rotating laminated nano cantilever beam

Mechanics of Advanced Materials and Structures ISSN: 1537-6494 (Print) 1537-6532 (Online) Journal homepage: http://www.tandfonline.com/loi/umcm20 Nonlocal nonlinear bending and free vibration analysis of a rotating laminated nano cantilever beam K. Preethi, P. Raghu, A. Rajagopal & J. N. Reddy To cite this article: K. Preethi, P. Raghu, A. Rajagopal & J. N. Reddy (2017): Nonlocal nonlinear bending and free vibration analysis of a rotating laminated nano cantilever beam, Mechanics of Advanced Materials and Structures, DOI: 10.1080/15376494.2016.1278062 To link to this article: http://dx.doi.org/10.1080/15376494.2016.1278062 Accepted author version posted online: 11 Jan 2017. Published online: 11 Jan 2017. Submit your article to this journal Article views: 36 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=umcm20 Download by: [103.232.241.5] Date: 10 April 2017, At: 21:33 MECHANICS OF ADVANCED MATERIALS AND STRUCTURES http://dx.doi.org/./.. ORIGINAL ARTICLE Nonlocal nonlinear bending and free vibration analysis of a rotating laminated nano cantilever beam K. Preethia , P. Raghua , A. Rajagopala , and J. N. Reddyb a Department of Civil Engineering, Indian Institute of Technology Hyderabad, Hyderabad, Telangana, India; b Department of Mechanical Engineering, Texas A&M University, College Station, Texas, USA ABSTRACT ARTICLE HISTORY In this article, we present the nonlocal, nonlinear inite element formulations for the case of nonuniform rotating laminated nano cantilever beams using the Timoshenko beam theory. The surface stress efects are also taken into consideration. Nonlocal stress resultants are obtained by employing Eringen’s nonlocal diferential model. Geometric nonlinearity is taken into account by using the Green Lagrange strain tensor. Numerical solutions of nonlinear bending and free vibration are presented. Parametric studies have been carried out to understand the efect of nonlocal parameter and surface stresses on bending and vibration behavior of cantilever beams. Also, the efects of angular velocity and hub radius on the vibration behavior of the cantilever beam are studied. Received  December  Accepted  December  1. Introduction The properties of any material at the macroscopic scale are inluenced by the inhomogeneities present at the microscopic scale and nanoscale. This points to the need for incorporating micromotions in continuum mechanical formulations [1, 2]. There has been considerable focus toward the development of generalized continuum theories [3], which accounts for the inherent microstructure in such natural and engineering materials (see [4, 5]).The notion of generalized continua uniies several extended continuum theories that account for such a size dependence due to the underlying microstructure of the material. A systematic overview and detailed discussion of generalized continuum theories have been given by Bazant and Jirasek [6]. These theories can be categorized as gradient continuum theories (see, for instance, the works of Mindilin et al. [7–9], Toupin [10], Steinmann et al. [1, 11–13], Casterzene et al. [14], Fleck et al. [15, 16], and Askes et al. [17–19]), microcontinuum theories (see works by Eringen [3, 20, 21], Steinmann et al. [22, 23], and nonlocal continuum theories (see works by Eringen [24], Jirasek [25], Reddy [26], and others [27–30]). In some of the earlier works, the higher-order gradient theory for inite deformation has been elaborated (see, for instance, [14, 31–33]) within classical continuum mechanics in the context of homogenization approaches. A comparison of various higher-order gradient theories can be found in [15]. A more detailed formulation of gradient approach in spatial and material setting has been presented in [22], and an overview of nonlocal theories of continuum mechanics can be found in [34]. The nonlocal formulations can be of integral-type with weighted spatial averaging or by implicit gradient models which are categorized as strongly nonlocal, while weakly nonlocal theories include, for instance, explicit gradient models [6]. In KEYWORDS Timoshenko beam; nonlocal parameter; surface stress; geometric nonlinearity; finite element analysis; numerical results this work, we consider a strongly nonlocal problem. The Timoshenko beam can be considered as a speciic one-dimensional version of a Cosserat continuum. Reddy [26] reformulated various beam theories such as Euler–Bernoulli, Timoshenko, Reddy, and Levinson beam theory using Eringen’s nonlocal diferential constitutive model. The analytical solutions for bending, buckling, and free vibrations were also presented in [26]. Various shear deformation beam theories were also reformulated in recent works by Reddy [35] using nonlocal diferential constitutive relations. Similar works have been carried out to study bending, buckling, and free vibration of nanobeams by Aydogdu [27] and Civalek [28]. Classical continuum mechanics takes exclusively the bulk into account, nevertheless, neglecting possible contributions from the surface of the deformable body. However, surface efects play a crucial role in the material behavior, the most prominent example being surface tension . A mathematical framework was irst developed by Gurtin [36] to study the mechanical behavior of material surfaces. The efect of surface stress on wave propagation in solids has also been studied by Gurtin [37]. The tensorial nature of surface stress was established using the force and moment balance laws. Bodies whose boundaries are material surfaces are discussed and the relation between surface and body stress examined in a recent work by Steinmann [38] and by Hamilton [39]. The surface efects has been applied to modeling two- [40] and three-dimensional continua in the frame work of inite element method (FEM) [41, 42]. Similar studies on static analysis of nano beams using nonlocal inite element models have been conducted by Mahmoud [43]. Eringen’s nonlocal elasticity theory has also been applied to bending, buckling, and vibration of nano beams using the CONTACT A. Rajagopal [email protected] Department of Civil Engineering, Indian Institute of Technology Hyderabad, Hyderabad , Telangana, India. Color versions of one or more of the figures in this article can be found online at www.tandfonline.com/umcm. ©  Taylor & Francis Group, LLC 2 K. PREETHI ET AL. Timoshenko beam theory (see [44–47]). Analytical solutions for beam bending problems for diferent boundary conditions were derived using the nonlocal elasticity theory and the Timoshenko beam theory by Wang et al. [48]. A inite element framework for nonlocal analysis of beams has also been made in a recent work by Sciarra et al. [49]. Nonlocal elastic rod models have been developed to investigate the small-scale efect on axial vibrations of the nano rods by Aydogdu [50] and Adhikari et al. [51]. Free vibration analysis of microtubules based on nonlocal theory and the Euler–Bernoulli beam theory was carried out by Civalek et al. [28]. Free vibration analysis of functionally graded carbon nanotubes with various thicknesses, based on the Timoshenko beam theory, has been investigated to obtain numerical solutions using Diferential Quadrature Method (DQM) by Janghorban et al. [52] and others [53–55]. Analytical study on the nonlinear free vibration of functionally graded nano beams incorporating surface efects has been presented in [56–58]. The efect of surface stresses on bending properties of metal nanowires is presented in [59]. Free vibration analysis of rotating nano cantilevers using the nonlocal theory and the Euler– Bernoulli beam theory has been carried out in [60] and [61]. The focus of this work is on nonlocal nonlinear formulation together with surface efects for static and free vibration analysis of rotating layered nano cantilever beams using Timoshenko beam theory. A nonlocal nonlinear inite element formulation for the case of nonuniform rotating isotropic and laminated nano cantilever beams using the Timoshenko beam theory is presented. The surface stress efects and the geometric nonlinearity is taken into account by using the Green Lagrange strain tensor. Numerical results are presented to bring out the parametric efects of nonlocal parameter and surface stresses on bending and vibration behavior of layered nano cantilever beams. lattice dynamics with nonlocal results [24]. For example Kernel function for two-dimensional (2D) problems has the form: K(|x|) = (2π 2 l 2 τ 2 )−1 K0 (|x|/lτ ), τ = e0 a/l, (3) where K0 is the modiied Bessel function, a and l are internal and external characteristic lengths, e0 is the material constant which is deined by the experiment. In the nonlocal linear elasticity, equations of motion can be obtained from nonlocal balance law: σi j, j + fi = ρ ü (4) where i, j take symbols x, y, z, and fi , ρ and ui are the components of the body force, mass density and displacement vector, respectively. By substituting Eq. (1) into Eq. (4), the integral form of nonlocal constitutive equation is obtained. Because solving an integral equation is more diicult than a diferential equation, Eringen [24] proposed a diferential form of nonlocal constitutive equation as: (1 − τ 2 l 2 ∇ 2 )σi j = ti j (5) Eq. (5) is more convenient than the integral relation (1) to apply to various linear elasticity problems. 3. Laminated beams 3.1. Classical Timoshenko beam theory In the Timoshenko beam theory, the efects of shear deformation are also considered. Distribution of transverse shear stress is assumed to be constant through the thickness. The displacement ield is given by: u(x, z, t ) = [u(x, t ) + zϕx ] êx + w(x, t )êz 2. Nonlocal theories In classical elasticity, stress at a point is a function of strain at that point, whereas in nonlocal elasticity, stress at a point is a function of strains at all points in the continuum. In nonlocal theories, forces between the atoms and internal length scale are considered in the constitutive equation. Nonlocal theory was irst introduced by Eringen [3]. According to Eringen, the stress ield at a point x in an elastic continuum not only depends on the strain ield at that point but also on the strains at all other points of the body. Eringen attributed this fact to the atomic theory of lattice dynamics and experimental observation on phonon dispersion. The nonlocal stress tensor σ at a point x in the continuum is expressed as Eringen expressed a constitutive model that expresses the nonlocal stress tensor at a point x as:  (1) σi j = K(|x′ − x|, τ )ti j (x′ )dv (x′ ) v where the volume intergral in Eq. (1) is over the region v occupied by the body and ti j is the Hookean stress tensor at point x deined as: ti j = ci jkl εkl (2) and the kernel function K(|x′ − x|, τ ) represents the nonlocal modulus, |x′ − x| is the distance and τ is the material constant that depends on internal and external characteristic lengths. The Kernel function can be obtained by matching the (6) The nonzero components of Lagrangian strain tensor can be written as:      du 1 dw 2 dϕx + εxx = +z dx 2 dx dx (0) (1) = εxx + z εxx   1 dw εxz = ϕx + 2 dx     ∂u 2 ∂w 2 ∂w 1 1 + εzz = + = ϕx2 ∂z 2 ∂z ∂z 2 (7a) (7b) (7c) and εzz is positive and nonzero. Its contribution is through a material length scale. Therefore, for a laminate layer, following stress-strain relationship is used: σxx = Q̄11 (εxx − ςxx T ) + αεzz σxz = Ks Q̄55 γxz σzz = αεxx + βεzz (8a) σ s = τ 0 + E s εxx (8b) where α and β are material length scale parameters, Ks is shear correction factor and: Q̄11 = Q11 cos4 θ + 2(Q12 + 2Q66 ) sin2 θ cos2 θ + Q22 sin4 θ (9) MECHANICS OF ADVANCED MATERIALS AND STRUCTURES Q̄55 = Q55 cos2 θ + Q44 sin2 θ (10) ςxx = ς1 cos2 θ + ς2 sin2 θ ν12 E2 E1 Q11 = , Q12 = , Q66 = G12 1 − ν12 ν21 1 − ν12 ν21 E2 , Q55 = G13 , Q44 = G23 Q22 = 1 − ν12 ν21 (11) ρA2 (L − x)(L + x − 2r) 2 (13) (14) Therefore, the stress resultants can be written as: N Nxx = k=1 Mxx zk+1  σ s ds bσxx dz + zk Ŵ ρA2 + (L − x)(L + x − 2r) 2  N zk+1 bσxx z dz + σ s z ds = k=1 N Nzz = k=1 N Nxz = k=1 zk zk+1  zk+1 N 2 (zk+1 − zk2 )b, k=1 1 2 N (k) (k) 2 ςxx (zk+1 − zk2 )b Q̄11 k=1 3.2. Nonlocal Timoshenko beam theory Using Eq. (5), the nonlocal stress resultants in terms of strains can be written as: nl d 2 Nxx (0) (1) + Ãεxx + B̃εxx + C̃εzz + 2τ 0 (b + h) − H̃ T dx2 ρA2 (L − x)(L + x − 2r) (18) + 2 nl d 2 Mxx nl (1) (0) ˜ zz − Õ T Mxx =μ + Jε (19) + D̃εxx + B̃εxx dx2 (0) nl + F̃εzz (20) = C̃εxx Nzz nl =μ Nxx (15a) (15b) bσzz dz (15c) bσxz dz (15d) nl d 2 Nxz + G̃γxz 2 dx (21) 3.3. Equations of motion Let us consider the transversely applied point load on the cantilever beam as Dirac’s delta function given as: P = Q0 δ(x − x p ) zk zk Using Eqs. (7) and (8), the stress resultants in Eq. (15) can be written as: (0) Nzz = C̃εxx + F̃εzz (16c) Nxz = G̃γxz (16d) (22) where Q0 is the point load applied at the point x p on the beam. By using the principle of virtual work, the equations of motion for cantilever beam using Timoshenko beam theory can be obtained as: d2 u dNxx + fx = m0 2 dx dt   d dw d2 w Nxz + Nxx + fz + P = mo 2 dx dx dt dMxx d2 ϕ − (Nxz + Nzz ϕx ) = m1 2 dx dt (0) (1) Nxx = Ãεxx + B̃εxx + C̃εzz + 2τ 0 (b + h) − H̃ T ρA2 (L − x)(L + x − 2r) (16a) + 2 (1) (0) ˜ zz − Õ T Mxx = B̃εxx + Jε (16b) + D̃εxx (23) (24) (25) where: mi = where:  ρzi dA A and fx and fz are the axially and transversely distributed forces, respectively. Manipulating the equations of motion and using Eqs. (18)–(21), the following relations are obtained: N (k) (zk+1 − zk )b + 2E s (b + h) , Q̄11 à = k=1 B̃ = Õ = (17d) nl Nxz =μ Ŵ  1 2 (12) where E1 , E2 , ν12 , G12 , G13 and G23 are six independent engineering constants and θ is the orientation of the laminate layer. The axial force due to rotation of a cantilever beam is given as: Nax = J˜ = 3 1 2 1 D̃ = 3 N (k) 2 Q̄11 (zk+1 − zk2 )b (17a) k=1 N (k) 3 Q̄11 (zk+1 − zk3 )b + k=1 E s   h3 bh2 + , 6 2 Q̄55 (zk+1 − zk )b (17b) k=1 N N α(zk+1 − zk )b, C̃ = β(zk+1 − zk )b, F̃ = N (k) (k) ςxx (zk+1 − zk )b Q̄11 H̃ = k=1 (0) (1) Ãεxx + B̃εxx + C̃εzz + 2τ 0 (b + h)   dw d3 u d fx + μm0 −μ dx dxdt 2 dx × k=1 k=1 nl Mxx (17c) d fx dx d3 u ρA2 − H̃ T + (L − x)(L + x − 2r) dxdt 2 2 (26a) 3 d ϕ (0) (1) ˜ zz − Õ T + μm1 = B̃εxx + D̃εxx + Jε dxdt 2 d d2 w +μm0 2 − μ fz − μP − μ dx  dt +μm0 N G̃ = Ks nl (0) (1) = Ãεxx + B̃εxx + C̃εzz + 2τ 0 (b + h) − μ Nxx 4 K. PREETHI ET AL. −μ d dx +μ d dx   dw ρA2 (L − x)(L + x − 2r) 2 dx    (0) C̃εxx (26b) + F̃εzz ϕx   d2 w d4 w d 2 fz d  G̃γ − μ − μm = + fz + P xz 0 dt 2 dx2 dt 2 dx dx2 d  (0) (1) Ãεxx + B̃εxx + C̃εzz + 2τ 0 (b + h) + dx   dw d3 u d fx + μm0 −μ dx dxdt 2 dx    ρA2 d dw −H̃ T + + (L − x)(L + x − 2r) dx 2 dx d 3  (0) (1) −μ 3 Ãεxx + B̃εxx + C̃εzz + 2τ 0 (b + H ) dx   dw d3 u d fx −μ + μm0 dx dxdt 2 dx    3 ρA2 dw d −H̃ T + −μ 3 (L − x)(L + x − 2r) dx 2 dx (28)  d 4 ϕx d 2 ϕx d  (0) (1) ˜ zz − Õ T + Jε m1 2 − μm1 2 2 = B̃εxx + D̃εxx dt dx dt dx     d 2  (0) (0) + F̃εzz + μ 2 C̃εxx −G̃γxz − ϕx C̃εxx + F̃εzz ϕx dx (29) −H̃ T + nl (0) Nzz = C̃εxx + F̃εzz m0 (26c) d fz dx  (0) (1) Ãεxx + B̃εxx + C̃εzz + 2τ 0 (b + h) nl Nxz = G̃γxz − μ −μ d2 dx2   dw d fx d3 u −μ + μm0 dx dxdt 2 dx    ρA2 dw d2 −H̃ T + (L − x)(L + x − 2r) −μ 2 dx 2 dx (26d) By substituting the expressions for nonlocal stress resultants (26) back in the equations of motion (23)–(25), we obtain the equilibrium equation for nonlocal Timoshenko beam theory including surface stress efects as: d4 u d2 u − μm0 2 2 2 dt dx dt   d d fx (0) (1) 0 = Ãεxx + B̃εxx + C̃εzz + 2τ (b + h) − μ dx dx   ρA2 d −H̃ T + (L − x)(L + x − 2r) + fx + dx 2 (27) m0 3.4. Finite element formulation The principle of virtual work for the Timoshenko beam has the form:  l nl (0) nl (1) nl nl 0= [Nxx δεxx + Mxx δεxx + Nxz δγxz + Nzz δεzz 0 − fx δu − fz δw − Pδw + m0 üδu + m0 ẅδw + m1 ϕ̈δϕ] dx −Q1 δu(0) − Q4 δu(l) − Q2 δw(0) − Q5 δw(l) − Q3 δϕ(0) −Q6 δϕ(0) (30) After substituting the expressions for stress resultants from Eq. (26) into Eq. (30), we obtain: T  l d3 u d fx (0) (1) + μm0 0= − H̃ T + B̃εxx + C̃εzz + 2τ 0 (b + H ) − μ Ãεxx dx dxdt 2 0 0    ρA2 d3 ϕ d2 w (1) (0) + (L − x)(L + x − 2r) δεxx + μm1 + μm − μ f − μP δεxx 0 z 2 dxdt 2 dt 2   (0) (1) (1) ˜ zz − Õ T δεxx + D̃εxx + Jε + B̃εxx    dw d3 u d d fx (1) (0) (1) 0 δεxx + μm0 −μ Ãεxx + B̃εxx + C̃εzz + 2τ (b + H ) − μ dx dx dxdt 2 dx    ρA2 dw d (1) −H̃ T + δεxx (L − x)(L + x − 2r) −μ dx 2 dx     d3 w d fz d  (0) (1) C̃εxx + F̃εzz ϕx δεxx + G̃γxz + μm0 δγxz −μ +μ dx dxdt 2 dx    dw d2 d3 u d fx (0) (1) δγxz + μm0 −μ 2 + B̃εxx + C̃εzz + 2τ 0 (b + H ) − μ Ãεxx dx dx dxdt 2 dx    ρA2 d2 dw δγxz −H̃ T + −μ 2 (L − x)(L + x − 2r) dx 2 dx   (0) + F̃εzz δεzz − fx δu − fz δw − Pδw + C̃εxx  +m0 üδu + m1 ẅδw + m1 ϕ¨x δϕx dx [−Q1 δu(xa ) − Q4 δu(xb ) − Q2 δw(xa ) − Q5 δw(xb ) − Q3 δϕ(xa ) − Q6 δϕ(xb )] dT MECHANICS OF ADVANCED MATERIALS AND STRUCTURES The underlined expressions in the above equation do not allow us to construct a quadratic functional. So, after omitting the underlined expressions in Eq. (31), it can be equivalently written into the following three equations:  T  l  (0) (1) Ãεxx + B̃εxx + C̃εzz + 2τ 0 (b + H ) 0 0  dδu d fx d3 u −μ + μm0 dx dxdt 2 dx   dδu ρA2 (L − x)(L + x − 2r) + −H̃ T + 2 dx  (32) 0 expressed as: ⎤⎧ K 11 K 12 K 13 ⎨ ⎣K 21 K 22 K 23 ⎦ ⎩ K 31 K 32 K 33 ⎧ 1⎫ ⎨F ⎬ = F2 ⎩ 3⎭ F T  l (0) (1) Ãεxx + B̃εxx + C̃εzz + 2τ 0 (b + H ) 0  d fx dw dδw d3 u −μ + μm0 2 dx dxdt dx dx   dw dδw ρA2 (L − x)(L + x − 2r) + −H̃ T + 2 dx dx   d3 w d fz dδw −μ + G̃γxz + μm0 2 dxdt dx dx + [Q2 δw(xa ) − Q5 δw(xb )] dT = 0   l d3 w d fz δϕx −μ G̃γxz + μm0 dxdt 2 dx 0 0   d 3 ϕx d2 w dδϕx + μm1 + μm0 2 − μ fz − μP 2 dxdt dt dx   dδϕ x (1) (0) ˜ zz − Õ T + B̃εxx + Jε + D̃εxx dx   (0) + F̃εzz ϕx δϕx + m1 ϕ¨x δϕx dx + C̃εxx =  =  Ki13j =  Ki21j =  Ki11j Ki12j à l 0 Ki22j = (33) Ki23j = Ki31j = Ki32j = Ki33j = (34) The generalized displacements (ū, w̄, ϕ¯x ) are approximated using the Lagrange interpolation functions: 1 (1) j ψ j (x) l 0 l (1) à 0 dw dψi(2) dψ j dx dx dx dx (2) dψ (2) dψ j G̃ i dx dx 0     (2) dψ (2) 1 dw 2 j 2 dψi dx (37) + + C̃ (ϕx ) à 2 dx dx dx  l  l (3) dψ (2) dw dψi(2) dψ j B̃ G̃ i ψ j(3) dx + dx dx dx dx 0 0   l (1) (1) (3) l dψ j dψ dψ j B̃ i dx + dx C̃ϕx ψi(3) dx dx dx 0 0  l  l (2) dψ j(2) 1 dw dψi(3) dψ j dx + B̃ dx G̃ψi(3) dx dx 0 2 dx dx 0  l (3) dψ (3) dψ (3) dψ j 1 ˜ x i ψ j(3) + G̃ψi(3) ψ j(3) + Jϕ D̃ i dx dx 2 dx 0    2 1 dw + + F̃ (ϕx )2 ψi(3) ψ j(3) dx C̃ 2 dx  l (1) Mi11j = m0 ψi(1) ψ j(1) + μm0 dψi(1) dψ j dx dx (1) Mi21j = μm0 (35b) Mi22j dw dψi(2) dψ j dx dx dx (2) n 2 (2) j ψ j (x) = j=1 p 3 (3) j ψ j (x)By ϕ¯x (x) = dψi(1) dψ j dx dx dx (2) (35a) j=1 w̄(x) = (36) 1 dw dψi(1) dψ j dx à 2 dx dx dx  l (3) dψ (1) dψ j dψ (1) 1 B̃ i dx C̃ϕx i ψ j(3) dx + 2 dx dx dx 0 m ū(x) = ⎤⎧ ⎫ M 13 ⎨ ¨ 1 ⎬ M 23 ⎦ ¨ 2 ⎩ ¨ 3⎭ M 33 (1) l 0 T + [−Q3 δϕx (xa ) − Q6 δϕx (xb )] dT = 0 M 12 M 22 M 32 α Miαβ j , and force coeicients Fi (α, β = 1, 2, 3) are deined as follows: − fz δw − Pδw + m0 ẅδw dx  ⎫ ⎬ ⎡ 11 M 2 + ⎣M 21 3⎭ M 31 1 ⎡ where the stifness coeicients Kiαβ j , mass matrix coeicients − fx δu + m0 üδu dx + [−Q1 δu(xa ) − Q4 δu(xb )] dT  5 (35c) j=1 substituting Eq. (35) for ū, w̄ and ϕ¯x , and putting δ ū = ψi1 , δ w̄ = ψi2 , δ ϕ¯x = ψi3 into the weak form statements (32)–(34), the inite element model of the Timoshenko beam can be Mi32j = Mi33j = Mi12j = Fi1 = dψ (2) dψ j (38) + μm0 i dx dx dψ j(2) dψ (3) + μm0 i ψ j(2) μm0 ψi(3) dx dx (3) dψ (3) dψ j + m1 ψi(3) ψ j(3) μm1 i dx dx 0 , Mi13j = 0 , Mi23j = 0 , Mi31j = 0   l dψi(1) d fx dψi(1) (1) 0 − 2τ (b + h) fx ψi + μ dx dx dx dx 0 m0 ψi(2) ψ j(2) 6 K. PREETHI ET AL.   l dψi(1) ρA2 (L − x)(L + x − 2r) dx H̃ T − + 2 dx 0 +Q1 ψi(1) (0) + Q4 ψi(1) (l)    l d fz d fx dw dψi(2) (2) (2) 2 Fi = fz ψi + Pψi + μ + dx dx dx dx 0  dw dψi(2) −2τ 0 (b + h) dx dx dx   l ρA2 dw dψi(2) H̃ T − (L−x)(L+x−2r) dx + 2 dx dx 0 Nzz = Nxz = (39) A σzz dA (42c) σxz dA (42d) A Using Eqs. (7) and (8), the stress resultants in Eq. (42) can be written as: (0) Nxx = Ãεxx + C̃εzz + 2τ 0 (b + h) − EAς T ρA2 (L − x)(L + x − 2r) 2 (1) = D̃εxx + Mxx +Q2 ψi(2) (0) + Q5 ψi(2) (l)    l  dψi(3) dψi(3) dψi(3) d fz (3) 3 Fi = μ fz + Õ T dx +P + ψ dx dx dx i dx 0 +Q3 ψi(3) (0) + Q6 ψi(3) (l)  (43c) Nxz = G̃γxz (43d) where: à = EA + 2E s (b + h) The displacement ield and corresponding strains are the same as given in (6) and (7). For an isotropic material, following stress-strain relationship is used: C̃ = αA (40) Nax = ρA2 (L − x)(L + x − 2r) 2 A Ŵ Figure . Rotating nano laminated cantilever beam.  3 bh h + 6 2 2 (46) F̃ = βA (47) G̃ = Ks GA (48) The equations of motion of the cantilever beam using Timoshenko beam theory can now be obtained as: dNxx d2 u + fx = m0 2 dx dt   d dw d2 w Nxz + Nxx + fz + P = mo 2 dx dx dt (41) where ρ is the mass density,  is the angular velocity of rotation and r is the hub radius as shown in Figure 1. The stress resultants can be written as:  ρA2 Nxx = σxx dA + σ s ds + (L − x)(L + x − 2r) 2 Ŵ A (42a)  Mxx = (42b) σxx z dA + σ s z ds (44) (45) D̃ = EI + E s σxx = Eεxx + αεzz − Eς T εxx where α and β are material length scale parameters, and E, G, Ks and ς are Young’s moduli, shear moduli, shear correction factor and co-eicient of thermal expansion, respectively. The axial force due to rotation of a cantilever beam is given as: (43b) (0) + F̃εzz Nzz = C̃εxx 4. Reduction to the case of isotropic beams σxz = GKs γxz σzz = αεxx + βεzz σ s = τ 0 + E s εxx (43a) dMxx d2 ϕ − (Nxz + Nzz ϕx ) = m1 2 dx dt (49) (50) (51) where: mi =  ρzi dA A and fx and fz are the axially and transversely distributed forces, respectively. MECHANICS OF ADVANCED MATERIALS AND STRUCTURES The generalized displacements (ū, w̄, ϕ¯x ) are approximated using Lagrange interpolation functions: (1) Mi21j = μm0 dw dψi(2) dψ j dx dx dx (2) m 1 (1) j ψ j (x) ū(x) = (52a) Mi22j = m0 ψi(2) ψ j(2) + μm0 j=1 n (52b) p 3 (3) j ψ j (x) = μm0 ψi(3) (52c) j=1 By substituting Eq. (52) for ū, w̄ and ϕ¯x , and putting δ ū = ψi1 , δ w̄ = ψi2 , δ ϕ¯x = ψi3 into the weak form statements as discussed in previous sections, the inite element model of the Timoshenko beam can be expressed as: ⎤⎧ ⎫ ⎤⎧ ⎫ ⎡ 11 ⎡ 11 M K M 12 M 13 ⎨ ¨ 1 ⎬ K 12 K 13 ⎨ 1 ⎬ 2 ⎣K 21 K 22 K 23 ⎦ + ⎣M 21 M 22 M 23 ⎦ ¨ 2 ⎩ ¨ 3⎭ ⎩ 3⎭ M 31 M 32 M 33 K 31 K 32 K 33 ⎧ 1⎫ ⎨F ⎬ (53) = F2 ⎩ 3⎭ F = à 0 1 dw dψi(1) dψ j dx à 2 dx dx dx Ki13j = l dψ (1) 1 C̃ϕx i ψ j(3) dx 2 dx Ki21j =  Ki22j = Ki23j = Ki31j = Ki32j = Ki33j = 0 à dw dψi(2) dψ j dx dx dx dx 20.30 3 1 4 2 5 Nonlocal parameter, (in nm2 ) (a) (2) (1) Mi11j 20.32 0 dψ (2) dψ j G̃ i dx dx 0     (2) dψ (2) 1 dw 2 j 2 dψi + dx + C̃ (ϕx ) à 2 dx dx dx  l dψ (2) G̃ i ψ j(3) dx dx 0  l dψ j(1) dx C̃ϕx ψi(3) dx 0  l dψ j(2) (3) dx G̃ψi dx 0  l (3) dψi(3) dψ j D̃ + G̃ψi(3) ψ j(3) dx dx 0     dw 2 1 2 + + F̃ (ϕx ) ψi(3) ψ j(3) dx C̃ 2 dx l 20.34 (1) l 0  20.36 20.28 (2) l 0  dψi(1) dψ j dx dx dx dψ (1) dψ j = m0 ψi(1) ψ j(1) + μm0 i dx dx (54) End deflection, w (in nm) Ki12j  (1) l dψi(3) (2) ψj dx +Q1 ψi(1) (0) + Q4 ψi(1) (l) α Miαβ j and force coeicients Fi (α, β = 1, 2, 3) are deined as follows:  dx + μm0 (55) dψ (3) dψ j + m1 ψi(3) ψ j(3) Mi33j = μm1 i dx dx Mi12j = 0, Mi13j = 0, Mi23j = 0, Mi31j = 0   l dψi(1) d fx dψi(1) (1) 1 0 Fi = fx ψi + μ dx − 2τ (b + h) dx dx dx 0  (1)  l dψi ρA2 EAς T − (L−x)(L+x−2r) dx + 2 dx 0 where the stifness coeicients Kiαβ j , mass matrix coeicients Ki11j = dψ j(2) dψi(2) dψ j dx dx (3) j=1 ϕ¯x (x) = Mi32j End deflection, w (in nm) 2 (2) j ψ j (x) w̄(x) = 7 41.2 41.0 40.8 40.6 0 2 3 1 Nonlocal parameter, 5 4 (in nm2 ) (b) Figure . Plot of nonlocal parameter vs. end deflection of the cantilever beam subjected to point load (Q0 = 10 N) at the end (for L/H = 20) for (a) isotropic beam and (b) laminated beam. 8 K. PREETHI ET AL. Fi2 =  l 0 fz ψi(2) + −2τ 0 (b + h) Pψi(2)  d fx dw d fz + +μ dx dx dx  dw dψi(2) dx dx  dψi(2) dx dx   l ρA2 dw dψi(2) (L−x)(L+x−2r) dx EAς T − + 2 dx dx 0 +Q2 ψi(2) (0) + Q5 ψi(2) (l)   l  dψi(3) dψi(3) d fz (3) 3 Fi = +P + ψ μ fz dx dx dx dx i 0 +Q3 ψi(3) (0) + Q6 ψi(3) (l) (56) both isotropic and laminated beam are carried out in the second example. Clamped-free (C-F) boundary conditions are considered in each example. Sinusoidal distribution of load with the intensity q0 is used. Numerical implementation is made after developing a MATLAB code for the Timoshenko beam inite element as discussed in the previous section. For the static bending analysis of the beam, the following cases are considered for the parametric study, namely (a) the efect of nonlocal parameter μ, (b) the efect of surface modulus Es and (c) the efect of surface tension parameter τ on the nonlinear behavior of both isotropic and laminated beam. For the free vibration analysis, the following cases are considered for the parametric study, namely (a) the variation of fundamental frequency ratio with aspect ratio for diferent values of nonlocal parameter μ, (b) the efect of surface modulus Es on the variation 5. Numerical results We will present numerical examples to demonstrate the application of the above nonlinear nonlocal formulation in this section. Static bending behavior of both isotropic and laminated beam are studied in the irst example. Free vibration analysis of End deflection, w(innm) End deflection, w (in nm) 8 6 4 2 0 2 0 4 Es 0 Es 13 N m Es 3N m 6 6 8 4 2 0 17N m 0 2 0 4 10 6 8 Intensity of distributed load, q0 (in N m) (a) 10 Intensity of distributed load, q0 (in N m) (a) 40 End deflection, w(innm) End deflection, w (in nm) 40 30 20 10 0 0 2 4 6 Es 0 Es 13 N m Es 3N m 8 1010 Intensity of distributed load, q0 (in N m) (b) Figure . Plot of load vs. end deflection for different values of surface parameter Es for (a) isotropic beam and (b) laminated beam. 30 20 0 17N m 10 0 10 6 2 8 0 4 Intensity of distributed load, q0 (in N m) (b) Figure . Plot of load vs. end deflection for different values of surface tension parameter τ for (a) isotropic beam and (b) laminated beam. MECHANICS OF ADVANCED MATERIALS AND STRUCTURES C-F beam: u(x=0) = 0, w(x=0) = 0, φ(x=0) = 0 5.1. Example 1: Static bending analysis The material properties of the isotropic beam are taken as: Elastic modulus E = 17.73 × 1010 N/m2 and Poisson’s ratio ν = 0.27. The material properties of the laminated beam are taken as: E11 = 140 × 109 N/m2 , E22 = 10 × 109 N/m2 and ν12 = 0.3. Four-layered cross ply laminate (0/90/0/90) is considered. The boundary condition read as: To study the efect of nonlocal parameter μ on the nonlinear behavior of the beam, the nonlocal parameter μ is varied from 0 to 5 nm2 . The plot of nonlocal parameter μ versus the central delection w is shown in Figure 2. It is observed that for both isotropic beam and laminated beam, with the increase in the nonlocal parameter μ there is a decrease in the bending behavior. Frequency ratio 0.985 0.980 0 1 nm2 2 nm2 0.975 3 nm2 To study the efect of the surface modulus Es on the nonlocal nonlinear behavior, the surface modulus values of 0 N/m, 13 N/m and −3 N/m are taken. The plot of intensity of distributed load versus central transverse delection is shown in Figure 3. For the positive value of surface modulus Es , there is an increase in stifness of the beam, and hence a reduction in the delection of the beam. Negative values of Es tends to decrease the stifness of the beam and hence results in reduced delection. To study the efect of surface tension τ on the bending behavior of the beam, a surface tension value of τ = 1.7 N is taken for analysis. A plot of intensity of distributed load versus center delection for diferent values of τ is presented in Non-dimensional natural frequency ( 1010 ) of fundamental frequency with aspect ratio of both isotropic and laminated beam: 2.5 2.0 50 Frequency ratio 0.98 0 1 nm2 2 nm2 3 nm2 4 nm2 5 nm2 0.90 0.88 10 20 Es 3N m 1.0 0.5 0 10 (a) 1 0.92 13N m 5 nm2 (a) 0.94 Es 4 nm 40 20 30 Aspect ratio, L H 0.96 0 40 20 30 Aspect ratio, L H 40 30 Aspect ratio, L H 50 (b) Figure . Plot of aspect ratio L/H vs. frequency ratio for different values of nonlocal parameter μ for (a) isotropic beam and (b) laminated beam. Non-dimensional natural frequency ( 1010 ) 10 Es 1.5 2 0.970 9 50 1.5 1.0 Es 0 Es 13N m Es 3N m 0.5 0 10 40 20 30 Aspect ratio, L H 50 (b) Figure . Plot of aspect ratio L/H vs. nondimensional natural frequency for different values of surface modulus Es for (a) isotropic beam and (b) laminated beam. K. PREETHI ET AL. 5.2. Example 2: Free vibration analysis In this example, a beam with aspect ratio L/H varying from 10 to 50 is considered for the nonlocal nonlinear free vibration analysis. The material properties for isotropic and laminated beam are taken the same as in the previous example. C-F boundary condition is considered. To study the efect of nonlocal parameter on the variation of frequency ratio with the aspect ratio of the beam, various nonlocal parameters μ from 0 to 5 nm2 are taken. The frequency ratio is deined as: Frequency ratio = ωnl (with nonlocal efect) ωnl (without nonlocal efect) The plot of frequency ratio versus aspect ratio L/H of isotropic and laminated beams is presented in Figure 5. It is Non dimensional natural frequency 2.90 2.85 2.80 0 1nm22 2nm2 3nm 4nm2 5nm2 2.75 2.70 2.90 2.85 2.80 0 0.2 1 0. 4 0. 6 0. 8 Non dimensional angular velocity Non-dimensional natural frequency 1010 (a) 5.6 5.2 0 1 nm2 2 nm2 3 nm2 4 nm2 5 nm2 4.8 4.4 4.0 0 2.90 0.4 0.2 0. 6 0. 8 Non-dimensional angular velocity 1 (b) Figure . Plot of nondimensional angular velocity vs. nondimensional natural frequency for different values of nonlocal parameter for (a) isotropic beam and (b) laminated beam. 0 1 nm2 2 nm2 3 nm2 4 nm2 5 nm2 2.85 2.80 2.75 1 2.75 2.70 observed that with the increase in the nonlocal parameter μ, there is a decrease in the natural frequency of vibration of the beam. This trend is the same for both isotropic beam and laminated beam. The efect of surface modulus Es on the nonlinear natural frequency versus aspect ratio is studied. The plots for both isotropic and laminated beams are presented in Figure 6. Positive values of Es stifen the beam and thus result in higher frequencies. Negative values of Es have the opposite efect and decrease the frequencies. Surface tension τ has no efect on the vibration characteristics of the beam. Figure 7 shows the variation of natural frequency with nondimensional angular velocity λ (see Eq. 57) for diferent values on nonlocal parameter μ. It is observed that as the dimensionless Non-dimensional natural frequency ( 1010 ) Figure 4. The surface tension τ stifens the beam and reduces the delections. 2 4 5 3 Hub radius (in nm) 6 (a) Non-dimensional natural frequency ( 1010 ) 10 5.0 0 1 nm22 2 nm2 3 nm2 4 nm2 5 nm 4.8 4.6 4.4 4.2 4.0 1 2 3 4 5 Hub radius (in nm) 6 (b) Figure . Plot of hub radius vs. nondimensional natural frequency for different values of nonlocal parameter μ for (a) isotropic beam and (b) laminated beam. MECHANICS OF ADVANCED MATERIALS AND STRUCTURES angular velocity increases, the natural frequency also increases. It is also seen that with the increase in nonlocality, there is an increase in natural frequencies of vibration. λ4 = ρAL4 2  EI (57) Non-dimensional natural frequency ( 1010 ) The variation of natural frequency with the variation of hub radius for diferent nonlocal parameter values is presented in Figure 8. It is seen that natural frequency decreases with increasing hub radius. It is also seen that with the increase in nonlocality, there is an increase in natural frequencies of vibration. The variation of natural frequency with the variation of dimensionless angular velocity for diferent surface modulus Es values is presented in Figure 9. Natural frequencies of vibration are higher for positive value of Es and lower for negative values of Es . 4.5 3.5 Es Es Es 3.0 1 2 3 0 13N m 3N m 4 5 Non-dimensional angular velocity Non-dimensional natural frequency ( 1010 ) (a) 5.0 4.8 Es Es Es 4.6 5.0 0 13N m 3N m 4.5 4.0 3.5 0 6. Summary and conclusions The efects of nonlocal parameter and surface stress on nonlinear bending and vibration characteristics of beams are studied using the Timoshenko beam theory and Eringen’s nonlocal diferential model together with the Gurtin and Murdoch surface elasticity theory. Simpliied Green–Lagrange strain tensor is used to model geometric nonlinearity. The FEM is used to solve the resulting nonlinear equations. Parametric studies are carried out to investigate the inluence of nonlocal parameter (μ), surface parameters (Es and τ ), hub radius (r) and angular velocity (λ). 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