Non-linear theories of beams and plates accounting for moderate rotations and material length scales

International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Contents lists available at ScienceDirect

International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm

Non-linear theories of beams and plates accounting for moderate rotations and material length scales J.N. Reddy n, A.R. Srinivasa Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 18 February 2014 Received in revised form 5 June 2014 Accepted 7 June 2014

The primary objective of this paper is to formulate the governing equations of shear deformable beams and plates that account for moderate rotations and microstructural material length scales. This is done using two different approaches: (1) a modified von Kármán non-linear theory with modified couple stress model and (2) a gradient elasticity theory of fully constrained finitely deforming hyperelastic cosserat continuum where the directors are constrained to rotate with the body rotation. Such theories would be useful in determining the response of elastic continua, for example, consisting of embedded stiff short fibers or inclusions and that accounts for certain longer range interactions. Unlike a conventional approach based on postulating additional balance laws or ad hoc addition of terms to the strain energy functional, the approaches presented here extend existing ideas to thermodynamically consistent models. Two major ideas introduced are: (1) inclusion of the same order terms in the strain– displacement relations as those in the conventional von Kármán non-linear strains and (2) the use of the polar decomposition theorem as a constraint and a representation for finite rotations in terms of displacement gradients for large deformation beam and plate theories. Classical couple stress theory is recovered for small strains from the ideas expressed in (1) and (2). As a part of this development, an overview of Eringen's non-local, Mindlin's modified couple stress theory, and the gradient elasticity theory of Srinivasa–Reddy is presented. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Beams and plates Discrete peridynamics Gradient elasticity Material length scales Modified couple stress theory

1. An overview of theories with material length scales 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 (or length scale) is comparable to the size of the secondary constituents. Examples of such materials are provided by nematic elastomers and carbon nanotube composites [1] and environment resistent coatings made of CNT reinforced materials [2,3]. In addition, the flexoelectric effect [4], 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. Most structural systems involve the use of rods, beams, plates, and shells. They are also commonly used in micro- and nano-scale

n

Corresponding author. Tel.: þ 1 979 862 2417; fax: þ 1 979 862 3989. E-mail address: [email protected] (J.N. Reddy).

devices, that is, MEMS and NEMS. 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 [5,6]. All beam and plate theories based on the classical elasticity theory do not account for the microstructure-dependent size effects. Therefore, the conventional beam and plate theories are not capable of predicting the size effects, that is, their response may be influenced by the microstructural parameters. Thus, it is useful to develop modified theories of beams and plates that account for size effects and geometric non-linearity. The present study is focused on formulating beam and plate theories with aforementioned effects. The following sections provide a background for the present study. 1.2. Eringen's non-local elasticity 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 nonlocal 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.

http://dx.doi.org/10.1016/j.ijnonlinmec.2014.06.003 0020-7462/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: J.N. Reddy, A.R. Srinivasa, Non-linear theories of beams and plates accounting for moderate rotations and material length scales, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.06.003i

J.N. Reddy, A.R. Srinivasa / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

Non-local parameter

Such non-local elasticity was initiated in the works of Eringen [7– 9] and Eringen and Edelen [10]. According to Eringen [7,8], the state of stress σ at a point x in an elastic continuum not only depends on the strain field ε 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 non-local stress tensor σ at point x is expressed as Z σ ¼ Kðjx0  xj; τÞtðx0 Þ dx0 ð1Þ Ω

where t(x) is the classical, macroscopic stress tensor at point x and the kernel function Kðjx0  xj; τÞ represents the non-local modulus, jx0  xj being the distance (in the Euclidean norm) and τ is a material parameter that depends on internal and external characteristic lengths (such as the lattice spacing and wavelength, respectively). The macroscopic stress t at a point x in a Hookean solid is related to the strain ε at the point by the generalized Hooke's law ð2Þ

where C is the fourth-order elasticity tensor and : denotes the ‘double-dot product’ (see Reddy [11]). The constitutive equations (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 [8] proposed an equivalent differential model as ð1  τ2 ℓ2 ∇2 Þσ ¼ t;

τ¼

e0 a ℓ

ð3Þ

where e0 is a material constant, and a and ℓ are the internal and external characteristic lengths, respectively. The non-local 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, etc. The use of non-local elasticity to study size-effects in micro and nanoscale structures was first carried out by Peddieson et al. [12]. 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 non-local elasticity to show the small-scale effects on buckling of MWCNTs under axial compression [13] and radial pressure [14]. Wang [15] and Wang and Varadan [16] have studied wave propagation in carbon nanotubes (CNTs) with non-local 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 non-local 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 et al. [17] formulated a non-local Timoshenko beam theory, neglecting the non-local effect in writing the shear stress– strain relation. Reddy [18,19] and Reddy and Pang [20] have formulated nonlocal versions of various beam and plate theories, and presented numerical solutions of bending, vibration, and buckling of beams. The results show that the non-local parameter μ ¼ τ2 ℓ2 ¼ e20 a2 has the effect of softening the beam, and thus predict larger deflections and lower buckling loads and vibration frequencies (see Figs. 1 and 2; taken from [18]). There are numerous other papers that study the response of nanosystems using theories that are based on Eringen's

Transverse deflection, Fig. 1. Non-dimensional center deflection w vs. the non-local parameter μ for a simply supported beam under uniformly distributed load of intensity q0 (L denotes the length and h is the height of the beam); taken from [18].

Non-local parameter

tðxÞ ¼ CðxÞ : εðxÞ

Buckling load/Fundamental frequency Fig. 2. Non-dimensional critical buckling load N cr and non-dimensional natural frequency ω vs. the non-local parameter μ for a simply supported beam (L denotes the length and h is the height of the beam); taken from [18].

differential model (see [21–24], and references therein). Reddy [18,19] pointed out that Eringen's differential model does not conform to the normal structural mechanics formulations in that the resulting equations are not derivable from a strain energy potential for the cases in which the von Kármán non-linearity along with kinetic energy are accounted for (see Reddy and ElBorgi [25] for details). Also, the force boundary conditions associated with the non-local beams, when expressed in terms of the displacement variables, contain the non-local parameter. Thus, there is a need to reexamine Eringen's differential model closely for its suitability in accounting for non-local elasticity. 1.3. Modified couple stress theories Theories of micro-structured media have been developed dating back to the 1960s. There is a large body of literature on small deformation couple stress theories with constrained microrotation, beginning with the early works of Mindlin, Toupin, Green, Naghdi, and Rivlin [26–32]. The classical couple stress elasticity theory of Koiter [27] contains four material length scale

Please cite this article as: J.N. Reddy, A.R. Srinivasa, Non-linear theories of beams and plates accounting for moderate rotations and material length scales, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.06.003i

J.N. Reddy, A.R. Srinivasa / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Ω

1 Homogeneous Beam, Pinned-pinned connected beam E = 14.4 GPa, ν = 0.38 h=17.6 μ m l/h = 0 L= 20h, b=2h 0.8

Transversedeflection(inμm)

parameters. The higher-order Euler–Bernoulli beam model developed by Papargyri-Beskou et al. [33] is based on the gradient elasticity theory with surface energy and it involves four elastic constants. In view of the difficulties in determining microstructure dependent length scale parameters (see Yang and Lakes [34]; Lam et al. [5]; and Maranganti and Sharma [35]), models that involve only one material length scale parameter were sought. One such model has been developed for the Bernoulli–Euler beam theory by Park and Gao [36,37] using a modified couple stress theory proposed by Yang et al. [38] and Mindlin's model [26] that contains only one material length scale parameter. The modified couple stress model used by Park and Gao [36,37] was based on the ad hoc idea of modifying the strain energy potential U to include an additional term due to couple stress: Z Z h=2 δU ¼ ðσ : δε þ m : δχ Þ dz dx dy ð4Þ

3

l/h=0.2

0.6

0.4

l/h=0.6

0.2

l/h=1

 h=2

where σ is the symmetric part of the stress tensor, m is the deviatoric part of the couple stress tensor, ε is the strain tensor, and χ is the curvature tensor 1 2

ε ¼ ½ð∇uÞT þ ð∇uÞ;

1 2

χ ¼ ½ð∇ωÞT þ ð∇ωÞ

ð5Þ

and ω is the rotation vector 1 2

ω¼ ∇u

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nondim. distance along length ( x / L ) Fig. 3. Center transverse deflection w vs. normalized distance x=L for various ratios of the material length scale parameter ℓ to beam thickness (or height) h for a simply supported beam under uniformly distributed load of intensity q0 (L denotes the length of the beam; a generalized third-order beam theory is used [56]).

ð6Þ

and u is the displacement vector. When dynamics are studied, the kinetic energy K remained unchanged, while the virtual work done by external forces V included the work done by body couple vector c (in addition to the work done by body forces and surface tractions). The constitutive equations relating σ to ε and m to χ are assumed, for isotropic materials, to be of the form

σ ¼ 2μ ε þ λI trðεÞ;

Linear solution - dotted line Nonlinear soution - lines with symbols

0

m ¼ 2μ ℓ2 χ

ð7Þ

The starting point of such a theory is the notion of a strain energy function that depends not only upon the Cauchy–Green Stretch tensor with cartesian components C AB but also the components of the rotation gradient RiA;B , where the comma denotes partial derivative with respect to the reference position. Invariance requirements imply that the strain energy should be of the form

Ψ ¼ Ψ^ ðC AB ; RiA RiB;C Þ

where μ and λ are the Lamé constants (λ ¼ Eν=ð1 þ νÞð1  2νÞ and 2μ ¼ E=ð1 þ νÞ, with E being Young's modulus and ν Poisson's ratio), ℓ is a material length scale parameter measuring the effect of couple stress, and trðεÞ denotes the trace of the strain tensor (εkk). Mindlin [26] interpreted ℓ to be the square root of the ratio of the moduli of curvature to the shear. In recent years numerous papers that use Eqs. (4)–(7) in formulating the classical and shear deformable beam and plate theories have appeared (see Park and Gao [36,37], Ma et al. [39– 41], Reddy [42], Reddy and Berry [43], Santos and Reddy [44,45], Reddy and Kim [46], Ke and Wang [47,48], Gao et al. [49], Roque et al. [50,51], Arbind and Reddy [52], Simsek and Reddy [53,54], Kim and Reddy [55], and Arbind et al. [56], among several others). The beam and plate theories using the modified couple stress theory [i.e., using Eqs. (4)–(7)] predict bending deflections that decrease in magnitude as the ratio of ℓ to the beam or plate thickness h is increased (see Fig. 3 from [56]). Thus, the modified couple stress theories represent beam and plate models as stiffer compared to their conventional counterparts.

The above equation reveals that the constrained cosserat theory presented by Srinivasa and Reddy [57] is a special kind of a strain gradient theory with a complicated dependence on the strain gradients. This is a generalization of small deformation couple stress theories, such as the one considered in Section 1.3. To illustrate the gradient elasticity model of Srinivasa and Reddy [57], we consider moderate rotation beams according to the Euler–Bernoulli kinematics with rotation gradient dependence. The displacement field is of the form

1.4. The specialized gradient elasticity model of Srinivasa and Reddy

u ¼ ðuðxÞ þ zθx Þe^ x þ we^ z ;

As stated in Section 1.3, all previous approaches, especially the modified couple stress approach, are based on postulating the equations directly for small deformation elasticity. In particular, there was no corresponding finite deformation couple stress theory with constrained rotations, until the recent work of Srinivasa and Reddy [57] developed such a finite deformation gradient elasticity theory for a fully constrained finitely deforming hyperelastic cosserat continuum where the directors are constrained to rotate with the body rotation.

where x-axis is taken along the length of the beam and z-axis transverse to it, with the xz-plane as the plane of bending (about the y-axis), and u is the displacement vector, u is the axial displacement, w the transverse displacement, and e^ x and e^ z are unit vectors along the x- and z-axes, respectively. We assume that pffiffiffi 2 2 ðdu=dxÞ is of order ϵ, ðdw=dxÞ is of order ϵ, and L0 ðd w=dx Þ is of order ϵ, where L0 is the intrinsic length scale associated with the higher gradient theory and ϵ⪡1 is a small parameter (see Reddy [58] and Schmidt and Reddy [59]).

Now the compatibility conditions in terms of the deformation gradient for finite deformation, which take the form F iA;B ¼ 0 together with the polar decomposition theorem, whichpstates that ffiffiffi F iB ¼ RiA U AB where R is proper orthogonal and U ¼ C is symmetric and positive definite, can be used to show that 1 ; RiA RiB ;C ¼ ðΓ PQC U  1 ;AC  U BQ ;C ÞU BQ

θx ¼ 

dw dx

1 2

Γ ABC ¼ ðC AB;C þ C CA:B  C BC;A Þ

ð8Þ

Please cite this article as: J.N. Reddy, A.R. Srinivasa, Non-linear theories of beams and plates accounting for moderate rotations and material length scales, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.06.003i

J.N. Reddy, A.R. Srinivasa / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4

The Lagrangian strain tensor E, the rotation R, and its gradient are of the form (omitting terms of order ϵ1:5 and ϵ2)1 "  2 # du dθx 1 dw 1 2 e^ x e^ x þ θx e^ z e^ z ; þz þ E dx 2 dx 2 dx   1 dw  θx ðe^ z e^ x  e^ x e^ z Þ W¼ ð9Þ 2 dx 1 R  IþWþ W " 2   # 2 1 dw dw W ¼ 1  ðe^ z e^ x  e^ x e^ z Þ ðe^ x e^ x þ e^ z e^ z Þ þ 2 dx dx

Ω  R T  ∇R  ∇W ¼ e^ x

2

d w dx

2

ðe^ z e^ x  e^ x e^ z Þ

ð10Þ

1 2

Ψ ¼ ½a1 E2xx þ a2 E2zz þ a3 ðθ0x Þ2 þ 2a4 Exx Ezz þ 2a5 Exx θ0x þ 2a6 Ezz θ0x  ð12Þ where ðÞ0 denotes the derivative of the enclosed quantity with respect to x, and a1 through a6 are material parameters; some of them (a1, a2, and a4) are elastic constants and others (a3, a5, and a6) are material length scales. The form above introduces a length pffiffiffiffiffiffiffiffiffiffiffiffi scale L0 : ¼ a3 =a1 which is the characteristic material length scale of the problem. The generalized nominal stresses associated with this energy function are ∂Ψ 0 Sxx ¼ ¼ a1 Exx þ a4 Ezz þ a5 θx ∂Exx

ð13Þ

∂Ψ 0 ¼ a4 Exx þ a2 Ezz þ a6 θx Szz ¼ ∂Ezz

ð14Þ

ð15Þ

where (Sxx, Szz) are the components of the second Piola–Kirchhoff stress tensor. We note that unlike conventional Euler–Bernoulli beam theory, there is an extra term m, which is the couple moment due to the curvature dependence of the strain energy. Now by assuming that the beam has a thickness h in the z direction and width b in the y direction, we can obtain the governing equations directly from the principle of virtual work [61], resulting in a set of equations of the following form: ð0Þ

dM xx ¼ f ðxÞ dx



d M xx

ð1Þ

2

2

dx



ð16Þ

  d ð0Þ dw ðM ð0Þ ¼ qðxÞ þ M Þ xx zz dx dx

ð17Þ

where f(x) and q(x) are the axial and transverse external loads on the beam, respectively, the former being in the x direction and the ð1Þ ð1Þ latter being in the z direction, M ¼ M xx þ M, and M ðiÞ xx , i ¼ 0; 1, are the stress resultants Z h=2 Z h=2 M ð0Þ Sxx dz; M ð0Þ Szz dz xx ¼ b zz ¼ b  h=2

Z

h=2  h=2

M ð0Þ zz ¼ b

M ð1Þ xx ¼ b Z M¼b

Z

h=2  h=2

Z

h=2  h=2

h=2  h=2

M¼b

zSxx dz;

h=2

ð18Þ

m dz  h=2

" Szz dz ¼ a4 A

zSxx dz ¼ a1 I "

m dz ¼ a5 A

 2 # du 1 dw 1 dθx 2 þ þ a2 A θ x þ a6 A dx 2 dx 2 dx

dθ x dx

ð20Þ

ð21Þ

 2 # du 1 dw 1 dθ x 2 þ þ a6 Aθx þ a3 A dx 2 dx 2 dx

ð22Þ

3

where A¼ bh is the area and I ¼ bh =12 is the moment of inertia of the beam cross section. In contrast to the conventional von Kármán beam theory based on the Euler–Bernoulli kinematics, the boxed terms in Eqs. (19)–(21) indicate the rotation gradient or curvature effect. The boundary conditions at the ends x ¼0 and z ¼0 and the following geometric or force boundary conditions: Geometric : u; Force : M ð0Þ xx ;

∂Ψ 0 0 ¼ a5 E xx þa6 Ezz þa3 θ x ∂θx



Z

We note that the equilibrium equations involve the transverse shear effect due to the moderate rotation as is to be expected. Eqs. (16) and (17) are formally identical to the classical von Kármán ð0Þ equations for a beam [with the exception of M zz in Eq. (17)]; the role of the rotation gradient dependence will become apparent only in the stress resultant–displacement relations, as shown next. The stress resultants in Eqs. (16) and (17) are given in terms of the displacements as "  2 # Z h=2 du 1 dw 1 dθ x 2 M ð0Þ þ ð19Þ ¼ b S dz ¼ a A þ a4 Aθx þ a5 A xx 1 xx dx 2 dx 2 dx  h=2

ð11Þ

We note that the strain is of order ϵ while the rotation is of order pffiffiffi ϵ. Furthermore, the rotation does not coincide with that of the linearized theory (since dw=dx is moderate). The rotation gradient dependent strain energy potential is of the form that is correct to the order ϵ2 (remembering that pffiffiffi θx ¼  dw=dx is of order ϵ while d2 w=dx2 is of order ϵ), that is

m¼

M ð1Þ xx ¼ b

w;

dw dx

ð0Þ Q  ðM ð0Þ xx þ M zz Þ

ð23Þ ð1Þ

dw dM xx þ ; dx dx

ð1Þ

M xx

ð24Þ

The governing equations of the modified couple stress theory are obtained as a special case by setting a5 ¼ a6 ¼ 0 and a2 ¼ 2Gℓ2 , as will be shown in the sequel. 1.5. Present study The present study has the objective of formulating the governing equations of shear deformable beams and plates that account for moderate rotations and microstructural material length scales. This is done using (1) a modified von Kármán non-linear theory and (2) the gradient elasticity theory of Srinivasa and Reddy [57]. In particular, generalized Euler–Bernoulli and Timoshenko beam theories are developed, and the connection between the two approaches is discussed. Finite element models are also formulated for the two beam models. In the following sections, we shall use the notation (σ xx , σ zz , σ xz ) for the components of the second Piola–Krichoof stress tensor components (Sxx , Szz , Sxz ) and (εxx , εzz , 2εxz ¼ γ xz ) for the simplified Green–Lagrange strain tensor components (Exx , Ezz , 2Exz ).

2. Generalized beam theories with material length scales 2.1. The Euler–Bernoulli beam theory

 h=2

1 If terms of the order ϵ1:5 are included, we will have non-zero E13; see Reddy and Mahaffey [60].

2.1.1. Generalized von Kármán EBT The displacement field of the Euler–Bernoulli beam theory is uðx; z; tÞ ¼ ux ðx; z; tÞe^ x þuz ðx; z; tÞe^ z

ð25Þ

Please cite this article as: J.N. Reddy, A.R. Srinivasa, Non-linear theories of beams and plates accounting for moderate rotations and material length scales, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.06.003i

J.N. Reddy, A.R. Srinivasa / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

δθx : Mð1Þ xx

with ux ðx; z; tÞ ¼ uðx; tÞ þzθx ;

∂w uz ðx; z; tÞ ¼ wðx; tÞ; θx   ∂x

ð26Þ

The simplified Green–Lagrange strain tensor is (i.e., using the order-of-magnitude assumption and retaining terms of order ϵ) E  ε ¼ εxx e^ x e^ x þ εzz e^ z e^ z

ð27Þ

where

εzz ¼ εð0Þ zz

ð1Þ εxx ¼ εð0Þ xx þ zεxx ;



∂u 1 ∂w þ ∂x 2 ∂x

2 ;

εð1Þ xx ¼

M ð1Þ xx ¼ ∂θ x ; ∂x

1 2

2 εð0Þ zz ¼ θ x

ð29Þ

Thus, the generalized von Kármán non-linear strains include εð0Þ zz , requiring us to use the two-dimensional stress–strain relations. Most beam theories that include the classical von Kármán nonlinearity omit the non-linear term due to the transverse normal strain so that one can use one-dimensional constitutive relations. The equations of motion can be obtained using Hamilton's principle (see Reddy [61]): Z t2 0¼ ð  δK þ δU þ δV Þ dt ð30Þ t1

where δK is the virtual kinetic energy, δU is the virtual strain energy, and δV is the virtual work done by external forces: Z L _ δwÞ _ dx δK ¼ ðm0 u_ δu_ þ m2 θ_ x δθ_ x þ m0 w ð31Þ 0

δU ¼

Z

L 0

δV ¼ 

    ð0Þ ∂δu ∂w ∂δw ð1Þ ∂δθ x þ þ M xx þ M ð0Þ M xx zz θ x δθ x dx ∂x ∂x ∂x ∂x

Z

L 0

ðf δu þ q δwÞ dx

ð33Þ

where L is the length of the beam, (m0 ; m2 ) are the mass inertias Z ðm0 ; m2 Þ ¼ ρð1; z2 Þ dA ð34Þ A

Various stress resultants used in Eq. (32) are defined as Z Z Z ð1Þ ð0Þ σ xx dA; M xx ¼ z σ xx dA; M zz ¼ σ zz dA M ð0Þ xx ¼ A

A

ð35Þ

A

and qðx; tÞ ¼ ½qt ðtÞ þ qb ðtÞ

ð36Þ

Here ρ is the mass density, qt and qb denote the distributed load for top and bottom surfaces, respectively, and f is the axial force per unit length. Substituting δU, δV, and δK into the Hamilton's principle, Eq. (30), performing integration-by-parts with respect to t as well as x to relieve the generalized displacements δu and δw of any differentiations, and using the fundamental lemma of calculus variations, we obtain the following equations of motion: ∂M ð0Þ xx

m0

∂ u ¼ f ðx; tÞ  ∂x ∂t 2

ð37Þ

m0

  ∂2 w ∂4 w ∂ ∂2 M ð1Þ ð0Þ ∂w xx ðM ð0Þ   m2 2  ¼ qðx; tÞ xx þ M zz Þ 2 2 ∂x ∂x2 ∂t ∂t ∂x ∂x

ð38Þ

2

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

δu : Mð0Þ xx ð0Þ δw : ðMð0Þ xx þ M zz Þ

ð1Þ ∂w ∂M xx ∂3 w þ þ m2 2 ∂x ∂x ∂t ∂x

M ð0Þ zz

Z A

z σ xx dA ¼ Ið2μ þ λÞ

dθ x dx

ð41Þ

"

Z ¼ A

σ zz

 2 #  2 du 1 dw 1 dw þ dA ¼ Aλ þ Að2μ þ λÞ dx 2 dx 2 dx

ð42Þ

Comparing the results in Eqs. (40)–(42) to those in Eqs. (19)–(21), we note the following correspondence: a1 ¼ a2 ¼ 2μ þ λ;

a4 ¼ λ;

a3 ¼ a5 ¼ a6 ¼ 0

ð43Þ

Thus, the material length scale parameters, a3, a5, and a6, are missing in the modified theory because couple stress effect is not accounted for in the virtual strain energy δU of Eq. (32). 2.1.2. Generalized von Kármán EBT with modified couple stress The modified couple stress theory uses the virtual strain energy δU [see Eq. (4)] Z LZ δU ¼ ðσ xx δεxx þ m δχ Þ dA dx 0 L

A

   ∂δu ∂w ∂δw ∂δθx þ þ M ð1Þ M ð0Þ ¼ xx xx ∂x ∂x ∂x ∂x 0   2 1 ∂δθx ∂ δw ð0Þ þM zz  θx δθx þ M dx 2 ∂x ∂x2 Z

ð32Þ

ð39Þ

We note that the equations of motion derived here reduce to the equilibrium equations in Eqs. (16) and (17) for the static case. Due to the use of two-dimensional elasticity constitutive relations, in the present case we have "  2 #  2 Z du 1 dw 1 dw ð0Þ þ M xx ¼ σ xx dA ¼ Að2μ þ λÞ ð40Þ þ Aλ dx 2 dx 2 dx A

ð28Þ

and

εð0Þ xx ¼

5

ð44Þ

where m is the deviatoric part of the couple stress and curvature [see Eqs. (5) and (6)]   ∂ω 1 ∂ ∂w χ¼ y¼ θx  2 ∂x ∂x ∂x

χ is the ð45Þ

The resulting equations of motion are the same as those in Eqs. ð1Þ ð1Þ (37) and (38), with M ð1Þ xx replaced by M xx ¼ M xx þ M. The modified couple stress theory discussed in [36–56] is based on the ideas presented above, with the constitutive relation for M given by Eq. (7). Thus, the Euler–Bernoulli beam theory presented in this section is a special case of that proposed by Srinivasa and Reddy [57] for a5 ¼ a6 ¼ 0 and a3 ¼ 2μℓ2 . 2.2. The Timoshenko beam theory 2.2.1. Generalized von Kármán TBT with modified couple stress The displacement field of the Timoshenko beam theory is given by Eq. (25) with ux ðx; z; tÞ ¼ uðx; tÞ þ zϕx ðx; tÞ;

uz ðx; z; tÞ ¼ wðx; tÞ

ð46Þ

where ϕx ðx; tÞ represents the rotation of a transverse normal line about the y-axis. The simplified Green–Lagrange strain tensor components are ð1Þ εxx ¼ εð0Þ xx þ zεxx ;

ð0Þ γ xz ¼ γ xz ;

εzz ¼ εð0Þ zz

ð47Þ

with

εð0Þ xx ¼

  ∂u 1 ∂w 2 þ ; ∂x 2 ∂x

ð1Þ εxx ¼

∂ ϕx ; ∂x

γ ð0Þ xz ¼ ϕx þ

∂w ; ∂x

1 2

2 εð0Þ ð48Þ zz ¼ ϕx

Following the ideas similar to those presented for the Euler– Bernoulli beam theory, we obtain the following equations of motion of the TBT (compare with Reddy [42] and Reddy and

Please cite this article as: J.N. Reddy, A.R. Srinivasa, Non-linear theories of beams and plates accounting for moderate rotations and material length scales, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.06.003i

J.N. Reddy, A.R. Srinivasa / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

6

Mahaffey [60]): ð0Þ ∂M xx

∂ u þ m0 2 ¼ f  ∂x ∂t   ∂ 1 ∂2 M ∂2 w ð0Þ ∂w M ð0Þ  þ m0 2 ¼ q  xz þ M xx 2 ∂x ∂x 2 ∂x ∂t 2

ð0Þ M ð0Þ xz þ M zz ϕx 

where M ð0Þ xx

M ð1Þ xx

σ xx

¼ A

Z

 2 # du 1 dw 1 þ dA ¼ Að2μ þ λÞ þ Aλ ðϕx Þ2 dx 2 dx 2

z σ xx

¼ A

Z

m¼ ð50Þ

ð51Þ

dϕ dA ¼ Ið2μ þ λÞ x dx

  dw z σ xz dA ¼ AK s μ ϕx þ dx A

¼ A

σ zz

 2 # du 1 dw 1 þ dA ¼ Aλ þ Að2μ þ λÞðϕx Þ2 dx 2 dx 2

ð52Þ

∂Ψ ¼ a7 γ xz þ a8 εxx þ a9 εzz þ a10 χ ∂γ xz

ð63Þ

∂Ψ ¼ a5 εxx þ a6 εzz þ a3 χ þ a10 γ xz ∂χ

ð64Þ

where the parameters a8 and a9 are zero for an isotropic or orthotropic material, while a7 is equal to the shear modulus μ. Thus, the Timoshenko beam theory contains an additional material length scale (a10) due to the presence of the transverse shear strain γxz. The resultants in the gradient elasticity theory of Timeoshenko beams are given by Mð0Þ xx

Z

h=2

¼b

 h=2

σ xx

"     # ∂u 1 ∂w 2 1 a5 ∂ ϕ x ∂ 2 w 2 þ a4 Aϕx þ A dz ¼ a1 A  2 þ 2 ∂x ∂x 2 ∂x 2 ∂x

ð53Þ

ð54Þ

ð65Þ Mð0Þ zz ¼ b

Z

"

h=2  h=2

σ zz dz ¼ a4 A

 ∂u 1 ∂w þ ∂x 2 ∂x

2 #

  1 a6 ∂ ϕ x ∂ 2 w 2 þ a2 Aϕx þ A  2 2 ∂x 2 ∂x

ð66Þ

"

Z

Z M¼

ð49Þ

"

Z

M ð0Þ xz ¼ K s M ð0Þ zz

∂M ð1Þ 1 ∂M ∂2 ϕ xx þ m2 2 x ¼ 0  2 ∂x ∂x ∂t

σ xz ¼

m dA ¼ A

αd



2 dx

ϕx 

dw dx

ð55Þ

Z

ð56Þ

1 2 ð0Þ ∂w 1 ∂M þ δw : Mð0Þ þ M xz xx ∂x 2 ∂x 1 M: δϕx : Mð1Þ þ xx 2

M ð0Þ xz ¼ bK s

M¼b

  1 dw  ϕx ðe^ z e^ x  e^ x e^ z Þ 2 dx   1 d dw  ϕx ðe^ z e^ x  e^ x e^ z Þ   χ e^ x ðe^ x e^ z  e^ z e^ x Þ Ω  ∇W ¼ e^ x 2 dx dx

h=2  h=2



σ xz dz ¼ a7 K s A ϕx þ "

h=2  h=2

m dz ¼ a5 A

 ∂u 1 ∂w þ ∂x 2 ∂x

xb

"

d δw 2



2

ð1Þ

M xx þ

# dδ w dw ð0Þ ðM xx þ M ð0Þ Þ  δ w q dx zz dx dx

dx  δwðxa ÞQ 2  δwðxb ÞQ 5  δθx ðxa ÞQ 3  δθx ðxb ÞQ 6

where

ð1Þ M xx

¼

ð1Þ M xx þM

1 Ψ ¼ ½a1 ε2xx þ a2 ε2zz þa3 ðχ Þ2 þ 2a4 εxx εzz þ 2a5 εxx χ þ 2a6 εzz χ 2 þa7 γ 2xz þ a8 εxx γ xz þa9 εzz γ xz þ a10 γ xz χ 

" # ð1Þ dM xx ð0Þ ð0Þ þ ðM xx þ M zz Þ Q2 ¼  dx x " # a ð1Þ dM xx ð0Þ þ ðM xx Q5 ¼ þ M ð0Þ zz Þ dx

∂Ψ ¼ a1 εxx þ a4 εzz þ a5 χ þ a8 γ xz ∂εxx

ð61Þ

σ zz ¼

∂Ψ ¼ a4 εxx þ a2 εzz þ a6 χ þ a9 γ xz ∂εzz

ð62Þ

 M ð0Þ xx ðxa Þ;

ð71Þ

and Qi are defined by

Q1 ¼

σ xx ¼

  1 a3 ∂ ϕx ∂ 2 w 2  2 þ a6 Aϕx þ A 2 2 ∂x ∂x ð69Þ

Then we choose the strain energy potential to be

Then the constitutive relations are given by

2 #

In this section we present the finite element models of the strain gradient theory of Srinivasa and Reddy [57] as applied to the Euler–Bernoulli beam theory. The discussion is limited to the static case, although the time-dependent case is a straightforward extension of the development here (see Reddy [62]). We begin with the virtual work statements over a typical element domain (xa, xb):  Z xb  dδ u ð0Þ M xx  δu f dx  δuðxa ÞQ 1  δuðxb ÞQ 4 ð70Þ 0¼ dx xa

xa

ð60Þ

   ∂w a10 ∂ϕx ∂2 w þ A  2 ∂x 2 ∂x ∂x

3.1. The Euler–Bernoulli beam theory

0¼

ð59Þ

ð67Þ

3. Finite element models of the strain gradient beams

Z

W¼

Z

∂ ϕx ∂x

  ∂w : þ a10 A ϕx þ ∂x

ð57Þ

2.2.2. Gradient elasticity model of TBT The governing equations of the Timoshenko beam theory according to the gradient elasticity model [57] are the same as in Eqs. (49)–(51); however, m, and hence M, are related to the generalized displacements and rotation gradients differently than that in Eqs. (15) and (22). To establish the new relations, we begin with the Lagrangian strain tensor E and the gradient of the rotation for the Timoshenko beam theory as "  2 # du dϕx 1 dw E þz e^ x e^ x þ dx dx 2 dx   1 ∂w 1 2 ðe^ x e^ z þ e^ z e^ x Þ þ ϕx e^ z e^ z þ ϕx þ ð58Þ 2 ∂x 2

zσ xx dz ¼ a1 I

ð68Þ Z

ð1Þ δϕx : Mxx þ M

h=2  h=2



and Ks denotes the shear correction factor and α is a material length scale parameter. The natural boundary conditions are

δu : Mð0Þ xx ;

M ð1Þ xx ¼ b

Q 4 ¼ M ð0Þ xx ðxb Þ

ð72Þ

ð73Þ

xb

ð1Þ

Q 3 ¼ M xx ðxa Þ;

ð1Þ

Q 6 ¼  M xx ðxb Þ

ð74Þ

The weak forms in terms of the displacements (u, w) are

Please cite this article as: J.N. Reddy, A.R. Srinivasa, Non-linear theories of beams and plates accounting for moderate rotations and material length scales, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.06.003i

J.N. Reddy, A.R. Srinivasa / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Z 0¼

xb

"

xa

# " ( )  2 # dδ u du 1 dw 1 dθx 2 a1 A þ þ a4 Aθx þ a5 A  δu f dx dx dx 2 dx 2 dx

 δuðxa ÞQ 1  δuðxb ÞQ 4 xb

"

d δw 2

(

"

 δwðxa ÞQ 2  δwðxb ÞQ 5  δθx ðxa ÞQ 3  δθx ðxb ÞQ 6

ð76Þ

Axial displacement u and transverse displacement w are approximated using linear and Hermite cubic interpolations, respectively, 2

uðxÞ  ∑ Δj ψ j ðxÞ; 1

j¼1

4

wðxÞ  ∑ ΔJ φJ ðxÞ 2

ð77Þ

J¼1

where ψ j ðxÞ are the linear polynomials, φJ ðxÞ are the Hermite cubic 1 1 polynomials, (Δ1 , Δ2 ) are the nodal values of u at xa and xb, 2 respectively, and ΔJ (J¼ 1, 2, 3, 4) are the nodal values associated with w

Δ ¼ wðxa Þ; 2 1

Δ ¼ wðxb Þ; 2 3

Δ ¼ θx ðxa Þ; 2 2

Δ ¼ θx ðxb Þ 2 4

where Z K 11 ij ¼

dψ i dψ j dx dx dx " # Z xb 2 d φJ ða1 þ a4 Þ dw dφJ dψ i  a K 12 ¼ A þ dx 5 iJ 2 2 dx dx dx xa dx " # Z xb 2 d φ dw dφI dψ j dx A  a5 2 I þ ða1 þ a4 Þ K 21 Ij ¼ dx dx dx xa dx ( " # Z xb 2 2 d φJ Aða5 þ a6 Þ dw dφJ d φI 22 ða1 I þ a3 AÞ 2  K IJ ¼ 2 2 dx dx xa dx dx " #) 2 d φJ dw dφI ða1 þ a2 þ2a4 Þ dw dφJ  ða5 þ a6 Þ 2 þA dx dx dx 2 dx dx dx Z xb f ψ i dxþ Q 1 ψ i ðxa Þ þ Q 4 ψ i ðxb Þ F 1i ¼ xa     Z xb dφ dφ F 2I ¼ qφI dx þ Q 2 φI ðxa Þ þ Q 5 φI ðxb Þ þ Q 3  I þ Q 6  I dx xa dx xb xa xb

a1 A

xa

ð80Þ 3.2. The Timoshenko beam theory The weak forms for the gradient elasticity theory of the Timoshenko beam can be derived using Eqs. (49)–(51) (set the time derivative terms to zero) as  Z xb  dδu ð0Þ M xx  δu f dx  δuðxa ÞQ 1  δuðxb ÞQ 5 0¼ ð81Þ dx xa Z 0¼

xb xa

"

xa

#   2 dδ w dw ð0Þ 1 d δw ð0Þ M xz þ M M  δw q dx  dx dx xx 2 dx2

 δwðxa ÞQ 2  δwðxb ÞQ 6  δθx ðxa ÞQ 3  δθx ðxb ÞQ 7

ð82Þ

    dδϕx 1 ð1Þ ð0Þ M xx þ M þ δϕx ðM xz þ M ð0Þ ϕ Þ dx x zz 2 dx ð83Þ

where Qi are defined by Q 1 ¼  M ð0Þ xx ðxa Þ;

ð0Þ Q 5 ¼ M xx ðxb Þ

ð84Þ

  dw 1 dM ð0Þ þ M xz þ M ð0Þ ; xx dx 2 dx xa   ð0Þ dw 1 dM þ M ð0Þ Q6 ¼ xz þ M xx dx 2 dx xb

Q2 ¼ 

1 Q 3 ¼  Mðxa Þ; 2

ð85Þ

1 Q 7 ¼ Mðxb Þ 2

1 Q 4 ¼ M ð1Þ xx ðxa Þ þ Mðxa Þ; 2

ð86Þ

  1 ð1Þ Q 8 ¼  M xx ðxb Þ þ Mðxb Þ 2

ð87Þ

The weak forms in terms of the generalized displacements (u, w, ϕx ) are obtained using Eqs. (65)–(69): ( " !)  2 # Z xb " 2 dδ u du 1 dw 1 a5 dϕx d w 2 a1 þ  2 A þ a4 ðϕx Þ þ 0¼ dx dx 2 dx 2 2 dx xa dx   δu f dx

ð78Þ

where θx   ðdw=dxÞ. By substituting Eq. (77) for u and w and putting δu ¼ ψ i and δw ¼ φi into the virtual work statements in Eqs. (75) and (76), the finite element equations can be obtained as " #( ) ( ) K11 K12 F1 Δ1 ð79Þ ¼ or KΔ ¼ F 2 K21 K22 F2 Δ

xb

 δϕx ðxa ÞQ 4  δϕx ðxb ÞQ 8

ð75Þ

)  2 # du 1 dw 1 dθ x dθ x 2 þ a þ a 0¼  a A A θ þ a A I þ 5 6 3 1 x 2 dx 2 dx 2 dx dx xa dx " # (  2 dδ w dw du 1 dw 1 dθx 2 a1 A þ þ þ a4 A θ x þ a5 A dx dx dx 2 dx 2 dx " )  2 # du 1 dw 1 dθ x 2 þ þ a4 A þ a2 Aθx þ a6 A  δw qdx dx 2 dx 2 dx Z

Z 0¼

7

ð88Þ

   dδ w dw a7 K s ϕx þ A dx dx xa ( "  2 #! 2 a10 dϕx d w dw du 1 dw a1 þ  2þ þ dx dx 2 dx 2 dx dx !) ( "  2 # 2 2 1 a5 dϕx d w A d δw du 1 dw þ  2 a þ a4 ðϕx Þ2 þ  5 2 2 dx2 dx 2 dx 2 dx dx ) !   2 1 2 a3 dϕx d w dw  δw qg dx  2 þ a10 ϕx þ þ a6 ϕx þ 2 dx 2 dx dx

Z 0¼

 δuðxa ÞQ 1  δuðxb ÞQ 5

xb



 δwðxa ÞQ 2  δwðxb ÞQ 6  δθx ðxa ÞQ 3  δθx ðxb ÞQ 7 (

ð89Þ

(

"  2 # Z xb dδϕx dϕx A dδϕx du 1 dw 1 2 þ a6 ϕx þ þ a5 a1 I 0¼ dx 2 dx 2 dx dx 2 dx xa !  )    2 a3 dϕx d w dw dw þ Aδϕx a7 K s ϕx þ  2 þ a10 ϕx þ þ dx dx 2 dx dx ! ( "  2 # 2 a10 dϕx d w du 1 dw 1 þ þ  2 þ a4 þ a2 ðϕx Þ2 dx 2 dx 2 2 dx dx !) 2 a6 dϕx d w  2 ϕx g dx  δϕx ðxa ÞQ 4  δϕx ðxb ÞQ 8 ð90Þ þ 2 dx dx Due to the fact that both w and θx are primary variables in the weak forms (along with u and ϕx), we approximate w using the Hermite cubic interpolation functions, while u and ϕx are approximated using the Lagrange interpolation functions (linear or higher). In general, one may use different order functions for u and ϕx, as implied by the following interpolations: m

uðxÞ  ∑ Δj ψ ð1Þ j ðxÞ; j¼1

1

4

wðxÞ  ∑ ΔJ φJ ðxÞ; J¼1

2

n

ϕx ðxÞ  ∑ Δ3j ψ ð2Þ j ðxÞ j¼1

ð91Þ Substitution of Eq. (91) into the weak forms in Eqs. (88)–(90, we obtain the following finite element model: 2 11 38 1 9 8 1 9 K K12 K13 > <Δ > = > = 6 21 7 2 ¼ F2 or KΔ ¼ F ð92Þ 4K K22 K23 5 Δ > > > : 3; : 3> ; K31 K32 K33 F Δ

Please cite this article as: J.N. Reddy, A.R. Srinivasa, Non-linear theories of beams and plates accounting for moderate rotations and material length scales, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.06.003i

J.N. Reddy, A.R. Srinivasa / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

8

textbook by Reddy [64] and the paper by Reddy and Kim [46] on modified couple stress theory of plates (only for certain developments).

where Z K 11 ij ¼

Aa1 xa

¼

K 13 ij ¼ K 21 Ij ¼ K 22 IJ ¼

ð1Þ j

dψ dψ dx dx dx ð1Þ i

! 2 dψ ð1Þ a1 dw dφJ a5 d φJ i  A dx dx 2 dx dx 2 dx2 xa ð2Þ ! Z xb dψ ð1Þ a4 a dψ A i ϕx ψ jð2Þ þ 5 j dx dx 2 2 dx xa ! ð1Þ Z xb 2 a5 d φI dw dφI dψ j dx A  þ a 1 dx dx 2 dx2 dx xa " !#   Z xb ( 2 2 dφJ a10 d φJ dw a1 dw 2 dφJ a5 d φJ dφI K s a7   A þ dx 2 dx dx dx 2 dx2 dx 2 dx2 xa Z

K 12 iJ

xb

xb

" #) 2 2 dφJ 1 d φI a5 dw dφJ a3 d φJ   þ a10 dx 2 dx2 2 dx dx 2 dx2 dx " ð2Þ ð2Þ !# Z xb ( dφI a10 dψ j dw a4 a5 dψ j ð2Þ K s a7 ψ ð2Þ þ A þ ϕ ψ þ K 23 x Ij ¼ j j dx 2 dx 2 dx 2 dx xa "

" ð2Þ ð2Þ # ð2Þ

dψ ið2Þ dψ j A dψ i a6 a3 dψ j þ ϕx þa10 ψ ð2Þ þ j 2 dx dx dx 2 2 dx xa  

ð2Þ a2 2 1 dψ dx þ Aψ ið2Þ K s a7 þ ϕx ψ jð2Þ þ ða10 þa6 ϕx Þ 2 2 dx Z xb ð1Þ ð1Þ F 1i ¼ f ψ ð1Þ i dxþ Q 1 ψ i ðxa Þ þQ 5 ψ i ðxb Þ xa     Z xb dφ dφ qφI dx þ Q 2 φI ðxa Þ þ Q 6 φI ðxb Þ þ Q 3  I þ Q 7  I F 2I ¼ dx xa dx xb xa

K 33 ij ¼

Z

xb

The first-order shear deformation plate theory is based on the displacement field u ¼ ðuðx; yÞ þ zϕx Þe^ x þðvðx; yÞ þzϕy Þe^ y þ we^ z

ð94Þ

(u, v, w) are the displacements along the coordinate lines of a material point on the xy-plane, which is taken to be the undeformed midsurface of the plate, and (ϕx , ϕy ) are the rotations of a transverse normal about the y and x axes, respectively. The approximated Green–Lagrange strains are 9 8 ð0Þ 9 8 ð1Þ 9 8 8 ð0Þ 9 8 8 2 2 ε 9 ε 9 > εxx > > εxx > > εzz > > 12ðϕx þ ϕy Þ > > > > > > = > = > = > = > = > = < xx > < zz > < < < < ð0Þ ð1Þ εyy ¼ εyy γ xz ¼ γ ð0Þ ϕx þ ∂w þ z εyy ; ¼ xz ∂x > > > > > > > > > > > > :γ ; > ; > ; : γ yz ; > ; > ; : γ ð0Þ > : γ ð1Þ > : γ ð0Þ > : ϕ þ ∂w > xy xy

y

yz

xy

ð2Þ #) j

2

1 d φI a6 a3 dψ ϕ þ a10 ψ ð2Þ dx j þ 2 dx 2 dx2 2 x ! ð1Þ Z xb ð2Þ dψ j a5 dψ i ð2Þ þ a dx ¼ A ϕ ψ K 31 4 x i ij 2 dx dx xa #  Z xb ( dψ ð2Þ " 2 dφJ a 3 d φJ 1 a5 dw j K 32 þ a  ¼ A 10 iJ 2 dx 2 dx dx 2 dx2 xa " !#) 2 2 dφJ a10 d φJ a4 dw dφJ a6 d φJ   K a þ ϕ þ ψ ð2Þ dx s 7 x i dx 2 dx2 2 dx dx 2 dx2



4.2. Displacement and strain fields

(

a1 I

F 3i ¼ Q 4 ψ ið2Þ ðxa Þ þ Q 8 ψ ið2Þ ðxb Þ

ð93Þ

This completes the development of finite element models of the gradient elasticity theories of the Euler–Bernoulli and Timoshenko beams. We note that the element stiffness matrices in Eqs. (79) and (92) are not symmetric (because Kαβ a Kβα ). However, it can be shown that the tangent stiffness matrices associated with these models (needed in Newton's iteration scheme to solve the non-linear equations) are symmetric (see Reddy [63]). 4. Modified von Kármán first-order plate theory 4.1. Preliminary comments Here we present gradient elasticity model based on the generalized von Kármán non-linear theory with the first-order shear deformation kinematics. The plan is to use the order-of-magnitude assumption to simplify the Green–Lagrange strain tensor components and invoke the assumption that the strain energy potential is a function of the simplified Green–Lagrange strain components as well as the gradients of the rotations θx ¼ ∂w=∂x and θy ¼  ∂w=∂y to derive the equations of motion. In the interest of brevity, details are not included, and the readers may refer to the

∂y

ð95Þ with 8 ð0Þ 9 ε > > > = < xx >

εð0Þ yy

> > > ; : γ ð0Þ > xy

¼

8 > > > > <

∂u 1 ∂w 2 ∂x þ 2 ∂x 2 ∂v 1 ∂w ∂y þ 2 ∂y

9 > > > > =

> > > > > > > ∂v ∂w ∂w > ; : ∂u ∂y þ ∂x þ ∂x ∂y

;

8 ð1Þ 9 ε > > > = < xx >

εð1Þ yy

> > > ; : γ ð1Þ > xy

¼

8 > > > > <

∂ϕx ∂x ∂ ϕy ∂y

9 > > > > =

> > > > > ϕx ∂ ϕy > > ; : ∂∂y þ ∂x >

ð96Þ

In view of the displacement field in Eq. (94), the components of the rotation vector ω and curvature tensor χ Eqs. (5) and (6) take the form (with ω1 ¼ ωx , ω2 ¼ ωy , ω3 ¼ ωz , χ 11 ¼ χ xx , χ 22 ¼ χ yy , χ 12 ¼ χ 21 ¼ χ xy , and so on)     1 ∂w 1 ∂w  ϕy  ωð0Þ  ωyð0Þ ωx ¼ ; ω ¼ ϕ  y x x 2 ∂y 2 ∂x " !  # 1 ∂v ∂ϕy ∂u ∂ϕ ð1Þ þz þz x  ωð0Þ ωz ¼ ð97Þ  z þ z ωz 2 ∂x ∂y ∂x ∂y   1 ∂ ∂w  ϕy  χ ð0Þ xx 2 ∂x ∂y   1 ∂ ∂w ð0Þ  χ yy χ yy ¼ ϕx  2 ∂y ∂x ! 1 ∂ ϕy ∂ ϕx ð0Þ  χ zz ¼  χ zz 2 ∂x ∂y      1 ∂ ∂w ∂ ∂w  ϕy þ  χ ð0Þ χ xy ¼ ϕx  xy 2 ∂y ∂y ∂x ∂x "  ! #  1 ∂ ∂v ∂u ∂ ∂ϕy ∂ϕx ð0Þ  þz  χ xz ¼ þ zχ ð1Þ  χ xz xz 2 ∂x ∂x ∂y ∂x ∂x ∂y      1 ∂ ∂v ∂u ∂ ∂θ y ∂θ x ð1Þ  þz   χ ð0Þ χ yz ¼ yz þ zχ yz 2 ∂y ∂x ∂y ∂y ∂x ∂y

χ xx ¼

where   1 ∂ ∂v ∂u  ; χ ¼ 2 ∂x ∂x ∂y   1 ∂ ∂v ∂u  ; χ ð0Þ yz ¼ 2 ∂y ∂x ∂y ð0Þ xz

χ

ð1Þ xz

χ

ð1Þ yz

1 ∂ ∂ ϕy ∂ ϕx  ¼ 2 ∂x ∂x ∂y

ð98Þ

!

! 1 ∂ ∂ϕy ∂ϕx  ¼ : 2 ∂y ∂x ∂y

ð99Þ

4.3. Equations of motion and boundary conditions The equations of motion are obtained using Hamilton's principle in Eq. (30), where δK, δU, and δV are determined from the

Please cite this article as: J.N. Reddy, A.R. Srinivasa, Non-linear theories of beams and plates accounting for moderate rotations and material length scales, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.06.003i

J.N. Reddy, A.R. Srinivasa / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

following expressions: Z Z h=2 ∂u ∂δu dz dx dy δK ¼ ρ  ∂t Ω  h=2 ∂t Z _ δϕ _ þ θ_ δθ_ Þ dx dy _ þ m2 ðϕ ¼ ½m0 ðu_ δu_ þ v_ δvÞ y y x x

δv :

ð100Þ

Ω

δU ¼

Z Z

h=2

Ω

 h=2

ð0Þ

∂Myz ∂Mð0Þ xz þ 2 ∂x ∂y

1 ð0Þ ð0Þ M xy nx þM yy ny 

δw :

ðσ xx δεxx þ σ yy δεyy þ σ zz δεzz þ σ xy δγ xy þ σ xz δγ xz þ σ yz δγ yz

9

! ð111Þ

nx

    ∂w ð0Þ ∂w ð0Þ ∂w ð0Þ ∂w ð0Þ ð0Þ ð0Þ M xx þ M xy nx þ M xy þ M yy ny þM xz nx þM yz ny ∂x ∂y ∂x ∂y ! ! ð0Þ ð0Þ ∂Mð0Þ 1 ∂Mð0Þ 1 ∂Myy ∂Mxy xy xx þ þ  ð112Þ ny þ nx 2 2 ∂x ∂y ∂y ∂x

! ∂Mð1Þ 1 ∂Mð1Þ yz ð1Þ ð1Þ xz þ mxx δχ xx þ myy δχ yy þ mzz δχ zz þ mxy δχ xy þ mxz δχ xz þ myz δχ yz Þdz dx dy δϕx : M xx þ nx þM xy ny þ ny 2 ∂x ∂y Z " 1 1 ð0Þ ð0Þ ðiÞ ðiÞ ð0Þ ð0Þ ð0Þ ð0Þ ð1Þ ðiÞ ðiÞ ðiÞ þ ðMxy nx þMyy ny  Mzz ny Þ ∑ ðM ðiÞ ¼ xx δεxx þM yy δεyy þ M xy δγ xy Þ þ M zz δεzz þ M xz δγ xz 2 Ω i¼0 ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ þ M ð0Þ yz δγ yz þ Mxx δχ xx þ Myy δχ yy þ Mzz δχ zz þ Mxy δχ xy # 1

þ ∑

i¼0

δV ¼ 

Z Ω

ðMðiÞ xz

δχ

ðiÞ ðiÞ xz þ Myz

δχ

ðiÞ yz Þ

ð101Þ

dx dy

ðf x δu þf y δv þqδwÞ dx dy

ð102Þ

where the superposed dot on a variable indicates time derivative, e.g., u_ ¼ ∂u=∂t, and mi (i ¼ 0; 1) are the mass moments of inertia Z mi ¼

h 2

 2h

ρðzÞi dz ) m0 ¼ ρh;

m2 ¼

ρh3 12

 h=2

and (f x , f y , q) are the distributed forces along the x, y, and z coordinates, respectively. The equations of motion are ! ð0Þ ð0Þ ∂Mð0Þ ∂M ð0Þ ∂M 1 ∂ ∂Mxz ∂2 u yz þ δu : xx þ xy þ ð105Þ þ f x ¼ m0 2 2 ∂y ∂x ∂y ∂x ∂y ∂t

δv :

∂M ð0Þ xy

δw :

∂x

þ

ð0Þ ∂M yy

∂y

ð0Þ



∂Myz 1 ∂ ∂Mð0Þ xz þ 2 ∂x ∂x ∂y

! þ f y ¼ m0

∂2 v ∂t 2

ð107Þ ! ð1Þ ð0Þ ð0Þ ∂M ð1Þ ∂M 1 ∂Mxy ∂Myy ∂Mð0Þ ð0Þ zz þ  δϕx : xx þ xy  ðM xz þ M ð0Þ ϕ Þ þ x zz 2 ∂x ∂y ∂x ∂y ∂y ! ð1Þ ð1Þ 2 ∂M 1 ∂ ∂Mxz ∂ ϕ yz þ ð108Þ þ ¼ m2 2 x 2 ∂y ∂x ∂y ∂t ∂M ð1Þ xy ∂x

þ

ð1Þ ∂M yy

∂y

ð0Þ  ðM ð0Þ yz þ M zz ϕy Þ  ð1Þ

∂Myz 1 ∂ ∂Mð1Þ xz þ  2 ∂x ∂x ∂y

! ¼ m2

ð0Þ

ð0Þ ∂Mxy ∂Mzz 1 ∂Mð0Þ xx þ  2 ∂x ∂y ∂x

∂ 2 ϕy ∂t 2

nx ð114Þ

The strain energy potential is assumed to be of the form 1h Ψ ¼ c11 ε2xx þ c22 ε2yy þ c33 ε2zz þ c12 εxx εyy þ c13 εxx εzz þ c23 εyy εzz 2 þa11 χ 2xx þ a22 χ 2yy þ a33 χ 2zz þ a44 χ 2yz þ a55 χ 2xz þ a12 χ xx χ yy þ a13 χ xx χ zz þ a23 χ yy χ zz þb11 εxx χ xx þ b12 εxx χ yy þ b13 εxx χ zz þ b21 εyy χ xx þ b22 εyy χ yy þ b23 εyy χ yy i þ b31 εzz χ xx þ b32 εzz χ yy þ b33 εzz χ yy

ð115Þ

where cij, aij and bij are the material parameters, some of which are the usual material parameters and the others are the length scale parameters (cij are the usual macroscale material coefficients (which can be expressed in terms of the engineering material constants), while aij and bij are the possible material length scale parameters). The normal stresses and the couple stresses can derived from the relations (i; j ¼ 1; 2; 3)

σ ij ¼

∂Ψ ; ∂εij

mij ¼

!

mxz ¼

∂Ψ ; ∂χ xz

∂Ψ ∂χ ij

myz ¼

ð116Þ

mxx ¼

∂Ψ ; ∂χ xx

myy ¼

ð117Þ

∂Ψ ; ∂χ yy

mzz ¼

∂Ψ ∂χ zz

ð118Þ

are the “bending” couple stresses. 4.4. Finite element model The use of the constitutive relations resulting from Eq. (116) in Eq. (104) yields the stress resultants in terms of strains and, subsequently, in terms of the generalized displacements (u, v, w, θx , θy , ϕx ϕy ). Therefore, the statement of the Hamilton's principle, Eq. (30), can be expressed in terms of the generalized displacements. Then the associated finite element model can be derived using the following approximations: m

ð109Þ

∂Ψ ∂χ yz

are the “drilling” couple stresses and

uðx; yÞ  ∑ Δj ψ ð1Þ j ðx; yÞ; 1

j¼1 12

The secondary (or force) variables of the theory are ! ð0Þ ∂Myz 1 ∂Mð0Þ ð0Þ xz þ δu : Mð0Þ n þ M n þ ny xx x xy y 2 ∂x ∂y

!

In particular, ð106Þ

    ð0Þ ∂M ð0Þ ∂ ∂w ð0Þ ∂w ð0Þ ∂ ∂w ð0Þ ∂w ð0Þ ∂M xz yz M xx þ M xy þ M xy þ M yy þ þ ∂x ∂x ∂y ∂y ∂x ∂y ∂x ∂y ! ! ð0Þ ð0Þ ∂Mð0Þ 1 ∂ ∂Mð0Þ 1 ∂ ∂Myy ∂Mxy ∂2 w xy xx þ þ  þ þ q ¼ m0 2 2 ∂y 2 ∂x ∂x ∂y ∂y ∂x ∂t

δϕy :

ð1Þ

ð1Þ ∂Myz 1 ∂Mxz þ 2 ∂x ∂y

1 ð0Þ ð0Þ ð1Þ  ðMxx nx þMxy ny  Mzz nx Þ 2

ð103Þ

ðkÞ M ðkÞ ij and Mij are the thickness-integrated stress resultants (with the following change in notation for brevity of writing the ðiÞ ðiÞ ðiÞ expressions: M ðiÞ 11 ¼ M xx , M 22 ¼ M yy , and so on) Z h=2 Z h=2 M ðkÞ ðzÞk σ ij dz; MðkÞ ðzÞk mij dz; k ¼ 0; 1 ð104Þ ij ¼ ij ¼  h=2

1 2

ð1Þ ð1Þ δϕy : Mð1Þ xy nx þ M yy ny  cz nx 

ð113Þ

m

vðx; yÞ  ∑ Δj ψ jð1Þ ðx; yÞ 2

j¼1

wðx; yÞ  ∑ ΔJ φJ ðx; yÞ 3

J¼1

ð110Þ

n

ϕx ðx; yÞ  ∑ Δ4j ψ ð2Þ j ðx; yÞ; j¼1

n

ϕy ðx; yÞ  ∑ Δ5j ψ jð2Þ ðx; yÞ

ð119Þ

j¼1

Please cite this article as: J.N. Reddy, A.R. Srinivasa, Non-linear theories of beams and plates accounting for moderate rotations and material length scales, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.06.003i

J.N. Reddy, A.R. Srinivasa / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

10

where ψ ð1Þ and ψ ið2Þ are the Lagrange interpolation functions and i φi are the Hermite interpolation functions (see Reddy [63] for details). Substitution of Eq. (119) into the statement of Hamilton's principle, with δK, δU, and δV given by Eqs. (10)–(102)?, results in the following finite element model: 2 11 3 M M12 M13 M14 M15 6 21 7 6M M22 M23 M24 M25 7  6 31 7 1 € 2 € 3 € 4 € 5 32 33 34 35 7 6M € Δ Δ Δ Δ M M M M Δ 6 7 6 41 7 4M M42 M43 M44 M45 5 M51

M52 M53 2 11 K K12 6 21 6K K22 6 31 6 þ6 K K32 6 41 4K K42 51 K K52

M54

M55

13

14

K

K

K23

K24

33

K34

43

K44 K54

K

K K53

38 1 9 8 1 9 K > > > > >Δ > >F > > > > > > 2> > 7> > > > > > > K25 7> F2 > Δ = = < < 7 3 35 7 3 ¼ F K 7 Δ > > > > 4> > 4> 7> > > > > > >F > K45 5> > > > > > Δ5 > > > ; : : 55 5; K F Δ

The beam and plate theories (and their finite element models) presented herein should be tested against some physical evidence of such material length scale effects. Computer implementation of these models and their relation to discrete peridynamics [65] are being studied by the authors [66]. Extensions of Eringen's model and the gradient elasticity model to shear deformable shells are awaiting attention (see Ke et al. [67] and Amabili and Reddy [68]).

Acknowledgments The authors gratefully acknowledge the support of this work by the National Science Foundation under Grant no. CMMI-1000790.

15

ð120Þ

where the mass coefficients Mαβ , stiffness coefficients Kαβ , and force coefficients Fα (for α, β ¼ 1, 2, 3, 4, 5) can be identified following the ideas presented in Section 3 for beams. Although algebraically complex, the expressions for these coefficients can be determined in terms of the interpolation functions and the material parameters aij, bij, and cij [see Eq. (115)].

5. Summary and remarks 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 (or length scale) is comparable to the size of the secondary constituents has spurned interest in recent years in developing structural theories that have the ability to capture material length scale effects. In this paper, beam and plate theories that account for such material length scales are presented using the gradient elasticity model developed by the authors [57]. The theories account for shear deformation and von Kármán non-linearity. The models developed herein contain, as special cases, those developed using the modified couple stress theory of Mindlin [26]. An equivalence between the modified couple stress theory and strain-gradient theory of the authors is also established. In addition, the finite element models of the derived beam and plate theories are also developed for completeness. Several comments are in order on the transverse normal strain εzz, which is included based on the “order-of-magnitude” assumption. Of course, such a term can be neglected in beams, especially if one-dimensional constitutive relations are to be invoked. How2 ever, as discussed herein, the transverse normal strain εzz ¼ ϕx is not due to Poisson's effect but it is due to a material length scale. The fact that it is always positive also indicates that it is not due to Poisson's effect but due to material length scale that contributes to “stiffening” of the material. It should be noted that there is no displacement variable associated with the stretching or shortening of a transverse normal (i.e., thickness stretch is not explicitly included in the theory) in the expansion of w. Consequently, there is no linear terms in εzz. The constitutive relations proposed herein are based on strain energy potential and the second law of thermodynamics. Since εzz is not a part of the conventional constitutive relations, it is not expected to contribute to the thickness locking in the associated finite element models; the term appears only when the corresponding material length scales are included. It is expected that these terms will only have secondary effects on the response.

References [1] M. Cadek, J.N. Coleman, K.P. Ryan, V. Nicolosi, G. Bister, A. Fonseca, J.B. Nagy, K. Szostak, F. Beguin, W.J. Blau, Reinforcement of polymers with carbon nanotubes: the role of nanotube surface area, Nano Lett. 4 (2004) 353–356. [2] M. Kaempgen, G.S. Duesberg, S. Roth, Transparent carbon nanotube coatings, Appl. Surf. Sci. 252 (2005) 425–429. [3] W.X. Chen, J.P. Tu, L.Y. Wang, H.Y. Gan, Z.D. Xu, X.B. Zhang, Tribological application of carbon nanotubes in a metal-based composite coating and composites, Carbon 41 (2003) 215–222. [4] G. Catalan, A. Lubk, A.H.G. Vlooswijk, E. Snoeck, C. Magen, A. Janssens, G. Rispens, G. Rijnders, D.H.A. Blank, B. Noheda, Flexoelectric rotation of polarization in ferroelectric thin films, Nat. Mater. 10 (2011) 963–967. [5] D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang, P. Tong, Experiments and theory in strain gradient elasticity, J. Mech. Phys. Solids 51 (2003) 1477–1508. [6] A.W. McFarland, J.S. Colton, Role of material microstructure in plate stiffness with relevance to microcantilever sensors, J. Micromech. Microeng. 15 (2005) 1060–1067. [7] A.C. Eringen, Nonlocal polar elastic continua, Int. J. Eng. Sci. 10 (1972) 1–16. [8] A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys. 54 (1983) 4703–4710. [9] A.C. Eringen, Nonlocal Continuum Field Theories, Springer-Verlag, New York, 2002. [10] A.C. Eringen, D.G.B. Edelen, On nonlocal elasticity, Int. J. Eng. Sci. 10 (1972) 233–248. [11] J.N. Reddy, Introduction to Continuum Mechanics with Applications, Cambridge University Press, New York, 2013. [12] J. Peddieson, G.G. Buchanan, R.P. McNitt, Application of nonlocal continuum models to nanotechnology, Int. J. Eng. Sci. 41 (1972) 305–312. [13] Y.Q. Zhang, G.R. Liu, J.S. Wang, Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression, Phys. Rev. B 70 (2004), pp. 205430-1–195404-6. [14] Y.Q. Zhang, G.R. Liu, X.Y. Xie, Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity, Phys. Rev. B 71 (2005), pp. 195404-1–195404-7. [15] Q. Wang, Wave propagation in carbon nanotubes via nonlocal continuum mechanics, J. Appl. Phys. 98 (2005), pp. 124301-1–124301-6. [16] Q. Wang, V.K. Varadan, Vibration of carbon nanotubes studied using nonlocal continuum mechanics, Smart Mater. Struct. 15 (2006) 659–666. [17] C.M. Wang, Y.Y. Zhang, S.S. Ramesh, S. Kitipornchai, Buckling analysis of microand nano-rods/tubes based on nonlocal Timoshenko beam theory, J. Phys. D 39 (2006) 3904–3909. [18] J.N. Reddy, Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci. 45 (2007) 288–307. [19] J.N. Reddy, Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates, Int. J. Eng. Sci. 48 (2010) 1507–1518. [20] J.N. Reddy, S.D. Pang, Nonlocal continuum theories of beams for the analysis of carbon nanotubes, Appl. Phys. Lett. 103 (2008), pp. 023511-1–023511-16. [21] L.J. Sudak, Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics, J. Appl. Phys. 94 (2003) 7281–7287. [22] S. Adali, Variational principles for transversely vibrating multi-walled carbon nanotubes based on nonlocal Euler–Bernoulli beam models, Nano Lett. 9 (2009) 1737–1741. [23] N. Challamel, C.M. Wang, The small length scale effect for a non-local cantilever beam: a paradox solved, Nanotechnology 19 (2008), pp. 3457031–345703-6. [24] N. Challamel, Z. Zhang, C.M. Wang, J.N. Reddy, Q. Wang, T. Michelitsch, B. Collet, On non-conservativeness of Eringens nonlocal elasticity in beam mechanics: correction from a discrete-based approach, Arch. Appl. Mech., http://dx.doi.org/10.1007/s00419-014-0862-x, published online, 23 May 2014. [25] J.N. Reddy, S. El-Borgi, Eringen's nonlocal theories of beams accounting for moderate rotations, Int. J. Engng. Sci. 82 (2014), 159-177. [26] R.D. Mindlin, Influence of couple-stresses on stress concentrations, Exp. Mech. 3 (1963) 1–7. [27] W.T. Koiter, Couple-stresses in the theory of elasticity: I and II, K. Ned. Akad. Wet. (R. Neth. Acad. Arts Sci.) B67 (1964) 17–44.

Please cite this article as: J.N. Reddy, A.R. Srinivasa, Non-linear theories of beams and plates accounting for moderate rotations and material length scales, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.06.003i

J.N. Reddy, A.R. Srinivasa / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎ [28] R.A. Toupin, Theories of elasticity with couple-stress, Arch. Ration. Mech. Anal. 17 (1964) 85–112. [29] R.D. Mindlin, Second gradient of strain and surface-tension in linear elasticity, Int. J. Solids Struct. 1 (1965) 417–438. [30] A.E. Green, P.M. Naghdi, R.S. Rivlin, Directors and multipolar displacements in continuum mechanics, Int. J. Eng. Sci. 2 (1965) 611–620. [31] A.E. Green, P.M. Naghdi, Micropolar and director theories of plates, Q. J. Mech. Appl. Math. 20 (1967) 183–199. [32] A.E. Green, P.M. Naghdi, A unified procedure for construction of theories of deformable media. II. Generalized continua, Proc. R. Soc. Lond. Ser. A: Math. Phys. Sci. 448 (1995) 357–377. [33] S. Papargyri-Beskou, K.G. Tsepoura, D. Polyzos, D.E. Beskos, Bending and stability analysis of gradient elastic beams, Int. J. Solids Struct. 40 (2003) 385–400. [34] J.F.C. Yang, R.S. Lakes, Experimental study of micropolar and couple stress elasticity in compact bone in bending, J. Biomech. 15 (1982) 91–98. [35] R. Maranganti, P. Sharma, A novel atomistic approach to determine straingradient elasticity constants: tabulation and comparison for various metals, semiconductors, silica, polymers and the(ir) relevance for nanotechnologies, J. Mech. Phys. Solids 55 (2007) 1823–1852. [36] S.K. Park, X.L. Gao, Bernoulli–Euler beam model based on a modified couple stress theory, J. Micromech. Microeng. 16 (2006) 2355–2359. [37] S.K. Park, X.L. Gao, Variational formulation of a modified couple stress theory and its application to a simple shear problem, Z. Angew. Math. Phys. 59 (2008) 904–917. [38] F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, Int. J. Solids Struct. 39 (2002) 2731–2743. [39] H.M. Ma, X.L. Gao, J.N. Reddy, A microstructure-dependent Timoshenko beam model based on a modified couple stress theory, J. Mech. Phys. Solids 56 (2008) 3379–3391. [40] H.M. Ma, X.L. Gao, J.N. Reddy, A nonclassical Reddy–Levinson beam model based on a modified couple stress theory, Int. J. Multiscale Comput. Eng. 8 (2010) 167–180. [41] H.M. Ma, X.L. Gao, J.N. Reddy, A non-classical Mindlin plate model based on a modified couple stress theory, Acta Mech. 220 (2011) 217–235. [42] J.N. Reddy, Microstructure-dependent couple stress theories of functionally graded beams, J. Mech. Phys. Solids 59 (2011) 2382–2399. [43] J.N. Reddy, Jessica Berry, Modified couple stress theory of axisymmetric bending of functionally graded circular plates, Compos. Struct. 94 (2012) 3664–3668. [44] J.V. Araujo dos Santos, J.N. Reddy, Vibration of Timoshenko beams using nonclassical elasticity theories, Shock Vib. 19 (2012) 251–256. [45] J.V. Araujo dos Santos, J.N. Reddy, Free vibration and buckling analyses of Timoshenko beams with couple stress and Poisson's effects, Int. J. Appl. Mech. 4 (2012), pp. 1250056-1–1250056-28. [46] J.N. Reddy, J. Kim, A nonlinear modified couple stress-based third-order theory of functionally graded plates, Compos. Struct. 94 (2012) 1128–1143. [47] L.L. Ke, Y.S. Wang, Z.D. Wang, Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory, Compos. Struct. 94 (2012) 2038–2047. [48] L.L. Ke, Y.S. Wang, Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory, Smart Mater. Struct. 21 (2012), pp. 025018-1–025018-12.

11

[49] X.L. Gao, J.X. Huang, J.N. Reddy, A non-classical third-order shear deformation plate model based on a modified couple stress theory, Acta Mecc. 224 (2013) 2699–2718. [50] C.M.C. Roque, D.S. Fidalgo, A.J.M. Ferreira, J.N. Reddy, A study of a microstructure-dependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method, Compos. Struct. 96 (2013) 532–537. [51] C.M.C. Roque, A.J.M. Ferreira, J.N. Reddy, Analysis of Mindlin micro plates with a modified couple stress theory and a meshless method, Appl. Math. Model. 37 (2013) 4626–4633. [52] A. Arbind, J.N. Reddy, Nonlinear analysis of functionally graded microstructure-dependent beams, Compos. Struct. 98 (2013) 272–281. [53] M. Simsek, J.N. Reddy, Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory, Int. J. Eng. Sci. 64 (2013) 37–53. [54] M. Simsek, J.N. Reddy, A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory, Compos. Struct. 101 (2013) 47–58. [55] J. Kim, J.N. Reddy, Analytical solutions for bending, vibration, and buckling of FGM plates using a couple stress-based third-order theory, Compos. Struct. 103 (2013) 86–98. [56] A. Arbind, J.N. Reddy, A. Srinivasa, Modified couple stress-based third-order theory for nonlinear analysis of functionally graded beams, Lat. Am. J. Solids Struct. 11 (2014) 459–487. [57] A.R. Srinivasa, J.N. Reddy, A model for a constrained, finitely deforming, elastic solid with rotation gradient dependent strain energy, and its specialization to von Kármán plates and beams, J. Mech. Phys. Solids 61 (2013) 873–885. [58] J.N. Reddy, A small strain and moderate rotation theory of elastic anisotropic plates, J. Appl. Mech. 54 (1987) 623–626. [59] R. Schmidt, J.N. Reddy, A refined small strain and moderate rotation theory of elastic anisotropic shells, J. Appl. Mech. 55 (1988) 611–617. [60] J.N. Reddy, Patrick Mahaffey, Generalized beam theories accounting for von Kármán nonlinear strains with application to buckling and post-buckling, J. Coupled Syst. Multiscale Dyn. 1 (2013) 120–134. [61] J.N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, 2nd edition, John Wiley & Sons, New York, 2002. [62] J.N. Reddy, An Introduction to the Finite Element Method, McGraw–Hill, New York, 2006. [63] J.N. Reddy, An Introduction to Nonlinear Finite Element Analysis, 2nd edition, Oxford University Press, Oxford, UK, 2015. [64] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells, Theory and Analysis, 2nd edition, CRC Press, Boca Raton, FL, 2004. [65] S.A. Silling, Reformulation of elasticity theory for discontinuities and longrange forces, J. Mech. Phys. Solids 48 (2000) 175–209. [66] J.N. Reddy, A. Srinivasa, A. Arbind, P. Khodabakhshi, On nonlocal elasticity and peridynamics with applications to beams and plates, in: Proceedings of the 6th ECCOMAS Thematic Conference on Smart Structures and Materials (SMART2013), 24–26 June 2013, Politecnico di Torino, Torino, Italy. [67] L.L. Ke, Y.S. Wang, J.N. Reddy, Thermo-electro-mechanical vibration of sizedependent piezoelectric cylindrical nanoshells under various boundary conditions, Compos. Struct., 116 (2014), 626-636. [68] M. Amabili, J.N. Reddy, A new non-linear higher-order shear deformation theory for large-amplitude vibrations of laminated doubly curved shells, Int. J. Non-Linear Mech. 45 (2010) 409–418.

Please cite this article as: J.N. Reddy, A.R. Srinivasa, Non-linear theories of beams and plates accounting for moderate rotations and material length scales, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.06.003i