International Journal of Engineering & Technology Sciences Volume 03, Issue 05, Pages , 2015

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1 International Journal of Engineering & Technology Sciences Volume 3, Issue 5, Pages , 25 Size-Dependent Behavior of Functionally Graded Micro-Beams, Based on the Modified Couple Stress Theory lireza Babaei a, *, hmad Ghanbari b, Farid Vakili-Tahami c a Mechatronic Research aboratory, Department of Mechanical Engineering, University of Tabriz, Tabriz, Iran b Mechatronic Research aboratory, Department of Engineering-Emerging Technologies, University of Tabriz, Tabriz, Iran c Department of Mechanical Engineering, University of Tabriz, Tabriz, Iran * Corresponding author. Tel.: ; address: babaeiar.mech.eng@gmail.com b s t r a c t Keywords: Functionally graded material micro-beam, Modified couple stress theory, Free vibration, Non-dimensional frequency, Euler-Bernoulli beam theory, ccepted:4 October25 In this article a micro-beam model based on the Euler-Bernoulli beam theory with consideration to the size dependent effects in functionally graded material (FGM) is studied. The mechanical properties of the micro beam are assumed to vary continuously through the thickness of the beam; this variation is assumed according to a power law. The equation of motion is derived based on the modified couple stress theory and using Hamilton s principle. n exact analytical solution is presented for free vibration behaviour of the proposed model. By comparison of the obtained results to the benchmark results available in the technical literature a good level of accuracy is obtained. lso the effects of the power index, material length scale parameter (the non-classical parameter) and slenderness ratio on the dynamic response of FG micro-beams are studied. cademic Research Online Publisher. ll rights reserved.. Introduction Micro-technology is chiefly concerned with fabrication of structures at micro-scale, most of these components are used in technological applications such as micro-electromechanical systems (MEMS), Nano-electromechanical systems (NEMS), atomic force microscopes (FMs), sensors, actuators and biosensors [6, 7, 8, 9, 2, 2]. Functionally graded materials (FGMs) are a kind of advanced composites. These graded materials belong to specific group of multiphase composites which possess intentionally smooth spatial variations of the volume fractions of the materials constituting the structure [3,,, 2, 3, 4]. s a result FGMs are inhomogeneous materials at which material properties can be considered as functions of the spatial coordinates. FGMs contain beneficial traits like improved stress dispensing, enhanced thermal resistance, higher toughness in fracture. lso for some laminated composites, there is a serious problem related to the great shear stresses originating due to the spanning from one surface to the other surface with superfluous different mechanical properties; this serious disadvantage can be obviated by exerting the continuous smooth variations of the constituents. Because of these variations and other benefits, FGMs are being used widely in diverse engineering areas such as biomechanics, optoelectronics, high temperature technologies and micro-technology [7, 8, 9]. One of the most interesting geometrical models used in these technologies are beams with different boundary conditions. Beams used in MEMS, NEMS and FMs, have the thickness in the order of microns and sub-microns, they are generally classified in the group of small- 364 P a g e

2 lireza Babaei et al. / International Journal of Engineering and Technology Sciences (IJETS) 3 (4): , 25. scale structures. Such structural elements exhibit size-dependent behaviour, which cannot be predicted by the classical continuum theories. In fact the non-classical elasticity theory should be able to consider the intrinsic micro-structural length scale parameter into the analysis. One of the most famous non-classical theories is modified couple stress theory. In this theory which is generated from the existence of rotational gradient of the strain tensor; a single material length scale parameter is needed. This parameter relates the symmetric part of the curvature tensor to the couple stress components [2, 4, 23]. Modified couple stress and other non-classical elasticity theories are used in various studies to analyse the static and dynamic responses of microstructural problems. Park and Gao (26) studied the static response of an Euler Bernoulli beam and interpreted the outcomes of an epoxy polymeric beam bending test [6]. Yong et al. (22) obtained the governing equation, boundary and initial conditions of an Euler Bernoulli beam using the modified coupled stress theory and the variationbased approach [5]. They declared that the natural frequencies of the beam are influenced by the size effects. lso, the difference between the natural frequencies obtained by the classical continuum theory and those calculated by the modified couple stress theory are significant. sghari et al. (2) used the modified couple stress theory to analyse the size-dependent behaviour of the FG Euler- Bernoulli micro-beams [5]. ghazadeh, Cigeroglu and Dag (24) studied the free vibration and static behaviour of small scale FG micro-beams using different beam theories and the modified couple stress theory []. lso they assumed the varying length scale parameter and its effects upon the tip deflection and frequency characteristics. nsari, Sahmani et al. (24) researched on the free vibration behaviour of FG micro-beams using the strain gradient theory and the Timoshenko beam theory [22]. They tried to simulate a model more realistic to the thick micro-beams. i et al. (23) used the modified strain gradient theory for the bending and free vibration analysis of FG piezoelectric beam, in which the electrical voltage effects upon the natural frequency is reported [24]. In this paper, a small-scale model of Euler- Bernoulli beam made up of functionally graded materials is studied for the size-dependent free vibration analysis. The methodology is based on the modified couple stress theory. The ratio of the beam length to the beam thickness influence (slenderness ratio s effect), volume fraction profiles of the constituent material phases and gradient index (power quantity) upon the natural frequency of the simply supported micro-beam are reported. The governing equation and boundary conditions are obtained using the Hamilton s principle and an exact solution procedure is applied for the results. The only similar article is the one done by sghari et al. (2) [5] which has some differences like; the metal constituent used in composition is different, they have used a parametric shape for bending and additive stiffness but we used nonparametric shape, their results are reported according to ratio of frequency but the results of this paper are in the form of non-dimensional frequency, their comparison base is the classical results but our base for comparison is the non-local elasticity theory, this paper is more comprehensive due to covering the response of the system with first five modes of vibration while they studied just the first mode and finally different boundary conditions are chosen in this paper. lso the solution procedures are different. 2. Functionally graded materials functionally graded micro-beam of length, width b and thickness h is shown in Figure. The Cartesian coordinates are set like x = x, y = x 2, z = x 3. It is supposed that the micro-beam is made up of two dissimilar materials, Steel (metal constituent) and lumina (non-metal constituent), so that the effective mechanical properties vary through the thickness direction (x 3 ). Based on the rule of mixtures, the effective properties (P) can be obtained as: P = P a V a + P s V s () Where P a and P s are the effective mechanical properties, also V a and V s are the volume fractions of the constituents. Volume fractions are related to each other by a constraint equation like: V a + V s = (2) 365 P a g e

3 lireza Babaei et al. / International Journal of Engineering and Technology Sciences (IJETS) 3 (4): , 25. In this study the effective material properties of the FG micro-beam are defined by the power-law form. The volume fraction of the second material (alumina) is defined as: V a = ( x 3 h + 2 )k (3) Where k is the power-law exponent, which shows the material variation contour throughout the beam thickness. Using Equations (2) and (3), the effective material properties of the FG micro-beam take the form: P(x 3 ) = (P a P s ) ( x 3 h + 2 ) k + P s (4) Now Equation (4) will describe the Young s modulus, modulus of rigidity and density of the structure as: E(x 3 ) = (E a E s ) ( x 3 h + 2 ) k + E s (5) G(x 3 ) = (G a G s ) ( x 3 h + 2 ) k + G s (6) ρ(x 3 ) = (ρ a ρ s ) ( x 3 h + 2 ) k + ρ s (7) From Equations (5-7) it is obvious that at the upper and lower surfaces, one can reach pure non-metal and metal constituent, as follows: If x 3 = + h 2, then E = E a, G = G a, ρ = ρ a If x 3 = h 2, then E = E s, G = G s, ρ = ρ s ccording to the modified couple stress theory total strain energy of a linear-homogenous continuum is defined as: U s = 2 (σ ij: ε ij + m ij : χ ij ) dv V (8) The integrand function denotes the strain density function. Where σ ij designates the classic stress tensor, ε ij is the classic strain tensor, m ij represents the deviator part of the couple stress tensor and χ ij stands for the symmetric curvature tensor. The tensors ε ij and χ ij are defined by Equations (9), (). ε ij = 2 (u ij + u ji ) (9) χ ij = 2 (e ipqε qj,p + e jpq ε qi,p ) () u in equation (9) is the displacement vector; e ipq in Equation () denotes the alternating tensor and comma refers to differentiation. Constitutive relations regarding the classic stress tensor and the deviatoric part of the couple stress tensor is to be written in the following form: σ ij = 2με ij + λδ ij ε kk () m ij = 2μl 2 χ ij (2) Where λ and μ are two classical ame parameters. lso ν is the Poisson s ratio. By means of two constraint equations, the number of elastic parameters essential to explain the mechanical properties of a structure will reduce to two parameters, the constraint equations mentioned above are: λ = Eν (+ν)( 2ν), G = E 2(+ν) (3) Fig. : Schematic of the micro FG beam. 3. Mathematical formulation l In Equation (2) is the material length scale parameter. Existence of this non-classic parameter is the proof of capturing the size-dependency in the structure which can be realized as an elastic parameter. 3.. Modified couple stress theory 366 P a g e

4 lireza Babaei et al. / International Journal of Engineering and Technology Sciences (IJETS) 3 (4): , Euler-Bernoulli model for a micro-beam The general displacement components in an Euler- Bernoulli beam can be represented by Park and Gao (26) [6]: u (x, x 3, t) = x 3 w (4) u 2 (x, x 3, t) = (5) u 3 (x, x 3, t) = w(x, t) (6) It is clear that the displacement component along the beam length is proportional to the minus beam slope or rotation angle, the component across the beam width is neglected due to the lateral vibration case. Finally the displacement component along the thickness direction is shown by w, which is a function of x and time. By utilizing these displacement fields and Equations (9)-(2), one can derive the elements of ε ij, χ ij, σ ij and m ij as follows: ε = x 3 2 (7) χ 2 = χ 2 = 2 2 (8) σ = Ex 3 2 (9) m 2 = m 2 = Gl 2 2 w 2 (2) The other components of the classic and nonclassic stresses and strain tensors are zero. By substitution of Equations (7)-(2) into the Equation (8) one can obtain: Us = 2 2 w [( 2 2 ) 2 (E(x 3 )x G(x 3)l 2 )]ddx (2) T = 2 ρ(x 3)( w t )2 ddx 3.3. Equation of motion (22) Hamilton s principle which seems to be the most advanced variation-based principle of mechanics considers the entire motion of the system between two instants, thus it is an integral principle and the problems of dynamics are reduced to the scalar integral. Based on the Hamilton's principle, the governing dynamic equation of the beam containing initial conditions and boundary conditions can be determined by using the following variation-based equation: t 2 δ [ (T Us)dt] = (23) t By exerting the variation operator to the energy terms, the following expressions will be generated: δus = [(E(x 3 )x G(x 3)l 2 ) ( 2 w 2 ) 2 2 δw] ddx (24) δus 2 = [(S + S 2 ) 2 2 ]δwdx (25) Where S and S 2 are defined such as follows: S = E(x 3 )(x 3 ) 2 d (26) S 2 = G(x 3 )l 2 d (27) nd variation of the kinetic energy is such like Equation (28): w δt = ρ(x 3 )I t t δwdx (28) fter calculating the potential energy, total kinetic energy of the Euler- Bernoulli beam can be expressed as: In which I is mass inertia, defined as: I = ρ(x 3 )d (29) 367 P a g e

5 lireza Babaei et al. / International Journal of Engineering and Technology Sciences (IJETS) 3 (4): , 25. By substituting Equations (25)-(28) into Equation (23) the dynamic equation is changed into the double integral as follows: t t 2 w { [ρ(x 3 )I t t δw (S 2 + S 2 ) 2 w 2 2 δw]dx } dt (3) By integration by parts on Equation (3), Equation (3) is concluded: t t 2 { [ I t 2 (S + S 2 ) 4 w 4 ] dx } dt t 2 + {[(S + S 2 ) 2 w δw] 2 t [(S + S 2 ) 3 w 3 δw] }dt (3) Eventually dynamic-vibration equation for a FG Euler-Bernoulli beam is obtained from Equation (3) as follows: I t 2 + (S + S 2 ) 4 w 4 = (32) ccording to the mechanical properties distribution from Equations (5)-(7), S, S 2 and I are calculated as follows: S = E(x 3 )(x 3 ) 2 d = bh 3 k 2 + k + 2 ( 4(k + )(k + 2)(k + 3) (E a E s ) + 2 E s) (33) S 2 = G(x 3 )l 2 d = bhl 2 ( k + (G a G s ) + G s ) (34) I = ρ(x 3 )d = bh( k + (ρ a ρ s ) + ρ s ) (35) 3.4. Solution procedure For the case of free vibration, the transverse displacement holds a harmonic-wise variation with respect to time, on the other hand due to homogeneity of the governing equation, the method of separation of variables can be used: w(x, t) = Y(x )e iωt (36) Where ω denotes the natural frequency of vibration, and i =. Y(x ) lso shows the transverse vibration amplitude. Using Equation (37) gives the following Equation: d 4 Y dx 4 I S + S 2 ω 2 Y = (38) Now we introduce auxiliary parameter: I S + S 2 ω 2 = β 4 (39) The general solution for Equation (38) is in the form: Y(x ) = C cosh βx + C 2 sinh βx + C 3 cos βx + C 4 sin βx (4) For a simply-supported beam the boundary conditions are: w x =, =, 2 x =, = Exerting the boundary conditions gives a homogeneous system of equations which can be set in a matrix form like following: []{C} = (4) For such a like system, we expect the determinant of the coefficient matrix to be zero which gives the frequency equation in the simple form: Sin β n l = (42) By calculating frequency of transverse vibration, non-dimensional frequency can be obtained for first five modes of vibration. 368 P a g e

6 lireza Babaei et al. / International Journal of Engineering and Technology Sciences (IJETS) 3 (4): , 25. Table : Mechanical properties of FGM constituents. Properties Steel lumina ρ 78 (Kg m 3 ) 396 (Kg m 3 ) E 2 (GPa) 39 (GPa) ν.3.24 l h Table 2: Comparison of non-dimensional fundamental natural frequency (Classic theory) ω n k = k = et al. (22) k =. k =. et al. (22) k =. 2 k =. 2 et al. (22) 5 n = n = n = n = n = n = n = n = n = n = Table 3: Comparison of non-dimensional fundamental natural frequency (Classic theory) l h ω n k =. 5 k =. 5 et al. (22) k = k = et al. (22) k = 2 k = 2 et al. (22) 5 n = n = n = n = n = n = n = n = n = n = P a g e

7 lireza Babaei et al. / International Journal of Engineering and Technology Sciences (IJETS) 3 (4): , 25. Table 4: Comparison of non-dimensional fundamental natural frequency (Classic theory) l h ω n k = 5 k = 5 et al. (22) k = k = et al. (22) 5 n = n = n = n = n = n = n = n = n = n = Table 5: Variation of non-dimensional fundamental natural frequency (modified couple stress theory) l h ω n k =. k =. k =. 2 k =. 5 2 n = n = n = n = n = n = n = n = n = n = n = n = 2 n = 3 n = 4 n = Table 6: Variation of non-dimensional fundamental natural frequency (modified couple stress theory) l h ω n k = k = 2 k = 5 k = 2 n = n = n = n = n = n = n = n = n = n = n = n = 2 n = 3 n = 4 n = P a g e

8 lireza Babaei et al. / International Journal of Engineering and Technology Sciences (IJETS) 3 (4): , Conclusion In the present study, free vibration analysis of micro-fg beams is established. Non-classical constitutive terms are applied to the strain density function. Equations of motion are obtained using the Hamilton s principle. The new model predicts the frequencies a bit larger than the classical theory. Results show that the modified parameter plays a major role in the dynamic behaviour of micro-structures. dditionally it can be realized that using the proper power-law index for design and fabrication of functionally graded materials is vital; such that dynamic behaviour can be enhanced by taking a profoundly fastidious look. cknowledgments This work has been accomplished in the Mechatronic Research aboratory of the University of Tabriz. References [] R ghazadeh, E Cigeroglu, S Dag. Static and free vibration analyses of small-scale functionally graded beams possessing a variable length scale parameter using different beam theories. European Journal of Mechanics-/Solids 24; 46: [2] M Simsek. Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He s variational method. Composite Structures 24; 2: [3] O Rahmani, O Pedram. nalysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory, International Journal of Engineering Science 23; 77: [4] R nsari, R Gholami, S Sahmani. Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory, Composite Structures 2; 94(): [5] M sghari, M hmadian, T Kahrobaiyan, MH Rahaeifard, On the size-dependent behaviour of functionally graded micro-beams. Materials & Design 2; 3(5): [6] H Rokni, Milani, S, RJ Seethaler, Sizedependent vibration behaviour of functionally graded CNT-Reinforced polymer microcantilevers: Modeling and optimization. European Journal of Mechanics-/Solids 25; 49: [7] N Fleck, GM Muller, MF shby, JW Hutchinson, Strain gradient plasticity: theory and experiment. cta Metall Mater 992; 42(2): [8] JS Stolken, G Evans, Microbend test method for measuring the plasticity length scale. cta Mater 998; 46(4): 59 5 [9] CM Chong, DCC am, Strain gradient plasticity effect in indentation hardness of polymers. Mater Res 999; 4(): 43 [] W McFarland, JS Colton, Role of material microstructure in plate stiffness with relevance to microcantilever sensors. Micromech Microeng 25; 6 7. [] EC ifantis, Strain gradient interpretation of size effects. International Journal of Fracture 995; 95: [2] WT Koiter, Couple-stresses in the theory of elasticity, I and II. Proc K Ned kad Wet B 964; 67: [3] RD Mindlin, HF Tierstenm, Effects of couplestresses in linear elasticity. rch Ration Mech nal 962; (): [4] R Toupin, Elastic materials with couplestresses. rch Ration Mech nal 962; (): [5] FCM Yang,.C.M. Chong, D.C.C. am, P. Tong, Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures 22; 39(): P a g e

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