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Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity
Physica E 88 (2017) 194–200
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Physica E journal homepage: www.elsevier.com/locate/physe
Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity
MARK
⁎
A. Norouzzadeh, R. Ansari
Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran
A R T I C L E I N F O
A BS T RAC T
Keywords: Integral constitutive equations Pure numerical approach Nano-scaled Timoshenko beam Finite element analysis Boundary conditions Paradox
Stress-strain relation in Eringen's nonlocal elasticity theory was originally formulated within the framework of an integral model. Due to difficulty of working with that integral model, the differential model of nonlocal constitutive equation is widely used for nanostructures. However, paradoxical results may be obtained by the differential model for some boundary and loading conditions. Presented in this article is a finite element analysis of Timoshenko nano-beams based on the integral model of nonlocal continuum theory without employing any simplification in the model. The entire procedure of deriving equations of motion is carried out in the matrix form of representation, and hence, they can be easily used in the finite element analysis. For comparison purpose, the differential counterparts of equations are also derived. To study the outcome of analysis based on the integral and differential models, some case studies are presented in which the influences of boundary conditions, nonlocal length scale parameter and loading factor are analyzed. It is concluded that, in contrast to the differential model, there is no paradox in the numerical results of developed integral model of nonlocal continuum theory for different situations of problem characteristics. So, resolving the mentioned paradoxes by means of a purely numerical approach based on the original integral form of nonlocal elasticity theory is the major contribution of present study.
1. Introduction Nowadays, it is quite clear that classical continuum theories are unable to give precise and comprehensive predictions of the phenomena at micro- and nano-scale. Size-dependency, effects of surface energies and dispersion characteristics of wave propagation are examples of important issues which are neglected in the classical continuum mechanics [1–5]. It refers to the fact that there is no internal material length parameter in the classical theories and they are scale-free. As an example on this area, one can mention the recent paper of Carcaterra [6] in which an equivalent theory of Euler beam is proposed to discover the quantum effects on the flexural vibration of nano-beams. Although the main body of attempts began in 1960's, the development of higher-order/size-dependent elasticity theories are considered as the contribution of many researchers over more than a century. From fundamental studies on this field, one can mention papers by Mindlin [7–10], Toupin [11,12], Kroner [13,14] and works by Eringen [15–18]. Further investigations in this field lead to the improvement of introduced theories and as a result, the well-known nonlocal elasticity [17–19], couple stress/strain gradient [7,11,20,21] and modified couple stress/strain gradient [22,23] theories have been employed in
⁎
Corresponding author.
http://dx.doi.org/10.1016/j.physe.2017.01.006 Received 12 October 2016; Accepted 3 January 2017 Available online 16 January 2017 1386-9477/ © 2017 Elsevier B.V. All rights reserved.
literature extensively. Moreover, it is reported that the effects of surface layers are considerable in micro/nano and thin structures which ratios of surface to volume are high [24,25]. Among proposed models, the surface elasticity theory of Gurtin and Murdoch [26,27] is attracted a lot of attention. Javili et al. [28] showed that, in contrast to the surface elasticity, consideration of surface and curve energies require the first and second derivatives of the deformation gradient, respectively. Thickness of surface and the flexural resistance are taken into account in their presented higher-gradient surface elasticity theory. Nonlocal elasticity theory indicates that the stress component of a body at reference point depends on the strain field on the entire domain (not only that point) which is against on what classical/local theories are based on. Eringen formulated this nonlocal stress-strain relationship concept by means of integral constitutive equation.
σ ( x) =
∫v
k ( x −x , κ ) Cε (x) dv (x)
(1)
Because of the adversity of dealing with integro-partial differential equations, differential form of above relation is presented for a specific kind of kernel which is assumed to be a Green function of linear differential operator [18].
(1 − κ 2∇2 ) σ = Cε
(2)
Physica E 88 (2017) 194–200
A. Norouzzadeh, R. Ansari
Here, it is worth mentioning that nonlocality concept is also seen in the so-called peridynamics continuum theory. According to [29,30], the modern peridynamics theory [31] was perceived and even formulated in Piola's works. This theory presents a nonlocal model of force interactions; i.e., the force applied on a material particle of a continuum depends on the deformation of particles at a finite difference. As the original model of nonlocal elasticity, the idea of force interactions in peridynamics is implemented by means of an integral operator [32]. This integral form and the corresponding integro-differential equations of motion provide an appropriate tool to analyze the problems with singularities and discontinuity which attract a lot of attention in fracture mechanics [33,34]. In addition of conceiving peridynamics, dell’Isola and co-authors pointed out that Piola's works in 19th century were the cornerstone of the concepts of nonlocal and higher-order gradient continuum theories. For more information, the reader can follow the detailed discussion in [29,30]. The notable point is that due to the incapability of Cauchy stress formulation to model the internal action of nonlocal continuum, the generalized concept of stress is developed in nonlocal elasticity theories. dell’Isola et al. [35] also showed that there is a need for at least Nth gradient continuum theory to generalize the Cauchy representation of contact interactions between sub-bodies. They utilized the principle of virtual powers (or virtual works) to generalize the Cauchy formulations for Nth gradient Piola continuum with the corresponding suitable boundary conditions [29]. Trace of literature shows that the differential form of nonlocal elasticity is widely used to detect size-dependent characteristics of small-scale structures in different static and dynamic problems. For instance, Ansari et al. [36] studied the vibration behaviour of singlelayered graphene sheets (SLGSs) based on the nonlocal elasticity and Mindlin plate theories. The Molecular Dynamic (MD) simulations are also established to obtain the appropriate values of nonlocal parameters corresponding to different boundary conditions. Narendar and Gopalakrishnan [37] investigated the thermo-mechanical pre-buckling characteristics of single-walled carbon nanotubes (SWCNTs) resting on Winkler medium. They incorporate the nonlocal continuum and Timoshenko beam theories to achieve the critical buckling temperatures. Also, Rahmani and Pedram [38] presented the effects of small scales in functionally graded materials (FGMs) based on the nonlocal Timoshenko beam model. Simsek [39] employed the nonlocal strain gradient theory (NLSGT) with von-Karman nonlinearities to propose a size-dependent FGM Euler-Bernoulli beam model. The Galerkin's method and a novel Hamiltonian approach are used to find a closedform solution of the free vibration problem. Despite of the wide usage of differential form of nonlocal elasticity theory, there are some contradictions in predicted results of this model in certain occasions. It is reported [40–42] that in cantilever beams/ plates, a surprising stiffening behaviour is seen with increase of nonlocal parameter. This is while, the softening responses are observed for other boundary conditions and higher vibrational modes of a cantilever body [43]. Also it is clear that in case of point loading, there is no impacts of material length scale i.e. no nonlocal and sizedependent effects. To tackle this issue, some attempts are made in recent years including the studies of Challamel and co-workers [42– 44], Polizzotto [45] and paper of Pisano and Fuschi [46]. In the present paper, Eringen's nonlocal elasticity theory with integral form of constitutive relations is applied to analyze the mechanical characteristics of nano-beams. Integral model of nonlocal continuum theory is used with the Timoshenko beam model to detect the small scale effects without the mentioned paradoxical behaviour. The content of paper is organised as follows: the governing equations of motion are obtained in matrix-vector form of both integral and differential models in Sections 2 and 3, respectively. The matrix-vector form of representation provides a suitable tool for the implementation of the finite element method (FEM). The minimum total potential energy principle is adopted to derive the equations of static bending
Fig. 1. Schematic view of nano-beam.
problem of nano-beams. Section 4 is dedicated to discuss about numerical case studies in which the responses of nano-beams predicted by local elasticity, integral and differential models of nonlocal theories are compared for different boundary and load conditions. As it is expected, the present integral model does not suffer from inconsistence results of differential form of nonlocal elasticity theory. 2. Integral model of nonlocal elasticity A schematic view of the nano-beam under consideration is shown in Fig. 1. It is assumed that the axial direction of nano-beam (with length L ) is located on the x1 axis; also width b and thickness h directions are x2 and x3 axes, respectively. Material properties are consist of Young modulus E , Poisson's ratio ν and internal length parameter a . Furthermore, the transverse distributed static load P is applied on nano-beam. 2.1. Derivation of formulation Based on the Eringen's nonlocal elasticity theory, the constitutive relations of beam-type structures are in the general form as follows
tij (x , x3) = λδij εkk (x , x3)+2μεij (x , x3) = Cijkl εkl (x , x3)
σij (x, x3) =
(3)
∫x k ( x−x , κ ) tij (x , x3) dx
(4)
in which, tij and σij are the components of local and nonlocal stress tensors, respectively and εij expresses the strain tensor component. δij denote Kronecker delta and Cijkl is the fourth order elastic tensor consists of λ and μ , the Lame's classical constants, in which λ = Eν /1 − ν 2,μ = E /2(1 + ν ). Also, x = x1 signals the reference point in which the stress component σij is related to all points on domain represented by x =x1. The nonlocal modulus or kernel k is a function of nonlocal parameter κ and neighborhood distance x − x in which κ = e0 a and e0 is a material constant. In the present study, it is assumed that nano-beam is made from the linear isotropic materials. In this manner, the strain energy of beam and its variation can be written, respectively, as
1 2 1 = 2
Πs =
δ Πs =
∫A ∫x σij (x, x3) εij (x, x3) dxdA ⎞
⎛
∫A ∫x ⎜⎝∫x k ( x−x , κ ) tij (x , x3) dx ⎟⎠ εij (x, x3) dxdA
∫A ∫x σij (x, x3) δεij (x, x3) dxdA
=
⎛
(5) ⎞
∫A ∫x ⎜⎝∫x k ( x−x , κ ) tij (x , x3) dx ⎟⎠ δεij (x, x3) dxdA (6)
Now, displacement field of Timoshenko beam is introduced as
u1 (x , x3) = u (x )+x3 ψ (x ),
u3 (x , x3) = w (x )
(7)
where u and w are the axial and transverse deflections, respectively and ψ shows the rotation angle of the cross section with respect to the vertical direction. Above relations can be written in following form
u = Pq,
u (x , x3) = [ u1 u3 ]T
⎡1 0 x3 ⎤ P (x3) = ⎢ ⎥ , q (x ) = [ u w ψ ]T ⎣0 1 0 ⎦
(8)
The constitutive relation of Eq. (4) is expressed in matrix-vector form as 195
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σ (x, x3) =
∫x
k ( x−x , κ ) Cε (x , x3) dx
(9)
⎡ λ +2μ 0 ⎤ C=⎢ ⎥, ks μ ⎦ ⎣ 0
⎡ σxx (x, x3)⎤ σ (x, x3) = ⎢ ⎥, ⎣ σxz (x, x3) ⎦
⎡ εxx (x , x3) ⎤ ε (x , x3) = ⎢ ⎥ ⎣ 2εxz (x , x3)⎦
(23)
∼ ∼ ∼ N (ξ ) = I3 ⊗ N (ξ ), N (ξ ) = [ N1 (ξ ) … Nn (ξ )] N (ξ ) = I3 ⊗ N (ξ ), ∼ N (ξ ) = [ N1 (ξ ) … Nn (ξ )]
(24)
in which, d and d denote the nodal values in ξ - and ξ -coordinates respectively. Also, Ni is the shape function of ith node, ⊗ signals the Kronecker delta product and Ip is the p -by- p unit matrix. So, the strain vectors are achieved as
(10) where, σ is the stress vector at reference point. C indicates the elastic matrix and ε is the strain vector of beam. Also, ks denotes shear correction factor. By considering the strain-displacement relationship as
1 ⎛ ∂u ∂uj ⎞ ⎟ εij = ⎜ i + 2 ⎝ ∂xj ∂xi ⎠
q = N (ξ ) d , q = N (ξ ) d
ε (ξ ) = PB (ξ ) d,
ε (ξ ) = PB (ξ ) d,
(25)
δ ε (ξ ) = PB (ξ ) δ d
(26)
B (ξ ) = EN (ξ ), B (ξ ) = EN (ξ ) (11)
With respect to Eq. (18), the stress vector of an element is determined
one has (12)
ε (x , x3) = P (x3) Eq (x )
σe = Sd, S =
∫ξ k CPBJ dξ
(27)
in which the matrix P is determined in Eq. (8) and
where J denotes Jacobian in ξ -coordinates. In this stage, it is useful to define the following resultant
⎡ ∂/∂x 0 0 ⎤ E = ⎢ 0 ∂/∂x 1 ⎥ ⎢⎣ ⎥ 0 0 ∂/∂x ⎦
(13)
ˆe = N
Also, Eq. (5) and Eq. (6) can be rewritten as
Πs =
1 2
δ Πs =
ˆ , ˆ e = Sd N
∫A ∫x εT (x, x3) σ (x, x3) dxdA
(15)
δ ε (x, x3) = P (x3) Eδ q (x )
E = E ∂/∂x →∂/∂x,
ˆ, S→,Sˆ →
Πs =
k ( x−x , κ ) CPEqdx
∫x
⎞ k CPEqdx ⎟ dxdA ⎠
⎛ ⎜ ⎝
⎞ k CPEqdx ⎟ dxdA ⎠
∫A ∫x
⎛ qTETPT ⎜ ⎝
∫A ∫x
δ qTETPT
1 2
δ Πs =
∫x
∫x
(17)
Πs, e =
δ Πw =
∫x
F = [ Fu Fw Fψ ]T
δ qTFdx
=
∫A PTCPdA
(29)
d→
(30)
∫A ∫ξ dTBTPTJdξdA
∫ξ
=
1 2
ˆ Jdξ ∫ξ dTBT
(31)
ˆ Jdξ = δ dTKs, e δ dTBT
(32)
(18)
where J is the Jacobian in ξ -coordinates and the stiffness matrix of element Ks, e is achieved as
(19)
Ks, e =
∫ξ BTˆ Jdξ
(33)
Also, the external work of element and its variation from Eq. (21) and Eq. (22) are captured as
(20)
Furthermore, the external work applied on nano-beam is achieved
∫x qTFdx,
1 2
δ Πs, e =
δ Πw, e = δ dTFe Fe =
as
1 Πw = 2
∫ξ k CBJ dξ , C
According to Eq. (19) and Eq. (20), the strain energy of element and its variation are presented as
(16)
So, the stress vector, strain energy and its variation are obtained in terms of displacement field components as follows
σ (x, x3) =
(28)
By performing the assemblage process in ξ -coordinates, the stress components at reference point x are obtained. So, one has the total matrices and vectors as
in which the strain vector at reference point ε is
q (x ) = [ u (x ) w (x ) ψ (x )]T ,
Sˆ =
(14)
∫A ∫x δ εT (x, x3) σ (x, x3) dxdA
⎡ εxx (x, x3) ⎤ ε (x, x3) = ⎢ ⎥ = P (x3) Eq (x ), ⎣ 2εxz (x, x3)⎦
∫A PTσe dA
∫ξ
N TFJdξ
(34)
After assemblage procedure in ξ -coordinates, the assembled matrices and vectors are
(21)
d→ ,
(22)
Ks, e→s ,
Fe→
(35)
where Fu and Fw exhibit the distributed load on axial and lateral directions, respectively, and Fψ is the distributed moment.
and by choosing the same nodes for FEM local coordinate systems (ξ = ξ ), one has = . So, the strain energy and external work of Timoshenko nano-beam are specified as
2.2. Finite element analysis
δ Πs = δ T s δ Πw =
Now, the Finite Element Method (FEM) is adopted to solve the integral formulation of nonlocal beam elasticity. As it is observed from foregoing formulation, at a given point of x , the stress tensor should be calculated in entire domain of x . So, unlike the finite element analysis of conventional problems, two procedures of assemblage are required to determine the strain energy of structure. The local coordinate system of elements corresponding to x and x domains are denoted by ξ and ξ , respectively. In an element, the displacement components of beam are interpolated by appropriate shape functions as
(36)
δ T
Employing the minimum δ (Πs−Πw) = 0 leads to
(37) total
potential
s =
energy
principle (38)
3. Differential model of nonlocal elasticity It would be helpful to present the governing differential equations of motion and the corresponding finite element formulation of the 196
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Ks, e d = Fe
conventional differential model of nonlocal elasticity theory. This makes it possible to directly compare the different natures of two models of nonlocal theory.
→
s =
(55)
4. Selected results 3.1. Derivation of formulation In this section, some numerical examples have been brought to investigate the results predicted by the integral and differential models of nonlocal elasticity theory. Also, the effects of different important parameters on the bending responses of nano-beams are analyzed through the following discussion. It is known that inconsistent results appear by employing differential model of nonlocal elasticity in analysis of cantilever beams. The main paradoxes are: 1) deflection does not affected by nonlocal parameter in case of point loading; 2) the stiffness of structure is increased by nonlocal parameter in distributed loading conditions. According to this important issue, the present study aims to resolve the above-mentioned paradoxical behaviors by implementation of the original integral model of nonlocal continuum theory. In this regard, a purely numerical approach with no simplification assumptions, is considered herein. First, some comparisons are made to validate the proposed formulation and solution methodology of present paper. Ref. [47] studied the bending problem of Euler-Bernoulli beam with using the integral form of nonlocal elasticity. They presented the analytical solution of integral form of nonlocal elasticity with specific kernel function in which satisfied the boundary and loading conditions proposed by [48]. To validate present investigation based on finite element analysis of integral formulation of nonlocal nano-scale Timoshenko beam theory, the analysis of cantilever beam is carried out with assumption of L / h = 25, ν = 0, ks = 5/6 and results are compared with those reported by Ref. [47]. Also, Fu = Fψ = 0, Fw = P and three types of transverse loading conditions including a) point load, b) uniform distributed and c) triangular loads are considered. Dimensionless lateral deflection is defined as W = wEI / qL4 where I denotes the second moment of inertia and q has the following values: a) q = P / L , b) q = P , c) q = L . The kernel function is considered as [47]
Eringen stated that for a specific kernel function, the integral constitutive relations of Eq. (4) can be transformed to the differential model as
(1 − μ ∇2 ) σij = λδij εkk +2μεij = Cijkl εkl
(39)
or in matrix-vector form as
σ = [ σxx σxz ]T
(1 − μ ∇2 ) σ = Cε,
(40)
∇2
κ2
∂ 2/∂x 2 .
= in which μ = and Laplacian operator in 1D problem is By taking into consideration of Eq. (10) and Eq. (12), the resultant of stress vector is written as ˆ = N
∫A PTσ
dA
(41)
ˆ = CEq (1 − μ ∇2 ) N
(42)
where C is introduced in Eq. (29). So, the strain energy, external work and their variations have the following form
Πs =
1 2
Πw =
∫
∫A ∫x εTσdxdA
=
1 2
∫A ∫x qTETPTσdxdA
=
1 2
∫x qTETNˆ dx
(43)
qTFdx
x
(44)
δ Πs =
∫x δ qTETNˆ dx
(45)
δ Πw =
∫
δ qTFdx (46)
x
By means of minimum total potential energy principle, one has
∫x (δ qTETNˆ −δ qTF) dx
=
∫x (δ qTETCEq−(1 − μ ∇2) δ qTF) dx
=0
(47)
k ( x−x , κ ) =
and the variation of equivalent energy terms are defined as
δ Πs =
∫x
δ qTETCEqdx
δ Πw =
∫
(1 − μ ∇2 ) δ qTFdx
From Fig. 2, it is seen that the above mentioned two paradoxes are solved by applying the integral model of nonlocal continuum theory. In all three cases, the maximum deflection of cantilever nano-beam is increased significantly with increase of nonlocal parameter. In addition of the trend of changes, a very good agreement is observed with reported results of Ref. [47]. For rest of paper, it is assumed that nano-beam is made of Al in which is subjected to the uniform distributed load. The material and geometrical properties are as follows
(48)
(49)
x
3.2. Finite element analysis The conventional finite element implementation is carried out to detect the equations of motion of nano-scale Timoshenko beam based on the differential model of nonlocal elasticity. So,
∼ N = I3 ⊗ N,
E = 68. 5GPa,
∼ N = [ N1 … Nn ]
(52)
From Eq. (48) and Eq. (49), one has
δ Πs, e = δ dTKs, e d ,
δ Πw, e = δ dTFe , Fe =
Ks, e =
∫ ξ
∫ξ
BT CBJdξ
ks = 5/6,
L /h = 12,
b /h = 1
and dimensionless deflection is w = Also, the nonlocal parameter is fixed at κ = 1nm [49]. Here, some explanations are provided about the computational features of present approach. As it is stated further, the assemblage process of the stress resultant must be carried out in each point of the Gaussian quadrature scheme. So, if m times function evaluation is needed in a conventional finite element analysis (like differential model), this number is raised to m2 times in assemblage procedure of integral model. It is clear that the computational run time of this approach is higher than differential model which came back to different nature of constitutive relations. Besides of what above, the “dynamic rate of convergence” of elements is another new aspect of this approach. It is related to the neighborhood distance concept in kernel function such that with increase of structure scales, higher number of elements should be
(51)
ε = PBd, B = EN
ν = 0. 35,
WEI / PL4 .
(50)
q = Nd
1 − x−x e κ 2κ
(53)
(1 − μ ∇2 ) N TFJdξ (54)
and finally by adopting the assemblage process, the following set of equations are achieved 197
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Fig. 2. Comparison of integral and differential models of nonlocal elasticity in predicting the maximum deflection of cantilever nano-beam, a) point load at end; b) uniform distributed load; c) triangular distributed load.
employed to cover the domain properly and give small values of x − x . To show this issue, convergence trends are tabulated in Table 1 corresponding to different thickness-to-nonlocal parameter ratios. In present study, the higher order elements (with 8 nodes) are applied in which are able to satisfy appropriately the neighboring distance of points as well as being locking free. Depicted in Figs. 3–5 are variation of maximum deflection of cantilever, clamped-pinned and pinned-pinned nano-beams with thickness-to-nonlocal parameter ratios, respectively. For comparison purpose, the graphs of predicted results of classical/local continuum
Table 1 Convergence study of finite element analysis with different values of thickness-tononlocal parameter ratios. h κ
5 10 15 20
Number of elements 20
40
60
80
100
120
0.1158 0.1071 0.0989 0.0905
0.1174 0.1123 0.1089 0.1055
0.1177 0.1134 0.1111 0.1092
0.1179 0.1137 0.1119 0.1108
0.1179 0.1139 0.1121 0.1113
0.1179 0.1139 0.1122 0.1115
Fig. 3. Maximum deflection of cantilever nano-beam versus thickness ratios based on the integral and differential nonlocal elasticity as well as local elasticity.
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Fig. 4. Maximum deflection of clamped-pinned nano-beam versus thickness ratios based on the integral and differential nonlocal elasticity as well as local elasticity.
Fig. 5. Maximum deflection of pinned-pinned nano-beam versus thickness ratios based on the integral and differential nonlocal elasticity as well as local elasticity.
bending of Timoshenko nano-beams. The integral model of nonlocal continuum theory was employed to study the small scales effects. The formulations of problem for integral and differential models were obtained in the matrix-vector form of representation. Then, finite element analysis was carried out to solve the governing equations of motion. Some numerical examples for various boundary conditions and different loadings were given to realize the differences of results of two models. It was reported that differential model of nonlocal continuum theory leads to the unexpected inconsistent results in certain types of boundary and loading conditions. But, it was observed that there is no paradox in the behaviour of cantilever nano-beams with the implementation of integral model of nonlocal elasticity theory. The softening responses were reported by the increase of nonlocal parameter, in all types of loading conditions. Also, it was concluded that the effects of material length parameters are negligible in nano-structures with larger scales. Hence, present work showed that the integral model of nonlocal elasticity can be analyzed in the general case without simplifying assumptions and predicts no contradictory results. This solution strategy is capable of being straightforwardly used in similar problems.
and differential model of Eringen's nonlocal elasticity are presented, too. It is observed that, in all cases, by increase of nano-beam scales, reported responses of nonlocal theories tend to converge to those of predicted by the classical elasticity theory. Fig. 3 demonstrates the existing paradox of differential model in cantilever beams, again. It is found that the convergence of integral model to the local elasticity theory occurred at higher ratios in compare with differential model. Also, it can be concluded that small-scale effects in present model are significantly higher than predicted by differential model. Fig. 4 demonstrates that employing both integral and differential forms of nonlocal elasticity in analysis of clamped-pinned nano-beams lead to similar trend of changes. But, differential model underestimates the maximum deflection of nan-beam as well as the ratio of convergence to the local theory. Fig. 5 shows that the integral and differential models of nonlocal continuum theories produce approximately the same responses in analysis of pinned-pinned nano-scale beams. Consequently, there is no remarkable error to use differential model in analyzing nano-beams with this type of boundary conditions. 5. Conclusion Investigated in this paper, was a size-dependent analysis of static 199
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200
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Physica E journal homepage: www.elsevier.com/locate/physe
Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity
MARK
⁎
A. Norouzzadeh, R. Ansari
Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran
A R T I C L E I N F O
A BS T RAC T
Keywords: Integral constitutive equations Pure numerical approach Nano-scaled Timoshenko beam Finite element analysis Boundary conditions Paradox
Stress-strain relation in Eringen's nonlocal elasticity theory was originally formulated within the framework of an integral model. Due to difficulty of working with that integral model, the differential model of nonlocal constitutive equation is widely used for nanostructures. However, paradoxical results may be obtained by the differential model for some boundary and loading conditions. Presented in this article is a finite element analysis of Timoshenko nano-beams based on the integral model of nonlocal continuum theory without employing any simplification in the model. The entire procedure of deriving equations of motion is carried out in the matrix form of representation, and hence, they can be easily used in the finite element analysis. For comparison purpose, the differential counterparts of equations are also derived. To study the outcome of analysis based on the integral and differential models, some case studies are presented in which the influences of boundary conditions, nonlocal length scale parameter and loading factor are analyzed. It is concluded that, in contrast to the differential model, there is no paradox in the numerical results of developed integral model of nonlocal continuum theory for different situations of problem characteristics. So, resolving the mentioned paradoxes by means of a purely numerical approach based on the original integral form of nonlocal elasticity theory is the major contribution of present study.
1. Introduction Nowadays, it is quite clear that classical continuum theories are unable to give precise and comprehensive predictions of the phenomena at micro- and nano-scale. Size-dependency, effects of surface energies and dispersion characteristics of wave propagation are examples of important issues which are neglected in the classical continuum mechanics [1–5]. It refers to the fact that there is no internal material length parameter in the classical theories and they are scale-free. As an example on this area, one can mention the recent paper of Carcaterra [6] in which an equivalent theory of Euler beam is proposed to discover the quantum effects on the flexural vibration of nano-beams. Although the main body of attempts began in 1960's, the development of higher-order/size-dependent elasticity theories are considered as the contribution of many researchers over more than a century. From fundamental studies on this field, one can mention papers by Mindlin [7–10], Toupin [11,12], Kroner [13,14] and works by Eringen [15–18]. Further investigations in this field lead to the improvement of introduced theories and as a result, the well-known nonlocal elasticity [17–19], couple stress/strain gradient [7,11,20,21] and modified couple stress/strain gradient [22,23] theories have been employed in
⁎
Corresponding author.
http://dx.doi.org/10.1016/j.physe.2017.01.006 Received 12 October 2016; Accepted 3 January 2017 Available online 16 January 2017 1386-9477/ © 2017 Elsevier B.V. All rights reserved.
literature extensively. Moreover, it is reported that the effects of surface layers are considerable in micro/nano and thin structures which ratios of surface to volume are high [24,25]. Among proposed models, the surface elasticity theory of Gurtin and Murdoch [26,27] is attracted a lot of attention. Javili et al. [28] showed that, in contrast to the surface elasticity, consideration of surface and curve energies require the first and second derivatives of the deformation gradient, respectively. Thickness of surface and the flexural resistance are taken into account in their presented higher-gradient surface elasticity theory. Nonlocal elasticity theory indicates that the stress component of a body at reference point depends on the strain field on the entire domain (not only that point) which is against on what classical/local theories are based on. Eringen formulated this nonlocal stress-strain relationship concept by means of integral constitutive equation.
σ ( x) =
∫v
k ( x −x , κ ) Cε (x) dv (x)
(1)
Because of the adversity of dealing with integro-partial differential equations, differential form of above relation is presented for a specific kind of kernel which is assumed to be a Green function of linear differential operator [18].
(1 − κ 2∇2 ) σ = Cε
(2)
Physica E 88 (2017) 194–200
A. Norouzzadeh, R. Ansari
Here, it is worth mentioning that nonlocality concept is also seen in the so-called peridynamics continuum theory. According to [29,30], the modern peridynamics theory [31] was perceived and even formulated in Piola's works. This theory presents a nonlocal model of force interactions; i.e., the force applied on a material particle of a continuum depends on the deformation of particles at a finite difference. As the original model of nonlocal elasticity, the idea of force interactions in peridynamics is implemented by means of an integral operator [32]. This integral form and the corresponding integro-differential equations of motion provide an appropriate tool to analyze the problems with singularities and discontinuity which attract a lot of attention in fracture mechanics [33,34]. In addition of conceiving peridynamics, dell’Isola and co-authors pointed out that Piola's works in 19th century were the cornerstone of the concepts of nonlocal and higher-order gradient continuum theories. For more information, the reader can follow the detailed discussion in [29,30]. The notable point is that due to the incapability of Cauchy stress formulation to model the internal action of nonlocal continuum, the generalized concept of stress is developed in nonlocal elasticity theories. dell’Isola et al. [35] also showed that there is a need for at least Nth gradient continuum theory to generalize the Cauchy representation of contact interactions between sub-bodies. They utilized the principle of virtual powers (or virtual works) to generalize the Cauchy formulations for Nth gradient Piola continuum with the corresponding suitable boundary conditions [29]. Trace of literature shows that the differential form of nonlocal elasticity is widely used to detect size-dependent characteristics of small-scale structures in different static and dynamic problems. For instance, Ansari et al. [36] studied the vibration behaviour of singlelayered graphene sheets (SLGSs) based on the nonlocal elasticity and Mindlin plate theories. The Molecular Dynamic (MD) simulations are also established to obtain the appropriate values of nonlocal parameters corresponding to different boundary conditions. Narendar and Gopalakrishnan [37] investigated the thermo-mechanical pre-buckling characteristics of single-walled carbon nanotubes (SWCNTs) resting on Winkler medium. They incorporate the nonlocal continuum and Timoshenko beam theories to achieve the critical buckling temperatures. Also, Rahmani and Pedram [38] presented the effects of small scales in functionally graded materials (FGMs) based on the nonlocal Timoshenko beam model. Simsek [39] employed the nonlocal strain gradient theory (NLSGT) with von-Karman nonlinearities to propose a size-dependent FGM Euler-Bernoulli beam model. The Galerkin's method and a novel Hamiltonian approach are used to find a closedform solution of the free vibration problem. Despite of the wide usage of differential form of nonlocal elasticity theory, there are some contradictions in predicted results of this model in certain occasions. It is reported [40–42] that in cantilever beams/ plates, a surprising stiffening behaviour is seen with increase of nonlocal parameter. This is while, the softening responses are observed for other boundary conditions and higher vibrational modes of a cantilever body [43]. Also it is clear that in case of point loading, there is no impacts of material length scale i.e. no nonlocal and sizedependent effects. To tackle this issue, some attempts are made in recent years including the studies of Challamel and co-workers [42– 44], Polizzotto [45] and paper of Pisano and Fuschi [46]. In the present paper, Eringen's nonlocal elasticity theory with integral form of constitutive relations is applied to analyze the mechanical characteristics of nano-beams. Integral model of nonlocal continuum theory is used with the Timoshenko beam model to detect the small scale effects without the mentioned paradoxical behaviour. The content of paper is organised as follows: the governing equations of motion are obtained in matrix-vector form of both integral and differential models in Sections 2 and 3, respectively. The matrix-vector form of representation provides a suitable tool for the implementation of the finite element method (FEM). The minimum total potential energy principle is adopted to derive the equations of static bending
Fig. 1. Schematic view of nano-beam.
problem of nano-beams. Section 4 is dedicated to discuss about numerical case studies in which the responses of nano-beams predicted by local elasticity, integral and differential models of nonlocal theories are compared for different boundary and load conditions. As it is expected, the present integral model does not suffer from inconsistence results of differential form of nonlocal elasticity theory. 2. Integral model of nonlocal elasticity A schematic view of the nano-beam under consideration is shown in Fig. 1. It is assumed that the axial direction of nano-beam (with length L ) is located on the x1 axis; also width b and thickness h directions are x2 and x3 axes, respectively. Material properties are consist of Young modulus E , Poisson's ratio ν and internal length parameter a . Furthermore, the transverse distributed static load P is applied on nano-beam. 2.1. Derivation of formulation Based on the Eringen's nonlocal elasticity theory, the constitutive relations of beam-type structures are in the general form as follows
tij (x , x3) = λδij εkk (x , x3)+2μεij (x , x3) = Cijkl εkl (x , x3)
σij (x, x3) =
(3)
∫x k ( x−x , κ ) tij (x , x3) dx
(4)
in which, tij and σij are the components of local and nonlocal stress tensors, respectively and εij expresses the strain tensor component. δij denote Kronecker delta and Cijkl is the fourth order elastic tensor consists of λ and μ , the Lame's classical constants, in which λ = Eν /1 − ν 2,μ = E /2(1 + ν ). Also, x = x1 signals the reference point in which the stress component σij is related to all points on domain represented by x =x1. The nonlocal modulus or kernel k is a function of nonlocal parameter κ and neighborhood distance x − x in which κ = e0 a and e0 is a material constant. In the present study, it is assumed that nano-beam is made from the linear isotropic materials. In this manner, the strain energy of beam and its variation can be written, respectively, as
1 2 1 = 2
Πs =
δ Πs =
∫A ∫x σij (x, x3) εij (x, x3) dxdA ⎞
⎛
∫A ∫x ⎜⎝∫x k ( x−x , κ ) tij (x , x3) dx ⎟⎠ εij (x, x3) dxdA
∫A ∫x σij (x, x3) δεij (x, x3) dxdA
=
⎛
(5) ⎞
∫A ∫x ⎜⎝∫x k ( x−x , κ ) tij (x , x3) dx ⎟⎠ δεij (x, x3) dxdA (6)
Now, displacement field of Timoshenko beam is introduced as
u1 (x , x3) = u (x )+x3 ψ (x ),
u3 (x , x3) = w (x )
(7)
where u and w are the axial and transverse deflections, respectively and ψ shows the rotation angle of the cross section with respect to the vertical direction. Above relations can be written in following form
u = Pq,
u (x , x3) = [ u1 u3 ]T
⎡1 0 x3 ⎤ P (x3) = ⎢ ⎥ , q (x ) = [ u w ψ ]T ⎣0 1 0 ⎦
(8)
The constitutive relation of Eq. (4) is expressed in matrix-vector form as 195
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A. Norouzzadeh, R. Ansari
σ (x, x3) =
∫x
k ( x−x , κ ) Cε (x , x3) dx
(9)
⎡ λ +2μ 0 ⎤ C=⎢ ⎥, ks μ ⎦ ⎣ 0
⎡ σxx (x, x3)⎤ σ (x, x3) = ⎢ ⎥, ⎣ σxz (x, x3) ⎦
⎡ εxx (x , x3) ⎤ ε (x , x3) = ⎢ ⎥ ⎣ 2εxz (x , x3)⎦
(23)
∼ ∼ ∼ N (ξ ) = I3 ⊗ N (ξ ), N (ξ ) = [ N1 (ξ ) … Nn (ξ )] N (ξ ) = I3 ⊗ N (ξ ), ∼ N (ξ ) = [ N1 (ξ ) … Nn (ξ )]
(24)
in which, d and d denote the nodal values in ξ - and ξ -coordinates respectively. Also, Ni is the shape function of ith node, ⊗ signals the Kronecker delta product and Ip is the p -by- p unit matrix. So, the strain vectors are achieved as
(10) where, σ is the stress vector at reference point. C indicates the elastic matrix and ε is the strain vector of beam. Also, ks denotes shear correction factor. By considering the strain-displacement relationship as
1 ⎛ ∂u ∂uj ⎞ ⎟ εij = ⎜ i + 2 ⎝ ∂xj ∂xi ⎠
q = N (ξ ) d , q = N (ξ ) d
ε (ξ ) = PB (ξ ) d,
ε (ξ ) = PB (ξ ) d,
(25)
δ ε (ξ ) = PB (ξ ) δ d
(26)
B (ξ ) = EN (ξ ), B (ξ ) = EN (ξ ) (11)
With respect to Eq. (18), the stress vector of an element is determined
one has (12)
ε (x , x3) = P (x3) Eq (x )
σe = Sd, S =
∫ξ k CPBJ dξ
(27)
in which the matrix P is determined in Eq. (8) and
where J denotes Jacobian in ξ -coordinates. In this stage, it is useful to define the following resultant
⎡ ∂/∂x 0 0 ⎤ E = ⎢ 0 ∂/∂x 1 ⎥ ⎢⎣ ⎥ 0 0 ∂/∂x ⎦
(13)
ˆe = N
Also, Eq. (5) and Eq. (6) can be rewritten as
Πs =
1 2
δ Πs =
ˆ , ˆ e = Sd N
∫A ∫x εT (x, x3) σ (x, x3) dxdA
(15)
δ ε (x, x3) = P (x3) Eδ q (x )
E = E ∂/∂x →∂/∂x,
ˆ, S→,Sˆ →
Πs =
k ( x−x , κ ) CPEqdx
∫x
⎞ k CPEqdx ⎟ dxdA ⎠
⎛ ⎜ ⎝
⎞ k CPEqdx ⎟ dxdA ⎠
∫A ∫x
⎛ qTETPT ⎜ ⎝
∫A ∫x
δ qTETPT
1 2
δ Πs =
∫x
∫x
(17)
Πs, e =
δ Πw =
∫x
F = [ Fu Fw Fψ ]T
δ qTFdx
=
∫A PTCPdA
(29)
d→
(30)
∫A ∫ξ dTBTPTJdξdA
∫ξ
=
1 2
ˆ Jdξ ∫ξ dTBT
(31)
ˆ Jdξ = δ dTKs, e δ dTBT
(32)
(18)
where J is the Jacobian in ξ -coordinates and the stiffness matrix of element Ks, e is achieved as
(19)
Ks, e =
∫ξ BTˆ Jdξ
(33)
Also, the external work of element and its variation from Eq. (21) and Eq. (22) are captured as
(20)
Furthermore, the external work applied on nano-beam is achieved
∫x qTFdx,
1 2
δ Πs, e =
δ Πw, e = δ dTFe Fe =
as
1 Πw = 2
∫ξ k CBJ dξ , C
According to Eq. (19) and Eq. (20), the strain energy of element and its variation are presented as
(16)
So, the stress vector, strain energy and its variation are obtained in terms of displacement field components as follows
σ (x, x3) =
(28)
By performing the assemblage process in ξ -coordinates, the stress components at reference point x are obtained. So, one has the total matrices and vectors as
in which the strain vector at reference point ε is
q (x ) = [ u (x ) w (x ) ψ (x )]T ,
Sˆ =
(14)
∫A ∫x δ εT (x, x3) σ (x, x3) dxdA
⎡ εxx (x, x3) ⎤ ε (x, x3) = ⎢ ⎥ = P (x3) Eq (x ), ⎣ 2εxz (x, x3)⎦
∫A PTσe dA
∫ξ
N TFJdξ
(34)
After assemblage procedure in ξ -coordinates, the assembled matrices and vectors are
(21)
d→ ,
(22)
Ks, e→s ,
Fe→
(35)
where Fu and Fw exhibit the distributed load on axial and lateral directions, respectively, and Fψ is the distributed moment.
and by choosing the same nodes for FEM local coordinate systems (ξ = ξ ), one has = . So, the strain energy and external work of Timoshenko nano-beam are specified as
2.2. Finite element analysis
δ Πs = δ T s δ Πw =
Now, the Finite Element Method (FEM) is adopted to solve the integral formulation of nonlocal beam elasticity. As it is observed from foregoing formulation, at a given point of x , the stress tensor should be calculated in entire domain of x . So, unlike the finite element analysis of conventional problems, two procedures of assemblage are required to determine the strain energy of structure. The local coordinate system of elements corresponding to x and x domains are denoted by ξ and ξ , respectively. In an element, the displacement components of beam are interpolated by appropriate shape functions as
(36)
δ T
Employing the minimum δ (Πs−Πw) = 0 leads to
(37) total
potential
s =
energy
principle (38)
3. Differential model of nonlocal elasticity It would be helpful to present the governing differential equations of motion and the corresponding finite element formulation of the 196
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A. Norouzzadeh, R. Ansari
Ks, e d = Fe
conventional differential model of nonlocal elasticity theory. This makes it possible to directly compare the different natures of two models of nonlocal theory.
→
s =
(55)
4. Selected results 3.1. Derivation of formulation In this section, some numerical examples have been brought to investigate the results predicted by the integral and differential models of nonlocal elasticity theory. Also, the effects of different important parameters on the bending responses of nano-beams are analyzed through the following discussion. It is known that inconsistent results appear by employing differential model of nonlocal elasticity in analysis of cantilever beams. The main paradoxes are: 1) deflection does not affected by nonlocal parameter in case of point loading; 2) the stiffness of structure is increased by nonlocal parameter in distributed loading conditions. According to this important issue, the present study aims to resolve the above-mentioned paradoxical behaviors by implementation of the original integral model of nonlocal continuum theory. In this regard, a purely numerical approach with no simplification assumptions, is considered herein. First, some comparisons are made to validate the proposed formulation and solution methodology of present paper. Ref. [47] studied the bending problem of Euler-Bernoulli beam with using the integral form of nonlocal elasticity. They presented the analytical solution of integral form of nonlocal elasticity with specific kernel function in which satisfied the boundary and loading conditions proposed by [48]. To validate present investigation based on finite element analysis of integral formulation of nonlocal nano-scale Timoshenko beam theory, the analysis of cantilever beam is carried out with assumption of L / h = 25, ν = 0, ks = 5/6 and results are compared with those reported by Ref. [47]. Also, Fu = Fψ = 0, Fw = P and three types of transverse loading conditions including a) point load, b) uniform distributed and c) triangular loads are considered. Dimensionless lateral deflection is defined as W = wEI / qL4 where I denotes the second moment of inertia and q has the following values: a) q = P / L , b) q = P , c) q = L . The kernel function is considered as [47]
Eringen stated that for a specific kernel function, the integral constitutive relations of Eq. (4) can be transformed to the differential model as
(1 − μ ∇2 ) σij = λδij εkk +2μεij = Cijkl εkl
(39)
or in matrix-vector form as
σ = [ σxx σxz ]T
(1 − μ ∇2 ) σ = Cε,
(40)
∇2
κ2
∂ 2/∂x 2 .
= in which μ = and Laplacian operator in 1D problem is By taking into consideration of Eq. (10) and Eq. (12), the resultant of stress vector is written as ˆ = N
∫A PTσ
dA
(41)
ˆ = CEq (1 − μ ∇2 ) N
(42)
where C is introduced in Eq. (29). So, the strain energy, external work and their variations have the following form
Πs =
1 2
Πw =
∫
∫A ∫x εTσdxdA
=
1 2
∫A ∫x qTETPTσdxdA
=
1 2
∫x qTETNˆ dx
(43)
qTFdx
x
(44)
δ Πs =
∫x δ qTETNˆ dx
(45)
δ Πw =
∫
δ qTFdx (46)
x
By means of minimum total potential energy principle, one has
∫x (δ qTETNˆ −δ qTF) dx
=
∫x (δ qTETCEq−(1 − μ ∇2) δ qTF) dx
=0
(47)
k ( x−x , κ ) =
and the variation of equivalent energy terms are defined as
δ Πs =
∫x
δ qTETCEqdx
δ Πw =
∫
(1 − μ ∇2 ) δ qTFdx
From Fig. 2, it is seen that the above mentioned two paradoxes are solved by applying the integral model of nonlocal continuum theory. In all three cases, the maximum deflection of cantilever nano-beam is increased significantly with increase of nonlocal parameter. In addition of the trend of changes, a very good agreement is observed with reported results of Ref. [47]. For rest of paper, it is assumed that nano-beam is made of Al in which is subjected to the uniform distributed load. The material and geometrical properties are as follows
(48)
(49)
x
3.2. Finite element analysis The conventional finite element implementation is carried out to detect the equations of motion of nano-scale Timoshenko beam based on the differential model of nonlocal elasticity. So,
∼ N = I3 ⊗ N,
E = 68. 5GPa,
∼ N = [ N1 … Nn ]
(52)
From Eq. (48) and Eq. (49), one has
δ Πs, e = δ dTKs, e d ,
δ Πw, e = δ dTFe , Fe =
Ks, e =
∫ ξ
∫ξ
BT CBJdξ
ks = 5/6,
L /h = 12,
b /h = 1
and dimensionless deflection is w = Also, the nonlocal parameter is fixed at κ = 1nm [49]. Here, some explanations are provided about the computational features of present approach. As it is stated further, the assemblage process of the stress resultant must be carried out in each point of the Gaussian quadrature scheme. So, if m times function evaluation is needed in a conventional finite element analysis (like differential model), this number is raised to m2 times in assemblage procedure of integral model. It is clear that the computational run time of this approach is higher than differential model which came back to different nature of constitutive relations. Besides of what above, the “dynamic rate of convergence” of elements is another new aspect of this approach. It is related to the neighborhood distance concept in kernel function such that with increase of structure scales, higher number of elements should be
(51)
ε = PBd, B = EN
ν = 0. 35,
WEI / PL4 .
(50)
q = Nd
1 − x−x e κ 2κ
(53)
(1 − μ ∇2 ) N TFJdξ (54)
and finally by adopting the assemblage process, the following set of equations are achieved 197
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Fig. 2. Comparison of integral and differential models of nonlocal elasticity in predicting the maximum deflection of cantilever nano-beam, a) point load at end; b) uniform distributed load; c) triangular distributed load.
employed to cover the domain properly and give small values of x − x . To show this issue, convergence trends are tabulated in Table 1 corresponding to different thickness-to-nonlocal parameter ratios. In present study, the higher order elements (with 8 nodes) are applied in which are able to satisfy appropriately the neighboring distance of points as well as being locking free. Depicted in Figs. 3–5 are variation of maximum deflection of cantilever, clamped-pinned and pinned-pinned nano-beams with thickness-to-nonlocal parameter ratios, respectively. For comparison purpose, the graphs of predicted results of classical/local continuum
Table 1 Convergence study of finite element analysis with different values of thickness-tononlocal parameter ratios. h κ
5 10 15 20
Number of elements 20
40
60
80
100
120
0.1158 0.1071 0.0989 0.0905
0.1174 0.1123 0.1089 0.1055
0.1177 0.1134 0.1111 0.1092
0.1179 0.1137 0.1119 0.1108
0.1179 0.1139 0.1121 0.1113
0.1179 0.1139 0.1122 0.1115
Fig. 3. Maximum deflection of cantilever nano-beam versus thickness ratios based on the integral and differential nonlocal elasticity as well as local elasticity.
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Fig. 4. Maximum deflection of clamped-pinned nano-beam versus thickness ratios based on the integral and differential nonlocal elasticity as well as local elasticity.
Fig. 5. Maximum deflection of pinned-pinned nano-beam versus thickness ratios based on the integral and differential nonlocal elasticity as well as local elasticity.
bending of Timoshenko nano-beams. The integral model of nonlocal continuum theory was employed to study the small scales effects. The formulations of problem for integral and differential models were obtained in the matrix-vector form of representation. Then, finite element analysis was carried out to solve the governing equations of motion. Some numerical examples for various boundary conditions and different loadings were given to realize the differences of results of two models. It was reported that differential model of nonlocal continuum theory leads to the unexpected inconsistent results in certain types of boundary and loading conditions. But, it was observed that there is no paradox in the behaviour of cantilever nano-beams with the implementation of integral model of nonlocal elasticity theory. The softening responses were reported by the increase of nonlocal parameter, in all types of loading conditions. Also, it was concluded that the effects of material length parameters are negligible in nano-structures with larger scales. Hence, present work showed that the integral model of nonlocal elasticity can be analyzed in the general case without simplifying assumptions and predicts no contradictory results. This solution strategy is capable of being straightforwardly used in similar problems.
and differential model of Eringen's nonlocal elasticity are presented, too. It is observed that, in all cases, by increase of nano-beam scales, reported responses of nonlocal theories tend to converge to those of predicted by the classical elasticity theory. Fig. 3 demonstrates the existing paradox of differential model in cantilever beams, again. It is found that the convergence of integral model to the local elasticity theory occurred at higher ratios in compare with differential model. Also, it can be concluded that small-scale effects in present model are significantly higher than predicted by differential model. Fig. 4 demonstrates that employing both integral and differential forms of nonlocal elasticity in analysis of clamped-pinned nano-beams lead to similar trend of changes. But, differential model underestimates the maximum deflection of nan-beam as well as the ratio of convergence to the local theory. Fig. 5 shows that the integral and differential models of nonlocal continuum theories produce approximately the same responses in analysis of pinned-pinned nano-scale beams. Consequently, there is no remarkable error to use differential model in analyzing nano-beams with this type of boundary conditions. 5. Conclusion Investigated in this paper, was a size-dependent analysis of static 199
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