Isogeometric vibration analysis of small-scale Timoshenko...

Scientia Iranica F (2018) 25(3), 1864{1878

Sharif University of TechnologyScientia Iranica

Transactions F: Nanotechnologyhttp://scientiairanica.sharif.edu

Isogeometric vibration analysis of small-scaleTimoshenko beams based on the most comprehensivesize-dependent theory

A. Norouzzadeha, R. Ansaria;�, and H. Rouhib

a. Department of Mechanical Engineering, University of Guilan, Rasht, P.O. Box 3756, Iran.b. Department of Engineering Science, Faculty of Technology and Engineering, East of Guilan, University of Guilan, P.C.

44891-63157, Rudsar-Vajargah, Iran.

Received 21 October 2017; received in revised form 29 January 2018; accepted 16 April 2018

KEYWORDSNonlocal elasticitytheory;Strain gradientelasticity theory;Timoshenkonanobeam.

Abstract. By taking the nonlocal and strain gradient e�ects into account, the vibrationalbehavior of Timoshenko micro- and nano-beams is studied in this paper based on a novelsize-dependent model. The nonlocal e�ects are captured using both di�erential and integralformulations of Eringen's nonlocal elasticity theory. Moreover, the strain gradient in uencesare incorporated into the model according to the most general form of strain gradienttheory, which can be reduced to simpler strain gradient-based theories such as modi�edstrain gradient and modi�ed couple stress theories. Hamilton's principle is employed toderive the variational form of governing equations. The isogeometric analysis (IGA) isthen utilized for the solution approach. Comprehensive results for the e�ects of small-scale and nonlocal parameters on the natural frequencies of beams under various typesof boundary conditions are given and discussed. It is revealed that using the di�erentialnonlocal strain gradient model for computing the fundamental frequency of cantileversleads to paradoxical results, and one must recourse to the integral nonlocal strain gradientmodel to obtain consistent results.© 2018 Sharif University of Technology. All rights reserved.

1. Introduction

The vibrational analysis of small-scale beam-typestructures is of great importance due to the re-lated applications, especially in micro- and nano-electromechanical systems (MEMS and NEMS).Among theoretical approaches used to study the me-chanical behaviors of micro- and nano-beams, contin-uum models based upon the Euler-Bernoulli, Timo-shenko, and other beam theories are extensively used.However, the classical continuum models have not the

*. Corresponding author. Tel./Fax: +98 13 33690276E-mail address: r [email protected] (R. Ansari)

doi: 10.24200/sci.2018.5267.1177

capability of considering small-scale e�ects. The reasoncan be found in the local nature of classical elasticityand the lack of length scale parameters in such theories.Therefore, some non-classical or size-dependent elas-ticity theories have been developed. There are manyresearch e�orts which con�rm the necessity of usinggeneralized continuum models (e.g., [1-5]).

The nonlocality is an important small-scale in u-ence on the mechanics of nanostructures. Accordingto the nonlocal elasticity theory, the stress tensor ata reference point in the body depends not only on thestrain tensor at that point, but also on the strain tensorat all other points of the body. This theory was �rstdeveloped by Kr�oner [6], Krumhansl [7], and Kunin [8].Moreover, its �rst formal introduction was related tothe papers of Eringen [9] and Eringen and Edelen [10].

A. Norouzzadeh et al./Scientia Iranica, Transactions F: Nanotechnology 25 (2018) 1864{1878 1865

Based on the original version of nonlocal theory,which is in the integral form, kernel functions areused so as to consider the nonlocal e�ects. In 1983,Eringen [11] formulated the di�erential form of thetheory by Green function of linear di�erential operatoras the kernel function. Since working with the resultingdi�erential equations is simpler than working withintegro-partial di�erential equations derived from theintegral form of the theory, the di�erential form ofnonlocal elasticity theory is extensively employed byresearchers. Refs. [12-19,20-26,27-32] can be men-tioned as examples of the di�erential nonlocal modelapplied to the problems of nanoplates, nanobeams, andnanoshells, respectively. In this regard, Norouzzadehand Ansari [19] employed the nonlocal continuumalong with surface elasticity theory to present a size-dependent analysis of the vibration characteristicsof low-dimensional rectangular and circular plates.Sedighi et al. [25] utilized the Euler-Bernoulli beammodel to investigate the stability of NEMS with theconsideration of conductivity, surface energy, and non-local e�ects. Also, Civalek and Demir [26] proposeda simple nonlocal beam element to capture the sizeimpacts on the buckling behavior of protein micro-tubules.

However, using the di�erential nonlocal model insome problems may lead to paradoxical results, andrecourse must be made to the original (integral) formof the nonlocal theory. A well-known paradox occurswhen this model is used for computing fundamentalfrequencies and de ections of beams under clamped-free boundary conditions. It is generally accepted thatincreasing the nonlocal parameter has a decreasinge�ect on the sti�ness of structure. But, the �rst naturalfrequency and de ection of cantilevers respectivelyincreases and decreases as the nonlocal parameter getslarger. Therefore, capturing the nonlocal e�ects withinthe framework of integral form of nonlocal elasticityhas recently attracted the attention of several researchworkers [33-39].

The Strain Gradient Theory (SGT) is anothernon-classical continuum theory to consider the size-dependent behavior of small-scale structures throughusing length scale parameters. The most general formof SGT was proposed by Mindlin [40,41]. Basedupon �rst- and second-order Mindlin's SGT, the �rstand second derivatives of the strain tensor e�ectiveon the strain energy density are taken into account,respectively. In the constitutive relations of �rst-orderSGT, there are �ve material length scale parameters(in addition to the classical material constants) tocharacterize the size-dependent behavior of structure.There are also some simpli�ed forms of SGT suchas the modi�ed SGT of Lam et al. [42], SGT ofAifantis [43], and modi�ed couple stress theory of Yanget al. [44]. Tracing the development of literature in this

area demonstrates the contribution of many researchersin recent years, e.g. [45-57]. Among them, Chen andLi [47] proposed the composite laminated Timoshenkobeam model on the basis of modi�ed couple stresstheory and anisotropic constitutive equations. Also, byconsidering thermal and mechanical loadings, Sadeghiet al. [51] studied the thick-walled Functionally Graded(FG) small-scale cylinders using the strain gradientelasticity. Tadi Beni [54] developed the size-dependentgeometrically nonlinear formulation of FG piezoelectricbeams and detected the responses of bending, buckling,and free vibration problems of such structures. To doso, he used the consistent model of couple stress theoryproposed by Hadjesfandiari and Dargush [58].

Recently, simultaneous capture of nonlocal andstrain gradient e�ects by a uni�ed model, called thenonlocal strain gradient model, has attracted muchattention. According to the nonlocal strain gradienttheory, the stress tensor at a reference point in thebody depends on the strain and strain gradient tensorsat all points of the body. In this regard, Narendar andGopalakrishnan [59] developed nonlocal strain gradientmodels. They studied the ultrasonic wave propagationof nano-rods by taking the nonlocality and strain gra-dient e�ects into account simultaneously. Some otherresearchers developed nonlocal strain gradient modelsto address various problems of small-scale structures.The reader is referred to [60-65] as some recentlypublished papers on this issue.

In the present paper, a novel nonlocal straingradient model is developed to study the free vibrationsof small-scale beams. To consider the nonlocal e�ect,both di�erential and integral forms of Eringen's nonlo-cal theory are applied. In addition, the strain gradientin uences are incorporated into the model based on themost general form of strain gradient formulation [40],which encompasses the modi�ed strain gradient andcouple stress theories (MSGT and MCST). The beamis also modeled according to the Timoshenko beamtheory accounting for shear deformation e�ect. First,the variational formulations are obtained for integralnonlocal strain gradient and di�erential nonlocal straingradient models using Hamilton's principle. Apartfrom the novelty of work in its formulation (generalform of nonlocal elasticity in conjunction with themost general form of SGT), the solution approachis another novel aspect of the current research; theobtained governing equations are solved through anisogeometric analysis (IGA). IGA [66,67] is a powerfulnumerical method whose sources of inspiration arethe Finite Element Method (FEM) and the ComputerAided Design (CAD). Construction of exact geometryvia Non-Uniform Rational B-Splines (NURBS) andeasy implementation of geometry re�nement tools arethe main advantages of IGA. Finally, comprehensivenumerical results are presented in order to investigate

1866 A. Norouzzadeh et al./Scientia Iranica, Transactions F: Nanotechnology 25 (2018) 1864{1878

the strain gradient/nonlocal e�ects on the vibrationsof beams subject to di�erent end conditions.

2. Derivation of variational formulation

2.1. Integral nonlocal strain gradient modelTo take the strain gradient e�ects into account, SGTaccording to [40] is employed. By considering thefollowing relations:

sij =@ �w@eij

; tijk =@ �w@�ijk

; (1)

and the classical stress tensor as:

sij (x) = ��ijekk (x) + 2�eij (x) = Aijklekl (x) ; (2)

the components of double stress tensors tijk can beobtained through the following formula:

tijk (x) =12a1 (�ij�kll (x) + 2�jk�lli (x) + �ik�jll (x))

+ 2a2�jk�ill (x)+a3 (�ij�llk (x)+�ik�llj (x))

+ 2a4�ijk (x) + a5 (�jki (x) + �kji (x))

= Aijklmn�lmn (x) ; (3)

where �w is the strain energy density. Also, eij and�ijk denote the strain and second-order deformationgradient tensors, respectively. The classical elastictensor, Aijkl, includes classical Lame's parametersgiven by:

� =E�

1� �2 ; � =E

2 (1 + �); (4)

in which E and � stand for Young's modulus andPoisson's ratio, respectively.

In addition to classical Lame's parameters, thesixth-order Aijklmn elastic tensor contains �ve non-classical constants (a1; a2; :::; a5), which are used tocapture the size e�ects. The theory used herein is themost general form of SGT with the capability of beingreduced to other simpli�ed theories. For instance, themodi�ed strain gradient model of Lam et al. [42] canbe retrieved by the following relations:

a1 = ��l22 � 4

15l21

�; a2 = �

�l20 � 1

15l21 � 1

2l22

�;

a3 = ��

415l21 +

12l22

�; a4 = �

�13l21 + l22

�;

a5 = ��

23l21 � l22

�: (5)

Moreover, the present model can be reduced to the

modi�ed couple stress model of Yang et al. [44] viathese substitutions:

a1 = a4 = �2a2 = �2a3 = �a5 = �l2; (6)

or the strain gradient model proposed by Aifantis [43]is obtained through:

a1 = a3 = a5 = 0; a2 =12�l2; a4 = �l2; (7)

where l0, l1, and sl2 (or l) are material length scaleparameters, which can be determined based on ex-perimental tests. As a well-known example, Lamet al. [42] presented an investigation into the straingradient behavior of micro-scale beams. Park andGao [68] deduced that by considering the results of Lamet al. [42] in case of l0 = l1 = l2 = l and isotropic epoxymaterial properties, the length scale parameter can bedetermined as l = 17:6 �m.

Now, the formulation of nonlocal elasticity isintroduced. The constitutive relations based uponthe original integral form of Eringen's nonlocal the-ory [9,10] for a 1D structure are expressed as:

Sij (�x) =Zx

k (jx� �xj ; �) sij (x) dx

=Zx

k (jx� �xj ; �)Aijklekl (x) dx; (8)

Tijk (�x) =Zx

k (jx� �xj ; �) tijk (x) dx

=Zx

k (jx� �xj ; �) Aijklmn�lmn (x) dx: (9)

Here, Sij and Tijk are the nonlocal stress tensors ofthe Cauchy and double stress tensors, respectively.Based on these relations, the stress component at areference point, �x, is obtained from the strains of allpoints in domain x. The nonlocality is provided usingthe kernel k that is a function of the neighborhooddistance jx � �xj and nonlocal parameter � = e0�a.According to [11], the ratio of internal (�a) to external(�l) characteristic length, i.e. �a=�l, and material constant(e0) are the e�ective parameters to determine thenonlocal stress-strain relationship. Making comparisonwith atomic lattice dynamics, Eringen calibrated theconstant as e0 �= 0:39 to achieve a perfect match.Also, in [12,16,27,31], Ansari et al. used the results ofMolecular Dynamics (MD) simulations to propose theappropriate values of nonlocal parameter in di�erentproblems of carbon nanotubes and graphene sheets.

Now, the Timoshenko beam theory is utilized tomodel the beam. Based on this theory, the displace-


ment �eld is introduced as:

u1 (x; z) = u (x) + z (x) ; u3 (x; z) = w (x) : (10)

The formulation of the paper is obtained in the matrix-vector form, which is suitable to use in FEM and IGA.To this end, the following relation is given for thedisplacement vector:

u = pq; (11)

u (x; z) =�u1 (x; z) u3 (x; z)

�T ;p (z) =

�1 0 z0 1 0

�;

q (x) =�u (x) w (x) (x)

�T : (12)

The strain and strain gradient tensors are also writtenas:

eij (x) =12

�@ui (x)@xj

+@uj (x)@xi

�; (13)

�ijk (x) = (ejk (x));i =12

�@uj (x)@xk

+@uk (x)@xj

�;i;(14)

which can be rewritten in the following vector forms:

e (x; z) =�exx2exz

�= P1 (z) Eq (x) ;

� (x; z) =

24 �xxx�zxx2�xxz

35 = P2 (z) Hq (x) ; (15)

where the coe�cient matrices (P1;P2) and di�erenti-ation ones (E;H) are:

P1 (z) =�1 0 z0 1 0

�;

P2 (z) =

241 0 z 00 0 0 10 1 0 0

35 ; (16)

E =

24@=@x 0 00 @=@x 10 0 @=@x

35 ;H =

2664@2=@x2 0 0

0 @2=@x2 @=@x0 0 @2=@x2

0 0 @=@x

3775 : (17)

From Eqs. (8) and (9), stress vectors in the referencepoint �S and �T are achieved according to the constitu-tive relations shown in Box I. In which A and A show

the elastic matrices associated with the classical andstrain gradient theories.

Hamilton's principle is used to derive the equa-tions of motion. This principle is formulated as:

�t2Zt1

(W � T ) dt = 0; (21)

whereW and T denote total strain and kinetic energies,respectively.

The strain energy of the beam occupying volumeV can be written as:

W =12

ZV

�wdV =12

ZA

Z�x

�Sij (�x) eij (�x)

+ Tijk (�x) �ijk (�x)�d�xdA; (22)

in which A is the cross section area. Consequently,the strain energy and its variation in the matrix-vectorformat are derived as:

W =12

ZA

Z�x

��eT (�x; z) �S (�x; z)

+ ��T (�x; z) �T (�x; z)�d�xdA; (23)

�W =ZA

Z�x

��eT (�x; z) �S (�x; z)

+ ��T (�x; z) �T (�x; z)�d�xdA: (24)

The strain and strain gradient vectors at referencepoint (�e; ��) and corresponding variations are also pre-sented as:

�e (�x; z) =�exx (�x; z)2exz (�x; z)

�= P1 (z) �E�q (�x) ;

��e (�x; z) = P1 (z) �E��q (�x) ; (25)

�� (�x; z) =

24 �xxx (�x; z)�zxx (�x; z)2�xxz (�x; z)

35 = P2 (z) �H�q (�x) ;

�� (�x; z) = P2 (z) �H��q (�x) ; (26)

�q (�x) =�u (�x) w (�x) (�x)

�T ;�E = Ej @=@x! @=@�x;

�H = Hj @=@x! @=@�x: (27)


�S (�x; z) =Zx

k (jx� �xj ; �) Ae (x; z) dx; �T (�x; z) =Zx

k (jx� �xj ; �) A� (x; z) dx; (18)

�S (�x; z) =�Sxx (�x; z)Sxz (�x; z)

�; �T (�x; z) =

24Txxx (�x; z)Tzxx (�x; z)Txxz (�x; z)

35 ; (19)

A =��+ 2� 0

0 ks�

�; A =

242 (a1 + a2 + a3 + a4 + a5) 0 00 2 (a2 + a4) 1

2 (a1 + 2a5)0 1

2 (a1 + 2a5) 12 (a3 + 2a4 + a5)

35 : (20)

Box I

Via performing the following analytical integrations:

C =ZA

PT1 AP1dA; C =

ZA

PT2 AP2dA; (28)

one arrives at:

W =12

Z�x

��qT �ET

Zx

kCEqdx

+ �qT �HTZx

kCHqdx�d�x; (29)

�W =Z�x

��qT �ET

Zx

kCEqdx

+ ��qT �HTZx

kCHqdx�d�x: (30)

Besides, the kinetic energy is determined in the follow-ing manner:

T =12

ZV

� _ui (�x) _ui (�x) dV; (31)

where � signals the mass density of beam. From thefollowing matrix-vector form of representation:

�u = p�q; �u (�x; z) =�u1 (�x; z) u3 (�x; z)

�T ; (32)

the kinetic energy and its variation are obtained as:

T =12

ZA

Z�x

��uT �u d�xdA =12

Z�x

�qT� �q d�x; (33)

t2Zt1

�Tdt = �t2Zt1

Z�x

��qT� �q d�xdt; (34)

in which:

� = �ZA

pTpdA: (35)

2.2. Di�erential nonlocal strain gradient modelAccording to the di�erential model of Eringen's nonlo-cal elasticity theory [11], the di�erential counterpartsof Eqs. (8) and (9) are given by:�

1� �2r2�Sij = ��ijekk + 2�eij = Aijklekl; (36)�1� �2r2�Tijk =

12a1 (�ij�kll + 2�jk�lli + �ik�jll)

+ 2a2�jk�ill + a3 (�ij�llk + �ik�llj)

+ 2a4�ijk + a5 (�jki + �kji)

= Aijklmn�lmn; (37)

where the 1D Laplacian operator is r2 = @[email protected] matrix-vector form of Eqs. (36) and (37) is

written as:�1� �2r2�S = Ae;

�1� �2r2�T = A�; (38)

in which:

S =�Sxx Sxz

�T ; T =�Txxx Tzxx Txxz

�T :(39)

From Subsection 2.1, the vectors e, � and matrices A,A in Eqs. (15) and (20), respectively, can also be found.

For the strain energy one has:

W =12

ZA

Zx

(Sijeij + Tijk�ijk) dxdA; (40)

or:


W =12

ZA

Zx

�eTS + �TT

�dxdA

=12

ZA

Zx

�qTETPT

1 S + qTHTPT2 T�dxdA

=12

Zx

qT �ETs + HTt�dx; (41)

where s and t denote the classical and strain gradientresultant vectors, respectively, given as:

s =ZA

PT1 SdA;

�1� �2r2� s = CEq; (42)

t =ZA

PT2 TdA;

�1� �2r2� t = CHq; (43)

in which C and C are presented in Eq. (28). Therefore,in the di�erential model, the variation of strain energyis achieved as follows:

�W =Zx

�qT �ETs + HTt�dx: (44)

Now, considering:

T =12

Zx

_qT� _qdx;

t2Zt1

�Tdt = �t2Zt1

Zx

�qT� q dxdt: (45)

and using Hamilton's principle lead to:t2Zt1

Zx

�qT �ETs + HTt + �q�dxdt

=t2Zt1

Zx

�qT��

ETCE + HTCH�

q

+ ��1� �2r2�q

�dxdt = 0: (46)

Regarding Eq. (46), the variations of equivalentstrain and kinetic energies are obtained as:

t2Zt1

�Wdt =t2Zt1

Zx

�qT�ETCE + HTCH

�qdxdt;

(47)

t2Zt1

�Tdt = �t2Zt1

Zx

�qT��1� �2r2�q dxdt: (48)

3. IGA

IGA is used in this work for the solution procedure.Using IGA has several advantages: In this type ofanalysis, through suitable solution properties such asgenerating uni�ed geometry in a patch and imple-menting e�cient re�nement approaches, the di�cultyof signi�cant run time does not exist. Moreover, animportant advantage of using IGA for a nonlocal straingradient model is that the order of basis functions canbe increased and gradients in SGT can be obtained inthis approach readily. Hence, in contrast to FEM, thereis no need to use Hermite functions.

3.1. Integral nonlocal strain gradient modelFor this model, to perform IGA, it is required thata novel solution procedure be developed. In otherwords, at any given point of �x, the assemblage of stressvectors should be performed in the entire domain ofx. Then, by the outer assemblage in �x domain, thetotal sti�ness matrix is obtained. In a patch, thedisplacement components of beam are interpolated byappropriate shape functions. The corresponding localcoordinates of x and �x domains are indicated by � and��, respectively. Considering NURBS basis function ofith knot as Ni, one has:

q = N (�) d; N (�) = I3 N (�) ;

N (�) =�N1 (�) : : : Nn (�)

�; (49)

�q = �N�� d; �N

��

= I3 N��;

N��

=�N1��

: : : Nn��; (50)

where d and �d are the control variables in �-and��-coordinates, respectively. Furthermore, is theKronecker product and Ip denotes the p-by-p identitymatrix.

Thus, the strain and strain gradient vectors andtheir variations are determined in a typical patch as:

e (�) = P1B (�) d; �e��

= P1 �B�� d;

��e��

= P1 �B��d; (51)

� (�) = P2D (�) d; ��

= P2 �D�� d;

��

= P2 �D��d; (52)

where:

B (�) = EN (�) ; �B��

= �E �N��; (53)

D (�) = HN (�) ; �D��

= �H �N��: (54)

The following relations are used in order to obtain the


stress vectors corresponding to the classical and straingradient theories:

M = Rd; M = Rd; (55)

R =Z�

kCBJd�; R =Z�

kCDJd�; (56)

in which J is the determinant of Jacobian transfor-mation matrix. By performing the inner assemblageprocess in �-coordinate, one can denote the totalmatrices of R, R, and vector d by R, R, and d,respectively. Accordingly, the strain energy of thepatch and its variation are determined as:

We =12

Z��

��dT �BTRd + �dT �DTRd

��Jd��; (57)

�We =Z��

��dT�

�BTR+ �DTR�

d �Jd�� = ��dTKs;ed;(58)

where �J is the counterpart of J in ��-coordinates. Thesecant sti�ness matrix of the patch, Ks;p, is achievedas:

Ks;p =Z��

��BTR+ �DTR

��Jd��: (59)

In addition, the variation of kinetic energy of the patchcan be expressed as:

t2Zt1

�Tpdt = �t2Zt1

��dTMp �d dt; (60)

here Mp stands for the inertia matrix given by:

Mp =Z��

�NT��N �Jd��: (61)

By assembling the sti�ness and inertia matrices, thetotal matrices and vectors ofKs,M, and �d are obtained.If one chooses the same local coordinate for � and ��,one �nds that d = �d. Hence:

t2Zt1

�Wdt =t2Zt1

�dTKsddt; (62)

t2Zt1

�Tdt = �t2Zt1

�dTMddt: (63)

Finally, using Hamilton's principle results in:

Md +Ksd = 0: (64)

3.2. Di�erential nonlocal strain gradient modelFor the di�erential nonlocal strain gradient model, theconventional assemblage process of IGA is performed.Thus, one has:

q=Nd; N=I3 N; N=�N1 : : : Nn

�; (65)

e = P1Bd; � = P2Dd; (66)

B = EN; D = HN: (67)

By inserting the presented interpolation into the equiv-alent energies in Eqs. (47) and (48), the variations ofstrain energy and kinetic energy are found as:

t2Zt1

�Wpdt =t2Zt1

�dTKs;pddt; (68)

t2Zt1

�Tpdt = �t2Zt1

�dTMp d dt; (69)

where the sti�ness and inertia matrices of the patch areexpressed as:

Ks;p =Z�

�BTCB + DTCD

�Jd�; (70)

Mp =Z�

NT��1� �2r2�NJd�: (71)

At last, by �nding total matrices and vectors, theequations of motion are derived as Eq. (64).

4. Results and discussion

In this section, selected numerical results are pre-sented to simultaneously analyze the e�ects of nonlocaland strain gradient parameters on the free vibrationresponse of small-scale beams under clamped-free,pinned-pinned, and clamped-pinned end conditions.The results are generated for both integral nonlocalstrain gradient and di�erential nonlocal strain gradientmodels. The following e�ective material and geometri-cal properties are considered:

� = 0:35; ks = 5=6; L=h = 15:

The non-dimensional frequency is introduced as:

= !Lp�= (�+ 2�):

Also, the standard kernel function is used as:

k (jx� �xj ; �) =1

2�e� jx��xj

� :


First, some comparisons are made to examinethe integrity of the present formulation as well asto verify the proposed solution approach. Initially,the advantages of using IGA in the nonlocal straingradient theory are clari�ed. As expressed in Section 3,according to the piecewise continuous polynomialsbasis functions in the standard NURBS-based IGA, theconsistency of deformation gradients has been satis�edin the entire domain of the patch. However, Hermiteshape functions and additional Degrees Of Freedom(DOFs) should be considered in the �nite elementanalysis of stain gradient theory. In this regard, Table 1is provided to compare the �rst three frequencies ofpinned and clamped beams determined by the presentisogeometric analysis and �nite element analysis of [69].The results are produced by using the classical, modi-�ed couple stress, and modi�ed strain gradient theoriesin di�erent thickness-to-length scale ratios. Ansari etal. [69] employed 3-node beam element with Hermite

shape functions including four DOFs at each node(w; ;w0; 0). The excellent agreement of results inTable 1 shows the e�ciency of the present standardIGA models of nonlocal strain gradient elasticity.

Also, it would be interesting to compare theoutcome of the present size-dependent theory withthe Molecular Dynamics (MD) simulations. Dueto the con�rmed accuracy of MD analysis of small-scale structures, one can calibrate the nonlocal andlength scale parameters through such comparison. Tothis end, Ref. [70] is considered and single-walledcarbon nanotubes are modelled via the present size-dependent Timoshenko beam theory. According tothe integral nonlocal modi�ed strain gradient theory,di�erent combinations of nonlocal and length scaleparameters can be found, which produce the sameresults of MD. For instance, two sets of (�; l) are shownin Figure 1 at which, corresponding to clamped-freeand clamped-clamped end conditions, the frequencies

Table 1. Comparison of the �rst three frequencies of pinned-pinned and clamped-clamped nano-beams versus thenon-dimensional small-scale parameter based on di�erent strain gradient theories.

hl Theory Present (IGA) Ref. [69] (FEA)

1st mode 2nd mode 3rd mode 1st mode 2nd mode 3rd mode

Pinned-pinned

1CT 1.6379 6.2015 12.9054 1.6379 6.2015 12.9054

MCST 2.9660 11.0772 22.7934 2.9661 11.0772 22.7934MSGT 4.4269 14.1670 26.9350 4.4269 14.1670 26.9351

5CT 0.3276 1.2403 2.5811 0.3276 1.2403 2.5811

MCST 0.3424 1.2969 2.7008 0.3424 1.2969 2.7008MSGT 0.3756 1.4054 2.8868 0.3757 1.4055 2.8868

10CT 0.1638 0.6202 1.2905 0.1638 0.6202 1.2905

MCST 0.1657 0.6274 1.3058 0.1657 0.6274 1.3058MSGT 0.1702 0.6424 1.3323 0.1702 0.6424 1.3324

20CT 0.0819 0.3101 0.6453 0.0819 0.3101 0.6453

MCST 0.0821 0.3110 0.6472 0.0821 0.3110 0.6472MSGT 0.0827 0.3129 0.6506 0.0828 0.3129 0.6506

Clamped-clamped

1CT 3.4810 8.7805 15.6511 3.4810 8.7805 15.6511

MCST 6.2225 15.5963 27.8798 6.2225 15.5963 27.8798MSGT 8.2014 18.9531 33.4258 8.2014 18.9531 33.4259

5CT 0.6962 1.7561 3.1302 0.6962 1.7561 3.1302

MCST 0.7298 1.8491 3.3138 0.7298 1.8491 3.3138MSGT 0.8199 2.0304 3.5756 0.8199 2.0305 3.5756

10CT 0.3481 0.8781 1.5651 0.3481 0.8781 1.5651

MCST 0.3529 0.8929 1.5975 0.3529 0.8929 1.5975MSGT 0.3718 0.9318 1.6531 0.3718 0.9319 1.6531

20CT 0.1741 0.4390 0.7826 0.1741 0.4390 0.7826

MCST 0.1749 0.4424 0.7910 0.1749 0.4424 0.7911MSGT 0.1804 0.4538 0.8068 0.1804 0.4538 0.8068


Figure 1. Fundamental frequencies of clamped-free and clamped-clamped nanotubes versus the length-to-diameter ratiobased on MD simulations [70] and the present integral nonlocal modi�ed strain gradient model.

Figure 2. Variation of non-dimensional fundamentalfrequency of clamped-free nano-beam versus thenon-dimensional small-scale and nonlocal parametersbased on di�erential and integral model of nonlocalmodi�ed strain gradient theory.

of NLSGT perfectly match the ones of moleculardynamics.

In Figures 2-4, the �rst dimensionless naturalfrequency of beams is plotted against dimensionlessnonlocal and small-scale parameters correspondingto clamped-free, pinned-pinned, and clamped-pinnedboundary conditions, respectively. The dimensionlessnonlocal parameter is de�ned as nonlocal parameter-to-length ratio (�=l), and dimensionless small scale pa-rameter is de�ned as material length scale parameter-to-length ratio (l=L). The results of these �guresare generated based on two models: integral nonlocalstrain gradient and di�erential nonlocal strain gradientmodels. Also, the results from the classical elasticitytheory (CT) are presented for the comparison purpose.Additionally, the strain gradient model is based on themodi�ed strain gradient theory with the assumptionof l0 = l1 = l2 = l. The e�ects of length scale andnonlocal parameters can be studied here simultane-

Figure 3. Variation of non-dimensional fundamentalfrequency of pinned-pinned nano-beam versus thenon-dimensional small-scale and nonlocal parametersbased on di�erential and integral model of nonlocalmodi�ed strain gradient theory.

Figure 4. Variation of non-dimensional fundamentalfrequency of clamped-pinned nano-beam versus thenon-dimensional small-scale and nonlocal parametersbased on di�erential and integral model of nonlocalmodi�ed strain gradient theory.


ously. It is �rst observed that the strain gradient e�ectis more pronounced than nonlocality e�ect. Increas-ing the length scale parameter leads to the increasein frequency for all types of boundary conditions.Whereas, the nonlocal e�ect on the free vibrationsof beams is dependent on the selected model. Inthe case of cantilever, the frequency slightly increasesby increasing the dimensionless nonlocal parameterbased on the di�erential nonlocal model, while thefrequency decreases as �=L gets larger based on the in-tegral nonlocal model. For other boundary conditions,increasing the nonlocal parameter has a decreasingin uence on the frequency of structure based on bothmodels. However, the decrease in frequency using theintegral model is more prominent than that based onthe di�erential model, especially for clamped-pinnedbeam. It can also be found that there are some speci�cvalues of nonlocal and length scale parameters at whichthe predictions of integral nonlocal strain gradientmodel become identical to those of the classical model.

Figures 5-7 show the variations of the �rst andsecond natural frequencies of beams with the di-mensionless nonlocal parameter according to �ve mod-els. These models are the classical model, di�erentialnonlocal model (Di�. NL), integral nonlocal model(Int. NL), di�erential nonlocal modi�ed strain gradientmodel (Di�. NLSGT), and integral nonlocal modi�edstrain gradient model (Int. NLSGT). Figures 5-7 aregiven for clamped-free, pinned-pinned, and clamped-pinned beams, respectively. Figure 5 reveals that basedon the di�erential nonlocal model (and di�erentialnonlocal modi�ed strain gradient model), the �rst

natural frequency of clamped-free beam increases withincrease in nonlocal parameter, which is a paradox.Such a paradox is not observed for the second modeof vibration as reported in [71]. In contrast, increasingthe nonlocal parameter has a decreasing e�ect on thenatural frequencies of cantilever based on the integralnonlocal model (and integral nonlocal modi�ed straingradient model). It should be noted that the decreasein second frequency based on the integral model is morepronounced than that based on the di�erential model.Also, Figures 6 and 7 indicate that using the di�erentialnonlocal model does not lead to paradoxical resultsfor pinned-pinned and clamped-pinned beams. As canbe expected, the combined nonlocal strain gradientmodels predict the vibration frequencies larger thantheir nonlocal counterparts do due to the sti�eninge�ects of strain gradient terms. The results also showthat at a given value of �=L, the smallest frequencybelongs to the integral nonlocal model and the largestone is computed based upon the di�erential nonlocalmodi�ed strain gradient model.

In Figures 8-10, the �rst frequency of beams withdi�erent boundary conditions is plotted versus thedimensionless small-scale parameter based on variousstrain gradient models including the modi�ed couplestress model (MCST), Aifantis's strain gradient model(SGT), the modi�ed strain gradient model (MSGT),and their combinations with the integral nonlocalmodel. It is seen that based on all models, thefrequency increases as l=L becomes larger. It is alsoobserved that the di�erence between the results of mod-els tends to diminish with decreasing the dimensionless

Figure 5. Variations of non-dimensional frequencies of the 1st and 2nd modes of clamped-free nano-beam versus thenon-dimensional nonlocal parameter based on di�erential and integral model of nonlocal continuum and correspondingnonlocal modi�ed strain gradient theory (l=L = 0:005).


Figure 6. Variations of non-dimensional frequencies of the 1st and 2nd modes of pinned-pinned nano-beam versus thenon-dimensional nonlocal parameter based on di�erential and integral model of nonlocal continuum and correspondingnonlocal modi�ed strain gradient theory (l=L = 0:005).

Figure 7. Variations of non-dimensional frequencies of the 1st and 2nd modes of clamped-pinned nano-beam versus thenon-dimensional nonlocal parameter based on di�erential and integral model of nonlocal continuum and correspondingnonlocal modi�ed strain gradient theory (l=L = 0:005).

small-scale parameter. Another �nding is that thesti�ening e�ect of strain gradient terms is the highestbased on the modi�ed strain gradient theory and isthe lowest based on the modi�ed couple stress theory.Furthermore, integral nonlocal strain gradient modelscan have underestimation or overestimation comparedto the classical theory depending on the value of lengthscale parameter.

5. Conclusion

In this article, the vibrational characteristics of small-scale Timoshenko beams were studied based on thenonlocal strain gradient theory. To accomplish thisgoal, Eringen's nonlocal theory in conjunction withthe most general strain gradient theory was utilized.The beam model was formulated according to both


Figure 8. Variation of non-dimensional fundamentalfrequency of clamped-free nano-beam versus thenon-dimensional small-scale parameter based on straingradient theories and corresponding integral model ofnonlocal strain gradient theories (�=L = 0:02).

Figure 9. Variation of non-dimensional fundamentalfrequency of pinned-pinned nano-beam versus thenon-dimensional small-scale parameter based on straingradient theories and corresponding integral model ofnonlocal strain gradient theories (�=L = 0:02).

original (integral) and di�erential forms of nonlocaltheory. Also, the adopted strain gradient theory wasthe most general one with additional non-classical ma-terial constants used for capturing small-scale e�ects.Hamilton's principle was used in order to obtain thegoverning equations of motion. Moreover, the govern-ing equations were derived in the vector-matrix form,which could be readily employed in the �nite elementor isogeometric analyses. IGA was then applied for thesolution procedure. The e�ects of nonlocal parameterof nonlocal theory and length scale parameter of straingradient theories on the frequencies of pinned-pinned,clamped-free, and clamped-pinned nano-beams wereinvestigated in detail. In addition, comparisons were

Figure 10. Variation of non-dimensional fundamentalfrequency of clamped-pinned nano-beam versus thenon-dimensional small-scale parameter based on straingradient theories and corresponding integral model ofnonlocal strain gradient theories (�=L = 0:02).

made between the predictions of di�erential/integralnonlocal models, strain gradient models, and nonlocalstrain gradient models. Some important �ndings of thestudy can be summarized as follows:

� The strain gradient in uence is more prominentthan the nonlocal e�ect. Also, by the increase inlength scale parameter, beams under all kinds ofboundary conditions vibrate with larger frequencies.The nonlocal e�ect in the di�erential model is de-pendent on the selected end conditions and vibrationmode;

� Using the di�erential nonlocal model to calculatethe fundamental frequencies of cantilevers leads toparadoxical results and recourse must be madeto the integral formulation of Eringen's nonlocaltheory. Such a paradox does not exist at highermodes and for other boundary conditions;

� The smallest and largest frequencies are obtainedbased upon the integral nonlocal model and thedi�erential nonlocal modi�ed strain gradient model,respectively. Furthermore, comparison betweendi�erent strain gradient models reveals that the sti�-ening e�ect of strain gradient terms is the highestaccording to the modi�ed strain gradient theory andis the lowest based upon the modi�ed couple stresstheory;

� The nonlocal and strain gradient e�ects cancel eachother out for some speci�c values of nonlocal andlength scale parameters;

� The standard NURBS-based IGA provides the con-sistency of higher gradients in di�erent types ofstrain gradient and nonlocal - strain gradient the-ories. This leads to achieving more e�cient solution


strategy than the �nite element analysis, whichrequires Hermite shape functions and additionalDOFs;

� With appropriate set of size-dependent materialconstants, the integral nonlocal modi�ed straingradient theory matches the molecular dynamicssimulation.

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Biographies

Amir Norouzzadeh received his BSc and MSc de-grees in Automotive and Mechanical Engineering, in2012 and 2014, from Iran University of Science andTechnology and University of Guilan, Iran, respec-tively. He is currently a PhD candidate in MechanicalEngineering at the University of Guilan, Iran. Hisresearch interests include computational mechanics,generalized continuum mechanics, �nite element, andisogeometric analysis.

Reza Ansari received his PhD degree, in 2008, fromthe University of Guilan, Iran, where he is currentlyfaculty member in the Department of MechanicalEngineering. He was also a visiting fellow at Wol-longong University, Australia, during 2006-2007. Hehas authored numerous refereed journal papers andbook chapters. His research interests include computa-tional nano- and micro-mechanics, advanced numericaltechniques, nonlinear analyses, and prediction of themechanical behavior of smart composite/FGM shell-type structures.

Hessam Rouhi received his PhD degree in MechanicalEngineering, in 2016, from the University of Guilan,Iran. He is currently Faculty Member in the Depart-ment of Engineering Science, Faculty of Technologyand Engineering, East of Guilan, University of Guilan,Iran. His research interests include computationalmicro- and nano-mechanics and the mechanical behav-ior of structures, including buckling and vibration.

Isogeometric vibration analysis of small-scale Timoshenko...

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