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Dynamic stiffness and eigenvalues of nonlocal nano beams

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Dynamic stiffness and eigenvalues of nonlocal nano beams - new methods for dynamic analysis of nano-scale structures. This lecture gives a review and proposed new techniques.

Text of Dynamic stiffness and eigenvalues of nonlocal nano beams

  • 1. Dynamic stiffness and eigenvalues of nonlocalnano-beamsS. Adhikari1, T. Murmu2,1College of Engineering, Swansea University, Swansea UKTwitter: @ProfAdhikari2School of Engineering, University of the West of Scotland, Glasgow, UKCST2014The Twelfth International Conference on Computational StructuresTechnology, Naples, Italy, September 2014

2. Outline of this talk1 Introduction2 Bending vibration of undamped nonlocal beamsEquation of motionNatural frequencies3 Finite element modelling of nonlocal dynamic systemsElement stiffness matrixElement mass matrix4 Bending vibration of damped nonlocal beamsEquation of motion5 Dynamic stiffness matrixDynamic shape functionsClosed-form expressions of the elements6 Numerical illustrations7 Conclusions 3. Nanoscale structures(a) DNA (b) Zinc Oxide ( ZnO ) nanowire( c ) Boron Nitride nanotube ( BNNT )(d) Protein Nanoscale systems have length-scale in the order of O(109)m.Nanoscale systems, such as those fabricated from simple and complexnanorods, nanobeams1 and nanoplates have attracted keen interestamong scientists and engineers. 4. Nanoscale structuresExamples of one-dimensional nanoscale objects include (nanorod andnanobeam) carbon nanotubes2, zinc oxide (ZnO) nanowires and boronnitride (BN) nanotubes, while two-dimensional nanoscale objects includegraphene sheets3 and BN nanosheets4.These nanostructures are found to have exciting mechanical, chemical,electrical, optical and electronic properties.Nanostructures are being used in the field of nanoelectronics,nanodevices, nanosensors, nano-oscillators, nano-actuators,nanobearings, and micromechanical resonators, transporter of drugs,hydrogen storage, electrical batteries, solar cells, nanocomposites andnanooptomechanical systems (NOMS).Understanding the dynamics of nanostructures is crucial for thedevelopment of future generation applications in these areas. 5. Continuum mechanics at the nanoscaleExperiments at the nanoscale are generally difficult at this point of time.On the other hand, atomistic computation methods such as moleculardynamic (MD) simulations5 are computationally prohibitive fornanostructures with large numbers of atoms.Continuum mechanics can be an important tool for modelling,understanding and predicting physical behaviour of nanostructures.Although continuum models based on classical elasticity are able topredict the general behaviour of nanostructures, they often lack theaccountability of effects arising from the small-scale.To address this, size-dependent continuum based methods69 aregaining in popularity in the modelling of small sized structures as theyoffer much faster solutions than molecular dynamic simulations forvarious nano engineering problems.Currently research efforts are undergoing to bring in the size-effectswithin the formulation by modifying the traditional classical mechanics. 6. Nonlocal continuum mechanicsOne popularly used size-dependant theory is the nonlocal elasticitytheory pioneered by Eringen10, and has been applied to nanotechnology.Nonlocal continuum mechanics is being increasingly used for efficientanalysis of nanostructures viz. nanorods11,12, nanobeams13,nanoplates14,15, nanorings16, carbon nanotubes17,18, graphenes19,20,nanoswitches21 and microtubules22. Nonlocal elasticity accounts for thesmall-scale effects at the atomistic level.In the nonlocal elasticity theory the small-scale effects are captured byassuming that the stress at a point as a function of the strains at all pointsin the domain:ij (x) =ZV(|x x|, )tijdV(x)where (|x x|, ) = (222)K0(px x/)Nonlocal theory considers long-range inter-atomic interactions and yieldsresults dependent on the size of a body.Some of the drawbacks of the classical continuum theory could beefficiently avoided and size-dependent phenomena can be explained bythe nonlocal elasticity theory. 7. Bending vibration of carbon nanotubesFigure: Bending vibration of an armchair (5, 5), (8, 8) double-walled carbon nanotube(DWCNT) with pinned-pinned boundary condition. 8. Bending vibration of nanobeamsFor the bending vibration of a nonlocal damped beam, the equation ofmotion can be expressed byEI4V(x, t)x4 + m1 (e0a)2 2x22V(x, t)t2=1 (e0a)2 2x2{F(x, t)} (1)In the above equation EI is the bending rigidity, m is mass per unit length,e0a is the nonlocal parameter, V(x, t) is the transverse displacement andF(x, t) is the applied force.Considering the free vibration, i.e., setting the force to zero, andassuming harmonic motion with frequency V(x, t) = v(x) exp [it ] (2)from (1) we haveEId4vdx4 m2v (e0a)2 d2vdx2= 0 (3)ord4vdx4 + b4(e0a)2 d2vdx2 b4v = 0 (4) 9. Bending vibration of nanobeamsHereb4 =m2EI(5)To obtain the characteristic equation, we assumev(x) = exp [x] (6)Substituting this in Eq. (4) we obtain4 + b4(e0a)22 b4 = 0 (7)or 2 = b2b2(e0a)2 q4 + b4(e0a)4/2 (8) 10. Bending vibration of nanobeamsDefining = b2(e0a)2 (9)the two roots can be expressed as2 = 2, 2 (10)Here = brp4 + 2 + /2 (11)and = brp4 + 2 /2 (12)Therefore, the four roots of the characteristic equation can be expressedas = i,i, , (13)where i = p1. 11. Boundary conditionsThe displacement function within the beam can be expressed by linearsuperposition asv(x) =X4j=1cj exp[jx] (14)Here the unknown constants cj need to be obtained from the boundaryconditions.Using Eq. (14), the natural frequency of the system can be obtained byimposing the necessary boundary conditions23. For example, thebending moment and shear force are given by:Bending moment at x = 0 or x = L:EId2v(x)dx2 = 0 (15)Shear force at x = 0 or x = L:EId3v(x)dx3 +m!2(e0a)2 dv(x)dx= 0 ord3v(x)dx3 + b4(e0a)2 dv(x)dx= 0 (16) 12. Pinned-pinned nonlocal beamsUndamped nonlocal natural frequencies of pinned-pinned nonlocalbeams can be obtained11 asj =2q j1 + 2j (e0a)2rEImwhere j = j/L, j = 1, 2, (17)Asymptotic behaviour: For higher order modes j ! 1 we can show thatj !j(e0a)rEImj = 1, 2, (18)Unlike conventional local beams where frequencies increase as squareof the mode count j, for nonlocal beams the frequencies increase linearlywith j1.1Lei, Y., Murmu, T., Adhikari, S. and Friswell, M. I., Asymptotic frequencies of dampednonlocal beams and plates, Mechanics Research Communication, to appear. 13. Finite element method for nonlocal beamsConventional finite element method for nonlocalbeams 14. Nonlocal element matricesWe consider an element of length e with bending stiffness EI and massper unit length m.12 l e Figure: A nonlocal element for the bending vibration of a beam. It has two nodesand four degrees of freedom. The displacement field within the element isexpressed by cubic shape functions.This element has four degrees of freedom and there are four shapefunctions. 15. Element stiffness matrixThe shape function matrix for the bending deformation24 can be given byN(x) = [N1(x),N2(x),N3(x),N4(x)]T (19)whereN1(x) = 1 3x22e+ 2x33e, N2(x) = x 2x2e+x32e,N3(x) = 3x22e 2x33e, N4(x) = x2e+x32e(20)Using this, the stiffness matrix can be obtained using the conventionalvariational formulation25 asKe = EIZ e0d2N(x)dx2d2NT (x)dx2dx =EI3e12 6e 12 6e6e 42e6e 22e12 6e 12 62e6e 22e6e 42e(21) 16. Nonlocal element mass matrixThe mass matrix for the nonlocal element can be obtained asMe = mZ e0N(x)NT (x)dx + m(e0a)2Z e0dN(x)dxdNT (x)dxdx=me420156 22e 54 13e22e 42e13e 32e2e2e54 13e 156 22e13e 322e 4+e0ae2 me3036 3e 36 3e3e 42e3e 2e36 3e 36 3e3e 2e3e 42e(22)For the special case when the beam is local, the mass matrix derivedabove reduces to the classical mass matrix24,25 as e0a = 0. 17. Bending vibration of a double-walled carbon nanotubeA double-walled carbon nanotube (DWCNT) is considered.An armchair (5, 5), (8, 8) DWCNT with Youngs modulus E = 1.0 TPa,L = 30 nm, density = 2.3103 kg/m3 and thickness t = 0.35 nm is usedThe inner and the outer diameters of the DWCNT are respectively0.68nm and 1.1nm.We consider pinned-pinned boundary condition.For the finite element analysis the DWCNT is divided into 100 elements.The dimension of each of the system matrices become 200 200, that isn = 200. 18. Nonlocal natural frequencies of DWCNTe02 4 6 8 10 12 14 16 18 20400350300250200150100500/w1Normalised natural freqency: ljFrequency number: je0a=2.0nma=1.5nme0a=1.0nme0a=0.5nmlocalanalyticalfinite elementFirst 20 undamped natural frequencies for the axial vibration of SWCNT. 19. Equation of motion of damped nonlocal beamsFor the bending vibration of a nonlocal damped beam, the equation ofmotion can be expressed byEI4V(x, t)x4 + m1 (e0a)2 2x22V(x, t)t2+bc15V(x, t)x4t+bc2V(x, t)t=1 (e0a)2 2x2{F(x, t)} (23)In the above equation EI is the bending rigidity, m is mass per unit length,e0a is the nonlocal parameter, V(x, t) is the transverse displacement andF(x, t) is the applied force.The constantbc1 is the strain-rate-dependent viscous damping coefficientandbc2 is the velocity-dependent viscous damping coefficient.Following the damping convention in dynamic analysis23, we considerstiffness and mass proportional damping. Therefore, we express thedamping constants asbc1 = 1(EI) and bc2 = 2(m) (24)where 1 and 2 are stiffness and mass proportional damping factors. 20. Damped vibration of nonlocal beamsConsidering the free vibration, i.e., setting the force to zero, andassuming harmonic motion with frequency V(x, t) = v(x) exp [it ] (25)from Eq. (23) we haveEId4vdx4 m2d2vv (e0a)2 dx2+ ibc1d4vdx4 + ibc2v = 0 (26)Using the damping factors, from Eq. (26) we haveEI (1 + i1)d4vdx4 + m2(e0a)2 d2vdx2 m2 (1 i2/) v = 0 (27)ord4vdx4 + b4(e0a)2 d2vdx2 b4v = 0 (28)where we define band introduce asb4 =m2EI (1 + i1)and = (1 i2/) (29) 21. Damped vibration of nonlocal beamsTo obtain the characteristic equation, we assumev(x) = exp [x] (30)Substituting this in Eq. (28) we obtain4 + b4(e0a)22 b4 = 0 (31)Defining = b2(e0a)2 (32)the two roots can be expressed as2 = 2, 2 (33) 22. Damped vibration of nonlocal beamsThe expressions of and are given by = brp4 + 2 + /2 (34)and = brp4 + 2 /2 (35)Therefore, the four roots of the characteristic equation can be expressedas = i,i, , (36)where i = p1. 23. Nonlocal element matrixWe consider an element of length L with bending stiffness EI and massper unit length m.12 l e Figure: A nonlocal element for the bending vibration of a beam. It has two nodesand four degrees of freedom. The displacement field within the element isexpressed by complex frequency dependent functions. 24. Dynamic shape functionsThe undimmed dynamic stiffness matrix can be obtained using thedisplacement and force boundary conditions using Banerjees method26We propose an approach using complex shape functions which issuitable for damped systemsSimilarly to the classical finite element method, assume that thefrequency-dependent displacement within an element is interpolated fromthe nodal displacements asve(x, ) = NT (x, )bve() (37)Here bve() 2 Cn is the nodal displacement vector N(x, ) 2 Cn is thevector of frequency-dependent shape functions and n = 4 is the numberof the nodal degrees-of-freedom.Suppose the sj (x, ) 2 C, j = 1, , 4 are the basis functions whichexactly satisfy Eq. (28). It can be shown that the shape function vectorcan be expressed asN(x, ) =