A stress-driven local-nonlocal mixture model for Timoshenko nano-beams
Raffaele Barretta, Andrea Caporale, S. Ali Faghidian, Raimondo Luciano, Francesco Marotti de Sciarra, Carlo Maria Medaglia
AA stress-driven local-nonlocal mixture model forTimoshenko nano-beams Ra aele Barretta a , Andrea Caporale b , S. Ali Faghidian c , RaimondoLuciano b , Francesco Marotti de Sciarra a , Carlo Maria Medaglia d a Department of Structures for Engineering and Architecture, University of NaplesFederico II, e-mails: [email protected] - [email protected] b Department of Civil and Mechanical Engineering, University of Cassino and SouthernLazio, e-mail: [email protected] - [email protected] c Department of Mechanical Engineering, Science and Research Branch, Islamic AzadUniversity, Tehran, Iran, e-mail: [email protected] d Link Campus University, e-mail: [email protected]
Abstract
A well-posed stress-driven mixture is proposed for Timoshenko nano-beams.The model is a convex combination of local and nonlocal phases and circum-vents some problems of ill-posedness emerged in strain-driven Eringen-likeformulations for structures of nanotechnological interest. The nonlocal partof the mixture is the integral convolution between stress field and a bi-exponential averaging kernel function characterized by a scale parameter.The stress-driven mixture is equivalent to a di erential problem equippedwith constitutive boundary conditions involving bending and shear fields.Closed-form solutions of Timoshenko nano-beams for selected boundary andloading conditions are established by an e ective analytical strategy. Thenumerical results exhibit a sti ening behavior in terms of scale parameter. Key words:
Integral elasticity, local/nonlocal stress-driven mixture,stubby nano-beams, nanomaterials, NEMS. . Introduction
The local continuum theory fails to capture size e ects in nanodevices,such as Nano-Electro-Mechanical Systems (NEMS) [1, 2, 3], and is not ableto describe the behavior of structures characterized by an external overalllength not much greater than the material internal characteristic length.The scientific literature suggests various approaches (atomistic, strain gra-dient, Eringen nonlocal theory and so on) to be considered when the localcontinuum theory is inadequate, see e.g. [4, 5, 6, 7, 8, 9, 10] and referencescited therein. In this paper, an innovative stress-driven nonlocal mixture isproposed for Timoshenko nano-beams. Such a model is combination of lo-cal and nonlocal elastic phases and di ers from Eringen-like [11, 12, 13, 14],strain gradient [15, 16, 17, 18, 19], couple stress models [20, 21] and otherapproaches available in literature [22, 23, 24, 25, 26, 27, 28, 29, 30]. Theneed of the proposed stress-driven approach with mixture of local and non-local phases derives from a recent research [31] on behavior of the strain-driven nonlocal model for bounded nanostructures, where the bending fieldis expressed as convolution of elastic curvature with an averaging kernel as-suming an exponential expression. The associated elastostatic problem canbe solved provided that the bending moment field satisfies suitable consti-tutive boundary conditions (CBC). This verification usually fails in casesof applicative interest, arising an ill-posedness of nonlocal nonlocal prob-lems in nanomechanics. In order to overcome this di culty, Romano andBarretta [32, 33] proposed a stress-driven nonlocal integral model, wherethe bending field is placed in the proper position of input variable so that2he constitutive law is evaluated by convolution between bending field andan averaging kernel. Strain-driven mixtures of local and nonlocal materiallaws are also considered in literature [34, 35, 36, 37, 38, 39, 40]: the localelastic fraction of the mixture induces well-posedness [31]. However, thisbeneficial e ect does not hold for vanishing local fractions, see e.g. [41].On the contrary, the stress-driven theory does not su er the limiting be-havior of strain-driven formulations as the local fraction tends to zero. Thestress-driven model has been adopted in various problems such as bendingof functionally graded nano-beams [42, 43] and nonlocal thermoelastic be-havior of nano-beams [44]. Such a model provides a sti ening structuralbehavior in accordance with experimental evidences [15, 45].In this paper, a stress-driven local-nonlocal mixture defined by convexlycombining local and nonlocal phases is adopted and presented in Section 2.In Section 3, it is assumed that the kernel function appearing in the integralconvolution is a bi-exponential function. Under this assumption, the inte-gral formulation in Section 2 is equivalent to a di erential problem equippedwith suitable constitutive boundary conditions. In Sections 4.1 and 4.2, pro-cedures for obtaining closed-form solutions of Timoshenko nano-beams areillustrated by applying integral and di erential formulations. Such proce-dures provide general closed-form solutions, which are valid for any kindof boundary conditions and external loads applied to the nano-beam andare evaluated for some fundamental schemes in Appendix A. Closed-formsolutions may have long expressions because contain the e ects of both theflexural and shear displacements. For this reason, numerical solutions havebeen presented in Section 5 for selected stubby nano-beams.3 . Mixture stress-driven integral model for Timoshenko beams The mixture stress-driven integral model (MStreDM) is applied to Tim-oshenko nano-beams. MStreDM is the generalization to stubby beams ofa previous constitutive version formulated in [42] for Bernoulli-Euler nano-beams in which the flexural curvature field ‰ ( x ) is expressed by the followingtwo-phase law: ‰ ( x ) = – M ( x ) EI + (1 ≠ – ) ⁄ L Â ( x ≠ t, L c ) M ( t ) EI dt, (1)with
L > nano-beam length; EI and M stand for local bending sti nessand moment, respectively;  is an averaging kernel dependent on the smallscale coe cient L c > . In the proposed MstreDM for Timoshenko beams,the shear deformation is convex combination of local and nonlocal phases,with the nonlocal phase given by Romano-Barretta stress-driven integrallaw. The shear deformation, evaluated as di erence between the slope ◊ ( x ) of the beam deformed center-line and the rotation „ ( x ) , is expressed by: ◊ ( x ) ≠ „ ( x ) = – T ( x ) Ÿ GA + (1 ≠ – ) ⁄ L  ( x ≠ t, L c ) T ( t ) Ÿ GA dt, (2)with GA local shear sti ness and Ÿ shear factor. In (1) and (2), bendingmoment M and shear force T are equilibrated. The phase parameter – belongs to [0 , . Integral equations (1) and (2) are coupled through thekinematic condition ‰ ( x ) = „ Õ ( x ) and the static condition M Õ ( x ) = ≠ T ( x ) .Figure 1 shows the meaning of some kinematic variables involved in theadopted nano-beam model. 4 . Mixture stress-driven di erential law for Timoshenko beams Assuming the following exponential expression for the function  :  ( x, L c ) = 12 L c exp ≠ | x | L c , (3)it will be proven that the problem defined by integral equations (1) and(2) admits an equivalent di erential form subject to constitutive boundaryconditions. Barretta et al. [42] have demonstrated that the nonlocal consti-tutive law (1) is equivalent to the following second-order di erential equationwhen  ( x, L c ) is assumed equal to the Helmholtz kernel (3): ‰ ÕÕ ( x ) ≠ L c ‰ ( x ) = – M ÕÕ ( x ) EI ≠ L c M ( x ) EI , (4)where x œ [0 , L ] and bending curvature ‰ ( x ) satisfies the following consti-tutive boundary conditions (CBCs) Y______]______[ ‰ Õ (0) ≠ L c ‰ (0) = – M Õ (0) EI ≠ – L c M (0) EI , ‰ Õ ( L ) + 1 L c ‰ ( L ) = – M Õ ( L ) EI + – L c M ( L ) EI . (5)In (5), derivatives are evaluated at the end points of the nano-beam, suchas ‰ Õ (0) © d ‰ dx | x =0 . Next, we demonstrate that the non-local constitutivelaw (2) is also equivalent to a second-order di erential equation subject toconstitutive boundary conditions. Let us split the integral in (2) by setting ◊ ( x ) ≠ „ ( x ) = – T ( x ) Ÿ GA + (1 ≠ – )( “ ( x ) + “ ( x )) (6)5ith Y______]______[ “ ( x ) = ⁄ x  ( x ≠ t, L c ) T ( t ) Ÿ GA dt = ⁄ x L c exp t ≠ xL c T ( t ) Ÿ GA dt, “ ( x ) = ⁄ Lx  ( x ≠ t, L c ) T ( t ) Ÿ GA dt = ⁄ Lx L c exp x ≠ tL c T ( t ) Ÿ GA dt. (7)In (7), the dependence of “ and “ on L c is omitted for lightening theexpressions in the next equations. Taking the first derivative, we get Y______]______[ “ Õ ( x ) = 1 L c + T ( x )2 Ÿ GA ≠ “ ( x ) , “ Õ ( x ) = 1 L c ≠ T ( x )2 Ÿ GA + “ ( x ) (8)and, from (8), di erentiating (6) yields the relation: ◊ Õ ( x ) ≠ „ Õ ( x ) = – T Õ ( x ) Ÿ GA + 1 ≠ – L c ( “ ( x ) ≠ “ ( x )) . (9)Di erentiating again, from (8) we get ◊ ÕÕ ( x ) ≠ „ ÕÕ ( x ) = – T ÕÕ ( x ) Ÿ GA + 1 ≠ – L c ≠ T ( x ) Ÿ GA + “ ( x ) + “ ( x ) , (10)which can be rewritten as ◊ ÕÕ ( x ) ≠ „ ÕÕ ( x ) = – T ÕÕ ( x ) Ÿ GA + 1 ≠ – L c ≠ T ( x ) Ÿ GA + 1 L c ◊ ( x ) ≠ „ ( x ) ≠ – T ( x ) Ÿ GA (11)6r, more concisely, as ◊ ÕÕ ( x ) ≠ ◊ ( x ) L c = – T ÕÕ ( x ) Ÿ GA ≠ L c T ( x ) Ÿ GA + „ ÕÕ ( x ) ≠ „ ( x ) L c . (12)Eqs. (2) and (12) provide the Bernoulli-Euler kinematic relation „ ( x ) = ◊ ( x ) when the shear sti ness GA is infinitely large. We get the constitutiveboundary conditions for (12) by evaluating (6) at the boundary points x = 0 and x = L of the nano-beam axis: (1 ≠ – ) “ (0) = ◊ (0) ≠ „ (0) ≠ – T (0) Ÿ GA , (1 ≠ – ) “ ( L ) = ◊ ( L ) ≠ „ ( L ) ≠ – T ( L ) Ÿ GA (13)and then by substituting (13) in (9) evaluated at the same boundary points: ◊ Õ (0) ≠ L c ◊ (0) = – T Õ (0) Ÿ GA ≠ – L c T (0) Ÿ GA + „ Õ (0) ≠ L c „ (0) , ◊ Õ ( L ) + 1 L c ◊ ( L ) = – T Õ ( L ) Ÿ GA + – L c T ( L ) Ÿ GA + „ Õ ( L ) + 1 L c „ ( L ) . (14)Concluding, the integral formulation governed by equations (1) and (2) ad-mits the di erential formulation represented by the di erential equations(4) and (12) subject to the constitutive boundary conditions (5) and (14),respectively. The integral formulation provides the same solution given bythe di erential formulation. 7 . Nonlocal solution procedures Next, we describe the procedures for obtaining the solution of the Tim-oshenko nano-beams by applying both the integral formulation explainedin Section 2 and the the di erential formulation explained in Section 3.These procedures provide general solutions valid for any kinds of boundaryconditions and external loads applied to the nano-beam. Specifically, theconsidered external loads are a transverse distributed load q y ( x ) , a concen-trated load F or a concentrated couple m , but other load such as distributedcouples may also be introduced in the formulation. The procedure based onthe integral formulation is explained in the following Section 4.1 whereasthe procedure based on the di erential form is described in Section 4.2. Solving the equilibrium di erential equation M ÕÕ ( x ) = q y ( x ) , the bendingmoment is given by M ( x ) = ⁄ x ( x ≠ s ) q y ( s ) ds + A x + A , (15)resulting an expression in terms of the two integration constants A and A .Setting f ( x ) = 1 EI ⁄ x ( x ≠ s ) q y ( s ) ds + A x + A , (16)and taking into account (1) and (15), the bending curvature ‰ ( x ) is ‰ ( x ) = – f ( x ) + (1 ≠ – ) ⁄ L  ( x ≠ t, L c ) f ( t ) dt. (17)8he evaluation of (17) provides the curvature ‰ ( x ) again in terms of the twointegration constants A and A . Then, solving the kinematic compatibilitydi erential equation ‰ ( x ) = „ Õ ( x ) , the cross-sectional rotation is „ ( x ) = ⁄ x ‰ ( s ) ds + A , (18)which, by virtue of (17), results an expression in terms of the three integra-tion constants A , A and A . Solving the equilibrium di erential equation M Õ ( x ) = ≠ T ( x ) and taking into account (15), the shear force is given by T ( x ) = ≠ ⁄ x q y ( s ) ds ≠ A . (19)Setting h ( x ) = ≠ Ÿ GA x q y ( s ) ds + A , (20)and taking into account (2) and (19), the slope ◊ ( x ) of the transverse dis-placements is ◊ ( x ) ≠ „ ( x ) = – h ( x ) + (1 ≠ – ) ⁄ L  ( x ≠ t, L c ) h ( t ) dt, (21)which now is evaluated with rotation „ ( x ) given by (18), providing a newexpression of ◊ ( x ) . Solving the kinematic compatibility di erential equation ◊ ( x ) = v Õ ( x ) with ◊ ( x ) given by (21), the transverse displacement is v ( x ) = ⁄ x ◊ ( s ) ds + A , (22)resulting an expression in terms of the four integration constants A , A , A and A to be determined by using the canonical boundary conditions. There-9ore, the last step is determining the above-mentioned integration constantsby solving a system of four linear boundary conditions in the unknowns A , A , A and A . These boundary conditions are obtained by imposing thatdisplacement v ( x ) , rotation „ ( x ) , shear T ( x ) and/or bending moment M ( x ) assume suitable values at the boundary points x = 0 and x = L of thenano-beam. Observing the structure of the previous equations, it is alsoclear that the integration constants have the following meaning: A = ≠ T (0) , A = M (0) , A = „ (0) , A = v (0) . (23) erential procedure In this section, the solution of the Timoshenko nano-beam is obtained byapplying the di erential formulation in Section 3, which involves the solutionof two di erential equations with the corresponding constitutive boundaryconditions. The first step is solving the second-order di erential equation(4) subject to the CBCs (5); taking into account (15) and (16), this task isequivalent to solve the following di erential equation ‰ ÕÕ ( x ) ≠ L c ‰ ( x ) = – f ÕÕ ( x ) ≠ L c f ( x ) (24)with x œ [0 , L ] , subject to the following CBCs Y______]______[ ‰ Õ (0) ≠ L c ‰ (0) = – f Õ (0) ≠ – L c f (0) , ‰ Õ ( L ) + 1 L c ‰ ( L ) = – f Õ ( L ) + – L c f ( L ) . (25)10he solution ‰ ( x ) of (24) is used to evaluate the cross-sectional rotation „ ( x ) = ⁄ x ‰ ( s ) ds + A , (26)which results an expression in terms of the three integration constants A , A and A . The second step is solving the second-order di erential equation(12) subject to the CBCs (14); taking into account (19) and (20), this taskis equivalent to solve the following di erential equation ◊ ÕÕ ( x ) ≠ ◊ ( x ) L c = – h ÕÕ ( x ) ≠ h ( x ) L c + „ ÕÕ ( x ) ≠ „ ( x ) L c , (27)with „ ( x ) given by (26), subject to the following CBCs Y______]______[ ◊ Õ (0) ≠ L c ◊ (0) = – h Õ (0) ≠ – L c h (0) + „ Õ (0) ≠ L c „ (0) , ◊ Õ ( L ) + 1 L c ◊ ( L ) = – h Õ ( L ) + – L c h ( L ) + „ Õ ( L ) + 1 L c „ ( L ) . (28)Finally, the solution ◊ ( x ) of (27) is used to evaluate the displacement v ( x ) = ⁄ x ◊ ( s ) ds + A , (29)which is expressed in terms of the integration constants A , A , A and A to be determined by using the boundary conditions described in Section 4.1.
5. Solutions of Timoshenko nano-beams
The procedures described in Sections 4.1 and 4.2 provide closed-formsolutions for kinematic and static entities (such as v ( x ) , „ ( x ) , T ( x ) , etc.) of11he Timoshenko nano-beam problem. Closed-form solutions v ( x ) and ◊ ( x ) have long expressions because contain the e ects of both the flexural andshear deformations. This leads to represent the solution of the Timoshenkonano-beams by means of graphics in this section, avoiding the writing oflong symbolic expressions for v ( x ) , ◊ ( x ) , etc. Closed-form solutions forsome fundamental problems are obtained with the proposed methods andare reported in Appendix A. The following geometric dimensionless variablesare introduced: › = xL , ⁄ = L c L . (30)Integral equation (1) shows that kinematic variable ‰ ( x ) is proportional tothe external load contained in M ( x ) and inversely proportional to the bend-ing sti ness EI . Analogously from (2), kinematic variable “ ( x ) = „ ( x ) ≠ ◊ ( x ) is proportional to the external load contained in T ( x ) and inversely propor-tional to the shear sti ness GA . This allows adopting dimensionless kine-matic variables for representing the solution of nano-beams. To this end,the shear sti ness GA is expressed in terms of the flexural sti ness EI byintroducing the dimensionless ratio — = Ÿ GA L EI . (31)The expression of the dimensionless transverse displacement v ú ( › ) used tographically represent the solution depends on the kind of load applied to the12ano-beam and is given by v ú ( › ) = Y_____________]_____________[ v ( x ) EImL if a concentrated couple m is applied ,v ( x ) EIF L if a transverse concentrated force F is applied ,v ( x ) EIqL if a transverse uniform load q y ( x ) = q is applied . (32)The dimensionless displacements defined in (32) are usually adopted forbeams subject to pure flexural deflection without shear deformation; butsubstituting Ÿ GA = — EIL in the closed-form expressions of v ( x ) , accordingto (31), displacements v ú ( › ) defined in (32) also result dimensionless forbeams exhibiting shear deformations in conformity with the Timoshenkomodel.In the proposed nonlocal theory, the parameter ⁄ plays an important role.Setting ⁄ = 0 provides the classical local Timoshenko beam solution , whichcan also be recovered by solving the problem with – = 1 , whatever value of ⁄ is adopted. Next, v ú LT ( › ) denotes the dimensionless transverse displacementof the local Timoshenko beam (LT). The nonlocal Timoshenko beam (NLT)occurs when ⁄ > and – ” = 1 . Another significant solution is obtainedby setting ⁄ æ + Œ : the corresponding dimensionless displacement v ú ( › ) isdenoted by v ú T Œ ( › ) ; from (1) and (2), it results v ú T Œ ( › ) = – v ú LT ( › ) , see alsoAppendix A. In this paper, setting a parameter equal to zero (or + Œ ) also means evaluating a limit,i.e. finding the solution as that parameter tends to zero (or + Œ ). See also Appendix A. ness ratio — = Ÿ GA L EI influences the magnitude of the flexural andshear deformations. As — tends to + Œ the shear deformation tends tovanish with respect to the flexural deformation. Therefore, setting — = + Œ provides the Bernoulli-Euler beam, which is local (LBE) if ⁄ = 0 and isnonlocal (NLBE) if ⁄ > and – ” = 1 . The mixture stress-driven formulationsfor Bernoulli-Euler nano-beams has been treated in greater detail in [42].Analogously to the Timoshenko beam, v ú LBE ( › ) denotes the dimensionlesstransverse displacement of the local Bernoulli-Euler beam and v ú BE Œ ( › ) = – v ú LBE ( › ) denotes the dimensionless transverse displacement obtained bysetting ⁄ = + Œ for the NLBE.Figure 2 shows dimensionless displacement v ú against › for a cantileversubject to a transverse concentrated load at the free end point. The curvesof the figure corresponding to the local and nonlocal Timoshenko beams aredenoted by LT and NLT, respectively. In the same figure, the curves cor-responding to local and nonlocal Bernoulli-Euler beams are also reportedand denoted by LBE and NLBE, respectively. For nonlocal beams, themixture parameter – is assumed equal to 0.5. For the Timoshenko beams,the sti ness ratio — is assumed equal to 4. The comparison between Tim-oshenko and Bernoulli-Euler beams in Figure 2 highlights the increment ofdisplacement v ú ( › ) due to the shear deformation e ect characterizing theTimoshenko model. The curves of the figure refer to several values of ⁄ varying in the set S = { , . , . , . , . , . , + Œ } . The parameter ⁄ hasthe e ect of reducing the displacement v ú ( › ) , that is a larger ⁄ involves asmaller displacement v ú ( › ) for a given value of – . Therefore, v ú T Œ ( › ) and v ú LT ( › ) ’ › œ [0 , are lower and upper bounds, respectively, for the Timo-14henko dimensionless displacement v ú ( › ) corresponding to any ⁄ œ S . Ananalogous observation holds for the Bernoulli-Euler beams.In the proposed mixture formulation, the transition from local to nonlocalmodels is governed by the parameter – . As the nonlocal model is sti erthan the local model, the parameter – has the e ect of increasing the dis-placement v ú ( › ) , that is a larger – involves a larger displacement v ú ( › ) for agiven value of ⁄ . Figures 3-10 show dimensionless displacement v ú against › for di erent kinds of loads and boundary conditions. Table 1 shows theboundary conditions of the generic nano-beam considered in Figures 2-10:specifically, the first column of the table reports the name of the nano-beamand the figure where the nano-beam is investigated; the second column ofthe table reports the corresponding boundary conditions. The transversedistributed load is absent except in nano-beams C-q , SS-q , CS-q and
CC-q ,where a uniformly transverse distributed load q is applied. In Figures 3-10,the curves corresponding to the local and nonlocal Timoshenko beams aredenoted by LT and NLT, respectively. In the same figures, the curves corre-sponding to local and nonlocal Bernoulli-Euler beams are denoted by LBEand NLBE, respectively. Moreover, the mixture parameter – is assumedequal to 0.5 for nonlocal beams and the sti ness ratio — is assumed equalto 4 for the Timoshenko beams. The considerations made for Figure 2 arealso valid for Figures 3-4, 6, 8-10. Specifically, v ú T Œ ( › ) and v ú LT ( › ) ’ › œ [0 , are lower and upper bounds, respectively, for the Timoshenko dimensionlessdisplacement v ú ( › ) corresponding to any ⁄ œ S . An analogous observationholds for the Bernoulli-Euler beams. The closed-form expressions of the di-mensionless transverse displacement v ú T Œ ( › ) are evaluated in Appendix A15nd are reported in Table 2 for the nano-beams considered in Figures 2-10.Finally, Table 3 reports the exact solutions v ú ( › ) for – = 0 . , — = 4 , ⁄ = 0 . and › = 0 , . , . , . , . , .
6. Conclusions
An innovative stress-driven elastic mixture has been developed for Tim-oshenko nano-beams. The model has been formulated by convexly com-bining local and nonlocal phases and does not su er the limit behavior ofthe strain-driven mixture as the local fraction tends to zero. Both inte-gral and di erential formulations of the mixture stress-driven model havebeen provided and relevant procedures for obtaining closed-form solutions(reported in Appendix A for some simple structural schemes) have beenillustrated. Finally, extensive numerical solutions have been presented forstubby beams. In agreement with experimental evidences [46], a sti eningstructural response has been enlightened for increasing values of the scaleparameter. Acknowledgment - Financial supports from the Italian Ministry ofEducation, University and Research (MIUR) in the framework of the ProjectPRIN 2015 “COAN 5.50.16.01” - code 2015JW9NJT - and from the researchprogram ReLUIS 2018 are gratefully acknowledged.16ame of the nano-beam Boundary conditions
C-F (Fig. 2) v (0) = 0 , „ (0) = 0 , M ( L ) = 0 , T ( L ) = F C-q (Fig. 3) v (0) = 0 , „ (0) = 0 , M ( L ) = 0 , T ( L ) = 0 SG-F (Fig. 4) v (0) = 0 , M (0) = 0 , „ ( L ) = 0 , T ( L ) = F SS-m (Fig. 5) v (0) = 0 , M (0) = 0 , v ( L ) = 0 , M ( L ) = m SS-q (Fig. 6) v (0) = 0 , M (0) = 0 , v ( L ) = 0 , M ( L ) = 0 CS-m (Fig. 7) v (0) = 0 , „ (0) = 0 , v ( L ) = 0 , M ( L ) = m CS-q (Fig. 8) v (0) = 0 , „ (0) = 0 , v ( L ) = 0 M ( L ) = 0 CG-F (Fig. 9) v (0) = 0 , „ (0) = 0 , „ ( L ) = 0 , T ( L ) = F CC-q (Fig. 10) v (0) = 0 , „ (0) = 0 , v ( L ) = 0 „ ( L ) = 0 Table 1: Fundamental schemes with corresponding boundary conditions. v ú T Œ ( › ) C-F (Fig. 2) ≠ › – + › – + – ›— C-q (Fig. 3) › – ≠ › – + › – + ≠ › – / ›–— SG-F (Fig. 4) ≠ › – + ›– + – ›— SS-m (Fig. 5) › – ≠ ›– SS-q (Fig. 6) › – ≠ › – + ›– + ≠ › – + ›– — CS-m (Fig. 7) ( › – ≠ › – ) — +6 › – ≠ ›– — +12 CS-q (Fig. 8) – ( › ≠ › +3 › ) — +6 ( › ≠ › ≠ › +5 › ) — ≠ › +72 › — +144 — CG-F (Fig. 9) ≠ › – + › – + – ›— CC-q (Fig. 10) › – ≠ › – + › – + ≠ › – + ›– — Table 2: v ú T Œ ( › ) for di erent types of nano-beams. C - F C - q S G - F SS - m SS - q C S - m C S - q C G - F CC - q . . . . - . . - . . . . . . . . - . . - . . . . . . . . - . . - . . . . . . . . - . . - . . . . . . . . . T a b l e : E x a c t s o l u t i o n s v ú ( › ) f o r – = . , — = a nd ⁄ = . . . Appendix: Closed-form solutions for Timoshenko nano-beams Next, closed-form solutions for Timoshenko nano-beams are provided byapplying the integral procedure of Section 4.1 or, equivalently, the di eren-tial procedure of Section 4.2. For the most nano-beams considered in thiswork, the dimensionless transverse displacement v ú may be expressed by thefollowing formula: v ú ( › , – , — , ⁄ ) = F ( › , – , ⁄ ) + 1 — H ( › , – , ⁄ ) , (33)or, equivalently, the transverse displacement may be represented as v ( x, – , EI, Ÿ GA, L, L c , L ) = L EI ˆ F ( x, – , L, L c ) + L Ÿ GA ˆ H ( x, – , L, L c ) , (34)where L is one of the loads m , F and q . Basically, the right-hand sideof (33) or (34) is the sum of two addends: the first addend represents theflexural deflection and the second addend is the shear deformation. For-mulae (33) and (34) occur not only in statically determinate (or isostatic)structures, such as C-F , C-q , SG-F , SS-q and
SS-m , but also in some stati-cally indeterminate (or hyperstatic) structures. In Table 2, these hyperstaticstructures are
CG-F and
CC-q . In contrast, the dimensionless displacement v ú of the CS-m and
CS-q nano-beams can not be represented with formula(33) as the sum of two terms, one independent on — and the other inverselyproportional to — . In fact, the dimensionless displacement v ú of the CS-q nano-beam exhibits the following limit (the closed-form expression of v ú is20oo long and is not reported): lim ⁄ æ + Œ v ú ( › , – , — , ⁄ ) = – — + 144 — Ë1 › ≠ › + 3 › — ++6 › ≠ › ≠ › + 5 › — ≠ › + 72 › È . (35)See also the following subsections, where the dependence of v ú on – and — in the left-hand side of (33) is omitted for the sake of brevity. A.1. Cantilever subject to a concentrated force at the free end (C-F)
The closed-form expression of v ( x ) is used in the following relation inorder to obtain the dimensionless transverse displacement v ú ( › ) = v ( x ) EIF L , (36)which depends on the dimensionless parameters – , — , ⁄ and › . Highlightingthe dependence of v ú ( › ) on ⁄ , we rename v ú ( › ) to v ú ( › , ⁄ ) , helping us topresent the following closed-form expressions of dimensionless displacements.The dimensionless transverse displacement of the cantilever is v ú ( › , ⁄ ) = F CF ( › , ⁄ ) + 1 — H CF ( › , ⁄ ) , (37)where F CF ( › , ⁄ ) has been found in [42] and H CF ( › , ⁄ ) is given by H CF ( › , ⁄ ) = › + 12 (1 ≠ – ) Ë ⁄ e ≠ ›⁄ ≠ + e ≠ ⁄ ≠ ≠ e ( › ≠ ⁄ ≠ ≠ ⁄ È . (38)21he maximum dimensionless transverse displacement of the cantilever isgiven by v ú (1 , ⁄ ) = F CF (1 , ⁄ ) + 1 — H CF (1 , ⁄ ) , (39)where F CF (1 , ⁄ ) = 1 / – ≠ e ≠ ⁄ ≠ ⁄ ≠ ⁄ + e ≠ ⁄ ≠ ⁄ + ⁄ / , (40) H CF (1 , ⁄ ) = 1 + (1 ≠ – ) e ≠ ⁄ ≠ ⁄ ≠ ⁄ . (41)Other significant dimensionless displacements are: v ú T Œ ( › ) = lim ⁄ æ + Œ v ú ( › , ⁄ ) = ≠ › – + 12 › – + – ›— , (42) v ú LT ( › ) = lim ⁄ æ + v ú ( › , ⁄ ) = ≠ › + 12 › + ›— , (43) v ú LT (1) = 1 / — ≠ . (44)Entities v ú T Œ ( › ) in (42) and v ú LT ( › ) in (43) are defined in Section 5. v ú T Œ ( › ) is also reported in Table 2 for di erent types of nano-beams. The simpleformula (44) allows explaining easily the use of the parameter — in dimen-sional analysis for the transition from dimensionless quantities to dimen-sional quantities. In fact, multiplying both sides of (44) by F L EI gives F L EI v ú LT (1) = F L EI + F L — EI . (45)22he transverse displacement of the local Timoshenko beam is v LT ( x ) = F L EI v ú LT ( › ) . Moreover assuming Ÿ GA = — EIL , relation (45) becomes v LT ( L ) = F L EI + F L Ÿ GA , (46)which is the well-known formula of the maximum transverse displacementof a Timoshenko cantilever subject to a point load at the free end. Thedimensional analysis for more complicated formulas such as (39) is done ina similar way.
A.2. Cantilever subject to a uniformly distributed load (C-q)
Following the steps in Appendix A.1, the dimensionless transverse dis-placement is v ú ( › ) = v ( x ) EIqL , (47)which is renamed v ú ( › , ⁄ ) . The maximum dimensionless transverse displace-ment is v ú (1 , ⁄ ) = F Cq (1 , ⁄ ) + 1 — H Cq (1 , ⁄ ) , (48)where F Cq (1 , ⁄ ) = 18 + 14 ( – ≠ e ≠ ⁄ ≠ ⁄ ≠ ⁄ + ⁄ , (49) H Cq (1 , ⁄ ) = 12 Ë ≠ – ) e ≠ ⁄ ≠ ⁄ ≠ ⁄ . (50)Notice that shear contribution H Cq (1 , ⁄ ) in (50) for cantilever with dis-tributed load is equal to half of the shear contribution H CF (1 , ⁄ ) in (41) for23antilever with point load. Other significant solutions are v ú T Œ ( › ) = lim ⁄ æ + Œ v ú ( › , ⁄ ) = 124 › – ≠ › – + 14 › – + ≠ › – / ›–— , (51) v ú LT ( › ) = lim ⁄ æ + v ú ( › , ⁄ ) = 124 › ≠ › + 14 › + ≠ › / ›— , (52) v LT ( L ) = qL EI v ú LT (1) = 18 qL EI + 12 qL Ÿ GA . (53)
A.3. Simply supported beam with uniformly distributed load (SS-q)
The maximum dimensionless transverse displacement is v ú , ⁄ = F SSq , ⁄ + 1 — H SSq , ⁄ , (54)where F SSq (1 / , ⁄ ) =5 /
384 + ( – ≠ ⁄ + ⁄ / e ≠ (2 ⁄ ) ≠ ≠ e ≠ ⁄ ≠ / ≠ / + ⁄ / , (55) H SSq (1 / , ⁄ ) =1 / ≠ – ) ⁄ + ⁄ / e ≠ (2 ⁄ ) ≠ ≠ e ≠ ⁄ ≠ / ≠ ⁄ / ≠ ⁄ / , (56)24ther significant solutions are v ú T Œ ( › ) = lim ⁄ æ + Œ v ú ( › , ⁄ ) = 124 › – ≠ › – + 124 ›– + ≠ › – / ›– / — , (57) v ú LT ( › ) = lim ⁄ æ + v ú ( › , ⁄ ) = 124 › ≠ › + 124 › + ≠ › / › / — , (58) v LT L = qL EI v ú LT = 5384 qL EI + 18 qL Ÿ GA . (59)
A.4. Nano-beam with a clamped end and a guided end subject to a pointforce (CG-F)
Following the steps in Appendix A.1, the maximum dimensionless trans-verse displacement is v ú (1 , ⁄ ) = F CGF (1 , ⁄ ) + 1 — H CGF (1 , ⁄ ) , (60)where F CGF (1 , ⁄ ) =1 /
12 + ( – ≠ e ≠ ⁄ ≠ ⁄ ≠ ⁄ + e ≠ ⁄ ≠ ⁄ + e ≠ ⁄ ≠ ⁄ / ⁄ / , (61) H CGF (1 , ⁄ ) = 1 + (1 ≠ – ) e ≠ ⁄ ≠ ⁄ ≠ ⁄ . (62)25otice that the shear contribution H CGF (1 , ⁄ ) in (62) is equal to the shearcontribution H CF (1 , ⁄ ) in (41). Other significant solutions are v ú T Œ ( › ) = lim ⁄ æ + Œ v ú ( › , ⁄ ) = ≠ › – + 14 › – + – ›— , (63) v ú LT ( › ) = lim ⁄ æ + v ú ( › , ⁄ ) = ≠ › + 14 › + ›— , (64) v LT ( L ) = F L EI v ú LT (1) = F L EI + F L Ÿ GA . (65)
Acknowledgment - Financial supports from the Italian Ministry ofEducation, University and Research (MIUR) in the framework of the ProjectPRIN 2015 “COAN 5.50.16.01” - code 2015JW9NJT - and from the researchprogram ReLUIS 2018 are gratefully acknowledged.
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Figure 1: Kinematic variables defining the deformation of the beam. igure 2: Cantilever with concentrated load at the free end ( C-F ): v ú against › for — = 4 .Figure 3: Cantilever with uniformly distributed load ( C-q ): v ú against › for — = 4 . igure 4: Beam with a simply supported end and a guided end subject to transverseconcentrated load ( SG-F ): v ú against › for — = 4 .Figure 5: Simply supported beam with concentrated couple at the right end ( SS-m ): v ú against › for — = 4 . igure 6: Simply supported beam with uniformly distributed load ( SS-q ): v ú against › for — = 4 .Figure 7: Beam with a clamped end and a simply supported end subject to a concentratedcouple ( CS-m ): v ú against › for — = 4 . igure 8: Beam with a clamped end and a simply supported end with uniformly distributedload ( CS-q ): v ú against › for — = 4 .Figure 9: Beam with a clamped end and a guided end subject to transverse concentratedload ( CG-F ): v ú against › for — = 4 . igure 10: Fixed beam with uniformly distributed load ( CC-q ): v ú against › for — = 4 ..