Variational nonlocal gradient elasticity for nano-beams

International Journal of Engineering Science 143 (2019) 73–91

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Variational nonlocal gradient elasticity for nano-beams Raffaele Barretta∗, Francesco Marotti de Sciarra Department of Structures for Engineering and Architecture, University of Naples Federico II, via Claudio 21, Naples 80125, Italy

a r t i c l e

i n f o

Article history: Received 10 April 2019 Accepted 17 June 2019

Keywords: Nonlocal gradient elasticity Nanobeams Well-posedness CBC

a b s t r a c t In the paper (Zaera, Serrano, & Fernández-Sáez, 2019) it is claimed that the nonlocal strain gradient theory (NSGT) leads to ill-posed structural problems. This conclusion was motivated by observing that Constitutive Boundary Conditions (CBC) conflict with non-standard Kinematic and Static Higher-Order Boundary Conditions (KHOBC) - (SHOBC). In the present study, it is shown that no ill-posedness holds if NSGT is established by an adequate variational formulation, with appropriate test fields. KHOBC and SHOBC have nothing to do with the proper mathematical formulation and thus they have not to be prescribed, while standard kinematic and static boundary conditions and CBC have to be imposed to close the relevant nonlocal gradient problem. This conclusion follows from a well-posed abstract variational scheme conceived for nonlocal gradient inflected beams. The treatment provides as special cases most of the size-dependent models adopted in Engineering Science to assess size-effects in nanostructures, such as NSGT, strain-driven and stress-driven local-nonlocal elasticity approaches. Additionally, a well-posed Nonlocal Stress Gradient (NStressG) model is presented, coupling the stress-driven nonlocal strategy (Romano and Barretta, 2017) with the stress gradient elasticity. The presented methodology is elucidated and validated by investigating the structural behavior of a variety of inflected nano-beams of nanotechnological interest, such as sensors and actuators. NStressG is able to predict both softening and hardening nonlocal responses and, unlike the special stress gradient elasticity model, leads to well-posed structural problems in Nanomechanics. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Modelling and analysis of scale phenomena in micro- and nano-structures is a topic widely investigated in Engineering Science, being crucial for design and optimization of modern devices of nanotechnological interest, such as viscoelastic nanoresonators (Rahmanian & Hosseini-Hashemi, 2019), MEMS capacitors (Samaali et al., 2019), magnetorheological elastomer sandwich cantilever MEMS actuators (Akhavan, Ghadiri, & Zajkani, 2019), piezoelectric energy harvesters (Theng et al., 2019), long lifecycle MEMS beams (Zhang et al., 2019), MEMS silicon beams in transmission electron microscopes (LobatoDauzier et al., 2019), micro/nanoelectromechanical beam resonators (Gusso, Viana, Mathias, & Caldas, 2019), resonant MEMS magnetometers (Liu, Liang, & Xiong, 2019), micro resonant pressure sensors (Fu & Xu, 2019), MEMS arches (Ouakad & Sedighi, 2019), ring-beam (Ye, Zhou, Jin, Yu, & Wang, 2019) and multi-layer (Lu, Fan, & Morita, 2019) piezoelectric actuators, MEMS resonators under random mass disturbances (Qiao, Xu, Sun, & Zhang, 2019).

∗

Corresponding author. E-mail addresses: [email protected] (R. Barretta), [email protected] (F. Marotti de Sciarra).

https://doi.org/10.1016/j.ijengsci.2019.06.016 0020-7225/© 2019 Elsevier Ltd. All rights reserved.

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Such a objective is nowadays conveniently achieved by exploiting nonlocal and gradient continuum formulations of elasticity theory, based on the pioneering contributions by Rogula (1965, 1982), Kröner (1967) and Eringen (1972, 1983, 1987, 2002); Mindlin (1965, 1964). A survey on classical size-dependent constitutive models can be found in Bažant and Jirásek (2002). Adoption of a constitutive theory which leads to mathematically well-posed structural problems and that is able to significantly capture the size effects in nanostructures of applicative interest is a challenging issue, often not adequately addressed in literature. An emblematic example is the strain-driven nonlocal integral model of elasticity, successfully exploited by Eringen for problems involving fractured materials, screw dislocations and surface waves (Eringen, 1983). This theory is not applicable to nanostructures due to conflict between constitutive and equilibrium requirements (Romano & Barretta, 2016; Romano, Barretta, Diaco, & Marotti de Sciarra, 2017). As discussed in (Romano, Barretta, & Diaco, 2017) and Romano, Luciano, Barretta, and Diaco (2018), these difficulties have been partially bypassed by resorting to a mixture of strain-driven local and nonlocal phases formulated by Eringen (1972) and applied to nanostructures in several recent papers (see e.g. Fernández-Sáez & Zaera, 2017; Khaniki, 2019; Pisano & Fuschi, 2003; Wang, Zhu, & Dai, 2016; Zhu, Wang, & Dai, 2017). Ill-posedness of Eringen integral model in structural mechanics can be fully bypassed by resorting to the stress-driven nonlocal integral approach presented by Romano and Barretta (2017), effectively used to solve dynamic, thermal and buckling problems of nanomechanics (Apuzzo, Barretta, Luciano, Marotti de Sciarra, & Penna, 2017; Barretta, Diaco et al., 2018; Barretta, Fabbrocino, Luciano, Marotti de Sciarra, & Ruta, 2019; Barretta, Faghidian, & Luciano, 2018; Barretta, Faghidian, & ˇ Marotti de Sciarra, 2019; Barretta, Luciano, Marotti de Sciarra, & Ruta, 2018; Barretta, Canadija, Feo et al., 2018; Barretta, ˇ Luciano, & Marotti de Sciarra, 2018; Oskouie, Ansari, & Rouhi, 2018). Canadija, In recent years, nonlocal strain gradient elasticity developed in Lim, Zhang, and Reddy (2015) by coupling the strain gradient and Eringen’s strain-driven integral models has been exploited in a large number of papers in order to evaluate size effects in nanostructures for Nano-Electro-Mechanical Systems (NEMS) (see e.g. Barati, 2017; S¸ ims¸ ek, 2016; Ebrahimi & Barati, 2018; Ghayesh & Farajpour, 2018; 2019; Li, Li, & Hu, 2016; Li, Tang, & Hu, 2018; Li, Li, Hu, Ding, & Deng, 2017; Lu, Guo, & Zhao, 2017a; 2017b; Sahmani & Aghdam, 2018; Shafiei & She, 2018; She, Yuan, Karami, Ren, & Xiao, 2019; She, Yuan, & Ren, 2018; Xu, Wang, Zheng, & Ma, 2017; Xu, Zheng, & Wang, 2017; Xu, Zhou, & Zheng, 2017; Zhu & Li, 2017). In these papers, the considered structural problems are solved by prescribing unnecessary non-standard higher-order kinematic and static boundary conditions which, instead, should be replaced with the natural constitutive boundary conditions (CBC) established by Barretta and Marotti de Sciarra (2018) and exploited by Apuzzo, Barretta, Faghidian, Luciano, and Marotti de Sciarra (2018, 2019). In a recent paper (Zaera, Serrano, & Fernández-Sáez, 2019) it is claimed that both non-standard and constitutive boundary conditions are mandatory and that therefore nonlocal strain gradient structural problems admit no solution. Motivated by the need of amending this unsupported conclusion, we will show (in agreement with the previous outcomes in (Barretta & Marotti de Sciarra, 2018)) that, if the nonlocal gradient elasticity model is properly formulated in variational terms, the non-classical boundary conditions to be prescribed are of constitutive kind and that the non-standard (kinematic and static) boundary conditions have nothing to do with the nonlocal gradient theory of elasticity. Accordingly, the corresponding nonlocal gradient nanostructural problems of applicative interest are well-posed and can be advantageously investigated to assess size effects in modern devices, such as nanosensors and nanoactuators. The plan is the following. Preliminary assumptions and basic equations governing the kinematics and equilibrium of Bernoulli–Euler nanobeams under flexure are recalled in Section 2. The proper mathematical model of nonlocal strain gradient elasticity is obtained in Section 4 by specializing a novel abstract variational formulation presented in Section 3 that also provides, in particular, most of strain-driven and stress-driven elasticity laws available in literature. In addition, an effective stress-driven version of nonlocal strain gradient elasticity for inflected nanobeams is also inferred in Section 5 by the general well-posed abstract variational methodology. Selected case-sudies of nanotechnological interest are examined and compared in detail in Section 6. Closing remarks are provided in Section 7. 2. Kinematics and statics of inflected nano-beams In this section, some kinematic and static notions of Bernoulli–Euler beams are briefly recalled. No reference is made to constitutive aspects. Let us consider a beam of length L bent by an equilibrated transverse force system F composed of a distributed loading q: [0, L]→ and of boundary concentrated forces F0 and FL and couples M0 and ML . The equilibrated stress in a Bernoulli–Euler beam is a bending moment field M: [0, L]→ fulfilling the virtual work condition1

 F , δv  =



with

 F , δv  := 1

L

0



M (x ) (∂x2 δv )(x ) dx , L

0

q(x ) δv(x ) dx + ML (∂x δv )(L ) + M0 (∂x δv )(0 ) + FL δv(L ) + F0 δv(0 ) ,

The symbol ∂xn denotes nth differentiation along the beam axis x ∈ [0, L]⊆ .

(1)

(2)

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virtual work of the force F , for any transversal virtual displacement field δ v: [0, L]→ fulfilling homogeneous kinematic boundary conditions. The scalar function χδv := ∂x2 δv : [0, L] →  in the virtual work condition Eq. (1) is the kinematically compatible flexural curvature associated with the transversal virtual displacement δ v: [0, L]→ . Integrating by parts Eq. (1), a standard localization procedure provides the equivalent equilibrium differential problem

∂x2 M = q , in [0, L] ,

(3)

equipped with the (standard) boundary conditions

⎧ ⎪ ⎪−M (0 ) · (∂x δv )(0 ) = M0 · (∂x δv )(0 ) , ⎪ ⎨ M (L ) · (∂ δv )(L ) = M · (∂ δv )(L ) , x x L ⎪ ∂x M (0 ) · δv(0 ) = F0 · δv(0 ) , ⎪ ⎪ ⎩ −∂x M (L ) · δv(L ) = FL · δv(L ) .

(4)

3. Variational nonlocal gradient elasticity In this section, the nonlocal gradient elasticity for Bernoulli–Euler nano-beams is formulated by a consistent variational formulation which leads to well-posed structural problems of engineering interest. Let us preliminarily recall the definition of local elastic stiffness (assumed to be uniform along the beam axis) which associates the bending moment field M: [0, L]→ with the elastic flexural curvature χ : [0, L]→



k :=



E y2 dA ,

(5)

where  is the cross-section, E is the Euler-Young elastic modulus and y the flexural axis. To formulate a rather general formulation of nonlocal gradient elasticity, including as special cases the known models adopted in mechanics of nano-structures, we introduce an abstract size-dependent constitutive relation which relates two variables: a source field s: [0, L]→ and an output field f: [0, L]→ . Nonlocal Strain-driven Gradient (NstrainG) and Nonlocal Stress-driven Gradient (NstressG) models of elasticity are obtained by setting 1. NstrainG: s = χ and f = k−1 M , 2. NstressG: s = M and f = k χ . For conciseness sake, we adopt the following symbolism for the integral convolution between an averaging kernel φ λ and a source field s

 L    φc ( x − ξ ) · s ( ξ ) d ξ , φc ∗ s x = φc ∗ s (x ) :=



(6)

0

where x and ξ are points of the structural interval [0, L] and c is the length-scale parameter describing nonlocal effects. Following the seminal treatment by Eringen (1983), the kernel φ λ will be chosen in a such way that the following properties are met 1. Positivity and symmetry

φc ( x − ξ ) = φ c ( ξ − x ) ≥ 0 .

(7)

2. Normalisation



+∞

−∞

3. Impulsivity

φc (x ) dx = 1 .



lim+

c→0

+∞ −∞

φc ( x − ξ ) · s ( ξ ) d ξ = s ( x ) ,

(8)

(9)

for any continuous test source field s: → . The abstract formulation of nonlocal gradient elasticity is assumed to be governed by the following potential

R(s ) :=

1 2



L 0



 

α s2x + (1 − α ) φc ∗ s x sx + l 2 (φc ∗ ∂x sx )∂x sx dx ,

(10)

where α ∈ [0; 1] is a mixture parameter introduced by Eringen (1972) and l ∈ [0; +∞[ is a gradient length parameter introduced by Mindlin (1964). The nonlocal gradient response f , at a point x of the nanobeam domain [0, L] , associated with the source field s is provided by variational condition

 f, δ s  :=



L 0

f (x )δ s(x )dx =  dR(s ), δ s  ,

(11)

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for any virtual source field δ s ∈ C01 ([0, L];  ) . The r.h.s. of Eq. (11)  dR(s ), δ s  is the directional derivative d of the elastic energy R , evaluated at the source field s , along a virtual source field δ s . Remark 3.1. The choice of assuming virtual source fields δ s ∈ C01 ([0, L];  ) , having compact support in the structural domain, is motivated by the fact that the abstract differential condition Eq. (11) leads to well-posed nonlocal gradient formulations of elasticity. It is worth recalling that, in this framework, the boundary values δ sL := δ s(L) and δ s0 := δ s(0) are vanishing. The explicit expression of the output field f in terms of the source field s consists of two steps as follows. 1. Evaluation of the directional derivative of the elastic energy R Eq. (10) and integration by parts

 dR(s ), δ s  =

 

=

L 0



 

α sx + (1 − α ) φc ∗ s x δ sx + l 2 (φc ∗ ∂x s )x ∂x (δ s )x dx

 L  

∂x (φc ∗ ∂x s )x δ sx dx α sx + (1 − α ) φc ∗ s x δ sx dx − l 2 0 0   + l 2 ( φ c ∗ ∂ x s ) L δ s L − ( φc ∗ ∂ x s ) 0 δ s 0 . L



(12)

2. Prescription of the variational condition Eq. (11) and, as noted in Remark 3.1, vanishing of the virtual sources on the boundary δ sL = δ s0 = 0 in order to get, upon localization, the sought nonlocal gradient relation

f (x ) = α s (x ) + (1 − α )



   φc ∗ s ( x ) − l 2 ∂ x φc ∗ ∂ x s ( x ) .

(13)

Eq. (13) is an integro-differential law relating source s and output f fields. Following Eringen (1983) and assuming the special bi-exponential kernel

φc (x ) :=

1 2c





exp − |xc| ,

(14)

we get the following noteworthy result. Proposition 3.1. The nonlocal strain gradient integro-differential relation Eq. (12) is equivalent to the differential law

1 1 f (x ) − ∂x2 f (x ) = 2 s(x ) − c2 c



α+

l2 c2



∂x2 s(x ) ,

(15)

equipped with the constitutive boundary conditions (CBC)

⎧ ⎨∂x f (0 ) − ⎩∂x f (L ) +

1 c

f (0 ) = − αc s(0 ) +

1 c

f (L ) = αc s(L ) +





α+

α+

2

l c2

l2 c2





∂x s ( 0 ) ,

∂x s ( L ) .

(16)

The proof of Prop. 3.1 can be carried out by following the strategy exploited by Barretta and Marotti de Sciarra (2018) in the special framework of the purely nonlocal strain gradient model for elastic nano-beams. 4. Nonlocal strain Gradient (NstrainG) elasticity The nonlocal strain gradient (NstrainG) model of elasticity for inflected nanobeams, being of strain-driven kind, is obtained from the abstract formulation Eq. (13) by setting s = χ and f = k−1 M

M (x ) = k



 

  α χ ( x ) + ( 1 − α ) φc ∗ χ ( x ) − l 2 ∂ x φc ∗ ∂ x χ ( x ) .

(17)

Proposition 3.1 provides the equivalent differential problem

1 1 M (x ) − ∂x2 M (x ) = 2 k χ (x ) − k c2 c



α+

l2 c2



∂x2 χ (x ) ,

(18)

with the Constitutive Boundary Conditions (CBC)

 ⎧ ⎨∂x M (0 ) − 1c M (0 ) = −k αc χ (0 ) + k α + cl 22 ∂x χ (0 ) ,  ⎩∂x M (L ) + 1 M (L ) = k α χ (L ) + k α + l 22 ∂x χ (L ) . c c c

It is worth noting that the following special cases can be recovered by NstrainG.

(19)

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• Modified nonlocal strain gradient model. Setting α = 0 in Eq. (17), we get the differential equation (Lim et al., 2015)

1 1 l2 2 M ( x ) − ∂ M ( x ) = k χ ( x ) − k x c2 c2 c2

∂x2 χ (x ) ,

(20)

with CBC (Barretta & Marotti de Sciarra, 2018)

2 ∂x M (0 ) − 1c M (0 ) = k cl 2 ∂x χ (0 ) ,

(21)

∂x M (L ) + 1c M (L ) = k cl 2 ∂x χ (L ) . 2

Remark 4.1. Unlike the claims in the recent paper by Zaera, Serrano, & Fernández-Sáez (2019), the nonlocal strain gradient model of elasticity is governed by the differential problem Eq. (20) which has to be solved by prescribing the constitutive boundary conditions Eq. (21). Such a result is a direct consequence of the variational nonlocal gradient equation Eq. (11) in Section 3. The (non-standard) higher-order kinematic and static boundary conditions (KHOBC) (SHOBC), widely and improperly adopted in literature (see e.g. Li et al., 2016, Eq. (60)) and (see e.g. Xu, Wang et al., 2017, Eq. (31)5 ), have nothing to do with the well-posed mathematical model of nonlocal strain gradient elasticity illustrated in Section 3, which provides an effective approach to assess size effects in nanostructures. • Purely nonlocal strain-driven model. Setting α = 0 and l = 0 in Eq. (17), we get the differential equation (Eringen, 1983)

1 1 M (x ) − ∂x2 M (x ) = 2 k χ (x ) , c2 c

(22)

with CBC (Romano, Barretta, Diaco, Marotti de Sciarra et al., 2017)

 ∂x M (0 ) − 1c M (0 ) = 0 ,

(23)

∂x M (L ) + 1c M (L ) = 0 . • Two-phase local/nonlocal strain-driven model. Setting l = 0 in Eq. (17), we get the differential equation (Eringen, 1972; 1987)

1 1 M (x ) − ∂x2 M (x ) = 2 k χ (x ) − k α ∂x2 χ (x ) , c2 c

(24)

with CBC (Wang et al., 2016)

 ∂x M (0 ) − 1c M (0 ) = −k αc χ (0 ) + k α ∂x χ (0 ) , ∂x M (L ) + 1c M (L ) = k αc χ (L ) + k α ∂x χ (L ) .

(25)

5. Nonlocal stress Gradient (NstressG) elasticity The nonlocal stress gradient (NstressG) model of elasticity for inflected nanobeams, being of stress-driven kind, is obtained from the abstract formulation Eq. (13) by setting s = M and f = k χ

   

χ (x ) = k−1 α M (x ) + (1 − α ) φc ∗ M (x ) − l 2 ∂x φc ∗ ∂x M (x ) .

(26)

Proposition 3.1 provides, the equivalent differential problem

1 1 χ (x ) − ∂x2 χ (x ) = 2 k−1 M (x ) − k−1 c2 c



α+

l2 c2



∂x2 M (x ) ,

(27)

with the Constitutive Boundary Conditions (CBC)

 ⎧ ⎨∂x χ (0 ) − 1c χ (0 ) = −k−1 αc M (0 ) + k−1 α + cl 22 ∂x M (0 ) ,  ⎩∂x χ (L ) + 1 χ (L ) = k−1 α M (L ) + k−1 α + l 22 ∂x M (L ) . c c c

(28)

It is worth noting that the following special cases can be recovered by NstressG. • Purely nonlocal stress-driven model (Romano & Barretta, 2017). Setting α = 0 and l = 0 in Eq. (26), we get the differential equation

1 1 χ (x ) − ∂x2 χ (x ) = 2 k−1 M (x ) , c2 c

(29)

with the Constitutive Boundary Conditions (CBC)

 ∂x χ (0 ) − 1c χ (0 ) = 0 , ∂x χ (L ) + 1c χ (L ) = 0 .

(30)

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• Two-phase local/nonlocal stress-driven model (Barretta, Fabbrocino, Luciano, & Marotti de Sciarra, 2018). Setting l = 0 in Eq. (26), we get the differential equation

1 1 χ (x ) − ∂x2 χ (x ) = 2 k−1 M (x ) − α k−1 ∂x2 M (x ) , c2 c

(31)

with the Constitutive Boundary Conditions (CBC)

 ∂x χ (0 ) − 1c χ (0 ) = −k−1 αc M (0 ) + k−1 α ∂x M (0 ) , ∂x χ (L ) + 1c χ (L ) = k−1 αc M (L ) + k−1 α ∂x M (L ) .

(32)

6. Nonlocal gradient nano-structural problems and case-studies The variationally consistent nonlocal strain and stress gradient models of elasticity presented in Sections 4 (NstrainG) and 5 (NstressG) are adopted in this section to investigate the size-dependent behaviour of four structural schemes of engineering interest: cantilever, simply supported, clamped pinned and clamped clamped nano-beams. The cantilever beam is assumed to be subjected to a concentrated force FL , at the axial abscissa x = L (free cross-section), which describes the structural scheme of a nano-actuator. The other nano-beams are assumed to be subjected to a uniformly distributed loading q . The relevant elastostatic problems associated with NstrainG and NstressG are formulated in terms of the transverse displacement field v: [0, L]→ , whose linearized flexural curvature is expressed by the Bernoulli–Euler differential condition of kinematic compatibility

χv = ∂x2 v : [0, L] →  .

(33)

In absence of thermal distortions, the kinematically compatible flexural curvature χ v coincides with the elastic flexural curvature χ which is constitutively related to the bending moment M by means of the integral relations Eqs. (17) and (26), governing NstrainG and NstressG respectively. The ordinary differential equations (ODE) governing the elastic equilibrium of NstrainG and NstressG inflected beams are respectively obtained by replacing in Eqs. (18) and (27) the elastic flexural curvature field χ with the second derivative of the transverse displacement ∂x2 v , by taking the second axial derivative and enforcing the differential equilibrium condition Eq. (3). 1. NstrainG ODE

1 1 q(x ) − ∂x2 q(x ) = 2 k ∂x4 v(x ) − k c2 c



α+

l2 c2



∂x6 v(x ) ,

(34)

The sixth-order NstrainG ODE (34) is univocally solved by prescribing two Constitutive Boundary Conditions (CBC) Eq. (19), which are independent of the considered boundary kinematic constraints, and four Standard (kinematic/static) Boundary Conditions (SBC) specialized below for the examined case-studies. CBC Eq. (19) are explicitly expressed in terms of transverse displacements by replacing the elastic curvature χ with

∂x2 v

 ⎧ ⎨∂x M (0 ) − 1c M (0 ) = −k αc ∂x2 v(0 ) + k α + cl 22 ∂x3 v(0 ) ,  ⎩∂x M (L ) + 1 M (L ) = k α ∂ 2 v(L ) + k α + l 22 ∂ 3 v(L ) , x c c x c

(35)

and observing that the expression of the equilibrated bending moment M is obtained by prescribing in Eq. (18) the differential condition of equilibrium Eq. (3)

M (x ) = c2 q(x ) + k ∂x2 v(x ) − k c2



α+

l2 c2



∂x4 v(x ) .

(36)

The derivative of the bending moment field is therefore given by

 l2 ∂x M (x ) = c2 ∂x q(x ) + k ∂x3 v(x ) − k c2 α + 2 ∂x5 v(x ) . c

(37)

2. NstressG ODE

1 4 1 ∂ v(x ) − ∂x6 v(x ) = 2 k−1 q(x ) − k−1 c2 x c



α+

l2 c2



∂x2 q(x ) ,

(38)

The sixth-order NstressG ODE (38) is univocally solved by prescribing two Constitutive Boundary Conditions (CBC) Eq. (28), which are independent of the considered boundary kinematic constraints, and four Standard (kinematic/static) Boundary Conditions (SBC) specialized below for the examined case-studies.

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79

Fig. 1. Cantilever nano-beam under concentrated load at the free end: non dimensional transverse displacement of the free end v∗ (1) vs. gradient parameter λl , with nonlocal parameter λc = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG.

Fig. 2. Cantilever nano-beam under concentrated load at the free end: non dimensional transverse displacement of the free end v∗ (1) vs. nonlocal parameter λc , with gradient parameter λl = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG.

CBC Eq. (28) are explicitly expressed in terms of transverse displacements by replacing the elastic curvature χ with

∂x2 v

 ⎧ ⎨∂x3 v(0 ) − 1c ∂x2 v(0 ) = −k−1 αc M (0 ) + k−1 α + cl 22 ∂x M (0 ) ,  ⎩∂ 3 v(L ) + 1 ∂ 2 v(L ) = k−1 α M (L ) + k−1 α + l 22 ∂x M (L ) , x c x c c

(39)

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Fig. 3. Cantilever nano-beam under concentrated load at the free end: non dimensional transverse displacement of the free end v∗ (1) vs. gradient parameter λl and nonlocal parameter λc , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG.

Fig. 4. Cantilever nano-beam under concentrated load at the free end: non dimensional transverse displacement of the free end v∗ (1) vs. gradient parameter λl and nonlocal parameter λc , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStressG.

and observing that the expression of the equilibrated bending moment M is obtained by prescribing in Eq. (27) the differential condition of equilibrium Eq. (3)

M (x ) = c2



α+

l2 c2



q(x ) + k ∂x2 v(x ) − c2 k ∂x4 v(x ) .

(40)

The derivative of the bending moment field is therefore given by

 l2 ∂x M (x ) = c2 α + 2 ∂x q(x ) + k ∂x3 v(x ) − c2 k ∂x5 v(x ) . c

(41)

The four additional Standard (kinematic/static) Boundary Conditions (SBC) to be prescribed to solve the ODEs above Eqs. (34) and (38) are formulated for the following simple structural schemes of engineering interest, by specializing the analysis carried out in Section 2.

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81

Fig. 5. Simply supported nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. gradient parameter λl , with nonlocal parameter λc = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG.

Fig. 6. Simply supported nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. nonlocal parameter λc , with gradient parameter λl = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG.

• Cantilever beam under concentrated load FL at the free end. The displacement solution field v: [0, L]→ has to fulfill the (standard) essential kinematic boundary conditions

v(0 ) = 0 , and (∂x v )(0 ) = 0 .

(42)

Accordingly, the transversal virtual displacement fields δ v: [0, L]→ have to fulfill the homogeneous boundary conditions

δv(0 ) = 0 , and (∂x δv )(0 ) = 0 ,

(43)

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Fig. 7. Simply supported nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. gradient parameter λl and nonlocal parameter λc , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG.

Fig. 8. Simply supported nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. gradient parameter λl and nonlocal parameter λc , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStressG.

so that the boundary relations Eq. (4) provide the (standard) natural static boundary conditions

M (L ) = 0 ,

and

− (∂x M )(L ) = FL .

(44)

• Simply supported beam under uniformly distributed load q. The displacement solution field v: [0, L]→ has to fulfill the (standard) essential kinematic boundary conditions

v (0 ) = 0 , v (L ) = 0 .

(45)

Accordingly, the transversal virtual displacement fields δ v: [0, L]→ have to fulfill the homogeneous boundary conditions

δv(0 ) = 0 , δv(L ) = 0 ,

(46)

so that the boundary relations Eq. (4) provide the (standard) natural static boundary conditions

M (0 ) = 0 ,

M (L ) = 0 .

(47)

• Clamped pinned beam under uniformly distributed load q. The displacement solution field v: [0, L]→ has to fulfill the (standard) essential kinematic boundary conditions

v(0 ) = 0 , (∂x v )(0 ) = 0 , v(L ) = 0 .

(48)

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Fig. 9. Clamped pinned nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. gradient parameter λl , with nonlocal parameter λc = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG.

Fig. 10. Clamped pinned nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. nonlocal parameter λc , with gradient parameter λl = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG.

Accordingly, the transversal virtual displacement fields δ v: [0, L]→ have to fulfill the homogeneous boundary conditions

δv(0 ) = 0 , (∂x δv )(0 ) = 0 , δv(L ) = 0 ,

(49)

so that the boundary relations Eq. (4) provide the single (standard) natural static boundary condition

M (L ) = 0 .

(50)

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Fig. 11. Clamped pinned nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. gradient parameter λl and nonlocal parameter λc , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG.

Fig. 12. Clamped pinned nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. gradient parameter λl and nonlocal parameter λc , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStressG.

• Clamped clamped beam under uniformly distributed load q. The displacement solution field v: [0, L]→ has to fulfill the (standard) essential kinematic boundary conditions

v(0 ) = 0 , (∂x v )(0 ) = 0 , v(L ) = 0 , (∂x v )(L ) = 0 .

(51)

Accordingly, the transversal virtual displacement fields δ v: [0, L]→ have to fulfill the homogeneous boundary conditions

δv(0 ) = 0 , (∂x δv )(0 ) = 0 , δv(L ) = 0 , (∂x δv )(L ) = 0 ,

(52)

so that the boundary relations Eq. (4) are verified. As a consequence, no (standard) natural static boundary conditions have to be prescribed. Before providing numerical solutions of the structural problems described above, we conveniently define the nondimensional nonlocal and gradient parameters, respectively by

λc :=

c , L

λl :=

l . L

(53)

Furthermore, the following non-dimensional transverse displacement field v∗ : [0, 1]→ is introduced as

v∗ :=

K v, qL4

(54)

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85

Fig. 13. Clamped clamped nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. gradient parameter λl , with nonlocal parameter λc = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG.

Fig. 14. Clamped clamped nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. nonlocal parameter λc , with gradient parameter λl = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG.

except for the cantilever beam that is assumed to be given by

v∗ :=

K v. F L3

(55)

The maximum displacement v∗ (1) of the cantilever nano-beam under concentrated load at the free end, associated with NStrainG and NStressG, is numerically evaluated and collected in Tables 1 and 2 for several values of the characteristic parameters λc and λl and of the mixture parameter α . A stiffening response is exhibited by NStrainG for increasing gradient parameter λl (see Fig. 1) and a softening behaviour is observed for increasing nonlocal parameter λc (see Fig. 2).

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Fig. 15. Clamped clamped nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. gradient parameter λl and nonlocal parameter λc , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG.

Fig. 16. Clamped clamped nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. gradient parameter λl and nonlocal parameter λc , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStressG. Table 1 Cantilever nano-beam under concentrated load at the free end: non dimensional transverse displacement of the free end v∗ (1) vs. gradient parameter λl , with the nonlocal parameter λc = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG. v∗ (1)

λl +

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

NStrainG

NStressG

α =0

α =0.5

α =1.0

α =0

α =0.5

α =1.0

1.08333 0.97534 0.86180 0.76428 0.69260 0.64293 0.60871 0.58475 0.56756 0.55494 0.54545

0.46003 0.45720 0.44917 0.43743 0.42415 0.41114 0.39945 0.38944 0.38106 0.37415 0.36845

0.33333 0.33200 0.32823 0.32264 0.31601 0.30905 0.30230 0.29606 0.29048 0.28559 0.28135

0.15758 0.15894 0.16300 0.16976 0.17924 0.19142 0.20630 0.22390 0.24420 0.26720 0.29292

0.24546 0.24681 0.25087 0.25764 0.26711 0.27929 0.29418 0.31177 0.33207 0.35508 0.38079

0.33333 0.33469 0.33875 0.34551 0.35499 0.36717 0.38205 0.39965 0.41995 0.44295 0.46867

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Table 2 Cantilever nano-beam under concentrated load at the free end: non dimensional transverse displacement of the free end v∗ (1) vs. nonlocal parameter λc , with gradient parameter λl = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG.

λc

v∗ (1) NStrainG

0+ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

NStressG

α =0

α =0.5

α =1.0

α =0

α =0.5

α =1.0

0.27373 0.33804 0.40711 0.48095 0.55956 0.64293 0.73108 0.82399 0.92167 1.02412 1.13134

0.27373 0.30734 0.33758 0.36480 0.38926 0.41114 0.43068 0.44807 0.46354 0.47731 0.48958

0.27373 0.28285 0.29126 0.29843 0.30431 0.30905 0.31287 0.31594 0.31842 0.32045 0.32211

0.45833 0.38434 0.31719 0.26329 0.22242 0.19142 0.16748 0.14860 0.13339 0.12092 0.11053

0.45833 0.40884 0.36335 0.32688 0.29961 0.27929 0.26388 0.25191 0.24242 0.23474 0.22841

0.45833 0.43334 0.40951 0.39047 0.37680 0.36717 0.36027 0.35523 0.35145 0.34855 0.34629

Table 3 Simply supported nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. gradient parameter λl , with nonlocal parameter λc = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG. v∗ (1/2)

λl +

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

NStrainG

NStressG

α =0

α =0.5

α =1.0

α =0

α =0.5

α =1.0

0.04427 0.03118 0.02146 0.01665 0.01429 0.01302 0.01228 0.01181 0.01149 0.01127 0.01112

0.01741 0.01684 0.01544 0.01383 0.01238 0.01123 0.01034 0.00968 0.00917 0.00879 0.00849

0.01302 0.01278 0.01213 0.01127 0.01037 0.00954 0.00883 0.00824 0.00776 0.00738 0.00706

0.00674 0.0 070 0 0.00775 0.0 090 0 0.01076 0.01302 0.01578 0.01905 0.02281 0.02708 0.03185

0.00988 0.01013 0.01089 0.01214 0.01390 0.01616 0.01892 0.02218 0.02595 0.03022 0.03499

0.01302 0.01327 0.01403 0.01528 0.01704 0.01930 0.02206 0.02532 0.02909 0.03336 0.03813

Table 4 Simply supported nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. nonlocal parameter λc , with gradient parameter λl = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG. v∗ (1/2)

λc 0+ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

NStrainG

NStressG

α =0

α =0.5

α =1.0

α =0

α =0.5

α =1.0

0.01065 0.01075 0.01103 0.01151 0.01217 0.01302 0.01406 0.01529 0.01671 0.01832 0.02012

0.01065 0.00997 0.00975 0.00996 0.01049 0.01123 0.01207 0.01296 0.01383 0.01466 0.01543

0.01065 0.00935 0.00885 0.00886 0.00915 0.00954 0.00995 0.01033 0.01067 0.01097 0.01122

0.04427 0.03592 0.02689 0.02039 0.01602 0.01302 0.01088 0.00929 0.00807 0.00712 0.00636

0.04427 0.03639 0.02821 0.02246 0.01869 0.01616 0.01439 0.01310 0.01213 0.01139 0.01080

0.04427 0.03687 0.02953 0.02453 0.02136 0.01930 0.01790 0.01691 0.01619 0.01566 0.01524

87

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Table 5 Clamped pinned nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. gradient parameter λl , with nonlocal parameter λc = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG.

λl

v∗ (1/2) NStrainG

0+ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

NStressG

α =0

α =0.5

α =1.0

α =0

α =0.5

α =1.0

0.04187 0.02746 0.01638 0.01045 0.00719 0.00523 0.00395 0.00308 0.00246 0.00201 0.00166

0.00866 0.00823 0.00717 0.00592 0.00477 0.00384 0.00311 0.00254 0.00210 0.00176 0.00149

0.37596 0.00506 0.00465 0.00411 0.00354 0.0 030 0 0.00254 0.00215 0.00182 0.00156 0.00134

0.0 0 085 0.0 0 099 0.00143 0.00219 0.00331 0.00485 0.00685 0.00934 0.01235 0.01591 0.02001

0.00312 0.00327 0.00373 0.00451 0.00565 0.00718 0.00913 0.01154 0.01443 0.01782 0.02174

0.00521 0.00536 0.00584 0.00664 0.00780 0.00933 0.01126 0.01362 0.01644 0.01974 0.02355

Table 6 Clamped pinned nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. nonlocal parameter λc , with gradient parameter λl = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG. v∗ (1/2)

λc +

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

NStrainG

NStressG

α =0

α =0.5

α =1.0

α =0

α =0.5

α =1.0

0.00114 0.00162 0.00227 0.00309 0.00407 0.00523 0.00655 0.00804 0.00971 0.01155 0.01357

0.00114 0.00154 0.00203 0.00260 0.00321 0.00384 0.00445 0.00502 0.00555 0.00603 0.00645

0.00114 0.00147 0.00184 0.00225 0.00264 0.0 030 0 0.00332 0.00358 0.00381 0.0 040 0 0.00415

0.03007 0.02191 0.01433 0.00953 0.00665 0.00485 0.00368 0.00287 0.00230 0.00188 0.00157

0.03007 0.02281 0.01590 0.01149 0.00884 0.00718 0.00610 0.00536 0.00483 0.00445 0.00415

0.03007 0.02365 0.01735 0.01329 0.01085 0.00933 0.00833 0.00765 0.00717 0.00682 0.00655

Table 7 Clamped clamped nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. gradient parameter λl , with nonlocal parameter λc = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG. v∗ (1/2)

λl 0+ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

NStrainG

NStressG

α =0

α =0.5

α =1.0

α =0

α =0.5

α =1.0

0.03906 0.02392 0.01271 0.00717 0.00446 0.0 030 0 0.00214 0.00160 0.00124 0.0 0 099 0.0 0 080

0.00481 0.00455 0.00391 0.00316 0.00249 0.00196 0.00155 0.00125 0.00102 0.0 0 084 0.0 0 070

0.00260 0.00252 0.00231 0.00203 0.00173 0.00146 0.00122 0.00102 0.0 0 086 0.0 0 073 0.0 0 063

0.0 0 020 0.0 0 027 0.0 0 049 0.0 0 085 0.00135 0.0 020 0 0.00280 0.00373 0.00482 0.00604 0.00741

0.00141 0.00149 0.00172 0.00211 0.00266 0.00337 0.00423 0.00525 0.00642 0.00775 0.00924

0.00260 0.00269 0.00293 0.00334 0.00391 0.00465 0.00555 0.00662 0.00785 0.00924 0.01079

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Table 8 Clamped clamped nano-beam under uniformly distributed load: non dimensional midpoint deflection v∗ (1/2) vs. nonlocal parameter λc , with gradient parameter λl = 0.5 , evaluated by α = 0 , α = 0.5 and α = 1.0 in NStrainG and NStressG. v∗ (1/2)

λc +

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

NStrainG

NStressG

α =0

α =0.5

α =1.0

α =0

α =0.5

α =1.0

0.0 0 024 0.0 0 046 0.0 0 084 0.00138 0.00210 0.0 030 0 0.00408 0.00534 0.00678 0.00842 0.01023

0.0 0 024 0.0 0 045 0.0 0 076 0.00114 0.00155 0.00196 0.00235 0.00270 0.00301 0.00327 0.00351

0.0 0 024 0.0 0 043 0.0 0 069 0.0 0 096 0.00123 0.00146 0.00165 0.00180 0.00193 0.00203 0.00211

0.01823 0.01199 0.0 070 0 0.00430 0.00284 0.0 020 0 0.00148 0.00114 0.0 0 090 0.0 0 073 0.0 0 060

0.01823 0.01300 0.00835 0.00569 0.00422 0.00337 0.00283 0.00247 0.00223 0.00205 0.00192

0.01823 0.01395 0.00959 0.00697 0.00551 0.00465 0.00411 0.00376 0.00351 0.00334 0.00321

A softening response is instead exhibited by NStressG for increasing gradient parameter λl (see Fig. 1) and a stiffening behaviour is observed for increasing nonlocal parameter λc (see Fig. 2). It is worth noting that, unlike NStrainG, the plots associated with NStressG do not present physically unmotivated changes of concavity. 3-D evidences of the structural responses corresponding to NStrainG and NStressG are provided in Figs. 3 and 4 respectively. All the comments above for the nanoactuator apply mutatis mutandis to all other case-studies. In particular, numerical evidences and corresponding plots are reported in: • Tables 3 and 4 and Figs. 5–8 for simply supported nano-beams subjected to uniformly distributed loading; • Tables 5 and 6 and Figs. 9–12 for clamped pinned nano-beams subjected to uniformly distributed loading; • Tables 7 and 8 and Figs. 13–16 for clamped pinned nano-beams subjected to uniformly distributed loading. It also interesting underline that the variationally consistent strain gradient model of elasticity (SGT) is a special case of NStrainG by setting a vanishing mixture parameter α = 0 . The numerical results for the nano-actuator reported in Tables 1 and 2 extend therefore those recently contributed by Barretta and Marotti de Sciarra (2018). 7. Conclusions The outcomes of the present study may be summarized as follows. • The nonlocal gradient theory of elasticity for inflected nano-beams has been developed by making recourse to a consistent variational constitutive formulation equipped with suitably chosen test curvature fields. The associated elastic equilibrium problem has been shown to be governed by a sixth-order differential equation, involving the nonlocal transverse displacement field, whose solution requires prescription of uniquely standard essential and natural (kinematic and static) boundary conditions of the classical Bernoulli–Euler beam theory and of constitutive boundary conditions. • Unlike the claims in the recent paper (Zaera, Serrano, & Fernández-Sáez, 2019), it has been proven that the nonstandard kinematic and static higher-order boundary conditions, widely and improperly resorted to in nonlocal strain gradient mechanics of nanostructures, have nothing to do with the proper mathematical formulation of nonlocal gradient elasticity and therefore they do not conflict with the natural constitutive boundary conditions. The nonlocal gradient elasticity model leads to well-posed problems of Nanomechanics, that are generally defined in bounded domains. • The contributed variational constitutive formulation has been developed in abstract terms in order to obtain a consistent unified model of nonlocal gradient elasticity which is able to provide both strain-driven and stress-driven methodologies. Moreover, the well-known nonlocal, local/nonlocal, strain gradient, stress gradient and nonlocal strain gradient models have been recovered as special cases. • The well-posed stress-driven nonlocal model of elasticity presented by Romano and Barretta (2017) has been shown to be a particular case of the novel Nonlocal Stress Gradient (NStressG) of elasticity which advantageously is able to simulate also softening structural responses. Accordingly, NStressG can be useful for design and optimize structural components of ground-breaking MEMS and NEMS. • The strain-driven and stress-driven nonlocal gradient models have been applied to selected nano-beams of applicative interest, such as nano-sensors and nano-actuators. Size effects have been numerically evaluated for cantilever, simply supported, clamped pinned and clampled clamped nano-beams under a variety of loading systems. Strain-driven

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