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Fractional viscoelastic models with non-singular kernels
Accepted Manuscript
Fractional viscoelastic models with non-singular kernels Jianmin Long , Rui Xiao , Wen Chen PII: DOI: Reference:
S0167-6636(18)30101-7 10.1016/j.mechmat.2018.07.012 MECMAT 2905
To appear in:
Mechanics of Materials
Received date: Revised date: Accepted date:
4 February 2018 13 July 2018 16 July 2018
Please cite this article as: Jianmin Long , Rui Xiao , Wen Chen , Fractional viscoelastic models with non-singular kernels, Mechanics of Materials (2018), doi: 10.1016/j.mechmat.2018.07.012
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ACCEPTED MANUSCRIPT Highlights Fractional viscoelastic models with non-singular kernels are introduced. Analytical expressions are derived for typical viscoelastic properties. The performance of new definitions of fractional derivative in describing viscoelastic behaviors is discussed.
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ACCEPTED MANUSCRIPT Fractional viscoelastic models with non-singular kernels
Jianmin Longa,b, Rui Xiaoa,*, Wen Chena,* a
Institute of Soft Matter Mechanics, College of Mechanics and Materials,
b
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Hohai University, Nanjing 210098, China State Key Laboratory for Strength and Vibration of Mechanical Structures,
Xi’an Jiaotong University, Xi’an 710049, China
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Abstract: New definitions of fractional derivative with non-singular kernels are proposed recently. In the present paper, we apply the classical and new definitions of fractional derivative to four fractional viscoelastic models, namely, fractional
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Maxwell model, fractional Kelvin-Voigt model, fractional Zener model and fractional Poynting-Thomson model. For each fractional viscoelastic model, the stress relaxation
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modulus, creep compliance and dynamic modulus are derived analytically under the classical and new fractional derivative definitions. The performance of these models
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under different fractional derivative definitions is further compared. The results show
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that the fractional Zener model and fractional Poynting-Thomson model are equivalent in all conditions. Compared with the classical fractional derivative
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definition with a power function kernel, the fractional derivative definition with a logarithmic function kernel can be used to describe the ultraslow creep and relaxation behaviors. However, the performance of fractional Maxwell model with the exponential function kernel is close to that of integer-order Maxwell model. Fractional Maxwell model and fractional Zener model with the Mittag-Leffler *
Corresponding authors. E-mail addresses: [email protected] (R. Xiao), [email protected] (W. Chen)
ACCEPTED MANUSCRIPT function kernel do not provide accurate descriptions of the stress relaxation modulus at shortest time and the storage modulus at highest frequency. Thus, specific modification is needed when applying the new definitions of fractional derivative to viscoelasticity.
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Keywords: Fractional derivative; Viscoelastic model; Non-singular kernel.
1. Introduction
Viscoelastic materials, such as rubbers, glasses, concretes and biological
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materials, exhibit both elastic and viscous characteristics when undergoing deformation (Lakes, 2009). For elastic response, the stress-strain relationship is rate-independent. For viscous behaviors, the stress depends on deformation rate
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representing as a stiffer response at a larger loading rate. The viscoelastic response shows a dependence on the deformation history (Ferry, 1980). Fractional calculus,
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which contains long-time memory and global correlation information, has been
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proven to be a powerful tool to characterize the viscoelastic behaviors of materials (Koeller, 1984).
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In fractional viscoelastic models, a fractional dashpot is proposed to describe the dependence of viscous response on deformation history, while an elastic spring is
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used to represent the time-independent elastic response. Through different combinations of elastic springs and fractional dashpots, various fractional viscoelastic models have been constructed, such as the fractional Maxwell model, the fractional Kelvin-Voigt model and the fractional Zener model. Complex fractional models containing more than three elements have also been developed by Arikoglu (2014) and Lei et al. (2018). These models have been successfully applied to describe the
ACCEPTED MANUSCRIPT viscoelastic behaviors of various material systems. For example, Carrera et al. (2017) applied fractional Maxwell model to non-Newtonian fluids. Carmichael et al. (2015) adopted fractional Zener model to describe the mechanical properties of living cells. Lei et al. (2018) demonstrated that four-element fractional models could describe the stress response of amorphous thermoplastics at various temperatures and strain rates.
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The fractional derivative models have been used to describe various physical processes, such as heat transfer (Jiang and Qi, 2012), diffusion (Ying et al., 2018), wave propagation (Kumar and Gupta, 2015), viscoelasticity (Mainardi, 2010; Di Paola et al., 2011) and viscoplasticity (Sumelka and Nowak, 2018). However, the
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most widely used Riemann-Liouville fractional derivative and Caputo fractional derivative employ a power function kernel, which has singularity and leads to computational complexity. To overcome this disadvantage, two new definitions of
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fractional derivative with non-singular kernels have been proposed, i.e., the fractional derivative with the exponential function kernel (Caputo and Fabrizio, 2015) and the
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fractional derivative with the Mittag-Leffler function kernel (Atangana and Baleanu,
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2016). To describe the ultraslow physical phenomenon such as ultraslow diffusion and ultraslow creep, the Hadamard fractional derivative with the logarithmic function
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kernel has also been developed (Beghin et al., 2014; Garra et al., 2017). These new fractional derivatives have been applied in electromagnetism (Caputo and Fabrizio,
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2016) and anomalous diffusion (Sun et al., 2017). However, up to now, the performance of these new definitions of fractional derivative in describing viscoelastic behaviors is unclear. And whether they have obvious advantages compared with the classical definitions remains an open question. In this work, an analytical study is conducted to compare the performance of the fractional viscoelastic models with classical and new definitions of fractional
ACCEPTED MANUSCRIPT derivative. In Sect. 2, four fractional derivative definitions, i.e., the classical Riemann-Liouville fractional derivative, the fractional derivatives with the exponential function kernel and Mittag-Leffler function kernel, and the Hadamard fractional derivative, are briefly introduced. Sect. 3 summarizes the constitutive relationships of four fractional viscoelastic models (fractional Maxwell model, Kelvin-Voigt
model,
fractional
Zener
model
and
fractional
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fractional
Poynting-Thomson model). The analytical solutions of relaxation modulus, creep compliance, storage modulus and loss modulus of the above four fractional viscoelastic models with different kernels are then derived in Sect. 4. The
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performance of fractional viscoelastic models with different kernels is compared in the following section. The advantages and disadvantages of different fractional
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derivative definitions are also discussed in detail.
2. Definitions of fractional derivative with different kernels
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The classical Riemann-Liouville fractional derivative, which has a power
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function kernel, is defined as (Kilbas et al., 2006)
D f t
1 d t f x dx, 1 dt 0 t x
(1)
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where Γ(x) is the Eulerian Gamma function, α is the order of the time fractional
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derivative (0<α<1). The Riemann-Liouville fractional derivative is singular when t=x. To avoid such
singularity, several other definitions have been proposed. Caputo and Fabrizio (2015) proposed the fractional derivative with an exponential function kernel, defined as CF
D f t
t x 1 d t f x exp dx. 0 1 dt 1
(2)
ACCEPTED MANUSCRIPT The fractional derivative with a Mittag-Leffler function kernel has also been proposed (Atangana and Baleanu, 2016), which has the following form, AB
1 d t D f t f x M 1 dt 0
t x dx, 1
(3)
al., 2014)
xk . k 0 k 1
M x
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where Mα(x) is the single parameter Mittag-Leffler function, defined as (Gorenflo et
(4)
The fractional derivative with a logarithmic function kernel, also known as the
H
D f t
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Hadamard fractional derivative, was proposed by Beghin et al. (2014) as
1 d t 1 t dx . 1 t 0 f x ln 1 dt 1 x 1 x
(5)
It should be pointed out that the left terminal of fractional derivatives must not
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necessarily be zero. In the present paper, zero is chosen just for convenience, which
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does not affect the subsequent results.
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3. Fractional viscoelastic models
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The rheological representative of viscoelastic models can be constructed by different combinations of elastic spring and viscous dashpot. The corresponding
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fractional viscoelastic models can be obtained by replacing the integer-order dashpot with fractional-order dashpot. In the following subsections, we briefly summarize the constitutive relationships of four commonly used fractional viscoelastic models, i.e., fractional Maxwell model, fractional Kelvin-Voigt model, fractional Zener model and fractional Poynting-Thomson model.
3.1. Fractional Maxwell model
ACCEPTED MANUSCRIPT The rheological representative of fractional Maxwell model is shown in Fig. 1(a), which is consisted of a spring in series with a fractional dashpot. The relationship between stress σ and strain ε of fractional Maxwell model can be expressed as
d d E , dt dt
(6)
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where E is the elastic modulus of the spring, and τ is the relaxation time of the fractional dashpot.
3.2. Fractional Kelvin-Voigt model
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The fractional Kelvin-Voigt model is composed by a spring in parallel with a fractional dashpot (Fig. 1(b)). The constitutive relationship has the following form, (7)
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dae e s + = . dt a t a Et a
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3.3. Fractional three-element models
The fractional three-element models include the fractional Zener model and the
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fractional Poynting-Thomson model. The fractional Zener model is composed of an elastic spring and a fractional Maxwell model in parallel (Fig. 1(c)). The stress is
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related with the strain as
d d E1 E E 1 2 , dt dt
(8)
where E1 is the elastic modulus of the spring in the first branch, E2 is the elastic modulus of the spring in the branch of the fractional Maxwell element, and τ is the relaxation time of the fractional dashpot. While the fractional Poynting-Thomson model is consisted of a spring and a fractional Kelvin-Voigt element, which are arranged in series (Fig. 1(d)). The corresponding constitutive relationship is
ACCEPTED MANUSCRIPT represented as d 1 d 1 1 , dt ˆ Eˆ1 dt Eˆ1 Eˆ 2 ˆ
(9)
where 𝐸̂1 and 𝐸̂2 are the corresponding elastic moduli of the two springs, 𝜏̂ is the relaxation time of the fractional dashpot.
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Though the fractional Zener model and fractional Poynting-Thomson model are constructed from different combinations of elastic spring and fractional dashpot, it is found that the two models are actually equivalent. If the material constants E1, E2 and τα of fractional Zener model in Eq. (8) are replaced by 𝐸̂1 𝐸̂2/(𝐸̂1 +𝐸̂2 ), 𝐸̂12 /(𝐸̂1+𝐸̂2 )
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and 𝐸̂2 𝜏̂ 𝛼 /(𝐸̂1 +𝐸̂2 ) respectively, the obtained constitutive relationship is identical to Eq. (9). Thus, in the following sections, we only present the results of the fractional
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Maxwell model, fractional Kevin-Voigt model and fractional Zener model.
3.4. General remarks
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In the above subsections, the fractional viscoelastic models are developed by
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replacing dashpot of integer order with dashpot of fractional order. Though this approach has been widely used, the underlying physical mechanism is not well
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established. Mainardi et al. (2010) provided a physical interpretation of fraction Zener model through fractional diffusion. Pfitzenreiter (2008) developed a structural model
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to explain the fractional derivatives in stress strain relationships of polymers. However, so far the physical interpretation of fractional viscoelastic models is far from settled. Nevertheless, fractional models are still widely used because it has the advantage to use a limited set of parameters to describe complex phenomena. For fractional viscoelastic models, the physical interpretation for the fractional order α is important. One possible explanation is that fractional order α represents the
ACCEPTED MANUSCRIPT distribution of the relaxation processes. A smaller α indicates materials have a broad distribution of relaxation processes. This may be related with the structure of materials, such as molecular weight distribution of polymers.
4. The analytical solutions of the relaxation modulus and creep compliance
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In the following subsections, we derive the stress relaxation modulus, creep compliance and dynamic properties of the above four models with different kernels. We will use the fractional Maxwell model as an example. The procedures to obtain these properties of the other three models are similar. Tables 1-3 summarize the
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obtained stress relaxation modulus, creep compliance, storage modulus and loss modulus of fractional Maxwell model, fractional Kelvin-Voigt model and fractional Zener model. The mechanical properties of fractional Poynting-Thomson model can
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be obtained easily by replacing the corresponding model parameters from fractional
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Zener model as discussed in Sect. 3.3.
4.1. The stress relaxation modulus
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In a stress relaxation test, a constant strain ε=ε0 is applied and the stress is
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measured. The stress relaxation modulus is defined as G(t)=σ(t)/ε0. To obtain the stress relaxation modulus of fractional Maxwell model with power function kernel,
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we apply the Laplace transform to Eq. (6) and obtain the following relationship, s s
s Es s ,
(10)
where 𝜎̃(s) and 𝜀̃(s) represent the Laplace transform of σ(t) and ε(t), respectively. The Laplace transform of relaxation modulus can then be derived as
s s 1 G s E . s s s 1/
(11)
ACCEPTED MANUSCRIPT By applying the inverse Laplace transform to the above equation, the relaxation modulus of fractional Maxwell model with power function kernel is obtained as t GMRL t EM
.
(12)
In the above derivation process, the property (Gorenflo et al., 2014)
s 1 , s
(13)
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L M t ; s
is used, where L denotes the Laplace transform, λ is an arbitrary positive real number. Using the same procedure for the fractional Maxwell model with exponential
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function kernel leads to the following results,
s s s E , 1 / s 1 / s
(14)
s E . s s 1 s
(15)
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G s
The relaxation modulus can then be obtained using the inverse Laplace
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transform,
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GMCF t
E t exp 1 1
.
(16)
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For fractional Maxwell model with Mittag-Leffler function kernel, Eq. (3) is
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substituted into Eq. (6). Then the Laplace transform is applied, which yields,
s s s E . 1 / s 1 / s
(17)
The relaxation modulus can then be derived as G s
GMAB t
s E s 1 , s s 1 s
(18)
E t M . 1 1
(19)
ACCEPTED MANUSCRIPT For stress relaxation, ε=ε0; by using the definition (5), Eq. (6) with logarithmic function kernel can be rewritten as H
D t
t E 0 ln 1 t . 1
(20)
The stress can then be derived as
by using the property (Beghin et al., 2014) D M ln 1 t
ln 1 t M ln 1 t . 1
(22)
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H
(21)
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ln 1 t ,
t 0 EM
In Eq. (22), γ is an arbitrary positive real number.
The relaxation modulus of fractional Maxwell model with logarithmic function kernel is then obtained as
ln 1 t .
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t EM 0
(23)
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GMH t
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4.2. The creep compliance
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The Laplace transform of the creep compliance can be estimated from the below
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equation,
J s
1 . s G s 2
(24)
The creep compliance can then be obtained through inverse Laplace transform. The obtained creep compliances of fractional Maxwell model with power function kernel, exponential function kernel and Mittag-Leffler function kernel are shown as following respectively,
ACCEPTED MANUSCRIPT 1 t 1 , E 1
J MRL t
J
CF M
(25)
1 t , t E
J MAB t
(26)
1 t / . E
(27)
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We derive the creep compliance of fractional Maxwell model with logarithmic function kernel directly from Eq. (6). For creep, σ=σ0; by using the definition (5), Eq. (6) can be rewritten as D t
0 1 ln 1 t . E 1
The strain can be derived as
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H
ln 1 t t 1 , E 1
(29)
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0
(28)
by using the following equaiton (Beghin et al., 2014),
D ln 1 t 1 .
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(30)
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The creep compliance is then obtained as
t 1 ln 1 t 1 . 0 E 1
(31)
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J MH t
4.3. The dynamic properties Understanding the dynamic response of the viscoelastic models is important for
the application of these models. In a dynamic test, a sine-shaped strain or stress is applied. The corresponding stress or strain is then measured to give the dynamic properties, such as the storage modulus and loss modulus. Here we assume the strain has the form ε=ε0eiωt, where ω is the angular frequency. The complex modulus can be
ACCEPTED MANUSCRIPT estimated from the relaxation modulus using the following relationship, G iG i .
(32)
Using the above equation and Eq. (11), we can obtain the complex modulus of the fractional Maxwell model with power function kernel as
i . E 1 i
(33)
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G
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The storage modulus is the real part of G* MRL (ω), and loss modulus is the imaginary part of G*MRL (ω). Rearranging the real part and imaginary part of G*MRL(ω)
RL 1M
G
RL 2M
RL M
cos / 2 , E 2 2 cos / 2 1
(34)
RL M
sin / 2 . E 2 2 cos / 2 1
(35)
Re G
Im G
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G
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leads to the following analytical solution for the storage modulus and loss modulus,
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The procedures to obtain storage modulus and loss modulus of the fractional Maxwell model with exponential function kernel and Mittag-Leffler function kernel
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are the same as above. The analytical equations can be found in Table 1. As shown in
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Eq. (32), the complex modulus is deduced from the Laplace transform of the stress relaxation modulus. For fractional Maxwell model with logarithmic function kernel,
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the Laplace transform of the relaxation modulus does not have an analytical form. Therefore, the dynamic properties of fractional Maxwell model with logarithmic function kernel do not have the corresponding analytical forms. Besides, with 0<α<1, the storage moduli and loss moduli of fractional Maxwell model under different fractional derivative definitions are positive for all frequencies, as shown by the equations in Table 1. That is, the thermodynamic constraints are satisfied for fractional Maxwell model (Pritz, 2003). It can be easily concluded that
ACCEPTED MANUSCRIPT fractional
Kelvin-Voigt
model,
fractional
Zener
model
and
fractional
Poynting-Thomson model with different kernels all satisfy the thermodynamic constraints, as shown by the equations in Tables 2 and 3.
5. Results and discussions
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In this section, we compare the performance of fractional viscoelastic models with different kernels. As shown in Tables 1-3, the mechanical performance of each model depends on elastic modulus of each spring, relaxation time of each fractional dashpot and fractional order. To be simple, in the following part, the fractional order
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and relaxation time are set as typical values of α=0.5 and τ=10 s respectively if not specified elsewhere. The mechanical response of fractional Maxwell model under four fractional derivative definitions are shown in Fig. 2. It can be seen from Figs. 2(a) and
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2(b) that the relaxation moduli (creep compliances) of the Mittag-Leffler function kernel and logarithmic function kernel decrease (increase) more slowly than that of
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the power function kernel when the fractional order is the same. The logarithmic
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functional kernel results in the slowest relaxation or creep rate among four fractional derivative definitions, which can be used to characterize the ultraslow relaxation
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(creep) of viscoelastic materials. When time is small, the relaxation modulus and creep compliance of the logarithmic function kernel are the same as that of classical
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power function kernel. The fractional Maxwell model with exponential function kernel exhibits exponential relaxation and linear creep, which have the same forms as that of the integer-order Maxwell model. The relaxation moduli and creep compliances of exponential function kernel and Mittag-Leffler function kernel do not approach the elastic modulus E and its inverse 1/E, respectively, when time t tends to 0. These observations demonstrate that the fractional Maxwell model with the
ACCEPTED MANUSCRIPT exponential function kernel and Mittag-Leffler function kernel is problematic to describe the relaxation and creep behaviors of viscoelastic materials. This also applies to the dynamic properties. The storage moduli of fractional Maxwell model using these two kernels do not reach the elastic modulus of the spring at highest frequency (Fig. 2 (c)). Fig. 2(d) shows that the maximum value of loss modulus of the
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exponential function kernel is about twice of that of the other two kernels since the performance of fractional Maxwell model with the exponential function kernel is close to that of the integer-order Maxwell model.
Fig. 3 shows the mechanical resposne of fractional Kelvin-Voigt model. For
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relaxation modulus and creep compliance (Figs. 3(a) and 3(b)), similar conclusions can be drawn as that for fractional Maxwell model. The creep compliance of the exponential function kernel exhibits an exponential increase with time instead of a
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linear increase compared with fractional Maxwell model. For dynamic properties, unlike that of the power function kernel, the storage moduli of the exponential
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function kernel and Mittag-Leffler function kernel have finite values with increasing
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frequency (Fig. 3(c)). Similar to the fractional Maxwell model, the maximum value of loss modulus of the exponential function kernel is about twice of that of the
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Mittag-Leffler function kernel (Fig. 3(d)). The mechanical properties of fractional Zener model are illustrated in Fig. 4,
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where the ratio between the elastic moduli of two springs is set as E1/E2=0.01 to represent a tremendous modulus change with time or frequency, which is commonly observed in polymers. A comparison between Figs. 4(a)-4(d) and Figs. 2(a)-2(d) reveals that the performance of fractional Zener model is close to that of fractional Maxwell model under all four fractional derivative definitions. Figs. 5 and 6 plot the relaxation modulus of fractional Maxwell model to
ACCEPTED MANUSCRIPT compare the performance of different kernels for different α and τ. It can be seen from Figs. 5(a) and 5(c) that, for a given τ, the relaxation moduli of the power function kernel and Mittag-Leffler function kernel decrease firstly more slowly and then more rapidly with increasing α. While the relaxation modulus of the logarithmic function kernel decreases more slowly with increasing α over the entire time period (Fig. 5(d)).
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The relaxation modulus of the exponential function kernel has the same variation trend with that of the logarithmic function kernel (Fig. 5(b)). For a given α, Fig. 6 shows that the relaxation moduli of four kernels decrease more slowly over the entire time period as τ increasing. The curves of the power function kernel for different τ can
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overlap together through translation (Fig. 6(a)), since t scales with τ in this case (Eq. (12)). In addition, Figs. 5 and 6 show again that the relaxation moduli of the exponential function kernel and Mittag-Leffler function kernel do not approach the
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elastic modulus of the spring when t tends to 0. A larger deviation between initial relaxation moduli and elastic modulus can be observed with a smaller value of α or τ.
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The results of fractional viscoelastic models are also compared to the
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experimental results. Firstly, the fractional Maxwell model with different kernels is applied to describe the experimental measured creep compliance of high strength
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self-compacting concrete (Maia and Figueiras, 2012). The model parameters are identified for different kernels, as shown in Table 4. It can be seen from Fig. 7 that the
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power function kernel and Mittag-Leffler function kernel perform similarly in this case. The fractional Maxwell model with the logarithmic function kernel provides the best description of the ultraslow creep behavior. The fractional Maxwell model with the exponential function kernel totally failed to describe the experimental data. As shown in Table 2, the Kelvin-Voigt models with all different kernels do not have an immediate elastic response, which is not consistent with the experimental
ACCEPTED MANUSCRIPT observations in Fig. 7. Thus, the fractional Kelvin-Voigt models are not used. The fractional Zener model with different kernels is applied to describe the stress relaxation modulus and creep compliance of lightly vulcanized Hevea rubber (Ferry, 1980). The model parameters are identified from the experimental data of stress relaxation modulus (Fig. 8(a)) and listed in Table 5. And then the fractional model
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together with the identified parameters is used to predict the creep behavior (Fig. 8(b)). It can be seen from Fig. 8 that the fractional model with the logarithmic function kernel exhibits exactly the same performance as the power function kernel. The fractional Zener models with the exponential function kernel and Mittag-Leffler
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function kernel fail to describe the experimental data, because they do not provide the accurate stress relaxation modulus at shortest time, as shown in Fig. 4(a). We do not use the experimental measured dynamic properties because the fractional Zener model
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with the exponential function kernel and Mittag-Leffler function kernel does not predict the right storage modulus at highest frequency, while no analytical solutions
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exist for the storage modulus when using the logarithmic function kernel.
6. Conclusions
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In summary, the performance of different definitions of fractional derivative in four fractional viscoelastic models is examined. The analytical solutions of the stress
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relaxation modulus, creep compliance and dynamic modulus are obtained. It is found that the fractional Poynting-Thomson model and fractional Zener model are equivalent under all the four fractional derivative definitions. The fractional Zener model has similar performance as fractional Maxwell model. The fractional derivative with the logarithmic function kernel can be used to characterize the ultraslow relaxation and creep. The fractional Maxwell model with the exponential function
ACCEPTED MANUSCRIPT kernel yields an exponential form of the stress relaxation modulus and a linear form of the creep compliance, which are the same as that of the integer-order Maxwell model. The fractional Maxwell model and fractional Zener model with the Mittag-Leffler function kernel do not provide accurate descriptions of the stress relaxation modulus at shortest time and the storage modulus at highest frequency. These results
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demonstrate that the fractional derivatives with the exponential function kernel and Mittag-Leffler function kernel may be problematic in describing the viscoelastic behaviors of materials. The classical fractional derivative is still a relative good choice for viscoelastic models. Extra efforts should be made to apply the new fractional
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derivatives to viscoelasticity.
Acknowledgments
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The authors acknowledge the supports from the National Natural Science Foundation of China (Grant No. 11702081), the Fundamental Research Funds for
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Central Universities, Hohai University (Grant No. 2016B01414), China Postdoctoral
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Science Foundation (Grant No. 2017M621605) and Open Project of State Key Laboratory for Strength and Vibration of Mechanical Structures (Grant No.
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SV2017-KF-19). We also would like to acknowledge the useful discussions with
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Dumitru Baleanu, Hongguang Sun and Yingjie Liang.
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Sun, H.G., Hao, X.X., Zhang, Y., Baleanu, D., 2017. Relaxation and diffusion models with non-singular kernels. Physica A: Stat. Mech. Appl. 468, 590-596.
Ying, Y.P., Lian, Y.P., Tang, S.Q., Liu, W.K., 2018. Enriched reproducing kernel particle method for fractional advection-diffusion equation. Acta Mech. Sin. 34, 515-527.
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Figure captions
Figure 1. Rheological representative of fractional viscoelastic models: (a) fractional Maxwell model, (b) fractional Kelvin-Voigt model, (c) fractional Zener model and (d)
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fractional Poynting-Thomson model.
ACCEPTED MANUSCRIPT Figure 2. Mechanical properties of fractional Maxwell model: (a) relaxation modulus,
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(b) creep compliance, (c) storage modulus and (d) loss modulus.
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Figure 3. Mechanical properties of fractional Kelvin-Voigt model: (a) relaxation
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modulus, (b) creep compliance, (c) storage modulus and (d) loss modulus.
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Figure 4. Mechanical properties of fractional Zener model: (a) relaxation modulus, (b)
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creep compliance, (c) storage modulus and (d) loss modulus.
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Figure 5. Relaxation modulus of fractional Maxwell model for different α: (a) power
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function kernel, (b) exponential function kernel, (c) Mittag-Leffler function kernel
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and (d) logarithmic function kernel. Here fractional order and relaxation time are set
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as typical values of α=0.3, 0.5, 0.7, 1 and τ=10 s, respectively.
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Figure 6. Relaxation modulus of fractional Maxwell model for different τ: (a) power
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function kernel, (b) exponential function kernel, (c) Mittag-Leffler function kernel
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and (d) logarithmic function kernel. Here fractional order and relaxation time are set
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as typical values of α=0.5 and τ=1 s, 10 s, 100 s, 1000 s, respectively.
Figure 7. Comparison of fractional Maxwell model with experimental measured creep
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compliance of high strength self-compacting concrete.
Figure 8. Comparison of fractional Zener models with experimental data of lightly
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vulcanized Hevea rubber: (a) relaxation modulus, (b) creep compliance.
Table captions
Table 1. Mechanical properties of fractional Maxwell model. Power function
Property
Exponential function
Mittag-Leffler function
Logarithmic function
E 1 t M 1
ln 1 t EM
1 1 t / 1 E
ln 1 t 1 1 E 1
1 E cos / 2 2 1 2 2 2 1 cos / 2
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Kernel
Relaxation modulus
t EM
E 1 t exp 1
Creep compliance
1 t 1 E 1
1 1 t 1 E
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E
AC
Storage modulus
Loss modulus
cos / 2
2 1 2 cos / 2
E 2 1
2 1 2 2
E sin / 2
E sin / 2 2 1 2 cos / 2
E
2 1 2 2
2 1 2 2 2 1 cos / 2
NA
ACCEPTED MANUSCRIPT Table 2. Mechanical properties of fractional Kelvin-Voigt model. Kernel
Power function
Exponential function
Mittag-Leffler function
Logarithmic function
Relaxation modulus
t E 1 1
t E 1 exp 1 1
t E 1 M 1 1
ln 1 t E 1 1
Creep compliance
1 1 t E M
1 1 1 E t exp 1
1 1 1 t E M 1
1 1 ln 1 t E M
2 1 E 1 2 2 2 1
2 1 cos / 2 E 1 2 2 2 1 2 1 cos / 2
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E sin / 2 2 1 2 2 2 1 cos / 2
NA
1 E cos / 2
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Storage modulus
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Property
E
E
sin / 2
1 2 2
2
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PT
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Loss modulus
ACCEPTED MANUSCRIPT Table 3. Mechanical properties of fractional Zener model. Replacing E1, E2 and τα in this table by 𝐸̂1 𝐸̂2 /(𝐸̂1 +𝐸̂2), 𝐸̂12 /(𝐸̂1 +𝐸̂2 ) and 𝐸̂2 𝜏̂ 𝛼 /(𝐸̂1 +𝐸̂2) respectively leads to the mechanical properties of fractional Poynting-Thomson model. Kernel
Power function
Exponential function
Mittag-Leffler function
E2 1 t exp 1
E2 1 t M 1
Logarithmic function
Property
Relaxation modulus
E1
t M
Creep compliance
E2 1 E E 1 2 E1 t M E1 E2
1 E1
1 E1
E2 1 E 2 E 1 1 E1 t exp E2 E1 1
E2 1 E 2 E 1 1 E1 t M E2 E1 1
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1 E1
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Loss modulus
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1 cos / 2
E1 E2 2
M
1 1
2
2
2
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E2 sin / 2 2 1 2 cos / 2
ln 1 t M
1 E1
E2 1 E E 1 2 E1 M E1 E2 ln 1 t
Storage modulus
E1 E2
E1 E2
E1 E2 cos / 2 2 1 2 cos / 2
E1
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E1 E2
2 1 2 2 2 1 cos / 2
NA
E2 sin / 2
E2
2 1 2 2
2 1 2 2 2 1 cos / 2
NA
ACCEPTED MANUSCRIPT Table 4. Model parameters identified from the experimental data of ultraslow creep of high strength self-compacting concrete. The elastic modulus of the material is E=39.9 GPa. α
τ (s)
residuals
Power function
0.14
23.76
8.85E-5
Exponential function
0.79
848.15
1.96E-2
Mittag-Leffler function
0.25
2.70
1.68E-4
Logarithmic function
0.58
3.40
1.56E-5
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kernel
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Table 5. Model parameters identified from the stress relaxation modulus of lightly vulcanized Hevea rubber. The two elastic moduli of the material are E1=1.62 MPa and E2=1.175 GPa.
α
log τ
residuals
0.7707
-7.9374
0.3648
1.0000
-7.4348
5.4478
Mittag-Leffler function
0.9907
-3.7786
33.2983
Logarithmic function
0.7707
-7.9374
0.3648
Power function
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CE
PT
ED
Exponential function
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kernel
Fractional viscoelastic models with non-singular kernels Jianmin Long , Rui Xiao , Wen Chen PII: DOI: Reference:
S0167-6636(18)30101-7 10.1016/j.mechmat.2018.07.012 MECMAT 2905
To appear in:
Mechanics of Materials
Received date: Revised date: Accepted date:
4 February 2018 13 July 2018 16 July 2018
Please cite this article as: Jianmin Long , Rui Xiao , Wen Chen , Fractional viscoelastic models with non-singular kernels, Mechanics of Materials (2018), doi: 10.1016/j.mechmat.2018.07.012
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Highlights Fractional viscoelastic models with non-singular kernels are introduced. Analytical expressions are derived for typical viscoelastic properties. The performance of new definitions of fractional derivative in describing viscoelastic behaviors is discussed.
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ACCEPTED MANUSCRIPT Fractional viscoelastic models with non-singular kernels
Jianmin Longa,b, Rui Xiaoa,*, Wen Chena,* a
Institute of Soft Matter Mechanics, College of Mechanics and Materials,
b
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Hohai University, Nanjing 210098, China State Key Laboratory for Strength and Vibration of Mechanical Structures,
Xi’an Jiaotong University, Xi’an 710049, China
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Abstract: New definitions of fractional derivative with non-singular kernels are proposed recently. In the present paper, we apply the classical and new definitions of fractional derivative to four fractional viscoelastic models, namely, fractional
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Maxwell model, fractional Kelvin-Voigt model, fractional Zener model and fractional Poynting-Thomson model. For each fractional viscoelastic model, the stress relaxation
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modulus, creep compliance and dynamic modulus are derived analytically under the classical and new fractional derivative definitions. The performance of these models
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under different fractional derivative definitions is further compared. The results show
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that the fractional Zener model and fractional Poynting-Thomson model are equivalent in all conditions. Compared with the classical fractional derivative
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definition with a power function kernel, the fractional derivative definition with a logarithmic function kernel can be used to describe the ultraslow creep and relaxation behaviors. However, the performance of fractional Maxwell model with the exponential function kernel is close to that of integer-order Maxwell model. Fractional Maxwell model and fractional Zener model with the Mittag-Leffler *
Corresponding authors. E-mail addresses: [email protected] (R. Xiao), [email protected] (W. Chen)
ACCEPTED MANUSCRIPT function kernel do not provide accurate descriptions of the stress relaxation modulus at shortest time and the storage modulus at highest frequency. Thus, specific modification is needed when applying the new definitions of fractional derivative to viscoelasticity.
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Keywords: Fractional derivative; Viscoelastic model; Non-singular kernel.
1. Introduction
Viscoelastic materials, such as rubbers, glasses, concretes and biological
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materials, exhibit both elastic and viscous characteristics when undergoing deformation (Lakes, 2009). For elastic response, the stress-strain relationship is rate-independent. For viscous behaviors, the stress depends on deformation rate
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representing as a stiffer response at a larger loading rate. The viscoelastic response shows a dependence on the deformation history (Ferry, 1980). Fractional calculus,
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which contains long-time memory and global correlation information, has been
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proven to be a powerful tool to characterize the viscoelastic behaviors of materials (Koeller, 1984).
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In fractional viscoelastic models, a fractional dashpot is proposed to describe the dependence of viscous response on deformation history, while an elastic spring is
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used to represent the time-independent elastic response. Through different combinations of elastic springs and fractional dashpots, various fractional viscoelastic models have been constructed, such as the fractional Maxwell model, the fractional Kelvin-Voigt model and the fractional Zener model. Complex fractional models containing more than three elements have also been developed by Arikoglu (2014) and Lei et al. (2018). These models have been successfully applied to describe the
ACCEPTED MANUSCRIPT viscoelastic behaviors of various material systems. For example, Carrera et al. (2017) applied fractional Maxwell model to non-Newtonian fluids. Carmichael et al. (2015) adopted fractional Zener model to describe the mechanical properties of living cells. Lei et al. (2018) demonstrated that four-element fractional models could describe the stress response of amorphous thermoplastics at various temperatures and strain rates.
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The fractional derivative models have been used to describe various physical processes, such as heat transfer (Jiang and Qi, 2012), diffusion (Ying et al., 2018), wave propagation (Kumar and Gupta, 2015), viscoelasticity (Mainardi, 2010; Di Paola et al., 2011) and viscoplasticity (Sumelka and Nowak, 2018). However, the
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most widely used Riemann-Liouville fractional derivative and Caputo fractional derivative employ a power function kernel, which has singularity and leads to computational complexity. To overcome this disadvantage, two new definitions of
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fractional derivative with non-singular kernels have been proposed, i.e., the fractional derivative with the exponential function kernel (Caputo and Fabrizio, 2015) and the
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fractional derivative with the Mittag-Leffler function kernel (Atangana and Baleanu,
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2016). To describe the ultraslow physical phenomenon such as ultraslow diffusion and ultraslow creep, the Hadamard fractional derivative with the logarithmic function
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kernel has also been developed (Beghin et al., 2014; Garra et al., 2017). These new fractional derivatives have been applied in electromagnetism (Caputo and Fabrizio,
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2016) and anomalous diffusion (Sun et al., 2017). However, up to now, the performance of these new definitions of fractional derivative in describing viscoelastic behaviors is unclear. And whether they have obvious advantages compared with the classical definitions remains an open question. In this work, an analytical study is conducted to compare the performance of the fractional viscoelastic models with classical and new definitions of fractional
ACCEPTED MANUSCRIPT derivative. In Sect. 2, four fractional derivative definitions, i.e., the classical Riemann-Liouville fractional derivative, the fractional derivatives with the exponential function kernel and Mittag-Leffler function kernel, and the Hadamard fractional derivative, are briefly introduced. Sect. 3 summarizes the constitutive relationships of four fractional viscoelastic models (fractional Maxwell model, Kelvin-Voigt
model,
fractional
Zener
model
and
fractional
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fractional
Poynting-Thomson model). The analytical solutions of relaxation modulus, creep compliance, storage modulus and loss modulus of the above four fractional viscoelastic models with different kernels are then derived in Sect. 4. The
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performance of fractional viscoelastic models with different kernels is compared in the following section. The advantages and disadvantages of different fractional
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derivative definitions are also discussed in detail.
2. Definitions of fractional derivative with different kernels
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The classical Riemann-Liouville fractional derivative, which has a power
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function kernel, is defined as (Kilbas et al., 2006)
D f t
1 d t f x dx, 1 dt 0 t x
(1)
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where Γ(x) is the Eulerian Gamma function, α is the order of the time fractional
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derivative (0<α<1). The Riemann-Liouville fractional derivative is singular when t=x. To avoid such
singularity, several other definitions have been proposed. Caputo and Fabrizio (2015) proposed the fractional derivative with an exponential function kernel, defined as CF
D f t
t x 1 d t f x exp dx. 0 1 dt 1
(2)
ACCEPTED MANUSCRIPT The fractional derivative with a Mittag-Leffler function kernel has also been proposed (Atangana and Baleanu, 2016), which has the following form, AB
1 d t D f t f x M 1 dt 0
t x dx, 1
(3)
al., 2014)
xk . k 0 k 1
M x
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where Mα(x) is the single parameter Mittag-Leffler function, defined as (Gorenflo et
(4)
The fractional derivative with a logarithmic function kernel, also known as the
H
D f t
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Hadamard fractional derivative, was proposed by Beghin et al. (2014) as
1 d t 1 t dx . 1 t 0 f x ln 1 dt 1 x 1 x
(5)
It should be pointed out that the left terminal of fractional derivatives must not
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necessarily be zero. In the present paper, zero is chosen just for convenience, which
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does not affect the subsequent results.
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3. Fractional viscoelastic models
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The rheological representative of viscoelastic models can be constructed by different combinations of elastic spring and viscous dashpot. The corresponding
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fractional viscoelastic models can be obtained by replacing the integer-order dashpot with fractional-order dashpot. In the following subsections, we briefly summarize the constitutive relationships of four commonly used fractional viscoelastic models, i.e., fractional Maxwell model, fractional Kelvin-Voigt model, fractional Zener model and fractional Poynting-Thomson model.
3.1. Fractional Maxwell model
ACCEPTED MANUSCRIPT The rheological representative of fractional Maxwell model is shown in Fig. 1(a), which is consisted of a spring in series with a fractional dashpot. The relationship between stress σ and strain ε of fractional Maxwell model can be expressed as
d d E , dt dt
(6)
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where E is the elastic modulus of the spring, and τ is the relaxation time of the fractional dashpot.
3.2. Fractional Kelvin-Voigt model
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The fractional Kelvin-Voigt model is composed by a spring in parallel with a fractional dashpot (Fig. 1(b)). The constitutive relationship has the following form, (7)
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dae e s + = . dt a t a Et a
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3.3. Fractional three-element models
The fractional three-element models include the fractional Zener model and the
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fractional Poynting-Thomson model. The fractional Zener model is composed of an elastic spring and a fractional Maxwell model in parallel (Fig. 1(c)). The stress is
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related with the strain as
d d E1 E E 1 2 , dt dt
(8)
where E1 is the elastic modulus of the spring in the first branch, E2 is the elastic modulus of the spring in the branch of the fractional Maxwell element, and τ is the relaxation time of the fractional dashpot. While the fractional Poynting-Thomson model is consisted of a spring and a fractional Kelvin-Voigt element, which are arranged in series (Fig. 1(d)). The corresponding constitutive relationship is
ACCEPTED MANUSCRIPT represented as d 1 d 1 1 , dt ˆ Eˆ1 dt Eˆ1 Eˆ 2 ˆ
(9)
where 𝐸̂1 and 𝐸̂2 are the corresponding elastic moduli of the two springs, 𝜏̂ is the relaxation time of the fractional dashpot.
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Though the fractional Zener model and fractional Poynting-Thomson model are constructed from different combinations of elastic spring and fractional dashpot, it is found that the two models are actually equivalent. If the material constants E1, E2 and τα of fractional Zener model in Eq. (8) are replaced by 𝐸̂1 𝐸̂2/(𝐸̂1 +𝐸̂2 ), 𝐸̂12 /(𝐸̂1+𝐸̂2 )
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and 𝐸̂2 𝜏̂ 𝛼 /(𝐸̂1 +𝐸̂2 ) respectively, the obtained constitutive relationship is identical to Eq. (9). Thus, in the following sections, we only present the results of the fractional
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Maxwell model, fractional Kevin-Voigt model and fractional Zener model.
3.4. General remarks
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In the above subsections, the fractional viscoelastic models are developed by
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replacing dashpot of integer order with dashpot of fractional order. Though this approach has been widely used, the underlying physical mechanism is not well
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established. Mainardi et al. (2010) provided a physical interpretation of fraction Zener model through fractional diffusion. Pfitzenreiter (2008) developed a structural model
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to explain the fractional derivatives in stress strain relationships of polymers. However, so far the physical interpretation of fractional viscoelastic models is far from settled. Nevertheless, fractional models are still widely used because it has the advantage to use a limited set of parameters to describe complex phenomena. For fractional viscoelastic models, the physical interpretation for the fractional order α is important. One possible explanation is that fractional order α represents the
ACCEPTED MANUSCRIPT distribution of the relaxation processes. A smaller α indicates materials have a broad distribution of relaxation processes. This may be related with the structure of materials, such as molecular weight distribution of polymers.
4. The analytical solutions of the relaxation modulus and creep compliance
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In the following subsections, we derive the stress relaxation modulus, creep compliance and dynamic properties of the above four models with different kernels. We will use the fractional Maxwell model as an example. The procedures to obtain these properties of the other three models are similar. Tables 1-3 summarize the
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obtained stress relaxation modulus, creep compliance, storage modulus and loss modulus of fractional Maxwell model, fractional Kelvin-Voigt model and fractional Zener model. The mechanical properties of fractional Poynting-Thomson model can
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be obtained easily by replacing the corresponding model parameters from fractional
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Zener model as discussed in Sect. 3.3.
4.1. The stress relaxation modulus
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In a stress relaxation test, a constant strain ε=ε0 is applied and the stress is
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measured. The stress relaxation modulus is defined as G(t)=σ(t)/ε0. To obtain the stress relaxation modulus of fractional Maxwell model with power function kernel,
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we apply the Laplace transform to Eq. (6) and obtain the following relationship, s s
s Es s ,
(10)
where 𝜎̃(s) and 𝜀̃(s) represent the Laplace transform of σ(t) and ε(t), respectively. The Laplace transform of relaxation modulus can then be derived as
s s 1 G s E . s s s 1/
(11)
ACCEPTED MANUSCRIPT By applying the inverse Laplace transform to the above equation, the relaxation modulus of fractional Maxwell model with power function kernel is obtained as t GMRL t EM
.
(12)
In the above derivation process, the property (Gorenflo et al., 2014)
s 1 , s
(13)
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L M t ; s
is used, where L denotes the Laplace transform, λ is an arbitrary positive real number. Using the same procedure for the fractional Maxwell model with exponential
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function kernel leads to the following results,
s s s E , 1 / s 1 / s
(14)
s E . s s 1 s
(15)
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G s
The relaxation modulus can then be obtained using the inverse Laplace
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transform,
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GMCF t
E t exp 1 1
.
(16)
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For fractional Maxwell model with Mittag-Leffler function kernel, Eq. (3) is
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substituted into Eq. (6). Then the Laplace transform is applied, which yields,
s s s E . 1 / s 1 / s
(17)
The relaxation modulus can then be derived as G s
GMAB t
s E s 1 , s s 1 s
(18)
E t M . 1 1
(19)
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D t
t E 0 ln 1 t . 1
(20)
The stress can then be derived as
by using the property (Beghin et al., 2014) D M ln 1 t
ln 1 t M ln 1 t . 1
(22)
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H
(21)
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ln 1 t ,
t 0 EM
In Eq. (22), γ is an arbitrary positive real number.
The relaxation modulus of fractional Maxwell model with logarithmic function kernel is then obtained as
ln 1 t .
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t EM 0
(23)
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GMH t
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4.2. The creep compliance
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The Laplace transform of the creep compliance can be estimated from the below
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equation,
J s
1 . s G s 2
(24)
The creep compliance can then be obtained through inverse Laplace transform. The obtained creep compliances of fractional Maxwell model with power function kernel, exponential function kernel and Mittag-Leffler function kernel are shown as following respectively,
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J MRL t
J
CF M
(25)
1 t , t E
J MAB t
(26)
1 t / . E
(27)
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We derive the creep compliance of fractional Maxwell model with logarithmic function kernel directly from Eq. (6). For creep, σ=σ0; by using the definition (5), Eq. (6) can be rewritten as D t
0 1 ln 1 t . E 1
The strain can be derived as
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H
ln 1 t t 1 , E 1
(29)
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0
(28)
by using the following equaiton (Beghin et al., 2014),
D ln 1 t 1 .
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(30)
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The creep compliance is then obtained as
t 1 ln 1 t 1 . 0 E 1
(31)
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J MH t
4.3. The dynamic properties Understanding the dynamic response of the viscoelastic models is important for
the application of these models. In a dynamic test, a sine-shaped strain or stress is applied. The corresponding stress or strain is then measured to give the dynamic properties, such as the storage modulus and loss modulus. Here we assume the strain has the form ε=ε0eiωt, where ω is the angular frequency. The complex modulus can be
ACCEPTED MANUSCRIPT estimated from the relaxation modulus using the following relationship, G iG i .
(32)
Using the above equation and Eq. (11), we can obtain the complex modulus of the fractional Maxwell model with power function kernel as
i . E 1 i
(33)
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G
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The storage modulus is the real part of G* MRL (ω), and loss modulus is the imaginary part of G*MRL (ω). Rearranging the real part and imaginary part of G*MRL(ω)
RL 1M
G
RL 2M
RL M
cos / 2 , E 2 2 cos / 2 1
(34)
RL M
sin / 2 . E 2 2 cos / 2 1
(35)
Re G
Im G
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G
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leads to the following analytical solution for the storage modulus and loss modulus,
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The procedures to obtain storage modulus and loss modulus of the fractional Maxwell model with exponential function kernel and Mittag-Leffler function kernel
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are the same as above. The analytical equations can be found in Table 1. As shown in
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Eq. (32), the complex modulus is deduced from the Laplace transform of the stress relaxation modulus. For fractional Maxwell model with logarithmic function kernel,
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the Laplace transform of the relaxation modulus does not have an analytical form. Therefore, the dynamic properties of fractional Maxwell model with logarithmic function kernel do not have the corresponding analytical forms. Besides, with 0<α<1, the storage moduli and loss moduli of fractional Maxwell model under different fractional derivative definitions are positive for all frequencies, as shown by the equations in Table 1. That is, the thermodynamic constraints are satisfied for fractional Maxwell model (Pritz, 2003). It can be easily concluded that
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Kelvin-Voigt
model,
fractional
Zener
model
and
fractional
Poynting-Thomson model with different kernels all satisfy the thermodynamic constraints, as shown by the equations in Tables 2 and 3.
5. Results and discussions
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In this section, we compare the performance of fractional viscoelastic models with different kernels. As shown in Tables 1-3, the mechanical performance of each model depends on elastic modulus of each spring, relaxation time of each fractional dashpot and fractional order. To be simple, in the following part, the fractional order
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and relaxation time are set as typical values of α=0.5 and τ=10 s respectively if not specified elsewhere. The mechanical response of fractional Maxwell model under four fractional derivative definitions are shown in Fig. 2. It can be seen from Figs. 2(a) and
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2(b) that the relaxation moduli (creep compliances) of the Mittag-Leffler function kernel and logarithmic function kernel decrease (increase) more slowly than that of
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the power function kernel when the fractional order is the same. The logarithmic
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functional kernel results in the slowest relaxation or creep rate among four fractional derivative definitions, which can be used to characterize the ultraslow relaxation
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(creep) of viscoelastic materials. When time is small, the relaxation modulus and creep compliance of the logarithmic function kernel are the same as that of classical
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power function kernel. The fractional Maxwell model with exponential function kernel exhibits exponential relaxation and linear creep, which have the same forms as that of the integer-order Maxwell model. The relaxation moduli and creep compliances of exponential function kernel and Mittag-Leffler function kernel do not approach the elastic modulus E and its inverse 1/E, respectively, when time t tends to 0. These observations demonstrate that the fractional Maxwell model with the
ACCEPTED MANUSCRIPT exponential function kernel and Mittag-Leffler function kernel is problematic to describe the relaxation and creep behaviors of viscoelastic materials. This also applies to the dynamic properties. The storage moduli of fractional Maxwell model using these two kernels do not reach the elastic modulus of the spring at highest frequency (Fig. 2 (c)). Fig. 2(d) shows that the maximum value of loss modulus of the
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exponential function kernel is about twice of that of the other two kernels since the performance of fractional Maxwell model with the exponential function kernel is close to that of the integer-order Maxwell model.
Fig. 3 shows the mechanical resposne of fractional Kelvin-Voigt model. For
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relaxation modulus and creep compliance (Figs. 3(a) and 3(b)), similar conclusions can be drawn as that for fractional Maxwell model. The creep compliance of the exponential function kernel exhibits an exponential increase with time instead of a
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linear increase compared with fractional Maxwell model. For dynamic properties, unlike that of the power function kernel, the storage moduli of the exponential
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function kernel and Mittag-Leffler function kernel have finite values with increasing
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frequency (Fig. 3(c)). Similar to the fractional Maxwell model, the maximum value of loss modulus of the exponential function kernel is about twice of that of the
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Mittag-Leffler function kernel (Fig. 3(d)). The mechanical properties of fractional Zener model are illustrated in Fig. 4,
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where the ratio between the elastic moduli of two springs is set as E1/E2=0.01 to represent a tremendous modulus change with time or frequency, which is commonly observed in polymers. A comparison between Figs. 4(a)-4(d) and Figs. 2(a)-2(d) reveals that the performance of fractional Zener model is close to that of fractional Maxwell model under all four fractional derivative definitions. Figs. 5 and 6 plot the relaxation modulus of fractional Maxwell model to
ACCEPTED MANUSCRIPT compare the performance of different kernels for different α and τ. It can be seen from Figs. 5(a) and 5(c) that, for a given τ, the relaxation moduli of the power function kernel and Mittag-Leffler function kernel decrease firstly more slowly and then more rapidly with increasing α. While the relaxation modulus of the logarithmic function kernel decreases more slowly with increasing α over the entire time period (Fig. 5(d)).
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The relaxation modulus of the exponential function kernel has the same variation trend with that of the logarithmic function kernel (Fig. 5(b)). For a given α, Fig. 6 shows that the relaxation moduli of four kernels decrease more slowly over the entire time period as τ increasing. The curves of the power function kernel for different τ can
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overlap together through translation (Fig. 6(a)), since t scales with τ in this case (Eq. (12)). In addition, Figs. 5 and 6 show again that the relaxation moduli of the exponential function kernel and Mittag-Leffler function kernel do not approach the
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elastic modulus of the spring when t tends to 0. A larger deviation between initial relaxation moduli and elastic modulus can be observed with a smaller value of α or τ.
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The results of fractional viscoelastic models are also compared to the
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experimental results. Firstly, the fractional Maxwell model with different kernels is applied to describe the experimental measured creep compliance of high strength
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self-compacting concrete (Maia and Figueiras, 2012). The model parameters are identified for different kernels, as shown in Table 4. It can be seen from Fig. 7 that the
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power function kernel and Mittag-Leffler function kernel perform similarly in this case. The fractional Maxwell model with the logarithmic function kernel provides the best description of the ultraslow creep behavior. The fractional Maxwell model with the exponential function kernel totally failed to describe the experimental data. As shown in Table 2, the Kelvin-Voigt models with all different kernels do not have an immediate elastic response, which is not consistent with the experimental
ACCEPTED MANUSCRIPT observations in Fig. 7. Thus, the fractional Kelvin-Voigt models are not used. The fractional Zener model with different kernels is applied to describe the stress relaxation modulus and creep compliance of lightly vulcanized Hevea rubber (Ferry, 1980). The model parameters are identified from the experimental data of stress relaxation modulus (Fig. 8(a)) and listed in Table 5. And then the fractional model
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together with the identified parameters is used to predict the creep behavior (Fig. 8(b)). It can be seen from Fig. 8 that the fractional model with the logarithmic function kernel exhibits exactly the same performance as the power function kernel. The fractional Zener models with the exponential function kernel and Mittag-Leffler
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function kernel fail to describe the experimental data, because they do not provide the accurate stress relaxation modulus at shortest time, as shown in Fig. 4(a). We do not use the experimental measured dynamic properties because the fractional Zener model
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with the exponential function kernel and Mittag-Leffler function kernel does not predict the right storage modulus at highest frequency, while no analytical solutions
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exist for the storage modulus when using the logarithmic function kernel.
6. Conclusions
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In summary, the performance of different definitions of fractional derivative in four fractional viscoelastic models is examined. The analytical solutions of the stress
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relaxation modulus, creep compliance and dynamic modulus are obtained. It is found that the fractional Poynting-Thomson model and fractional Zener model are equivalent under all the four fractional derivative definitions. The fractional Zener model has similar performance as fractional Maxwell model. The fractional derivative with the logarithmic function kernel can be used to characterize the ultraslow relaxation and creep. The fractional Maxwell model with the exponential function
ACCEPTED MANUSCRIPT kernel yields an exponential form of the stress relaxation modulus and a linear form of the creep compliance, which are the same as that of the integer-order Maxwell model. The fractional Maxwell model and fractional Zener model with the Mittag-Leffler function kernel do not provide accurate descriptions of the stress relaxation modulus at shortest time and the storage modulus at highest frequency. These results
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demonstrate that the fractional derivatives with the exponential function kernel and Mittag-Leffler function kernel may be problematic in describing the viscoelastic behaviors of materials. The classical fractional derivative is still a relative good choice for viscoelastic models. Extra efforts should be made to apply the new fractional
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derivatives to viscoelasticity.
Acknowledgments
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The authors acknowledge the supports from the National Natural Science Foundation of China (Grant No. 11702081), the Fundamental Research Funds for
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Central Universities, Hohai University (Grant No. 2016B01414), China Postdoctoral
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Science Foundation (Grant No. 2017M621605) and Open Project of State Key Laboratory for Strength and Vibration of Mechanical Structures (Grant No.
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SV2017-KF-19). We also would like to acknowledge the useful discussions with
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Dumitru Baleanu, Hongguang Sun and Yingjie Liang.
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ACCEPTED MANUSCRIPT non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20, 763-769. Beghin, L., Garra, R., Macci, C., 2014. Correlated fractional counting processes on a finite-time interval. J. Appl. Probab. 52, 263-279.
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Caputo, M., Fabrizio, M., 2015. A new definition of fractional derivative without
Caputo, M., Fabrizio, M., 2016. Applications of new time and spatial fractional derivatives with exponential kernels. Progr. Fract. Differ. Appl. 2, 1-11.
Carmichael, B., Babahosseini, H., Mahmoodi, S.N., Agah, M., 2015. The fractional
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viscoelastic response of human breast tissue cells. Phys. Biol. 12, 046001.
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Ferry, J.D., 1980. Viscoelastic Properties of Polymers, 3rd ed. John Wiley & Sons,
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Garra, R., Mainardi, F., Spada, G., 2017. A generalization of the Lomnitz logarithmic
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creep law via Hadamard fractional calculus. Chaos Soliton. Fract. 102, 333-338.
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Jiang, X.Y., Qi, H.T., 2012. Thermal wave model of bioheat transfer with modified Riemann-Liouville fractional derivative. J. Phys. A: Math. Theor. 45, 485101. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., 2006. Theory and Applications of
ACCEPTED MANUSCRIPT Fractional Differential Equations. Elsevier, Singapore. Koeller, R.C., 1984. Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 299-307. Kumar, R., Gupta, V., 2015. Wave propagation at the boundary surface of an elastic and thermoelastic diffusion media with fractional order derivative. Appl. Math.
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Model. 39, 1674-1688.
Lakes, R., 2009. Viscoelastic Materials. Cambridge University Press, Cambridge.
Lei, D., Liang, Y.J., Xiao, R., 2018. A fractional model with parallel fractional Maxwell elements for amorphous thermoplastics. Physica A: Stat. Mech. Appl.
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490, 465-475.
Maia, L., Figueiras, J., 2012. Early-age creep deformation of a high strength self-compacting concrete. Constr. Build. Mater. 34, 602-610.
College Press, London.
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Mainardi, F., 2010. Fractional Calculus and Waves in Linear Viscoelasticity. Imperial
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Pfitzenreiter, T. 2008, Where do fractional derivatives in stress strain relations come
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from? A structural approach. Z. Angew. Math. Mech. 88, 540-551. Pritz, T., 2003. Five-parameter fractional derivative model for polymeric damping
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materials. J. Sound Vib. 265, 935-952. Sumelka, W., Nowak, M., 2018. On a general numerical scheme for the fractional
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plastic flow rule. Mech. Mater. 116, 120-129.
Sun, H.G., Hao, X.X., Zhang, Y., Baleanu, D., 2017. Relaxation and diffusion models with non-singular kernels. Physica A: Stat. Mech. Appl. 468, 590-596.
Ying, Y.P., Lian, Y.P., Tang, S.Q., Liu, W.K., 2018. Enriched reproducing kernel particle method for fractional advection-diffusion equation. Acta Mech. Sin. 34, 515-527.
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Figure captions
Figure 1. Rheological representative of fractional viscoelastic models: (a) fractional Maxwell model, (b) fractional Kelvin-Voigt model, (c) fractional Zener model and (d)
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fractional Poynting-Thomson model.
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(b) creep compliance, (c) storage modulus and (d) loss modulus.
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Figure 3. Mechanical properties of fractional Kelvin-Voigt model: (a) relaxation
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modulus, (b) creep compliance, (c) storage modulus and (d) loss modulus.
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Figure 4. Mechanical properties of fractional Zener model: (a) relaxation modulus, (b)
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creep compliance, (c) storage modulus and (d) loss modulus.
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Figure 5. Relaxation modulus of fractional Maxwell model for different α: (a) power
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function kernel, (b) exponential function kernel, (c) Mittag-Leffler function kernel
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and (d) logarithmic function kernel. Here fractional order and relaxation time are set
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as typical values of α=0.3, 0.5, 0.7, 1 and τ=10 s, respectively.
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Figure 6. Relaxation modulus of fractional Maxwell model for different τ: (a) power
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function kernel, (b) exponential function kernel, (c) Mittag-Leffler function kernel
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and (d) logarithmic function kernel. Here fractional order and relaxation time are set
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as typical values of α=0.5 and τ=1 s, 10 s, 100 s, 1000 s, respectively.
Figure 7. Comparison of fractional Maxwell model with experimental measured creep
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compliance of high strength self-compacting concrete.
Figure 8. Comparison of fractional Zener models with experimental data of lightly
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vulcanized Hevea rubber: (a) relaxation modulus, (b) creep compliance.
Table captions
Table 1. Mechanical properties of fractional Maxwell model. Power function
Property
Exponential function
Mittag-Leffler function
Logarithmic function
E 1 t M 1
ln 1 t EM
1 1 t / 1 E
ln 1 t 1 1 E 1
1 E cos / 2 2 1 2 2 2 1 cos / 2
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Kernel
Relaxation modulus
t EM
E 1 t exp 1
Creep compliance
1 t 1 E 1
1 1 t 1 E
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E
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Storage modulus
Loss modulus
cos / 2
2 1 2 cos / 2
E 2 1
2 1 2 2
E sin / 2
E sin / 2 2 1 2 cos / 2
E
2 1 2 2
2 1 2 2 2 1 cos / 2
NA
ACCEPTED MANUSCRIPT Table 2. Mechanical properties of fractional Kelvin-Voigt model. Kernel
Power function
Exponential function
Mittag-Leffler function
Logarithmic function
Relaxation modulus
t E 1 1
t E 1 exp 1 1
t E 1 M 1 1
ln 1 t E 1 1
Creep compliance
1 1 t E M
1 1 1 E t exp 1
1 1 1 t E M 1
1 1 ln 1 t E M
2 1 E 1 2 2 2 1
2 1 cos / 2 E 1 2 2 2 1 2 1 cos / 2
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E sin / 2 2 1 2 2 2 1 cos / 2
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1 E cos / 2
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Storage modulus
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Property
E
E
sin / 2
1 2 2
2
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Loss modulus
ACCEPTED MANUSCRIPT Table 3. Mechanical properties of fractional Zener model. Replacing E1, E2 and τα in this table by 𝐸̂1 𝐸̂2 /(𝐸̂1 +𝐸̂2), 𝐸̂12 /(𝐸̂1 +𝐸̂2 ) and 𝐸̂2 𝜏̂ 𝛼 /(𝐸̂1 +𝐸̂2) respectively leads to the mechanical properties of fractional Poynting-Thomson model. Kernel
Power function
Exponential function
Mittag-Leffler function
E2 1 t exp 1
E2 1 t M 1
Logarithmic function
Property
Relaxation modulus
E1
t M
Creep compliance
E2 1 E E 1 2 E1 t M E1 E2
1 E1
1 E1
E2 1 E 2 E 1 1 E1 t exp E2 E1 1
E2 1 E 2 E 1 1 E1 t M E2 E1 1
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1 E1
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Loss modulus
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1 cos / 2
E1 E2 2
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1 1
2
2
2
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E2 sin / 2 2 1 2 cos / 2
ln 1 t M
1 E1
E2 1 E E 1 2 E1 M E1 E2 ln 1 t
Storage modulus
E1 E2
E1 E2
E1 E2 cos / 2 2 1 2 cos / 2
E1
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E1 E2
2 1 2 2 2 1 cos / 2
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E2 sin / 2
E2
2 1 2 2
2 1 2 2 2 1 cos / 2
NA
ACCEPTED MANUSCRIPT Table 4. Model parameters identified from the experimental data of ultraslow creep of high strength self-compacting concrete. The elastic modulus of the material is E=39.9 GPa. α
τ (s)
residuals
Power function
0.14
23.76
8.85E-5
Exponential function
0.79
848.15
1.96E-2
Mittag-Leffler function
0.25
2.70
1.68E-4
Logarithmic function
0.58
3.40
1.56E-5
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kernel
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Table 5. Model parameters identified from the stress relaxation modulus of lightly vulcanized Hevea rubber. The two elastic moduli of the material are E1=1.62 MPa and E2=1.175 GPa.
α
log τ
residuals
0.7707
-7.9374
0.3648
1.0000
-7.4348
5.4478
Mittag-Leffler function
0.9907
-3.7786
33.2983
Logarithmic function
0.7707
-7.9374
0.3648
Power function
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Exponential function
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kernel