Free Vibration Analysis of Triclinic Nanobeams Based on the Di ﬀ erential Quadrature Method

: In this work, the nonlocal strain gradient theory is applied to study the free vibration response of a Timoshenko beam made of triclinic material. The governing equations of the problem and the associated boundary conditions are obtained by means of the Hamiltonian principle, whereby the generalized di ﬀ erential quadrature (GDQ) method is implemented as numerical tool to solve the eigenvalue problem in a discrete form. Di ﬀ erent combinations of boundary conditions are also considered, which include simply-supports, clamped supports and free edges. Starting with some pioneering works from the literature about isotropic nanobeams, a convergence analysis is ﬁrst performed, and the accuracy of the proposed size-dependent anisotropic beam model is checked. A large parametric investigation studies the e ﬀ ect of the nonlocal, geometry, and strain gradient parameters, together with the boundary conditions, on the vibration response of the anisotropic nanobeams, as useful for practical


Introduction
In the past decades, different analytical and numerical approaches have been applied in literature to study the structural response of even more complicated structures [1][2][3][4][5][6]. Based on the literature, it is well known that many analytical solutions based on the Navier approximations cannot satisfy the governing equations of the problem, such that numerical approaches are usually required. In this context, the differential quadrature method (DQM) has been increasingly applied in several works and demanding applications as a powerful and efficient numerical method [7][8][9][10][11][12], due to its beneficial properties. This method, indeed, is user-friendly for different engineering problems and it features a high accuracy even with few grid points and a reduced computational effort, (see [7][8][9][10][11]). In most cases, the DQM has been used to study the dynamic, static or stability response of structures such as beams, plates, and shells, whereas any application of the DQM in literature has focused to nanostructures made of anisotropic materials, as typically occurs in the actual nature of materials due to their mechanical properties, with different elastic components in each direction. If an isotropic behavior is related to a certain uniformity along all the orientations, on the other hand, anisotropy refers to situations where properties vary systematically. For example, a triclinic material features different properties in different directions, with 21 elastic constants and three components of the propagation vector [13]. Due to the complexity of anisotropic material models, a large amount of simplification in most works in the literature is generally based on isotropic material assumptions. To date, only a few studies have focused on the mechanical response of anisotropic structures. More specifically, in [14][15][16] the authors

Theory and Formulation
In what follows, the nonlocal strain gradient theory [36] is applied to account for both the nonlocal stress field and the strain gradient effects, by means of two small-scale parameters. This theory defines the stress field as σ ij − l 2 1 σ ij,mm = C ijkl (ε kl − l 2 2 ε kl,mm ) where σ ij and ε ij are the stress and strain tensors; C ijkl refers to the elastic properties' matrix, while l 1 and l 2 denote the internal length scales to be determined experimentally or numerically by means of microscopic models, e.g., the MD simulations. A triclinic nanobeam of length L, width b, thickness h is shown in Figure 1. In the current study, it is assumed that the stress components depend on both the longitudinal and transverse shear strains, i.e., where ε xx , γ xz denote the longitudinal and transverse shear strains, respectively, whereas σ xx , τ xz stand for the axial and shear stress, respectively. Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 17 In a context where the Timoshenko beam theory has been largely applied to model isotropic structures, herein the theory is extended to handle triclinic beams. According to the proposed continuum model, the displacement field takes the following form where w refers to the transverse displacements; ψ stands for the rotation of the cross-section and t denotes the time. According to the small deformations assumption, the constitutive relations for the anisotropic nanobeam are defined as follows (9) cij being the elastic components of the triclinic material, defined as [13] The governing Equations of the problem are determined through the Hamiltonian principle as follows U and T being the strain and kinetic energy, respectively. More specifically, the strain energy has the following form where In a context where the Timoshenko beam theory has been largely applied to model isotropic structures, herein the theory is extended to handle triclinic beams. According to the proposed continuum model, the displacement field takes the following form where w refers to the transverse displacements; ψ stands for the rotation of the cross-section and t denotes the time. According to the small deformations assumption, the constitutive relations for the anisotropic nanobeam are defined as follows c ij being the elastic components of the triclinic material, defined as [13] c 11 = 98.84 × 10 9 Pa c 15 = c 51 = 1.05 × 10 9 Pa c 55 = 21.10 × 10 9 Pa The governing Equations of the problem are determined through the Hamiltonian principle as follows U and T being the strain and kinetic energy, respectively. More specifically, the strain energy has the following form where Appl. Sci. 2019, 9, 3517 4 of 17 And κ is the shear correction factor which depends on the material properties. By using the nonlocal strain gradient theory relations (Equation (1)) and by mathematical manipulation with Equations (13) and (14), the following relations can be obtained where In addition, the kinetic energy is expressed as follows By substituting Equations (12) and (20) into Equation (11), the following equations are obtained, when the coefficients of dw and dψ are assumed to be null, i.e., Thus, the introduction of Equations (15) and (16) into Equations (21) and (22), respectively, yields to the following expressions The governing equations of a nonlocal strain gradient triclinic beam with a continuous variation in thickness, are obtained by substituting Equations (23) and (24) into Equations (21) and (22) as follows, In the current study, a combination of simply supported, clamped, and free edges is investigated, which satisfy the following conditions 1.
Simply supported/Simply supported (SS)

Generalized Differential Quadrature Method (GDQM)
In what follows, the GDQM is proposed as a numerical method to solve the equations of motion to free the above-mentioned vibration problem of nanobeams, due to its fast convergence and accuracy as largely demonstrated in the literature for different demanding applications [47][48][49][50][51][52][53]. The GDQM discretizes the partial derivative of a function with respect to a variable by a weighted linear sum of function values at all grid points in that direction. This approximation yields the following relation [7], for i = 1, 2 . . . , N ζ ; j = 1, 2 . . . , N η ; and r = 1, 2 . . . , N ζ − 1. According to this technique, two important factors should be considered, namely, the appropriate distribution of grid points and weighting coefficients for discretization purposes.
As far as the first key aspect is concerned, different distributions could be selected, involving both uniform or not uniform discretizations, whose numerical performances have been largely discussed and compared in literature [54,55]. In this research the Chebyshev-Gauss-Lobatto sampling point rule is selected, due its high accuracy and fast convergence, and defined by the following relation As far as the weighting coefficients are concerned, the following expressions for the first and second derivatives are considered where L ζ is the length of the domain along the ζ-direction and The weighting coefficients of second-forth order derivative can be obtained as follows Before applying this numerical approach, it is worth mentioning the large variety of versions available in literature. For example, Zhu et al. [56] developed a new Crank-Nicolson type DQM to discretize the 2D space-fractional advection-diffusion equations based on a set of cubic B-splines. Dahiya and Mittal [57] presented a modified cubic B-spline DQM to solve numerically a three-dimensional non-linear diffusion problem, and the pertaining equations. Eftekhari [58] proposed a combined differential quadrature-integral quadrature procedure, to handle singular functions, and possible related drawbacks.

Implementation of the GDQM
The combination of simply supported, clamped and free triclinic nanobeams are here discretized into N grid points (i = 1, 2 . . . , N). Considering the GDQM, the equations of motion for the nanobeam at the i-th grid point can be discretized as where µ = l 1 and l = l 2 . To find the unknown frequencies, the above equations can be written in the following form, where the subscripts d and b represent, respectively, the domain and boundary points related to the stiffness and mass matrices. Considering the eigenvalue and eigenvector system, the natural frequencies will be computed as To obtain a non-trivial solution of Equation (35), the determinant of the coefficient matrix must be enforced equal to zero, namely After computing the eigenvalues from Equation (36), the system frequencies can be easily obtained.

Numerical Results
In this section a triclinic nanobeam is considered with length L = 36.8 nm, and thickness depending on its length. A preliminary convergence analysis is performed between our model and predictions from Ref. [7] based on the application of the DQM (see Table 1). Then, using the present size-dependent model for anisotropic materials, the convergence of the model is studied for triclinic nanobeams with different boundary conditions (see Table 2). Based on these two tables, a fast convergence of the results can be observed, even with a reduced number of grid points. This justifies the selection of the limited number of grid points N = 19, as done henceforth within the numerical investigation.  Table 2. Convergence analysis of the size-dependent triclinic beam model (L/h = 100, l = µ = 1 nm 2 ).

Simply Supported Clamped-Simply Supported Clamped-Clamped Clamped-Free
Node To validate the numerical size-dependent methodology of the present work, the first-two dimensionless natural frequencies of the nanostructure are compared to predictions by Eltehar [59] for different values of the nonlocal parameter, based on the Euler Bernoulli beam theory (EBBT), see Table 3. A systematic study is performed to check for the sensitivity of the response for different boundary conditions, with a clear good agreement between the two different approaches, and a general increase of the natural frequencies while moving to a clamped nanostructure at both sides.
More specifically, in Table 4 the first four non-dimensional frequencies of a simply supported triclinic nanobeam are summarized for a different length-to-thickness ratio (L/h), nonlocal parameter µ, and strain gradient l. The same results are also represented in the 3D plots of Figure 2. An increased mode number enables higher values of the frequency, which, in turn reduce for increasing nonlocal parameters, and increase with the strain gradient parameters. In addition, the small-scale parameter seems to affect the response especially for higher frequencies rather than the lower ones. For lower mode numbers, any meaningful impact can be observed for a varying length-to-thickness ratio, whereas a visible increase of the natural frequency can be observed by changing the length-to-thickness ratio, for higher modes of vibration (see e.g., the results associated to the forth mode of vibration), while keeping fixed the strain gradient and the nonlocal parameter. parameter seems to affect the response especially for higher frequencies rather than the lower ones. For lower mode numbers, any meaningful impact can be observed for a varying length-to-thickness ratio, whereas a visible increase of the natural frequency can be observed by changing the length-to-thickness ratio, for higher modes of vibration (see e.g., the results associated to the forth mode of vibration), while keeping fixed the strain gradient and the nonlocal parameter.   Thus, the same systematic study is repeated for a clamped-simply nanobeam, whose results are listed in Table 5 and are depicted in Figure 3, with the aim of understanding the role of the nonlocal parameter, the strain gradient parameter, and the nondimensional geometrical ratio L/h, in its vibration response. Based on a comparative evaluation of the response between the present case (clamped-simply supports) and the simply-supported case, a general increase in frequency is observed with respect to the previous example, due to the clamped boundary condition enforced on one side, and the general increase in stiffness of the structures. Moreover, an increasing nonlocality µ yields a decreasing structural stiffness, together with a general decrease in the fundamental frequencies. At the same time, an increase in the strain gradient l enables an increase in frequency, independently of the length-to-thickness ratio. Similar considerations can be repeated for the whole vibration modes here analyzed. frequencies. At the same time, an increase in the strain gradient l enables an increase in frequency, independently of the length-to-thickness ratio. Similar considerations can be repeated for the whole vibration modes here analyzed.  As a further boundary condition, a fully clamped triclinic nanobeam is analyzed under the same geometry and mechanical assumptions. The results are summarized in Table 6 along with the plots in Figure 4. As expected, an overall increase in stiffness is observed, because of the clamped boundary condition at both sides of the structure. Note also that an increase in the strain gradient parameter l, and nonlocal parameter μ, cause a general increase and decrease of the fundamental L/h=50 L/h=100 L/h=20 L/h=50 L/h=100 L/h=20 As a further boundary condition, a fully clamped triclinic nanobeam is analyzed under the same geometry and mechanical assumptions. The results are summarized in Table 6 along with the plots in Figure 4. As expected, an overall increase in stiffness is observed, because of the clamped boundary condition at both sides of the structure. Note also that an increase in the strain gradient parameter l, and nonlocal parameter µ, cause a general increase and decrease of the fundamental frequencies, respectively, in line with the previous examples. As also reported in the pioneering work on the topic [60], the fundamental frequency computed according to the MD is always lower than predictions based on the classical continuum elasticity theory. This behavior is consistent with our findings for nonlocal clamped nanobeams. The last combination of boundary conditions analyzed herein, accounts for a clamped-free triclinic nanobeam, whose parametric vibration response is listed in Table 7 and represented in Figure 5, in terms of the first natural frequencies, while varying the strain gradient parameter l, the nonlocal parameter µ, and the geometrical ratio L/h. Based on the results, note that the clamped-free nanobeam exhibits a different behavior compared to the structural response for the other boundary conditions. Except for the first frequency, the other frequencies reduce for increasing values of µ, and increase for an increasing value of l. The contrary occurs for the first frequency, which decreases for an increasing strain gradient parameter l, and increases by increasing the nonlocal parameter µ.
Remarkably, these results are perfectly in line with the findings of Eltaher et al. [59] for a nonlocal cantilever beam. Due to the higher flexibility of the free structure at one side, the lowest values of natural frequencies are registered and compared to all the other examples previously discussed.     Finally, the last parametric investigation compares the response of the triclinic nanobeam under the assumption of constant, linear or quadratic variation in thickness. Table 8 summarizes the results in terms of the first-three non-dimensional natural frequencies, for different nonlocal and strain L/h=50 L/h=100 L/h=20 L/h=50 L/h=100 L/h=20 L/h=20 L/h=20 L/h=20 L/h=20 Figure 5. Variation of the first four natural frequencies for different L/h and small-scale parameters (clamped-free cantilever nanobeam). Table 7. Effect of the L/h and small-scale parameters on the natural frequencies for a clamped-free cantilever triclinic nanobeam.
Finally, the last parametric investigation compares the response of the triclinic nanobeam under the assumption of constant, linear or quadratic variation in thickness. Table 8 summarizes the results in terms of the first-three non-dimensional natural frequencies, for different nonlocal and strain gradient parameters and boundary conditions. Based on the results in Table 8, a variation in the power-law index q yields a different vibration response. This is clearly affected by the combined values of power-law index and mode numbers. Moreover, it is worth noticing that the small-scale parameters do not affect significantly the response, for different power-law indexes, which is of great interest for design purposes. Table 8. Effect of the L/h and thickness variation on the first-three natural frequencies of a triclinic nanobeam for different boundary conditions.

Conclusions
In this paper, the free vibration of size-dependent nanobeams made of triclinic material has been investigated. The equations of motion and the associated boundary conditions have been handled by means of the Hamiltonian principle and the Timoshenko beam theory in the context of a nonlocal strain gradient theory. The GDQM has been applied as numerical tool to solve the problem under different boundary conditions assumptions. First, a convergence study verifies successfully the accuracy of the proposed formulation against the available literature. It follows a systematic investigation aimed at checking for the sensitivity of the structural response to small-scale parameters, geometrical dimensions, or possible variations in thickness. According to the parametric results, it is concluded that, the fundamental frequencies increase as the strain gradient parameter increases and nonlocal parameter decreases for all boundary conditions, except for the first mode (in the only case of clamped-free nanobeams). The structural sensitivity to the small-scale parameters becomes much pronounced for higher modes rather than the lower ones. Moreover, the thickness variation impact depends on the vibrational modes and boundary conditions. The highest frequency of the nanobeam is reached always for clamped-clamped boundary conditions, for the same nonlocal parameters and geometrical assumptions. A higher flexibility of the nanostructures is gradually permitted moving from clamped-simply supports, to simply-simply supports, and clamped-free supports. The last combination of boundary conditions yields the lowest values of the vibrational frequency.

Conflicts of Interest:
The authors declare no conflict of interest.