Computational Mechanics (2021) 68:697–708https://doi.org/10.1007/s00466-021-01987-6

ORIG INAL PAPER

Formulation and experimental validation of space-fractionalTimoshenko beammodel with functionally gradedmaterials effects

Paulina Stempin1 ·Wojciech Sumelka1

Received: 20 October 2020 / Accepted: 30 January 2021 / Published online: 13 March 2021© The Author(s) 2021

AbstractIn this study, the static bending behaviour of a size-dependent thick beam is considered including FGM (Functionally GradedMaterials) effects. The presented theory is a further development and extension of the space-fractional (non-local) Euler–Bernoulli beam model (s-FEBB) to space-fractional Timoshenko beam (s-FTB) one by proper taking into account sheardeformation. Furthermore, a detailed parametric study on the influence of length scale and order of fractional continua fordifferent boundary conditions demonstrates, how the non-locality affects the static bending response of the s-FTB model.The differences in results between s-FTB and s-FEBB models are shown as well to indicate when shear deformations need tobe considered. Finally, material parameter identification and validation based on the bending of SU-8 polymer microbeamsconfirm the effectiveness of the presented model.

Keywords Timoshenko beam · Non-local model · Fractional calculus · Microbeam bending

AbbreviationsFGM Functionally graded materialss-FEBB Space-fractional Euler–Bernoulli beam

models-FTB Space-fractional Timoshenko beamSE Scale effectRVE Representative volume elementBD Body dimensionsLS Length scaleFDO Fractional differential operatorCTB Classical Timoshenko beam

1 Introduction

The first works on scale effect (SE), which manifest depen-dence on the answer of the system (e.g. deformation of amaterial body - the crux of this paper) in relation to itsdimensions, date to the investigations of Leonardo da Vinci

In Honor of Professor Tomasz Łodygowski on the Occasion of His70th Birthday.

B Wojciech Sumelkawojciech.sumelka@put.poznan.pl

1 Institute of Structural Analysis, Poznan University ofTechnology, Piotrowo 5 Street, 60-965 Poznan, Poland

at the very beginning of XVI century [1,2]. Since that timeboth experimental techniques [3–7] and theoretical concepts[8–13] for SE analysis were considerably developed andmoreover SE phenomena were utilised successfully in themodern industry [14–18]. The main message is that from thestandpoint of mechanics, which constitutes the central pointof presented considerations, one can say that each structure,due to the complexity of material over different scales ofobservation, reveals SE. On the other hand, it is clear thatthe strength of SE phenomena is different depending on theanalysed case and proportional to the ratio of the represen-tative volume element (RVE) of a specific material to bodydimensions (BD).

From the theoretical side, when constructing a mathemat-ical model for the description of mechanical phenomena,one should choose certain mathematical objects proper to theexperimental observation scale [19–22]. Herein, we operateon meso/macro level, therefore for SE modelling a phe-nomenological approach is used, thus in consequence RVEto BD ratio is mapped utilising so-called length scale (LS)parameter (it is clear that LS meaning is different dependingon certain theory [15,23–26]). To be precise, the developeds-FTB theory is defined in the framework of space-FractionalContinuum Mechanics (s-FCM) [27,28], where LS is intro-duced through the fractional differential operator (FDO) andfurthermore, an additional parameter which controls SE is

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698 Computational Mechanics (2021) 68:697–708

introduced, namely order of FDO. Finally, because of thecomplex nature of mechanical properties through the beamthickness FGM concept is also used [29–32].

This paper is a continuation of previous studies focusedon the development of non-local beam models [27,28]. Thecurrently formulated space-Fractional Timoshenko beammodel, compared to the space-Fractional Euler–Bernoullibeam model, takes into account the shear effect to extendmodelling range for thick beams. The definition of rotationhas been therefore extended by additional rotation resultingfrom non-local shear deformations. Hence, the appropri-ate governing equations and enriched numerical algorithmare provided. The discussion also includes a study of theinfluence of non-locality parameters for different bound-ary conditions on the static beam response to bending, acomparison study between s-FTB and s-FEBB models, andidentification and validation of s-FTB model parameters forthe experimental data of the SU-8 polymer microcantileverbending test (including FGM effects).

The paper is structured as follows. Section 2 deals withthe s-FTB definition. Section 3 is devoted to the numericalscheme and parametric study. Section 4 provides experimen-tal validation and finally Sect. 5 concludes the paper.

2 Theory

The space-Fractional Timoshenko beam (s-FTB) theory is anextension of the space-Fractional Euler–Bernoulli beam (s-FEBB) theory [28] to include shear deformation and make itsuitable for thick beams as mentioned in the introduction. Asin the previous study, the non-locality is taken into account,based on the fractional elasticity concept [33,34], by the fol-lowing definition of small strain

�εi j = 1

2�α−1f

(α

Dx jui (x) + α

Dxiu j (x)

), (1)

where ui are the components of the displacement vector, x

is a spatial variable, whereas the termα

D( . ) denotes theRiesz–Caputo fractional derivative [35],

α

Dx jui (x) = 1

2

�(2 − α)

�(2)

(Cx j−� f

Dαx j ui (x)+(−1)n C

x j Dαx j+� f

ui (x)), (2)

with the left-side and right-side Caputo derivatives

Cx j−� f

Dαx j ui (x) = 1

�(n − α)

∫ x j

x j−� f

u(n)i (τ )

(x j − τ)α−n+1 dτ , (3)

Cx j D

αx j+� f

ui (x) = −1

�(n − α)

∫ x j+� f

x j

u(n)i (τ )

(x j − τ)α−n+1 dτ , (4)

where � is an Euler gamma function, n = [α] + 1, and [.]denotes the integer part of a real number, α ∈ (0, 1] is theorder of FDO, and � f is LS i.e. the surrounding affectingthe considered material point. The concept of variable lengthscale [36] � f = � f (x), as function decreasing at the bound-aries, has been kept. These two parameters α and � f areregarded as associated with microstructure [37] and respon-sible for SE mapping.

In the presented study one considers the static bendingbehavior in the x1x3-plane, therefore, keeping the assump-tion that the cross-section is infinitely rigid in its own planeand remains plane after deformation. In consequence, thedisplacement field takes the form

u1(x1, x2, x3) = x3�2, u2(x1, x2, x3) = 0, u3(x1, x3, x3) = u3.

(5)

where u3 = u3(x1) is the rigid body translation of the cross-section in 3-rd axis direction of the coordinate system and�2 = �2(x1) is the rigid body rotation (positive keeping theright-hand rule). Rotation �2 depends on the Riesz–Caputofractional derivativewith the proportionality factor �α−1

f and,by comparison to the already developed s-FEBB [28], isextended by an additional rotation due to the fractional shear

deformation�γ 13,

�2 = −�α−1f

α

Dx1u3 + �

γ 13. (6)

Herein, it should be emphasised that on one hand side theassumption Eq. (6) reflects the influence ofmicrostructure onbeam cross section rotation, but simultaneously acts as a con-sistency condition.Namely, it allows to obtain proper relation

between the component of the fractional Cauchy strain�ε13

and the fractional shear deformation�γ 13. The last statement

causes that in limit case when s-FTB reduces to s-FEBB�ε13 = 0 which is fundamental for s-FEBB [28].

Next, using Eqs (1) and (6) the nonzero fractional Cauchystrains are�ε11 = x3�

α−1f

[− α

Dx1

(�α−1f

α

Dx1u3

)+ α

Dx1

�γ 13

],

�ε13 = �

ε31 = 1

2

[(− �α−1

f

α

Dx1u3 + �

γ 13

) α

Dx3x3 + α

Dx1u3

]�α−1f = 1

2�γ 13.

(7)

It is so because in Eq. (72) �α−1f

α

Dx3x3 = 1, then

�ε13 = 1

2�γ 13.

The corresponding stresses are

σ11 = x3�α−1f

[− α

Dx1

(�α−1f

α

Dx1u3

)+ α

Dx1

�γ 13

]E(x2, x3),

σ13 = G�ε13, (8)

where E(x2, x3) is Young’s modulus, G = G(x2, x3) =E(x2,x3)2(1+ν)

is Kirchhoff modulus and ν is Poisson’s ratio. Based

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Computational Mechanics (2021) 68:697–708 699

on the above results, the bending moment M2 and the shearforce V3 can be expressed as

M2 = M2(x1) =∫Ax3σ11 dA and V3 = V3(x1) =

∫A

σ13 dA. (9)

and, by introducing Eq. (8), as

M2 =∫Ax3σ11dA = �α−1

f

[− α

Dx1

(�α−1f

α

Dx1u3

)+ α

Dx1

�γ 13

](E I )∗, (10)

V3 =∫A

σ13dA = k(GA)∗ �γ 13, (11)

where k is the shear correction factor, (E I )∗ = ∫A E(x2, x3)

x23 dA and (GA)∗ = ∫A G(x2, x3) dA. The relationship

among V3, M2 and distributed load p3 = p3(x1) (for thederivation of following formulas from the principle of vir-tual work see Appendix and [28]) is

V3 = α

Dx1

(M2�

α−1f

), (12)

p3 = − α

Dx1

(V3�

α−1f

). (13)

Moreover, the shear force V3, by using Eqs. (10) and (12),can be also expressed as

V3 = α

Dx1

{�2α−2f

[− α

Dx1

(�α−1f

α

Dx1u3

)+ α

Dx1

�γ 13

]}(E I )∗. (14)

Finally, substituting Eq. (14) in Eq. (13), and comparingEq. (14) and Eq. (11) we obtain the fractional equations gov-erning the bending of s-FTB

⎧⎪⎨⎪⎩

α

Dx1

{�α−1f

α

Dx1

[�2α−2f

α

Dx1

(�α−1f

α

Dx1u3

)]}(E I )∗ = p3 + α

Dx1

[�α−1f

α

Dx1

(�2α−2f

α

Dx1

�γ 13

)](E I )∗,

α

Dx1

[�2α−2f

α

Dx1

(�α−1f

α

Dx1u3

)]E I − α

Dx1

(�2α−2f

α

Dx1

�γ 13

)(E I )∗ + k(GA)∗ �

γ 13 = 0.(15)

It is worth noting that when the shear deformation is

neglected, i.e.�γ 13 ≈ 0, the rotation Eq. (6) takes the form

�2 = −�α−1f

α

Dx1u3, (16)

and consequently, Eq. (15) reduces to equation governing thebending of a s-FEBB model [28]

α

Dx1

{�α−1f

α

Dx1

[�2α−2f

α

Dx1

(�α−1f

α

Dx1u3

)]}(E I )∗ = p3. (17)

Moreover, for α = 1, the rotations Eqs. (6) and (16) takethe classical form for Timoshenko and Euler–Bernoulli beamtheories, respectively

�2 = −du3dx1

+ γ13 and �2 = −du3dx1

, (18)

and the governing equations Eqs. (15)1 and (17) also reduceto the classical local descriptions

d4u3dx41

(E I )∗ = p3 + d3γ13dx31

(E I )∗ andd4u3dx41

(E I )∗ = p3,

(19)

Timoshenko and Euler–Bernoulli beam theories, respec-tively.

3 Numerical study

3.1 Discretization

Equation (15) has been solved utilising the numericalmethod[38]. The beam was discretized in n intervals of length h(see Fig. 1). The trapezoidal rule [38–40] was applied to

approximate Caputo fractional derivativesα

Dx1

( . )i at the node

xi1 by the sum of first-order derivatives at nodes xi−m1 ÷ xi+m

1(see the Fig. 1) with appropriate weight coefficients markedas B, Ca and Cb, according to the Eq. (20)

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700 Computational Mechanics (2021) 68:697–708

α

Dx1

( . )i = h1−αA[B( . )′i−m +

i−1∑ja=i−m+1

Ca( . )′ja + 2( . )′i +i+m−1∑jb=i+1

Cb( . )′jb + B( . )′i+m

], (20)

where

m = mi = (� f )i/h ≥ 2, A = �(2 − α)

2�(2)�(3 − α), B = (m − 1)2−α − (m + α − 2)m1−α,

Ca = (i − ja + 1)2−α − 2(i − ja)2−α + (i − ja − 1)2−α,

Cb = ( jb − i + 1)2−α − 2( jb − i)2−α + ( jb − i − 1)2−α, (21)

and (� f )i = � f (xi1).Based on the Eq. (20), the numerical representation of

the distributed load Eq. (15), shear force Eq. (14), bendingmoment Eq. (10) and rotation Eq. (6) at node xi1 are given by

Di (E I )∗ = p3(xi1) + Ci (E I )∗, Ci (E I )∗ − Bi (E I )∗ + k(GA)∗ �

γ 13(xi1) = 0,

V3(xi1) = (−Ci + Bi )(E I )∗, or V3(x

i1) = k(GA)∗ �

γ 13(xi1),

M2(xi1) = (�α−1

f )i (−Bi + Ai )(E I )∗, �2(xi1) = −(�α−1

f )iAi + �γ 13(x

i1),

(22)

where

Di = α

Dx1

{(�α−1

f )iα

Dx1

[(�2α−2

f )iα

Dx1

((�α−1

f )iα

Dx1u3(x

i1)

)]}= α

Dx1

[(�α−1

f C)i

]= h1−αA

[B(�α−1

f C)′i−m

+i−1∑

ja=i−m+1

Ca(�α−1f C)′ja + 2(�α−1

f C)′i +i+m−1∑jb=i+1

Cb(�α−1f C)′jb + B · (�α−1

f C)′i+m

],

(23)

Ci = α

Dx1

[(�2α−2

f )iα

Dx1

((�α−1

f )iα

Dx1u3(x

i1)

)]= α

Dx1

[(�2α−2

f B)i

]= h1−αA

[B(�2α−2

f B)′i−m

+i−1∑

ja=i−m+1

Ca(�2α−2f B)′ja + 2(�2α−2

f B)′i +i+m−1∑jb=i+1

Cb(�2α−2f B)′jb + B(�2α−2

f B)′i+m

],

(24)

Bi = α

Dx1

[(�α−1

f )iα

Dx1u3(x

i1)

]= α

Dx1

[(�α−1

f A)i

]= h1−αA

[B(�α−1

f A)′i−m

+i−1∑

ja=i−m+1

Ca(�α−1f A)′ja + 2(�α−1

f A)′i +i+m−1∑jb=i+1

Cb(�α−1f A)′jb + B(�α−1

f A)′i+m

],

(25)

Ai = α

Dx1u3(x

i1) = h1−αA

[Bu′

i−m +i−1∑

ja=i−m+1

Cau′ja + 2u′

i +i+m−1∑jb=i+1

Cbu′jb + Bu′

i+m

](26)

Ci = α

Dx1

[(�α−1

f )iα

Dx1

((�2α−2

f )iα

Dx1

�γ 13(x

i1)

)]= α

Dx1

[(�α−1

f B)i

]= h1−αA

[B(�α−1

f B)′i−m

+i−1∑

ja=i−m+1

Ca(�α−1f B)′ja + 2(�α−1

f B)′i +i+m−1∑jb=i+1

Cb(�α−1f B)′jb + B(�α−1

f B)′i+m

],

(27)

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Computational Mechanics (2021) 68:697–708 701

Fig. 1 Discretization of theanalysed beam of length L—homogeneous grid: real nodesx01 ÷ xn1 ; fictitious nodesx−81 ÷ x−1

1 and xn+11 ÷ xn+8

1

Bi = α

Dx1

[(�2α−2

f )iα

Dx1

�γ 13(x

i1)

]= α

Dx1

[(�2α−2

f A )i

]= h1−αA

[B(�2α−2

f A )′i−m

+i−1∑

ja=i−m+1

Ca(�2α−2f )′jaA ′

ja + 2(�2α−2f A )′i +

i+m−1∑jb=i+1

Cb(�2α−2f A )′jb + B(�2α−2

f A )′i+m

],

(28)

Ai = α

Dx1

�γ 13(x

i1) = h1−αA

[Bγ ′

i−m +i−1∑

ja=i−m+1

Caγ ′ja + 2γ ′

i +i+m−1∑jb=i+1

Cbγ ′jb + Bγ ′

i+m

]. (29)

It should be pointed out that the first derivatives ( . )′ inEqs. (23 ÷ 29) are approximated by forward, backward orcentral difference formulae at the relevant nodes accordingto Eq. (30) and Table 1,

( . )′i =[

− ( . )i+N1−N2 + ( . )i+N1+N2

] 1

2N2h, (30)

where N1 and N2 determine which finite difference schemehas been applied (see Table 1).

One should conclude that similarly to Eq. (19), for α = 1Eq. (22) returns for N1 = 0 and N2 = 1

2 to the classicalcentral difference scheme (A = 1

2 , B = Ca = Cb = 0)

�i = −−ui−1/2 + ui+1/2

h+ γi , (31)

Mi =(

− ui−1 − 2ui + ui+1

h2+ −γi−1/2 + γi+1/2

h

)(E I )∗, (32)

Vi =(

− −ui−3/2 + 3ui−1/2 − 3ui+1/2 + ui+3/2

h3+ γi−1 − 2γi + γi+1

h2

)(E I )∗, (33)

and

ui−2 − 4ui−1 + 6ui − 4ui+1 + ui+2

h4(E I )∗ = pi + −γi−3/2 + 3γi−1/2 − 3γi+1/2 + γi+3/2

h3(E I )∗, (34)

where �i = �2(xi1), Mi = M2(xi1), Vi = V3(xi1), pi =p3(xi1), γi = γ13(xi1) and ui = u3(xi1).

Applied numerical methods have resulted in fictitiousnodes (x−8

1 ÷ x−11 and xn+1

1 ÷ xn+81 ) outside the beam in

addition to real nodes (x01 ÷ xn1 ) - see Fig. 1. The applicationof the variable length scale � f (x), which is decreasing at theboundaries [36], results in only 8 fictitious nodes (x−8

1 ÷x−11 ,

xn+11 ÷xn+8

1 ) on each side of the beam. These points are elim-inated in final set of equations, by the analogy to the approachpresented in [36], by equating of the finite difference approx-imation

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702 Computational Mechanics (2021) 68:697–708

Table 1 The applied finitedifference schemes evaluated atnode xi1 where N1 and N2determine which finitedifference scheme has beenapplied (see also Eq. (30) andFig. 1)

Forward Backward Central CentralN1 = 1

2 , N1 = − 12 , N1 = 0, N1 = 0,

N2 = 12 N2 = 1

2 N2 = 12 N2 = 1

u′i x i1 = x−8

1 xi1 = xn+81 xi1 = x−7.5

1 ÷ xn+7.51 xi1 = x−7

1 ÷ x21 ; xn−21 ÷ xn+7

1

(�α−1f A)′i x i1 = x−6

1 xi1 = xn+61 xi1 = x−5

1 ÷ xn+51

(�2α−2f B)′i

i1 = x−4

1 xi1 = xn+41 xi1 = x−3.5

1 ÷ xn+3.51 xi1 = x−3

1 ÷ xn+31

(�α−1f C)′i x i1 = x−2

1 xi1 = xn+21 xi1 = x−1

1 ÷ xn+11

�γ

′i x i1 = x−6

1 xi1 = xn+61 xi1 = x−5.5

1 ÷ xn+5.51 xi1 = x−5

1 ÷ x21 ; xn−21 ÷ xn+5

1

(�2α−2f A )′i x i1 = x−4

1 xi1 = xn+41 xi1 = x−3

1 ÷ xn+31

(�α−1f B)′i x i1 = x−2

1 xi1 = xn+21 xi1 = x−1

1 ÷ xn+11

• with the central and forward schemes for the fourth order derivative of displacement at nodes x−61 ÷ x−1

1 and for the thirdorder derivative of strain at nodess x−4

1 ÷ x11 ,

ui−2 − 4ui−1 + 6ui − 4ui+1 + ui+2

h4= ui − 4ui+1 + 6ui+2 − 4ui+3 + ui+4

h4, for x−6

1 ÷ x−11 ,

−�γ i−2 + 2

�γ i−1 − 2

�γ i+1 + �

γ i+2

2h3= −�

γ i + 3�γ i+1 − 3

�γ i+2 + �

γ i+3

h3, for xi1 = x−4

1 ÷ x11 ;(35)

• with the central and backward schemes for the fourth order derivative of displacement at nodes xn+11 ÷ xn+6

1 and for thethird order derivative of strain at nodes xn−1

1 ÷ xn+41

ui−2 − 4ui−1 + 6ui − 4ui+1 + ui+2

h4= ui−4 − 4ui−3 + 6ui−2 − 4ui−1 + ui

h4, for xi1 = xn+1

1 ÷ xn+61 ,

−�γ i−2 + 2

�γ i−1 − 2

�γ i+1 + �

γ i+2

2h3= −�

γ i−3 + 3�γ i−2 − 3

�γ i−1 + �

γ i

h3, for xi1 = xn−1

1 ÷ xn+41 .

(36)

3.2 Parametric study

This section highlights the influence of the material param-eters α and � f on the bending behaviour of the beam andcompares the fractional and classical approaches to show theability of taking into account the SE. A comparison of s-FTB and s-FEBB is provided as well to emphasize the needof considering the shear effect when the beam is thick inrelation to its length. The examples of beam with follow-ing data are thereby considered: beam length L = 100 μm,

width a = 10 μm and height b = 50 μm of rectangularcross-section, homogenized Young’s modulus E = 10 GPa,Poisson’s ratio ν = 0.2 and h = 0.1 μm. The shear correc-tion factor for rectangular cross-section is assumed k = 5/6.In each of the examples, the beam was loaded with a con-centrated force: at the mid-span for simply supported, fixed,and propped cantilever schemes, and at the free end for acantilever scheme (see Fig. 2). The point load P = 10 μN is

introduced by equivalent continuous load [28,41]

p3(ξ) = k1k22 tanh(k1/2)

1

cosh2[k1(ξ − L1)]P

L, (37)

where k1 = 100 and k2 are dimensionless parameters,k2 = (L1 + 1)−200 + 1 for L1 ∈ 〈0.0; 0.5] and k2 =(−L1+2)−200+1 for L1 ∈ (0.5; 1.0], ξ = x1/L is a dimen-sionless coordinate and L1 is a point load position in relationto the beam length (L1 = 0.5 for load at the midpoint andL1 = 1.0 for load at the end of beam). The boundary condi-tions are summarised in Table 2. The effect of non-localitywas investigated for the following parameters:α ∈ {0.8, 0.6}and �max

f ∈ {0.001L, 0.10L, 0.2L} with a symmetric dis-tribution that is smoothly decreasing at the boundaries (seeFig. 3), described by the following function [42]

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Computational Mechanics (2021) 68:697–708 703

� f (x1) =

⎧⎪⎨⎪⎩

�maxf ζ 3

1 (10 − 15ζ1 + 6ζ 21 ) for 0 ≤ x1 ≤ η1�

maxf

�maxf for η1�

maxf < x1 < L − η2�

maxf

�maxf ζ 3

2 (10 − 15ζ2 + 6ζ 22 ) for L − η2�

maxf ≤ x1 ≤ L

, (38)

where ζ1 = x1/(η1�maxf ), ζ2 = (L − x1)/(η2�max

f ) andfor symmetric case η1 = 1.2, η2 = 1.2.

The results of this study, presented in Fig. 4, show theeffect of a small scale on the deflection. It can be observedthat parameters α and � f influence the stiffness of the beam.With a decrease in α or an increase in �max

f , the deflection ofthe simply supported and cantilever beam increases,while forpropped cantilever andfixedbeams the stiffeningor softeningeffect is observed for certain values of these parameters. Forlength scale � f , small in relation to L (�max

f = 0.001L), theeffect of non-locality disappears and the results are identicalto the classic local formulation, as expected.

Moreover, in the above examples, the length to heightratio is L/b = 2, and it is visible that for this beam geom-etry, the deflections predicted by s-FTB and s-FEBB differsignificantly. Beam deflections predicted by s-FTB and s-FEBB theories in the wider range of L/b = 2 ÷ 26 arecompared in Fig. 5. The maximum deflections are higherfor s-FTB than the s-FEBB model, and similarly to the clas-sic approach, the differences are noticeable for thick beamsand negligible for slender beams. Moreover, the inclusion ofthe non-locality makes the shear effect more significant thanin the classic approach. It is especially visible for the fixedbeam scheme, where, for example, for the ratio L/b = 2,the maximum deflection for the Timoshenko beam is about3.85 times greater than for the Euler–Bernoulli beam in theclassical approach (α = 1) and 4.85 in fractional approachwith α = 0.6 and �max

f = 0.1L . Therefore, the shear effectfor small scale thick beam definitely should not be ignoredand should even be considered for a higher ratio L/b thanin local theory. However, for a higher L/b ratio, the ratio of

deflections us−FT B

us−FEBB reaches a value of 1.0. For this reason, forlong beams, the s-FTB model can be reduced to the s-FEBBmodel without losing the correctness of the results - which isanalogical to local Timoshenko and Euler–Bernoulli beams.

4 Experimental validation

To show the effectiveness of the developed model, it wasvalidated based on the microcantilever bending experimentpresented in [43]. The tested samples were made of a SU-8 polymer with the following material parameters: the bulkvalue of Young’s modulus Eb = 6.9 GPa and Poisson’s ratioν = 0.22. These cantilevers had a length in the range ofL = 80 ÷ 850 μm and a rectangular cross-section with a

width of a = 82 μm and a = 122 μm, and a height ofb = 8.4 μm and b = 14.4 μm. These data allow for theanalysis of non-local effects depending not only on the cross-section dimensions but also on the length of the beam. Asshown in Fig. 8, decreasing the length of the beam causes adecrease in modulus of elasticity compared to the bulk value.Conversely, the decrease of the cross-section dimensions ismanifested by the increase of Young’s modulus. Moreover,we took this increase into account by analogy to the resultsobtained for iPP+0.2% PACS polymer samples. Figure 6shows the distribution of elastic modulus in cross-sectionmeasured for samples from the nanoindentation test, whileYoung’s modulus from the tensile test is 1.75 ± 0.1 GPa(results derived from [44]). This map of Young’s modulusindicates that it is not constant in the whole cross-section,but there is a core with other characteristics. Based on thisinformation, we include the change of elastic modulus inx3 direction by the core-shell (CS) model (see Fig. 7) witha stiffer core (Ec = f Eb) of thickness c and a bulk shell(Eb), where f means the ratio of the core Young’s modu-lus Es and bulk Young’s modulus Eb. The stiffness of thecore-shell model can be expressed by the effective bendingstiffness

Ee Ie = Ec Ic + Eb Ib, (39)

where Ee is the effective Young’s modulus and Ie, Ic and Ibare the moments of inertia of the effective (homogeneous)cross-section, the core, and the bulk shell respectively,

Ie = ab3

12, Ic = ac3

12, Ib = ab3 − ac3

12. (40)

Then, the effective Young’s modulus is

Ee = Eb

(1 + ( f − 1)

c3

b3

). (41)

It was found that for SU-8 polymer rectangular samplesthe height of core zone is c = 7.0 μm, which has a Youngmodulus f = 1.91 times higher than the bulk value, regard-less of sample size. Subsequently, using the Eq. (41), theeffective elastic modulus is 7.62 GPa for sample of dimen-sions a = 122 μm, b = 14.4 μm and 10.52 GPa forsample of dimensions a = 82 μm, b = 8.4 μm. Then,the parameters of s-FTB model were identified as α = 0.75and �max

f = 5 μm with asymmetric distribution (cf. Fig. 3)described by Eq. (38), where η1 = 1.2, η2 = 2.0.

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704 Computational Mechanics (2021) 68:697–708

Fig. 2 Static schemes of thebeams used in the parametricstudy under arbitrary transverseload: a simply supported, b fixedand c propped cantilever—withmid-span point load; and dcantilever—with end point load

(a) (b)

(c) (d)

Fig. 3 Assumed distributions ofthe length scale � f (x1) alongthe beam length: symmetric caseused in parametric study andasymmetric case used invalidation [see also Eq. (38)]

Table 2 Boundary conditions applied for the static schemes presented in Fig. 2 (see also Eq. (30) and Fig. 1)

Beam type Boundary conditions

Simply supported u3(x01 ) = 0, M2(x01 ) = 0, u0 = 0, −B0 + A0 = 0

u3(xn1 ) = 0, M2(xn1 ) = 0 un = 0, −Bn + An = 0

Fixed u3(x01 ) = 0, �2(x01 ) = 0 u0 = 0, −(�α−1f )0A0 + (−C0 + B0)

(E I )∗k(GA)∗ = 0

u3(xn1 ) = 0, �2(xn1 ) = 0 un = 0, −(�α−1f )nAn + (−Cn + Bn)

(E I )∗k(GA)∗ = 0

Propped cantilever u3(x01 ) = 0, �2(x01 ) = 0 u0 = 0, −(�α−1f )0A0 + (−C0 + B0)

(E I )∗k(GA)∗ = 0

u3(xn1 ) = 0, M2(xn1 ) = 0 un = 0, −Bn + An = 0

Cantilever u3(x01 ) = 0, �2(x01 ) = 0 u0 = 0, −(�α−1f )0A0 + (−C0 + B0)

(E I )∗k(GA)∗ = 0

M2(xn1 ) = 0, V3(xn1 ) = 0 −Bn + An = 0, −Cn + Bn = 0

The validation results are shown in Fig. 8, where the elas-tic modulus was calculated according to Timoshenko beamtheory for a cantilever with a point load at the free end

E =( FL3

3I+ 2(1 + ν) FL

kA

) 1

u3. (42)

It is observed, that the s-FTB model results agree very wellwith the measurement data. In addition, the results fromthe classical Timoshenko beam (CTB) model are plotted inFig. 8, demonstrating that the classical approach is not suit-able for describing the behavior of small scale beams.

5 Conclusions

In the presented work, the space-Fractional Timoshenkobeam theory has been developed from space-FractionalEuler–Bernoulli beam theoryby including the shear deforma-tion. The effect of non-locality for bending of different beamtypeswas studied and the results of the space-Fractional Tim-oshenko and space-Fractional Euler–Bernoulli beam theorieswere compared. The validation of the model was also pro-vided. From these analyzes, the following conclusions aredrawn:

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Computational Mechanics (2021) 68:697–708 705

(a) (b)

(c) (d)

Fig. 4 Deflection of beam predicted by s-FTB and s-FEBB models for: a the simply supported, b the fixed and c the propped cantilever schemeswith mid-span point load; and d the cantilever scheme with end point load, for α ∈ {0.8, 0.6}, �max

f ∈ {0.001L, 0.10L, 0.20L} and symmetric � fdistribution (cf. Fig. 3)

• the numerical algorithm has been enriched with the sheareffect while keeping the possibility of including anyboundary conditions, any transverse load, and variablelength scale,

• parameters α and � f control the stiffness of the fractionalbeam,

• the shear effect is more significant in non-local beams,therefore it should be considered even with more slenderbeams than in a local approach,

• the current model is adequate for modeling small scaleshort beams,

• the validation confirms the applicability of the presentedmodel.

Acknowledgements This work is supported by the National ScienceCentre, Poland under Grant No. 2017/27/B/ST8/00351.

Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing, adap-tation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons licence, and indi-cate if changes were made. The images or other third party materialin this article are included in the article’s Creative Commons licence,unless indicated otherwise in a credit line to the material. If materialis not included in the article’s Creative Commons licence and yourintended use is not permitted by statutory regulation or exceeds thepermitted use, youwill need to obtain permission directly from the copy-right holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Appendix

The equilibrium equations of fractional beam are derivedfrom the principle of virtual work δU = δW , where δU isthe variation in the internal energy

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706 Computational Mechanics (2021) 68:697–708

Table 3 Boundary conditions applied for the static schemes presented in Fig. 2

Beam type Boundary conditions

Simply supported u3(x01 ) = 0, M2(x01 ) = �α−1f (x01 )

[− α

Dx1

(�α−1f (x01 )

α

Dx1u3(x01 )

)+ α

Dx1

�γ 13(x

01 )

](E I )∗ = 0

u3(xn1 ) = 0, M2(xn1 ) = �α−1f (xn1 )

[− α

Dx1

(�α−1f (xn1 )

α

Dx1u3(xn1 )

)+ α

Dx1

�γ 13(x

n1 )

](E I )∗ = 0

Fixed u3(x01 ) = 0, �2(x01 ) = −�α−1f (x01 )

α

Dx1u3(x01 ) + �

γ 13(x01 ) = 0,

u3(xn1 ) = 0, �2(xn1 ) = −�α−1f (xn1 )

α

Dx1u3(xn1 ) + �

γ 13(xn1 ) = 0

Propped cantilever u3(x01 ) = 0, �2(x01 ) = −�α−1f (x01 )

α

Dx1u3(x01 ) + �

γ 13(x01 ) = 0,

u3(xn1 ) = 0, M2(xn1 ) = �α−1f (xn1 )

[− α

Dx1

(�α−1f (xn1 )

α

Dx1u3(xn1 )

)+ α

Dx1

�γ 13(x

n1 )

](E I )∗ = 0

Cantilever u3(x01 ) = 0, �2(x01 ) = 0,

M2(xn1 ) = �α−1f (xn1 )

[− α

Dx1

(�α−1f (xn1 )

α

Dx1u3(xn1 )

)+ α

Dx1

�γ 13(x

n1 )

](E I )∗ = 0

V3(xn1 ) = α

Dx1

{�2α−2f (xn1 )

[− α

Dx1

(�α−1f (xn1 )

α

Dx1u3(xn1 )

)+ α

Dx1

�γ 13(x

n1 )

]}(E I )∗ = 0

Fig. 5 Comparison of maximum deflection for s-FTB and s-FEBBmodels for: the simply supported scheme, the fixed scheme and thepropped cantilever scheme with mid-span point load and the cantileverschemewith end point load, for α = 0.6, �max

f = 0.10 L and symmetric� f distribution

Fig. 6 Local elasticitymodulusmap fromnanoindentation test for poly-mer (iPP+0.2% PACS) sample, reprinted from [44] with permission ofJohn Wiley and Sons

c

c

Fig. 7 Core-shell model of rectangular sample consisting of: the stiffercore of aYoung’smodulus Ec and thickness of c; shell of a bulkYoung’smodulus Eb

Fig. 8 The comparison of experimental measurements [43] (elasticmodulus) of SU-8 polymer microcantilevers loaded at the free end, vs.the results of s-FTB model and CTB model

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Computational Mechanics (2021) 68:697–708 707

δU =∫V

δεT σdV =∫L

∫A(δ

�ε11σ11 + 2δ

�ε13σ13) dA dL =

=∫L

∫A[x3�α−1

f σ11α

Dx1

δ�2 + (δ�2α

Dx3x3 + α

Dx1

δu3)�α−1f σ13] dA dL

(43)

and introducing Eqs. (10), (11) and �α−1f

α

Dx3x3 = 1,

δU =∫L[M2�

α−1f

α

Dx1

δ�2+ (δ�2+�α−1f

α

Dx1

δu3)V3] dL, (44)

and δW is the variation in the external work

δW =∫Lp3 δu3 dL. (45)

To obtain the equilibrium equations, we look for a minimumof the functional J

δJ = δU − δW

=∫L[(M2�

α−1f

α

Dx1

δ�2 + δ�2V3) + (�α−1f V3

α

Dx1

δu3 − p3δu3)] dL = 0,

(46)

which, using the fractional Euler–Lagrange equation [45,46],provides

V3 − α

Dx1

(M2�

α−1f

)= 0 → V3 = α

Dx1

(M2�

α−1f

), (47)

p3 + α

Dx1

(V3�

α−1f

)= 0 → p3 = − α

Dx1

(V3�

α−1f

). (48)

The boundary conditions for selected beam types are pre-sented in Table 3.

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