Special Issue Article
Advances in Mechanical Engineering 2017, Vol. 9(10) 1–12 Ó The Author(s) 2017 Characterizing the rheological DOI: 10.1177/1687814017699765 behaviors of non-Newtonian fluid via a journals.sagepub.com/home/ade viscoelastic component: Fractal dashpot
Xianglong Su, Wen Chen and Wenxiang Xu
Abstract Based on the fractal derivative, a robust viscoelastic element—fractal dashpot—is proposed to characterize the rheolo- gical behaviors of non-Newtonian fluid. The mechanical responses of the fractal dashpot are investigated with different strains and stresses, which are compared with the existing dashpot models, including the Newton dashpot and the Abel dashpot. The results show that as the derivative order is between 0 and 1, the viscoelastic behavior of the fractal dash- pot is similar to that of the Abel dashpot. However, the fractal dashpot has a high computational efficiency compared with the Abel dashpot. On the other hand, the fractal dashpot can be reduced to the Newton dashpot when the deriva- tive order equals to 1. As an extension of fractal dashpot, a fractal Bingham model is also introduced in this study. The accuracy of proposed fractal models is verified by the relevant rheological experimental data. Moreover, the obtained parameters can not only provide quantitative insights into both the viscoelasticity and the relative strength of rheopexy and thixotropy, but also quantitatively distinguish shear thinning and thickening phenomena.
Keywords Viscoelastic, fractal derivative, dashpot, non-Newtonian fluid, rheology
Date received: 5 December 2016; accepted: 22 February 2017
Academic Editor: Praveen Agarwal
Introduction models.10–12 But nevertheless, these rheological models involving a number of material parameters are Unlike Newtonian fluid, non-Newtonian fluid exhibits restricted to specific materials or rheological situations complex rheological phenomena, such as creep, time-, and are difficult to be applied in other cases. and shear-dependent viscosity. It is usually called shear thinning when the apparent viscosity decreases with increased stress (shear rate), and otherwise is called 1 shear thickening. Some relevant rheological models Institute of Soft Matter Mechanics, College of Mechanics and Materials, and experimental researches can be found in the litera- Hohai University, Nanjing, China ture.2–4 The pseudoplastic material with a limiting yield 5 Corresponding authors: stress is usually characterized by the Bingham model Wen Chen, Institute of Soft Matter Mechanics, College of Mechanics and 6,7 and other generalized models. In addition, it is also Materials, Hohai University, Nanjing 211100, China. widely observed in non-Newtonian fluid that viscosity Email: [email protected] decreases (thixotropic) or increases with time (rheo- pexy). Thixotropic8 and rheopexy9 behaviors have been Wenxiang Xu, Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 211100, China. extensively predicted through a variety of relevant Email: [email protected]
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In the recent decades, fractional calculus is found to usually take a scale transform T = ta in equation (1), be an excellent mathematical instrument to characterize and then a form of normal derivative can be obtained viscoelastic behaviors13–20 with fewer parameters. GWS Blair21 proposed a viscoelastic model to connect the df (t) f (t) f (t0) = lim ð2Þ ideal Hookean and Newtonian components via the dT T!T 0 T T 0 fractional derivative approach, called the Abel dash- The relation of fractal derivative and normal deriva- pot.22 Based on the idea of the component model, some tive can be expressed as follows fractional viscoelastic models23,24 were also developed, 25 including fractional Maxwell model and fractional df (t) f 0(t) Kelvin model.26 On the other hand, the fractional con- = ð3Þ dta a ta 1 stitutive models have been introduced to describe the motion of non-Newtonian fluid.27,28 However, the glo- In a sense, the fractal derivative can be considered as bal property of fractional calculus requires considerably a modification of the first derivative. huge computational costs and memory requirements in 29 its numerical simulation. Definition 2. Fractional derivative in Caputo form39 Alternatively, fractal and local fractional deriva- (other definitions and applications of fractional deriva- tives as local operators were proposed to model com- tive can be seen in the literature40–43) plex behaviors of fractal materials.30–34 Based on fractal space–time transforms, the Hausdorff fractal ðt 30 dbf (t) 1 f (n)(t)dt derivative was proposed to analyze the anomalous = , diffusion process. It has been successfully applied in dtb G(n b) (t t)b n + 1 ð4Þ 0 anomalous diffusion,30,35,36 oscillation,37 heat genera- tion,38 and viscoelastic models.29 It is observed that (n = ½b + 1, n 1\b\n, t.t0) the fractal derivative models are mathematically sim- where b is the order of fractional derivative, and G( )is pler and computationally much more efficient than Gamma function. When the derivative order b varies the fractional derivative models. from 0 to 1, the fractional derivative can be simplified In this article, applying the Hausdorff fractal deriva- as tive, a generalized rheological model is proposed and is called the fractal dashpot. This article is organized as ðt dbf (t) 1 f 0(t) follows. In section ‘‘Fractal derivative and operator,’’ = dt ð5Þ the definitions of fractal and fractional derivative are dtb G(1 b) (t t)b t given. Section ‘‘Fractal dashpot’’ introduces the fractal 0 dashpot and examines the strain and stress responses. From the above two definitions, it can be seen that In addition, we investigate the main characteristics of the fractal derivative is a local operator without convo- the fractal dashpot compared with the Abel dashpot lution integral, which is quite different from the integral and the Newton dashpot. In section ‘‘Fractal compo- definition of the fractional derivative. nent models,’’ a fractal Bingham model is suggested as an extension of the fractal dashpot. Section Fractal dashpot ‘‘Numerical results and discussion’’ applies the fractal model and the generalized models to fit experimental The Newtonian fluid can be described by Newton’s law data. Finally, some conclusions are presented in section de ‘‘Conclusion.’’ s = m ð6Þ dt where s is the shear stress, e is the shear strain, and m Fractal derivative and operator is the shear viscosity. In the component system, compo- 44 In this article, we employ the definition of Hausdorff nent following this formula is called Newton dashpot. fractal derivative defined by Chen.30 In this article, we replace the normal derivative in the Newton dashpot with the above-mentioned fractal derivative, the constitutive relation of fractal dashpot Definition 1. Fractal derivative can be obtained as
df (t) f (t) f (t0) de(t) = lim a ð1Þ s(t)=h a.0 ð7Þ dta t!t0 ta t0 dta where a is the order of fractal derivative, represented a where h is the material parameter and a is the order of fractal measure; t denotes the coordinate in time. We fractal dashpot. Su et al. 3
Table 1. The comparisons of Newton dashpot, Abel dashpot, and fractal dashpot.
Newton dashpot Abel dashpot Fractal dashpot
Symbol Constitutive relation de dbe de s = m s = j s = h dt dtb dta The range of order None 0 Creep compliance and relaxation modulus The creep compliance and relaxation modulus of Newton dashpot can be written as t J(t)= ð8Þ m G(t)=m d(t) ð9Þ where d(t) is the Dirac delta function. Replacing t in (equations (8) and (9)) with ta, the creep compliance and relaxation modulus of the fractal dashpot can be obtained, which can be seen in Table 1. The main characteristics of Abel dashpot22 are also pre- sented in Table 1, which is a viscoelastic component based on fractional derivative. Loading and unloading numerical experiments are Figure 1. The stress in the loading and unloading processes. conducted, as shown in Figure 1, where the stress keeps stable from time t1 = 1 to time t2 = 2 and then remov- ing the stress. The results of creep and recovery for these three dashpots are shown in Figure 2. From Figure 2, we can see that at a small value of a, the strain of fractal dashpot changes quickly both on the loading and unloading processes, which exhibits the high elasticity of fractal dashpot. On the other hand, with the order a approaching to 1, the strain of fractal dashpot is close to that of Newton dashpot (a special case of fractal dashpot with order equals to 1), which shows the high viscosity of fractal dashpot. In addition, it can be seen that fractal dashpot and Abel dashpot are almost similar in viscoelastic property with respect to the creep and recovery processes. The strain response under dynamic load Figure 2. The creep and recovery curves of fractal dashpot, A dynamic load is set to further study the strain Newton dashpot, and Abel dashpot. response characteristics of fractal dashpot. For simpli- city, the dynamic load is set as a form of sinusoidal function where s^ is the amplitude and f is the frequency. According to the constitutive relation in equation (7), s(t)=s^ sin 2p ft ð10Þ the strain response of fractal dashpot can be written as 4 Advances in Mechanical Engineering Table 2. The CPU time for fractal dashpot and Abel dashpot (AMD A6-3420M, RAM 2.74 GB, 32 bit windows 7 and MATLAB 2010b). Fractal dashpot Abel dashpot Derivative order 0.1 0.5 0.9 0.1 0.5 0.9 Dt = 0:01 s 0.1258 s 0.1356 s 0.015067 s 0.5029 s 0.5004 s 0.501391 s Dt = 0:001 s 0.1333 s 0.1333 s 0.2240 s 483.6862 s 483.4233 s 469.9336 s Figure 4. The strain responses of fractal dashpot, Abel Figure 3. The strain responses of fractal dashpot, Abel dashpot, and Newton dashpot under dynamic load at derivative dashpot, and Newton dashpot under dynamic load at derivative order 0.5. order 0.9. ðt s^ a e(t)= ta 1 sin 2pf tdt ð11Þ h 0 Similarly, the strain responses of Newton dashpot and Abel dashpot are presented as s^ e(t)= ð1 cos 2p ftÞð12Þ 2p f m and ðt s^ sin 2pf tdt e(t)= ð13Þ j G(b) (t t)1 b 0 For the sake of convenience, all the material para- Figure 5. The strain responses of fractal dashpot, Abel meters, amplitude, and frequency of stress are set as 1, dashpot, and Newton dashpot under dynamic load at derivative that is, h = m = j = f = s^ = 1. The strain responses of order 0.1. three dashpots are shown in Figures 3–5. From these three figures, we can find that fractal dashpot has the fractal dashpot displays a higher computational efficiency same cycle and phase with Abel dashpot and Newton compared with Abel dashpot. dashpot. Although the amplitude of strain for Newton dashpot keeps stable, both fractal dashpot and Abel dash- pot damp with time. Furthermore, the amplitude damps The stress response of fractal dashpot under more quickly with smaller derivative order both for frac- tal dashpot and Abel dashpot. It is worth mentioning that power-law strain rate the computational time of fractal dashpot is much less Let strain rate keep as a constant, that is, e_ = c, the than Abel dashpot, as shown in Table 2, which means stress response of fractal dashpot can be written as Su et al. 5 Figure 6. The stress response of fractal dashpot at constant Figure 7. The stress response of fractal dashpot with strain strain rate. rate at time-linear strain rate. h c t1 a s = ð14Þ a h 1 (a 1) 1 + 1 (1 a) s = cn e_ ðÞn ð17Þ For the sake of convenience, the material parameter a and strain rate are set as 1, that is, c = h = 1. The stress Equation (17) can be used to characterize shear thicken response is shown in Figure 6. phenomenon with a\1 and shear thinning phenom- 45 The apparent viscosity of non-Newtonian fluid is enon with a.1. generally defined by the ratio of stress to strain rate. In this situation, the apparent viscosity can be written as The stress response of fractal dashpot under h t1 a dynamic strain hb = ð15Þ a A dynamic strain is set to further study the stress It is easy to find that the stress response under con- response characteristics of fractal dashpot. Similarly, stant strain rate is a form of power-law function. From we set the strain as a sinusoidal function for simplicity Figure 6, the stress is found to decrease with time when the derivative order is big than 1 and it keeps increasing e(t)=^e sin 2p ft ð18Þ when the order is less than 1. Apparent viscosity where ^e is the amplitude of strain. The stress response changes the same way as stress response, which means of fractal dashpot can be written as in this situation fractal dashpot is suitable to character- ize the time-dependent viscosity for non-Newtonian 2p f ^eh 9 8 s = t1 a cos 2p ft ð19Þ fluid, including ‘‘rheopexy’’ and ‘‘thixotropy’’ a phenomena. Strain rate is set as a linear function of time e_ = c t, The stress responses of Abel dashpot and Newton and the stress response of fractal dashpot can be written dashpot are presented as as ðt a 1 2p f ^ej cos 2p f t h c 2 a s = e_ ð16Þ s = b dt ð20Þ a G(1 b) (t t) 0 Figure 7 shows the stress response with strain rate and (h = c = 1). From Figure 7, we can see that the stress is shown as s = 2p f ^em cos 2p ft ð21Þ a lower convex function when 0 \ a \ 1 and an upward convex function when a . 1. These variations of stress For the sake of convenience, all the material para- over shear rate agree with the shear thickening and shear meters, amplitude, and frequency of stress are set as 1, thinning phenomena1 for non-Newtonian fluid. that is, h = m = j = f = s^ = 1. The stress responses of In general, the stress response of fractal dashpot three dashpots are shown in Figures 8 and 9. under power-law strain rate (e_ = c tn) can be written From Figures 8 and 9, it can be seen the stress as amplitude of fractal dashpot increases with time, but 6 Advances in Mechanical Engineering Figure 10. The schematic diagram of fractal Bingham model. Figure 11. The schematic diagram of fractal Maxwell model. model is introduced. The schematic diagram of fractal Bingham model is shown in Figure 10. Figure 8. The stress responses of fractal dashpot, Abel The constitutive relation of fractal Bingham model dashpot, and Newton dashpot under dynamic strain at can be written as follows derivative order 0.7. e = 0, s\ss de ð22Þ s = s + h , s.s s dta s where ss represents the yield stress. The stress response of fractal Bingham model under constant strain rate e_ = c is written as h c t1 a h ca s = s + or s = s + e1 a ð23Þ s a s a Under power-law strain rate e_ = c tn, the stress response of fractal Bingham model is presented as h c n + 1 a s = ss + t or a ð24Þ h 1 (a 1) 1 + 1 (1 a) s = s + cn e_ ðÞn s a Figure 9. The stress responses of fractal dashpot, Abel where n is a positive number as well as n ¼. 1. dashpot, and Newton dashpot under dynamic strain at Cai et al.29 introduced fractal Maxwell model and derivative order 0.5. fractal Kelvin model to characterize the creep phenom- ena for viscoelastic materials. The schematic diagram of fractal Maxwell model is shown in Figure 11. The decreases with the increased derivative order. On the constitutive relation is presented as29 other hand, the stress amplitudes of Abel dashpot and Newton dashpot show an independent relation with 1 ds s de time and derivative orders. In addition, stress response + = ð25Þ E dta h dta of fractal dashpot has the same phase with Newton dashpot, which is different from that of Abel dashpot. where E is the elastic modulus of spring. The creep modulus of the fractal Maxwell model is Fractal component models 1 1 J = + ta ð26Þ In order to extend the use of fractal dashpot, it can be E h taken parallel and series combinations with spring-pot and Saint-Venant’s body. Based on the idea of the clas- Numerical results and discussion sical component system, these models can be called fractal component models. In this section, we apply some relevant experimental By taking a combination of fractal dashpot in paral- data to validating the practicability of fractal dashpot lel with Saint-Venant’s body, the fractal Bingham and its extended fractal component models. Su et al. 7 Figure 12. The creep experimental data of PMMA and the simulation results of fractal dashpot (a) as well as the relative error of strain (b). Figure 13. The tension creep experimental data of PMR-15 and simulation result of fractal dashpot (a) as well as the relative error of strain (b). Apply fractal dashpot and fractal Maxwell model into Table 3. The parameters from fitting creep experimental data creep behaviors of PMMA by fractal dashpot. Figures 12 and 13 have shown the creep experimental ha data of polymethyl methacrylate (PMMA)46 and PMR- 47 q = 42 MPa 4.65E + 03 0.071 15 (one kind of polyimides) as well as the simulation q = 53 MPa 4.11E + 03 0.0847 results of fractal dashpot. Here, a formula of relative q = 62 MPa 3.82E + 03 0.1074 error (Rerr) is introduced to analyze the accuracy of the q = 74 MPa 4.01E + 03 0.1466 proposed model q = 85 MPa 5.01E + 03 0.2128 PMMA: polymethyl methacrylate. y ^y Rerr = ð27Þ y in Tables 3 and 4. It can be seen that the higher the where y and ^y represent the experimental data and the stress is, the bigger the derivative order is. In addi- theoretical value, respectively. Figures 12(b) and tion, the best-fitting derivative orders are all between 13(b) show the relative errors between the theoretical 0 and 1, which means these two materials PMMA value and the experimental data. It can be seen that and PMR-15 clearly exhibit viscoelasticity during the fractal dashpot has a good accuracy with experi- creep. Furthermore, their viscosities are higher with mental data. The obtained parameters can be found larger stress. 8 Advances in Mechanical Engineering Figure 14. The creep experimental data of PP-G and simulation results by fractal Maxwell model (a) as well as the relative error of strain (b). Table 4. The parameters from fitting experimental data of Table 6. The parameters from fitting constant-strain rate PMR-15 by fractal dashpot. stress response by fractal dashpot. ah ah s = 14.1 MPa 0.0354 2.6739e + 003 e# =53 1023 s21 0.7319 200.7924 s = 21.15 MPa 0.0354 2.6906e + 003 e# = 2.5 3 1025s 21 0.2406 6.8749 s = 28.2 MPa 0.0365 2.2985e + 003 s = 32.9 MPa 0.0416 2.1201e + 003 at 175 °C. It can be seen that stress of PPP increases with time (or strain) at specific strain rate, which Table 5. The parameters from fitting creep experimental data reflects the increasing viscosity of PPP during the shear by fractal Maxwell model. process, namely rheopexy. Applying fractal dashpot into fitting the experimental data, the simulation result and E haparameters can be found in Figure 15 and Table 6. From Figure 15(b), it can be seen that stress response of fractal s = 32 MPa 17.9962 31.9897 0.6249 s = 35 MPa 21.6354 22.8882 0.5937 dashpot has a good accuracy with experimental data. s = 37 MPa 18.8202 14.4057 0.6568 We introduce a concept of the relative strength of s = 40 MPa 16.6189 3.1636 0.9948 rheopexy or thixotropy (r) by the ratio of change rates of apparent viscosity in two situations Figure 14(a) shows the creep experimental data of dh =dt h 1 a a b1 1 1 2 a2 a1 polypropylene copolymer containing short glass fibers r = = t ð28Þ dhb =dt h2 1 a2 a1 (PP-G)48 and simulation results of fractal Maxwell 2 model. And the best-fitting parameters are presented in If r . 1, it means that the rheopexy (or thixotropy) Table5.FromFigure14(b),fractalMaxwellmodel phenomenon in the first situation is more obvious than shows high accuracy. It can be seen from Table 5, the that in the second situation. If r \ 1, we can conclude derivative order is bigger when the creep stress is larger that the rheopexy (or thixotropy) in the second situation except for the situation in s =32MPa,whichmeansthe is more obvious. If r = 1, it means the rheopexy (or viscosity of PP-G is larger with increasing stress during thixotropy) has the same strength in the two situations. creep. It is worth mentioning that PP-G shows an initial Using equation (27) to calculate the relative strength elastic phase during creep, which is very suitable to char- (r) of rheopexy at two situations: e_ = 5 3 10 3 and acterize by fractal Maxwell model. e_ = 2:5 3 10 5, the result is written as follows Apply fractal dashpot into describing time-dependent r = 3:3896 3 t 0:4913 ð29Þ viscosity r is a monotonic decreasing function of time, and r . 1 Figure 15(a) shows the stress response of poly-para- when t \ 11.9971 s. It means that at the initial time, phenylene (PPP)49 under different constant strain rates apparent viscosity increases more quickly under high Su et al. 9 Figure 15. The stress response of PPP at constant strain rate and simulation results by fractal dashpot (a) as well as relative error of stress (b). Figure 16. The viscosity experimental data of WPS at constant strain rate and simulation results by fractal dashpot (a) as well as the relative error of apparent viscosity (b). strain rate. On the following time, apparent viscosity Table 7. The parameters from fitting viscosity experimental increases more quickly under low strain rate. It can be data by fractal dashpot. concluded that at the initial phase of the shear process, high strain rate is helpful in improving rheopexy of ah PPP. But with time going on, the lower strain rate Strain rate = 50 s21 0.9149 0.3370 makes the rheopexy stronger. 21 Figure 16(a) has shown the apparent viscosity Strain rate = 300 s 1.0116 0.2443 experimental data50 of waxy potato starch (WPS) at constant strain rates. Applying equation (15) to fitting these experimental data, the simulation results and exhibits a rheopexy at small strain rate while a thixo- obtained parameters can be found in Figure 16(a) and tropy at large strain rate. Table 7. The relative error of fitting curves and experi- mental data are also presented in Figure 16(b). The 22 maximal relative error is less than 5 3 10 , which Apply fractal Bingham model to simulating shear test illustrates a high accuracy of fractal dashpot with experimental data. From Table 7, we can see that the of muddy clay derivative order is less than 1 at 50 strain rate and is Yin et al.51 conducted the pulling sphere test of muddy larger than 1 at 300 strain rate, which means WPS clay at different shear rate functions: e_ = 0:015 s 1, 10 Advances in Mechanical Engineering Figure 17. The stress-time experimental data at e_ = 0:015 s 1 (a), e_ = 0:375310 3 t s 1 (b), and e_ = 1:215310 4 t2 s 1 (c) as well as the simulation results by the fractal Bingham model and the fractional Bingham model. shear thinning phenomenon with 1 \ a \ 2. It is neces- Table 8. The parameters from fitting experimental data by the fractal Bingham model. sary to mention that compared with the fractional Bingham model, the fractal Bingham model has the ss hasame behavior in characterizing the shear-dependent viscosity for non-Newtonian fluid. e_ = 0:015 s 1 5.2077 21.3312 0.5778 e_ = 0:375310 3 t s 1 5.9740 36.3644 0.7116 e_ = 1:215310 4 t2 s 1 5.2594 359.2366 1.2813 Conclusion As a new viscoelastic component, the fractal dashpot e_ = 0:375 3 10 3 t s 1, and e_ = 1:215 3 10 4 t2 s 1. The was proposed in this article using the fractal derivative experimental data and the simulation results by fractal modeling approach. The creep compliance and relaxa- Bingham model and fractional Bingham model51 are tion modulus of the fractal dashpot were derived. shown in Figure 17(a)–(c). Through a series of loading experiments, the fractal As shown in Figure 17(a), stress response at constant dashpot demonstrated similar viscoelastic characteris- strain rate has yield stress and exhibits time-dependent tics to the Abel dashpot when the derivative order var- viscosity. From Figure 17(b) and (c), it can be seen that ies from 0 to 1, while the fractal dashpot has a clear the muddy clay behaves as shear thickening fluid with advantage of computational efficiency over the Abel time-linear strain rate and a shear thinning property dashpot. The Newton dashpot was a special case of the can be shown when the strain rate accelerates. This con- fractal dashpot with derivative order equivalent to 1. clusion can also be deduced from the fitting parameters On the other hand, the fractal dashpot was found in Table 8, where the derivative order is less than 1 at suitable to characterize the time- and shear-dependent linear strain rate and bigger than 1 at accelerating strain viscosity of non-Newtonian fluid under constant and rate. It comfirms again that the fractal dashpot can power-law strain rates. As an extension of the fractal describe shear thickening property with 0 \ a \ 1 and dashpot, a fractal Bingham model was also introduced. Su et al. 11 The proposed fractal models were validated with 9. Oates KMN, Krause WE, Jones RL, et al. Rheopexy of high accuracy compared with rheological experimental synovial fluid and protein aggregation. J R Soc Interface data of non-Newtonian fluid. The fitted derivative 2011; 3: 167–174. orders increased with the increasing stress during creep. 10. 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