An equivalence between generalized Maxwell model and fractional Zener model

Accepted Manuscript

An Equivalence between Generalized Maxwell Model and Fractional Zener Model Rui Xiao, Hongguang Sun, Wen Chen PII: DOI: Reference:

S0167-6636(16)30109-0 10.1016/j.mechmat.2016.06.016 MECMAT 2607

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Mechanics of Materials

Received date: Revised date: Accepted date:

5 March 2016 1 June 2016 23 June 2016

Please cite this article as: Rui Xiao, Hongguang Sun, Wen Chen, An Equivalence between Generalized Maxwell Model and Fractional Zener Model, Mechanics of Materials (2016), doi: 10.1016/j.mechmat.2016.06.016

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ACCEPTED MANUSCRIPT Highlights

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• We present a comparison of the performance of the generalized Maxwell model and fractional Zener model to describe viscoelasticity. • The discrete cumulative relaxation spectrum of the generalized Maxwell model forms a staircase approximation of that of the fractional Zener model. • The results show that the two models are quantitatively equivalent under various loading conditions.

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An Equivalence between Generalized Maxwell Model and Fractional Zener Model Rui Xiao a,∗ Hongguang Sun a Wen Chen a of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, Jiangsu 210098, China.

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a Institute

Abstract

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Both classic rheological models and fractional derivative models have been widely adopted to model the viscoelastic behaviors of materials. In this work, we present a detailed comparison of the performance of the generalized Maxwell model and fractional Zener model. We first describe a method to determine the parameters of the generalized Maxwell model from the fractional Zener model based on the equivalence of complex modulus in the frequency domain of the two models. The two models are then applied to investigating the stress response under constant strain rate, stress relaxation, cyclic and random loading conditions. The simulation results of the two models show excellent quantitatively equivalence. This finding can provide insight into choosing the more suitable model for specific conditions.

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1 Introduction

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Key words: Viscoelasticity, rheological model, fractional model

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Viscoelasticity represents that the behavior of materials is intermediate between linear solids and viscous liquid (Ferry 1980). When deformed, most of polymers and biological tissues exhibit this time-dependent viscous behavior represented as a stiffer stress response at a higher loading rate and a more compliant response at a lower loading rate. To understand the origin of viscoelasticity, various physical-based models have been proposed (Rouse Jr 1953, De Gennes 1979, Doi and Edwards 1988, Li et al. 2015). For example, the Rouse model (Rouse Jr 1953) was used to explain the properties of unentangled polymer solutions and melts. The reptation model (De Gennes 1979, Doi and Edwards 1988) was used to explain the relaxation and viscosity of entangled polymeric materials. However, for engineering applications, the most widely used method to describe the viscoelasticity is based on rheological models. ∗ Corresponding author. Email address: [email protected] (Rui Xiao).

Preprint submitted to Elsevier

24 June 2016

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The viscoelastic rheological models contain the elastic components modeled as springs and the viscous components modeled as dashpots (Ferry 1980). Based on the arrangement of these components, various models have been developed, such as the Maxwell model, the Kelvin-Vogit model, the Zener model and more complex generalized Maxwell model. The generalized Maxwell model contains an elastic spring in parallel with multiple Maxwell models to represent the relaxation occurring at a broad distribution of time. This model has been successfully applied to studying various viscoelastic solids (Del Nobile et al. 2007, Kaufman et al. 2008, Yu et al. 2014, Xiao et al. 2015a). For example, Del Nobile et al. (2007) has used the generalized Maxwell model to fit the experimental data of five different food matrices. Kaufman et al. (2008) has applied the generalized Maxwell model to studying the stress relaxation of hydrogels, while Yu et al. (2014) and Xiao et al. (2015a) have employed this model to describe the shape-memory behaviors of amorphous polymers.

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Though the classic viscoelastic models can well describe the experimental results, they typically involve excessive number of parameters. It is shown that the viscoelastic models can be generalized into fractional derivative models (Koeller 1984, Bagley and Torvik 1986, Schiessel et al. 1995). In recent years, fractional models have been widely adopted in the field of diffusion (Wu 2012, Wang et al. 2010, Zhao and Sun 2011), heat transfer (Jiang and Qi 2012), chaos (Baleanu et al. 2015, Wu and Baleanu 2015) and nonlocal elasticity (Tarasov 2014). However, the most extensive application of fractional models still lies in the field of linear viscoelasticity (Mainardi 2010). The general procedure to obtain the fractional viscoelastic model is through replacing the derivative of order 1 of the dashpot with the fractional derivative of order between 0 and 1. Through this process, various fractional derivative models can be obtained, such as fractional Maxwell model, fractional Kelvin-Vogit model, fractional Zener model (Mainardi 2010) and more complex model as shown in Arikoglu (2014). These models have been widely adopted to describe the relaxation and creep behaviors of elastomers (Di Paola et al. 2011) and natural materials (Cataldo et al. 2015), dynamic behavior of biological tissue (Kohandel et al. 2005) and other solids (Rossikhin and Shitikova 2010), and visco-elastic Euler-Bernoulli beam (Di Paola et al. 2013). Fan et al. (2015) and Yu et al. (2015) have developed numerical algorithm to obtain the model parameters for fractional derivative models. Though both rheological models and fractional models can be applied to describe the viscoelastic behaviors, limited work has been done to compare their performance in detail.

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In this work, we present a numerical study to compare the performance of generalized Maxwell model and fractional Zener model. The model descriptions are pesented in Section 2. The following section describes the procedure of obtaining the model parameters of generalized Maxwell model from an approximation between the dynamic modulus of the two models. Finally, we compare the performance of the two models under four different types of loading conditions: constant strain rate, stress relaxation, cyclic loading and a random loading condition.

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ACCEPTED MANUSCRIPT 2 Constitutive model

2.1

Generalized Maxwell model

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The rheological representative of generalized Maxwell model is shown in Figure 1.a, which is composed of a spring to describe the equilibrium elastic response and multiple Maxwell elements arranged in parallel to represent the viscoelastic response.

Eeq

Eeq

τ1

EN

τN (a)

Eneq

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E1

τ

(b)

Fig. 1. Rheological representative of a) generalized Maxwell model and b) fractional Zener model

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In each Maxwell element, the total strain of the spring ε ej and the dashpot ε vj should be equal to the strain ε in the elastic branch, which yields, ε = ε ej + ε vj , j = 1..N,

(1)

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where N is the total number of Maxwell elements.

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The total stress is given by,

N

σ = E ε + ∑ E j ε ej , eq

(2)

j

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where E eq is the equilibrium elastic modulus and E j is the modulus of the spring in jth Maxwell element.

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The following linear evolution equation is adopted for ε vj , ε˙ vj =

ε − ε vj τj

ε vj (t = 0) = 0,

,

(3)

where τ j is the relaxation time of the dashpot in jth Maxwell element. Equations (1)-(3) complete the generalized Maxwell model, which contains the following param
eq eters: equilibrium elastic modulus E and viscoelastic relaxation spectrum τ j , E j . 4

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Fractional Zener model

The 1D rheological representative of fractional Zener model is shown in Figure 1.b, which is consisted of an equilibrium elastic spring in parallel with a fractional damping Maxwell element. Similarly, the total strain in the non-equilibrium branch equals to that of the equilibrium branch, which gives, ε = ε e + ε v.

σ = E eq ε + E neq ε e ,

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The stress response is given by,

(4)

(5)

where E eq is the modulus of the equilibrium elastic spring and E neq is the modulus of the spring in the nonequilibrium fractional damping Maxwell element.

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The evolution of ε v in the fractional damping element can be described as (Haupt et al. 2000), dα ε v ε − ε v = , dt α τα α

ε v (t = 0) = 0,

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where ddt(·) α is the Riemann-Liouville fractional derivative with 0 < α < 1, which is defined as (Haupt et al. 2000, Nguyen et al. 2010),

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dα ( f ) 1 d = dt α Γ (1 − α) dt

Z t 0

f (s) ds, (t − s)α

f (t = 0) = 0.

(7)

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Γ (x) is the Eulerian Gamma function.

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Equations (4)-(6) complete the fractional Zener model.

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3 Relaxation spectrum of generalized Maxwell model

The procedures of obtaining viscoelastic relaxation spectrum of generalized Maxwell model from fractional derivative model have been discussed in detail in Haupt et al. (2000), Nguyen et al. (2010), Xiao et al. (2013) and Xiao and Nguyen (2015). Here we briefly summarized the main processes. The dynamic storage and loss modulus of the two models under a small sinusoidal oscillations can 5

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0 Egene (ω) = E eq + ∑ j

E j ω 2 τ 2j

0 Efrac (ω) = E eq +

, 2

1 + ω 2τ j

E neq ((ωτ)2α + (ωτ)α cos(απ/2) , 1 + (ωτ)2α + (ωτ)α cos(απ/2)

E neq (ωτ)α sin(απ/2) 00 , Efrac (ω) = 1 + (ωτ)2α + (ωτ)α cos(απ/2)

N

E j ωτ j 00 Egene (ω) = ∑ , 2 2 j 1+ω τj

(8)

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where ω is the angular frequency, E 0 (ω) denotes the storage modulus, and E” (ω) denotes the loss modulus. The viscoelastic spectra h (ν) can be calculated from the complex moduli E ∗ (iω) = E 0 (ω) + iE 00 (ω) using the inverse Stieltjes transform (Christensen 2003), E ∗ (iω) = iω

Z ∞ h (ν) 0

ν + iω

dν,

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where ν is the relaxation frequency, which is inverse of the relaxation time. Rν

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The cumulative relaxation spectra are defined as H (ν) = spectra of the two models,

0

h (u) du, which yields the cumulative

N

 Hgene (ν) = ∑ E j ν − ν j , j

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     (ντ)α + cos(απ) 1 E neq arctan −π −α , Hfrac (ν) = απ sin(απ) 2



 where ν − ν j = 1 for ν ≥ ν j and ν − ν j = 0 for ν < ν j .

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A power law distribution of relaxation frequencies is then assumed, ν 0j

0 = νmin



0 νmax 0 νmin

j−1  N−1

.

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0 and ν 0 are the upper and low limit of relaxation frequency. where νmax min

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To obtain the viscoelastic relaxation spectrum of the generalized Maxwell model, we assume the cumulative relaxation spectrum of the generalized Maxwell model forms a step-wise approximation of that of the fractional Zener model, which gives,
1 Hfrac (ν10 ) + Hfrac (ν20 ) , 2  1 Ej = Hfrac (ν 0j+1 ) − Hfrac (ν 0j−1 ) , 1 < j < N − 1, 2

E1 =

EN = E neq −

N−1

∑ E j. j

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(12)

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10

2

10 3

α=0.7 α=0.5 1

Storage modulus(MPa)

E j(MPa)

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10 0 10 -1

10 -3 -10 10

10 -5

10 0

τj (second)

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10 -2

10 2

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We choose E eq = 1 MPa, E neq = 999 MPa to represent a typical thermoset polymer. The relaxation time is chosen as: τ = 0.1 or τ = 0.01, and fractional order is chosen as α = 0.7 or α = 0.5. The shape of relaxation spectrum with the same fraction order remains unchanged. Thus, in the following we only plot the relaxation spectra with relaxation time τ = 0.1 (Figure 2-a). As shown, 0 = 10−5 /s the relaxation spectrum of α = 0.5 is much broader than that of α = 0.7. We chose νmin 0 5 0 −8 0 8 and νmax = 10 /s for α = 0.7, while νmin = 10 /s and νmax = 10 /s are chosen for α = 0.5. The number of Maxwell elements N is chosen as 200 to guarantee the storage modulus of the generalized Maxwell model showed excellent agreement with that of fractional Zener model (Figure 2-b).

10 5

10 0 10 -10

10 10

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10 -5

α=0.7

Fractional Zener model Generalized Maxwell model

10 0

10 5

10 10

ω

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Results and discussions

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Fig. 2. a) The viscoelastic relaxation spectra with τ = 0.1, b) comparison of the storage modulus between generalized Maxwell model and fractional Zener model.

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As shown in Section 3, the two models predict almost identical dynamic response under certain frequency. In the following, we perform a series of numerical simulations to compare the performance of the two models under various loading conditions.

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4.1 Numerical example I: constant strain rate

We first investigate the performance of the two models under constant strain rate. In the simulation, the strain is ramped to 0.1 under strain rate 0.0001/s, 0.001/s or 0.01/s. As shown in Figure 3, the simulation results of two models agree excellently regarding all different values of τ and α. To quantitatively describe the difference of the two models, we define the relative difference ∆ = |σgene − σ f rac |/|σgene |, where σgene and σ f rac are the stress of the generalized Maxwell model and fractional Zener model respectively. In all the cases, the relative differences are smaller than 7

ACCEPTED MANUSCRIPT 0.2% except for τ = 0.01, α = 0.7 and strain rate 0.0001/s, where the relative difference reaches 0.87%. These results suggest that the two models show negligible difference in describing the stress response at constant strain rate. 5

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Generalized Maxwell model Fractional Zener model

Generalized Maxwell model Fractional Zener Model

0.01/s

4

0.01/s

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0.001/s

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0.0001/s

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Generalized Maxwell model Fractional Zener Model

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Generalized Maxwell model Fractional Zener model

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Stress(MPa)

Stress(MPa)

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Fig. 3. Comparison of the stress response at constant strain rate of generalized Maxwell model and fractional Zener model, a) τ = 0.1, α = 0.7, b) τ = 0.01, α = 0.7, c) τ = 0.1, α = 0.5, d) τ = 0.01, α = 0.5.

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4.2 Numerical example II: stress relaxation

Following the real experimental setup and the discussion in Di Paola et al. (2014), the stress relaxation test is performed by an initial linear ramp and then the strain remains constant. Specifically, the strain is increased to a certain level (0.01, 0.03 or 0.05) at strain rate 0.01/s and held at that level for 60 seconds to allow stress relaxation. Figure 4 plots the stress response of the two models with different fractional order α = 0.5 and α = 0.7. In all the cases, the relative difference is smaller than 0.2%. From the results, we can see that a smaller fractional order results in a slower stress relaxation behavior indicating the relaxation occurs at a broader distribution of relaxation time. 8

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7.5

Generalized Maxwell model Fractional Zener model

Generalized Maxwell model Fractional Zener model

Stress(MPa)

Stress(MPa)

3

2

0.05

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1 0.03

0.01

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0

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Time (second)

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30

Time (second)

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Fig. 4. Comparison of the stress relaxation response of generalized Maxwell model and fractional Zener model, a) τ = 0.1, α = 0.7, b) τ = 0.1, α = 0.5.

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4.3 Numerical example III: cyclic test

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In the cyclic test, the strain is increased to 0.05 at strain rate 0.01/s, then decreased to 0 at the same stain rate and held for 5 seconds, while this process is repeated four times as shown in Figure 5. Figure 6 plots the corresponding stress response of the cyclic test as a function of time. As shown, the stress response of the two models shows good agreement even after several cycles. The relative difference does not show any increase with increasing cycles.

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Strain

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15

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Time (second)

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Fig. 5. Strain as a function of time in the cyclic test

4.4 Numerical example IV: random loading test

We further simulate the stress response under a random loading condition. In the simulation, the time step is chosen as ∆t = 0.01. At each time step, the strain is increased by a random number 9

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Generalized Maxwell model Fractional Zener model

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Generalized Maxwell model Fractional Zener model

3

Stress(MPa)

1.5

0

-1.5

-3

2.5

0

-2.5

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Time (second)

45

-5

60

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5

15

30

Time (second)

45

60

(b)

Fig. 6. Comparison of the stress response of generalized Maxwell model and fractional Zener model in the cyclic test, a) τ = 0.1, α = 0.7, b) τ = 0.1, α = 0.5.

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between -0.0001 and 0.0001. Figure 7-a shows the strain as a function of time during one random loading condition. The corresponding stress response of the two models with τ = 0.1 and α = 0.7 is plotted in Figure 7-b and c, which shows identical pattern. The simulation results of the last 80 steps are plotted in Figure 7-d. As shown, the two models show excellent agreement. All of the above results have demonstrated the two models are quantitatively equivalent at various loading conditions.

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4.5 General remarks

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From Section 2, we can see that the classic dashpot is a local component while the fractional dashpot is inherently a global component. There should be fundamental differences between the two models. However, our results clearly demonstrate that the two models show equivalences in various loading conditions. We are not quite clear about the physical mechanism behind this equivalence. However, this finding can been extremely useful for the engineering applications.

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Viscoelastic materials, especially polymers, usually exhibit broad relaxation spectrum, which requires long time to reach the equilibrium state. The simple rheological models, such as the Kelvin model and the Maxwell model, fail to describe the long-term relaxation response (Xiao et al. 2015b). To accurately describe the real materials behaviors, an excess number of parameters need to be determined, which could be time-consuming. The fractional models have the potential to overcome this disadvantages, which require fewer parameters to fully describe the experimental observations. In this work, we clearly show that the three parameter fractional Zener model has the ability to represent a similar response of generalized Maxwell model with multiple parameters. However, the fractional models exhibit nonlocal properties, which need take into account all the previous deformation history. The fractional viscoelastic models may be difficult to be applied for long-term loading conditions. In this case, the classic rheological models involving local calcula10

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Generalized Maxwell model

-0.005

Stress(MPa)

-0.01

0

-0.25

-0.5

0

50

100

Time (second)

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(a) 0.5

50

Time (second)

100

(b)

Generalized Maxwell model Fractional Zener model

Fractional Zener model

0.2

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0.25

Stress(MPa)

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50

Time (second)

(c)

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(d)

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Fig. 7. The random loading test: a) strain as a function of time, b) stress response of generalized Maxwell model, c) stress response of the fractional Zener model, d) comparison the stress response of the two models.

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tions may be more practical to use. Especially, in this work we demonstrate that the generalized Maxwell model and fractional Zener model are numerically equivalent. To our knowledge, no existing work has demonstrated this equivalence. Thus, this work could arouse the interest of researchers to explore the equivalence or non-equivalence of using rheological models and fractional derivative models. Based on the excellent agreement of the two models, this work may also provide some insight into the numerical algorithm for solving the fractional differential equations. For fractional model, the physical meaning of fractional order is far from settled. From the Section 3, we can see the fractional order of fractional Zener model has a clear physical meaning, which represents the breadth of relaxation spectrum. A smaller fractional order means the relaxation behavior occurs in a broader time region, which may directly relate with the microstructure of materials. Thus, this work can also shed light on the physical significance behind fractional viscoelastic models. 11

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The generalized Maxwell model and fractional Zener model have been widely used to model viscoelastic solids. The fractional Zener model is based on fractional calculus and exhibits global correlation, while the generalized Maxwell model exhibits only local property. In this work, we have compared the performance of the two models under various loading conditions. Our results clearly demonstrate that the two models are quantitatively equivalent regarding all different values of relaxation time and fractional order. This finding could potentially assist choosing the suitable model for practical applications. We also find that a smaller fractional order represents the relaxation occurring at a broader distribution of time, which results in a slower relaxation behavior. This finding can shed light on understanding the physical meaning of fractional order.

Acknowledgments

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This work is supported by the National Natural Science Foundation of China (Grant Nos. 11502068, 11572112) and the Fundamental Research Funds for Central Universities, Hohai University (Grant No. 2016B01414).

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