c_{10} h^{3}}, K_p =k_p \frac{L^{2}}{c_{10} h^{3}},K_{NL} =k_{nl} \frac{L^{4}}{c_{10} h}. \end{aligned}$$
(49)

5 Results and discussion

In this Section, geometrically nonlinear vibration characteristics of higher-order shear deformation hyper-elastic beams have been studied and figured out. The Ishihara hyper-elastic material model, having three material parameters, has been considered. In Sect. 3, it was discussed that the Yeoh model is impotent to consider shear deformation strains, so it is not suitable for dynamic modeling of a higher-order beam model. In Sect. 4, the closed-form of nonlinear vibration frequency of the beam was obtained. The effects of hyper-elastic material parameters, hyper-elastic material type, nonlinear elastic medium, slenderness ratio and boundary conditions on the vibration frequency of the hyper-elastic beam will be studied in detail (Fig. 1).

Fig. 1
figure 1

Geometry of the hyper-elastic beam resting on an elastic foundation

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Fig. 2
figure 2

Validation of the nonlinear vibration frequency for a hyper-elastic beam

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The nonlinear vibration frequency validation of a hyper-elastic beam with rectangular cross section based on the Yeoh model is given in Fig. 2 and with that of Barforooshi and Mohammadi [21], and an excellent agreement can be seen between the two curves. The thickness of the hyper-elastic beam is considered as \(h=0.65\). For this validation, the effect of shear deformation has been discarded due to the reason that in the paper of Barforooshi and Mohammadi [21], classical beam theory is used.

Fig. 3
figure 3

Normalized vibration frequency versus dimensionless amplitude for silicon rubber and unfilled natural rubber \((L/h=10)\)

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The nonlinear vibration frequency versus dimensionless amplitude of a hyper-elastic beam based on the Ishihara hyper-elastic model is plotted in Fig. 3 for both silicon rubber and unfilled natural rubber. Both S–S and C–C boundary conditions have been considered. First of all, it must be pointed out that an increase in vibration amplitude is corresponding to higher frequencies owing to the incorporation of nonlinear hardening impacts. However, the minimum frequency (natural frequency) of the tube can be obtained at \(\tilde{W}/h=0\). Also, it must be explained that the difference between two the curves is due to the different material parameters (\({c}_{10}\), \({c}_{01}\), \({c}_{20}\)) for silicon rubber and unfilled natural rubber. The slope of the two curves is mostly related to the material parameter \({c}_{20}\) because \({c}_{20}\) exists in nonlinear terms of the governing equation [\({G}_{{i}}\) in Eq. (38)]. Thus, the slope of the frequency curve for silicone rubber is higher than for natural rubber since silicone rubber has a larger value for \({c}_{20}\). Also note that unfilled natural rubber has less industrial application compared to silicone rubber a However, this Figure is presented to give some perception.

Fig. 4
figure 4

Nonlinear vibration frequency versus dimensionless amplitude based on various hyper-elastic material parameters \((L/h=20)\)

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As it is well known in the literature, obtaining hyper-elastic material parameters is a complex task since they are dependent on various conditions such as loading type and environment. When using optimization to obtain material models, it is important to note that the answer is an approximation within predefined tolerances of the correct one. In Fig. 4, the effect of hyper-elastic material parameters of the Ishihara model (\({c}_{10}\), \({c}_{01}\), \({c}_{20}\)) with \(\pm 10\% \) and \(\pm 20\% \) tolerance is presented. One can see that an increase in the \({c}_{10}\) the parameter reduces the nonlinear vibration frequency, but an increase in the \({c}_{20}\) parameter increases the nonlinear vibration frequency. This is due to the fact that an increase in the \({c}_{20}\) parameter increases the structural stiffness of the beam. Also, \({c}_{01}\) is a material parameter which is absent in nonlinear terms of the governing equation [Eq. (38)]. Thus, from Fig. 4 it is obvious that the effect of the \({c}_{01}\) material parameter is not sensible to an increase in vibration amplitude.

Fig. 5
figure 5

Nonlinear vibration frequency versus dimensionless amplitude based on various slenderness ratios

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The effect of slenderness ratio (\(L/h\)) on the nonlinear vibration behavior of hyper-elastic beams with S–S and C–C boundary conditions is presented in Fig. 5. The Ishihara model of hyper-elastic materials is considered in this Figure. It is known that beams are more flexible at larger slenderness ratios. Thus, obtained nonlinear vibration frequencies become lower with an increase in the slenderness ratio at constant normalized amplitudes (\(\tilde{W}/h\)). However, the obtained nonlinear vibration frequencies for various values of the slenderness ratio depend on the magnitude of the normalized amplitude. For smaller slenderness ratios, the nonlinear vibration frequency increases with a higher rate with respect to the normalized amplitude than higher the slenderness ratios or thinner beams. This is because the beam is stiffer at small slenderness ratios.

Fig. 6
figure 6

Nonlinear vibration frequency versus dimensionless amplitude based on various Winkler parameters (\({L}/{h}=10\), \({K}_{\mathrm{p}}=0\))

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Fig. 7
figure 7

Nonlinear vibration frequency versus dimensionless amplitude based on various Pasternak parameters (\({L}/{h}=10\), \({K}_{\mathrm{w}}=200\))

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Figures 6 and 7 show the dependency of the nonlinear free vibration characteristics of geometrically nonlinear hyper-elastic beams on the elastic medium parameters. It is clear from the Figure that an increase in Winkler (\({K}_{\mathrm{w}}\)) and Pasternak (\({K}_{\mathrm{P}}\)) elastic medium coefficients only increases the value of the vibration frequency and their influence is not dependent on the vibration amplitude. In fact, the nonlinear vibration frequency is increased with the increase in elastic medium parameters regardless of the magnitude of the vibration amplitude. It can be concluded that this observation is related to the increase in bending rigidity of the beam by considering the influence of the elastic medium.

Fig. 8
figure 8

Nonlinear vibration frequency versus dimensionless amplitude based on various nonlinear elastic medium parameters (\({L}/{h}=10\), \({K}_{\mathrm{w}}=200\), \({K}_{\mathrm{p}}=20\))

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The effect of the nonlinear parameter of an elastic medium (\({K}_{\mathrm{NL}}\)) on the vibration frequency of hyper-elastic beams is depicted in Fig. 8 when \({K}_{\mathrm{w}}=100\) and \({K}_{\mathrm{p}}=10\). It is evident from this Figure that the influence of the nonlinear elastic medium parameter (\({K}_{\mathrm{NL}}\)) on the vibrational frequency is ignorable near the zero vibration amplitude. Thus, its effects becomes more prominent at large vibration amplitudes. Actually, the nonlinear elastic medium coefficient cannot change the value of the natural frequency, the frequency at \(\tilde{W}/h=0\), because the natural frequency is independent of the nonlinear stiffness. However, as the magnitude of the vibration amplitude becomes higher, the influence of the nonlinear elastic medium parameter on the vibration the frequency becomes more prominent because of the enhancement of nonlinear hardening influences.

6 Conclusions

Based on a higher-order shear deformation beam theory, the present research dealt with the nonlinear vibration analysis of hyper-elastic beams resting on an elastic foundation. The beam was considered to be made of silicon rubber or unfilled natural rubber. The Ishihara model was used for the modeling of hyper-elastic material. It was discussed that the Yeoh model is not able to incorporate shear strain effects. An extended Hamiltonian approach was implemented to solve the nonlinear governing equation of the beam having cubic and quadratic nonlinearities. It was reported that the frequency curves for silicone rubber and natural rubber have different slopes because of the difference in their material parameters. Also, it was reported that an increase in \({c}_{10}\) and \({c}_{20}\) parameters, respectively, reduced and increased the nonlinear vibration frequency, Also, the effect of the \({c}_{01}\) material parameter was not sensible at the prescribed vibration amplitudes.