Auxetic Metamaterial Pre-twisted Helical Nanobeams: Vibrational Characteristics

The current investigation pertains to the vibrational characteristics of an out of plane helical nanobeam composed of auxetic material. This study marks the first instance of such an analysis. The Frenet-triad is a mathematical tool utilized to account for the impact of curvature, allowing for the dynamic alteration of the coordinate system and the precise definition of the location vector. In order to examine this model, the governing equations are formulated utilizing Timoshenko's beam theory and Eringen's nonlocal elasticity theory, and applying Hamilton's principle. The generalized differential quadrature method (GDQM) is employed to solve the governing equations under various boundary conditions. The present study validates the precision and effectiveness of the existing model through a comparison between the outcomes obtained from the simplified approach and the benchmark results reported in the literature. The findings reveal a satisfactory level of conformity between the two sets of results. Ultimately, the numerical outcomes are derived with a focus on exploring the impact of various factors such as inclination angle, auxetic rib length, curvatures, tortuosity, and pre-twists on the helical nanobeam. The findings of this work may be used as a benchmark for further investigations.


Introduction
The beam is a structural element that is oriented horizontally and is designed to bear lateral loads or torques.Curved beam structures find numerous applications in the fields of mechanical and civil engineering.Applications such as the design of bridges, machinery, industrial parts, and structures.The out of plane curved nanobeams have potential applications in the field of biology.Specifically, they can be utilized to model DNA, protein, and other biological or molecular structures using principles of mechanical engineering.The examination of the out of plane vibrational behavior of helical beams is deemed crucial and imperative due to advancements in modern engineering design and biotechnology innovation within organizations.In this regard, some researches did so.For instance, Davis et al. [1] conducted a study on the in-plane vibration of a curved beam, taking into account its constant curvature.Tung-Ming et al. [2] have presented a study on the vibrations of a curved beam.The researchers conducted an investigation into the natural frequency of curved beams, taking into account the effects of transverse shear, rotary inertia, and the opening angle of the arc.Kawakami and Sakiyama [3] conducted an investigation on the in-plane and out-of-plane vibrations of curved beams with variable sections.Huang et al. [4] conducted an investigation of the identical issue pertaining to non-circular curved beams, utilizing numerical Laplace transform techniques.The authors Lee and Chao [5] conducted an investigation on non-uniform beams, taking into account the constant radius of curvature.The dynamic stiffness matrix method was utilized by Huang et al. [6] to investigate the out-of-plane dynamic analysis of beams with varying arbitrary curvatures and cross-sections.The present study involves a re-examination of the out-of-plane vibration of curved beams by Lee and Chao [7], wherein they have provided precise solutions for the aforementioned beam.The variable curvature of curved beams was investigated by Lee et al. [8].Leung [9] is credited as the pioneer in the study of curved beams.The latest investigation on curved beams pertains to the study conducted by Liu et al. [10].The isogeometric method was employed to conduct an analysis of the free vibrations pertaining to the in-plane and out-of-plane vibrations of Timoshenko's curved beams.The current investigation involved a review of nano Timoshenko's straight beams to authenticate the findings.
The exceptional capabilities exhibited by metamaterials are attributed to their unique structure, which involves the integration of diverse materials with a repeated micro-unit structure.Although the Poisson's ratio of most materials is defined as positive, a particular procedure applied to the structure of metamaterials resulted in a negative Poisson's ratio.Auxetics are a type of metamaterial characterized by a negative or zero Poisson's ratio, derived from the Greek term "auxetikos" meaning "to grow."Materials with negative Poisson's ratio (NPR) [11] have been observed to possess improved properties such as resistance to compression and shear [12], energy absorption [13], sound insulation (acoustic energy) [14], synclastic behavior (elastic recoil) [15], anisotropy [16], high elasticity (in auxetic that follows Hooke's law) [17], and high damping resistance [18].Due to this distinctive characteristic, scholars conducted an inquiry into auxetic substances.For instance, Mukahal and Sobhy [19] conducted an analysis on the wave propagation and free vibration of curved beams with auxetic core, composed of FG graphene platelets, that are situated between two layers and are resting on a viscoelastic foundation.The authors Zhao et al. [20] conducted research on a beam made of auxetic metamaterial with a FG (functionally graded) composition.The beam's nonlinear free vibration behavior was found to be adjustable through the use of graphene origami.The authors in the study conducted by Ebrahimian et al. [21] utilized auxetic clamped-clamped resonators to achieve high-efficiency vibration energy harvesting under low-frequency excitation.Kushwaha et al. [22] conducted an investigation on the free vibration behavior of auxetic metamaterial structural composites based on PLA, utilizing the finite element method.Zhao et al. [23] conducted a study on the vibrational behavior of auxetic metamaterial beams enabled by FG graphene origami, utilizing machine learning techniques.
Prior studies have primarily concentrated on the in-plane vibration behavior of curved beams, with limited investigation into the out-of-plane vibration characteristics of such structures.Until now, the nonlocal theory has been utilized in numerous theories concerning straight beams and inplane curved beams.This article presents an investigation into the vibration behavior of out-ofplane curved beams utilizing the nonlocal theory made of auxetic materials, which has been developed for the first time.The application of numerical techniques is employed in the resolution of equations that are derived from Hamilton's principle.The GDQM is a highly effective numerical technique utilized for the resolution of vibration problems.

Auxetic Materials
Auxetic materials are formed by cells that are arranged in a honeycomb pattern.The mechanical properties of the honeycomb auxetic core layer, such as the Young's modulus, the shear modulus, the negative Poisson's ratio, and the thermal expansion coefficients, are believed to be influenced by the geometrical characteristics of the individual unit cell.

Auxetic core
The possible ramifications of the efficient mechanical properties of the auxetic core are as follows [24,25]: (1) (2) The equations presented in Eqs (1-5) offer the mechanical properties of the utilized auxetic core as shown in Fig. 1.The dimensions of the auxetic cell are represented by the variables , , and , which respectively denote the thickness of its ribs, the length of its horizontal ribs, and the length of its inclined vertical ribs.The stress-strain relationship for auxetic materials can be expressed as follows [24]: =  66    (10)

2.2.Nonlocal elasticity theory
As per the nonlocal elasticity theory, the stress tensor at a given reference point within an elastic continuum is influenced not only by the strain components at that particular location, but also by the strains of all neighboring regions.Thus, the stress tensor that is nonlocal at the point x can be expressed as [26][27][28]: The aforementioned equation system involves the classical, macroscopic stress tensor T(x), which is linked to the strain tensor through Hooke's law as demonstrated in Eq. (11).Additionally, the fourth-order elasticity tensor is denoted as C(x), while the strain tensor is represented by ε(x).
The nonlocal modulus is expressed as (| ′ − |, ).The expression |x′−x| denotes the magnitude of the displacement between two points, while τ is a material constant that is contingent upon the internal and external length parameters.Specifically, τ is defined as the product of the external length parameter,   , and the internal length parameter, a, divided by the characteristic length scale, L. The variable   represents a constant that is proportional to the specific material being studied.The variable a denotes an internal characteristic length, such as the length of bonds within the material.The variable L represents an external characteristic length, such as the wavelength of the material.While the resolution of the integral constitutive relation presented in Eq. ( 11) poses a challenge, Eringen proposed an equivalent relation in differential form, expressed as: The Laplacian operator is denoted by  2 .The equation represented as Eq. ( 11) can be restated in the following manner: − (  ) 2  2  = () (13) The helical beams' equation C(x)=Y is contingent upon the elastic modulus matrix denoted by Y, which comprises the elastic modules G and E.

2.3.Beam theory and displacement of a helical beam
A beam that exhibits both twist and curvature along its central axis is commonly referred to as an out-of-plane curved beam, as illustrated in Fig. 2. The geometric configuration of this model necessitates the selection of a coordinate system that varies at each instance in order to establish the positional vector.The Frenet triad system can be employed for the analysis of the out-of-plane curved beam.Timoshenko's assumptions.The initial segment pertains to the displacements at the centerline in conjunction with the local axes v, while the subsequent segment pertains to the rotations of the cross-section θ, as depicted in Fig. 3.As per Timoshenko's assumption, the definition of these two parts is restricted solely to the  3 axis.The vector of the primary displacement functions is denoted as r in Eq. ( 15) [29].
The subsequent procedure involves computing the derivative of the displacement vector of the curved beam that is out of plane, relative to the arc length denoted as  3 .The displacement gradient of an out-of-plane curved beam can be obtained by utilizing the Frenet frame and differentiating the displacement vector with respect to the arc length  3 .
The parameters τ, κ, and μ utilized in Eq. ( 16) correspond to tortuosity, curvature, and pre-twist rate, respectively.The equation is given as  = ( ) = ( ), where  represents the twist per length.The non-vanishing strain matrix is obtained through the application of Timoshenko's assumption.
The expression for e is given by e=  +  1   +  2   , where the strain matrix is decomposed into three matrices.
By incorporating the strain vector into the aforementioned matrix and taking into account Eringen's nonlocal equation, it is possible to express Eq. ( 18) as follows: The Eq. ( 19) can be expressed in terms of axial force and bent moment by defining axial force as  = ∫   and bent moment as  = ∫  .
Where symbols  1 and  2 are defined as:

2.4.Strain Energy
In order to obtain the governing equations in the initial stage, it is necessary to determine the strain energy.
The equation represented by Eq. ( 21) can be expressed in matrix form as follows: Deviation from Eq. ( 22) results in: It is noteworthy that in order to apply the variational method, matrix (1) must be decomposed into matrix (2).The dimensions of the constituent elements of the initial matrix are 3 by 12, while those of the subsequent matrix are 3 by 6. Due to the strain tensor that defined initially, the .
In which, components   are defined as:
When the external force performs work (W), it results in a governing equation where the work done is zero.
Whereas matrix [      ] in Eq.( 28) is: In order to proceed with the derivation of the nonlocal equation for out-of-plane curved beams and its application in the nanoscale, it is necessary to combine Eq. ( 28) with Eq. ( 20).In the first step, to carry out this process, we should determine the [

′′
] matrix in the Eq.Error!Reference source not found..By differentiation from Eq. Error!Reference source not found.,Eq. ( 30) can be obtained as: Eq. ( 20) can be expressed in the following manner as: Upon substitution of Eq. ( 30) into Eq.( 31), the resulting expression is: The differentiation of Eq. ( 32) and subsequent utilization of Eq. ( 28) results in the derivation of Eq. ( 33), as presented below. ( By taking into account Eq. ( 20), Eq. ( 33) can be expressed in the following manner: Moreover, through the process of differentiation from Eq. ( 29) and subsequently substituting the acquired outcomes into Eq.(34), we can derive the following:

2.6.Governing Equation
The governing equation for the vibration behavior of an out-of-plane curved nanobeam can be derived by extending Eq. ( 35), resulting in the following expression as shown in Eqs.(36a-f).

GDQM method
Various numerical methods can be employed to address the governing equations in accordance with the corresponding boundary conditions.The utilization of GDQ method has demonstrated efficacy in addressing diverse issues pertaining to vibration analysis and dynamical systems.The GDQM technique is a potent approach that can potentially serve as a resolution for partial differential equations, exhibiting notable attributes such as high precision, convergence, and efficiency [31,32].
As per the GDQM approach, the  − ℎ order derivative of a function () with respect to x at   can be expressed as: The variable N represents the total number of grid points, while the matrix C(n) denotes the corresponding weighting coefficients matrix, which can be derived as follows: The function M(x) is defined as follows: The derivation of the weighting coefficients for the  − ℎ order is presented.
The Chebyshev-Gauss-Lobatto method is a viable approach for determining the spatial arrangement of grid nodes within a given domain.
By expressing the governing equations in a matrix format as presented below.
The mass matrix is denoted by M, while the stiffness matrix is represented by K.In order to apply the GDQ method to solve nonlocal governing equations, it is necessary to express Eq.(38) in the following forms: Ultimately, through the utilization of Eq. ( 37) and the application of the eigenvalue equation in the format of Eqs.(36a-f), the comprehensive issue will be streamlined as follows: The boundary parameter and domain parameter are denoted by the indexes  and , respectively.The   and   parameters are established based on the stiffness matrix of the governing equations.The values of   and   are established based on the stiffness matrix of boundary equations.The stiffness matrix denoted by   is associated with the nodes situated within the region, and is obtained by multiplying their weighted coefficients with the displacement of the nodes located on the boundary.In the stiffness matrix of the governing equations, solely those components of the matrix that pertain to the nodes of the region in their respective rows and the boundary nodes in their respective columns shall be retained, while the remaining elements of the matrix shall be eliminated.The symbols   and   are established based on the mass matrix of the governing equations.Moreover, the values of   and   have been ascertained based on the mass matrix of boundary equations.The mass matrix denoted by   is associated with the nodes located within a particular region.The matrix is computed by multiplying the weighted coefficients of the nodes with their corresponding displacements within the domain.In the mass matrix of the governing equations, only the components of the matrix that correspond to the nodes of the region and their related domain nodes will be retained, while the remaining matrix elements will be eliminated.

Convergence investigation
It should be noted that the natural frequency of out-of-plane curved nanobeams is influenced by the quantity of grid points.The current study aims to examine the vibration characteristics of an out-of-plane helical nanobeam, marking the first investigation of its kind.In order to authenticate our research, we conducted a comparison of our findings with those of Leung [9] at the macro level, with the selection of ea=0 (as shown in Table 2).It is assumed that the values of τ and κ are both equal to zero.The nanobeam that was initially out of plane will undergo a transformation and become a straight nanobeam.The outcomes were verified by cross-referencing with the findings of M. A. Eltaher et.al [29] and J. N. Reddy [33] at the nanoscale, with τ =κ =0 being the selected parameters, as demonstrated in Table 3.

Results
The preceding segment involved the utilization of GDQM to determine the natural frequencies and mode shapes of a slender helical rod under varying conditions of curvatures, tortuosity, and pre-twists.This section presents the impact of different parameters on the natural frequency of outof-plane curved beams.The outcomes of altering one of the aforementioned parameters will be presented and compared.

5.1.The effect of the auxetic inclination angle
Table 4 has been generated to monitor the initial four mode shapes of the nanobeam, with the aim of investigating the impact of the auxetic inclination angle on the natural frequencies.Table 4 indicates that an increase in the auxetic inclination angle leads to a decrease in the nafrequencies, which can be attributed to a reduction in flexural stiffness.This phenomenon can primarily occur as a result of the Poisson's ratio values.

5.2.The effect of the rib length and auxetic thickness
Tables 5 and 6 have been presented to analyze the impact of varying vertical rib length and thickness of auxetic materials on the natural frequencies.The tables demonstrate that an increase in the length of the vertical rib results in a corresponding increase in the natural frequencies.In Figure 4, the influence of the vertical rib length on the stability of the system has been considered.As Figure 4

5.3.The effect of the non-local parameter
The results presented in Table 7 demonstrate that an augmentation of the nonlocal parameter leads to a reduction in the natural frequency.The reasoning behind this observation is that an increase in the nonlocal parameter leads to a proportional increase in a specific parameter.In other words, there has been a rise in the atomic bond length, leading to a reduction in the gravitational force within the atoms.As atomic mobility escalates, the rigidity of the beam would diminish.An increase in the nonlocal parameter and aspect ratio is observed to cause a corresponding increase in the e_0 parameter, which in turn leads to a decrease in the stiffness of the beam.

5.4.The effect of the curvatures and tortuosity
Table 8 illustrates that an augmentation in the beam's curvature leads to a reduction in its natural frequency.The underlying principle of this methodology is that an escalation in curvature (coupled with a reduction in the radius of curvature) results in a proportional augmentation of flexibility.
The empirical evidence suggests that a reduction in bodily stiffness leads to a concomitant reduction in frequency, as tortuosity levels increase.

Fig. 2 .
Fig. 2.An out of plane curved beam An out-of-plane curved beam was modeled utilizing Timoshenko's beam theory.The displacement parameter u is postulated to comprise of two components in accordance with

Figure 4 :
Figure 4: Dimensionless natural frequency of a helical beam versus rib length of auxetic material for a) first mode b) second mode

Figure 5 :
Figure 5: Dimensionless natural frequency of a helical beam versus thickness of auxetic material for a) first mode b) second mode

Table 1 .
Convergence review