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SurfaceandNonlocalEffecrsfornonlinearanalysisofTimoshenkobeams

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Surface and non-local effects for non-linear analysisof Timoshenko beams

Kasirajan Preethia, Amirtham Rajagopala,n, Junuthula Narasimha Reddy b

a Department of Civil Engineering, Indian Institute of Technology, Hyderabad, Indiab Department of Mechanical Engineering, Texas A&M University, College Stations, USA

a r t i c l e i n f o

Article history:Received 24 April 2015Received in revised form7 June 2015Accepted 10 June 2015Available online 19 June 2015

Keywords:Timoshenko beamNon-local parameterSurface effectsGeometric non-linearityFinite element analysis

a b s t r a c t

In this paper, we present a non-local non-linear finite element formulation for the Timoshenko beamtheory. The proposed formulation also takes into consideration the surface stress effects. Eringen's non-local differential model has been used to rewrite the non-local stress resultants in terms of non-localdisplacements. Geometric non-linearities are taken into account by using the Green–Lagrange straintensor. A C0 beam element with three degrees of freedom has been developed. Numerical solutions areobtained by performing a non-linear analysis for bending and free vibration cases. Simply supported andclamped boundary conditions have been considered in the numerical examples. A parametric study hasbeen performed to understand the effect of non-local parameter and surface stresses on deflection andvibration characteristics of the beam. The solutions are compared with the analytical solutions availablein the literature. It has been shown that non-local effect does not exist in the nano-cantilever beam(Euler–Bernoulli beam) subjected to concentrated load at the end. However, there is a significant effectof non-local parameter on deflections for other load cases such as uniformly distributed load andsinusoidally distributed load (Cheng et al. (2015) [10]). In this work it has been shown that for acantilever beam with concentrated load at free end, there is definitely a dependency on non-localparameter when Timoshenko beam theory is used. Also the effect of local and non-local boundaryconditions has been demonstrated in this example. The example has also been worked out for otherloading cases such as uniformly distributed force and sinusoidally varying force. The effect of the local ornon-local boundary conditions on the end deflection in all these cases has also been brought out.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The classical theory of hyperelasticity is used to solve a largenumber of problems in engineering, wherein the stress at a givenpoint uniquely depends on the current values and possibly also theprevious history of deformation and temperature at that pointonly. Deformation in this case is characterized by the deformationgradient or by an appropriate strain tensor, that is, it is fullydetermined by the first gradient of the displacement field. Inmodeling micro/nano structures where the size effect becomesprominent, for example, study of elastic waves when dispersioneffect is taken to account and the determination of stress at thecrack tip when the singularity of the solution is of concern, theclassical theory cannot model the material behavior accurately.

The inhomogeneities present in any material at the microscopicscale influence its properties at the macroscopic scale: materials suchas suspensions, blood flows, liquid crystals, porous media, polymeric

substances, solids with microcracks, dislocations, turbulent fluids withvortices, and composites point to the need for incorporating micro-motions in continuum mechanical formulations [13]. There has beenconsiderable focus towards the development of generalized continuumtheories [19] that account for the inherent microstructure in suchnatural and engineering materials (see [36,15]). The notion of general-ized continua unifies several extended continuum theories thataccount for such a size dependence due to the underlying micro-structure of the material. A systematic overview and detailed discus-sion of generalized continuum theories has been given by Bazant andJirasek [8]. These theories can be categorized as gradient continuumtheories (see works by Mindilin et al. [45–47], Toupin [68], Steinmannet al. [13,34,66,37], and Casterzene et al. [52], Fleck et al. [20,63], Askeset al. [3–5]), microcontinuum theories (see works by Eringen[18,16,19]), Steinmann et al. [35,28], and non-local continuum theories(see works by Eringen [17], Jirasek [33], Reddy [54], and others[7,12,51,10]). Recently, the higher order gradient theory for finitedeformation has been elaborated (for instance see [21,38,39,52]),within classical continuum mechanics in the context of homogeniza-tion approaches. A comparison of various higher order gradienttheories can be found in [20]. A more detailed formulation of

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/nlm

International Journal of Non-Linear Mechanics

http://dx.doi.org/10.1016/j.ijnonlinmec.2015.06.0060020-7462/& 2015 Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail address: rajagopal@iith.ac.in (R. Amirtham).

International Journal of Non-Linear Mechanics 76 (2015) 100–111

gradient approach in spatial and material setting has been presentedin [35].

Classical continuum mechanics takes exclusively the bulk intoaccount, nevertheless, neglecting possible contributions from thesurface of the deformable body. However, surface effects play a crucialrole in the material behavior, the most prominent example beingsurface tension. A mathematical framework was first developed byGurtin [23] to study the mechanical behavior of material surfaces. Theeffect of surface stress on wave propagation in solids has also beenstudied by Gurtin [24]. The tensorial nature of surface stress wasestablished using the force and moment balance laws. Bodies whoseboundaries are material surfaces are discussed and the relationbetween surface and body stress examined in a recent work bySteinmann [64] and by Hamilton [25]. The surface effects have beenapplied to modeling two [31] and three-dimensional continua in theframe work of finite element method (see [32,14]). Similar studies onstatic analysis of nanobeams using non-local finite element modelshave been done by Mahmoud [43].

The focus of this work is on non-local non-linear formulationtogether with surface effects for static and free vibration analysis ofTimoshenko beams. The non-local formulations can be of integral-type formulations with weighted spatial averaging or by implicitgradient models which are categorized as strongly non-local, whileweakly non-local theories include for instance explicit gradientmodels [8]. Herein we consider a strongly non-local problem. TheTimoshenko beam can be considered as a specific onedimensionalversion of a Cosserat continuum. Recently various beam theoriessuch as Euler–Bernoulli, Timoshenko, Reddy, and Levinson beamtheories were reformulated using Eringen's non-local differentialconstitutive model by Reddy [54]. The analytical solutions forbending, buckling and free vibrations were also presented in [54].Various shear deformation beam theories were also reformulated inrecent works by Reddy [55] using non-local differential constitutiverelations. Similar works have been done to study bending, bucklingand free vibration of nanobeams by Aydogdu [7], Civalek [12].

Eringen's non-local elasticity theory has also been applied tostudy bending, buckling and vibration of nanobeams usingTimoshenko beam theory (see [40,60,72,48]). Numerical solutionswere obtained by a meshless method. Two different collocationtechniques, global (RDF) and local (RDF-FD), were used with multi-quadrics radial basis functions by Roque et al. [58]. Static deforma-tion of micro- and nano-structures was studied using non-localEuler–Bernoulli and Timoshenko beam theory and explicit solutionshave been derived for deformations for standard boundary condi-tions by Wang et al. (see [71,70]). Analytical solutions for beambending problems for different boundary conditions were derivedusing non-local elasticity theory and Timoshenko beam theory byWang et al. [69]. Iterative non-local elasticity for Kirchhoff plates hasbeen presented in [62]. Thai et al. [67] developed a non-local sheardeformation beam theory with a higher order displacement fieldthat does not require shear correction factors. Some explicit solu-tions involving trigonometric expansions are also presented recentlyfor non-local analysis of beams [74]. A finite element framework fornon-local analysis of beams has also been made in a recent work bySciarra et al. [61]. Size effects on elastic moduli of plate likenanomaterials have been studied in [65].

Non-local elastic rodmodels have been developed to investigate thesmall-scale effect on axial vibrations of the nanorods by Aydogdu [6]and Adhikari et al. [1]. Free vibration analysis of microtubules based onnon-local theory and Euler–Bernoulli beam theory was done by Civaleket al. [12]. Free vibration analysis of functionally graded carbonnanotube with various thickness based on Timoshenko beam theoryhas been investigated to obtain numerical solutions using the Differ-ential Quadrature Method (DQM) by Janghorban et al. [30] and others(see [11,27,2]). Studies to understand thermal vibration of single wallcarbon nanotube embedded in an elastic medium using DQMhave also

been reported in [49]. The recent studies have been towards theapplication of non-local non-linear formulations for the vibrationanalysis of functionally graded beams [53]. Analytical study on thenon-linear free vibration of functionally graded nanobeams incorp-orating surface effects has been presented in [26,59,42]. The effect ofnon-local parameter, surface elasticity modulus and residual surfacestress on the vibrational frequencies of Timoshenko beam has beenstudied in [73,41]. The coupling between non-local effect and surfacestress effect for the non-linear free vibration case of nanobeams hasbeen studied in [29]. The effect of surface stresses on bending proper-ties of metal nanowires is presented in [75]. There has been someworks on transforming non-local approaches to gradient type formula-tions [9]. Semi-analytical approach for large amplitude free vibrationand buckling of non-local functionally graded beams has been reportedin [50].

In this paper, we present a non-local non-linear finite elementformulation for the Timoshenko beam theory. The proposedformulation takes into consideration the surface stress effects.Eringen's non-local differential model has been used to write thenon-local stress resultants. Geometric non-linearities are takeninto account by using Green–Lagrange strain tensor. Numericalsolutions are obtained by performing a non-linear analysis forbending and free vibration cases. Simply supported and clampedboundary conditions have been considered in the numericalexamples. A parametric study has been performed to understandthe effect of non-locality and surface stresses on deflection andvibration characteristics of the beam. The solutions are comparedwith the analytical solutions available in the literature. Thefollowing Section 2 gives a background on Eringen's non-localtheory. Section 3 gives the mathematical formulation for the non-local Timoshenko beam theory. The finite element formulationfor the Timoshenko beam theory is explained in Section 4. InSection 5 numerical examples are presented together with para-metric studies to demonstrate the effect of non-local and surfacestresses on the bending and vibration characteristics of the beam.

2. Non-local theories

In classical elasticity, stress at a point is a function of strain at thatpoint. Whereas in non-local elasticity, stress at a point is a functionof strains at all points in the continuum. In non-local theories, forcesbetween the atoms and internal length scale are considered in theconstitutive equation. Non-local theory was first introduced byEringen [19]. According to Eringen, the stress field at a point x inan elastic continuum not only depends on the strain field at thatpoint but also on the strains at all other points of the body. Eringenattributed this fact to the atomic theory of lattice dynamics andexperimental observation on phonon dispersion. The non-localstress tensor σ at a point x in the continuum is expressed as

σ ¼Z

Kðjx0 �xj ; τÞtðx0Þ dx0 ð1Þ

where tðxÞ is the classical macroscopic stress tensor at point x andthe kernel function Kðjx0 �xj ; τÞ represents the non-local modulus,jx0 �xj is the distance and τ is the material constant that dependson internal and external characteristic lengths.

Stress and strain at a point are related to each other by Hooke'slaw as

tðxÞ ¼ CðxÞ : εðxÞ ð2Þ

where t is the macroscopic stress tensor, ε is the strain tensor, C isthe fourth-order elasticity tensor and ‘:’ denotes double dotproduct. Eqs. (1) and (2) together form the non-local constitutiveequations of Hookean solid. Constitutive equations can also be

P. Kasirajan et al. / International Journal of Non-Linear Mechanics 76 (2015) 100–111 101

expressed in equivalent differential form as

ð1�τ2l2∇2Þσ ¼ t ð3Þ

τ¼ eoal

ð4Þ

where e0 is a material constant, and a and l are the internal andexternal characteristic lengths respectively.

3. Mathematical formulations

3.1. Classical Timoshenko beam theory

In Timoshenko beam theory, the effects of shear deformationare also considered. A linear distribution of transverse shear stressis assumed. The displacement field is given by

u¼ uðx; tÞþzφx ð5aÞ

w¼wðx; tÞ ð5bÞwhere φx is the measure of rotation of the beam cross-section. Thenon-zero components of Lagrangian strain tensor can be written as

εxx ¼dudx

þ12

dwdx

� �2" #

þzdφx

dx

� �

¼ εð0Þxx

� �þz εð1Þxx

� � ð6aÞ

εxz ¼12

φxþdwdx

� �ð6bÞ

εzz ¼ 12φ2

x ð6cÞ

Since ϵzz is positive and non-zero, two-dimensional Hooke's lawshould be used. But it makes the beam flexible by 38% [56]. So, itscontribution is through certain material length scale parameters (see[56,57]). Therefore, for an isotropic material, following stress–strainrelationship is used:

σxx ¼ Eεxxþαεzz ð7aÞ

σxz ¼ GKsγxz ð7bÞ

σzz ¼ αεxxþβεzz ð7cÞ

σs ¼ τ0þEsεxx ð7dÞwhere α and β are certain material length scale parameters (forexample, in the case of a soft material embedded with hardinclusions they may describe the geometry and distribution of thehard inclusions), and E, G and Ksð ¼ 5

6Þ are Young's moduli, shearmoduli, and shear correction factor, respectively. σs is the surfacestress in the direction of the length of the beam (see [23,22]). Thestress resultants can be written as

Nxx ¼ZAσxx dAþ

IΓσs ds ð8aÞ

Mxx ¼ZAσxxz dAþ

IΓσsz ds ð8bÞ

Nzz ¼ZAσzz dA ð8cÞ

Nxz ¼ZAσxz dA ð8dÞ

Using Eqs. (6) and (7), the stress resultants in Eq. (8) can be writtenas

Nxx ¼ ~Aεð0Þxx þ ~Cεzzþ2τ0 bþhð Þ ð9aÞ

Mxx ¼ ~Dεð1Þxx ð9bÞ

Nzz ¼ ~Cεð0Þxx þ ~Fεzz ð9cÞ

Nxz ¼ ~Gγxz ð9dÞwhere

~A ¼ EAþ2Es bþhð Þ ð10aÞ

~C ¼ αA ð10bÞ

~D ¼ EIþEsh3

6þbh2

2

" #ð10cÞ

~F ¼ βA ð10dÞ

~G ¼ KsGA ð10eÞ

3.2. Non-local Timoshenko beam theory

Using Eq. (3), the non-local stress resultants in terms of strainscan be written as

Nnlxx ¼ μ

d2Nnlxx

dx2þ ~Aεð0Þxx þ ~Cεzzþ2τ0 bþhð Þ ð11Þ

Mnlxx ¼ μ

d2Mnlxx

dx2þ ~Dεð1Þxx ð12Þ

Nnlzz ¼ ~Cεð0Þxx þ ~Fεzz ð13Þ

Nnlxz ¼ μ

d2Nnlxz

dx2þ ~Gγxz ð14Þ

3.3. Equations of motion

By using the principle of virtual work, the equations of motionfor the classical Timoshenko beam can be obtained as

dNxx

dxþ f x ¼m0

d2u

dt2ð15Þ

ddx

NxzþNxxdwdx

� �þ f z ¼mo

d2w

dt2ð16Þ

dMxx

dx� NxzþNzzφx

� ¼m1d2φ

dt2ð17Þ

where

mi ¼ZAρzi dA

and fx and fz are the axially and transversely distributed forces,respectively. The boundary conditions are

Geometric : u;w;φx

Force : Nxx;NxzþNxxdwdx

;Mxx

Considering the stress resultants in Eqs. (15)–(17) to be non-localand using the relations from (11) to (17), the following equationsfor non-local stress resultants are obtained:

Nnlxx ¼ ~Aεð0Þxx þ ~Cεzzþ2τ0 bþhð Þ�μ

df xdx

þμm0d3u

dxdt2ð18aÞ

P. Kasirajan et al. / International Journal of Non-Linear Mechanics 76 (2015) 100–111102

Mnlxx ¼ ~Dεð1Þxx þμm1

d3φ

dxdt2þμm0

d2w

dt2�μf z

�μddx

~Aεð0Þxx þ ~Cεzzþ2τ0 bþhð Þ�μdf xdx

þμm0d3u

dxdt2

!dwdx

" #

þμddx

~Cεð0Þxx þ ~Fεzz �

φx

h ið18bÞ

Nnlzz ¼ ~Cεð0Þxx þ ~Fεzz ð18cÞ

Nnlxz ¼ ~Gγxz�μ

df zdx

�μd2

dx2~Aεð0Þxx þ ~Cεzzþ2τ0 bþhð Þ�μ

df xdx

þμm0d3u

dxdt2

!dwdx

" #

ð18dÞIt is clearly seen in the above equation that the non-localityreduces the resistance offered by the beam to the external forces.By substituting the expressions for non-local stress resultants (18)back in the equations of motion (15) to (17), we obtain theequilibrium equation for non-local Timoshenko beam theoryincluding surface stress effects as

m0d2u

dt2�μm0

d4u

dx2dt2¼ ddx

~Aεð0Þxx þ ~Cεzzþ2τ0 bþhð Þ�μdf xdx

� �þ f x

ð19Þ

m0d2w

dt2�μm0

d4w

dx2dt2¼ ddx

~Gγxz �

�μd2f zdx2

þ f z

þ ddx

~Aεð0Þxx þ ~Cεzzþ2τ0 bþhð Þ�μdf xdx

þμm0d3u

dxdt2

!dwdx

" #

�μd3

dx3~Aεð0Þxx þ ~Cεzzþ2τ0 bþHð Þ�μ

df xdx

þμm0d3u

dxdt2

!dwdx

" #ð20Þ

m1d2φx

dt2�μm1

d4φx

dx2dt2¼ ddx

~Dεð1Þxx

�� ~Gγxz�φx

~Cεð0Þxx þ ~Fεzz �

þμd2

dx2~Cεð0Þxx þ ~Fεzz �

φx

h ið21Þ

4. Finite element formulation

The principle of virtual work for the Timoshenko beam has theform

0¼Z l

0½Nnl

xxδεð0Þxx þMnl

xxδεð1Þxx þNnl

xzδγxzþNnlzzδεzz

� f xδu� f zδwþm0 €uδuþm0 €wδwþm1 €φδφ� dx�Q1δuð0Þ�Q4δuðlÞ�Q2δwð0Þ�Q5δwðlÞ�Q3δφð0Þ�Q6δφð0Þ ð22Þ

After substituting the expressions for stress resultants fromEq. (18) into Eq. (22), we obtain

0¼Z T

0

Z l

0

~Aεð0Þxx þ ~Cεzzþ2τ0 bþHð Þ�μdf xdx

þμm0d3u

dxdt2

!δεð0Þxx

þ ~Dεð1Þxx þμm1d3φ

dxdt2þμm0

d2w

dt2�μf z

!δεð1Þxx

�μddx

~Aεð0Þxx þ ~Cεzzþ2τ0 bþHð Þ�μdf xdx

þμm0d3u

dxdt2

!dwdx

" #εð1Þxx

þμddx

~Cεð0Þxx þ ~Fεzz �

φx

h iδεð1Þxx þ ~Gγxzþμm0

d3w

dxdt2�μ

df zdx

!δγxz

�μ d2

dx2~Aεð0Þxx þ ~Cεzzþ2τ0 bþHð Þ�μ

df xdx

þμm0d3u

dxdt2

!dwdx

" #δγxz

þ ~Cεð0Þxx þ ~Fεzz �

δεzz� f xδu� f zδw

þm0 €uδuþm1 €wδwþm1 €φxδφx dx

� Q1δuðxaÞþQ4δuðxbÞþQ2δwðxaÞþQ5δwðxbÞþQ3δφðxaÞþQ6δφðxbÞ� �

dT

ð23ÞThe underlined expressions in the above equation do not allow usto construct a quadratic functional. So after omitting the under-lined expressions in Eq. (23), it can be equivalently written into thefollowing three equations:Z T

0

Z l

0

~Aεð0Þxx þ ~Cεzzþ2τ0 bþHð Þ�μdf xdx

þμm0d3u

dxdt2

!dδudx

"

� f xδuþm0 €uδu�dx

� Q1δuðxaÞþQ4δuðxbÞ� �

dT ¼ 0 ð24Þ

Z T

0

Z l

0

~Aεð0Þxx þ ~Cεzzþ2τ0 bþHð Þ�μdf xdx

þμm0d3u

dxdt2

!dwdx

dδwdx

þ ~Gγxzþμm0d3w

dxdt2�μ

df zdx

!dδwdx

� f zδwþm0 €wδw dx� Q2δwðxaÞþQ5δwðxbÞ� �

dT ¼ 0 ð25Þ

Z T

0

Z l

0

~Gγxzþμm0d3w

dxdt2�μ

df zdx

!δφx

þ ~Dεð1Þxx þμm1d3φx

dxdt2þμm0

d2w

dt2�μf z

!dδφx

dx

þ ~Cεð0Þxx þ ~Fεzz �

φxδφxþm1 €φxδφx dx

� Q3δφxðxaÞþQ6δφxðxbÞ� �

dT ¼ 0 ð26ÞThe generalized displacements ðu;w;φx Þ are approximated

using the Lagrange interpolation functions

uðxÞ ¼Xmj ¼ 1

Δ1j ψ

ð1Þj ðxÞ ð27aÞ

wðxÞ ¼Xnj ¼ 1

Δ2j ψ

ð2Þj ðxÞ ð27bÞ

φx ðxÞ ¼Xpj ¼ 1

Δ3j ψ

ð3Þj ðxÞ ð27cÞ

By substituting Eq. (27) for u, w and φx , and putting δu ¼ψ1i ,

δw ¼ψ2i , δφx ¼ψ3

i into the weak form statements (24)–(26), thefinite element model of the Timoshenko beam can be expressed as

K11 K12 K13

K21 K22 K23

K31 K32 K33

264

375 Δ1

Δ2

Δ3

8><>:

9>=>;þ

M11 M12 M13

M21 M22 M23

M31 M32 M33

264

375 €Δ

1 €Δ2 €Δ

3�

¼F1

F2

F3

8><>:

9>=>; ð28Þ

where the stiffness coefficients Kαβij , mass matrix coefficients Mαβ

ijand force coefficients Fαi (α;β¼ 1;2;3) are defined as follows:

K11ij ¼

Z l

0

~Adψ ð1Þ

i

dx

dψ ð1Þj

dxdx

K12ij ¼

Z l

0

12~Adwdx

dψ ð1Þi

dx

dψ ð2Þj

dxdx

K13ij ¼

Z l

0

12~Cφx

dψ ð1Þi

dxψ ð3Þ

j dx

K21ij ¼

Z l

0

~Adwdx

dψ ð2Þi

dx

dψ ð1Þj

dxdx

P. Kasirajan et al. / International Journal of Non-Linear Mechanics 76 (2015) 100–111 103

K22ij ¼

Z l

0

~Gdψ ð2Þ

i

dx

dψ ð2Þj

dx

þ12

~Adwdx

� �2

þ ~C ðφxÞ2( )

dψ ð2Þi

dx

dψ ð2Þj

dxdx ð29Þ

K23ij ¼

Z l

0

~Gdψ ð2Þ

i

dxψ ð3Þ

j dx

K31ij ¼

Z l

0

~Cφxψð3Þi

dψ ð1Þj

dxdx

K32ij ¼

Z l

0

~Gψ ð3Þi

dψ ð2Þj

dxdx

K33ij ¼

Z l

0

~Ddψ ð3Þ

i

dx

dψ ð3Þj

dxþ ~Gψ ð3Þ

i ψ ð3Þj

þ12

~Cdwdx

� �2

þ ~F ðφxÞ2( )

ψ ð3Þi ψ ð3Þ

j dx

M11ij ¼m0ψ

ð1Þi ψ ð1Þ

j þμm0dψ ð1Þ

i

dx

dψ ð1Þj

dx

M21ij ¼ μm0

dwdx

dψ ð2Þi

dx

dψ ð1Þj

dx

M22ij ¼m0ψ

ð2Þi ψ ð2Þ

j þμm0dψ ð2Þ

i

dx

dψ ð2Þj

dxð30Þ

M32ij ¼ μm0ψ

ð3Þi

dψ ð2Þj

dxþμm0

dψ ð3Þi

dxψ ð2Þ

j

M33ij ¼ μm1

dψ ð3Þi

dx

dψ ð3Þj

dxþm1ψ

ð3Þi ψ ð3Þ

j

M12ij ¼ 0; M13

ij ¼ 0; M23ij ¼ 0; M31

ij ¼ 0

F1i ¼Z l

0f xψ

ð1Þi þμ

df xdx

dψ ð1Þi

dx�2τ0ðbþhÞdψ

ð1Þi

dx

( )dx

þQ1ψð1Þi ð0ÞþQ4ψ

ð1Þi ðlÞ

F2i ¼Z l

0f zψ

ð2Þi þμ

df zdx

þdf xdx

dwdx

� �dψ ð2Þ

i

dx�2τ0ðbþhÞdw

dxdψ ð2Þ

i

dx

( )dx

þQ2ψð2Þi ð0ÞþQ5ψ

ð2Þi ðlÞ

F3i ¼Z l

0μ f z

dψ ð3Þi

dxþdf z

dxψ ð3Þ

i

" #dxþQ3ψ

ð3Þi ð0ÞþQ6ψ

ð3Þi ðlÞ ð31Þ

5. Numerical results

In this section we will present numerical examples to demon-strate the application of the above discussed non-local non-linearformulation. The first example deals with non-local non-linearbending analysis of beams and the second example deals withnon-local non-linear free vibration analysis of beams. Variousboundary conditions, such as both ends simply supported (S–S)and both ends clamped (C–C), are considered. Two cases of loaddistribution (uniformly distributed load and sinusoidally distributedload) with load intensity q0 are considered. Numerical implementa-tion is made after developing a MATLAB code for the Timoshenkobeam finite element as discussed in the previous section.

Based on the non-local non-linear analysis of the beam, aparametric study has been performed for all the boundary condi-tions considered. For the static bending analysis the followingcases are considered for the parametric study, namely (a) theeffect of non-local parameter μ, (b) the effect of surface modulus Esand (c) the effect of surface tension parameter τ on the non-linearbehavior of the beam. For the dynamic analysis the effect of non-local parameter μ on (a) the variation of fundamental frequencyratio with aspect ratio (b) surface frequency ratio versus amplituderatio (c) the effect of surface modulus Es on the variation offundamental frequency with aspect ratio of the beam are con-sidered for the parametric study.

5.1. Example 1: static bending analysis

In this example a beamwith aspect ratio L=H varying from 10 to100 is considered for the non-local non-linear bending analysis.The material properties of the beam are taken as elastic modulus,E¼ 17:73� 1010 N=m2 and Poisson's ratio ν¼ 0:27. The breadth Band height H of the beam are taken as 1 nm. Simply supported (S–S) and clamped (C–C) boundary conditions are considered. Takingthe symmetry of the beam (about x¼ L=2) in both cases, one halfof the beam is only considered for analysis. The boundary condi-tions for the two cases read as

S–S beam : wðx ¼ 0Þ ¼ 0; uðx ¼ L=2Þ ¼ 0; ϕðx ¼ L=2Þ ¼ 0

C–C beam : uðx ¼ 0Þ ¼ 0; wððx ¼ 0Þ ¼ 0; ϕðx ¼ 0Þ ¼ 0

For verification of the non-local non-linear analysis resultsobtained from the present study as a specific case the linear analysis

Table 1Variation of non-dimensionalized central deflection w for various L=H ratios andnon-local parameter μ (for L=H¼ 100).

μ Reddy [54] Thai [67] w ðpresentÞ

0 1.3134 1.3024 1.30241 1.4492 1.4274 1.42742 1.5849 1.5525 1.55243 1.7207 1.6775 1.67744 1.8565 1.8025 1.8024

Fig. 1. Plot of load versus transverse center deflection for C–C beam and S–S beamunder sinusoidally distributed load.

P. Kasirajan et al. / International Journal of Non-Linear Mechanics 76 (2015) 100–111104

results are compared with the analytical solutions given by Reddy[54] and with other methods available in the literature [67]. Table 1gives the values of dimensionless static deflection w ¼ 100�wmaxEI

q0l4 for

various values of non-local parameter μ ranging from 0 to 4. It isobserved that with increase in the non-local parameter there is anincrease in the dimensionless central deflection w . This clearlyshows a decrease in the stiffness with increase in the non-localparameter, and the trends and values obtained from present analysisare very close to the analytical results.

To study the effect of non-local parameter μ on the non-linearbehavior of the beam, the non-local parameter μ is varied from0 to 5 nm2. S–S and C–C boundary conditions are considered. Anon-linear non-local analysis is performed. The plot of intensity ofsinusoidally distributed load versus the central deflection w isshown in Fig. 1. It is observed that for both S–S and C–C boundaryconditions with increase in the non-local parameter μ there is anincrease in the non-linear behavior and the values of the centraldeflection w. As expected the deflections in simply supported caseare higher than those with clamped boundary conditions.

To study the effect of the surface modulus Es on the non-localnon-linear behavior, the surface modulus values of (Es) 0 N/m,13 N/m and �3 N/m are taken. The plot of intensity of sinusoidallydistributed load versus the central deflection w for various surfacemodulus is shown in Fig. 2(a) and (b). It is observed that for bothS–S and C–C boundary conditions the effect of considering thesurface effects results in an increased value of stiffness and hence a

reduction in the deflection of the beam. Positive value of Es tendsto increase the stiffness of the beam and hence results in reduceddeflection. On the contrary, negative value of Es reduces thestiffness of the beam and results in increased deflection. Asexpected, the deflections in simply supported case are higher thanthose with clamped boundary conditions.

To study the effect of surface tension τ on the non-local non-linear bending behavior, a surface tension value of τ¼ 1:7 N=m istaken for the analysis. It is observed that non-local effect tends torelax the stiffness of the beam resulting in increased deflection asseen from Fig. 1. On the contrary the surface tension (τ) stiffens thebeam and reduces the deflections. A plot of load versus transversecenter deflection for SS and CC beams for different values of τ ispresented in Fig. 3(a) and (b).

5.2. Example 2: dynamic analysis

In this example a beamwith aspect ratio L=H varying from 10 to100 is considered for the non-local non-linear free vibrationanalysis. The material properties of the beam are taken as elasticmodulus, E¼ 17:73� 1010 N=m2 and Poisson's ratio ν¼ 0:27. Thewidth B and height H of the beam both are taken as 1 nm. Simplysupported (S–S) and clamped (C–C) boundary conditions areconsidered. Taking the symmetry of the beam about x¼ L=2 inboth cases, one half of the beam is modeled.

Fig. 2. Plot of load versus transverse center deflection for various values of surface parameter (Es) for (a) S–S beam and (b) C–C beam under sinusoidally distributed load.

Fig. 3. Load versus transverse center deflection plot for different values of surface parameter (τ) for (a) SS beam and (b) CC beam under sinusoidally distributed load.

P. Kasirajan et al. / International Journal of Non-Linear Mechanics 76 (2015) 100–111 105

For a verification of the non-local non-linear free vibrationanalysis results obtained from the present study as a specific casethe linear and the non-linear analysis results are compared withthe analytical solutions given by Reddy [54] and with othermethods available in the literature [67]. Table 2 gives the values

of non-dimensionalized natural frequency ω ¼ω� l2ffiffiffiffiffim0EI

qfor var-

ious values of non-local parameter μ ranging from 0 to 4. It isobserved that with increase in the non-local parameter there is adecrease in the dimensionless natural frequency ω. This clearlyreflects a decrease in the stiffness with increase in the non-localparameter and the trends and values obtained from presentanalysis are very close to the analytical results.

Table 3 contains the values of frequency parameter (ratio ofnon-linear natural frequency to linear natural frequency) fordifferent amplitude ratio obtained from a non-linear analysis.The results are compared to those obtained from the DQM [44]and from the Ritz method [42]. There is an increase in frequencyparameter with increase in amplitude ratio.

To study the effect of non-local parameter on the variation offrequency ratio with the aspect ratio of the beam, various non-local parameters μ from 0 to 5 are considered. The frequency ratio

is defined as

Frequency ratio¼ ωnlðwith non� local effectÞωnlðwithout non� local effectÞ

The plot of frequency ratio versus aspect ratio L=H for S–S andC–C beams is presented in Fig. 4. It is observed that with theinclusion of non-local parameter there is a stiffening effect on thebeam, thereby resulting in lower values of natural frequencies. Forthe values of L=H less than 25, the difference in frequencies fordifferent values of μ is very prominent. The trend is same for bothS–S and C–C beams.

To study the effect of non-local parameter on the variation ofsurface frequency ratio with the aspect ratio of the beam , variousnon-local parameters μ from 0 to 5 are considered. The surfacefrequency ratio is defined as

Surface frequency ratio¼ ωnlðwith surface effectÞωnlðwithout surface effectÞ

The plot of surface frequency ratio versus amplitude ratio plotsfor both beam cases is presented in Fig. 5 for various values of μ.On the contrary to non-local effect, surface effect stiffens the beamwhich results in higher frequencies.

The effect of non-local parameter on non-linear natural fre-quency variation with aspect ratio is studied. The plots for bothS–S and C–C beam cases are presented in Fig. 6. Positive values ofEs stiffen the beam and thus resulting in higher frequencies.Negative values of Es have the opposite effect and decrease thefrequencies. Surface tension τ has no effect on the vibrationcharacteristics of the beam.

5.3. Example 3: cantilever beam example

In this example a cantilever beam with aspect ratio L=H varyingfrom 10 to 50 is considered for the non-local non-linear bendinganalysis. The material properties of the beam are taken as elasticmodulus, E¼ 17:73� 1010 N=m2 and Poisson's ratio ν¼ 0:27. Thebreadth B and height H of the beam are taken as 1 nm. The non-localparameter is varied from 0 to 5. In an earlier work [10] it has beenshown that non-local effect does not exist in the nano-cantileverbeam (Euler–Bernoulli beam) subjected to concentrated load at theend. However, it is shown in [10] that there is a significant effect ofnon-local parameter on deflections for other load cases such asuniformly distributed load and sinusoidally distributed load. In thisstudy example, we mainly concentrate on studying the behavior of

Table 2Variation of the dimensionless natural frequency ratio obtained from linear analysisfor various non-local parameter μ (for L=H¼ 100).

μ Reddy [54] Thai [67] ω (present)

0 9.8683 9.8679 9.86771 9.4147 9.4143 9.41412 9.0183 9.0180 9.01793 8.6682 8.6678 8.66764 8.3558 8.3555 8.3553

Table 3Variation of frequency parameter (ωnon�linear=ωlinear) with various amplitude ratiosfor a non-linear analysis.

Wm=r DQM [44] RM [42] Present

1 1.0630 1.0486 1.04652 1.2505 1.1794 1.17513 1.5987 1.3635 1.3626

Fig. 4. Frequency ratio versus aspect ratio L=H for (a) S–S beam and (b) C–C beam for different values of non-local parameter (μ).

P. Kasirajan et al. / International Journal of Non-Linear Mechanics 76 (2015) 100–111106

cantilever beams subjected to different load conditions namely apoint load at free end, uniformly distributed force and sinusoidallyvarying force. The beam has been modeled using the Timoshenkobeam formulation. The effect of the local or non-local boundaryconditions on the end deflection has also been brought out.

Let us consider the transversely applied point load case on thecantilever beam as dirac delta function given as

P ¼Q0δðx�xpÞ ð32Þwhere Q0 is the point load applied at the point xp on the beam.

Fig. 6. Non-linear natural frequency versus aspect ratio L=H for (a) S–S beam and (b) C–C beam for different values of surface parameter (Es).

Fig. 5. Surface frequency ratio versus amplitude ratio (Wm=r) for (a) S–S beam and (b) C–C beam for different values of non-local parameter (μ).

Fig. 7. (a) Non-local parameter versus end deflection of the cantilever beam subjected to point load (Q0 ¼ 10 N) at the end (for L=H¼ 50). (b) Aspect ratio versus ratio of enddeflection of the cantilever beam for different values of non-local parameter when subjected to the point load at the end (Q0 ¼ 10 N).

P. Kasirajan et al. / International Journal of Non-Linear Mechanics 76 (2015) 100–111 107

Fig. 8. (a) Non-local parameter versus end deflection of the cantilever beam subjected to sinusoidally varying load (q0 ¼ 10 N, L=H¼ 50). (b) Aspect ratio versus ratio of enddeflection of the cantilever beam for different values of non-local parameter when subjected to sinusoidally varying load (q0 ¼ 10 N).

Fig. 9. (a) Non-local parameter versus end deflection of the cantilever beam subjected to uniformly varying load (q0 ¼ 10 N, L=H¼ 50). (b) Aspect ratio versus ratio of enddeflection of the cantilever beam for different values of non-local parameter when subjected to uniformly varying load (q0 ¼ 10 N).

Fig. 10. Non-local parameter versus center deflection of the simply supported beam subjected to various load conditions (with q0 ¼ 10 N, L=H¼ 10) with (a) local boundaryconditions and (b) non-local boundary conditions.

P. Kasirajan et al. / International Journal of Non-Linear Mechanics 76 (2015) 100–111108

By using the principle of virtual work, the equations ofequilibrium for the Timoshenko beam can be obtained as

dNxx

dxþ f x ¼ 0 ð33Þ

ddx

NxzþNxxdwdx

� �þ f zþP ¼ 0 ð34Þ

dMxx

dx� NxzþNzzφx

� ¼ 0 ð35Þ

and, fx and fz are the axially and transversely distributed forces,respectively. The boundary conditions are

Geometric : u;w;φx

Force : Nxx;NxzþNxxdwdx

;Mxx

Manipulating Eqs. (33)–(35) and using Eringen's non-local differ-ential model, the following relations are obtained:

Nnlxx ¼ ~Aεð0Þxx þ ~Cεzzþ2τ0 bþhð Þ�μ

df xdx

ð36aÞ

Mnlxx ¼ ~Dεð1Þxx �μf z�μP

�μddx

~Aεð0Þxx þ ~Cεzzþ2τ0 bþhð Þ�μdf xdx

þμm0d3u

dxdt2

!dwdx

" #

þμddx

~Cεð0Þxx þ ~Fεzz �

φx

h ið36bÞ

Nnlzz ¼ ~Cεð0Þxx þ ~Fεzz ð36cÞ

Nnlxz ¼ ~Gγxz�μ

df zdx

�μd2

dx2~Aεð0Þxx þ ~Cεzzþ2τ0 bþhð Þ�μ

df xdx

þμm0d3u

dxdt2

!dwdx

" #

ð36dÞBy substituting the expressions for non-local stress resultants

(36) back in the equations of motion (33)–(35), we obtain theequilibrium equation for the non-local Timoshenko beam theoryincluding surface stress effects as

0¼ ddx

~Aεð0Þxx þ ~Cεzzþ2τ0 bþhð Þ�μdf xdx

� �þ f x ð37Þ

0¼ ddx

~Gγxz �

�μd2f zdx2

þ f zþP

þ ddx

~Aεð0Þxx þ ~Cεzzþ2τ0 bþhð Þ�μdf xdx

þμm0d3u

dxdt2

!dwdx

" #

�μd3

dx3~Aεð0Þxx þ ~Cεzzþ2τ0 bþHð Þ�μ

df xdx

þμm0d3u

dxdt2

!dwdx

" #ð38Þ

0¼ ddx

~Dεð1Þxx

�� ~Gγxz�φx

~Cεð0Þxx þ ~Fεzz �

þμd2

dx2~Cεð0Þxx þ ~Fεzz �

φx

h ið39Þ

Finite element formulation for Timoshenko beam with point load:The principle of virtual work for the Timoshenko beam has theform

0¼Z l

0½Nnl

xxδεð0Þxx þMnl

xxδεð1Þxx þNnl

xzδγxzþNnlzzδεzz

� f xδu� f zδw�Pδw� dx�Q1δuð0Þ�Q4δuðlÞ�Q2δwð0Þ�Q5δwðlÞ�Q3δφð0Þ�Q6δφðlÞ

ð40ÞAfter substituting the expressions for stress resultants from

Eq. (36) into Eq. (40) and by substituting the corresponding shapefunctions for virtual displacements, we obtain the finite elementmodel as before. The stiffness coefficients Kαβ

ij and mass coeffi-cients Mαβ

ij remain same. Force coefficients Fαi (α;β¼ 1;2;3) aregiven as

F1i ¼Z l

0f xψ

ð1Þi þμ

df xdx

dψ ð1Þi

dx�2τ0ðbþhÞdψ

ð1Þi

dx

( )dx

þQ1ψð1Þi ð0ÞþQ4ψ

ð1Þi ðlÞ ð41aÞ

F2i ¼Z l

0f zψ

ð2Þi þPψ ð2Þ

i þμdf zdx

þdf xdx

dwdx

� �dψ ð2Þ

i

dx�2τ0ðbþhÞdw

dxdψ ð2Þ

i

dx

( )dx

þQ2ψð2Þi ð0ÞþQ5ψ

ð2Þi ðlÞ ð41bÞ

F3i ¼Z l

0μ f z

dψ ð3Þi

dxþP

dψ ð3Þi

dxþdf z

dxψ ð3Þ

i

" #dxþQ3ψ

ð3Þi ð0ÞþQ6ψ

ð3Þi ðlÞ

ð41cÞAs seen in Eq. (41c), the point load function (P) is associated with

the non-local parameter (μ). Therefore, point load subjected at any

Fig. 11. Non-local parameter versus center deflection of the clamped–clamped beam subjected to various load conditions (with q0 ¼ 1 N, L=H¼ 10) with (a) local boundaryconditions and (b) non-local boundary conditions.

P. Kasirajan et al. / International Journal of Non-Linear Mechanics 76 (2015) 100–111 109

point in the beam brings out non-local effect. If the point load isapplied at the end, the terms highlighted will become the nodalquantities but will be still associated with non-local parameter.

Numerical results: In this example, cantilever beam case isstudied under the following three load cases have been consideredfor analysis: (a) uniformly varying load (UVL), (b) sinusoidallyvarying load (SVL), and (c) point load acting at the end of the beam(PL). The material properties taken are same as those for examplesin the paper. To study the effect of non-local parameter (μ) on thenon-linear behavior of the beam, the non-local parameter μ isvaried from 0 to 5 nm2. Fig. 7(a) shows the non-local parameterversus end deflection of the cantilever beam subjected to pointload Q0 ¼ 10 N at the end (for L=H ¼ 50). It is observed that there isa decrease in the deflection with increase in non-local parameterfor the cantilever beam. Fig. 7(b) shows the aspect ratio versusratio of end deflection of the cantilever beam for different valuesof non-local parameter when subjected to the point load at theend (Q0 ¼ 10 N). This clearly indicates that the non-local para-meter has a significant effect on deflection characteristics of thecantilever beam.

Fig. 8(a) shows the non-local parameter versus end deflection ofthe cantilever beam subjected to sinusoidally varying load q0 ¼ 10 Nat the end (for L=H¼ 50). It is observed that there is a decrease in thedeflection with increase in non-local parameter for the cantileverbeam. Fig. 8(b) shows the aspect ratio versus ratio of end deflection ofthe cantilever beam for different values of non-local parameter whensubjected to the sinusoidally varying load (q0 ¼ 10 N). This clearlyindicates that the non-local parameter has a significant effect ondeflection characteristics of the cantilever beam.

Fig. 9(a) shows the non-local parameter versus end deflection ofthe cantilever beam subjected to uniformly varying load (q0 ¼ 10 N)at the end (for L=H ¼ 50) . It is observed that there is a decrease inthe deflection with increase in non-local parameter for the canti-lever beam. Fig. 9(b) shows the aspect ratio versus ratio of enddeflection of the cantilever beam for different values of non-localparameter when subjected to uniformly varying load q0 ¼ 10 N. Thisclearly indicates that the non-local parameter has a significant effecton deflection characteristics of the cantilever beam.

5.4. Example 4: cantilever, simply supported and clamped beam withvarying boundary conditions

In [54], it is suggested that the boundary conditions for thenon-local beam theory will remain same as that of local theories.In [10], it is suggested that bending moment and shear force in theboundary conditions should adopt the corresponding non-localexpressions (i.e., the boundary conditions still keep non-local). Tostudy the effect of local and non-local boundary conditions, theexamples considered above were analyzed with both local andnon-local boundary conditions. Results vary with the choice ofboundary conditions (local or non-local). The choice of boundarycondition did not affect the bending behavior of the cantileverbeam. In the case of simply supported and clamped beams, theresults had a dependency on the boundary conditions. Fig. 10(a) and (b) shows the variation of deflection with the non-localparameter for both local and non-local boundary conditions forthe simply supported case.

Fig. 11(a) and (b) shows the variation of deflection with thenon-local parameter for both local and non-local boundary con-ditions for the clamped–clamped case. In both the cases it isobserved that when local boundary conditions are applied thedeflection is found to increase with non-local parameter and whennon-local boundary conditions are applied the deflection is foundto decrease with non-local parameter.

6. Summary and conclusions

Using Eringen's non-local differential model together withGurtin and Murdoch surface elasticity theory, the effect of non-local parameter and surface stress on non-linear bending andvibration characteristics of beams is formulated. Green's straintensor was used to model geometric non-linearity. The finiteelement method is used to solve the resulting non-linear equa-tions. Parametric studies are carried out to investigate the influ-ence of non-local parameter (μ) and surface parameters (Es and τ)on bending and vibration characteristics of beams. It is found thatthe non-local parameter as well as the positive values of surfaceparameters relaxes the stiffness of the beam and results in largerdeflections and lower frequencies. Negative values of Es decreasethe deflections and increase the frequencies. In the case of acantilever beam with concentrated load, the non-local effects arefound to significantly affect the results.

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