Vibration analysis of a bi-directional functionally graded nano beams with various boundary conditions using Ritz method

Nội dung text: Vibration analysis of a bi-directional functionally graded nano beams with various boundary conditions using Ritz method

1. Tuyển tập Hội nghị khoa học toàn quốc lần thứ nhất về Động lực học và Điều khiển Đà Nẵng, ngày 19-20/7/2019, tr. 310-317, DOI 10.15625/vap.2019000295 Vibration analysis of a bi-directional functionally graded nano beams with various boundary conditions using Ritz method LE Quoc-Cuong*, NGUYEN Ba-Duy and NGUYEN Ho-Quang * Binh Dương Vietnam – Korea College, 1 Dong Cay Viet Street, Thu Dau Mot City, Binh Duong Province, Viet Nam Thu Dau Mot University, 6 Tran Van On Street, Thu Dau Mot City, Binh Duong Province, Viet Nam E-mail: lecuong2109@gmail.com Abstract buckling, and free vibration for nonlinear micro beams in Vibration analysis of functionally graded nano beams with which a nonlinear model has been conducted within the different boundary conditions is presented in this paper. The context of non-classical continuum mechanics by first-order shear deformation theory and nonlocal elasticity introducing a material length-scale parameter. Pradhan theory are used to incorporate size small effect of functionally and Murmu [6] developed a single nonlocal beam model graded nano beams. Ritz-type analytical solution is used to solve the characteristic equations of motion for different to investigate the bending and vibration characteristics of boundary conditions. Numerical results are presented to a nano cantilever beam. Phadikar and Pradhan [7] compare with those from earlier works, and to investigate the presented finite element formulations for nonlocal elastic effects of span-to-height ratio, material parameter and scale Euler–Bernoulli beam and Kirchhoff plate. Finite element factor on the natural frequencies of functionally graded nano results for bending, vibration, and buckling for nonlocal beams. beam with four classical boundary conditions have been computed. Thai [8] proposed a nonlocal beam theory for Keywords: Vibration, Bi-directional functionally graded nano beams, Nonlocal elasticity theory, Various boundary conditions. bending, buckling and vibration of simply-supported isotropic nano beams using Navier solution. Thai and Vo [9] developed a nonlocal sinusoidal shear deformation 1. Introduction beam theory for bending, buckling and free vibration of simply-supported nano beams. For FG nano beams, the Potential application of FG nano beams in recent studies on static, buckling and vibration behaviours of FG years led to the development of the field of computational nano beams have been considered by many authors. nano mechanics. In practice, the classical continuum Eltaher et al. [10] presented free vibration analysis of theories fail to accurately predict the mechanical functionally graded (FG) size-dependent nano beams behaviour of nanostructures due to small dimensions of a using finite element method in which the size-dependent such structure. To overcome this adverse, Eringen [1, 2] FG nano beam has been investigated on the basis of the proposed size-dependent continuum theory known as the nonlocal continuum model. Results from this work nonlocal elasticity theory. According to this approach, the showed the significance of the material distribution stress at a reference point in an elastic continuum not profile, nonlocal effect, and boundary conditions on the only depends on the strain at the point but also on strains dynamic characteristics of nano beams. Ebrahimi and at every point of the body. Based on the nonlocal Salari [11] analyzed thermo-mechanical effects on elasticity of Eringen, many researches on static, buckling vibration of nonlocal temperature dependent FG nano and vibration of isotropic nano beams have been beams with various boundary conditions in which investigated, only some representative references are nonlocal Euler-Bernoulli beam theory has been used. cited. Reddy [3] reformulated local beam theory by using Eltaher et al. [12] presented static and buckling responses the nonlocal differential constitutive relations of Eringen of FG nano beams with different boundary conditions to study bending, vibration, and buckling behaviors of using finite element method. Ebrahimi and Salari [13] nano beams in which an analytical solution has been used nonlocal Timoshenko beam theory for analysis of obtained to bring out the effect of the nonlocal behavior thermal buckling and free vibration of FG nano beams in of nano beams. Aydogdu [4] proposed a generalized which Navier solution has been applied for analysis of nonlocal beam theory to study bending, buckling, and simply-supported FG nano beams. Based on free vibration of nano beams by using the nonlocal Timoshenko’s theory, Simsek and Yurtcu [14] analyzed constitutive equations of Eringen. Xia et al. [5] used the bending and buckling of simply supported FG nano differential quadrature method to study bending, post beams using Navier solution. Ebrahimi and Barati [15]
2. LE Quoc-Cuong, NGUYEN Ba-Duy, NGUYEN Ho-Quang p investigated effects of moisture and temperature on free xz 1 z vibration characteristics of simply supported FG nano xPzPP , cm m Lh 2 beams resting on elastic foundation by developing various refined beam theories. This topic has also where P(x.z) is material elastic moduli as Young modulus expanded to FG nano beams with various boundary conditions by Ebrahimi and Barati [16] using differential Exz , , Poisson’s ratio  xz, , mass density transform method. A literature review shows that the studies on behaviours of FG nano beams considered xz, at location z; PPPcm c m are material elastic effects of transverse shear deformation by using nonlocal first-order shear deformation beam theory (FOBT) and properties of ceramic and metal, respectively; ppxz, is nonlocal higher-order shear deformation beam theory power-law material parameter. (HOBT), and most of them studied FG simply-supported nano beams using Navier-type solution. Some of researches tried to solve FG nano beams with different boundary condition using finite element method and trigonometric series solution. Moreover, it also reveals that the number of researches considered effects of Fig. 1. Geometry of a functionally graded nano beams. normal strain on behaviours of FG nano beams are limited. Tounsi et al. [17] proposed a nonlocal beam 2.1. Kinetic and strain theory for analysis of stretching effect of isotropic nano The displacement field of FG nano beams is given beams. Ebrahimi and Barati [18] applied a nonlocal by: strain gradient elasticity theory to wave dispersion uxzt1(,,) uxt (,) z xt , (2) behavior of a size-dependent FG nano beam in thermal uxzt3(,,) wxt (,) environment in which the theory contains two scale where the comma indicates the partial differentiation with parameters corresponding to both nonlocal and strain respect to the coordinate subscript that follows; uxt(,), gradient effects and a quasi-3D sinusoidal beam theory considering shear and normal deformations is employed.  xt, , wxt(,)are axial displacement, rotation and A literature review on the behavior analysis of FG nano beams shows that most of previous works study FG transverse displacement at the mid-plan of the nano beam, nano beams with simple-supported boundary conditions, respectively. a number of researches investigated various boundary The strain field of nano beams is given by: conditions are still limited. (0) (1) xx uz,, xx z  x The objectives of this paper is to propose vibration (3) (0) analysis of FG nano beams with various boundary xz xz w, x conditions. The theory is based on Timoshenko’s and 2.2. Equations of motion Eringer’s nonlocal elasticity ones. Ritz solution method Lagrangian functional is used to derive the equations is used to solve characteristic equations of motion with of motion: various boundary conditions. Numerical results are  UK- (4) compared to the earlier works and to investigate the where U and K denote the strain energy and kinetic effects of material distribution through the beam energy, respectively. thickness, span-to-height ratio, scale length parameter The variation of strain energy U of system is given and boundary conditions on the natural frequencies of by: FG nano beams. UdV V x x  xz  xz 2. Theoretical formulation L 010 (5) NMQdxxx xx xz Consider a FG nano beam as in Fig. 1 with 0 rectangular section bh and length L . It is made of a where the stress resultants are defined as: mixture of isotropic ceramic and metal whose properties h/2 vary continuously in the beam thickness as follows: NMxx,1, z x bdz h/2 pxx (6) Pxz , e Pz (1) h/2 s with Qkbdz xz h/2
3. Vibration analysis of a bi-directional functionally graded nano beams with various boundary conditions using Ritz method where ks is shear correction factor which is supposed to Where A, B, D, Ds are the stiffness’s of FG nano beams be 5/6. which are defined by: The variation of kinetic energy K of system is written h/2 ABD,, 1,, z z2 E zbdz by: h/2 22 (14) KxzuudV ,  h/2 13 ssEz V Ak bdz L (7) h/2 22  z pxx 22 2 eIuIIuIwdx 022 10 Substituting Eqs. (10) into Eqs. (13) leads to the 0 expressions of stress resultants as follows: where dot-superscript denotes the differentiation with  respect to the time t; ρ is the mass density of each layer, NIuIAuBxxxxx  0, 1, ,  , and I0, I1, I2 are the inertia coefficients defined by:  MIwIuIBuDxxxxx  01,2,,   , (15) h/2 III,1,, zz2 zbdz (8) s 0, 1 2 QIwA 0, xx w , h/2 Substituting Eqs. (5) and (7) into Eq. (4) Substituting Eqs. (15) into Eq. (9) yields: L 1 L 010  eAuBuDApxx 222 2   s  NMQxx xx xz ,,,,xxxx 2 0 0 (9) s 2 px   eIuIIuIwdxx 222  2 2wAwIuI,,0,1,0,xx xxx u 02 10    Leads to the following equations of motion:  Iw01,2,, Iuxxx I (16)   NIuIxx,01  Iw0, xx w ,   22 2 MQIuIxx,12  (10) Iu I2 Iu Iw dx 0,x 2 1 0 QIw  ,0x 2.3. Ritz-type analytical solution 2.2. Nonlocal elasticity theory for FG nano beams Based on the Ritz method, the displacements u0, w0, θ Based on the Eringen’s nonlocal elasticity theory [2], are approximated in the following forms: nonlocal constitutive equations are expressed by: m it 2 uxt(,)  jj x ue 1 ij t ij (11) j 1 m 2 it where  denotes Laplacian operator;  ()ea0 is wxt(,)  jj x we (17) j 1 parameter of scale length that considers the influences of m (,)xt x eit small size on the response of nanostructures with e0 is a  jj j 1 constant appropriate to each material, a is an internal where uj, wj, θj are unknown values to be determined; characteristics length (e.g., lattice parameter, granular 2 i =-1; ω is natural frequency; ψj(x) and φj(x) are the shape distance) and tij are global stresses. The constitutive functions which are proposed in Table 1 for equations of FG nano beams are hence written under the simply-supported (S-S), clamped-clamped (C-C) and following expressions: clamped-free (C-F) boundary conditions (BC). These xxxx , Exz ,  x (12) shape functions satisfy the BCs given in Table 2. xz  xz, xxGxz ,  xz Table 1. Shape functions [20]. where Gxz ,,/22 Exz  z is shear modulus.  ()x ()x j j BCs jxL/ jxL/ Substituting Eqs. (3) into Eqs. (12) and then subsequent e e results into the stress resultants in Eqs. (6), the following nonlocal constitutive equations of stress resultants are S-S Lx 2 xL x defined as: C-F x 2 NNxxxxxx  ,,, AuB x MM  BuD 2 xxxxxx,,, (13) C-C xL x xLx2 QQ  As w ,,xx x
4. LE Quoc-Cuong, NGUYEN Ba-Duy, NGUYEN Ho-Quang 3. Numerical examples and discussions Table 2. Kinematic BCs of nano beams. The fundamental natural frequencies with respect to the BCs Position Value series number pz and px=0 for different boundary conditions are given in Table 3. It is observed that the S-S x=0 w 0 responses converge quickly for three boundary x=L w 0 conditions: pz = 12 with L/h=10 for vibration. Thus, these numbers of series terms will be used for vibration C-F x=0 uww 0, 0,,x 0, 0 analysis, respectively throughout the numerical examples. x=L Table 3. Convergence studies for fundamental frequencies of FG nano beams (p =1, p =0, μ=1(nm)2) C-C x=0 uww 0, 0,,x 0, 0 z x Numbers of series N x=L uww 0, 0,,x 0, 0 L/h BC 8 10 12 14 16 S-S 6.5836 6.5836 6.5836 6.5836 6.5836 Substituting Eqs. (17) into Eq. (16), the following 10 C-F 2.4193 2.4191 2.4190 2.4190 2.4190 characteristic equation is obtained: C-C 14.2664 14.2607 14.2569 14.2542 14.2522 K0K11 13 M0M 11 13 Fig. 2 illustrate the fundamental frequencies with 22 23 2 22 0KK  0M0 changing of the nonlocality parameter, material TTKKK13 23 33 T M0M 13 33 distribution at L/h=100 and the variation of boundary (18) conditions. It can be concluded that, the frequency M0M11 13 decreases with high rate where the power exponent in mm u0  22 23 range from 0 to 4 than that the power exponent in interval  0MM  w0  mm between 4 and 10. The frequency decreases as the TT13 23 33 MMM θ0  -12 mmm nonlocality parameter increased from 0 to 5.0*10 with where the components of stiffness matrix K and mass the same rate. matrix M are given by: LL 11pxxx 13 px KAeij  i,, x j x dxKBe,, ij  i ,, x j x dx 00 (S – S) LL 22sspxxx 23 px KAeij i,, x j x dxKAe,, ij  i , x j dx 00 LL 33 pxxx s px KDeij  i,, x j x dxAe  i j dx 00 (19) LL 11pxxx 13 px MIeij 01  i j dxMIe,, ij  i j dx 00 LL 22pxxx 33 px MIeij 02 i j dxMIe, ij  i j dx (C – C) 00 LL 11pxxx 13 px MIemij 0,,1,,  i x j x dxMIe, mij  i x j x dx 00 LL 22pxxx 23 px MIemij 0,,0 i x j x dxMIe,, mij  i j , x dx 00 L 33 pxx MIemij 2,,  i x j x dx 0 Fig. 2. The variation of the frequency with material (20) graduation for different nonlocality parameter Finally, the vibration responses of FG nano beams can be determined by solving Eq. (18).
5. Vibration analysis of a bi-directional functionally graded nano beams with various boundary conditions using Ritz method Fig 3 illustrates the frequency with the different slenderness ratio (L/h=10, 20, 50, 100). When the ratio of length and height is 10 then there is a difference than the other ratios. The ratio of length and height of 100 is almost identical to that of length and height of 50. In this study, we use the ratio L/h = 100 to study for another problem. (μ = 1(nm)2) Fig. 4. The variation of the frequency with material graduation for the different boundary conditions. Tables 4-6 figure out the effects of nonlocal parameter and material graduation on the frequencies of the simply supported (S-S), clamped – free (C-F) and clamped – clamped (C-C) beams, respectively. It is concluded that, 2 (C – C, μ = 1(nm) ) as the material graduation increases the frequencies Fig. 3. The variation of the frequency with material decreased. Also, as the nonlocal parameter increases the graduation for the different span-to-height ratio frequencies decreased, for the both cases. Fig 4 illustrates the nondimensional frequency with the In Tables 7-9 the effect of the power-law indexes (px, pz) different boundary conditions at material at graduation on the nondimensional natural frequencies is illustrated p=1, the nonlocality parameter μ=1(nm)2 and the constant for the nonlocality parameter μ=1(nm)2 and for the slenderness ratio (L/h=100). In this figure, it indicates slenderness ratio (L/h=100) of the simply supported (S-S), that the dimensionless frequency will be gradually clamped – free (C-F) and clamped – clamped (C-C) decreasing from C-C, S-S and C-F. The largest beams, respectively. It is concluded that, as the material nondimensional frequency decreases when the material graduation increases (px, pz) the frequencies decreased constant is between 0 and 2. Then, the nondimensional with S-S and C-F boundary conditions but opposite frequency tends to move vertically as the material results for C-C boundary conditions. These results are constant increases. also clearly shown in Figure 5. Table 4. Variation of the nondimensional first natural frequencies with respect to the material distribution and the span-to-height ratio of S-S FG nano beams. μ L/h Theory Material parameter pz (px=0) (nm)2 0 0.5 1 5 μ=0 20 Present 9.8281 7.7141 6.9670 5.9169 Eltaher [10] 9.8797 7.8061 7.0904 6.0025 50 Present 9.8629 7.7412 6.9916 5.9389 Eltaher [10] 9.8724 7.7998 7.0852 5.9990 μ =1 20 Present 9.3831 7.3647 6.6515 5.6492 Eltaher [10] 9.4238 7.4458 6.7631 5.7256 50 Present 9.4106 7.3862 6.6710 5.6666 Eltaher [10] 9.4172 7.4403 6.7583 5.7218 μ =2 20 Present 8.9932 7.0587 6.3751 5.4146 Eltaher [10] 9.0257 7.1312 6.4774 5.4837 50 Present 9.0153 7.0759 6.3907 5.4285 Eltaher [10] 9.0205 7.1269 6.4737 5.4808
6. LE Quoc-Cuong, NGUYEN Ba-Duy, NGUYEN Ho-Quang Table 5. Variation of the nondimensional first three natural frequencies of C-F FG nano beams (L/h=100, N=10) Material parameter pz (px=0) μ (nm)2 Theory 0 0.5 1 5 10 0 Present 3.5161 2.7597 2.4925 2.1173 2.0225 Eltaher [10] 3.5167 2.7600 2.4932 2.1168 2.0221 1 Present 3.5153 2.7606 2.4919 2.1168 2.0220 Eltaher [10] 3.5292 2.7693 2.5134 2.1268 2.0310 2 Present 3.5145 2.7584 2.4914 2.1163 2.0215 Eltaher [10] 3.5461 2.7841 2.5149 2.1360 2.0405 3 Present 3.5137 2.7578 2.4908 2.1158 2.0211 Eltaher [10] 3.5632 2.8019 2.5259 2.1458 2.0498 Table 6. Variation of the nondimensional first three natural frequencies of C-C FG nano beams (L/h=100, N=10). Material parameter pz (px=0) μ (nm)2 Theory 0 0.5 1 5 10 0 Present 22.3597 17.5498 15.8506 13.4636 12.8607 Eltaher [10] 22.3744 17.5613 15.8612 13.4733 12.8698 1 Present 21.0991 16.5604 14.9570 12.7047 12.1358 Eltaher [10] 21.1096 16.5686 14.9645 12.7116 12.1423 2 Present 20.0255 15.7177 14.1958 12.0583 11.5183 Eltaher [10] 20.0330 15.7235 14.2013 12.0633 11.5230 3 Present 19.0974 14.9892 13.5379 11.4995 10.9845 Eltaher [10] 19.1028 14.9934 13.5419 11.5032 10.9880 Table 7. Variation of the nondimensional natural frequencies of S-S FG nano beams (L/h=100, N=10, μ=1(nm)2). Material parameter pz Theory 0 0.5 1 5 10 Present px=0 9.4146 7.3892 6.6738 5.6691 5.4152 Eltaher [10] px=0 9.4162 7.4396 6.7577 5.7212 5.4384 Present px=0.5 9.3894 7.3735 6.6560 5.6539 5.4008 Present px=1 9.3144 7.3146 6.6028 5.6088 5.3576 Present px=5 7.1758 5.6351 5.0868 4.3210 4.1275 Present px=10 3.1093 2.4417 2.2041 1.8723 1.7885 Table 8. Variation of the nondimensional natural frequencies of C-C FG nano beams (L/h=100, N=10, μ=1(nm)2). Material pz Theory parameter 0 0.5 1 5 10 Present px=0 21.0991 16.5604 14.9570 12.7047 12.1358 Eltaher [10] px=0 21.1096 16.5686 14.9645 12.7116 12.1423 Present px=0.5 21.2197 16.6640 15.0424 12.7773 12.2051 Present px=1 21.5626 16.9332 15.2855 12.9838 12.4023 Present px=5 29.7022 23.3253 21.0555 17.8851 17.0841 Present px=10 48.6493 38.2046 34.4869 29.2940 27.9821
7. Vibration analysis of a bi-directional functionally graded nano beams with various boundary conditions using Ritz method Table 9. Variation of the nondimensional natural frequencies of C-F FG nano beams (L/h=100, N=10, μ=1(nm)2). Material parameter pz Theory 0 0.5 1 5 10 Present px=0 3.5153 2.7606 2.4919 2.1168 2.0220 Eltaher [10] px=0 3.5292 2.7693 2.5134 2.1268 2.0310 Present px=0.5 3.0097 2.3635 2.1335 1.8123 1.7312 Present px=1 2.5658 2.0149 1.8189 1.5451 1.4759 Present px=5 0.6277 0.4930 0.4450 0.3780 0.3611 Present px=10 0.0885 0.0695 0.0627 0.0533 0.0509 [3] Reddy, J.N., Nonlocal theories for bending, buckling and vibration of beams. International Journal of 60 Engineering Science, 2007. 45(2): p. 288-307. [4] Aydogdu, M., A general nonlocal beam theory: Its 50 C-C application to nanobeam bending, buckling and 40 S-S C-F vibration. Physica E: Low-dimensional Systems and 30 Nanostructures, 2009. 41(9): p. 1651-1655. 20 [5] Xia, W., L. Wang, and L. Yin, Nonlinear non-classical microscale beams: Static bending, postbuckling and 10 Normalized fundamental frequency fundamental Normalized free vibration. International Journal of Engineering 0 10 8 Science, 2010. 48(12): p. 2044-2053. 6 2 0 4 4 2 6 [6] Pradhan, S.C. and T. Murmu, Application of nonlocal 0 10 8 px pz elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever. Physica E: Fig. 5. The variation of the frequency with material Low-dimensional Systems and Nanostructures, 2010. graduation for the different boundary conditions (px 42(7): p. 1944-1949. 2 and pz, μ = 1(nm) ) [7] Phadikar, J.K. and S.C. Pradhan, Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates. Computational Materials Science, 2010. 49(3): p. 492-499. 4. Conclusions [8] Thai, H.-T., A nonlocal beam theory for bending, The free vibration analysis of bi-directional functionally buckling, and vibration of nanobeams. International Journal of Engineering Science, 2012. 52: p. 56-64. graded nano beam modeled according to Timoshenko [9] Thai, H.-T. and T.P. Vo, A nonlocal sinusoidal shear beam theory is presented. The size-dependent (nonlocal) deformation beam theory with application to bending, effect is introduced according to Eringen’s nonlocal buckling, and vibration of nanobeams. International elasticity model. The variational problem governing the Journal of Engineering Science, 2012. 54: p. 58-66. [10] Eltaher, M.A., S.A. Emam, and F.F. Mahmoud, Free axial and lateral deformations is derived using the vibration analysis of functionally graded virtual-work principle. Ritz method is used to size-dependent nanobeams. Applied Mathematics and approximate the axial and lateral displacements, Computation, 2012. 218(14): p. 7406-7420. respectively. The fundamental frequencies of a nano [11] Ebrahimi, F. and E. Salari, Thermo-mechanical vibration analysis of nonlocal temperature-dependent beam are investigated versus the nonlocal and FG nanobeams with various boundary conditions. material-distribution parameters for different BCs of nano Composites Part B: Engineering, 2015. 78: p. 272-290. beam. The obtained results show that, the [12] Eltaher, M.A., S.A. Emam, and F.F. Mahmoud, Static material-distribution profile may be manipulated to select and stability analysis of nonlocal functionally graded nanobeams. Composite Structures, 2013. 96: p. 82-88. a specific design frequency. It is also shown that, the [13] Ebrahimi, F. and E. Salari, Thermal buckling and free nonlocal parameter has a notable effect on the vibration analysis of size dependent Timoshenko FG fundamental frequencies of nano beam. nanobeams in thermal environments. Composite Structures, 2015. 128: p. 363-380. [14] Şimşek, M. and H.H. Yurtcu, Analytical solutions for 5. References bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam [1] Eringen, A.C., On differential equations of nonlocal theory. Composite Structures, 2013. 97: p. 378-386. elasticity and solutions of screw dislocation and [15] Ebrahimi, F. and M.R. Barati, A unified formulation surface waves. Journal of Applied Physics, 1983. for dynamic analysis of nonlocal heterogeneous 54(9): p. 4703-4710. nanobeams in hygro-thermal environment. Applied [2] Eringen, A.C., Nonlocal polar elastic continua. Physics A, 2016. 122(9): p. 792. International Journal of Engineering Science, 1972. 10(1): p. 1-16.
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