Eringen’s nonlocal theories of beams accounting for moderate rotations

International Journal of Engineering Science 82 (2014) 159–177

Contents lists available at ScienceDirect

International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci

Eringen’s nonlocal theories of beams accounting for moderate rotations J.N. Reddy a,⇑, Sami El-Borgi b,c a

Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA Mechanical Engineering Program, Texas A&M University at Qatar, Engineering Building, P.O. Box 23874, Education City, Doha, Qatar c Applied Mechanics and Systems Research Laboratory, Tunisia Polytechnic School, University of Carthage, B.P. 743, La Marsa 2078, Tunisia b

a r t i c l e

i n f o

Article history: Received 28 March 2014 Received in revised form 8 May 2014 Accepted 13 May 2014

Keywords: Beams Eringen’s differential model Material length scales Finite element models Numerical results

a b s t r a c t The primary objective of this paper is two-fold: (a) to formulate the governing equations of the Euler–Bernoulli and Timoshenko beams that account for moderate rotations (more than what is included in the conventional von Kármán strains) and material length scales based on Eringen’s nonlocal differential model, and (b) develop the nonlinear finite element models of the equations. The governing equations of the Euler–Bernoulli and Timoshenko beams are derived using the principle of virtual displacements, wherein the Eringen’s nonlocal differential model and modified von Kármán nonlinear strains are taken into account. Finite element models of the resulting equations are developed, and numerical results are presented for various boundary conditions, showing the effect of the nonlocal parameter on the deflections. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction 1.1. Background There has been increased interest in recent years in developing structural theories that have the ability to capture material length scale effects. This is primarily due to the need to model the structural response of a variety of new materials which are being developed that require the consideration of very small length scales over which the neighboring secondary constituents interact, especially when the spatial resolution is comparable to the size of the secondary constituents. Examples of such materials are provided by nematic elastomers and carbon nanotube composites (Cadek et al., 2004) and environment resistent coatings made of CNT reinforced materials (Chen et al., 2003; Kaempgen, Duesberg, & Roth, 2005). In addition, the flexoelectric effect (Catalan et al., 2011), which is a size dependent strain gradient effect on the polarization of ferroelectrics, induces piezo-electric response in non-piezoelectric materials at very small scales. Miniaturized beams are the basic structures used in several applications including microactuators, microswitches, biosensors, nanowires, nanoprobes, ultra thin films, and MEMS and NEMS (Hung & Senturia, 1999; Li, Bhushan, Takashima, Baek, & Kim, 2003; Moser & Gijs, 2007; Pei, Tian, & Thundat, 2004; Najar, Nayfeh, Abdel-Rahman, Choura, & El-Borgi, 2010). Arash and Wang (2012) showed that small length scale effect should be accounted for in the modeling of micro and nano structures because the material properties at the nanoscale are size-dependent. In the past, many studies related

⇑ Corresponding author. Tel.: +1 979 862 2417; fax: +1 979 862 3989. E-mail address: [email protected] (J.N. Reddy). http://dx.doi.org/10.1016/j.ijengsci.2014.05.006 0020-7225/Ó 2014 Elsevier Ltd. All rights reserved.

160

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

to microbeams were based on classical continuum theory which neglects size-effects. To overcome this problem nonlocal elasticity theories were used to develop size-dependent beam models. The use of nonlocal continuum mechanics has found successful applications in several areas which include fracture mechanics, lattice dispersion of elastic waves, mechanics of dislocations and wave propagation in mechanics (see Peddieson, Buchanan, & McNitt, 2003). Due to the small physical dimensions of these devices, microstructure-dependent size effects are often exhibited by structural elements used in various micro- and nano-scale devices (McFarland & Colton, 2005). Nonlocal theories that have been studied in the literature include Eringen’s nonlocal elasticity theory (Eringen, 1972; Eringen, 1983; Eringen, 2002; Eringen & Edelen, 1972), modified couple stress theory of Mindlin (1963), Koiter (1964), and Toupin (1964), and the strain gradient theory (Mindlin, 1965; Yang, Chong, Lam, & Tong, 2002; Lam, Yang, Chong, Wang, & Tong, 2003). Eringen’s nonlocal elasticity can be classified into a differential nonlocal form or an integral nonlocal form. Detailed review of both forms is discussed by Lim (2010). The strain gradient theory is a different form of the modified couple stress theory and they can be related to each other. A more complete review of the early developments can be found in the work of Srinivasa and Reddy (2013), and the relationship between the modified couple stress theory and the strain gradient theory can be found in the recent work of Reddy and Srinivasa (in press). 1.2. A review of literature Eringen’s nonlocal elasticity differential model was used to model size-effects in homogeneous micro and nano structures. Reddy (2007) developed analytical solutions for bending, buckling and vibration of beams using Euler–Bernoulli, Timoshenko, Reddy, and Levinson beam theories. Thai (2012) studied analytically the bending, buckling and vibration response of a Euler–Bernoulli nanobeam. Roque, Ferreira, and Reddy (2011) used a meshless method based on collocation with radial basis to study the static bending, buckling and free vibration behaviors of a Timoshenko nanobeam. Li, Lim, and Yu (2011) studied analytically using the perturbation method the free vibration, the steady-state resonance and stability of a vibrating nanobeam subjected to a variable axial load. Eltaher, Alshorbagy, and Mahmoud (2013) solved the vibration problem of a Euler-Bernoulli nanobeam using the finite element method. The previous studies are all linear problems. Reddy (2010) accounted for the von Kármán nonlinear strains and reformulated classical and shear deformation beam and plate theories. Similarly, Simsek (2014) proposed an analytical solution based on He’s method (He, 1999) to study the nonlinear free vibration problem of a Euler-Bernoulli nanobeam considering the von Karman nonlinear strains. Mousavi, Bornassi, and Haddadpour (2013) used the differential quadrature method to study the nonlinear pull-in instability of a nanoswitch modeled as a Euler–Bernoulli nanobeam subjected to electrostatic and intermolecular forces and having different boundary conditions. A limited number of investigators exploited the modified couple stress theory to model size-effects in homogeneous micro and nano structures. Park and Gao (2006) and Park and Gao (2008) investigated analytically the bending problem of a Euler–Bernoulli beam. Abdi, Koochi, Kazemi, and Abadyan (2011) examined the pull-in instability of an electrostatic cantilever nanobeam. Akgöz and Civalek (2011) studied analytically the buckling problem of an axially loaded microbeam. Rahaeifard, Kharobaiyan, Asghari, and Ahmadian (2011) considered the deflection and static pull-in voltage of microcantilevers. Xia, Wang, and Yin (2010) investigated analytically the bending, post-buckling and free vibration behaviors of microbeams while accounting for the von Kármán nonlinear strains. Few researchers used the strain gradient theory to model size-effects in homogeneous micro and nano structures. Ansari, Gholami, and Sahmani (2011) examined analytically the free vibration response of a Timoshenko graded microbeam. Akgöz and Civalek (2011) studied analytically the buckling problem of an axially loaded microbeam. Eltaher, Hamed, Sadoun, and Mansour (2014) investigated bending, vibration and buckling behaviors of a Euler–Bernoulli nanobeam by modeling sizedependency using a strain gradient theory derived from Eringen’s nonlocal model. The last two decades have witnessed investigators exploring the possibility of using functionally graded materials (FGMs) as a promising alternative to conventional homogenous coatings (see Koizumi, 1993; Erdogan, 1995). FGMs comprise of at least two-phase inhomogeneous particulate composites and are synthesized in such a way that the volume fractions of the constituents vary continuously along any desired spatial direction, resulting in materials having smooth variation of mechanical properties. Such enhancements in properties endow FGMs with qualities such as resilience to fracture through reduction in propensity for stress concentration. FGMs promise attractive applications in a wide variety of wear coating and thermal shielding problems such as gears, cams, cutting tools, high temperature chambers, furnace liners, turbines, microelectronics and space structures (see Reddy and his colleagues Praveen & Reddy, 1998; Reddy & Chin, 1998; Praveen, Chin, & Reddy, 1999; Reddy, 2000; Reddy, 2011 for the analysis of through-thickness, two-constituent FGM beams, plates, and shells). With the progress of technology and fast growth of the use of nanostructures, FGMs have found potential applications in micro and nano scale in the form of shape memory alloy thin films (see Lü, Lim, & Chen, 2009), electrically actuated actuators (see Zhang & Fu, 2012), microswitches (see Shariat, Liu, & Rio, 2013) and atomic force microscopes (AFMs) (see Kahrobaiyan, Asghari, Rahaeifard, & Ahmadian, 2010). An increasing number of researchers have started focusing on functionally graded micro and nanobeams using mainly Eringen’s nonlocal elasticity differential model. Using the finite element method, Eltaher and co-workers (Eltaher, Emam, & Mahmoud, 2012; Eltaher, Emam, & Mahmoud, 2013; Eltaher, Khairy, Sadoun, & Omar, 2014) studied the static bending, vibration and buckling behaviors of functionally graded Euler–Bernoulli and Timoshenko nanobeams. Using Navier’s

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

161

method, Uymaz (2013) solved analytically the free and forced vibration problem of a graded nanobeam using various classical and higher-order beam theories. A similar study is carried out by Rahmani and Pedram (2014) who exploited Navier’s method to study analytically the vibration behavior of a graded Timoshenko nanobeams. Nazemnezhad and HosseiniHashemi (2014) accounted for the nonlinear von Kármán strains and studied analytically using the method of multiple scales and Galerkin’s method the nonlinear free vibration of functionally graded Euler–Bernoulli nanobeams. Few investigators studied the response of functionally graded micro and nanobeams using nonlocal theories other than Eringen’s nonlocal model. Reddy and his colleagues (Reddy, 2011; Reddy & Berry, 2012; Arbind & Reddy, 2013; Simsek & Reddy, 2013; Simsek & Reddy, 2013; Reddy & Kim, 2012; Kim & Reddy, 2013; Arbind, Reddy, & Srinivasa, 2014) studied bending, vibration, and buckling of functionally graded Euler–Bernoulli and Timoshenko beams. Simsek and Reddy (2013) and Simsek and Reddy (2013) studied analytically the bending and vibration behaviors of graded microbeams using classical and higher-order beam theories along with the modified couple stress theory to model size-effects. Arbind and Reddy (2013) and Arbind et al. (2014) accounted for the von Kármán nonlinear strains to develop nonlinear finite element models for functionally graded Euler–Bernoulli and Timoshenko beams. The vast majority of two-constituent FGM studies employed either a power-law or exponential distribution of the materials. In the power-law model, which is more commonly used in bending, vibration, and buckling studies, a typical material property P is assumed to vary through the thickness according to the formula (see Praveen & Reddy, 1998; Reddy & Chin, 1998; Praveen et al., 1999; Reddy, 2000; Reddy, 2011)

 Pðz; TÞ ¼ ½P 1 ðTÞ  P 2 ðTÞf ðzÞ þ P 2 ðTÞ;

f ðzÞ ¼

1 z þ 2 h

n

where P 1 and P 2 are the material properties of the top (material 1) and bottom (material 2) faces of the beam or plate, respectively, n is the volume fraction exponent, and T is the temperature (i.e., the material properties can be possibly function of temperature). The exponential model, which is often employed in fracture studies, is based on the formula (see Kim & Paulino, 2002; Zhang, Savaidis, Savaidis, & Zhu, 2003)

   1 z ; Pðz; TÞ ¼ P 1 ðTÞ exp a  2 h

a ¼ log



 P 1 ðTÞ P 2 ðTÞ

A review of the literature indicates that most investigators have based their work related to micro and nanobeams on geometrically linear theories of beams. Few researchers considered geometric nonlinearity in the sense of the classical von Kármán strains, and this is particularly true for functionally graded nanobeams (see Reddy, 2011; Arbind & Reddy, 2013; Reddy & Kim, 2012; Arbind et al., 2014). If we assume that all displacement gradients with respect to the in-plane coordinates xa are small, that is (see Reddy, 1987; Schmidt & Reddy, 1988)

  @ua     @xb   OðÞ;

  @u3  pffiffiffi    @xa   Oð Þ;

  1; a; b ¼ 1; 2

we can neglect the terms involving squares and products of @ua =@xb , while retaining the squares of @u3 =@xa in the definition of the Green–Lagrange strain tensor. This leads to strain–displacement equations that contain not only the von Kármán nonlinear terms but also additional nonlinear terms that have not been accounted for in beam and plate theories. Thus, it is useful to develop modified theories of beams and plates that account for size effects and the geometric nonlinearity that is based on an order-of-magnitude assumption with respect to the magnitude of the linear strains. 1.3. Eringen’s nonlocal differential model Classical continuum theories are based on hyperelastic constitutive relations which assume that the stress at a point is a function of strains at that point. On the other hand, the non-local continuum mechanics assumes that the stress at a point is a function of strains at all points, at least in some neighborhood of the point, in the continuum. These theories contain information about the forces between atoms, and the internal length scale is introduced into the constitutive equations as a material parameter. Such nonlocal elasticity was initiated in the works of Eringen (1972), Eringen (1983), Eringen (2002) and Eringen and Edelen (1972). According to Eringen (1972) and Eringen (1983), the state of stress r at a point x in an elastic continuum not only depends on the strain field e at the point (hyperelastic case) but also on strains at all other points of the body. Eringen attributed this to the atomic theory of lattice dynamics and experimental observations on phonon dispersion. Thus, the nonlocal stress ten at point x is expressed as sor r

r ¼

Z

Kðjx0  xj; sÞ rðx0 Þ dx0

ð1Þ

X

where rðxÞ is the classical, macroscopic second Piola–Kirchhoff stress tensor at point x and the kernel function Kðjx0  xj; sÞ represents the nonlocal modulus, jx0  xj being the distance (in the Euclidean norm) and s is a material parameter that depends on internal and external characteristic lengths (such as the lattice spacing and wavelength, respectively). The macroscopic stress r at a point x in a Hookean solid is related to the strain e at the point by the generalized Hooke’s law

162

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

rðxÞ ¼ CðxÞ : eðxÞ

ð2Þ

where C is the fourth-order elasticity tensor and: denotes the ‘double-dot product’ (see Reddy, 2013). The constitutive Eqs. (1) and (2) together define the nonlocal constitutive behavior of a Hookean solid. Eq. (1) represents the weighted average of the contributions of the strain field of all points in the body to the stress field at point x. In view of the difficulty in using the integral constitutive relation, Eringen (1983) proposed an equivalent differential model as

  ¼ r; 1  l20 r2 r

l0 ¼ s2 ‘2 ¼ e20 a2

ð3Þ

where e0 is a material constant, and a and ‘ are the internal and external characteristic lengths, respectively. It is assumed that when the local stress tensor is expressed in terms of the displacement gradients through the generalized Hooke’s law, the displacements appearing on the right-hand side of Eq. (3) are the nonlocal displacements. The nonlocal theory of elasticity of Eringen has been used extensively in the last decade to study lattice dispersion of elastic waves, wave propagation in composites, dislocation mechanics, fracture mechanics, surface tension fluids, and so on. The use of nonlocal elasticity to study size-effects in micro and nanoscale structures was first carried out by Peddieson et al. (2003). They used the nonlocal elasticity to study the bending of micro and nanoscale beams and concluded that size-effects could be significant for nano structures. Zhang et al. used nonlocal elasticity to show the small-scale effects on buckling of MWCNTs under axial compression (Zhang, Liu, & Wang, 2004) and radial pressure (Zhang, Liu, & Xie, 2005). Wang (2005) and Wang and Varadan (2006) have studied wave propagation in carbon nanotubes (CNTs) with nonlocal Euler–Bernoulli and Timoshenko beam models. The small-scale effect on CNTs wave propagation dispersion relation is explicitly determined for different CNTs wave numbers and diameters by theoretical analyses and numerical simulations. The scale coefficient in nonlocal continuum mechanics is roughly estimated for CNTs from the obtained asymptotic frequency. The findings proved to be effective in predicting small-scale effect on CNTs wave propagation with a qualitative validation study based on the published experimental work. Wang, Zhang, Ramesh, and Kitipornchai (2006) formulated a nonlocal Timoshenko beam theory, neglecting the non-local effect in writing the shear stress–strain relation. Reddy (2007), Reddy (2010) and Reddy and Pang (2008) have formulated nonlocal versions of various beam and plate theories, and presented numerical solutions of bending, vibration, and buckling of beams. There are numerous other papers that study the response of nanosystems using theories that are based on Eringen’s differential model (see Sudak, 2003; Adali, 2009; Challamel & Wang, 2008; Challamel et al., in press, and references therein). The results show that the nonlocal parameter l0 ¼ s2 ‘2 ¼ e20 a2 has the effect of softening the stiffness for all boundary conditions except for cantilever, and thus predict larger deflections and lower buckling loads and vibration frequencies. The paradox surrounding cantilever boundary conditions has been discussed in the literature by Challamel et al. (in press), who offered a remedy for the special case of the cantilever beam by tweaking the boundary condition for the bending moment of a free end based on the nonlocal model. 1.4. Present study If Eringen’s differential constitutive model is used directly in the virtual work statement, one obtains equations of motion in terms of the nonlocal stress resultants, which do not conform to the normal structural mechanics formulations in that the resulting equations are not derivable from a strain energy potential, as noted by Reddy (2007) and Reddy (2010). In addition, it is not possible to express the nonlocal stress resultants in terms of the displacements, and the boundary conditions derived from the virtual work statement do not conform to the physical boundary conditions known from the conventional theories. Consequently, most researchers used the conventional equations of motion and boundary conditions, with local stress resultants replaced by nonlocal stress resultants, to study nonlocal beams. Many of these issues are not well understood by most researchers and thus there is a need to revisit Eringen’s nonlocal differential model. The simplified Green–Lagrange strain tensor (see Reddy, 2013) E  e that includes small strains but moderate rotations is commonly known as the von Kármán strain tensor, and the associated theories are termed von Kármán beam theories. Con2 ð0Þ ventional von Kármán nonlinear beam theories only account for ð1=2Þðdw=dxÞ in the membrane strain exx . In the modified von Kármán beam theories, as proposed by Reddy and Mahaffey (2013), we also account for term of the type ð1=2Þðhx Þ2 and ð1=2Þð/x Þ2 in ezz . To date Eringen’s nonlocal elasticity constitutive model with the modified von Kármán nonlinearity has not been formulated, although Reddy (2010) presented such a model for the case of conventional von Kármán nonlinearity. Further, no numerical results of beams based on Eringen’s nonlocal elasticity constitutive model and the von Kármán nonlinearity have appeared in the literature. The present study will address these issues. The present study is focused on: (1) formulating beam theories based on Eringen’s differential model with the modified von Kármán nonlinearity, (2) developing finite element models, and (3) presenting numerical results for various boundary conditions to bring out the effect of the nonlocal parameter on the deflections. In this paper, we consider straight beams of length L, height h, and width b. The x-coordinate is taken along the length of the beam with the z-coordinate along the thickness (the height) and the y-coordinate along the width of the beam. In a beam theory, all applied loads and geometry are such that the displacements (u1 ; u2 ; u3 ) along the coordinates (x; y; z) are only functions of the x and z coordinates, because of the introduction of the stress resultants: ðiÞ

Mij ¼

Z A

ðzÞi rij dA

ð4Þ

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

163

where rij are the Cartesian components of the conventional (i.e., local) Piola–Kirchhoff stress tensor. It is further assumed that the displacement u2 is identically zero, that is, bending is in the xz-plane. In the following, equations of motion of the Euler–Bernoulli and Timoshenko beam theories are derived using the principle of virtual displacements (see Reddy, 2002; Reddy, 2004) and accounting for the modified von Kármán nonlinearity. Since the principle of virtual work is independent of the constitutive relations, the equations of equilibrium expressed in terms of the stress resultants are valid for local as well as Eringen’s nonlocal models. The derived equations of the two beam theories are modified to account for the nonlocal effects by expressing the stress resultants in terms of the nonlocal parameter l0 . In general, the nonlocal theories derived do not admit a quadratic functional. Finally, the displacement finite element models of the two nonlocal beam theories are developed and the tangent matrix coefficients are derived. 2. The Euler–Bernoulli beam theory 2.1. Displacement and strain fields The Euler–Bernoulli beam theory (EBT) is based on the displacement field

^x þ wðxÞ e ^z ; u ¼ ½uðxÞ þ zhx e

hx  

dw dx

ð5Þ

where (u; w) are the axial and transverse displacements of the point (x; 0) on the mid-plane (i.e., z ¼ 0) of the beam, and ^x ; e ^z ) are unit vectors along the (x; z) coordinates. (e pffiffiffi 2 2 We assume that the axial strain ðdu=dxÞ is of order , ðdw=dxÞ is of order , and ‘ðd w=dx Þ is of order , where ‘ is the intrinsic length scale associated with the nonlocal theory and   1 is a small parameter. The Lagrangian strain tensor E is of the form (omitting terms of order 2 )

Ee¼

"  2 # du dhx 1 dw 1 ^x þ h2x e ^z e ^z ^x e þz þ e dx 2 dx 2 dx

ð6Þ

pffiffiffi We note that the strains exx is of the order  while the rotation hx is of the order . The constitutive equations relating r to e assumed, for isotropic materials, to be of the form

r ¼ 2l e þ kI trðeÞ

ð7Þ

where l and k are the Lamé constants (k ¼ Em=ð1 þ mÞð1  2mÞ and 2l ¼ E=ð1 þ mÞ, with E being Young’s modulus and m Poisson’s ratio) and trðeÞ denotes the trace of the strain tensor (ekk ). 2.2. Equations of equilibrium and boundary conditions The principle of virtual displacements has the form

0¼

Z

L

0

h i ð1Þ ð0Þ ð0Þ ð1Þ ð0Þ M ð0Þ xx dexx þ M xx dexx þ M zz dezz  fx du  fz dw dx

ð8Þ

where fx and fz are the distributed loads along the x and z axes, respectively, and ð0Þ exx ¼

 2 du 1 dw þ ; dx 2 dx

eð1Þ xx ¼

dhx ; dx

1 2

eð0Þ h2x zz ¼

ð9Þ

ð0Þ

Note that ezz – 0, implying that the transverse normals are extensible and the Poisson effect is not neglected. Thus, we must use two-dimensional constitutive relations based on Eq. (7). We obtain the following Euler equations in 0 < x < L: ð0Þ

dM xx þ fx ¼ 0 dx   2 ð1Þ d M xx d  ð0Þ ð0Þ dw þ fz ¼ 0 M þ þ M xx zz 2 dx dx dx

ð10Þ ð11Þ

The force boundary conditions are to specify the following expressions (when the corresponding displacements are not specified)

Geometric :u;

w;

ð0Þ Force :M xx ;

ð12Þ

hx ð1Þ  dMxx ð0Þ dw V x  M ð0Þ þ ; xx þ M zz dx dx

ð1Þ Mxx

ð13Þ

164

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

The conventional stress resultants can be expressed in terms of the displacements as

"  2 # du 1 dw 1 þ c13 Ah2x þ dx 2 dx 2 A "  2 # Z du 1 dw 1 ¼ rzz dA ¼ c13 A þ þ c33 Ah2x dx 2 dx 2 A Z dhx ¼ zrxx dA ¼ c11 I dx A

Mð0Þ xx ¼ Mð0Þ zz ð1Þ Mxx

Z

rxx dA ¼ c11 A

ð14Þ ð15Þ ð16Þ

where A is the area and I is the moment of inertia about the y-axis of the beam cross section, 3

A ¼ bh;

I¼

bh 12

ð17Þ

and

c11 ¼ c33 ¼ 2l þ k;

c13 ¼ k

ð18Þ

3. The Timoshenko beam theory 3.1. Displacement and strain fields The displacement field in the Timoshenko beam theory is

^x þ wðxÞ e ^z u ¼ ½uðxÞ þ z/x ðxÞe

ð19Þ

where /x is the independent rotation of the transverse line at x about the y axis. The Lagrangian strain tensor E is of the form (omitting terms of order 2 )

"  2 #   du d/x 1 dw 1 dw 1 ^x e ^x þ ^z þ e ^z e ^x Þ þ /2x e ^z e ^z ^x e Ee¼ þz þ /x þ ðe e dx 2 dx 2 dx 2 dx

ð20Þ

The Timoshenko beam theory requires shear correction factors to compensate for the error due to the constant state of shear stress predicted by the kinematics. The shear correction factor depends, in general, not only on the material and geometric parameters but also on the load and boundary conditions. The principle of virtual displacements for the Timoshenko beam is given by

0¼

Z

L

h

0

where ð0Þ exx ¼

i ð0Þ ð1Þ ð0Þ ð0Þ ð1Þ ð0Þ M xx deð0Þ xx þ M xx dexx þ M xz dcxz þ M zz dezz  fx du  fz dw dx

 2 du 1 dw þ ; dx 2 dx

eð1Þ xx ¼

d/x ; dx

cxz ¼ /x þ

dw ; dx

1 2

eð0Þ /2x zz ¼

ð21Þ

ð22Þ

The Euler equations are ð0Þ

dM xx þ fx ¼ 0 dx   d ð0Þ dw þ fz ¼ 0 M ð0Þ xz þ M xx dx dx ð1Þ  dM xx ð0Þ  M ð0Þ xz þ M zz /x ¼ 0 dx

ð23Þ ð24Þ ð25Þ

The boundary conditions involve specifying the following variables:

Geometric : u;

w;

ð0Þ Force : M xx ;

ð26Þ

/x dw V x  Mð0Þ þ Mð0Þ xx xz ; dx

ð1Þ Mxx

ð27Þ

The conventional stress resultants of the Timoshenko beam theory can be expressed in terms of the displacements as

Mð0Þ xx ¼

Z

rxx dA ¼ c11 A A

"  2 # du 1 dw 1 þ c13 A/2x þ dx 2 dx 2

ð28Þ

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

Mð0Þ zz ¼

rzz dA ¼ c13 A

A

"  2 # du 1 dw 1 þ c33 A/2x þ dx 2 dx 2

ð29Þ

Z

d/ zrxx dA ¼ c11 I x dx   Z dw ¼ K s rxz dz ¼ c44 K s A /x þ dx A

ð1Þ ¼ Mxx

Mð0Þ xz

Z

165

ð30Þ

A

ð31Þ

where K s is the shear correction factor and c44 ¼ l ¼ G, the shear modulus of the beam. 4. Nonlocal stress resultants 4.1. Introduction The constitutive equations in Eqs. (2) and (3) together define the nonlocal constitutive behavior of a Hookean solid. In this section, we derive the differential equations for the nonlocal stress resultants in terms of the strains, assuming that the  zz ). This process involves use of the differential nonlocal effect is negligible for the transverse normal stress (i.e., rzz ¼ r constitutive model in Eq. (3). As opposed to the linear algebraic equations between the stress resultants and strains in a local theory, the nonlocal theory results in differential relations involving the nonlocal stress resultants and the strains. In the following, we present these relations for homogeneous isotropic beams governed by the Euler–Bernoulli and Timoshenko beam theories. The nonlocal constitutive relation in Eq. (3), with Eq. (2) for the macroscopic stress, yield the following relations for various nonlocal stress components:

r xx  l0 r xz  l0

2  xx d r

¼ c11 exx þ c13 ezz

ð32Þ

xz ðl0 ¼ e20 a2 Þ ¼ c44 c 2 dx ¼ c13 exx þ c33 ezz

ð33Þ

2

dx 2  xz d r

r zz ¼ rzz

ð34Þ

where bar over a variable indicates that the variable is nonlocal. When the nonlocal parameter l0 is set to zero, we obtain the constitutive relations of the local theory. In both Euler–Bernoulli and Timoshenko beam theories, the axial force–strain relation is the same, and it is given by [integrating Eq. (32) over the beam cross sectional area] 2

Mð0Þ xx  l0

ð0Þ

d Mxx 2

dx

 ð0Þ ¼ A c11 exx þ c13 eð0Þ zz

ð35Þ

ð0Þ

where M xx denotes the nonlocal axial stress resultant. Since the x-axis is taken along the geometric centroid of the beam, we have

Z

z dA ¼ 0

A

4.2. The Euler–Bernoulli beam theory ð1Þ

ð0Þ

The constitutive relations for M xx and M zz in the EBT are given by 2

ð1Þ

d Mxx ð1Þ Mxx  l0 ¼ Ic11 eð1Þ xx 2 dx

 ð0Þ ð0Þ ð0Þ Mzz ¼ A c13 exx þ c33 ezz

ð36Þ ð37Þ ð1Þ ð0Þ xx ; xx ,

where I denotes the second moment of area about the y-axis, and e all nonlocal quantities.

e

ð0Þ zz

and e

are defined by Eq. (9), with bar placed on

4.3. The Timoshenko beam theory ð1Þ

ð0Þ

ð0Þ

The constitutive relations for M xx , M xz , and M zz in the TBT are given by 2

ð1Þ Mxx  l0

ð1Þ

d Mxx 2

dx

¼ Ic11 eð1Þ xx

ð38Þ

166

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177 2

ð0Þ

d M xz xz Mð0Þ ¼ AK s c44 c xz  l0 2 dx

 ð0Þ ð0Þ Mð0Þ zz ¼ A c 13 exx þ c 33 ezz where

ð0Þ xx ; xz ,

e

c

ð1Þ xx ,

e

and

ð0Þ zz

e

ð39Þ ð40Þ

are defined by Eq. (22) with a bar placed on all strain and displacement components.

5. Governing equations of the nonlocal theories of beams 5.1. Preliminary comments  ij using the differential conIf one chooses to replace local stress components rij with the nonlocal stress components r stitutive relation Eq. (3) in the virtual strain energy potential, it will result in equations of motion that are the same as in Eqs. ðiÞ

ðiÞ

2

ðiÞ

2

(10) and (11) for the EBT and Eqs. (23)–(25) for the TBT, with local stress resultants M ij replaced by M ij  l0 ðd M ij =dx Þ. ðiÞ

However, there is no conceivable way to replace the nonlocal stress resultants M ij in terms of the displacements (u; w; hx or /x ). In addition, it is not possible to recover the underlying strain energy potential from the virtual strain energy potential. Therefore, such a procedure is abandoned here. Instead, we assume that the equations of motion of the local theories in terms of the stress resultants also hold for the nonlocal theories with the local stress resultants replaced by the nonlocal stress resultants. Using the differential equations for the nonlocal stress resultants, namely, Eqs. (36)–(40), and the equations of motion, the nonlocal stress resultants are determined in terms of the displacements. The nonlocal stress resultants so derived are then substituted back into the equations of motion to obtain the governing equations of the nonlocal beam theories in terms of the displacements. 5.2. Equation for axial motion We begin with the equation of equilibrium governing the axial displacement u: ð0Þ

dMxx þ fx ¼ 0 dx

ð41Þ ð0Þ

Substituting for the first derivative of the axial force M xx from Eq. (41) into Eq. (35), we obtain

 df ð0Þ ð0Þ  l0 x Mð0Þ xx ¼ A c 11 exx þ c 13 ezz dx

ð42Þ

ð0Þ M xx .

Clearly, M ð0Þ Therefore specification of M ð0Þ xx – xx at the ends of a beam is not the same as the specification of we assume that l0 ¼ 0 at the boundary points. ð0Þ Substituting M xx from Eq. (42) into the equation of equilibrium in Eq. (41), we obtain

ð0Þ M xx ,

unless

2

 d

d f ð0Þ þ fx  l0 2x ¼ 0 Ac11 eð0Þ xx þ Ac 13 ezz dx dx

ð43Þ

Eqs. (41)–(43) are valid for the Euler–Bernoulli and Timoshenko beam theories. The axial equation of equilibrium of the  to u). conventional (i.e., local) beam theory can be obtained from Eq. (43) by setting l0 ¼ 0 (and changing u 5.3. The Euler–Bernoulli beam theory Since Eq. (11) is valid for nonlocal Euler–Bernoulli beam, we have 2

ð1Þ

d M xx 2

dx

þ

    d ð0Þ ð0Þ dw þ fz ¼ 0 Mxx þ M zz dx dx 2

ð1Þ

ð44Þ

2

ð1Þ

Substituting for d M xx =dx from Eq. (44) into Eq. (36) and solving for M xx , we obtain ð1Þ Mxx

   dw  d ð0Þ ð1Þ ¼ c11 Iexx  l0 f z þ M xx þ M ð0Þ zz dx dx   d dw dfx ð1Þ ð0Þ  þ c Þ e  l ¼ c11 Iexx  l0 fz  l0 þAðc Aðc11 þ c13 Þeð0Þ 13 33 zz 0 xx dx dx dx ð0Þ

ð1Þ

ð45Þ

ð0Þ

Substituting for M xx from Eq. (42), M xx from Eq. (45), and M zz from Eq. (40) into Eq. (44), we obtain



    2 3   d fz d dw dfx d dw dfx ð0Þ ð0Þ c11 Ieð1Þ þ l0 þ Aðc13 þ c33 Þezz  l0 þ l0 3 Aðc11 þ c13 Þexx xx  fz þ l0 2 dx dx dx dx dx dx dx dx

  d dw ð0Þ ¼ 0 Aðc11 þ c13 Þeð0Þ  xx þ Aðc 13 þ c 33 Þezz dx dx 2 d

2

ð46Þ

167

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

The equation of equilibrium of the conventional Euler–Bernoulli beam theory with modified von Kármán nonlinearity can be obtained from Eqs. (43) and (46) by setting l0 ¼ 0. 5.4. The Timoshenko beam theory ð0Þ

First, eliminating M xz between Eqs. (24) and (25), we obtain 2

d

ð1Þ M xx 2

dx

   d ð0Þ dw   fz ¼ M ð0Þ zz /x  M xx dx dx

ð47Þ ð1Þ

Substituting for the second derivative of M xx from Eq. (47) into Eq. (38), we obtain



    d ð1Þ ð0Þ dw   fz ¼ Ic11 exx þ l0 M ð0Þ zz /x  M xx dx dx    

  

d ð1Þ  x c13 eð0Þ þ c33 eð0Þ  dw A c11 eð0Þ þ c13 eð0Þ :l dfx  fz þ l0 A/ ¼ Ic11 exx xx zz xx zz 0 dx dx dx

ð1Þ Mxx

ð48Þ

ð0Þ

Next, substituting for the second derivative of M xz from Eq. (24), 2

2

dx

   dw dfz  M ð0Þ xx dx dx dx 2

ð0Þ

d M xz

¼

d

ð49Þ

2

into Eq. (39), we obtain

" Mð0Þ xz

xz  l0 ¼ K s Ac44 c

#     2   

dfz d dw dfx dfz ð0Þ dw ð0Þ ð0Þ    þ  l0 M Ac c  l e þ c e l  A c ¼ K s 44 xz 11 13 xx xx zz 0 0 2 2 dx dx dx dx dx dx dx 2

d

ð1Þ

ð0Þ

ð50Þ

ð0Þ

Now substituting for M xx from Eq. (48), M xz from Eq. (50), and M xx from Eq. (42) into Eqs. (24) and (25), we obtain



    2 3   



d d f d dw dfx d dw dfx ð0Þ ð0Þ xz Þ þ l0 2z þ l0 3  fz ¼ 0 ð51Þ ðK s Ac44 c A c11 eð0Þ A c11 eð0Þ xx þc 13 ezz  l0 xx þc 13 ezz  l0 dx dx dx dx dx dx dx dx 2

xz  K s Ac44 c



 

 d

ð1Þ  x c13 eð0Þ þc33 eð0Þ  l d A/  x c13 eð0Þ þc33 eð0Þ ¼ 0 Ic11 exx þA/ xx zz xx zz 0 2 dx dx

ð52Þ

This completes the development of the governing equations of nonlocal Euler–Bernoulli and Timoshenko beam theories in terms of the nonlocal displacements. 5.5. Special cases 5.5.1. The Euler–Bernoulli beam theory When the material length scale parameter l0 is set to zero in Eqs. (43) and (46), we obtain the following nonlinear dif and w)  associated with the Euler–Bernoulli beam theory accounting for ferential equations governing the displacements (u the modified von Kármán strains:

( "  2 #  2 ) d du 1 dw 1 dw Ac11 þ þ Ac13 þ fx ¼ 0 dx dx 2 dx 2 dx ! ( "  2 #  3 ) 2 2 d d w d dw du 1 dw 1 dw þ þ þ fz ¼ 0 Aðc c I þ c Þ þ Aðc þ c Þ 11 11 13 13 33 2 2 dx dx dx 2 dx 2 dx dx dx

ð53Þ ð54Þ

The conventional von Kármán beam equations are given by setting c11 ¼ E; c13 ¼ 0, and c33 ¼ 0 (or m ¼ 0) in Eqs. (53) and (54):

( "  2 #) d du 1 dw þ fx ¼ 0 EA þ dx dx 2 dx ! ( "  2 #) 2 2 d d w d dw du 1 dw EA EI þ þ þ fz ¼ 0 2 2 dx dx dx 2 dx dx dx

ð55Þ ð56Þ

The nonlocal linearized equations of equilibrium are obtained by setting c11 ¼ E; c13 ¼ 0; c33 ¼ 0, and the nonlinear terms to zero in Eqs. (43) and (46):

  2  d du d fx  f x þ l0 2 ¼ 0 EA dx dx dx ! 2 2 2  d d w d fz EI 2  fz þ l0 2 ¼ 0 2 dx dx dx 

ð57Þ ð58Þ

168

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

5.5.2. The Timoshenko beam theory When the material length scale parameter l0 is set to zero, we obtain the following nonlinear differential equations associated with the Timoshenko beam theory accounting for the modified von Kármán nonlinear strains:

( " )  2 # d du 1 dw 1 2 þ c13 /x  fx ¼ 0 c11 A þ dx dx 2 dx 2 ( " " !#)    2 # d dw d dw du 1 dw 1 2  fz   K s Ac44 A c11 /x þ þ þ c13 /x ¼0 dx dx dx dx dx 2 dx 2 " # !    2 2 dw d /x du 1 dw 1 þ c33 /2x ¼ 0  Ic11 þ A/ c þ K s Ac44 /x þ 13 x 2 dx dx 2 dx 2 dx

ð59Þ ð60Þ ð61Þ

The conventional von Kármán beam equations are given by setting c11 ¼ E; c13 ¼ 0, and c33 ¼ 0 to zero:

( "  2 #) d du 1 dw EA  þ  fx ¼ 0 dx dx 2 dx ( "     2 #) d dw d dw du 1 dw   fz ¼ 0  K s GA /x þ EA þ dx dx dx dx dx 2 dx   2 d / dw ¼0  EI 2x þ K s AG /x þ dx dx

ð62Þ ð63Þ ð64Þ

The nonlocal linearized equations of equilibrium are obtained by setting c11 ¼ E; c13 ¼ 0; c33 ¼ 0, and the nonlinear terms to zero in Eqs. (43), (51), and (52):

  2  d du d fx  fx þ l0 2 ¼ 0 EA dx dx dx    2  w d d d fz x þ  fz  l0 2 ¼ 0 K s GA /  dx dx dx     x  w d d/ d x þ þ K s GA / ¼0  EI dx dx dx



ð65Þ ð66Þ ð67Þ

6. Integral statements (or weak forms) 6.1. Preliminary comments In view of the fact that the nonlocal stress resultants are known in terms of the generalized (nonlocal) displacements, the statement of the principle of virtual displacements can be expressed in terms of the displacements for the nonlocal theories of beams. This facilitates the direct derivation of the equations of motion in terms of the generalized displacements of a beam theory using Hamilton’s principle. However, in general, it is not possible to construct the underlying quadratic functionals for nonlinear nonlocal beam theories. Such functionals may be constructed, even for the modified nonlinear theories of beams, under some restrictive conditions by an inverse procedure (see Oden & Reddy, 1982; Reddy, 1991), as discussed in this section. The integral statements developed here are used in Section 7 to develop the displacement finite element models. To write the integral statements, we replace the local stress resultants by the nonlocal stress resultants, which are then expressed in terms of the generalized displacements in the principle of virtual displacements of each beam theory. 6.2. The Euler–Bernoulli beam theory The statement of the principle of virtual displacements in Eq. (8) takes the form

Z

i h iL ð0Þ ð0Þ ð0Þ b ð0Þ  b  b ð1Þ  ð1Þ ð0Þ   M xx dexx þ M ð1Þ xx dexx þ M zz dezz  fx du  fz dw dx  M xx du  V x dw þ M xx dhx 0 0   Z L





 df ð0Þ  l0 x deð0Þ ð1Þ ð1Þ ð0Þ ð0Þ ð0Þ   fz dw  þ A c11 eð0Þ ¼ fx du xx þ c 13 ezz xx þ c 11 I exx  l0 fz dexx þ A c 13 exx þ c 33 ezz dezz dx 0 )   h iL  d dw dfx ð0Þ ð0Þ b ð0Þ du b x dw b ð1Þ dhx þV  þM dx  M deð1Þ l0 þ Aðc13 þ c33 Þezz  l0 Aðc11 þ c13 Þexx xx xx xx 0 dx dx dx

0¼

L

h

ð68Þ

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

169

where quantities with an hat denote specified stress resultants at the boundary points x ¼ 0; L. Also term involving l20 is neglected. It can be verified that the Euler equations associated with the integral statement in Eq. (68) are indeed the same as those in Eqs. (43) and (46). In addition, the following natural boundary conditions at x ¼ 0; L are obtained:

b ð0Þ  M ð0Þ ¼ 0; M xx xx

b x  V x ¼ 0; V

b ð1Þ  M ð1Þ ¼ 0 M xx xx

ð69Þ

where V x is as the same as V x defined in Eq. (14). Because of the underlined expression, it is clear from Eq. (68) that the integral statement does not allow us to construct a quadratic functional whose first variation is equal to the integral statement in Eq. (68). 6.3. The Timoshenko beam theory The statement of the principle of virtual displacements for the TBT is given by

Z

h i h iL ð1Þ ð1Þ ð0Þ  ð0Þ ð0Þ b ð0Þ  b  b ð1Þ  ð0Þ   M ð0Þ xx dexx þ M xx dexx þ M xz dcxz þ M zz dezz  fx du  fz dw dx  M xx du þ V x dw þ M xx d/x 0 0    Z L 





 df df x z ð0Þ ð1Þ ð1Þ ð0Þ  l0 ð0Þ ð0Þ xz  l0 xz þ A c13 eð0Þ dexx dc ¼ A c11 eð0Þ þ Ic11 exx  l0 fz dexx þ K s Ac44 c xx þ c 13 ezz xx þ c 33 ezz dezz dx dx 0  

  

d  x c13 eð0Þ þ c33 eð0Þ  dw A c11 eð0Þ þ c13 eð0Þ  l dfx deð1Þ   f z dw  þ l0 A/ fx du xx zz xx zz xx 0 dx dx dx )  

2 h iL  

d dw df b ð0Þ du b x dw b ð1Þ d/ x ð0Þ  l0 x dc xz dx  M þV  þM A c11 eð0Þ l0 2 ð70Þ xx þ c 13 ezz xx xx 0 dx dx dx

0¼

L

The natural boundary conditions for the TBT at x ¼ 0; L are

b ð0Þ  M ð0Þ ¼ 0; M xx xx

b x  V x ¼ 0; V

b ð1Þ  M ð1Þ ¼ 0 M xx xx

ð71Þ

where V x is as the same as V x defined in Eq. (27). Again, Eq. (70) does not allow us to construct a quadratic functional, because of the underlined expression, whose first variation is equal to the integral statement in Eq. (70). Note that all of the underlined terms in Eqs. (68) and (70) are highly nonlinear. Also, they contain higher-order derivatives of the generalized displacements, requiring higher-order approximations. For example, u must be approximated with quadratic polynomials and w must be approximated using fourth-order polynomials. In the interest of simplicity and the reasons stated, the underlined terms are not considered in the development of the finite element models. 7. Finite element models 7.1. The Euler–Bernoulli beam theory The statement in Eq. (68), without the underlined terms, is equivalent to the following two statements for a typical finite element Xe ¼ ðxa ; xb Þ:

" )  2 #  2  1 dw   1  ddu   du ddu dw dfx ddu  dx  Q 1 du  ðxa Þ  Q 4 du  ðxb Þ þ þ Ac13  l0  fx du dx 2 dx dx 2 dx dx dx dx xa ( "  2 #  2 ) Z xb ( 2  d2 dw   ddw   1 dw   d w dfx dw du 1 dw  þ c11 I 2 0¼ fz dw  l þ Aðc þ c Þ þ Aðc þ c Þ þ 11 13 13 33 0 2 dx 2 dx dx 2 dx dx dx xa dx dx ) 2    ddw   dw d dw  a Þ þ Q 5 dwðx  b Þ þ Q 3 dhx ðxa Þ þ Q 6 dhx ðxb Þ dx  Q 2 dwðx þ l0 fz 2 dx dx dx

0¼

Z

xb

(

Ac11

where hx  ðdw=dxÞ. c11 ¼ c11 þ c13 and c33 ¼ c13 þ c33 , the term involving

Q 1  M ð0Þ xx ðxa Þ; Q 2  V x ðxa Þ; h i ð1Þ Q 3   Mxx ; xa

ð0Þ Q 4  M xx ðxb Þ Q 5  V x ðxb Þ h i ð1Þ Q 6  Mxx

ð72Þ

ð73Þ

l20 is neglected, and Q i are defined by ð74Þ

xb

The axial and transverse displacements are approximated using linear and Hermite cubic interpolations, respectively (see Reddy, 2006):

 ðxÞ  u

2 X D1j wj ðxÞ; j¼1

 wðxÞ 

4 X J¼1

D2J uJ ðxÞ

ð75Þ

170

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

where wj ðxÞ are the linear polynomials, uJ ðxÞ are the Hermite cubic polynomials, (D11 ; D12 ) are the nodal values of u at xa and xb , respectively, and D2J (J ¼ 1; 2; 3; 4) are the nodal values associated with w:

 a Þ; D21 ¼ wðx

 b Þ; D23 ¼ wðx

D22 ¼ hx ðxa Þ;

D24 ¼ hx ðxb Þ

ð76Þ

 and w  and putting du  ¼ wi and dw  ¼ ui into the weak-form statements in Eqs. (72) and (73), By substituting Eq. (75) for u the finite element equations can be obtained as

"

K11

K12

K21

K22

#(

D1

)

D2

( ¼

F1

) or KD ¼ F

F2

ð77Þ

The stiffness coefficients K aijb and force coefficients F ai (a; b ¼ 1; 2) are defined as follows:

K 11 ij ¼ K 21 Ij ¼

Z

xb

Ac11 xa

Z

dwi dwj dx; dx dx

xb

K 12 iJ ¼

Z

xa

Aðc11 þ c13 Þ

 duI dwj dw dx dx dx dx

"

2

xa

xb

 dwi duJ Aðc11 þ c13 Þ dw dx dx dx dx 2

#  2  duI duJ Aðc11 þ 2c13 þ c33 Þ dw dx 2 2 2 dx dx dx xa dx dx  Z xb  dfx dwi dx þ Q 1 wi ðxa Þ þ Q 4 wi ðxb Þ F 1i ¼ fx wi þ l0 dx dx xa !#     Z xb " 2  duI d uI dfx dw du du F 2I ¼ fz uI þ l0 fz þ þ Q6  I dx þ Q 2 uI ðxa Þ þ Q 5 uI ðxb Þ þ Q 3  I 2 dx dx dx dx xa dx xb xa dx

K 22 IJ ¼

Z

xb

c11 I

2 d uI d uJ

þ

ð78Þ

7.2. The Timoshenko beam theory The statement in Eq. (70), without the underlined terms, is equivalent to the following three statements for a typical finite element Xe ¼ ðxa ; xb Þ:

Z

0¼

xb

(( Ac11

xa

Z

0¼

xb

" ) )  2 #  1 dw    du Ac13  2 ddu df ddu   l0 x  ðxa Þ  Q 4 du  ðxb Þ /x þ  fx du þ dx  Q 1 du 2 dx dx dx 2 dx dx

ð79Þ

(

( " ) )    2 #  ddw   1 dw   ddw    ddw  dw du Ac13  2 dw dfz ddw dfx dw   þ / K s Ac44 /x þ þ Ac11 þ  fz dw  l0  l0 2 x dx dx dx dx dx dx dx dx dx dx 2 dx

xa

 a Þ  Q 5 wðx  bÞ dx  Q 2 wðx ð80Þ ( ( " # ) )    2 Z xb  x dd/ x x    d/ Ac33  2   dd/ dfz   x þ dw d/  x þ Ac13 du þ 1 dw 0¼ / /x d/x  l0 fz Ic11 þ K s Ac44 /  l0 d/x þ dx dx 2 dx dx dx 2 x dx dx xa  x ðxa Þ  Q 6 /  x ðxb Þ dx  Q 3 /

ð81Þ

where Q i have the same meaning as defined in Eq. (74), with V x defined by Eq. (27).  x ) are approximated using Lagrange interpolation functions:  ; w;  / The generalized displacements (u

 ðxÞ  u

m X D1j wjð1Þ ðxÞ; j¼1

 wðxÞ 

n X

D2J wð2Þ j ðxÞ;

J¼1

 x ðxÞ  /

p X

D3J wð3Þ j ðxÞ

ð82Þ

J¼1 ð1Þ

ð2Þ

ð3Þ

 x , and putting du x ¼ w  ; w,  and /  ¼ wi ; dw  ¼ wi ; d/ By substituting Eq. (82) for u i (79)–(81), the finite element model of the TBT can be expressed as

2

K11

6 21 4K 31

K

K12 K22 32

K

into the weak-form statements in Eqs.

38 1 9 8 1 9 K13 > = > = 2 23 7 ¼ K 5 D F2 > : 3> ; > : 3> ; D K33 F

The stiffness coefficients K aijb and force coefficients F ai (a; b ¼ 1; 2; 3) are defined as follows:

ð83Þ

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

K 11 ij ¼

Z

c11 A

xa

Z

ð1Þ

xb

ð1Þ dwi dwj dx; dx dx

K 12 ij ¼

Z

xb

xa

ð2Þ

dwj  dwð1Þ Ac11 dw i dx 2 dx dx dx

Z xb ð1Þ ð1Þ dwj  dwð2Þ Ac13  dwi dw ð3Þ i /x ¼ wj dx; K 21 Ac11 dx ij ¼ dx dx 2 dx dx xa xa  2 # ð2Þ Z xb " ð2Þ  dwi dwj Ac11 dw ¼ K Ac þ dx K 22 s 44 ij 2 dx dx dx xa  Z xb   dwið2Þ ð3Þ Ac13  dw / ¼ K Ac þ w dx K 23 s 44 x ij dx 2 dx j xa Z xb ð1Þ dw  x wð3Þ j dx / ¼ Ac K 31 13 ij i dx xa  Z xb  ð2Þ dwj  Ac13  dw ð3Þ /x wi K s Ac44 þ K 32 dx ij ¼ dx 2 dx xa ! Z xb ð3Þ ð3Þ dwi dwj Ac33  2 ð3Þ ð3Þ ð3Þ ð3Þ 33 dx / w w K s Ac44 wi wj þ Ic11 þ K ij ¼ dx dx 2 x i j xa ! Z xb ð1Þ dfx dwi ð1Þ ð1Þ ð1Þ F 1i ¼ fx wi þ l0 dx þ Q 1 wi ðxa Þ þ Q 4 wi ðxb Þ dx dx xa #   Z xb "  dwið2Þ dfz dfx dw ð2Þ ð2Þ ð2Þ 2 dx þ Q 2 wi ðxa Þ þ Q 5 wi ðxb Þ fz wi þ l0 þ Fi ¼ dx dx dx dx xa ! Z xb ð3Þ dwi dfz ð3Þ ð3Þ ð3Þ 3 Fi ¼ dx þ Q 3 wi ðxa Þ þ Q 6 wi ðxb Þ l0 fz þ w dx dx i xa K 13 ij

171

xb

ð84Þ

7.3. Solution of nonlinear equations The nonlinear equations in Eqs. (77) and (83) are to be solved iteratively. Here we consider Newton’s iterative procedure (see Reddy, 2014). The linearized element equation at the beginning of the rth iteration is

Te ðDðr1Þ ÞdDðrÞ ¼ Re ðDðr1Þ Þ ¼ ðFe  Ke De Þ

ðr1Þ

ð85Þ

e

where the tangent stiffness matrix T associated with each beam element is calculated using the definition

Te 

@Re @De

For the finite element equations at hand, we have (see Reddy, 2014)

T ijab 

@Rai @ Dbj

¼ K ijab þ

nc X @K ac ik b k¼1 @ Dj

Dck 

@F ai @ Dbj

ð86Þ

The global incremental displacement vector DU at the rth iteration is obtained by solving the assembled equations (after the imposition of the boundary conditions) 1

DU ¼ ½TðUðr1Þ Þ Rðr1Þ

ð87Þ

and the total solution is computed from

UðrÞ ¼ Uðr1Þ þ DU

ð88Þ

7.3.1. The Euler–Bernoulli beam theory The tangent stiffness coefficients for this case are given by (n1 ¼ 2 and n2 ¼ 4, and a; b, and c take the values of 1 and 2) 11 T 11 ij ¼ K ij ;

T 22 IJ

12 21 T 12 T 21 iJ ¼ 2K iJ ; Ij ¼ K Ij  2 # Z xb " Z xb   du dw duI duJ df du du 22 ¼ K IJ þ Aðc11 þ c13 Þ þ Aðc11 þ 2c13 þ c33 Þ l0 x I J dx dx  dx dx dx dx dx dx dx xa xa

The tangent stiffness matrix is symmetric (i.e., T aijb ¼ T bjia ).

ð89Þ

172

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

7.3.2. The Timoshenko beam theory The tangent stiffness coefficients for the TBT are given by (n1 ¼ m, n2 ¼ n, and n3 ¼ p, and a; b, and c take the values of 1, 2, and 3) 11 T 11 ij ¼ K ij ;

T 22 ij T 23 ij T 33 ij

12 13 21 T 12 T 13 T 21 ij ¼ 2K ij ; ij ¼ 2K ij ; ij ¼ K ij " # ð2Þ  2 Z xb Z xb ð2Þ ð2Þ ð2Þ dwj   du dw 1 df dw dwj 22 2 dwi  ¼ K ij þ Ac11 þ Ac11 þ Ac13 /x l0 x i dx  dx 2 dx dx dx dx dx dx dx xa xa  ð2Þ Z xb    x dw dwi wð3Þ dx; T 31 ¼ K 31 ; T 32 ¼ T 23 ¼ K s Ac44 þ Ac13 / ij ij ij ji dx dx j xa )  2 # Z xb ( " du 1 dw  2 wð3Þ wð3Þ dx þ c33 / ¼ K 33 A c13 þ x ij þ i j dx 2 dx xa

ð90Þ

The tangent stiffness matrix of the TBT element is also symmetric. 8. Numerical results Here we consider several numerical examples. In all cases, the beam is assumed to be of length L ¼ 100, width b ¼ 1, and height h ¼ 1 (see Fig. 1). These values are used only for the purpose of numerically evaluating the parametric effects of boundary conditions, loads, and nonlocal parameter l0 . A mesh of 8 elements (with linear approximation of u and Hermite cubic approximation of w) for the EBT and 8 linear elements (with equal interpolation of all variables, u; w, and /x ) for the TBT are used in a half beam when symmetry exists and in full beam, otherwise. Two-point quadrature rule is used for all terms except the shear and nonlinear terms of the stiffness coefficients and one-point integration rule is used for shear and nonlinear terms to avoid shear and membrane locking (see Reddy, 2014). In the examples presented herein, only transverse load qðxÞ is used and f ðxÞ ¼ 0. 8.1. The effect of the constitutive relations First, we wish to investigate the effect of including ezz and hence the two-dimensional constitutive relations in Eq. (7) in place of rxx ¼ Eexx . Use of the two-dimensional stress–strain relations in effect amounts to replacing E with 2l þ k  gE, where

g¼

1m ð1 þ mÞð1  2mÞ

ð91Þ

For m ¼ 0:3, the value of g is g ¼ 1:346. That is, the beam stiffness will increase by 34.6%, which is unreasonably very high. This is possibly the reason why von Kármán did not account for ezz , although it is of the same magnitude as other nonlinear strains. Fig. 2 shows plots of the load q versus the center deflection wðL=2Þ for conventional von Kármán beam and modified (include ezz and hence use Eq. (7)) von Kármán beam, with two ends pinned and subjected to uniformly distributed load of intensity q. It is clear that the use of two-dimensional constitutive relations make the beam unduly stiff. Hence, in the remaining part of this section, we revert to the conventional von Kármán beams, that is, not include ezz and set c11 ¼ c33 ¼ E; c44 ¼ G, and c13 ¼ 0. 8.2. The effect of boundary conditions We begin with a pinned–pinned beam, which is not the same as a hinged–hinged beam for the nonlinear analysis purposes, because the former requires the specification of both u and w to be zero at the pinned point where as hinged point requires only w to be zero. Using the symmetry about x ¼ L=2, only half beam is modeled using 8 elements. The boundary

Fig. 1. Geometry of a rectangular cross section beam.

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

173

Fig. 2. The effect of using uniaxial versus plane elastic constitutive relations on the center deflection of a pinned–pinned beam under uniformly distributed load.

conditions used are: uð0Þ ¼ wð0Þ ¼ 0 and uðL=2Þ ¼ hx ðL=2Þ ¼ 0 (replace hx with /x for TBT). Fig. 3 shows the plots of load versus center deflection for the case of uniformly distributed load (UDL). The effect of the nonlocal parameter l0 for this boundary condition is to increase the deflection (i.e., increasing l0 results in increased deflection w). The results predicted by both beam theories are essentially the same for the scale used to plot the results. This is expected because the length-to-height ratio is 100 and, hence, the transverse shear strain is negligible. Next we consider with a clamped–clamped beam. Again, the symmetry about x ¼ L=2 is exploited to model only half of the beam using 8 elements. The boundary conditions used are: uð0Þ ¼ wð0Þ ¼ hx ð0Þ ¼ 0 and uðL=2Þ ¼ hx ðL=2Þ ¼ 0 (replace hx with /x for TBT). For the case of uniformly distributed load of intensity q, the nonlocal parameter l0 has no effect on the load–deflection response. Further investigation showed that the load vector due to l0 [see Eq. (78)] only contributes to the rotational (i.e., moments) components because

l0

Z

2

xb

q xa

d uI 2

dx

dx ¼ 0;

for I ¼ 1; 3 and q ¼ constant

ð92Þ

Fig. 3. Load versus center deflection for pinned–pinned, local and nonlocal, Euler–Bernoulli and Timoshenko beams under uniformly distributed transverse load.

174

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

and, since the rotational degrees of freedom are specified to be zero, the nonzero components (for I ¼ 2; 4) do not contribute to the solution. When the load is changed to

qðxÞ ¼ q0 sin

px L

ð93Þ

all components of the load vector are nonzero (in particular, they are not zero for I ¼ 1; 3), and the effect of l0 is felt, as can be seen from the results presented in Fig. 4. The effect of the nonlocal parameter l0 for this boundary condition is also to increase the deflection. During this study it is also found, as others have, that for beams involving a boundary point where both deflection w and rotation hx (or /x ) are not specified (e.g., a cantilevered beam), the effect of the nonlocal parameter l0 is to decrease the deflection (for any load type), as shown in Fig. 5. Also, a cantilever beam and propped cantilever beam do not exhibit nonlinearity (for any load) because u – 0 at the other end where it is not clamped, when the applied transverse load is assumed to remain vertical during the deformation (see Reddy, 2014).

Fig. 4. Load versus center deflection for clamped–clamped, local and nonlocal, Euler–Bernoulli and Timoshenko beams under uniformly distributed load (UDL) and sinusoidally distributed load (SSL).

Fig. 5. Load versus distance along the beam for clamped-free (i.e., cantilevered), local and nonlocal beams under uniformly distributed load (UDL).

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

175

Fig. 6. Load versus deflection for clamped–pinned, local and nonlocal beams under uniformly distributed load (UDL).

As a final example, we consider a clamped–pinned beam under uniformly distributed load. The results are presented in Fig. 6. The deflection at x ¼ 62:5 is used to show the effect of nonlinearity as well as the nonlocal parameter l0 on the deflection. For this case also, the nonlocal parameter has the effect of increasing the deflection. 9. Summary and conclusions Equations of equilibrium of the Euler–Bernoulli and Timoshenko beam theories are derived based on Eringen’s nonlocal differential constitutive model and modified von Kármán nonlinear strains. The principle of virtual displacements is used to derive the equations of equilibrium and associated boundary conditions in terms of the stress resultants, which are assumed to hold for the nonlocal beams. Then the nonlocal stress resultants are expressed in terms of the nonlocal displacements with the help of Eringen’s nonlocal differential constitutive relation, which are then substituted into the equations of equilibrium. Integral statements (weak forms) of the equations are also derived, nonlinear finite element models of the nonlocal beam theories are developed, and solution strategies are discussed using the Newton iterative method. Numerical results for bending response are presented to illustrate the parametric effect of boundary conditions and the influence of the nonlocal parameter. It is shown that for all beams, except for those beams for which both the transverse displacement and slope are not specified at a boundary point, the nonlocal parameter has the effect of increasing the deflections (i.e., make the beam more flexible). For clamped–clamped beams under UDL, the nonlocal parameter has no effect on the deflections. For the case of beams for which both the transverse displacement and slope are not specified at a boundary point (e.g., cantilevered beam), the effect of the nonlocal parameter is to make the beam behave as a stiff beam (i.e., the displacement predicted by nonlocal beams are lower than those predicted by conventional beam). In general, the effect of the nonlocal parameter on the deflections is found to depend on the boundary conditions and the nature of the load. Acknowledgements The first author gratefully acknowledges the support of this work by the Oscar S. Wyatt Endowed Chair. The second author is grateful to Texas A&M University at Qatar for providing the research funding and to the Tunisian Ministry of Higher Education and Scientific Research for the funding provided his laboratory. References Abdi, J., Koochi, A., Kazemi, A. S., & Abadyan, M. (2011). Modelling the effects of size dependence and dispersion forces on the pull-in instability of electrostatic cantilever NEMS using modified couple stress theory. Smart Materials and Structures, 20, 055011. Adali, S. (2009). Variational principles for transversely vibrating multi-walled carbon nanotubes based on nonlocal Euler–Bernoulli beam models. Nano Letters, 9, 1737–1741. Akgöz, B., & Civalek, Ö. (2011). Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. International Journal of Engineering Science, 49, 1268–1280. Ansari, R., Gholami, R., & Sahmani, S. (2011). Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory. Composite Structures, 94, 221–228. Arash, B., & Wang, Q. (2012). A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Computational Materials Science, 51, 303–313.

176

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

Arbind, A., & Reddy, J. N. (2013). Nonlinear analysis of functionally graded microstructure-dependent beams. Composite Structures, 98, 272–281. Arbind, A., Reddy, J. N., & Srinivasa, A. (2014). Modified couple stress-based third-order theory for nonlinear analysis of functionally graded beams. Latin American Journal of Solids and Structures, 11, 459–487. Cadek, M., Coleman, J. N., Ryan, K. P., Nicolosi, V., Bister, G., Fonseca, A., et al (2004). Reinforcement of polymers with carbon nanotubes: The role of nanotube surface area. Nano Letters, 4, 353–356. Catalan, G., Lubk, A., Vlooswijk, A. H. G., Snoeck, E., Magen, C., Janssens, A., et al (2011). Flexoelectric rotation of polarization in ferroelectric thin films. Nature Materials, 10, 963–967. Challamel, N., Zhang, Z., Wang, C. M., Reddy, J. N., Wang, Q., Michelitsch, T., & Collet, B. (in press). On non-conservativeness of Eringens nonlocal elasticity in beam mechanics: Correction from a discrete-based approach. Archives of Applied Mechanics. Challamel, N., & Wang, C. M. (2008). The small length scale effect for a non-local cantilever beam: A paradox solved. Nanotechnology, 19, 345703-1–3457036. Chen, W. X., Tu, J. P., Wang, L. Y., Gan, H. Y., Xu, Z. D., & Zhang, X. B. (2003). Tribological application of carbon nanotubes in a metal-based composite coating and composites. Carbon, 41, 215–222. Eltaher, M. A., Alshorbagy, A. E., & Mahmoud, F. F. (2013). Vibration analysis of Euler–Bernoulli nanobeams by using finite element method. Applied Mathematical Modeling, 37, 4787–4797. Eltaher, M. A., Emam, S. A., & Mahmoud, F. F. (2012). Free vibration analysis of functionally graded size-dependent nanobeams. Applied Mathematics and Computation, 218, 7406–7420. Eltaher, M. A., Emam, S. A., & Mahmoud, F. F. (2013). Static and stability analysis of nonlocal functionally graded nanobeams. Composite Structures, 96, 82–88. Eltaher, M. A., Hamed, M. A., Sadoun, A. M., & Mansour, A. (2014). Mechanical analysis of higher order gradient nanobeams. Applied Mathematics and Computation, 229, 260–272. Eltaher, M. A., Khairy, A., Sadoun, A. M., & Omar, Fatema-Alzahraa (2014). Static and buckling analysis of functionally graded Timoshenko nanobeams. Applied Mathematics and Computation, 229, 283–295. Erdogan, F. (1995). Fracture mechanics of functionally graded materials. Composites Engineering, 5, 753–770. Eringen, A. C. (1972). Nonlocal polar elastic continua. International Journal of Engineering Science, 10, 1–16. Eringen, A. C. (1983). On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54, 4703–4710. Eringen, A. C. (2002). Nonlocal continuum field theories. New York: Springer-Verlag. Eringen, A. C., & Edelen, D. G. B. (1972). On nonlocal elasticity. International Journal of Engineering Science, 10, 233–248. He, J. H. (1999). Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178, 257–262. Hung, E. S., & Senturia, S. D. (1999). Extending the travel range of analog-tuned electrostatic actuators. Journal of Microelectromechanics Systems, 8, 497–505. Kaempgen, M., Duesberg, G. S., & Roth, S. (2005). Transparent carbon nanotube coatings. Applied Surface Science, 252, 425–429. Kahrobaiyan, M. H., Asghari, M., Rahaeifard, M., & Ahmadian, M. T. (2010). Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory. International Journal of Engineering Science, 48, 1985–1994. Kim, J., & Paulino, G. H. (2002). Finite element evaluation of mixed mode stress intensity factors in functionally graded materials. International Journal for Numerical Methods in Engineering, 53, 1903–1935. Kim, J., & Reddy, J. N. (2013). Analytical solutions for bending, vibration, and buckling of FGM plates using a couple stress-based third-order theory. Composite Structures, 103, 8698. Koiter, W. T. (1964). Couple-stresses in the theory of elasticity: I and II. Koninklijke Nederlandse Akademie van Wetenschappen (Royal Netherlands Academy of Arts and Sciences), B67, 17–44. Koizumi, M. (1993). The concept of FGM. Ceramic Transactions, 34, 3–10. Lam, D. C. C., Yang, F., Chong, A. C. M., Wang, J., & Tong, P. (2003). Experiments and theory in strain gradient elasticity. Journal of Mechanics and Physics of Solids, 51, 1477–1508. Li, X., Bhushan, B., Takashima, K., Baek, C. W., & Kim, Y. K. (2003). Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques. Ultramicroscopy, 97, 481–494. Li, C., Lim, C. W., & Yu, J. L. (2011). Dynamics and stability of transverse vibrations of nonlocal nanobeams with a variable axial load. Smart Materials and Structures, 20, 015023. Lim, C. W. (2010). On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: Equilibrium, governing equation and static deflection. Applied Mathematics and Mechanics, 31(1), 37–54. Lü, C. F., Lim, C. W., & Chen, W. Q. (2009). Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory. International Journal of Solids and Structures, 46, 1176–1185. McFarland, A. W., & Colton, J. S. (2005). Role of material microstructure in plate stiffness with relevance to microcantilever sensors. Journal of Micromechanics and Microengineering, 15, 1060–1067. Mindlin, R. D. (1963). Influence of couple-stresses on stress concentrations. Experimental Mechanics, 3, 1–7. Mindlin, R. D. (1965). Second gradient of strain and surface-tension in linear elasticity. International Journal of Solids and Structures, 1, 217–238. Moser, Y., & Gijs, M. A. M. (2007). Miniaturized flexible temperature sensor. Journal of Microelectromechanical Systems, 16, 1349–1354. Mousavi, T., Bornassi, S., & Haddadpour, H. (2013). The effect of small scale on the pull-in instability of nano-switches using DQM. International Journal of Solids and Structures, 50, 1193–1202. Najar, F., Nayfeh, A. H., Abdel-Rahman, E. M., Choura, S., & El-Borgi, S. (2010). Global stability of microbeam-based electrostatic microactuators. Journal of Vibration and Control, 16, 721–748. Nazemnezhad, R., & Hosseini-Hashemi, S. (2014). Nonlocal nonlinear free vibration of functionally graded nanobeams. Composite Structures, 110, 192–199. Oden, J. T., & Reddy, J. N. (1982). Variational methods in theoretical mechanics (2nd ed.). NY: Springer-Verlag. Park, S. K., & Gao, X. L. (2006). Bernoulli-Euler beam model based on a modified couple stress theory. Journal of Micromechanics and Microengineering, 16, 2355–2359. Park, S. K., & Gao, X. L. (2008). Variational formulation of a modified couple stress theory and its application to a simple shear problem. Zeitschrift fr angewandte Mathematik und Physik (ZAMP), 59, 904–917. Peddieson, J., Buchanan, G. R., & McNitt, R. P. (2003). Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science, 41, 305–312. Pei, J., Tian, F., & Thundat, T. (2004). Glucose biosensor based on the microcantilever. Analytical Chemistry, 76, 292–297. Praveen, G. N., Chin, C. D., & Reddy, J. N. (1999). Thermoelastic analysis of functionally graded ceramic-metal cylinder. Journal of Engineering Mechanics, ASCE, 125(11), 1259–1266. Praveen, G. N., & Reddy, J. N. (1998). Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. Journal of Solids and Structures, 35(33), 4457–4476. Rahaeifard, M., Kharobaiyan, M. H., Asghari, M., & Ahmadian, M. T. (2011). Static pull-in analysis of microcantilevers based on the modified couple stress theory. Sensors and Actuators A, 171, 370–374. Rahmani, O., & Pedram, O. (2014). Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. International Journal of Engineering Science, 77, 55–70. Reddy, J. N. (1987). A small strain and moderate rotation theory of elastic anisotropic plates. Journal of Applied Mechanics, 54, 623–626. Reddy, J. N. (1991). Applied functional analysis and variational methods in engineering. NY: McGraw-Hill [reprinted by Krieger, Melbourne, FL, 1991]. Reddy, J. N. (2000). Analysis of functionally graded plates. International Journal for Numerical Methods in Engineering, 47, 663–684.

J.N. Reddy, S. El-Borgi / International Journal of Engineering Science 82 (2014) 159–177

177

Reddy, J. N. (2002). Energy principles and variational methods in applied mechanics (2nd ed.). New York: John Wiley & Sons. Reddy, J. N. (2004). Mechanics of laminated composite plates and shells. Theory and analysis (2nd ed., ). Boca Raton, FL: CRC Press. Reddy, J. N. (2006). An introduction to the finite element method. New York: McGraw–Hill. Reddy, J. N. (2007). Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45, 288–307. Reddy, J. N. (2010). Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. International Journal of Engineering Science, 48, 1507–1518. Reddy, J. N. (2011). A general nonlinear third-order theory of functionally graded plates. International Journal of Aerospace and Lightweight Structures, 1(1), 1–21. Reddy, J. N. (2011). Microstructure-dependent couple stress theories of functionally graded beams. Journal of Mechanics and Physics of Solids, 59, 2382–2399. Reddy, J. N. (2013). Introduction to continuum mechanics with applications. New York: Cambridge University Press. Reddy, J. N. (2014). An Introduction to nonlinear finite element analysis. Oxford, UK: Oxford University Press [2nd edition to appear with a print date of 2014]. Reddy, J. N., & Srinivasa, A. R. (in press). Nonlinear theories of beams and plates accounting for moderate rotations and material length scales. International Journal of Non-Linear Mechanics. Reddy, J. N., & Berry, Jessica (2012). Modified couple stress theory of axisymmetric bending of functionally graded circular plates. Composites Structures, 94, 3664–3668. Reddy, J. N., & Chin, C. D. (1998). Thermomechanical analysis of functionally graded cylinders and plates. Journal of Thermal Stresses, 26(1), 593–626. Reddy, J. N., & Kim, Jinseok (2012). A nonlinear modified couple stress-based third-order theory of functionally graded plates. Composite Structures, 94, 1128–1143. Reddy, J. N., & Mahaffey, Patrick (2013). Generalized beam theories accounting for von Kármán nonlinear strains with application to buckling and postbuckling. Journal of Coupled Systems and Multiscale Dynamics, 1, 120–134. Reddy, J. N., & Pang, S. D. (2008). Nonlocal continuum theories of beams for the analysis of carbon nanotubes. Applied Physics Letters, 103, 023511-1–02351116. Roque, C. M. C., Ferreira, A. J. M., & Reddy, J. N. (2011). Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method. International Journal of Engineering Science, 49, 976–984. Schmidt, R., & Reddy, J. N. (1988). A refined small strain and moderate rotation theory of elastic anisotropic shells. Journal of Applied Mechanics, 55, 611–617. Shariat, B. S., Liu, Y., & Rio, G. (2013). Modelling and experimental investigation of geometrically graded NiTi shape memory alloys. Smart Materials and Structures, 22, 025030. Simsek, M. (2014). Large amplitude free vibration of nanobeams with various boundary conditions based on nonlocal elasticity theory. Composites: Part B, 56, 621–628. Simsek, M., & Reddy, J. N. (2013). Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory. International Journal of Engineering Science, 64, 37–53. Simsek, M., & Reddy, J. N. (2013). A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory. Composite Structures, 101, 47–58. Srinivasa, A. R., & Reddy, J. N. (2013). A model for a constrained, finitely deforming, elastic solid with rotation gradient dependent strain energy, and its specialization to von Karman plates and beams. Journal of Mechanics and Physics of Solids, 61, 873–885. Sudak, L. J. (2003). Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. Journal of Applied Physics, 94, 7281–7287. Thai, H-T. (2012). A nonlocal beam theory for bending, buckling, and vibration of nanobeams. International Journal of Engineering Science, 52, 56–64. Toupin, R. A. (1964). Theories of elasticity with couple-stress. Archive for Rational Mechanics and Analysis, 17, 85–112. Uymaz, B. (2013). Forced vibration analysis of functionally graded beams using nonlocal Elasticity. Composite Structures, 105, 227–239. Wang, Q. (2005). Wave propagation in carbon nanotubes via nonlocal continuum mechanics. Journal of Applied Physics, 98, 124301-1–124301-6. Wang, Q., & Varadan, V. K. (2006). Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Materials and Structures, 15, 659–666. Wang, C. M., Zhang, Y. Y., Ramesh, S. S., & Kitipornchai, S. (2006). Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory. Journal of Physics D, 39, 3904–3909. Xia, W., Wang, L., & Yin, L. (2010). Nonlinear non-classical microscale beams: Static bending, postbuckling and free vibration. International Journal of Engineering Science, 48, 2044–2053. Yang, F., Chong, A. C. M., Lam, D. C. C., & Tong, P. (2002). Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39, 2731–2743. Zhang, J., & Fu, Y. (2012). Pull-in analysis of electrically actuated viscoelastic microbeams based on a modified couple stress theory. Meccanica, 47, 1649–1658. Zhang, Y. Q., Liu, G. R., & Wang, J. S. (2004). Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression. Physical Review B, 70, 205430. Zhang, Y. Q., Liu, G. R., & Xie, X. Y. (2005). Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity. Physical Review B, 71, 195404-1–195404-7. Zhang, Ch., Savaidis, A., Savaidis, G., & Zhu, H. (2003). Transient dynamic analysis of a cracked functionally graded material by a BIEM. Computational Materials Science, 26, 167–174.