Vibration analysis of timoshenko microbeams made of functionally graded materials on a winkler–pasternak elastic foundation

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In this work, the free vibration analysis of Timoshenko microbeams made of the Functionally Graded Material (FGM) on the Winkler–Paternak elastic foundation based on the Modified Coupled Stress Theory (MCST) is investigated. Material characteristics of the beam vary throughout the thickness according to the power distribution and are estimated though Mori–Tanaka, Hashin–Shtrikman and Voigt homogenization techniques.

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Nội dung Text: Vibration analysis of timoshenko microbeams made of functionally graded materials on a winkler–pasternak elastic foundation

Vietnam Journal of Mechanics, Vol. 46, No. 1 (2024), pp. 31 – 43 DOI: https:/ /doi.org/10.15625/0866-7136/20579 VIBRATION ANALYSIS OF TIMOSHENKO MICROBEAMS MADE OF FUNCTIONALLY GRADED MATERIALS ON A WINKLER–PASTERNAK ELASTIC FOUNDATION Tran Van Lien1 , Le Thi Ha2,∗ 1 Hanoi University of Civil Engineering, Hanoi, Vietnam 2 University of Transport and Communications, Hanoi, Vietnam E-mail: lethiha@utc.edu.vn Received: 21 January 2024 / Revised: 19 March 2024 / Accepted: 22 March 2024 Published online: 31 March 2024 Abstract. In this work, the free vibration analysis of Timoshenko microbeams made of the Functionally Graded Material (FGM) on the Winkler–Paternak elastic foundation based on the Modified Coupled Stress Theory (MCST) is investigated. Material characteristics of the beam vary throughout the thickness according to the power distribution and are es- timated though Mori–Tanaka, Hashin–Shtrikman and Voigt homogenization techniques. The Timoshenko microbeam model considering the length scale parameter is applied. The free vibration differential equations of FGM microbeams are established based on the Fi- nite Element Method (FEM) and Kosmatka’s shape functions. The influences of the size- effect, foundation, material, and geometry parameters on the vibration frequency are then analyzed. It is shown that the study can be applied to other FGMs as well as more complex beam structures. Keywords: FGM, microbeam, nondimensional frequency, MCST. 1. INTRODUCTION Functionally Graded Materials (FGMs) are inhomogeneous composites which have attracted considerable attention due to their novel thermo-mechanical properties that en- able them to be used in a wide range of applications in many industries such as aircrafts, biomedical products, space vehicles... Micro–Electro-Mechanical Systems (MEMS) are the new field in which FGMs have been utilized to achieve the desired performance. Micro-sized structures as plates, sheets, beams, and framed structures are widely used in MEMS devices, for example, electrically actuated micro electromechanical devices,
32 Tran Van Lien, Le Thi Ha atomic force microscopes... For this reason, microstructures made of FGMs are especially attracting more and more attention due to their various potential applications. The classical mechanical theories fail to satisfy the solution of the micro elements because it is not effort the size-effect in the micro-scale. So, the non-classical theories such as Modified Couple Stress Theory (MCST) [1] and Modified Strain Gradient Theory (MSGT) [2] must be used in the mechanics of the micro structures which effort the size- effect in the microstructure. Simsek and Reddy [3, 4] examined static bending and free vibration of FGM mi- crobeams based on the MCST and various higher order beam theories. Ansari et al. [5] investigated free vibration characteristics of FGM microbeams based on the MSGT and the Timoshenko beam theory (TBT). Kahrobaiyan et al. [6] developed a new compre- hensive microbeam element on the basis of the MCST. The shape functions of the new element are derived by solving the governing equations of MCST homogeneous Timo- shenko beams. Using the differential quadrature method, Ke and Wang [7] investigated the dynamic stability of FGM microbeams based on the MCST and the TBT. The material properties of FGM microbeams are assumed to vary in thickness direction and are esti- mated though Mori–Tanaka homogenization technique. Thai et al. [8] examined static bending, buckling and free vibration behaviors of size-dependent FGM sandwich mi- crobeams based on the MCST and the TBT. To avoid the use of a shear correction factor, equilibrium equations were used to compute the transverse shear force and shear stress. Using third-order shear deformation theory, Salamat-Talab et al. [9] investigated the static and dynamic analysis of the FGM microbeam based on the MCST. By the Rayleigh–Ritz ¨ method, Akgoz and Civalek [10] studied vibration responses of non-homogenous and non-uniform microbeams using the Bernoulli–Euler beam theory (EBT) and the MCST. The boundary conditions of the microbeam are considered as fixed at one end and free at the other end. It is taken into consideration that material properties and the cross section of the microbeam vary continuously along the longitudinal direction. Chen et al. [11] investigated the static and dynamic responses of bi-directional functionally graded mi- crobeams. The material properties vary along both thickness and axial directions. Shafiei et al. [12] investigated the size dependent nonlinear vibration behavior of imperfect uni- form and non-uniform FGM microbeams based on the MCST and the EBT. In this work, free vibration of FGM microbeams on a Winkler–Pasternak elastic foun- dation is studied based on the MCST, the TBT and the Mori–Tanaka, Hashin–Shtrikman and Voigt homogenization techniques. The governing equations of vibration for the TBT microbeam are derived by using the Finite Element Method (FEM) and Kosmatka’s shape functions. A detailed study is performed to investigate the influences of material, foun- dation parameter, dimensionless length scale parameter and slenderness ratio on the nat- ural frequencies of FGM microbeams.
Rayleigh–Ritz method, Akgöz et al. [10] studied vibration responses of non-homogenous and non- uniform microbeams using the Bernoulli–Euler beam theory (EBT) and the MCST. The boundary conditions of the microbeam are considered as fixed at one end and free at the other end. It is taken into consideration that material properties and the cross section of the microbeam vary continuously along the longitudinal direction. Chen et al. [11] investigated the static and dynamic responses of bi-directional functionally graded microbeams. The material properties vary along both thickness and axial directions. Shafie et al. [12] investigated the size dependent nonlinear vibration behavior of imperfect uniform and non-uniform FGM microbeams based on the MCST and the EBT. Vibration analysis of Timoshenko microbeams made of functionally graded materials on a Winkler–Pasternak ... is In this work, free vibration of FGM microbeams on a Winkler-Pasternak elastic foundation 33 studied based on the MCST, the TBT and the Mori–Tanaka, Hashin-Shtrikman and Voigt homogenization techniques. The governing equations of vibration for the TBT microbeam are derived 2. PROBLEM AND FORMULATION by using the Finite Element Method (FEM) and Kosmatka’s shape functions. A detailed study is performed to investigate the influences of material, foundation parameter, dimensionless length scale Consider an FGM microbeamthe natural frequencies of FGM microbeams.cross-section b × h parameter and slenderness ratio on of the length L and rectangular on the Winkler-Pasternak elastic foundation as shown in Fig. 1. It is assumed that the mate- 2. Problem and formulation rials at bottom surface (z = h/2) and top surface (z = h/2) of the microbeam are metals Consider an FGM microbeam of the length L and rectangular cross-section b ´ h on the Winkler- and ceramics, respectively. The shown in Fig.1. It ismaterial properties of the FGM microbeam Pasternak elastic foundation as local effective assumed that the materials at bottom surface (z = - can be calculatedsurface (zthe Mori–Tanaka, Hashin-Shtrikmanrespectively. The local effective h/2) and top using = h/2) of the microbeam are metals and ceramics, and Voigt homogenization material properties of the FGM microbeam can be calculated using the Mori–Tanaka, Hashin-Shtrikman techniques.Voigt homogenization techniques. and z Ec Gc rc nc h x Em Gm rm nm Kp Kw b Fig. 1: A FGM microbeam on a Winkler-Pasternak elastic foundation Fig. 1. A FGM microbeam on a Winkler-Pasternak elastic foundation According to the Mori–Tanaka homogenization technique [13], the effective bulk modulus K and According to G can be calculated by shear modulusthe Mori–Tanaka homogenization technique [13], the effective bulk K - Km V modulus K and shear modulusc G can be calculated=by G - Gm Vc = ; (1) K c - K m 1 + (1 - Vc )( K c - K m ) / ( K m + 4Gm / 3) Gc - Gm 1 + (1 - Vc )(Gc - Gm ) K − Km Vc = Gm + Gm ( 9 K m + 8Gm ) / (6 K m + 12Gm ) , Kc − Km 1 + (1 − Vc )(Kc − Km )/(Km + 4Gm /3) where the subscripts m and c denote metal and ceramic materials, respectively; V denotes the volume G − G materials. The variation of the volume fraction of constituents can be described by (1) fraction of the phasem = Vc , Gc − Gm a power function as follows (1 − Vc )( Gc − Gm ) 1+ Gm + 1 mz(ö9Km + 8Gm ) z/(6Km + 12Gm ) æ G n æ1 ö n Vc = ç + ÷ ; Vm = 1 - ç + ÷ ; (2) where the subscripts m and c denote 2 h ø and ceramic materials, respectively; V de- è metal è2 hø notes the volume fraction fraction index. Effective material properties of theof themicrobeam fraction of where n is the volume of the phase materials. The variation FGM volume such as Young’s modulus E, Poisson’s ratio n and mass density r can be determined as follows constituents can be described by a power function as follows 9 KG 3K - 2G E ( z) E( z) = 1 n ( z )z= n ; ; G( z) = ; n 1 zr ( z ) = rcVc + r mVm ; (3) Vc 3= + G + 2(3,K + Vm = 1 − + n (+ K G) 2(1 z )) , (2) 2 h 2 h where n is the volume fraction index. Effective material properties of the FGM mi- crobeam such as Young’s modulus E, Poisson’s ratio ν, shear modulus G and mass den- sity ρ can be determined as follows 9KG 3K − 2G E (z) E(z) = , ν(z) = , G (z) = , ρ(z) = ρc Vc + ρm Vm , (3) 3K + G 2(3K + G ) 2(1 + ν(z)) Hashin–Shtrikman have evaluated the effective bulk modulus K and shear modulus G as follows Vm Vm K = Kc + , G = Gc + . (4) 1 1 − Vm 1 (1 − Vm ) (Kc + 2Gc ) + + Km − Kc Kc + Gc Gm − Gc 2Gc (Kc + Gc )
34 Tran Van Lien, Le Thi Ha The Voigt estimate is a frequently used estimate effective material properties P such as E, G, ρ based on the case of a two-phase composite because of the simplicity of this estimate P = Pm Vm (z) + Pc Vc (z). (5) The displacements at a point on the cross-section of the Timoshenko beam can be represented as u( x, z, t) = u0 ( x, t) − (z − h0 )θ ( x, t), w( x, z, t) = w0 ( x, t), (6) where u0 ( x, t), w0 ( x, t) are the axial displacement, the deflection of a point on axis, respec- tively; θ is the angle of rotation of the cross-section around the y axis; h0 is the distance from the neutral axis to x-axis. The nonzero deformation and stress components using the MCST are obtained as follows ∂u0 ∂θ γxz 1 ∂w0 ε xx = − ( z − h0 ) , ε xz = = −θ , ∂x ∂x 2 2 ∂x ∂u0 ∂θ ∂w0 σxx = (λ + 2G ) − ( z − h0 ) , σxz = G − θ , σyy = σzz = λε xx , (7) ∂x ∂x ∂x 1 ∂ 2 w0 ∂θ 1 ∂ 2 w0 ∂θ χ xy = − 2 + , m xy = − Gl 2 2 + , 4 ∂x ∂x 2 ∂x ∂x where λ, G are the Lame’s coefficient and shear modulus determined from E and ν as follows ν (z) E (z) E (z) λ (z) = , G (z) = , (8) [1 + ν (z)] [1 − 2ν (z)] 2 [1 + 2ν (z)] l is the scale material parameter, and m xy , χ xy are components of the deviatoric couple stress m and curvature χ tensors, respectively. The strain energy U of the microbeam L 1 U= σxx ε xx + 2k s σxz ε xz + 2m xy χ xy dAdx 2 0 A L 2 2 1 ∂u0 ∂u0 ∂θ ∂θ (9) = A11 − 2A12 + A22 2 ∂x ∂x ∂x ∂x 0 2 2 ∂w0 1 ∂ 2 w0 ∂θ +k s A33 −θ + l 2 A33 2 + dx, ∂x 4 ∂x ∂x where ks is the shear correction factor and ( A11 , A12 , A22 ) = [λ(z) + 2G (z)] 1, z, z2 dA, A33 = G (z)dA, (10) A A
Vibration analysis of Timoshenko microbeams made of functionally graded materials on a Winkler–Pasternak ... 35 A11 , A12 , A22 and A33 are the rigidities. The kinetic energy T of the microbeam is then given by L 2 2 1 ∂u ∂w T= ρ (z) + dAdx 2 ∂t ∂t 0 A (11) L 2 2 2 1 ∂u0 ∂w0 ∂u0 ∂θ ∂θ ∂u0 ∂θ = I11 + − I12 + + I22 dx, 2 ∂t ∂t ∂t ∂t ∂t ∂t ∂t 0 where I 11 , I 12 and I 22 are the mass moments ( I11 , I12 , I22 ) = ρ(z) 1, z, z2 dA. (12) A The strain energy UF of the Winkler–Pasternak foundation L 2 1 dw UF = Kw w2 + K p dx, (13) 2 dx 0 where K w and K p define the spring and shear moduli of the Winkler–Pasternak elastic foundation. Using the FEM, the beam is assumed to be divided into numbers of two-node beam elements of length L. The vector of nodal displacements de for the element considering the transverse shear rotation θ as an independent variable contains six components as T d e = u i , wi , θ i , u j , w j , θ j , (14) where ui , wi , θi , u j , w j , θ j are the values of u0 , w0 and θ at the node i and at the node j, respectively. In Eq. (14) and hereafter, a superscript ‘T’ is used to denote the transpose of a vector or a matrix.    u u Nu     u0  N1 0 0 N2 0 0 w w w w T w0 =  0 N1 N2 0 N3 N4  u i wi θ i u j w j θ j =  Nw  de , θ θ θ θ 0 N1 N2 0 N3 N4 Nθ   θ (15)
36 Tran Van Lien, Le Thi Ha where Nu is the Lagrange’s linear shape function, Nw and Nθ are Kosmatka’s shape func- tions [14] u u N1 = 1 − x / L, N2 = x / L, w 1 x 3 x 2 x N1 = 1 + 2 −3 −φ , 1+φ L L L w L x 3 x 2 x N2 = 2 − (4 + φ ) + (2 + φ ) , 2 (1 + φ ) L L L w −1 x 3 x 2 x N3 = 2 −3 −φ , 1+φ L L L (16) w L x 3 x 2 x N4 = 2 + ( φ − 2) −φ , 2 (1 + φ ) L L L 6 x x x 3 x N1 = − θ 1− , N2 = 1 − θ 1− , L (1 + φ ) L L L (1 + φ ) L 6 x x x 3 x θ N3 = 1− θ , N4 = 1− 1− , L (1 + φ ) L L L (1 + φ ) L and 12A22 φ= . (17) k s A33 L2 The element stiffness and mass matrices of the microbeam element are obtained as follows T  Nu   u  L L ρA 0 0 N ke,b = T Be De Be dx, me =  Nw   0 ρA 0   Nw dx, (18) 0 0 Nθ 0 0 ρI Nθ where ∂Nu     ∂x    ∂Nθ    EA 0 0 0      ∂x   0 EI 0 0  Be =  ∂Nw , De =  . (19)    − Nθ   0 0 k s GA 0  ∂x 2 0 0 0 4GAl      1 ∂ 2 Nw ∂Nθ     +  4 ∂x2 ∂x
Vibration analysis of Timoshenko microbeams made of functionally graded materials on a Winkler–Pasternak ... 37 The stiffness matrices of the Winkler–Pasternak elastic foundation are obtained as follows Le 1 ke,w = Kw (Nw )T Nw dx, 2 0 (20) Le T w w 1 ∂N ∂N ke,p = Kp dx. 2 ∂x ∂x 0 Having the element stiffness and mass matrices derived, the equations of motion for the free vibration analysis can be written in the form ¨ MD + KD = 0, (21) where D, M, and K are the structural nodal displacement vector, mass and stiffness ma- trices obtained by assembling the element displacement vector de , the mass matrix me , and stiffness matrices ke , kew and kep over the total elements, respectively. 3. NUMERICAL RESULT AND DISCUSSION In this section, the free vibration of the FGM microbeam on the elastic foundation is studied. It is considered the FGM microbeams consist of the aluminum (Al) and the ceramic (SiC) with the material properties Em = 70 GP, νm = 0.3, ρm = 2702 kg/m3 for Al and Ec = 427 GP, νc = 0.17, ρc = 3100 kg/m3 for SiC. The material scale parameter is equal to l = 15 µm. The nondimensional frequencies µi are defined as follows ωi L2 ρm µi = . (22) h Em The Winkler–Paternak foundation coefficients are given in the nondimensional form K w L4 K p L2 bh3 kw = , kp = , I= . (23) Em I Em I 12 To validate the proposed FEM model, the results obtained from the present analysis are compared with the analytical solutions given by Ansari et al. [5]. In Table 1, the nondimensional fundamental frequencies of the simply supported microbeam with an aspect ratio L/h = 10 and the volume fraction index n = 2 obtained in the present paper are compared with the results by Ansari et al. A good agreement can be received for the different material length scale parameters. The above comparisons validate the reliability of the proposed FEM model.
The Winkler–Paternak foundation coefficients are given in the nondimensional form The Winkler–Paternak foundation coefficients are given in the nondimensional form 2 K L ; k = K ppL ; I = bh 3 K wwL44 3 K L2 bh k ww = k = ; kpp = ;I = (26) Em II Em Em II Em 12 12 To validate the proposed FEM model, the results obtained from the present analysis are compared To validate the proposed FEM model, the results obtained from the present analysis are with the analytical solutions given by Ansari Van al. [5].Thi Ha with the analytical solutions given by Ansari et Lien,[5]. In Table 1, the nondimensional fundamental 38 Tran et al. Le In Table 1, the nondimensional fundamental frequencies of the simply supported microbeam with an aspect ratio L/h=10 and the volume fraction frequencies of the simply supported microbeam with an aspect ratio L/h=10 and the volume index n=2 obtained in the present paper are compared with the results by Ansari et al. A good agreement index n=2 obtained in the present paper are compared with the results by Ansari et al. A good can be received 1.forthe differentofmaterial length frequency parameters above comparisons validate the Table Comparison fundamental for FGM microbeams can be receivedfor the differentmaterial length scale parameters. The above comparisons validate scale parameters. The (L/h = 10, n = 2) reliability of the proposed FEM model. reliability of the proposed FEM model. Tableh1. Comparison of 15 Table 1. Comparison offundamental 30 (µm) fundamentalfrequency parameters for FGM microbeams (L/h=10,n=2). frequency parameters for 60 microbeams (L/h=10,n=2). 45 FGM 75 90 hh Ansari et al. 15 15 0.7983 30 30 0.5100 45 45 0.4341 60 60 0.4041 75 75 0.3894 90 90 0.3811 Ansari et al Present (M) Ansari et al 0.7983 0.7655 0.7983 0.5100 0.5062 0.5100 0.4341 0.4349 0.4341 0.4041 0.4064 0.4041 0.3894 0.3924 0.3894 0.3811 0.3845 0.3811 Present (H) 0.7976 0.5213 0.4444 0.4135 0.3983 0.3897 Present (M) Present (M) 0.7655 0.7655 0.5062 0.5062 0.4349 0.4349 0.4064 0.4064 0.3924 0.3924 0.3845 0.3845 Present (H) Present (H) 0.7976 0.7976 0.5213 0.5213 0.4444 0.4444 0.4135 0.4135 0.3983 0.3983 0.3897 0.3897 a) b) Fig 2. Effects of theL/h ratios on the first three nondimensional frequencies of the FGM microbeams L/h ratios (b) of the Fig 2. Effects of the(a) h/l = 2 on the first three nondimensional frequenciesh/l = 4 FGM microbeams when kkw=50,kkp=30,n=2 using homogenization techniques Voigt (V), Hashin–Shtrikman (H) and Mori– when w=50, p=30, n=2 using homogenization techniques Voigt (V), Hashin–Shtrikman (H) Tanaka (M) and different ratiosh/l: a) h/l=2; b) h/l=4. Fig. 2. Effects of the L/h h/l: a) h/l=2; b) h/l=4. Tanaka (M) and different ratios ratios on the first three nondimensional frequencies of the FGM microbeams when k w = 50, k p = 30, n = 2 using homogenization techniques Voigt (V), Fig 22shows the variation of the first three nondimensional frequenciesratios h/l microbeams Fig shows the variation of the first three nondimensional frequencies of the FGM microbeams Hashin–Shtrikman (H) and Mori–Tanaka (M) and different of the FGM with Winkler–Paternak foundation coefficients kw=50, k =30 and the volume fraction index with Winkler–Paternak foundation coefficients kw=50, kpp=30 and the volume fraction index n=2 using homogenization techniques Voigt (V), Hashin–Shtrikman (H), Mori–Tanaka (M) and different homogenization techniques Voigt (V), Hashin–Shtrikman (H), Mori–Tanaka (M) and different ratios h/l=2 (Fig. 2a), h/l=4 the variation of the first three nondimensional using the Hashin-Shtrikman h/l=2 (Fig. 2 shows (Fig. 2b). It shows that nondimensional frequencies frequencies of the FGM Fig. 2a), h/l=4 (Fig. 2b). It shows that nondimensional frequencies using the Hashin-Shtrikman homogenizationtechnique are aa little higher than nondimensional frequencies using the Mori-Tanaka microbeams technique are little higher than nondimensional frequencies using the Mori-Tanaka homogenization with Winkler–Paternak foundation coefficients kw = 50, k p = 30 and the vol- homogenization index n = 2 using of them are smaller than nondimensional frequencies using the ume fraction technique, but both of them are smaller than nondimensional frequencies homogenization technique, but both homogenization techniques Voigt (V), Hashin–Shtrikman (H), Mori–Tanaka (M) and different ratios h/l = 2 (Fig. 2(a)), h/l = 4 (Fig. 2(b)). It shows that nondimensional frequencies using the Hashin-Shtrikman homogenization technique are a little higher than nondimensional frequencies using the Mori–Tanaka homogeniza- tion technique, but both of them are smaller than nondimensional frequencies using the Voigt homogenization technique, specially for the higher frequencies. Moreover, nondi- mensional frequencies using three homogenization techniques increase when the ratio L/h increases. However, “turning point” ratios L/h, at which the given nondimensional
Vibration analysis of Timoshenko microbeams made of functionally graded materials on a Winkler–Pasternak ... 39 frequency changes from the increase to the constant, are dependent on the given fre- quency and the ratios h/l: the higher the frequency the higher the “turning point” ratio L/h, the lower the ratio h/l the higher the “turning point” ratio L/h. ratio h/l graded materials Vibration analysis ofof the effects of the slenderness functionallyon the materials onnondimensional Fig. 3 shows Timoshenko microbeams made functionally graded first threeona aPasternak elastic Vibration analysis Timoshenko microbeams made ofof Pasternak elastic foundation frequency of FGM microbeams with Winkler–Paternak foundation coefficients kw = 50,7 7 foundation k p = 30 and the volume fraction index n = 5 using homogenization techniques Voigt (V), different ratios L/h = 10 (Fig. 3(a)), L/h = Voigt homogenization technique, specially (M) and higher frequencies. Moreover, nondimensional Hashin–Shtrikman (H), Mori–Tanaka Voigt homogenization technique, speciallyfor the higher frequencies. Moreover, nondimensional for the 50 (Fig. 3(b)). three homogenization first three nondimensional frequencies decrease when It can homogenization techniques increase when the ratioL/h increases. However, frequencies using three frequencies using be seen that techniques increase when the ratio L/h increases. However, “turning point” ratios L/h, atatMoreover, given nondimensional frequency frequency and increase to “turning point” ratios L/h, which the given nondimensionalthe secondchanges from the the thirdto the ratios h/l increase. which the the difference of frequency changes from the increase thefrequency are dependent on the given frequency and theh/l are higher. the frequency the higher the constant, dependent on the giventhe ratio L/h andratios h/l: the higher the frequency the higher constant, are remarkable when frequency and the ratios h/l: the higher the “turning point” ratio L/h, the lower the ratio h/l the higher the “turning point” ratio L/h. the “turning point” ratio L/h, the lower the ratio h/l the higher the “turning point” ratio L/h. a) b) Fig 3.3. Effects of the L/h = 10 ratios h/l on the first three nondimensional frequencies of the FGM Fig Effects of the slenderness ratios h/l on the first three nondimensional frequencies of the FGM (a) slenderness (b) L/h = 50 microbeams when kwkw=50,pkp=30, n=5 using homogenization techniques Voigt (V), Hashin–Shtrikman microbeams when =50, k =30, n=5 using homogenization techniques Voigt (V), Hashin–Shtrikman (H) and Effects of the (M) and different ratios on a)a) L/h=10; b)nondimensional frequencies of the Fig. 3. Mori–Tanaka slenderness ratios h/l L/h: L/h=10; b) L/h=50. (H) and Mori–Tanaka (M) and different ratios L/h: the first three L/h=50. FGM microbeams when k w = 50, k p = 30, n = 5 using homogenization techniques Voigt (V), Fig 3 shows the effects ofof the slenderness ratio h/l on the first three nondimensional frequency of Fig 3 shows the effects the slenderness ratio h/l on (M) first three nondimensional frequency of Hashin–Shtrikman (H) and Mori–Tanaka the and different ratios L/h FGM microbeams with Winkler–Paternak foundation coefficients kw=50, =30 and the volume fraction FGM microbeams with Winkler–Paternak foundation coefficients kw=50, kpkp=30 and the volume fraction index n=5 using homogenization techniquesvolume (V), Hashin–Shtrikman (H), Mori–Tanaka (M) and index n=5 using homogenization techniques Voigt Fig. 4 shows the effects of the Voigt (V), Hashin–Shtrikman (H), Mori–Tanaka (M) and fraction index n on first three nondimen- different ratios L/h=10 (Fig. 3a), L/h=50 (Fig. 3b). ItIt canbe seen that first three nondimensional different ratios L/h=10 (Fig. 3a), L/h=50 (Fig. 3b). can be seen that first three nondimensional frequencies decrease when the ratios h/l increase.with Winkler–Paternakof the second frequency and frequencies decrease when the ratios h/l increase. Moreover, the difference foundation coefficients sional frequencies of FGM microbeams Moreover, the difference of the second frequency and thekthird 50, k p = 30, are remarkable when the ratio L/h and h/l are higher. (V), Hashin–Shtrikman the third frequency h/l = 2 using homogenization techniques Voigt w = frequency are remarkable when the ratio L/h and h/l are higher. (H), Mori–Tanaka (M) and different ratios L/h: a) L/h = 10; b) L/h = 50. It can be Fig 4 4 shows the effects of the volume fraction index n on first three nondimensional frequencies Fig shows the effects of the volume fraction index n on first three nondimensional frequencies ofof FGMmicrobeams with Winkler–Paternak foundation coefficients kw=50, kp=30, fraction using seen that first three nondimensional frequencies decrease when the volume h/l=2 using FGM microbeams with Winkler–Paternak foundation coefficients kw=50, kp=30, h/l=2 in- homogenization techniques Voigt the difference between (H), Mori–Tanaka (M) and differentratios homogenization techniques Voigt (V), Hashin–Shtrikman the second frequency and the third dex n increases. Moreover, (V), Hashin–Shtrikman (H), Mori–Tanaka (M) and different ratios L/h: a)a) L/h=10; b) L/h=50. It can be seen ratio L/h andnondimensional frequencies decreasehigher. L/h: L/h=10; b) remarkable when the that first three the volume fraction index n arewhen the frequency are L/h=50. It can be seen that first three nondimensional frequencies decrease when the volume fraction index n n increases. Moreover, the difference between the second frequency and the third volume fraction index increases. Moreover, the difference between the second frequency and the third frequency are remarkable when the ratio L/h and the volume fraction index n n are higher. frequency are remarkable when the ratio L/h and the volume fraction index are higher. Fig 5 5shows the effects ofofthe Winkler elastic foundation coefficient kw w on first three Fig shows the effects the Winkler elastic foundation coefficient k on first three nondimensional frequencies ofof FGM microbeams with Paternak foundation coefficients p=30, n=2 and nondimensional frequencies FGM microbeams with Paternak foundation coefficients k kp=30, n=2 and L/h=10 using homogenization techniques Voigt (V), Hashin–Shtrikman (H), Mori–Tanaka (M) and L/h=10 using homogenization techniques Voigt (V), Hashin–Shtrikman (H), Mori–Tanaka (M) and different ratios h/l=1 (Fig. 5a), h/l=6 (Fig. 5b). ItItcan be seen that the value of the first three different ratios h/l=1 (Fig. 5a), h/l=6 (Fig. 5b). can be seen that the value of the first three nondimensional frequencies increase unremarkably, especially when the ratio h/l is small.
40 Tran Van Lien, Le Thi Ha 88 Tran Van Lien, Le Thi Ha Tran Van Lien, Le Thi Ha 88 Tran Van Lien, Le Thi Ha Tran Van Lien, Le Thi Ha a) b) Fig 4. Effects of the(a) L/h = fractionindex n on the first three nondimensional frequencies of the FGM Fig 4. Effects of thevolume fraction index n on the first three nondimensional frequencies of the FGM volume 10 (b) L/h = 50 a) a) microbeams when kkw=50,kk=30, h/l=2 using homogenization techniques Voigt (V), Hashin–Shtrikman microbeams when w=50, pp=30, h/l=2 using homogenization techniques Voigt (V), Hashin–Shtrikman b) b) (H)and4.Effects of the (M) and fraction index L/h: a)L/h=10;nondimensional frequencies of the FGM Fig. andEffects ofthe volume different ratios on onfirst first b) L/h=50. Fig Mori–Tanaka (M) and fraction index L/h: first three nondimensional frequencies of the FGM 4. Effects the volume different ratios n the L/h=10; b) L/h=50. (H) Fig4.Mori–Tanaka volume fraction index nnon thea)the three three nondimensional frequencies of the FGM microbeams kw=50,kk=30, h/l=2 p usinghomogenization techniques Voigt (V), Hashin–Shtrikman microbeams when kw=50, p p=30,50, k using homogenization techniques Voigt (V), Hashin–Shtrikman microbeams when when w = h/l=2 = 30, h/l = 2 using homogenization techniques Voigt (V), (H) and Mori–Tanaka (M) and differentand Mori–Tanaka (M) and different ratios L/h (H) and Mori–Tanaka (M) and differentratios L/h: a) L/h=10; b) L/h=50. Hashin–Shtrikman (H) ratios L/h: a) L/h=10; b) L/h=50. a) a) b) b) a) b) Fig 5. Effects of the Winkler foundation coefficients kW on the first three nondimensional frequencies of Fig 5. Effects of the Winkler foundation coefficients kW on the first three nondimensional frequencies of theFig 5. Effects of the (a) h/l kkfoundation coefficients kkWon the first three nondimensional frequencies of theFGM Effects of theWinkler ppfoundationL/h=10 using homogenization techniques6Voigt (V), Hashin– Fig 5. microbeams when ==20, n=2, coefficients W homogenization nondimensional frequencies of FGM microbeams Winkler 1 when =20, n=2, L/h=10 using on the first three (b) h/l = Voigt (V), Hashin– techniques Shtrikman(H) and Mori–Tanaka=20, n=2, different usinghomogenizationb)techniquesVoigt (V), Hashin– the FGM (H) and Mori–Tanaka=20, and L/h=10 using homogenizationtechniques Voigt (V), Hashin– the FGMmicrobeams when kkp (M) n=2, differentratios h/l: a) h/l=1; b)h/l=6. Shtrikman microbeams when p (M) and L/h=10 ratios h/l: a) h/l=1; h/l=6. Fig. 5. Effects ofandMori–Tanaka (M) and different ratioskh/l: a)the first b)h/l=6. Shtrikman (H) and Mori–Tanaka (M) and different ratios h/l: a)h/l=1; b) h/l=6. Shtrikman (H) the Winkler foundation coefficients w on h/l=1; three nondimensional frequen- cies of the6 shows the effects whenthe Paternak = 2, L/hfoundation coefficient k p on first three Fig 6 FGM microbeams of the p = 20, n elastic foundation coefficient kp on first three Fig shows the effects of k Paternak elastic = 10 using homogenization techniques nondimensional shows the of FGM microbeams elastic foundation coefficient kp p on w=50, n=0.5 nondimensionalfrequencies effects of microbeams with Winkler foundation coefficients kfirst three Fig 66 shows the effects of the (H) and Mori–Tanaka foundation coefficients kwh/l three Fig Voigt (V), Hashin–Shtrikman Paternak with Winkler (M) and differentkratios first n=0.5 frequencies of FGM the Paternak elastic foundation coefficient on =50, andL/h=10 using homogenization FGMmicrobeams with Winkler foundation (H), Mori–Tanaka (M) and and nondimensional frequencies of FGM microbeams with Hashin–Shtrikman coefficients kkw=50,n=0.5 nondimensionalhomogenization techniques Voigt (V), Winkler foundation (H), Mori–Tanaka (M) and L/h=10 using frequencies of techniques Voigt (V), Hashin–Shtrikman coefficients w=50, n=0.5 and L/h=10 using homogenization techniques Voigt (V), Hashin–Shtrikmannondimensional frequencies seen first three (H), Mori–Tanaka (M) and and L/h=10 using homogenization techniques Voigt (V), Hashin–Shtrikman (H), Mori–Tanaka (M) and different ratios h/l=2 (Fig. 6a), h/l=8 (Fig. 6b). It can be seen first three nondimensional frequencies different ratios h/l=2 (Fig. 6a), h/l=8 (Fig. 6b). ItItcan be seen first three nondimensional frequencies different ratios h/l=2 (Fig. 6a), h/l=8 (Fig. 6b). It can be seen first three nondimensional frequencies different ratios h/l=2 (Fig. 6a), h/l=8 (Fig. 6b). can be
Vibration analysis of Timoshenko microbeams made of functionally graded materials on a Winkler–Pasternak ... 41 Fig. 5 shows the effects of the Winkler elastic foundation coefficient kw on first three nondimensional frequencies of FGM microbeams with Paternak foundation coefficients Vibration analysisandof Timoshenko microbeams made functionally graded materials on on Pasternak elastic k pVibration = 2 of Timoshenko microbeams made of of functionally graded materials a a Pasternak elastic = 30, n analysis L/h = 10 using homogenization techniques Voigt (V), Hashin–Shtrikman foundation foundation (H), Mori–Tanaka (M) and different ratios h/l = 1 (Fig. 5(a)), h/l = 6 (Fig. 5(b)). It can be 9 9 seen that the value of the first three nondimensional frequencies increase unremarkably, increase when the thethe ratiocoefficients kp increase. Moreover, thethe difference between the second especially when foundation coefficients kp increase. Moreover, difference between the second increase when foundation h/l is small. frequency andand the third frequency are remarkable when the ratio h/lhigher. frequency the third frequency are remarkable when the ratio h/l is is higher. (a) h/l = 2 (b) h/l = 8 Fig FigEffects of the the Paternak foundation coefficients on on the first three nondimensional frequencies 6. 6. Effects of Paternak foundation coefficients kp kp the first three nondimensional frequencies of the the FGM microbeams with =20, n=0.5, L/h=10 and using homogenization techniques Voigt (V), of FGM microbeams with kw kw =20, n=0.5, L/h=10 and using homogenization techniques Voigt (V), Fig.Hashin–Shtrikman (H), Mori–Tanaka (M) and different ratios h/l=2 first three nondimensional fre- Hashin–Shtrikmanthe Paternak foundation coefficients k p h/l=2 (Fig. 6a), h/l=8 (Fig. 6b). 6. Effects of (H), Mori–Tanaka (M) and different ratios on the (Fig. 6a), h/l=8 (Fig. 6b). 4. Conclusions quencies of the FGM microbeams with k w = 20, n = 0.5, L/h = 10 and using homogenization 4. Conclusions techniques Voigt (V), Hashin–Shtrikman (H), Mori–Tanaka (M) and different ratios h/l In thisthis work, free vibration FGM microbeams on on the Winkler-Pasternak elastic foundation is In work, free vibration of of FGM microbeams the Winkler-Pasternak elastic foundation is studied based on the the MCST, the TBT and Mori–Tanaka, Hashin-Shtrikman and Voigt homogenization studied based on MCST, the TBT and Mori–Tanaka, Hashin-Shtrikman and Voigt homogenization techniques. The differential equations free vibration for the foundation coefficient k pby using three techniques. The differential equations theof free vibration for the TBT microbeam are derived byfirstthethe Fig. 6 shows the effects of of Paternak elastic TBT microbeam are derived on using FEM andand Kosmatka’s shape functions. The influences thewith Winklerthe volume fraction index, thew nondimensional frequencies of FGM microbeams the size-effect, foundation coefficients k FEM Kosmatka’s shape functions. The influences of of size-effect, the volume fraction index, the slenderness 0.5 ratio, L/hfoundation parameters on the firsttechniques Voigtfrequencies of of the FGM = slenderness and and = 10 using homogenization three nondimensional frequencies the FGM 50, n = ratio, and foundation parameters on the first three nondimensional (V), Hashin–Shtrikman microbeams were discussed in detail. The obtained numerical results allow one to make thethe following microbeams were discussed in detail. The obtained numerical results allow one to make following (H), Mori–Tanaka (M) and different ratios h/l = 2 (Fig. 6(a)), h/l = 8 (Fig. 6(b)). It can be remarks: remarks: seen first three nondimensional frequencies increase when the foundation coefficients k p + The material length scale parameter plays an an important role thethe frequencies of microbeams. + The material length scale parameter plays important role in in frequencies of microbeams. increase. Moreover, the difference between the second frequency and volume fraction index The nondimensional frequencies decrease when the slenderness ratios h/l and the the third frequency The nondimensional frequencies decrease when the slenderness ratios h/l and the volume fraction index n are increase. n remarkable when the ratio h/l is higher. increase. + There are are “turning point” ratios L/h,which thethe given nondimensional frequency changes from + There “turning point” ratios L/h, at at which given nondimensional frequency changes from the increase to to constant. These turning point ratios are dependent on the given frequency and the the increase the the constant. These turning point ratios are dependent on the given frequency and the 4. CONCLUSIONS ratios h/l andand L/h. ratios h/l L/h. + Nondimensional frequencies using thethe Hashin-Shtrikman homogenization technique arelittle In + Nondimensional frequencies using Hashin-Shtrikman homogenization technique are a a little this work, free vibration of FGM microbeams on the Winkler–Pasternak elastic higher than nondimensional frequencies using the Mori-Tanaka homogenization technique, but both are higher than nondimensional based on the MCST, the TBT and Mori–Tanaka, Hashin-Shtrikman foundation is studied frequencies using the Mori-Tanaka homogenization technique, but both are smaller than nondimensional frequencies using thethe Voigt homogenization technique, especially for the smaller than nondimensional frequencies using Voigt homogenization technique, especially for the higher frequencies. higher frequencies. + Effects of of the Winkler elastic foundation coefficientsw kon on first three nondimensional + Effects the Winkler elastic foundation coefficients k w first three nondimensional frequencies of of FGM microbeams fewer than one of of Pasternak elastic foundation coefficients kp frequenciesFGM microbeams are are fewer than one thethe Pasternak elastic foundation coefficients,kp, especially the the ratio is small. especially ratio h/l h/l is small.
42 Tran Van Lien, Le Thi Ha and Voigt homogenization techniques. The differential equations of free vibration for the TBT microbeam are derived by using the FEM and Kosmatka’s shape functions. The influences of the size-effect, the volume fraction index, the slenderness ratio, and foun- dation parameters on the first three nondimensional frequencies of the FGM microbeams were discussed in detail. The obtained numerical results allow one to make the following remarks: - The material length scale parameter plays an important role in the frequencies of microbeams. The nondimensional frequencies decrease when the slenderness ratios h/l and the volume fraction index n increase. - There are “turning point” ratios L/h, at which the given nondimensional frequency changes from the increase to the constant. These turning point ratios are dependent on the given frequency and the ratios h/l and L/h. - Nondimensional frequencies using the Hashin–Shtrikman homogenization tech- nique are a little higher than nondimensional frequencies using the Mori–Tanaka ho- mogenization technique, but both are smaller than nondimensional frequencies using the Voigt homogenization technique, especially for the higher frequencies. - Effects of the Winkler elastic foundation coefficients k w on first three nondimen- sional frequencies of FGM microbeams are fewer than one of the Pasternak elastic foun- dation coefficients k p , especially the ratio h/l is small. All the mentioned notices are a useful indication for vibration analysis of FGM mi- crostructures. The study can be applied to more complex microstructures. DECLARATION OF COMPETING INTEREST The authors declare that they have no known competing financial interests or per- sonal relationships that could have appeared to influence the work reported in this paper. FUNDING This research received no specific grant from any funding agency in the public, com- mercial, or not-for-profit sectors. REFERENCES [1] F. Yang, A. C. M. Chong, D. C. C. Lam, and P. Tong. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39, (2002), pp. 2731–2743. https:/ /doi.org/10.1016/s0020-7683(02)00152-x.
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