Torsion of functionally graded nonlocal viscoelastic circular nanobeams

Composites: Part B xxx (2015) xxx–xxx

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Torsion of functionally graded nonlocal viscoelastic circular nanobeams Raffaele Barretta a,⇑, Luciano Feo b, Raimondo Luciano c a

Department of Structures for Engineering and Architecture, University of Naples Federico II, via Claudio 21, 80125 Naples, Italy Department of Civil Engineering, University of Salerno, 84084 Fisciano (Sa), Italy c Department of Civil and Mechanical Engineering, University of Cassino and Southern Lazio, via G. Di Biasio 43, 03043 Cassino (FR), Italy b

a r t i c l e

i n f o

Article history: Received 17 November 2014 Accepted 9 December 2014 Available online xxxx Keywords: A. Nano-structures B. Elasticity C. Analytical modeling

a b s t r a c t The elastostatic problem of functionally graded circular nanobeams under torsion, with nonlocal elastic behavior proposed by ERINGEN, is preliminarily formulated. Exact solutions are detected for nanobeams with arbitrary axial gradations of elastic properties and radially quadratic distributions of shear moduli. Extension of the treatment to nonlocal viscoelastic composite circular nanobeams is then performed. An effective solution procedure based on LAPLACE transform is developed, providing a new correspondence principle in nonlocal viscoelasticity for functionally graded materials. Displacements, shear strains and stresses are established for nonlocal viscoelastic nanobeams made of periodic fiber-reinforced materials, with polymeric matrix described by a MAXWELL model connected in series with a VOIGT model. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Analysis of functionally graded beams under torsion is a research topic of major interest in engineering applications. Nevertheless, exact solutions are available only for special cross-sections and gradations of elastic properties. Computational strategies and effective homogenization techniques [1–18] are thus usually adopted in order to analyze and design such structures. An elegant solution procedure, based on a modified version of LEKHNITSKII formalism [19], was proposed in [20] for cylindrically anisotropic beams. Applications on laminates and novel solutions for circular cylindrical bars, also with angular symmetry, were investigated in [21,22]. The effects of material inhomogeneities on the torsional response of linearly elastic isotropic bars were assessed in [23]. Analytical stress solutions for composite cylinders were given in [24]. Further solutions, with special emphasis on end effects, were assessed in [25,26] for cylindrically anisotropic circular tubes and bars under thermal and mechanical loadings. Functionally graded beams with shear moduli, defined by positive functions of the Prandtl stress function of corresponding elastically homogeneous beams were analyzed in [27,28]. Inhomogeneous hollow cylinders made also of isotropic and incompressible linearly elastic materials were studied in [29–31]. However, in these contributions the constitutive behavior is elastically local, with cross-sectional inhomogeneities, see also [32–37]. An exception was dealt with in [38] still ⇑ Corresponding author. E-mail addresses: [email protected] (R. Barretta), [email protected] (L. Feo), [email protected] (R. Luciano).

for local cylinders, but axially graded. The motivation of the present manuscript is in answering the question: ’’Is it possible to detect new exact solutions for composite viscoelastic nonlocal nanobeams under torsion?’’ The conclusion is affirmative for circular nanobeams with radially quadratic distributions of shear moduli. The plan is the following. Basic notations, assumptions and equilibrium conditions governing circular beams are collected in Section 2. The nonlocal elastic equilibrium problem of functionally graded isotropic nonlocal nanobeams is formulated in Section 3. The closed-form expression of nonlocal stresses is provided in Section 4 by resorting to a SAINT-VENANT-type semi-inverse approach. These fields are then transformed in LAPLACE domain in Section 5 to solve viscoelastic nanobeams governed by the ERINGEN nonlocal law given in Section 6. Analytical solutions are detected in Section 7 for composite nonlocal viscoelastic cylinders made of periodic fiber-reinforced materials, with polymeric matrix represented by a MAXWELL model connected in series with a VOIGT model. 2. Preliminary assumptions and equilibrium conditions Let us consider a circular domain X of radius R describing the cross-section of a straight cantilever subjected to a torque M at the free-end. Body forces are assumed to vanish and the beam lateral mantle is considered to be traction-free [39]. We denote by r the radius vector in the cross-section plane pX originating at the centroid G and by k the unit vector of the beam z-axis thru G. R is the linear operator performing the rotation in pX by p=2 counterclockwise and V and V are the linear spaces of translations

http://dx.doi.org/10.1016/j.compositesb.2014.12.018 1359-8368/Ó 2014 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Barretta R et al. Torsion of functionally graded nonlocal viscoelastic circular nanobeams. Composites: Part B (2015), http://dx.doi.org/10.1016/j.compositesb.2014.12.018

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associated with the EUCLID 3-D space S and with pX . Following CLEBSH [40], we conjecture that the normal interactions between longitudinal fibers of the beam vanish so that the CAUCHY stress tensor Tðr; z; tÞ is expressed, in terms of the shear stress vector sðr; z; tÞ on the cross-section at the point ðr; zÞ, by

Tðr; z; tÞ ¼



O

s



sT 0

ðr; z; tÞ;

ð1Þ

where t stands for time. Due to the the absence of body forces, the CAUCHY differential condition of equilibrium gives

s0 ðr; z; tÞ ¼ o;

div sðr; z; tÞ ¼ 0;

ð2Þ

cðr; z; tÞ ¼ h0 ðz; tÞRr:

ð11Þ

Substituting Eqs. (7) and (11) in Eq. (6), the differential condition of nonlocal elastic kinematic compatibility is expressed as

sðr; z; tÞ  ðeo aÞ2 D2 sðr; z; tÞ ¼ lr ðrÞ la ðzÞ h0 ðz; tÞ Rr:

ð12Þ

Taking the z-derivative and the r-divergence we get the equations (

s0 ðr; z; tÞ  ðeo aÞ2 D2 s0 ðr; z; tÞ ¼ lr ðrÞ ðla ðzÞ h0 ðz; tÞÞ0 Rr; 0 ðdiv sÞðr; z; tÞ  ðeo aÞ2 ðdiv D2 sÞðr; z; tÞ ¼ ðla ðzÞ h0 ðz; tÞÞ div ðlr ðrÞ RrÞ: ð13Þ

with the prime ðÞ0 denoting partial derivative along the z-axis and div divergence with respect to the position vector r. Since the tractions on the lateral mantle are assumed to vanish, the CAUCHY boundary condition of equilibrium takes the form

Resorting to Eq. (8), we get rlr ðrÞ  Rr ¼ 0. It follows that, being div Rr ¼ 0, also divðlr ðrÞRrÞ ¼ rlr ðrÞ  Rr þ lr ðrÞdiv Rr ¼ 0. Recalling Eqs. (2)2 and (4), we conclude that Eq. (13)2 is identically fulfilled. Moreover, imposing the equilibrium Eq. (2)1 in Eq. (13)1, we get

sðr; z; tÞ  nðr; z; tÞ ¼ 0;

ðla ðzÞ h0 ðz; tÞÞ ¼ 0;

ð3Þ

where the dot  stands for inner product and n is the outward unit normal to the cross-section boundary @ X. Let us denote by D2 :¼ divr the LAPLACE operator with respect to the position vector r. As shown in the next section, an useful implication of Eq. (2) is the vanishing of the divergence of the Laplacian of shear stress field

ðdiv D2 sÞðr; z; tÞ ¼ 0:

ð4Þ

Indeed, a cartesian evaluation gives

div s ¼ si=i ¼ 0

)

div D2 s ¼ si=jji ¼ si=ijj ¼ 0;

ð5Þ

3. Nonlocal isotropic elasticity The shear stress sðr; z; tÞ is assumed to be related to the shear strain vector cðr; z; tÞ by the isotropic nonlocal elastic law conceived by ERINGEN [41]

ð7Þ

The transversal shear modulus is assumed to be radially inhomogeneous according to the quadratic rule

lr ðrÞ :¼ mkrk2 þ k; with m; k 2 R;

ð8Þ

where krk is the norm of the vector r. The shear strain vector at r.h.s. in Eq. (6) is evaluated by conjecturing that the displacement field of the beam under torsion takes the form

uðr; z; tÞ ¼ hðz; tÞRr;

ð9Þ

where h is the rotation function, about the z-axis, of cross-sections with respect to the clamp.1 The kinematically compatible deformation writes thus as

 Dðr; z; tÞ ¼ ðsym duÞðr; z; tÞ ¼

O

c

T

0

c

 ðr; z; tÞ;

with the shear strain vector given by 1

In SAINT-VENANT theory the rotation h is affine in the abscissa z [47,48].

la ðzÞ h0 ðz; tÞ ¼ bðtÞ;

with bðtÞ 2 R:

ð15Þ

Eq. (15) was obtained in [38] for radially homogeneous linearly elastic (local) beams, grading the material only along z. The torsional rotation hðz; tÞ is then evaluated by integrating Eq. (15) and by setting hð0Þ ¼ 0

hðz; tÞ ¼ bðtÞ

Z

z

1

la ðqÞ

dq:

ð16Þ

As pointed out in [38], the scalar function bðtÞ is computed by imposing the static equivalence condition around the z-axis

MðtÞ ¼

Z

Rr  sðr; z; tÞ dA:

ð17Þ

X

The explicit expression of the function bðtÞ is provided in the next section.

ð6Þ

where eo is a material constant, a is the internal length. The magnitude of eo is determined experimentally or approximated by matching the dispersion curves of plane waves with those of atomic lattice dynamics. For single walled carbon nanotubes [42–44] the length scale parameter c :¼ eo a is assessed to be smaller than 2.0 nm [45]. Nonlocal constitutive relations for functionally graded materials have been discussed in [46]. Let us assign the shear modulus in a separable form in r and z as

lðr; zÞ ¼ lr ðrÞ la ðzÞ:

ð14Þ

whence the z-constancy condition follows, so that we may set

0

where the symbol = stands for partial derivative and i; j 2 f1; 2g.

sðr; z; tÞ  ðeo aÞ2 D2 sðr; z; tÞ ¼ lðr; zÞ cðr; z; tÞ;

0

ð10Þ

4. Nonlocal elastic shear stresses The shear stress field, solution of the nonlocal elastostatic problem of functionally graded circular nanobeams under torsion formulated in Section 3, is given by the formula

sðr; z; tÞ ¼ bðtÞðlr ðrÞ þ 8 c2 mÞ Rr ¼ bðtÞðmkrk2 þ k þ 8 c2 mÞRr:

ð18Þ

Note that the the physical dimensions of the parameters m and k are ½FL4  and ½FL2  respectively. The length scale parameter c and the scalar bðtÞ have respectively the physical dimensions of a length and of the inverse of a length. Proof of Eq. (18) consists of two steps. (1) Check of equilibrium, described by Eqs. (2) and (3). (2) Check of nonlocal elastic kinematic compatibility Eq. (12). Let us preliminary provide a list of noteworthy identities

8 2 > > > div ðkrk RrÞ ¼ 0; > > 2 2 > > < rðkrk RrÞ ¼ 2Rr  r þ krk R; div ð2Rr  rÞ ¼ 6Rr; > > 2 > > > div ðkrk RÞ ¼ 2Rr; > > : D2 ðkrk2 RrÞ ¼ div rðkrk2 RrÞ ¼ 8Rr:

ð19Þ

Due to the z-independence of Eq. (18), the differential condition of equilibrium Eq. (2)1 is trivially verified. Eq. (2)2 follows from Eq. (19)1. Fulfillment of the boundary equilibrium Eq. (3) is a direct consequence of the orthogonality condition Rr  nðrÞ ¼ 0, being nðrÞ

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proportional to the position vector r. Elastic kinematic compatibility is checked by substituting Eq. (18) in Eq. (12) and taking Eq. (19)5 into account. The proof is thus completed. The evaluation of the parameter bðtÞ is carried out by inserting the expression Eq. (18) of the nonlocal shear stress field in Eq. (17)

MðtÞ ¼ bðtÞ

Z

2

2

r  ðmkrk þ k þ 8 c mÞr dA ¼ bðtÞC;

ð20Þ

X

where

C :¼ m

Z

krk4 dA þ ðk þ 8 c2 mÞ J G

X

¼m

pR6

þ ðk þ 8 c2 mÞ

p R4

3 2  pR4  2 ¼ 2 m R þ 3ðk þ 8 c2 mÞ 6

5. Nonlocal elasticity in Laplace domain The evaluation of exact solutions of nonlocal viscoelastic nanobeams under torsion formulated in Section 6, is carried out by mapping equations and results given in the previous sections into LAPLACE domain, as follows. Let us preliminarily recall the definition of LAPLACE transform of a well behaved time function

f ðsÞ ¼ L½f ðsÞ :¼ ð21Þ R

2

pR4

is the nonlocal torsional stiffness and J G :¼ X krk dA ¼ 2 is the polar moment of inertia of a circle of radius R. The time parameter bðtÞ is thus given by the formula

bðtÞ ¼

as depicted in Fig. 1. It is worth noting that for single walled carbon nanotubes the parameter c is estimated to be smaller than 2 nm [45].

MðtÞ : C

ð22Þ

Z

þ1

expðstÞf ðtÞdt;

ð25Þ

0

with s complex variable and exp exponential function.2 Recalling Eqs. (2) and (3), the equilibrium conditions in LAPLACE domain are formulated as

8 0  ðr; z; sÞ ¼ o; > : sðr; z; sÞ  nðr; z; sÞ ¼ 0:

ð26Þ

The isotropic nonlocal elastic law Eq. (12) rewrites as Remark 4.1. For radially homogeneous beams, the constitutive constant m vanishes, so that Eq. (22) takes the form

MðtÞ bðtÞ ¼ : k JG

ð23Þ

In this case, the torsional stiffness C ¼ k JG is purely local, being independent of the length scale parameter c characterizing the ERINGEN model. Under the further assumption that the beam is axially homogeneous, with la ðzÞ ¼ 1, Eq. (23) provides the known relationship between uniform twist h0 of a linearly elastic beam and twisting stiffness described by the product of the constant shear modulus and the polar moment of inertia [49]. Dependence of the torsional stiffness on the length scale parameter c is explicated by splitting Eq. (21) as

C ¼ JG

  2 m R2 þ k þ J G 8 c 2 m ¼ C L þ C N ; 3

ð24Þ

with C L and C N :¼ JG 8 m c2 local and nonlocal contributions, respectively. The nonlocal torsional stiffness C N is a linear function both with respect to the polar moment of inertia JG and to the constitutive parameter m. The nonlocality effect is of quadratic-type

sðr; z; sÞ  ðeo aÞ2 D2 sðr; z; sÞ ¼ lr ðrÞla ðzÞcðr; z; sÞ:

Transfomation of Eqs. (9) and (11) for displacements and shear strains gives

(

 ðr; z; sÞ ¼ hðz; sÞRr; u  cðr; z; sÞ ¼ h0 ðz; sÞRr:

ð28Þ

Shear stress solution field in the s-variable is got by transforming Eq. (18), defined in the time domain, as

 sðr; z; sÞ ¼ bðsÞð lr ðrÞ þ 8 c2 mÞRr ¼ la ðzÞh0 ðz; sÞðmkrk2 þ k þ 8 c2 mÞRr;

ð29Þ

where the scalar function

 ¼ l ðzÞh0 ðz; sÞ ¼ MðsÞ bðsÞ a C

ð30Þ

is the LAPLACE transform of Eq. (22). 6. Nonlocal viscoelasticity Nonlocal viscoelastic nanobeams under torsion can be formulated by considering the following ERINGEN-type law

sðr; z; tÞ  ðeo aÞ2 D2 sðr; z; tÞ ¼

4

ð27Þ

Z

t

lðr; z; t  sÞc_ ðr; z; sÞds;

ð31Þ

0

Specific nonlocal stiffness

3.5

with lðr; z; tÞ shear relaxation function. Differential and boundary conditions of equilibrium are described by Eqs. (2) and (3). Displacements and shear strains are provided by Eqs. (9) and (11). The viscoelastic equilibrium problem is solved by detecting a correspondence principle for functionally graded nonlocal materials. It is known that, for viscoelastic inhomogeneous materials, the correspondence principle remains valid if the relaxation moduli are assigned in a separable form in space and time [51,52]. This prescription is adopted in the present note also for ERINGEN-type nonlocal materials, by introducing the time-dependent shear modulus as

3 2.5 2 1.5 1 0.5 0 0

0.5

1

1.5

2

lðr; z; tÞ ¼ lr ðrÞ la ðzÞgðtÞ;

ð32Þ

Length scale parameter Fig. 1. Nonlocal stiffness C

N

per unit JG 8 m vs. length scale parameter c.

2

Some LAPLACE transforms referred to in this paper can be found in [50].

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with g relaxation function of time, assumed to be well behaved and positive. Exact solutions of nonlocal nanobeams are obtained by performing the LAPLACE transform of equations Eqs. (31) and (32) governing the viscoelastostatic problem

sðr; z; sÞ  ðeo aÞ2 D2 sðr; z; sÞ ¼ lr ðrÞla ðzÞsgðsÞcðr; z; sÞ:

la ðrÞ s gðsÞ;

!

ð34Þ

(

n1 X

GðtÞ ¼

MðtÞ ðmkrk2 þ k þ 8 c2 mÞRr: C

ð35Þ

ð36Þ

ð37Þ

The displacements and shear strains in LAPLACE domain, given by Eq. (28), are formulated in terms of torsional rotation  h and its derivative  h0 as follows. From Eq. (36) we get

h0 ðz; sÞ ¼ MðsÞ sgðsÞ

1 : la ðzÞC

ð38Þ

By taking the inverse LAPLACE transform of Eq. (36), the twist function in time domain is given by

h0 ðz; tÞ ¼

  1 MðsÞ ðtÞ: L1 sgðsÞ la ðzÞC

ð39Þ

Viscoelastic torsional rotations are got by integrating Eq. (39) along the abscissa z and prescribing the boundary condition hð0Þ ¼ 0

  Z z 1 1 MðsÞ ðtÞ L C sgðsÞ 0

hðz; tÞ ¼

1

la ðqÞ

dq:

ð40Þ

7. Example Let us consider a circular nanobeam clamped at one end and subjected to a torque at the other end, described by the following HEAVISIDE step function of time

MðtÞ ¼

0;

if t < 0;

a; if t > 0:

  x y

 ;

R¼

  expðdj tÞ cIj sinhðlj tÞ þ cIIj coshðv j tÞ ;

j¼1

a1 a 1:232 2 a3 0:803 a :¼ ¼ ; a4 3:737 a5 6:045 a

b1 0:015 b 0:0001320 2 b :¼ ¼ ; b3 0:0001347 b 0:01548

n1 ¼ 4;

ð46Þ

4

6

cI 0:356 ; c I :¼ 1I ¼ c2 0:317

cII 1:2594 ; c II :¼ 1II ¼ c2 1:0177

d1 0:00831 ; d :¼ ¼ 0:00865 d2 ð47Þ

1 0:00817 ; l :¼ ¼ 0:008511 2

v

v 1 0:008172 ; :¼ ¼ 0:008511 v

n2 ¼ 2:

2

ð48Þ Note that constitutive parameters m; k; c1 ; c2 , involved in Eq. (44), must be chosen in such a way that the shear modulus function lðx; y; zÞ is positive. Cartesian components of the nonlocal viscoelastic shear stress field are provided by Eq. (37)





sxz ðx; y; z; tÞ ¼  MðtÞ mðx2 þ y2 Þ þ k þ 8 m c2 y; C

 syz ðx; y; z; tÞ ¼ MðtÞ mðx2 þ y2 Þ þ k þ 8 m c2 x; C

ð49Þ

with the torsional stiffness C given by Eq. (21). Note that the nonlocal term sN ðr; z; tÞ :¼ 8 mc2 MðtÞ Rr of the shear stress field Eq. C (49) is radially linear. The shear stress field, depicted in Fig. 2, has the same form of the one of SAINT-VENANT (local) beam theory [49]. The displacement field, to within a rigid body motion, is provided by Eq. (9)

8 > < ux ðx; y; z; tÞ ¼ hðz; tÞy; uy ðx; y; z; tÞ ¼ hðz; tÞx; > : uz ðx; y; z; tÞ ¼ 0:

ð50Þ

Recalling the noteworthy formulae

A pair of cartesian axes fx; yg passing thru the centroid G of the circle X, of radius R, is considered. Cartesian descriptions of position vector and rotation tensor write thus as

r¼

n2 X

characterizing a wide class of homogenized constitutive laws of viscoelastic composite materials of engineering interest [6,53]. The relaxation function Eq. (45) has been adopted also in [50] to analyze functionally graded viscoelastic thin plates. Hereafter, we assume the following effective parameters for periodic fiber-reinforced composites with polymeric matrix represented by a MAXWELL model connected in series with a VOIGT model [54]4

(

Remark 6.1. Nonlocal elastic displacements and strains established in Section 3 are recovered by assuming gðtÞ ¼ 1 and recalling that gðsÞ ¼ 1=s.



ai expðbi tÞ þ

ð45Þ

The shear stress solution in time domain is obtained by applying the inverse LAPLACE transform

sðr; z; tÞ ¼

ð44Þ

where c1 ; c2 are constants and expðÞ is the exponential function. GðtÞ The nondimensional positive scalar function gðtÞ ¼ Gð0Þ is described by

i¼1

 Due to the replacement in Eq. (34), the scalar function bðsÞs gðsÞ is evaluated by analogy from Eq. (30)

MðsÞ  bðsÞs gðsÞ ¼ la ðzÞh0 ðz; sÞsgðsÞ ¼ : C

3

lr ðx; yÞ ¼ mðx2 þ y2 Þ þ k; la ðzÞ ¼ c1 expðc2 zÞ;

we get the nonlocal viscoelastic shear stress in LAPLACE domain

sðr; z; sÞ ¼ la ðzÞh0 ðz; sÞsgðsÞðlr ðrÞ þ 8 c2 mÞRr    gðsÞ mkrk2 þ k þ 8 c2 m Rr: ¼ bðsÞs

ð43Þ

with radial and axial inhomogeneities assigned as

ð33Þ

and resorting to the transformed elastic solution discussed in Section 5. Exploiting Eqs. (26), (28)2, (29) and (30) and performing the replacement

la ðrÞ

lðx; y; z; tÞ ¼ lr ðx; yÞla ðzÞgðtÞ;

0 1 1

0

 ;

ð42Þ

(

MðsÞ ¼ a 1s ; Rz 1 dq ¼ c11c2 ðexpðc2 zÞ  1Þ; 0 l ðqÞ

ð51Þ

a

the torsional rotation function Eq. (40) rewrites as

T

whence Rr ¼ ½y x . The constitutive behavior of the cylinder is assumed to be defined by the ERINGEN nonlocal viscoelastic law Eq. (31). The shear modulus, expressed by Eq. (32), takes the form

3

Eq. (44)2 was chosen in [38] to analyze an axially graded local elastic cylinder. The numerical parameters, defining the relaxation function, are given in such a way that the values gðtÞ are expressed in GPa. 4

Please cite this article in press as: Barretta R et al. Torsion of functionally graded nonlocal viscoelastic circular nanobeams. Composites: Part B (2015), http://dx.doi.org/10.1016/j.compositesb.2014.12.018

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systems (NEMS) [55]. Extension of the treatment to nonlocal viscoelastic circular nanobeams has been carried out for functionally graded materials which are separable in space and time. The relevant elastoviscoelastic problem has been solved by resorting to the method of LAPLACE transform. A new correspondence principle in ERINGEN nonlocal viscoelasticity theory is thus enlightened. Closed-form solutions of applicative interest have been established for periodic fiber-reinforced composites, with polymeric matrix described by a MAXWELL model connected in series with a VOIGT model. Acknowledgements

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fig. 2. Nonlocal shear stresses. Color spectrum kRrk ¼ x2 þ y2 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

This research was carried out in the framework of the DPC/ ReLUIS 2014 – AQ DPC/ReLUIS 2014–2016 project (theme: reinforced concrete structures) funded by the Italian Department of Civil Protection. References

Fig. 3. Inverse LAPLACE transform w vs. time t [hr].

hðz; tÞ ¼

a ðexpðc2 zÞ  1Þ c1 c2 C

L1



 1 ðtÞ; s3 gðsÞ

ð52Þ

Viscoelastic shear strains are expressed, in terms of the twist function, by

8 > < cxz ðx; y; z; tÞ ¼ h0 ðz; tÞy ¼ y a > : cyz ðx; y; z; tÞ ¼ h0 ðz; tÞx ¼ x a

h

i

expðc2 zÞ 1 L s2 g1ðsÞ c1 C

h

expðc2 zÞ 1 L s2 1gðsÞ c1 C

i

ðtÞ; ð53Þ

ðtÞ:

Closed-form expression of the inverse LAPLACE transform involved in Eqs. (50), (52) and (53)

wðtÞ :¼ L1



 1 ðtÞ s2 gðsÞ

ð54Þ

are conveniently evaluated by resorting to a symbolic computer code. The plot of Eq. (54) given in Fig. 3 has been computed with MATLAB. 8. Conclusions The elastic equilibrium problem of composite nonlocal elastic nanobeams under torsion has been formulated by an intrinsic approach. Exact solutions have been detected for circular nanobeams with radially quadratic distributions of shear moduli, extending a previous contribution by BATRA [38] for the special case of cross-sectional local elastic homogeneities. Nonlocality effects exhibited by displacement, deformation and CAUCHY stress solutions and by the torsional stiffness have been analytically evaluated in terms of the length scale parameter introduced by ERINGEN, providing thus effective results for the analysis of nanoelectromechanical

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Please cite this article in press as: Barretta R et al. Torsion of functionally graded nonlocal viscoelastic circular nanobeams. Composites: Part B (2015), http://dx.doi.org/10.1016/j.compositesb.2014.12.018