Nonlocal Euler-Bernoulli Beam Theories With Material Nonlinearity and Their Application to Single-walled Carbon Nanotubes

: The small-scale effect and the material nonlinearity significantly impact the mechanical properties of nanobeams. However, the combined effects of two factors have not attracted the attention of researchers. In the present paper, we proposed two new nonlocal theories to model mechanical properties of slender nanobeams for centroid locus stretching or inextensional effect respectively. Two new theories consider both the material nonlinearity and the small-scale effect induced by the nonlocal effect. The new models are used to analyze the static bending and the forced vibration for single-walled carbon nanotubes (SWCNTs). The results indicate that the stiffness softening effect induced by the material nonlinearity has more prominent impact than the nonlocal effect on SWCNT’s mechanical properties. Therefore, neglecting the material nonlinearity may cause qualitative mistakes.


Introduction
Although Nanobeams have immense potential applications in nanoelectromechanical systems, how to model the beams' mechanical properties is still an open question [1][2][3][4]. One of the most important properties is the nanobeam's elastic properties that are central to determine the beam's overall mechanical behaviors [5][6]. The small-scale effect and the nonlinearity are two critical factors that significantly affect the structure's mechanical properties at the nanoscale. The smallscale effect reveals that the mechanical properties and the geometric size are strongly related if the structure's geometric size down to the nanometer scale [7][8][9][10][11]. The nonlinearity includes the geometric and material nonlinearity [12][13][14][15][16]. There are a lot of researches of the nanobeam's smallscale effect [17][18][19][20][21]. However, the researchers have rarely paid attention to the material nonlinearity of nanobeams [22]. Moreover, the nonlinearity in the thermal-electro-mechanical coupling also is an important field [23][24]. Researchers usually modify the classical continuum mechanics to capture the small-scale effect through three different paths: the nonlocal stress gradient model ( ) 2 2 0 1 ij ij ea   −  =  [7], the strain gradient model ( ) 2 2 0 1 ij ij ea    = −   [9,[25][26], and the surface stress model [27]. Here ij  and ij  are the local stress and strain; ij  is the nonlocal stress; 0 e is a small-scale parameter; and a is the material characteristic length that is the Carbon-Carbon bonding length for SWCNTs. The stress or strain gradient model is widely used to study carbon nanotubes (CNTs) and graphene [28][29][30]. Nevertheless, determining the scale parameter of CNTs and graphene, 0 e , is still a controversial open question [5,31] because the quantum mechanics of mechanical properties has not been thoroughly proposed [32]. The material nonlinearity has restricted nonlinear elastic parameters' calculations through molecular dynamics (MD) and density functional theory (DFT). For example, the nonlinear elasticity of CNTs [16], graphene [14], and silicon nanowires [13] have attracted widespread attentions. Since the material nonlinearity may complicate mechanical models of CNTs, it has been neglected in most studies. From the existing researches, the stain's cubic terms of strain in the potential energy (corresponding to the quadratic terms in the stress-strain relationship) of graphene and SWCNTs significantly affect their mechanical properties [22,33]. The accurate understanding of the nanobeam's mechanical properties is the basis of applications. Therefore, it is necessary to consider both the small-scale effect and the material nonlinearity comprehensively. Under the Euler-Bernoulli assumption of displacements, the present paper will propose two theories, which include both the nonlocal effect and the material nonlinearity, to characterize accurately the nanobeam's mechanical properties for different boundary conditions.

Methods
We restrict our attention to slender beams in the present research. Hence the Euler-Bernoulli hypothesis is employed [34]: the cross-sections which are perpendicular to the centroid locus before deformation remain plane and perpendicular to the deformed locus and suffer no strain in their planes. Under this hypothesis, we may only consider the beam's longitudinal ( x -direction) strain component xx  , as shown in Fig. 1. Existing atomic calculations show that the potential energies of graphene and silicon materials contain at least the strain's cubic nonlinear terms [12][13].
Geometrically，an SWCNT can be viewed as a graphene sheet that has been rolled into a tube, so the stress-strain relationship of the big diameter SWCNTs may be consistent with graphene. This was confirmed by MD simulations [16]. For simplicity, we assume that the beam's strain is finite but small, so only cubic terms of the potential energy are kept, so the local longitudinal stress can be expressed as [12,[15][16]22]. Here, 0 xx  is the initial prestress, E and D are the second-order and third-order elastic coefficients, respectively. Following the nonlocal differential constitutive relationship, has ( ) . Then the nanobeam's nonlocal constitutive relationship with the material nonlinearity is written as In order to establish the corresponding nanobeam model, it is necessary to give the beam's strain-displacement relations according to different boundary constraints [34][35][36]. In this paper, we consider two conditions: (1) the two ends of beams are immovable along the x direction, such as a hinged-hinged beam, hence the effect of axial elongation needs to consider; (2) beams are inextensional, such as a simply supported beam. The present paper will establish motion equations for two conditions, respectively.

Model with axial stretching effect
If the two ends of a beam cannot move along the x -axis, such as a hinged-hinged beam or clamped-clamped beam, the bending deformations may induce the axial extension, and the axial strain is [34] 2 2 (2) Here, u and w are axial displacements of the beam in the x and y directions respectively, as shown in Fig. 1. Substituting Eq. (2) into the expression of local stress, has the local axial force and bending moment as, Here, A and I are the beam's cross-sectional area and the moment of inertia [34]; is the initial axial load at the ends, as shown in Fig. 1. Neglecting the quartic terms of w and noticing for slender beams [34][35], the force reduces to The equations of motion with the stretching effect can write as [ Here m is the mass of the beam per unit span, N and M are the nonlocal axial force and the nonlocal moment. Taking into account Eq. (4) and Eq. (5), the nonlocal constitutive Eq. (1) transforms into 2 22 Substituting the motion equations, Eq. (6) and Eq. (7), into the constitutive equations, Eq. (8) and Eq. (9), gets Differentiates Eq (10) twice with respect to x , then substitutes it into the Eq. (7), gets ( ) ,.
Substituting Eq. (13) into Eq. (12) and ignoring the inertia term 2 2 / t u   , gets the lateral motion equation as following.
Differentiates Eq. (13) with respect to x , then substitutes it into Eq. (6), gets Eq. (14) and Eq. (15) are the nanobeam's plane motion equations with the nonlocal nonlinear constitutive and the stretching effect. Their boundary conditions of displacements are identical with the classical beam theory. If bending motion is considered only, the inertia terms of Eq. (15) can be ignored [35][36], then the equation simplifies as here Where 1 C and 2 C are functions of time t , which can be determined by imposing boundary conditions on w . For a beam with two unmovable ends, gets [35][36] ( ) In Eq. (19), we add a linear damping term / C w t  . For a hinged-hinged beam, the boundary conditions are [34][35][36]

The model with the centroid locus inextensional effect
For a nanobeam with an end being fixed or hinged, the other end being free or sliding, the beam are inextensional, and the strain can be written as [34][35][36] Substituting Eq. (21) Analogous to the virtual work principle of the classical beam theory [34], the beam's motion equations with the nonlocal nonlinear constitutive are, For simplicity, ignores the M 's nonlinear term in Eq. (25), and gets ( ) Here the nonlinear terms only keep up to cubic terms in Eq. (32). For slender beams, the nonlinear inertia terms can be omitted, and Eq. (32) can is simplified to

Solutions of models
At this point, it is convenient to introduce dimensionless variables in Eq. (19) and Eq. (33). Lets Eq. (33) for inextensible beams can rewrite as For simplicity, we take a hinged-hinged beam as an example to discuss the stretching effect and take a simply supported beam to discuss the inextensible beams. So their normalized boundary conditions are identical: It is difficult to accurately solve nonlinear Eq.
It is clear from Eq. (40) that the linear coefficients of the two models are identical, but the nonlinear coefficients are different. Moreover, the nonlocal effect and the material nonlinearity couple in nonlinear terms. The formulae of 1 d demonstrates that the influence of the material nonlinearity will decrease accompanying the increase of beam's length, as an SWCNT example shown in Fig. 2.
Nevertheless, if the nonlocal effect is neglected, the material nonlinearity's influence on 2 d is invariant accompanying the length's change due to For an SWCNT not subjected to the end force, 0 0 N = , the nonlinear terms induced by the finite deformations and the nonlocal effect produces a hardening effect in two models due to 2 0   ; and the nonlinear terms induced by the material nonlinearity has a softening effect in two models due to 0 D  . It is interesting, for hinged-hinged SWCNTs, that the hardening effect will be smaller than the softening effect for the small length tubes, as shown in Fig.2. The following discussions will show that this competitive relationship significantly affects the mechanical behavior of SWCNTs.
Neglecting the inertia and damping terms in Eqs. (39), the equations to determine the static bending deformations of the beam's middle points are obtained as We will research the mechanical behaviors of hinged-hinged beams and simply supported beams according to Eq. (39) and Eq. (41). In order to solve the beams' vibrations, we rewrite Eq. (39) as The multiple scale method [35], which widely use to solve weak nonlinear differential equations for macro-structures [37][38][39], will be applied to solve Eq. (42). Lets and substitutes it into Eq. (43), then equates the coefficients of  and 3  on both sides, has The solution of Eq. (45) is here, cc means the complex conjugate. Substituting Eq. (47) into Eq. (46), gets here NST denotes non-secular terms [36]. When the load's frequency  approaches the nanobeam's modal frequency  , the beam will appear a relatively large amplitude response. Under this condition, lets   (52). If the real parts of eigenvalues are greater than zero, the solutions are unstable [36]. Hence the steady-state motions are unstable when Eq. (53) indicates that the nonlinear term affects the stability of the steady-state solution. So the nonlocal effect and the material nonlinearity also affect the solutions' stability.

Results and discussions
This section use a ( ) confusion [17,31]. However, if the vibration frequency in CNTs is in the terahertz range, a conservative estimate of the scale coefficient is 0 2nm ea [5]. In the present research, we take the SWCNT's scale coefficient . It is bigger than most values in the existing SWCNT's researches [5]. Using a big nonlocal coefficient helps compare the differences between the material nonlinearity and the nonlocal effect.

Hinged-hinged SWCNTs
It can be found from Eqs. the nonlinear parameter, 1 d , will change from positive to negative with the length's decrease, as shown in Fig. 2 and Eqs. (40). This means that the nonlinear terms in Eqs. (39) change from a soft spring to a hard spring. The soft spring has significant influence on the nonlinear mechanical properties of structures [35][36][37][38][39].
Here we take  Fig. (6). Since the softening effects will appear in the small beam's length, the influence of the material nonlinearity will increase with the decrease of the SWCNT's length. Ignoring the material nonlinearity may lead to evident errors for short SWCNTs, as shown in Fig 7 and Fig. 8. The material nonlinearity produces the softening effect in the NCM and NNCM at 6nm l = , and the softening effect makes their frequency-response curves deviate to the left. On the contrary, the CNM and the NNM are hard springs, so the frequency-response curves is skewed to the right, as shown in Fig. 7. We calculated Eq. (42) numerically through the Runge-Kutta method for ( ) ( ) , , 6,5,5 The results show that ignoring the material nonlinearity may seriously underestimate the vibration amplitudes, as shown in Fig. 8. The numerical calculations also show that perturbation solutions are precise.

Simply supported SWCNTs
Here we demonstrate the tube length's effect on the nonlinear coefficients 2 d at first, as shown in Fig. 9. It is found from Fig. 9 and Eqs. (40) that the material nonlinearity makes 2 0 d  for the NCM and NNCM. This indicates that both the nonlocal effect and the material nonlinearity have a stiffness softening effect on SWCNTs. The softening effect makes the static bending deformations of the NNM and the NNCM significantly larger than these of the CNM and the NNM, as shown in Fig. 10 obtained from Eq. (41). It is more important that the material nonlinearity significantly affects the dynamic behaviors of SWCNTs. For example, the response amplitudes of the NCM and the NNCM have jumps accompanying excitation amplitude's change, while the CNM and NNM do not produce amplitude's jumps for ( ) ( ) (51). The above results indicate that neglecting the material nonlinearity may produce qualitatively incorrect results, and that the material nonlinearity may more significantly impact the SWCNT's mechanical properties than the nonlocal effect. Here we implement numerical calculations for Eq. (42) through the Runge-Kutta method to check the perturbation solutions Eq. (51) for the simply supported SWCNT. The numerical simulations confirm the accuracy of the analytical solutions, as shown in Fig. 13.

Conclusions
The present paper combines the nonlocal effect and the material nonlinearity to suggest two new Euler-Bernoulli models for nanobeams: the integral-partial differential equation model for the axial stretching effect, and the partial differential equation model for the axial inextensional effect. The two models consider both the small-scale effect and the material nonlinearity of nanobeams. A ( ) 15,15 SWCNT is used as an example to research the static bending and the forced vibration for the hinged-hinged and simply supported nanobeams. The results show that the material nonlinearity significantly softens SWCNT's stiffness, and the softening effect more significantly impacts on the mechanical properties than the nonlocal effect. Under certain conditions, neglecting the material nonlinearity may induce noticeable mistakes.